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author:
- Siarhei Finski
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Analytic torsion for surfaces with cusps I.\
Compact perturbation theorem and anomaly formula.
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abstract: 'In this paper, an improved receiver based on diversity combining is proposed to improve the bit error rate (BER) performance of layered asymmetrically clipped optical fast orthogonal frequency division multiplexing (ACO-FOFDM) for intensity-modulated and direct-detected (IM/DD) optical transmission systems. Layered ACO-FOFDM can compensate the weakness of traditional ACO-FOFDM in low spectral efficiency, the utilization of discrete cosine transform in FOFDM system instead of fast Fourier transform in OFDM system can reduce the computational complexity without any influence on BER performance. The BER performances of layered ACO-FOFDM system with improved receiver based on diversity combining and DC-offset FOFDM (DCO-FOFDM) system with optimal DC-bias are compared at the same spectral efficiency. Simulation results show that under different optical bit energy to noise power ratios, layered ACO-FOFDM system with improved receiver has 2.86dB, 5.26dB and 5.72dB BER performance advantages at forward error correction limit over DCO-FOFDM system when the spectral efficiencies are 1 bit/s/Hz, 2 bits/s/Hz and 3 bits/s/Hz, respectively. Layered ACO-FOFDM system with improved receiver based on diversity combining is suitable for application in the adaptive IM/DD systems with zero DC-bias.'
author:
- Mengqi Guo
- Ji Zhou
- Xizi Tang
- Fan Hu
- Jia Qi
- Yaojun Qiao
- Aiying Yang
- Yueming Lu
date: 'Received: 11 March 2017 / Accepted: 8 July 2017'
title: 'An Improved Diversity Combining Receiver for Layered ACO-FOFDM in IM/DD Systems'
---
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Introduction
============
Owing to the explosive growth of bandwidth hungry services, high capacity and low cost optical transmission systems are developed to meet the increasingly updated demands. As a multicarrier modulation format, orthogonal frequency division multiplexing (OFDM) can provide relatively high transmission capacity due to its superiorities in high spectral efficiency and robustness against chromatic dispersion and polarization-mode dispersion [@OFDM_Armstrong; @OFDM_Shieh; @OFDM_Djordjevic; @A; @cost-effective; @and; @efficient]. The intensity-modulated and direct-detected (IM/DD) system has lower cost and power consumption compared with coherent system, it can be applied in many scenarios such as metropolitan area networks, access networks, data center interconnects and visible light communications [@metropolitan; @area; @network; @PON; @data; @center; @VLC]. The signal transmitted in IM/DD OFDM system must be real and positive. To obtain real signal, Hermitian symmetry is needed to constrain the input constellations of inverse fast Fourier transform (IFFT). To obtain positive signal, DC-offset OFDM (DCO-OFDM) and asymmetrically clipped optical OFDM (ACO-OFDM) systems are most commonly used two schemes to generate positive signal [@DC-biased; @ACO].
The performance of DCO-OFDM depends on the DC-bias level. If the DC-bias is not large enough, the remaining negative values are clipped at zero level, which can introduce clipping distortions. If the DC-bias is quite large, this large DC-bias inefficiently occupies lots of optical power. The optimal DC-bias depends on the constellation size, which limits the performance of DCO-OFDM in adaptive systems. ACO-OFDM needs zero DC-bias, all the negative values are clipped at zero level in spite of the constellation size, so that ACO-OFDM is more suitable in adaptive systems. However, in ACO-OFDM system, only odd subcarriers are used to carry the signal, which leads to the loss of spectral efficiency.
To improve the spectral efficiency of ACO-OFDM, different types of multilayered schemes based on ACO-OFDM have been proposed. Asymmetrically clipped DC biased optical OFDM (ADO-OFDM) system transmits ACO-OFDM on odd subcarriers and DCO-OFDM on even subcarriers simultaneously, but the even subcarriers still need DC-bias [@Comparison; @of; @ACO-OFDM]. Hybrid ACO-OFDM system utilizes ACO-OFDM on odd subcarriers and pulse-amplitude-modulation discrete-multi-tone (PAM-DMT) on even subcarriers, but the real components of even subcarriers are still useless [@Hybrid; @asymmetrically; @clipped]. The superposition of more than two layers are proposed further to increase the spectral efficiency. Layered ACO-OFDM system applies different kinds of ACO-OFDM to different layers [@Layered; @ACO-OFDM]. Augmented spectral efficiency discrete multitone (ASE-DMT) system uses PAM-DMT on imaginary components of all the subcarriers, and employs layered ACO-OFDM on the real components [@Augmenting].
It is worth noting that the performance of ACO-OFDM signal can be improved by extracting useful information from clipping noise on even subcarriers. A novel technique called diversity combining has been proposed in [@Diversity; @combining], it utilizes the information from both of the odd and even subcarriers, which can also be used in multilayered schemes based on ACO-OFDM to further reduce the effect of noise and achieve better bit error rate (BER) performance.
In recent years, fast OFDM (FOFDM) based on discrete cosine transform (DCT) has been investigated in the IM/DD optical communication systems to reduce the computational complexity [@fast-OFDM1; @fast-OFDM2; @Asymmetrically; @Clipped; @Optical; @Fast; @OFDM; @FOFDM]. If the inputs of DCT or inverse DCT (IDCT) are real values, the outputs are also real values. Therefore, the real transformation does not need Hermitian symmetry any more, and the one-dimensional modulation has lower computational complexity. For FOFDM system, the interval between subcarriers decreases to half of that in OFDM system, but signal on the positive frequency of FOFDM system has corresponding image on negative frequency, so the M-PAM FOFDM system has the same spectral efficiency and BER performance as M$^2$-QAM OFDM system [@Asymmetrically; @Clipped; @Optical; @Fast; @OFDM; @FOFDM]. Consequently, it is not difficult to find DCT is also very suitable for replacing FFT in multilayered schemes to reduce the computational complexity without any influence on BER performance [@ZTE].
In this paper, we firstly propose an improved receiver based on diversity combining technique in layered asymmetrically clipped optical FOFDM (ACO-FOFDM) for IM/DD optical transmission systems. Layered ACO-FOFDM system improves the spectral efficiency of traditional ACO-FOFDM system, and the utilization of DCT in FOFDM system instead of FFT in OFDM system can reduce the computational complexity without any influence on BER performance. Simulation results show that the layered ACO-FOFDM system with improved receiver not only has better BER performance than the system without improved receiver, but also has better BER performance than DCO-OFDM system with the optimal DC-bias and the same spectral efficiency.
Transmitter Structure of Layered ACO-FOFDM
==========================================
{width="14cm"}
{width="14cm"}
For traditional ACO-FOFDM system, only half of the subcarriers are used to carry the signal, which leads to the spectral efficiency of ACO-FOFDM is half of that of DCO-FOFDM with the same modulation format. In order to improve the spectral efficiency of ACO-FOFDM, layered ACO-FOFDM system is proposed, so that signal on different layers can be transmitted simultaneously. Fig. \[transmitter\] reveals the transmitter scheme of layered ACO-FOFDM system. The uppercase $L$ represents the total number of layers and the lowercase $l$ represents the $l$-th layer, these notations are employed throughout this paper.
The superposition of multiple layers is performed in frequency domain. Signal on the $1$-st layer is the same as that on the traditional ACO-FOFDM, the $l+1$-th layer occupies half number of subcarriers as the $l$-th layer does. As depicted in Fig. \[transmitter\], for layer 1, the original bits perform serial to parallel conversion and M-PAM mapping at first, then only odd subcarriers are utilized to carry the signal (index $2k+1, k=0,1,2,...,N/{2^1}-1$). The real-valued time domain signal is generated after IDCT, cyclic prefix is added and parallel to serial conversion is performed. Then, all the negative value of this odd-symmetry time domain signal is clipped at zero level. The clipping noise only falls on the even subcarriers (index $2k, k=0,1,2,...,N/{2^1}-1$), which is incapable to influence the signal on odd subcarriers. For layer 2, the odd subcarriers of remaining unused subcarriers on layer 1 are used (index $2\times(2k+1), k=0,1,2,...,N/{2^2}-1$), and the clipping noise falls on the even subcarriers of remaining unused subcarriers on layer 1 (index $2\times2k, k=0,1,2,...,N/{2^2}-1$). According to this rule, for layer $l$, only the ${2^{l-1}}\times(2k+1){\kern 2pt}(k=0,1,2,\ldots,N/{2^l}-1)$ subcarries are utilized to carry the signal, the clipping noise falls on the ${2^{l-1}}\times(2k){\kern 2pt}(k=0,1,2,\ldots,N/{2^l}-1)$ subcarries. Finally, unipolar time domain signals on different layers are generated, signals and clipping noise of the $l$-th layer are orthogonal to those on layers $1$ to $l-1$.
Afterwards, the unipolar time domain signals on different layers are added to transmit simultaneously. After digital-to-analog conversion (DAC) and low pass filter (LPF), the analog electrical signal is modulated to optical carrier.
Receiver Structure of Layered ACO-FOFDM
=======================================
Figure \[receiver\] reveals the receiver scheme of layered ACO-FOFDM system. The traditional layered ACO-FOFDM system only needs conventional receiver before M-PAM demapper. To improve the BER performance of layered ACO-FOFDM system, improved receiver is appended to conventional receiver.
Conventional Receiver
---------------------
For traditional ACO-FOFDM system, the received signal ${X_{odd}}$ on the odd subcarrier can be represented as ${X_{odd}} = \frac{1}{2}DCT({x_n})$ and the received signal ${X_{even}}$ on the even subcarrier can be represented as ${X_{even}} = \\ \frac{1}{2}DCT(\left| {{x_n}} \right|)$, $x_n$ means the time domain signal on subcarrier $n$, $DCT(\cdot)$ means the DCT operation. These results are also suitable for higher layers in layered ACO-FOFDM system. The conventional receiver structure is shown in Fig. \[conventional-receiver\]. Due to the asymmetric clipping operation of $l$-th layer brings clipping noise, which can influence the signal on higher layers, the first objective is to extract the signal $\tilde Y{_{odd}^{(1)}}$ on layer 1 from odd subcarriers, this operation can help us regenerate the clipping noise falling on even subcarriers. It is worth noting that the hard-decision [@SEE-OFDM] is applied to get the clipping noise, which means $\tilde Y{_{odd}^{(l)}}$ is the firm decision on the constellation values instead of the noisy constellation values with soft-decision. The hard-decision has better BER performance than soft-decision because more accurate constellation values are used to estimate the clipping noise. Then, the obtained signal on layer 1 can perform $N$-point IDCT, get the absolute value and perform $N$-point DCT, so that the clipping noise on even subcarriers of layer 1 is obtained. After subtracting the clipping noise generated by layer 1, the signal $\tilde Y{_{odd}^{(2)}}$ on layer 2 can be extracted. Afterwards, the demodulation process continues in a similar way for all the subsequent layers until the information at all layers is recovered.
Improved Receiver Based on Diversity Combining
----------------------------------------------
The improved receiver is added after the conventional receiver. In the improved receiver, diversity combining technique is applied in each layer to improve the BER performance. From conventional receiver the signals on the odd subcarriers of all the layers are obtained, so that the diversity combining technique in improved receiver can be performed on the highest layer $L$ at first. Fig. \[improved-receiver\] presents the improved receiver structure of layered ACO-FOFDM. If we take noise into consideration, the signals on odd and even subcarriers of layer $L$ can be represented as $$Y_{odd}^{(L)}=X_{odd}^{(L)'}+N_{odd}^{(L)}=\frac{1}{2}DCT\left({x_{n}}^{(L)'}\right) + {N_{odd}^{(L)}}
\label{eq1}$$ $$Y_{even}^{(L)}=X_{even}^{(L)'}+N_{even}^{(L)}=\frac{1}{2}DCT\left(\left| {{x_{n}}^{(L)'}} \right|\right) + {N_{even}^{(L)}}
\label{eq2}$$ where $N_{odd}^{(L)}$ and $N_{even}^{(L)}$ are the frequency domain noises on odd and even subcarriers of layer $L$, respectively. It should be noted that $Y_{odd}^{(L)}$ and $Y_{even}^{(L)}$ are just the received signals on odd and even subcarriers, hard-decision is only needed to get the clipping noise.
The diversity combining technique can be applied to make use of the signal on both odd and even subcarriers. If we separate the signal on odd and even subcarriers, load $Y_{odd}^{(L)}$ and $Y_{even}^{(L)}$ to only the odd and even subcarriers of two separate IDCTs, we can get the results that $$y_{n,odd}^{(L)} = IDCT(Y_{odd}^{(L)}) = \frac{1}{2}{{x_{n}}^{(L)'}} + {n_{n,odd}^{(L)}}
\label{eq3}$$ $$\left|y_{n,even}^{(L)}\right| = IDCT(Y_{even}^{(L)}) = \frac{1}{2}{\left| {x_{n}}^{(L)'}\right|} + {n_{n,even}^{(L)}}
\label{eq4}$$ where $n_{n,odd}^{(L)}$ and $n_{n,even}^{(L)}$ represent the time domain noises through the corresponding IDCT.
If we extract the polarity information from odd subcarriers to indicate the sign flipping of even subcarriers, we can obtain another useful signal from even subcarriers, $$y_{n,even,f}^{(L)} = \left\{ \begin{array}{l}
{\kern 8pt} \left| {y_{n,even}^{(L)}} \right|{\kern 3pt},{\kern 10pt} {y_{n,odd}^{(L)}} > 0\\
- \left| {y_{n,even}^{(L)}} \right|{\kern 3pt},{\kern 10pt} {y_{n,odd}^{(L)}} \le 0
\end{array} \right.
\label{eq5}$$
Then, the maximal-ratio combining technology can be employed as $${y_n}^{(L)} = (1-\alpha){y_{n,odd}^{(L)}}+\alpha{y_{n,even,f}^{(L)}}
\label{eq6}$$ The combining coefficient $\alpha$ is related to signal to noise ratio (SNR) of $y_{n,odd}^{(L)}$ and $y_{n,even,f}^{(L)}$, $$\alpha = \frac{{SNR_{even}}}{{SNR_{odd}+ SNR_{even}}}
\label{eq7}$$ where $SNR_{even}$ is the SNR of $y_{n,even,f}^{(L)}$ and $SNR_{odd}$ is the SNR of $y_{n,odd}^{(L)}$. The value of $\alpha$ is always a little bit less than 0.5, the reason is that in the sign flipping process, $y_{n,odd}^{(L)}$ is influenced by noise, there is a small chance that noise has opposite sign and larger amplitude compared with the corresponding signal, which can lead to sign flipping error in $y_{n,even,f}^{(L)}$. Hence $y_{n,even,f}^{(L)}$ always has less SNR value than $y_{n,odd}^{(L)}$. Finally, ${y_n}^{(L)}$ inputs DCT and the required signal is on the odd subcarriers of layer $L$.
Afterwards, we should perform diversity combining on layer $L-1$, but signal of layer $L$ falls on the even subcarriers of layer $L-1$. The influence from layer $L$ on the even subcarriers of layer $L-1$ can be eliminated as shown in Fig. \[improved-receiver\]. The hard-decision constellation value $\tilde X_{odd}^{(L)}$ inputs IDCT, after zero clipping and DCT, we can get all the signal on layer $L$. By subtracting it from even subcarriers of layer $L-1$, diversity combining technique can be conducted on layer $L-1$. Afterwards, the diversity combining process continues in a similar way for all the subsequent layers until the information at all layers is recovered.
Simulation Results and Discussion
=================================
![BER performance comparison between improved receiver with diversity combining and conventional receiver without diversity combining in 4PAM layered ACO-FOFDM with different layers[]{data-label="4layer"}](Fig4){width="\linewidth"}
The BER performance advantage of improved receiver is analyzed through simulation in additive white gaussian noise (AWGN) channel. The DCT size is 256 in all the following simulations. Fig. \[4layer\] reveals BER performance comparison between improved receiver with diversity combining and conventional receiver without diversity combining in layered ACO-FOFDM. The layer numbers are 2, 3 and 4 with 4PAM modulation on each layer. We can find with diversity combining the BER improvements are 1.76dB, 1.35dB and 1.02dB at forward error correction (FEC) limit (i.e., 1$\times10^{-3}$) in terms of $E_{b(elec)}/N_0$, $E_{b(elec)}/N_0$ denotes the ratio between electrical energy per bit and single-sided noise power spectral density. The BER performance advantages increase with the decreasing of FEC limit. Therefore, the improved receiver based on diversity combining can improve the BER performance of layered ACO-FOFDM system.
Afterwards, we make BER performance comparison among layered ACO-FOFDM system with or without improved receiver and DCO-FOFDM system at the same spectral efficiency. If the constellation sizes on different layers are the same, the spectral efficiency gap between layered ACO-FOFDM and DCO-FOFDM can never be eliminated because infinite number of layers are required. If arbitrary constellation size can be chosen on each layer, the spectral efficiency gap can be eliminated with only small number of layers. Taking practical implementation into consideration, large number of layers introduce more computational complexity, so we utilize 2-layer ACO-FOFDM system with proper selection of constellation sizes on different layers. The average signal power applied to each layer is proportion to the number of bits transmitted in each layer. The constellation size combination mode with the best BER performance for 2-layer ACO-FOFDM system is chosen according to Monte Carlo simulations, finding the optimal layer number and constellation size combination mode to get the best BER performance can be fulfilled in the future work.
![BER as a function of $E_{b(elec)}/N_0$ for DCO-FOFDM system and layered ACO-FOFDM system with or without diversity combining. $E_{b(elec)}/N_0$ denotes the ratio between electrical energy per bit and single-sided noise power spectral density. The optimal DC-biases of DCO-FOFDM system are 4.9dB, 7.2dB and 9.2dB for BPSK, 4PAM and 8PAM as investigated in [@FOFDM][]{data-label="elec"}](Fig5){width="\linewidth"}
Figure \[elec\] reveals the BER performance comparison among DCO-FOFDM system and 2-layer ACO-FOFDM system with or without diversity combining at different values of $E_{b(elec)}/N_0$. The spectral efficiencies are 1 bit/s/Hz (i.e., BPSK+4PAM), 2 bits/s/Hz (i.e., 8PAM+\
4PAM) and 3 bits/s/Hz (i.e., 16PAM+16PAM). For DCO-FOFDM system, the optimal DC-biases are set to get the best BER performance according to [@FOFDM]. From Fig. \[elec\] we can find at FEC limit the BER performance of 2-layer ACO-FOFDM system without diversity combining is only a little bit better than DCO-FOFDM system when the spectral efficiency is 2 bits/s/Hz or 3 bits/s/Hz, and its BER performance is worse than DCO-FOFDM system when the spectral efficiency is 1 bit/s/Hz. However, at FEC limit, when the spectral efficiencies are 1 bit/s/Hz, 2 bits/s/Hz and 3 bits/s/Hz, the 2-layer ACO-FOFDM system with diversity combining can achieve 1.37dB, 2.89dB and 3.01dB BER gains compared with DCO-FOFDM system, and the improved receiver with diversity combining obtains 2.08\
dB, 2.54dB and 2.01dB BER gains compared with the system without diversity combining.
The BER performance against $E_{b(elec)}/N_0$ depends on the ratio between electrical energy per bit and single-sided noise power spectral density, but the main system constraint is usually the average transmitted optical power [@DC-biased]. So we also investigate the BER performance comparison among DCO-FOFDM system and 2-layer ACO-FOFDM system with or without diversity combining at different values of $E_{b(opt)}/N_0$. The conversion from $E_{b(elec)}/N_0$ to $E_{b(opt)}/N_0$ is demonstrated as following.
For DCO-FOFDM system, with unity optical power the relationship between $E_{b(elec)}/N_0$ and $E_{b(opt)}/N_0$ can be derived as [@DC-biased; @Asymmetrically; @Clipped; @Optical; @Fast; @OFDM] $$\frac{E_{b(opt)}}{N_0}= \frac{k^2}{1+k^2}\frac{E_{b(elec)}}{N_0}
\label{eq8}$$ where $k$ is related to the DC-bias $B_{DC}$ that represented as $B_{DC}=10log_{10}(k^2+1)$ dB.
For 2-layer ACO-FOFDM system, the average electrical signal power and optical signal power can be expressed as [@Augmenting; @Asymmetrically; @Clipped; @Optical; @Fast; @OFDM], $$\begin{aligned}
&P_{elec}=E\left[x_n^2\right]=E\left[\left(\sum\limits_{l=1}^{2}{x_n^{(l)}}\right)^2\right]\nonumber\\
&{\kern 21pt}=E\left[\left({x_n^{(1)}}\right)^2\right]+E\left[\left({x_n^{(2)}}\right)^2\right]+
2E\left[x_n^{(1)}\right]E\left[x_n^{(2)}\right]\nonumber\\
&{\kern 21pt}=\frac{1}{2}\left({\sigma^{(1)}}\right)^2+\frac{1}{2}\left({\sigma^{(2)}}\right)^2+2\frac{\sigma^{(1)}}{\sqrt{2\pi}}\frac{\sigma^{(2)}}{\sqrt{2\pi}}
\label{eq9}\end{aligned}$$ $$\begin{aligned}
&P_{opt}=E\left[x_n\right]=E\left[\sum\limits_{l=1}^{2}{x_n^{(l)}}\right]=\frac{\sigma^{(1)}}{\sqrt{2\pi}}+\frac{\sigma^{(2)}}{\sqrt{2\pi}}
\label{eq10}\end{aligned}$$ where $\left({\sigma^{(l)}}\right)^2$ ($l=1,2$) is the variance of bipolar signal on layer $l$. Therefore, the ratio between $\left({\sigma^{(1)}}\right)^2$ and $\left({\sigma^{(2)}}\right)^2$ is the same as the ratio between average signal power of layer $1$ and layer $2$. Due to the average signal power applied to each layer is proportion to the number of bits transmitted in each layer, and half number of subcarriers in layer $2$ carry useful signal compared with layer $1$, the ratios between $\left({\sigma^{(1)}}\right)^2$ and $\left({\sigma^{(2)}}\right)^2$ for the spectral efficiency of 1 bit/s/Hz (i.e., BPSK+4PAM), 2 bits/s/Hz (i.e., 8PAM+4PAM) and 3 bits/s/Hz (i.e., 16PAM+16PAM) are 1, 3 and 2, which can be represented by the alphabet $p$. So for the case of $P_{opt}=1$[@DC-biased; @Comparison; @of; @ACO-OFDM], the relationship between $E_{b(elec)}/N_0$ and $E_{b(opt)}/N_0$ can be derived as $$\frac{E_{b(opt)}}{N_0}= \frac{(1+\sqrt{p})^2}{(1+p)\pi+2\sqrt{p}}\frac{E_{b(elec)}}{N_0}
\label{eq11}$$
![BER as a function of $E_{b(opt)}/N_0$ for DCO-FOFDM system and layered ACO-FOFDM system with or without diversity combining. $E_{b(opt)}/N_0$ denotes the ratio between optical energy per bit and single-sided noise power spectral density. The optimal DC-biases of DCO-FOFDM system are 4.9dB, 7.2dB and 9.2dB for BPSK, 4PAM and 8PAM as investigated in [@FOFDM][]{data-label="opt"}](Fig6){width="\linewidth"}
The BER performance comparison among DCO-FOFDM system and 2-layer ACO-FOFDM system with or without diversity combining at different values of $E_{b(opt)}/N_0$ is depicted in Fig. \[opt\]. The spectral efficiencies are 1 bit/s/Hz, 2 bits/s/Hz and 3 bits/s/Hz. The 2-layer ACO-FOFDM system without diversity combining is already able to achieve better BER performance than DCO-FOFDM system, with diversity combining more BER gains can be obtained. At FEC limit, the 2-layer ACO-FOFDM system with diversity combining can achieve 2.86dB, 5.26dB and 5.72dB BER gains compared with DCO-FOFDM system when the spectral efficiencies are 1 bit/s/Hz, 2 bits/s/Hz and 3 bits/s/Hz, and the system with diversity combining has the same BER gains as that in Fig. \[elec\] compared with the system without diversity combining. Therefore, layered ACO-FOFDM system with improved receiver has low-cost property through the use of DCT, and is suitable for application in adaptive IM/DD systems with zero DC-bias.
Conclusion
==========
In this paper, an improved receiver based on diversity combining in layered ACO-FOFDM system is proposed for IM/DD optical transmission systems. Layered ACO-FOFDM can compensate the weakness of traditional ACO-FOFDM in low spectral efficiency, the utilization of DCT instead of FFT can reduce the computational complexity without any influence on BER performance. At the transmitter, the superposition of multiple layers is performed in frequency domain to improve the spectral efficiency, the average signal power applied to each layer is proportion to the number of bits transmitted in each layer. At the receiver, conventional receiver obtains transmitted signal from lower layer to higher layer firstly, and improved receiver performs diversity combining technique from higher layer to lower layer to improve the BER performance. The BER performances of layered ACO-FOFDM system with improved receiver based on diversity combining and DC-offset FOFDM system with the optimal DC-bias are compared at the same spectral efficiency. Simulation results show that under different optical bit energy to noise power ratios, layered ACO-OFDM system with improved receiver has 2.86dB, 5.26dB and 5.72dB BER performance advantages at FEC limit over the DCO-FOFDM system when the spectral efficiencies are 1 bit/s/Hz, 2 bits/s/Hz and 3 bits/s/Hz, respectively. Layered ACO-FOFDM system with improved receiver based on diversity combining is suitable for application in the adaptive IM/DD systems with zero DC-bias.
This work was supported in part by National Natural Science Foundation of China (61427813, 61331010); National Key Research and Development Program (2016YFB0800302).
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abstract: 'Given surjective homomorphisms $R\to T\gets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the *connected sum* $R\#_TS$ is defined to be ring obtained by factoring out the diagonal image of $V$ in the fiber product $R\times_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R\#_TS$. This result is used to study how closely an artinian ring can be approximated by a Gorenstein ring mapping onto it. When $T$ is regular, it is shown that $R\#_TS$ almost never is a complete intersection ring. The proof uses a presentation of the cohomology algebra $\operatorname{Ext}^*_{R\#_kS}(k,k)$ as an amalgam of the algebras $\operatorname{Ext}^*_{R}(k,k)$ and $\operatorname{Ext}^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree $2$.'
address:
- 'Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.'
- 'Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.'
- 'Department of Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.'
author:
- 'H. Ananthnarayan'
- 'Luchezar L. Avramov'
- 'W. Frank Moore'
title: Connected sums of Gorenstein local rings
---
[^1]
Introduction {#introduction .unnumbered}
============
We introduce, study, and apply a new construction of local Gorenstein rings.
The starting point is the classical fiber product $R\times_TS$ of a pair of surjective homomorphisms ${\ensuremath{\varepsilon_R}}\colon R\to T\gets S\ {:}\,{\ensuremath{\varepsilon_S}}$ of local rings. It is well known that this ring is local, but until recently, little was known about its properties. In Proposition \[cmProd\] we show that if $R$, $S$, and $T$ are Cohen-Macaulay of dimension $d$, then so is $R\times_TS$, but this ring is Gorenstein only in trivial cases. When ${\ensuremath{\varepsilon_R}}={\ensuremath{\varepsilon_S}}$, D’Anna [@D] and Shapiro [@Sh] proposed and partly proved a criterion for $R\times_TR$ to be Gorenstein. We complete and strengthen their results in Theorem \[thm:danna\]: $R\times_TR$ Is Gorenstein if and only if $R$ is Cohen-Macaulay and $\operatorname{Ker}{\ensuremath{\varepsilon_R}}$ is a canonical module for $R$.
Our main construction involves, in addition to the ring homomorphisms ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$, a $T$-module $V$ and homomorphisms ${\ensuremath{\iota_R}}\colon V\to R$ of $R$-modules and ${\ensuremath{\iota_S}}\colon V\to S$ of $S$-modules, for the structures induced through ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$, respectively. When these maps satisfy ${\ensuremath{\varepsilon_R}}{\ensuremath{\iota_R}}={\ensuremath{\varepsilon_S}}{\ensuremath{\iota_S}}$, we define a *connected sum* ring by the formula $$R\#_TS=(R\times_TS)/\{({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(v))\mid v\in V\}\,.$$
In case $R$, $S$, and $T$ have dimension $d$ (for some $d\ge0$), $R$ and $S$ are Gorenstein, $T$ is Cohen-Macaulay, and $V$ is a canonical module for $T$, one can choose ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ to be isomorphisms onto $(0:\operatorname{Ker}({\ensuremath{\varepsilon_R}}))$ and $(0:\operatorname{Ker}({\ensuremath{\varepsilon_S}}))$, respectively. In Theorem \[gorenstein\] we prove that if ${\ensuremath{\varepsilon_R}}{\ensuremath{\iota_R}}={\ensuremath{\varepsilon_S}}{\ensuremath{\iota_S}}$ holds, then $R\#_TS$ is Gorenstein of dimension $d$.
Much of the paper is concerned with Gorenstein rings of this form.
As a first application, we study how efficiently an artinian local ring can be approximated by a Gorenstein artinian ring mapping onto it. One numerical measure of proximity is given by the Gorenstein colength of an artinian ring, introduced in [@A1]. We obtain new estimates for this invariant. We use them in the proof of Theorem \[gcl1\] to remove a restrictive hypothesis from a result of Huneke and Vraciu [@HV], describing homomorphic images of Gorenstein local rings modulo their socles.
When $d = 0$ and $T$ is a field, the construction of $R\#_T S$ mimics the expression for the cohomology algebra of a connected sum $M\#N$ of compact smooth manifolds $M$ and $N$, in terms of the cohomology algebras of $M$ and $N$; see Example \[manifolds\]. This analogy provides the name and the notation for connected sums of rings.
The topological analogy also suggests that connected sums may be used for classifying Cohen-Macaulay quotient rings of Gorenstein rings. The corresponding classification problem is, in a heuristic sense, dual to the one approached through Gorenstein linkage: Whereas linkage operates on the set of *Cohen-Macaulay quotients of a fixed Gorenstein ring* $R$, connected sums operate on the set of *Gorenstein rings with a fixed Cohen-Macaulay quotient ring* $T$.
This point of view raises the question of identifying those rings $Q$ that are *indecomposable*, in the sense that an isomorphism $Q\cong
R\#_TS$ implies $Q\cong R$ or $Q\cong S$. In Theorem \[regular\] we show that if $T$ is regular and $Q$ is complete intersection, then either $Q$ is indecomposable, or it is a connected sum of two quadratic hypersurface rings. The argument uses the structure of the algebra $\operatorname{Ext}^*_{R\#_TS}(T,T)$, when $R$ and $S$ are artinian and $T$ is a field. In Theorem \[connected\] we show that it is an amalgam of $\operatorname{Ext}^*_{R}(T,T)$ and $\operatorname{Ext}^*_{S}(T,T)$ over a polynomial $T$-subalgebra, generated by an element of degree $2$. The machinery for the proof is fine-tuned in Sections \[sec:Cohomology algebras\] and \[sec:Cohomology of fiber products\].
Fiber products {#sec:Pdcts}
==============
The *fiber product* of a diagram of homomorphisms of commutative rings $$\label{diagramProd}
\begin{gathered}
\xymatrixrowsep{1pc}
\xymatrixcolsep{1pc}
\xymatrix{
R
\ar[dr]^{{\ensuremath{\varepsilon_R}}}
\\
&{\quad}T{\quad}
\\
S
\ar[ur]_{{\ensuremath{\varepsilon_S}}}
}
\end{gathered}$$ is the subring of $R\times S$, defined by the formula $$\label{eq:Prod}
R\times_TS=\{(x,y)\in R\times S\mid {\ensuremath{\varepsilon_R}}(x)={\ensuremath{\varepsilon_S}}(y)\}\,.$$
If $R{\ensuremath{\xleftarrow}}{\alpha_R}A{\ensuremath{\xrightarrow}}{\alpha_S}S$ are surjective homomorphisms of rings, then for $T=R\otimes_AS$, ${\ensuremath{\varepsilon_R}}(r)=r\otimes1$, and ${\ensuremath{\varepsilon_S}}(s)=1\otimes s$ the map $a\mapsto(\alpha_R(a),\alpha_S(a))$ is a surjective homomorphism of rings $A\to R\times_TS$ with kernel $\operatorname{Ker}(\alpha_R)\cap\operatorname{Ker}(\alpha_S)$, whence $$\label{eq:presentation}
R\times_TS\cong A/(\operatorname{Ker}(\alpha_R)\cap\operatorname{Ker}(\alpha_S))\,.$$
In the sequel, the phrase *$(Q,{\ensuremath{\mathfrak q}},k)$ is a local ring* means that $Q$ is a commutative noetherian ring with unique maximal ideal ${\ensuremath{\mathfrak q}}$ and residue field $k=Q/{\ensuremath{\mathfrak q}}$.
*The following setup and notation are in force for the rest of this section:*
\[setupProd\] The rings in diagram are local: $(R,{\ensuremath{\mathfrak r}},k)$, $(S,{\ensuremath{\mathfrak s}},k)$, and $(T,{\ensuremath{\mathfrak t}},k)$.
The maps ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$ are surjective; set $I=\operatorname{Ker}({\ensuremath{\varepsilon_R}})$ and $J=\operatorname{Ker}({\ensuremath{\varepsilon_S}})$, and also $$P=R\times_TS\,.$$
Let $\eta$ denote the inclusion of rings $P\to R\times S$, and let $R\gets R\times S\to S$ be the canonical maps. Each (finite) module over $R$ or $S$ acquires a canonical structure of (finite) $P$-module through the composed homomorphisms of rings $R\gets P\to S$ (finiteness is preserved because these maps are surjective).
The rings and ideals above are related through exact sequences of $P$-modules $$\begin{gathered}
\label{eq:fiber1}
0{\ensuremath{\longrightarrow}}I\oplus J{\ensuremath{\longrightarrow}}R\oplus S{\ensuremath{\xrightarrow}}{\,{\ensuremath{\varepsilon_R}}\oplus{\ensuremath{\varepsilon_S}}\,} T\oplus T{\ensuremath{\longrightarrow}}0
\\
\label{eq:fiber2}
0{\ensuremath{\longrightarrow}}R\times_TS{\ensuremath{\xrightarrow}}{\,\eta\,} R\oplus S{\ensuremath{\xrightarrow}}{\,({\ensuremath{\varepsilon_R}},-{\ensuremath{\varepsilon_S}})\,} T{\ensuremath{\longrightarrow}}0
\end{gathered}$$
A length count in the second sequence yields the relation $$\label{eq:lengthProd}
\operatorname{\ell}(R\times_TS)+\operatorname{\ell}(T) =\operatorname{\ell}(R)+\operatorname{\ell}(S)\,.$$
For completeness, we include a proof of the following result; see [@Gr 19.3.2.1].
\[local1\] The ring $R\times_TS$ is local, with maximal ideal ${\ensuremath{\mathfrak p}}={\ensuremath{\mathfrak r}}\times_{\ensuremath{\mathfrak t}}{\ensuremath{\mathfrak s}}$.
The rings $R$ and $S$ are quotients of $P$, so they are noetherian $P$-modules. Thus, the $P$-module $R\oplus S$ is noetherian, and hence so is its submodule $P$.
If $(r,s)$ is in $P$, but not in ${\ensuremath{\mathfrak r}}\times_{\ensuremath{\mathfrak t}}{\ensuremath{\mathfrak s}}$, then $r$ is not in ${\ensuremath{\mathfrak r}}$, so $r$ is invertible in $R$. Since ${\ensuremath{\varepsilon_S}}$ is surjective, there exists $s'\in S$ with ${\ensuremath{\varepsilon_S}}(s')={\ensuremath{\varepsilon_R}}(r^{-1})$. One then has ${\ensuremath{\varepsilon_S}}(s's)={\ensuremath{\varepsilon_R}}(r^{-1}){\ensuremath{\varepsilon_R}}(r)=1$, so $a=s's$ is an invertible element of $S$. Now $(r^{-1},a^{-1}s')$ is in $P$, and it satisfies $(r^{-1},a^{-1}s')(r,s)=(r^{-1}r,a^{-1}s's)=(1,1)$.
For any sequence ${\ensuremath{\boldsymbol{x}}}$ of elements of $P$ and $P$-module $M$, we let $\operatorname{H}_n({\ensuremath{\boldsymbol{x}}},M)$ denote the $n$th homology module of the Koszul complex on ${\ensuremath{\boldsymbol{x}}}$ with coefficients in $M$.
\[local2\] When ${\ensuremath{\boldsymbol{x}}}$ is a $T$-regular sequence in $R\times_TS$ and ${\overline}M$ denotes $M/{\ensuremath{\boldsymbol{x}}}M$ for each $(R\times_TS)$-module $M$, there is an isomorphism of rings $${\overline}{R\times_TS}\cong{\overline}R\times_{{\overline}T}{\overline}S$$ and there are exact sequences of $({\overline}{R\times_TS})$-modules $$\begin{gathered}
\label{eq:fiberOv1}
0{\ensuremath{\longrightarrow}}{\overline}I\oplus{\overline}J{\ensuremath{\longrightarrow}}{\overline}R\oplus{\overline}S{\ensuremath{\xrightarrow}}{\,{\overline}{\ensuremath{\varepsilon_R}}\oplus{\overline}{\ensuremath{\varepsilon_S}}\,}
{\overline}T\oplus{\overline}T{\ensuremath{\longrightarrow}}0
\\
\label{eq:fiberOv2}
0{\ensuremath{\longrightarrow}}{\overline}{R\times_TS}{\ensuremath{\xrightarrow}}{\ {\overline}\eta\ }{\overline}R\oplus{\overline}S{\ensuremath{\xrightarrow}}{\,({\overline}{\ensuremath{\varepsilon_R}},-{\overline}{\ensuremath{\varepsilon_S}})\,}{\overline}T{\ensuremath{\longrightarrow}}0
\qedhere \end{gathered}$$
The sequence ${\ensuremath{\boldsymbol{x}}}$ is $R\times_TS$-regular if and only if it is $R$-regular and $S$-regular.
One has $\operatorname{H}_n({\ensuremath{\boldsymbol{x}}},T)=0$ for $n\ge1$, so induces an exact sequence of Koszul homology modules, which contains . It also gives an isomorphism $$\begin{aligned}
\operatorname{H}_1({\ensuremath{\boldsymbol{x}}},P)&\cong\operatorname{H}_1({\ensuremath{\boldsymbol{x}}},R)\oplus\operatorname{H}_1({\ensuremath{\boldsymbol{x}}},S)\,,
\end{aligned}$$ which shows that ${\ensuremath{\boldsymbol{x}}}$ is $P$-regular if and only if it is $R$-regular and $S$-regular.
The exact sequence of Koszul homology modules induced by contains the exact sequence , which, in turn implies the desired isomorphism of rings.
We relate numerical invariants of $P$ to the corresponding ones of $R$, $S$, and $T$.
\[invariants\] When $Q$ is a local ring and $N$ a finite $Q$-module, $\dim_QN$ denotes its Krull dimension and $\operatorname{depth}_QN$ its depth of $N$. Recall that if $P\to Q$ is a finite homomorphism of local rings, then one has $\dim_PN=\dim_QN$ and $\operatorname{depth}_PN=\operatorname{depth}_QN$.
We set $\dim Q=\dim_QN$ and $\operatorname{depth}Q=\operatorname{depth}_QQ$; thus, there are equalities $\dim Q=\dim_PQ$ and $\operatorname{depth}Q=\operatorname{depth}_PQ$.
Recall that $\operatorname{edim}Q$ denotes the *embedding dimension* of $Q$, defined to be the minimal number of generators of its maximal ideal.
\[local3\] The following (in)equalities hold: $$\begin{aligned}
\label{eq:local3.4}
\operatorname{edim}(R\times_TS)&\ge\operatorname{edim}R+\operatorname{edim}S-\operatorname{edim}T\,.
\\
\label{eq:local3.1}
\dim(R\times_TS)&=\max\{\dim R\,,\dim S\}
\ge\min\{\dim R\,,\dim S\}\ge\dim T\,.
\\
\label{eq:local3.2}
\operatorname{depth}(R\times_TS)&\ge\min\{\operatorname{depth}R\,,\operatorname{depth}S\,,\,\operatorname{depth}T+1\}\,.
\\
\label{eq:local3.3}
\operatorname{depth}T&\ge\min\{\operatorname{depth}R, \operatorname{depth}S, \operatorname{depth}(R \times_T S) -1\}\,.
\end{aligned}$$
Lemma \[local1\] gives an exact sequence of $P$-modules $$0\to{\ensuremath{\mathfrak p}}\to{\ensuremath{\mathfrak r}}\oplus{\ensuremath{\mathfrak s}}\to{\ensuremath{\mathfrak t}}\to 0$$ Tensoring it with $P/{\ensuremath{\mathfrak p}}$ over $P$, we get an exact sequence of $k$-vector spaces $${\ensuremath{\mathfrak p}}/{\ensuremath{\mathfrak p}}^2\to{\ensuremath{\mathfrak r}}/{\ensuremath{\mathfrak r}}^2\oplus{\ensuremath{\mathfrak s}}/{\ensuremath{\mathfrak s}}^2\to{\ensuremath{\mathfrak t}}/{\ensuremath{\mathfrak t}}^2\to 0$$ because we have ${\ensuremath{\mathfrak p}}{\ensuremath{\mathfrak r}}={\ensuremath{\mathfrak r}}^2$, ${\ensuremath{\mathfrak p}}{\ensuremath{\mathfrak s}}={\ensuremath{\mathfrak s}}^2$, and ${\ensuremath{\mathfrak p}}{\ensuremath{\mathfrak t}}={\ensuremath{\mathfrak t}}^2$, due to the surjective homomorphisms $R\gets P\to S\to T\gets R$. These maps also give $\min\{\dim R\,,\dim S\}\ge\dim T$ and $\dim P\ge\max\{\dim R\,,\dim S\}$, while the inclusion $\eta$ from yields $$\max\{\dim_PR\,,\dim_PS\}=\dim_P(R\oplus S)\ge\dim_P P\,.$$ For and , apply the Depth Lemma, see [@BH 1.2.9], to .
For a local ring $(Q,{\ensuremath{\mathfrak q}},k)$ and $Q$-module $N$, set $\operatorname{Soc}N=\{n\in N\mid {\ensuremath{\mathfrak q}}n=0\}$. When ${\ensuremath{\boldsymbol{x}}}$ is a maximal $N$-regular sequence, $\operatorname{rank}_k\operatorname{Soc}(N/{\ensuremath{\boldsymbol{x}}}N)$ is a positive integer that does not depend on ${\ensuremath{\boldsymbol{x}}}$, see [@BH 1.2.19], denoted $\operatorname{type}_QN$. Set $\operatorname{type}Q=\operatorname{type}_QQ$; thus, $Q$ is Gorenstein if and only if it is Cohen-Macaulay and $\operatorname{type}Q=1$.
We interpolate a useful general observation that uses fiber producs.
\[lem:socle\] Let $(Q,{\ensuremath{\mathfrak q}},k)$ be a local ring and $W$ a $k$-subspace of $(\operatorname{Soc}(Q)+{\ensuremath{\mathfrak q}}^2)/{\ensuremath{\mathfrak q}}^2$.
There exists a ring isomorphism $Q\cong B\times_kC$, where $(B,{\ensuremath{\mathfrak b}},k)$ and $(C,{\ensuremath{\mathfrak c}},k)$ are local rings, such that ${\ensuremath{\mathfrak c}}^2=0$ and ${\ensuremath{\mathfrak c}}\cong W$.
If $W=\operatorname{Soc}(Q)+{\ensuremath{\mathfrak q}}^2)/{\ensuremath{\mathfrak q}}^2$, then $\operatorname{Soc}(B)\subseteq{\ensuremath{\mathfrak b}}^2$.
When $\operatorname{Soc}(Q)$ is in ${\ensuremath{\mathfrak q}}^2$, set $B=Q$ and $C=k$. Else, pick in $\operatorname{Soc}Q$ a set ${\ensuremath{\boldsymbol{x}}}$ that maps bijectively to a basis of $W$, then choose in ${\ensuremath{\mathfrak q}}$ a set ${\ensuremath{\boldsymbol{y}}}\subset{\ensuremath{\mathfrak q}}$, so that ${\ensuremath{\boldsymbol{x}}}\cup{\ensuremath{\boldsymbol{y}}}$ maps bijectively to a basis of ${\ensuremath{\mathfrak q}}/{\ensuremath{\mathfrak q}}^2$. Set $B=Q/({\ensuremath{\boldsymbol{x}}})$ and $C=Q/({\ensuremath{\boldsymbol{y}}})$. One then has ${\ensuremath{\mathfrak q}}=({\ensuremath{\boldsymbol{x}}})+({\ensuremath{\boldsymbol{y}}})$, hence $B\otimes_QC\cong k$, and also $({\ensuremath{\boldsymbol{x}}})\cap({\ensuremath{\boldsymbol{y}}})=0$, so $Q\cong B\times_k
C$ by . The desired properties of $B$ and $C$ are verified by elementary calculations.
The next two results concern ring-theoretic properties of fiber products.
\[cmProd\] Assume that $T$ is Cohen-Macaulay, and set $d=\dim T$.
The ring $R\times_TS$ is Cohen-Macaulay of dimension $d$ if and only if $R$ and $S$ are.
When $R\times_TS$ is Cohen-Macaulay of dimension $d$ the following inequalities hold: $$\begin{aligned}
\operatorname{type}R+\operatorname{type}S
&\ge\operatorname{type}(R\times_TS)
\\
&\ge\max\{\operatorname{type}R+\operatorname{type}S-\operatorname{type}T,\operatorname{type}_RI+\operatorname{type}_SJ\} \,.
\end{aligned}$$ If, in addition, $I$ and $J$ are non-zero, then $R\times_TS$ is not Gorenstein.
The first assertion follows directly from Lemmas \[local2\] an \[local3\], so assume that $P$ is Cohen-Macaulay of dimension $d$. Choosing in $P$ an $(P\oplus T)$-regular sequence of length $d$, from we get an exact sequence of $k$-vector spaces $$0{\ensuremath{\longrightarrow}}\operatorname{Soc}({\overline}{P}){\ensuremath{\xrightarrow}}{\,\operatorname{Soc}{\overline}\eta\,}\operatorname{Soc}{\overline}R\oplus\operatorname{Soc}{\overline}S
{\ensuremath{\xrightarrow}}{\,(\operatorname{Soc}{\overline}{\ensuremath{\varepsilon_R}},-\operatorname{Soc}{\overline}{\ensuremath{\varepsilon_S}})\,}\operatorname{Soc}{\overline}T$$ It provides the inequalities involving $\operatorname{type}R$ and $\operatorname{type}S$. Formula gives ${\overline}{\ensuremath{\varepsilon_R}}(\operatorname{Soc}{\overline}I)=0= {\overline}{\ensuremath{\varepsilon_S}}(\operatorname{Soc}{\overline}J)$, so the sequence above yields ${\overline}\eta(\operatorname{Soc}{\overline}P)\supseteq\operatorname{Soc}{\overline}I\oplus\operatorname{Soc}{\overline}J$. When $I\ne0\ne J$ holds, we get ${\overline}I\ne0\ne{\overline}J$ by Nakayama’s Lemma. Since ${\overline}R$ and ${\overline}S$ are artinian, one has $\operatorname{Soc}{\overline}I\ne0\ne\operatorname{Soc}{\overline}J$, whence $\operatorname{type}P\ge 2$.
When ${\ensuremath{\varepsilon_R}}\colon R\to R/I$ is the canonical map and ${\ensuremath{\varepsilon_S}}={\ensuremath{\varepsilon_R}}$, the ring $R \bowtie I=R \times_{R/I}R$ has been studied under the name *amalgamated duplication of $R$ along $I$*. We complete and strengthen results of D’Anna and Shapiro:
\[thm:danna\] Let $R$ be a local ring, $d$ its Krull dimension, and $I$ a non-unit ideal.
The ring $R \bowtie I$ is Cohen-Macaulay if and only if $R$ is Cohen-Macaulay and $I$ is a maximal Cohen-Macaulay $R$-module.
The ring $R \bowtie I$ is Gorenstein if and only if $R$ is Cohen-Macaulay and $I$ is a canonical module for $R$, and then $R/I$ is Cohen-Macaulay with $\dim(R/I)=d-1$.
We start by listing those assertions in the theorem that are already known.
Assume that the ring $R$ is Cohen-Macaulay.
\[parts1\] If $I$ is a maximal Cohen-Macaulay module, then $R \bowtie I$ is Cohen-Macaulay: This is proved by D’Anna in [@D Discussion 10].
\[parts2\] If $I$ is a canonical module for $R$, then $R \bowtie I$ is Gorenstein: This follows from a result of Eisenbud; see [@D Theorem 12].
\[parts3\] If $R \bowtie I$ is Gorenstein *and $I$ contains a regular element*, then $I$ is a canonical module for $R$: In D’Anna’s proof of [@D Theorem 11], this is deduced from [@D Proposition 3]; the italicized part of the hypothesis does not appear in the statement of that proposition, but Shapiro [@Sh 2.1] shows that it is needed.
\[parts4\] If $R \bowtie I$ is Gorenstein and $\dim R=1$, then $I$ contains a regular element: This is proved by Shapiro, see [@Sh 2.4]; in the statement of that result it is also assumed that $R$ reduced, but this hypothesis is not used in the proof.
Set $P = R \bowtie I$ and $d=\dim R$; thus, $\dim P=d$ by .
We obtain the first assertion from a slight variation of the argument for \[parts1\]. The map $R\to R\times R$, given by $r\mapsto(r,r)$, defines a homomorphisms of rings $R\to P$ that turns $P$ into a finite $R$-module. Thus, $P$ is a Cohen-Macaulay ring if and only if it is Cohen-Macaulay as an $R$-module; see \[invariants\]. This module is isomorphic to $R\oplus I$, because each element $(r,s)\in P$ has a unique expression of the form $(r,r)+(0,s-r)$. It follows that $P$ is Cohen-Macaulay if and only if $R$ is Cohen-Macaulay and $I$ is a maximal Cohen-Macaulay $R$-module.
In view of \[parts2\], for the rest of the proof we may assume $P$ Gorenstein.
Set $T= R/I$. We have $\operatorname{depth}T \geq d - 1\ge0$ by and Proposition \[cmProd\]. By the already proved assertion, $R$ is Cohen-Macaulay with $\operatorname{depth}R=d$, so we can choose in $P$ a $T$-regular and $R$-regular sequence ${\ensuremath{\boldsymbol{x}}}$ of length $d-1 $; for each $P$-module $M$ set ${\overline}M=M/{\ensuremath{\boldsymbol{x}}}M$. By Lemma \[local2\], ${\ensuremath{\boldsymbol{x}}}$ is $P$-regular, ${\overline}I$ is an ideal in ${\overline}R$ and there are isomorphisms of rings ${\overline}T\cong {\overline}R/{\overline}I$ and ${\overline}{P}\cong {\overline}R \bowtie {\overline}I$. As ${\overline}R$ is Cohen-Macaulay with $\dim {\overline}R=1$ and ${\overline}P$ is Gorenstein, \[parts4\] shows that ${\overline}I$ contains an ${\overline}R$-regular element. This yield $\dim {\overline}T = 0$, hence $\dim T = d - 1$, so $T$ is Cohen-Macaulay. Since $R$ is Cohen-Macaulay as well, we have $\operatorname{grade}_RT=\dim R-\dim T=1$, so $I$ contains a regular element, and hence $I$ is a canonical module for $R$, due to \[parts3\].
Connected sums {#sec:ConnSum}
==============
A *connected sum diagram* of commutative rings is a commutative diagram $$\label{diagramSum}
\begin{gathered}
\xymatrixrowsep{1pc}
\xymatrixcolsep{2pc}
\xymatrix{
& R \ar[dr]^{{\ensuremath{\varepsilon_R}}}
\\
V \ar[ur]^{{\ensuremath{\iota_R}}} \ar[dr]_{{\ensuremath{\iota_S}}}
&& T
\\
& S \ar[ur]_{{\ensuremath{\varepsilon_S}}}
}
\end{gathered}$$ where $V$ is a $T$-module, ${\ensuremath{\iota_R}}$ a homomorphism of $R$-modules (with $R$ acting on $V$ through ${\ensuremath{\varepsilon_R}}$) and ${\ensuremath{\iota_S}}$ a homomorphism of $S$-modules (with $S$ acting on $V$ via ${\ensuremath{\varepsilon_S}}$).
Evidently, $\{({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(v))\in R\times S\mid v\in V\}$ is an ideal of $R\times_TS$. We define the *connected sum of $R$ and $S$ along the diagram* to be the ring $$\label{eq:Sum}
R\#_TS=(R\times_TS)/\{({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(v))\mid v\in V\}\,.$$ As in the case of fiber products, the maps in the diagram are suppressed from the notation, although the resulting ring does depend on them; see Example \[fermat\]. The choices of name and notation are explained in Example \[manifolds\].
*We fix the setup and notation for this section as follows:*
\[setupSum\] The rings in diagram are local: $(R,{\ensuremath{\mathfrak r}},k)$, $(S,{\ensuremath{\mathfrak s}},k)$ and $(T,{\ensuremath{\mathfrak t}},k)$.
The maps ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$ are surjective; set $I=\operatorname{Ker}({\ensuremath{\varepsilon_R}})$, $J=\operatorname{Ker}({\ensuremath{\varepsilon_S}})$, also $$P=R\times_TS \quad\text{and}\quad Q=R\#_TS\,.$$
The maps ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ are injective, so there are exact sequences of finite $P$-modules $$\begin{gathered}
\label{eq:injection1}
0{\ensuremath{\longrightarrow}}V\oplus V{\ensuremath{\xrightarrow}}{\,{{\ensuremath{\iota_R}}}\oplus{{\ensuremath{\iota_S}}}\,} R\oplus S{\ensuremath{\longrightarrow}}R/{\ensuremath{\iota_R}}(V)\oplus
S/{\ensuremath{\iota_S}}(V){\ensuremath{\longrightarrow}}0
\\
\label{eq:injection2}
0{\ensuremath{\longrightarrow}}V{\ensuremath{\xrightarrow}}{\,{\iota}\,} R\times_TS{\ensuremath{\xrightarrow}}{\,{\kappa}\,}R\#_TS{\ensuremath{\longrightarrow}}0
\end{gathered}$$ where $\iota\colon v\mapsto({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(v))$ and $\kappa$ is the canonical surjection.
A length count in , using formula , yields $$\label{eq:lengthSum}
\operatorname{\ell}(R\#_TS)+\operatorname{\ell}(T)+\operatorname{\ell}(V)=\operatorname{\ell}(R)+\operatorname{\ell}(S) \,.$$
\[trivial\] The ring $Q$ is local and we write $(Q,{\ensuremath{\mathfrak q}},k)$, unless $Q=0$. The condition $Q=0$ is equivalent to ${\ensuremath{\iota_R}}(V)=R$, and also to ${\ensuremath{\iota_S}}(V)=S$: This follows from the fact that $(P,{\ensuremath{\mathfrak p}},k)$ is a local ring with ${\ensuremath{\mathfrak p}}={\ensuremath{\mathfrak r}}\times_T{\ensuremath{\mathfrak s}}$, see Lemma \[local1\].
When $I=0$ one has $R\times_TS\cong S$, hence $R\#_TS\cong S/{\ensuremath{\iota_S}}(V)$.
\[regularSum\] If a sequence ${\ensuremath{\boldsymbol{x}}}$ in $R\times_TS$ is regular on $R/{\ensuremath{\iota_R}}(V)$, $S/{\ensuremath{\iota_S}}(V)$, $T$, and $V$, then it is also regular on $R$, $S$, $R\times_TS$, and $R\#_TS$, and there is an isomorphism $${\overline}{R\#_TS}\cong {\overline}R\#_{{\overline}T}{\overline}S$$ of rings, where ${\overline}M$ denotes $M/{\ensuremath{\boldsymbol{x}}}M$ for every $R\times_TS$-module $M$.
The sequence induces an exact sequence of Koszul homology modules $$\label{eq:injection}
0{\ensuremath{\longrightarrow}}\operatorname{H}_1({\ensuremath{\boldsymbol{x}}}, R)\oplus\operatorname{H}_1({\ensuremath{\boldsymbol{x}}}, S){\ensuremath{\longrightarrow}}0 {\ensuremath{\longrightarrow}}{\overline}V\oplus {\overline}V{\ensuremath{\xrightarrow}}{\,{\overline}{{\ensuremath{\iota_R}}}\oplus{\overline}{{\ensuremath{\iota_S}}}\,}{\overline}R\oplus {\overline}S$$ It follows that ${\ensuremath{\boldsymbol{x}}}$ is $R$-regular and $S$-regular. Lemma \[local2\] shows that it is also $P$-regular, so induces an exact sequence of Koszul homology modules $$0{\ensuremath{\longrightarrow}}\operatorname{H}_1({\ensuremath{\boldsymbol{x}}},Q){\ensuremath{\longrightarrow}}{\overline}V{\ensuremath{\xrightarrow}}{\ {{\overline}\iota}\ }{\overline}P{\ensuremath{\xrightarrow}}{\ {\overline}\kappa\ }
{\overline}Q{\ensuremath{\longrightarrow}}0$$ Note that ${\overline}\iota$ is equal to the composition of the diagonal map ${\overline}V\to{\overline}V\oplus{\overline}V$ and ${\overline}{{\ensuremath{\iota_R}}}\oplus{\overline}{{\ensuremath{\iota_S}}}\colon {\overline}V\oplus{\overline}V\to{\overline}R\oplus {\overline}S$. Both are injective, the second one by , so ${\overline}\iota$ is injective as well. We get $\operatorname{H}_1({\ensuremath{\boldsymbol{x}}},Q)=0$, so ${\ensuremath{\boldsymbol{x}}}$ is $Q$-regular. After identifying ${\overline}P$ and ${\overline}R\times_{{\overline}T}{\overline}S$ through Lemma \[local2\], we get ${\overline}Q\cong{\overline}R\#_{{\overline}T}{\overline}S$ from the injectivity of ${\overline}\iota$.
\[cmSum\] If the rings, $R/{\ensuremath{\iota_R}}(V)$, $S/{\ensuremath{\iota_S}}(V)$, $T$, and the $T$-module $V$ are Cohen-Macaulay of dimension $d$, then so are the rings $R$, $S$, $R\times_TS$, and $R\#_TS$.
The exact sequence implies that $R$ and $S$ are Cohen-Macaulay of dimension $d$. Proposition \[cmProd\] then shows that so is $P$; this gives $\dim Q\le d$. Let ${\ensuremath{\boldsymbol{x}}}$ be a sequence of length $d$ in $P$, which is regular on $(R/{\ensuremath{\iota_R}}(V)\oplus S/{\ensuremath{\iota_S}}(V)\oplus
T\oplus V)$. By Lemma \[regularSum\], it is also $Q$-regular, so $Q$ is Cohen-Macaulay of dimension $d$.
To describe those situations, where connected sums do not produce new rings, we review basic properties of Hilbert-Samuel multiplicities.
\[lem:multiplicity\] Let $(P,{\ensuremath{\mathfrak p}},k)$ be a Cohen-Macaulay local ring of dimension $d$.
When $k$ is infinite, the *multiplicity* $e(P)$ can be expressed as $$e(P)=\inf\{\operatorname{\ell}(P/{\ensuremath{\boldsymbol{x}}}P)\mid {\ensuremath{\boldsymbol{x}}}\text{ is a $P$-regular sequence
in }P\}\,;$$ see [@BH 4.7.11]. If $P\to P'$ is a surjective homomorphism of rings, and $P'$ is Cohen-Macaulay of dimension $d$, then by [@Sl Ch.1, 3.3] there exists in $P$ a sequence ${\ensuremath{\boldsymbol{x}}}$ that is both $P$-regular and $P'$-regular, and $e(P')=\operatorname{\ell}(P'/{\ensuremath{\boldsymbol{x}}}P')$ holds.
When $k$ is finite, one has $e_P(M)=e_{P[y]_{{\ensuremath{\mathfrak p}}[y]}}\big(M\otimes_P P[y]_{{\ensuremath{\mathfrak p}}[y]}\big)$.
The ring $P$ is regular if and only if if $e(P)=1$.
It is a quadratic hypersurface if and only if $e(P)=2$.
\[multiplicity\] Assume that the rings $R/{\ensuremath{\iota_R}}(V)$, $S/{\ensuremath{\iota_S}}(V)$, and $T$, and the $T$-module $V$, are Cohen-Macaulay, and their dimensions are equal.
When $R$ is regular one has $I=0$ and $R\#_TS\cong S/{\ensuremath{\iota_S}}(V)$.
When $R$ is a quadratic hypersurface and $I\ne0$, one has $R\#_TS\cong S$.
Set $d=\dim T$. By Proposition \[cmSum\], $P$, $Q$, $R$, and $S$ are Cohen-Macaulay of dimension $d$. Thus, every $P$-regular sequence is also regular on $Q$, $R$, and $S$.
When $R$ is regular it is a domain; $\dim R=\dim T$ implies $I=0$, so \[trivial\] applies.
Assume $I\ne0$ and $e(R)=2$. Tensoring, if necessary, the diagram with $P[x]_{{\ensuremath{\mathfrak p}}[x]}$ over $P$, we may assume that $k$ is infinite. By \[lem:multiplicity\], there is a $P$- and $R$-regular sequence ${\ensuremath{\boldsymbol{x}}}$ of length $d$ in $P$, such that $\operatorname{\ell}({\overline}R)=2$, where overbars denote reduction modulo ${\ensuremath{\boldsymbol{x}}}$. [From]{} and $I\ne0$ one gets $\operatorname{\ell}({\overline}T)=\operatorname{\ell}({\overline}R)-\operatorname{\ell}({\overline}I)\le1$. This implies $\operatorname{\ell}({\overline}T)=1= \operatorname{\ell}({\overline}V)$, so Lemma \[regularSum\] and give $\operatorname{\ell}({\overline}Q)=\operatorname{\ell}({\overline}S)$.
Setting $K=\operatorname{Ker}(Q\to S)$, one sees that the induced sequence $$0{\ensuremath{\longrightarrow}}{\overline}K{\ensuremath{\longrightarrow}}{\overline}Q{\ensuremath{\longrightarrow}}{\overline}S{\ensuremath{\longrightarrow}}0$$ is exact, due to the $S$-regularity of ${\ensuremath{\boldsymbol{x}}}$, hence ${\overline}K=0$, and thus $K=0$.
A construction of canonical modules sets the stage for the next result.
\[dualizing\] The ideal $(0:I)$ of $R$ is a $T$-module, which is isomorphic to $\operatorname{Hom}_R(T,R)$. Similarly, $(0:J)\cong\operatorname{Hom}_S(T,S)$ as $T$-modules. If $R$ and $S$ are Gorenstein, $T$ is Cohen-Macaulay, and all three rings have dimension $d$, then $(0:I)$ and $(0:J)$ are isomorphic $T$-modules, since both are canonical modules for $T$; see [@BH 3.3.7].
\[gorenstein\] Let $R$ and $S$ be Gorenstein local rings of dimension $d$, let $T$ be a Cohen-Macaulay local ring of dimension $d$ and $V$ a canonical module for $T$.
Let ${\ensuremath{\varepsilon_R}}$, ${\ensuremath{\varepsilon_S}}$, ${\ensuremath{\iota_R}}$, and ${\ensuremath{\iota_S}}$ be maps that satisfy the conditions in *\[setupSum\]* and, in addition, $${\ensuremath{\iota_R}}(V)=(0:I) \quad\text{and}\quad {\ensuremath{\iota_S}}(V)=(0:J) \,.$$
If $I\ne0$ or $J\ne0$, then $R\#_TS$ is a Gorenstein local ring of dimension $d$.
The condition $I\ne0$ is equivalent to $J\ne0$.
Indeed $I=0$ implies $R=(0:I)={\ensuremath{\iota_R}}(V)$, hence ${\ensuremath{\varepsilon_S}}{\ensuremath{\iota_S}}(V)={\ensuremath{\varepsilon_R}}{\ensuremath{\iota_R}}(V)=T$. In particular, for some $v\in V$ one has ${\ensuremath{\varepsilon_S}}{\ensuremath{\iota_S}}(v)=1\in T$, hence $S=S{\ensuremath{\iota_S}}(v)
\subseteq(0:J)$, and thus $J=0$. By symmetry, $J=0$ implies $I=0$.
The $T$-module $V$ is Cohen-Macaulay of dimension $d$, see [@BH 3.3.13]. The rings $R/(0:I)$ and $S/(0:J)$ have the same property, by [@PS 1.3]. Proposition \[cmSum\] now shows that the ring $Q$ is Cohen-Macaulay of dimension $d$.
Choose in $P$ an $(R/{\ensuremath{\iota_R}}(V)\oplus S/{\ensuremath{\iota_S}}(V)\oplus Q\oplus T\oplus
V)$-regular sequence ${\ensuremath{\boldsymbol{x}}}$ of length $d$. It suffices to show that $Q/{\ensuremath{\boldsymbol{x}}}Q$ is Gorenstein. The $T/{\ensuremath{\boldsymbol{x}}}T$-module $V/{\ensuremath{\boldsymbol{x}}}V$ is canonical, see [@BH 3.3.5], so reduction modulo ${\ensuremath{\boldsymbol{x}}}$ preserves the hypotheses of the theorem. In view of Lemma \[regularSum\], we may assume that all rings involved are artinian.
Now we have $\operatorname{Soc}V=Tu$ for some $u\in\operatorname{Soc}T$; see [@BH 3.3.13]. To prove that $Q$ is Gorenstein we show that $\kappa(\iota_R(u),0)$ generates $\operatorname{Soc}Q$. Write $q\in\operatorname{Soc}Q$ in the form $$q=\kappa(a,b)
\quad\text{with}\quad
(a,b)\in {\ensuremath{\mathfrak r}}\times_T{\ensuremath{\mathfrak s}}={\ensuremath{\mathfrak p}}\,.$$ As $S$ is Gorenstein, one has ${\ensuremath{\iota_S}}(u)\in\operatorname{Soc}S\subseteq J$. For every $i\in I$ this gives ${\ensuremath{\varepsilon_R}}(i)=0={\ensuremath{\varepsilon_S}}{\ensuremath{\iota_S}}(u)$. Thus, $(i,{\ensuremath{\iota_S}}(u))$ is in ${\ensuremath{\mathfrak p}}$, so $\kappa(i,{\ensuremath{\iota_S}}(u))\cdot q\in{\ensuremath{\mathfrak q}}\cdot q=0$ holds, hence $$(ia,0)=(i,{\ensuremath{\iota_S}}(u))\cdot(a,b)=({\ensuremath{\iota_R}}(x),{\ensuremath{\iota_S}}(x))$$ for some $x\in V$. Since ${\ensuremath{\iota_S}}$ is injective we get $x=0$, hence $ia=0$. As $i$ was arbitrarily chosen in $I$, this implies $a\in(0:I)$; that is, $a={\ensuremath{\iota_R}}(v)$ for some $v$ in $V$. By symmetry, we conclude $b={\ensuremath{\iota_S}}(w)$ for some $w\in V$. As a consequence, we get $$q=\kappa({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(w))
\quad\text{with}\quad
v,w\in V\,.$$
Pick any $t$ in ${\ensuremath{\mathfrak t}}$, then choose $r$ in ${\ensuremath{\mathfrak r}}$ and $s$ in ${\ensuremath{\mathfrak s}}$ with ${\ensuremath{\varepsilon_R}}(r)=t={\ensuremath{\varepsilon_S}}(s)$. Thus, $(r,s)$ is in ${\ensuremath{\mathfrak p}}$, hence $\kappa(r,s)$ is in ${\ensuremath{\mathfrak q}}$, whence $\kappa(r,s)\cdot q=0$. We then have $$\begin{aligned}
({\ensuremath{\iota_R}}(tv),{\ensuremath{\iota_S}}(tw))=(r{\ensuremath{\iota_R}}(v),s{\ensuremath{\iota_S}}(w))=(r,s)\cdot({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(w))=({\ensuremath{\iota_R}}(y),{\ensuremath{\iota_S}}(y))
\end{aligned}$$ for some $y\in V$. This yields ${\ensuremath{\iota_R}}(tv)={\ensuremath{\iota_R}}(y)$ and ${\ensuremath{\iota_S}}(tw)={\ensuremath{\iota_S}}(y)$, hence $tv=y=tw$, due to the injectivity of ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$; in other words, $t(v-w)=0$. Since $t$ was an arbitrary element of ${\ensuremath{\mathfrak t}}$, we get ${\ensuremath{\mathfrak t}}(v-w)=0$, hence $v=w+t'u$ for some $t'\in T$. Choosing $r'$ in $R$ and $s'$ in $S$ with ${\ensuremath{\varepsilon_R}}(r')=t'={\ensuremath{\varepsilon_S}}(s')$, we have $(r',s')\in P$ and $$\begin{aligned}
q&=\kappa({\ensuremath{\iota_R}}(w),{\ensuremath{\iota_S}}(w))+\kappa({\ensuremath{\iota_R}}(t'u),0)
\\
& =\kappa\big((r',s')\cdot({\ensuremath{\iota_R}}(u),0)\big)
\\
& =\kappa(r',s')\cdot\kappa({\ensuremath{\iota_R}}(u),0)
\end{aligned}$$ As $q$ can be any element of $\operatorname{Soc}Q$, we get $\operatorname{Soc}Q=
Q\cdot\kappa({\ensuremath{\iota_R}}(u),0)$, as desired.
Examples and variations {#Examples}
=======================
We collect examples to illustrate the hypotheses and the conclusions of results proved above, and review variants and antecedents of the notion of connected sum.
Seemingly minor perturbations of diagram may lead to non-isomorphic connected sum rings. Next we produce a concrete illustration. See also Example \[sah\] for connected sums that are not isomorphic *as graded algebras*.
\[fermat\] Over the field ${\ensuremath{\mathbb{Q}}}$ of rational numbers, form the algebras $$R={\ensuremath{\mathbb{Q}}}[x]/(x^3)\,,
\quad
S={\ensuremath{\mathbb{Q}}}[y]/(y^3)\,,
\quad\text{and}\quad
T={\ensuremath{\mathbb{Q}}}\,.$$ Letting both ${\ensuremath{\varepsilon_R}}\colon R\to T$ and ${\ensuremath{\varepsilon_S}}\colon S\to T$ be the canonical surjections, one gets $$R\times_TS={\ensuremath{\mathbb{Q}}}[x,y]/(x^3,xy,y^3)\,.$$
Set $V={\ensuremath{\mathbb{Q}}}$ and let ${\ensuremath{\iota_R}}\colon V\to R$ and ${\ensuremath{\iota_S}}\colon V\to S$ be the maps $q\mapsto qx$ and $q\mapsto qy$, respectively. The connected sum defined by these data is a local ring $(Q,{\ensuremath{\mathfrak q}},k)$ with $$Q={\ensuremath{\mathbb{Q}}}[x,y]/(x^2-y^2,xy)\,.$$
On the other hand, take the same maps ${\ensuremath{\varepsilon_R}}$, ${\ensuremath{\varepsilon_S}}$, and ${\ensuremath{\iota_S}}$ as above, and replace ${\ensuremath{\iota_R}}$ with the map $q\mapsto pq$, where $p$ is a prime number that is not congruent to $3$ modulo $4$. We then get as connected sum a local ring $(Q',{\ensuremath{\mathfrak q}}',{\ensuremath{\mathbb{Q}}})$ with $$Q'={\ensuremath{\mathbb{Q}}}[x',y']/(x'^2-py'^2,x'y')\,.$$
We claim that these rings are not isomorphic. In fact, more is true:
Every ring homomorphism $\kappa\colon Q'\to Q$ satisfies $\kappa({\ensuremath{\mathfrak q}}')
\subseteq {\ensuremath{\mathfrak q}}^2$.
Indeed, any ring homomorphism of ${\ensuremath{\mathbb{Q}}}$-algebras is ${\ensuremath{\mathbb{Q}}}$-linear, so $\kappa$ is a homomorphism of ${\ensuremath{\mathbb{Q}}}$-algebras. The images of $x'$ and $y'$ can be written in the form $$\begin{aligned}
\kappa(x')&=ax+by+cy^2
\\
\kappa(y')&=dx+ey+fy^2
\end{aligned}$$ for appropriate rational numbers $a$, $b$, $c$, $d$, $e$, and $f$. In $Q$ this gives equalities $$\begin{aligned}
(a^2+b^2)x^2=a^2x^2+b^2y^2=\kappa(x'^2)=\kappa(py'^2)=p(d^2x^2+e^2y^2)
=p(d^2+e^2)x^2
\end{aligned}$$
We need to show that the only rational solution of the equation $$\label{eq:fermat}
a^2+b^2=p(d^2+e^2)$$ is the trivial one. If not, then $a^2+b^2\ne0$. Clearing denominators, we may assume $a,b,c,d\in{\ensuremath{\mathbb{Z}}}$ and write $a^2+b^2=p^ig$ and $d^2+e^2=p^jh$ with integers $g,h,i,j\ge0$, such that $gh$ is not divisible by $p$. By Fermat’s Theorem on sums of two squares, see [@Sm §5.6], $i$ and $j$ must be even. This is impossible, as forces $i=j+1$.
Now we turn to graded rings and degree-preserving homomorphisms.
Recall that the *Hilbert series* of a graded vector space $D$ over a field $k$, with $\operatorname{rank}_kD_n<\infty$ for all $n\in{\ensuremath{\mathbb{Z}}}$ and $D_n=0$ for $n\ll0$, is the formal Laurent series $$H_D=\sum_{n>-\infty}\operatorname{rank}_k(D_n)z^n\in{\ensuremath{\mathbb{Z}}}[\![z]\!][z^{-1}]\,.$$
\[gradedProd\] Let $k$ be a field and assume that the rings $R$ and $S$ in diagram are commutative finitely generated ${\ensuremath{\mathbb{N}}}$-graded $k$-algebras with $R_0=k=S_0$, and the maps are homogeneous. Equation then can be refined to: $$\label{eq:hilbertProd}
H_{R\times_TS}=H_R+H_S-H_T\,,$$ and the obvious version of Theorem \[cmProd\] for graded rings holds as well.
Assume, in addition, that in the diagram all maps are homogeneous. Equation then can be refined to: $$\label{eq:hilbertSum}
H_{R\#_TS}=H_R+H_S-H_T-H_V\,.$$
Diligence is needed to state a graded analog of Theorem \[gorenstein\]. Recall that for each finite graded $T$-module $N$ one has $H_M=h_N/g_N$ with $h_N\in\mathbb Z[z^{\pm 1}]$ and $g_N\in\mathbb Z[z]$, and that the integer $a(N)=\deg(h_N)-\deg(g_N)$ is known as the *$a$-invariant* of $N$.
\[gorensteinAnlog\] Let $R{\ensuremath{\xrightarrow}}{{\ensuremath{\varepsilon_R}}} T{\ensuremath{\xleftarrow}}{{\ensuremath{\varepsilon_S}}}S$ be surjective homomorphisms of commutative ${\ensuremath{\mathbb{N}}}$-graded $k$-algebras of dimension $d$, with $R_0=k=S_0$. Assume that $R$ and $S$ are Gorenstein, $T$ is Cohen-Macaulay, and $V$ is a canonical module for $T$.
A connected sum diagram , with ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ isomorphisms of graded modules, exists if and only if $a(R)=a(S)$. When this is the case the graded algebra $R\#_TS$ is Gorenstein of dimension $d$, with $a(R\#_TS)=a(R)$ and $$\label{eq:hilbertSumA}
H_{R\#_TS}(z)=H_R(z)+H_S(z)-H_T(z)-(-1)^dz^{a(R)}\cdot H_T(z^{-1})\,.$$
[From]{} [@BH 4.4.5] one obtains $$H_{\operatorname{Hom}_R(T,R)}(z)
=z^{a(R)}\cdot H_{\operatorname{Hom}_R(T,R(a))}(z)
=(-1)^dz^{a(R)}\cdot H_T(z^{-1})\,,$$ and a similar formula with $S$ in place of $R$. Thus, $\operatorname{Hom}_R(T,R)
\cong\operatorname{Hom}_S(T,S)$ holds as *graded* $T$-modules if and only if $a(R)=a(S)$. In this case, Theorem \[gorenstein\] (or its proof) shows that $R\#_TS$ is Gorenstein, and formula yields .
Generation in degree $1$ does not transfer from $R$ and $S$ to $R\times_TS$ or $R\#_TS$:
\[nonstandard\] Set $T=k[z]/(z^2)$ and form the homomorphisms of $k$-algebras $$R=k[x]/(x^5)\to T\gets k[y]/(y^5)=S
\quad\text{with}\quad
x\mapsto z \gets y\,.$$ Choose $V=T$ and define homomorphisms $R{\ensuremath{\xleftarrow}}{{\ensuremath{\iota_R}}} V{\ensuremath{\xrightarrow}}{{\ensuremath{\iota_S}}}S$ by setting ${\ensuremath{\iota_R}}(1)=x^3$ and ${\ensuremath{\iota_S}}(1)=y^3$. The graded $k$-vector space $R\times_TS$ has a homogeneous basis $$\{(x^i,y^i)\}_{0\le i\le4}\cup\{(0,y^j)\}_{2\le j\le4}\,,$$ which yields $R\times_TS\cong k[u,v]/(u^5,uv^2,v^2-u^2v)$ with $\deg(u)=1$ and $\deg(v)=2$.
The canonical module of $T$ is isomorphic to $(x^3)\subset R$ and $(y^3)
\subset S$. Therefore, one gets $R \#_T S \cong k[u,v]/(v^2-u^2v,2uv-u^3)$, with degrees as above.
\[bimodules\] For the definition of connected sum given in to work in a non-commutative context, the only change needed is to require that the maps ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ in diagram be homomorphisms of $T$-*bimodules*.
When the maps in the diagram are homogeneous homomorphisms of rings and bimodules, the resulting connected sum is a graded ring.
Remark \[bimodules\] is used implicitly in the next two examples, which deal with graded-commutative, rather than commutative, $k$-algebras.
\[manifolds\] Let $M$ and $N$ be compact connected oriented smooth manifolds of the same dimension, say $n$. The connected sum $M\#N$ is the manifold obtained by removing an open $n$-disc from each manifold and gluing the resulting manifolds with boundaries along their boundary spheres through an orientation-reversing homeomorphism. The cohomology algebras with coefficients in a field $k$ satisfy $\operatorname{H}^*(M \#
N)\cong\operatorname{H}^*(M)\#_k\operatorname{H}^*(N)$, with $\varepsilon_{\operatorname{H}^*(M)}$ and $\varepsilon_{\operatorname{H}^*(N)}$ the canonical augmentations, $V=k$, and $\iota_{\operatorname{H}^*(M)}(1)$ and $\iota_{\operatorname{H}^*(N)}(1)$ the orientation classes.
What may be the earliest discussion of connected sums in a ring-theoretical context followed very closely the topological model:
\[sah\] Sah [@Sa] formed connected sums of *graded Poincaré duality algebras* along their orientation classes, largely motivated by the following special case:
A Poincaré duality algebra $R$ with $R_i=0$ for $i\ne0,1,2$ is completely described by the quadratic form $R_1\to k$, obtained by composing the map $x\mapsto x^2$ with the inverse of the orientation isomorphism $k{\ensuremath{\xrightarrow}}{\cong}R_2$. Such algebras are isomorphic if and only if the corresponding forms are equivalent, and the connected sum of two such algebras corresponds to the Witt sum of the corresponding quadratic forms.
Gorenstein colength {#Gorenstein colength}
===================
Let $(Q,{\ensuremath{\mathfrak q}},k)$ be an artinian local ring and $E$ an injective hull of $k$.
The *Gorenstein colength* of $Q$ is defined in [@A1] to be the number $$\operatorname{gcl}Q = \min\left\{\operatorname{\ell}(A)-\operatorname{\ell}(Q)\,\left|\,
\begin{gathered}
Q\cong A/I \text{ with $A$ an artinian}\\
\text{Gorenstein local ring}
\end{gathered}
\right\}\right..$$ One has $\operatorname{gcl}Q=0$ if and only if $Q$ is Gorenstein, and $$0\le \operatorname{gcl}Q\le\operatorname{\ell}(Q)<\infty\,,$$ as the trivial extension $Q\ltimes E$ is Gorenstein, see [@BH 3.3.6], and $\operatorname{\ell}(Q\ltimes E)=2\operatorname{\ell}(Q)$.
\[edim\] If $Q$ is a non-Gorenstein artinian local ring and $Q\to C$ is a surjective homomorphism with $C$ Gorenstein, then the following inequality holds: $$\operatorname{gcl}Q\ge\operatorname{edim}(Q)-(\operatorname{\ell}(Q)-\operatorname{\ell}(C))\,.$$
Let $A\to Q$ be a surjection with $(A,{\ensuremath{\mathfrak a}},k)$ Gorenstein and $\operatorname{\ell}(A)-\operatorname{\ell}(Q)=\operatorname{gcl}Q$. It factors through ${\overline}A\to Q$, where ${\overline}A=A/\operatorname{Soc}A$. Applying $\operatorname{Hom}_A(-,A)$ to the exact sequence $0\to{\ensuremath{\mathfrak a}}/{\ensuremath{\mathfrak a}}^2\to A/{\ensuremath{\mathfrak a}}^2\to A/{\ensuremath{\mathfrak a}}\to0$, one gets an exact sequence $$0\to(0:{\ensuremath{\mathfrak a}})_A\to(0:{\ensuremath{\mathfrak a}}^2)_A\to\operatorname{Hom}_A({\ensuremath{\mathfrak a}}/{\ensuremath{\mathfrak a}}^2,A)\to0$$ that yields $\operatorname{Soc}({\overline}A)\cong\operatorname{Hom}_A({\ensuremath{\mathfrak a}}/{\ensuremath{\mathfrak a}}^2,A)$. As ${\ensuremath{\mathfrak a}}$ annihilates ${\ensuremath{\mathfrak a}}/{\ensuremath{\mathfrak a}}^2$, the second module is isomorphic to $\operatorname{Hom}_k({\ensuremath{\mathfrak a}}/{\ensuremath{\mathfrak a}}^2,k)$. Set $K=\operatorname{Ker}({\overline}A\to C)$. Since $\operatorname{\ell}(\operatorname{Soc}(C))=1$, the inclusion $\operatorname{Soc}({\overline}A)/(K\cap(\operatorname{Soc}({\overline}A)) \subseteq\operatorname{Soc}(C)$ gives the second inequality below: $$\operatorname{\ell}(K)\ge\operatorname{\ell}(K\cap\operatorname{Soc}{\overline}A)\ge
\operatorname{\ell}(\operatorname{Soc}({\overline}A))-1=\operatorname{edim}A-1\ge\operatorname{edim}Q-1.$$ The desired inequality now follows from a straightforward length count: $$\operatorname{\ell}(A)-\operatorname{\ell}(Q)=(\operatorname{\ell}(K)+1)-(\operatorname{\ell}(Q)-\operatorname{\ell}(C))
\ge\operatorname{edim}Q-(\operatorname{\ell}(Q)-\operatorname{\ell}(C))\,.\qedhere$$
Rings of embedding dimension $1$ need separate consideration.
\[gorbysocle\] Let $(S,{\ensuremath{\mathfrak s}},k)$ be an artinian local ring with $\operatorname{edim}S\le1$.
The ring $S$ is Gorenstein, and one has $S\cong C/(x^n)$ with $(C,(x),k)$ a discrete valuation ring and $n=\operatorname{\ell}(S)$; thus, there is a surjective, but not bijective, homomorphism $B\to S$, where $B=C/(x^{n+1})$ is artinian, Gorenstein, with $\operatorname{\ell}(B)=\operatorname{\ell}(S)+1$.
\[CSandGC\] Let $(R,{\ensuremath{\mathfrak r}},k)$ and $(S,{\ensuremath{\mathfrak s}},k)$ be artinian local rings, with ${\ensuremath{\mathfrak r}}\ne0\ne{\ensuremath{\mathfrak s}}$.
1. When $R$ and $S$ are Gorenstein, there is an inequality $$\operatorname{gcl}(R \times_k S)\ge\operatorname{edim}R+\operatorname{edim}S-1\,;$$ equality holds if $\operatorname{edim}R=1=\operatorname{edim}S$.
2. When $R$ is not Gorenstein, there are inequalities $$1\le\operatorname{gcl}(R \times_k S) \leq
\begin{cases}
\operatorname{gcl}R &\text{if}\quad\operatorname{edim}S=1\,;
\\
\operatorname{gcl}R+\operatorname{gcl}S-1 &\text{if}\quad\operatorname{gcl}S\ge1\,.
\end{cases}$$
The ring $R\times_kS$ is not Gorenstein by Proposition \[cmProd\], hence $\operatorname{gcl}(R\times_kS)\ge1$.
\(1) The ring $R\#_kS$ is Gorenstein by Theorem \[gorenstein\], so apply Lemma \[edim\] to the homomorphism $R\times_kS\to R\#_kS$ and use $\operatorname{edim}(R\times_kS)=\operatorname{edim}R+\operatorname{edim}S$.
\(2) Choose a surjective homomorphism $A\to R$ with $A$ artinian Gorenstein and $\operatorname{\ell}(A)=\operatorname{\ell}(R)+\operatorname{gcl}R$. If $\operatorname{gcl}S\ge1$, let $B\to S$ be a surjective homomorphism with $B$ artinian Gorenstein and $\operatorname{\ell}(B)=\operatorname{\ell}(S)+\operatorname{gcl}S$; if $\operatorname{edim}S=1$, let $B\to S$ be the map described in \[gorbysocle\]. In both cases there is a commutative diagram $$\xymatrixrowsep{0.8pc}
\xymatrixcolsep{1.5pc}
\xymatrix{
& A
\ar@{->>}[r]
& R
\ar@{->>}[dr]
\\
k
\ar@{->}[ur]
\ar@{->}[dr]
& & & k
\\
& B
\ar@{->>}[r]
& S
\ar@{->>}[ur]
}$$ where two-headed arrows denote surjective homomorphisms of local rings, and the maps from $k$ are isomorphisms onto the socles of $A$ and $B$. Both compositions $R\gets k\to S$ are zero, so there is a surjective homomorphism $A\#_kB\to R\times_kS$.
In the following string the inequality holds because $A\#_kB$ is Gorenstein, see Theorem \[gorenstein\], and the first equality comes from formulas and : $$\begin{aligned}
\operatorname{gcl}(R\times_kS)
&\le \operatorname{\ell}(A\#_kB)-\operatorname{\ell}(R\times_kS)\\
&=(\operatorname{\ell}(A)+\operatorname{\ell}(B)-2)-(\operatorname{\ell}(R)+\operatorname{\ell}(S)-1)\\
&=\operatorname{gcl}R+(\operatorname{\ell}(B)-\operatorname{\ell}(S)-1)
\end{aligned}$$ The desired upper bounds now follow from the choice of $B$.
As a first application, we give a new, simple proof of a result of Teter, [@Te 2.2].
\[squareZero\] A local ring $(Q,{\ensuremath{\mathfrak q}},k)$ with ${\ensuremath{\mathfrak q}}^2 =0$ has $\operatorname{gcl}Q=1$ or $\operatorname{edim}Q\le1$.
The condition ${\ensuremath{\mathfrak q}}^2=0$ is equivalent to ${\ensuremath{\mathfrak q}}=\operatorname{Soc}Q$. Set $\operatorname{rank}_k{\ensuremath{\mathfrak q}}=s$.
One has $s=\operatorname{edim}Q$, so we assume $s\ge2$; we then have $\operatorname{gcl}Q\ge1$. Lemma \[lem:socle\] gives $Q\cong R\times_kS$ where $(R,{\ensuremath{\mathfrak r}},k)$ and $(S,{\ensuremath{\mathfrak s}},k)$ are local rings, ${\ensuremath{\mathfrak r}}^2=0$, $\operatorname{edim}R=s-1$, and $\operatorname{edim}S=1$. If $s=2$, then $\operatorname{edim}R=1$, hence $\operatorname{gcl}Q=1$ by Proposition \[CSandGC\](1). If $s\ge3$, then $\operatorname{gcl}R=1$ holds by induction, so Proposition \[CSandGC\](2) yields $\operatorname{gcl}Q=1$.
Note that the conditions $\operatorname{gcl}Q=1$ and $\operatorname{edim}Q\le1$ are mutually exclusive; one or the other holds if and only if $R$ is isomorphic to the quotient of some artinian Gorenstein ring by its socle, see \[gorbysocle\]. Such rings are characterized as follows:
\[teter\] Let $(Q,{\ensuremath{\mathfrak q}},k)$ be an artinian local ring and $E$ an injective envelope of $k$.
Teter [@Te 2.3, 1.1] proved that there exists an isomorphism $Q\cong A/\operatorname{Soc}(A)$, with $(A,{\ensuremath{\mathfrak a}},k)$ an artinian Gorenstein local ring, if and only if there is a homomorphism of $Q$-modules $\varphi\colon{\ensuremath{\mathfrak q}}\to\operatorname{Hom}_Q({\ensuremath{\mathfrak q}},E)$ satisfying $\varphi(x)(y)=\varphi(y)(x)$ for all $x,y\in{\ensuremath{\mathfrak q}}$.
His analysis includes the following observation: $E\cong \operatorname{Hom}_{A}(Q,A)$, so the exact sequence $0\to k\to A\to Q\to0$ induces an exact sequence $0\to E\to A\to k\to0$. It yields $E\cong{\ensuremath{\mathfrak a}}$, and thus a composed $Q$-linear surjection $E\cong{\ensuremath{\mathfrak a}}\to{\ensuremath{\mathfrak a}}/\operatorname{Soc}(A)={\ensuremath{\mathfrak q}}$.
Using Teter’s result, Huneke and Vraciu proved a partial converse:
\[HVthm\] If $\operatorname{Soc}Q\subseteq{\ensuremath{\mathfrak q}}^2$, $2$ is invertible in $Q$, and there exists an epimorphism $E\to{\ensuremath{\mathfrak q}}$, then $Q\cong A/\operatorname{Soc}(A)$ with $A$ Gorenstein; see [@HV 2.5].
We lift the restriction on the socle of $Q$.
\[gcl1\] Let $(Q,{\ensuremath{\mathfrak q}},k)$ be an artinian local ring, in which $2$ is invertible, and let $E$ be an injective hull of $k$.
If there is an epimorphism $E\to{\ensuremath{\mathfrak q}}$, then $Q\cong A/\operatorname{Soc}(A)$ with $A$ Gorenstein.
By Lemma \[lem:socle\], there is an isomorphism $Q\cong R\times_kS$, where $(R,{\ensuremath{\mathfrak r}},k)$ is a local ring with $\operatorname{Soc}(R)\subseteq{\ensuremath{\mathfrak r}}^2$ and $(S,{\ensuremath{\mathfrak s}},k)$ is a local ring with ${\ensuremath{\mathfrak s}}^2=0$. Choose a surjective homomorphism $P\to Q$ with $P$ Gorenstein and set $E_R=\operatorname{Hom}_P(R,P)$ and $E_S= \operatorname{Hom}_P(S,P)$. We then have $E\cong\operatorname{Hom}_P(Q,P)$ and surjective homomorphisms $$E_R\oplus E_S{\ensuremath{\xrightarrow}}{\,\alpha\,} E
{\ensuremath{\xrightarrow}}{\,\beta\,}{\ensuremath{\mathfrak q}}={\ensuremath{\mathfrak r}}\oplus{\ensuremath{\mathfrak s}}{\ensuremath{\xrightarrow}}{\,\gamma\,}{\ensuremath{\mathfrak r}}$$ where $\alpha$ is induced by the composition $Q\cong R\times_kS
\hookrightarrow R\oplus S$, $\beta$ comes from the hypothesis, and $\gamma$ is the canonical map. Note that $\operatorname{\ell}(E)=\operatorname{\ell}(Q)>
\operatorname{\ell}({\ensuremath{\mathfrak q}})=\operatorname{\ell}(\beta(E))$ implies $\operatorname{Ker}(\beta)\ne0$; since $\operatorname{\ell}(\operatorname{Soc}E)=1$, we get $\operatorname{Soc}E\subseteq\operatorname{Ker}(\beta)$.
One has ${\ensuremath{\mathfrak q}}^2\alpha(E_S)=\alpha({\ensuremath{\mathfrak q}}^2E_S)=0$. This gives ${\ensuremath{\mathfrak q}}\alpha(E_S)\subseteq\operatorname{Soc}(E)\subseteq\operatorname{Ker}(\beta)$, hence ${\ensuremath{\mathfrak q}}\beta\alpha(E_S)=\beta({\ensuremath{\mathfrak q}}\alpha(E_S))=0$, and thus $\beta\alpha(E_S)\subseteq\operatorname{Soc}{\ensuremath{\mathfrak q}}$. [From]{} here we get $$\gamma\beta\alpha(E_S)\subseteq\gamma(\operatorname{Soc}{\ensuremath{\mathfrak q}})
\subseteq\operatorname{Soc}{\ensuremath{\mathfrak r}}\subseteq\operatorname{Soc}R\subseteq{\ensuremath{\mathfrak r}}^2\,.$$ Using the inclusions above, we obtain a new string: $${\ensuremath{\mathfrak r}}=\gamma\beta\alpha(E_R\oplus E_S)
=\gamma\beta\alpha(E_R)+\gamma\beta\alpha(E_S)
\subseteq\gamma\beta\alpha(E_R)+{\ensuremath{\mathfrak r}}^2\,.$$ By Nakayama’s Lemma, $\gamma\beta\alpha$ restricts to a surjective homomorphism $E_R\to{\ensuremath{\mathfrak r}}$.
As $E_R$ is an injective envelope of $k$ over $R$, and $\operatorname{Soc}R$ is contained in ${\ensuremath{\mathfrak r}}^2$, we get $\operatorname{gcl}R=1$ or $\operatorname{edim}R\le1$ from Huneke and Vraciu’s theorem; see \[HVthm\]. On the other hand, we know from Lemma \[squareZero\] that $S$ satisfies $\operatorname{gcl}S=1$ or $\operatorname{edim}S\le1$, so from Proposition \[CSandGC\] we conclude that $\operatorname{gcl}Q=1$ or $\operatorname{edim}Q\le1$ holds.
Finally, we take a look at the values of $\operatorname{\ell}(A)-\operatorname{\ell}(Q)$, when $Q$ is fixed.
Let $Q$ be an artinian local ring; set $\operatorname{edim}Q=e$ and $\operatorname{gcl}Q=g$.
If $e\le 1$ or $g\ge1$, then for every $n\ge 0$ there is an isomorphism $Q\cong A/I$, with $A$ a Gorenstein local ring and $\operatorname{\ell}(A)-\operatorname{\ell}(Q)=g+n$.
Indeed, the case of $e=1$ is clear from \[gorbysocle\], so we assume $e\ge2$. When $g\ge1$, let $R\to Q$ be a surjective homomorphism with $R$ Gorenstein and $\operatorname{\ell}(R)=g$. For $S= k[x]/(x^{n+2})$, the canonical surjection $R\times_kS\to R\times_kk\cong R$ maps $\operatorname{Soc}(R)\oplus\operatorname{Soc}(S)$ to zero, and so factors through $R\#_kS$. Theorem \[gorenstein\] shows that this ring is Gorenstein, and formula yields $\operatorname{\ell}(R\#_kS) = g+(n+2)-2$.
Cohomology algebras {#sec:Cohomology algebras}
===================
Our next goal is to compute the cohomology algebra of a connected sum of artinian Gorenstein rings over their common residue field, in terms of the cohomology algebra of the original rings. The computation takes up three consecutive sections.
In this section we describe some functorial structures on cohomology.
\[pi\] Let $(P,{\ensuremath{\mathfrak p}},k)$ be a local ring and $\kappa\colon P\to Q$ is a surjective ring homomorphism.
Let $F$ be a minimal free resolution of $k$ over $P$. One then has $$\operatorname{Ext}^*_P(k,k)=\operatorname{Hom}_P(F,k)
\quad\text{and}\quad
\operatorname{Tor}^P_{*}(k,k)=F\otimes_Pk\,.$$
*Homological products* turns $\operatorname{Tor}^P_{*}(k,k)$ into a graded-commutative algebra with divided powers, see [@GL 2.3.5] or [@Av:barca 6.3.5]; this structure is preserved by the map $$\operatorname{Tor}^{\kappa}_*(k,k)\colon\operatorname{Tor}^P_*(k,k)\to\operatorname{Tor}^Q_*(k,k)\,.$$
*Composition products* turn $\operatorname{Ext}^*_P(k,k)$ into a graded $k$-algebra see [@GL Ch.II, §3], and the homomorphism of rings $\kappa$ induces a homomorphism of graded $k$-algebras $$\operatorname{Ext}^*_{\kappa}(k,k)\colon\operatorname{Ext}^*_{Q}(k,k)\to\operatorname{Ext}^*_{P}(k,k)\,.$$
For each $n\in{\ensuremath{\mathbb{Z}}}$, the canonical bilinear pairing $$\operatorname{Ext}^n_{P}(k,k)\times\operatorname{Tor}_n^P(k,k)\to k$$ given by evaluation is non-degenerate; we use it to identify the graded vector spaces $$\operatorname{Ext}^*_P(k,k)=\operatorname{Hom}_k(\operatorname{Tor}^P_*(k,k),k)\,.$$
Let $\pi^*(P)$ be the graded $k$-subspace of $\operatorname{Ext}^*_P(k,k)$, consisting of those elements that vanish on all products of elements in $\operatorname{Tor}^P_{+}(k,k)$ and on all divided powers $t^{(i)}$ of elements $t\in\operatorname{Tor}^P_{2j}(k,k)$ with $i\ge2$ and $j\ge1$. As $\pi^*(P)$ is closed under graded commutators in $\operatorname{Ext}^*_P(k,k)$, it is a graded Lie algebra, called the *homotopy Lie algebra* of $P$. The canonical map from the universal enveloping algebra of $\pi^*(P)$ to $\operatorname{Ext}^*_P(k,k)$ is an isomorphism; see [@Av:barca 10.2.1]. The properties of $\operatorname{Tor}^{\kappa}_*(k,k)$ and $\operatorname{Ext}^*_{\kappa}(k,k)$ show that $\kappa$ induces a homomorphism of graded Lie algebras $$\pi^*(\kappa)\colon\pi^*(Q)\to\pi^*(P)\,.$$
The maps $\operatorname{Tor}^{\kappa}_*(k,k)$, $\operatorname{Ext}^*_{\kappa}(k,k)$, and $\pi^*(\kappa)$ are functorial.
The next lemma can be deduced from [@Av:Golod 3.3]. We provide a direct proof.
\[iota\] Given a local ring $(P,{\ensuremath{\mathfrak p}},k)$ and an exact sequence of $P$-modules $$\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{
0 \ar[r]
& V \ar[r]^{\iota}
& P \ar[r]^{\kappa}
& Q \ar[r]
& 0
}$$ there is a natural exact sequence of $k$-vector spaces $$\xymatrixcolsep{2pc}
\xymatrix{
0\ar[r]
&\pi^1(Q) \ar[r]^-{\pi^1(\kappa)}
&\pi^1(P) \ar[r]
&\operatorname{Hom}_P(V,k)\ar[r]^-{{\widetilde}\iota}
&\pi^2(Q) \ar[r]^-{\pi^2(\kappa)}
&\pi^2(P)
}$$
The classical change of rings spectral sequence $$\mathrm{E}^{p,q}_2=\operatorname{Ext}^p_{Q}(k,\operatorname{Ext}^q_{P}(Q,k))
\underset p{\implies}\operatorname{Ext}^{p+q}_{P}(k,k),$$ see [@CE XVI.5.(2)${}_4$], yields a natural exact sequence of terms of low degree $$\label{eq:edge}
\begin{gathered}
\xymatrixcolsep{1.8pc}
\xymatrixrowsep{0.3pc}
\xymatrix{
&{\qquad\qquad}0\ar[r]
&\operatorname{Ext}^1_{Q}(k,k) \ar[rr]^-{\operatorname{Ext}^1_{\kappa}(k,k)}
&&\operatorname{Ext}^1_{P}(k,k)
\\
{\ }\ar[r]
&\operatorname{Ext}^1_P(Q,k) \ar[r]^-{\delta}
&\operatorname{Ext}^2_{Q}(k,k) \ar[rr]^-{\operatorname{Ext}^2_{\kappa}(k,k)}
&&\operatorname{Ext}^2_{P}(k,k)
}
\end{gathered}$$
Next we prove $\operatorname{Im}(\delta)\subseteq\pi^2(Q)$. Indeed, $\operatorname{Tor}^P_2(k,k)$ contains no divided powers, so $\pi^2(P)$ is the subspace of $k$-linear functions vanishing on $\operatorname{Tor}^Q_1(k,k)^2$. Dualizing the exact sequence above, one obtains an exact sequence $$\xymatrixcolsep{1.8pc}
\xymatrixrowsep{0.3pc}
\xymatrix{
\ar[r]
&\operatorname{Tor}_2^{P}(k,k) \ar[rr]^-{\operatorname{Tor}_2^{\kappa}(k,k)}
&&\operatorname{Tor}_2^{Q}(k,k) \ar[rr]^-{\operatorname{Hom}_k(\delta,k)}
&&\operatorname{Tor}_1^P(Q,k)
\\
\ar[r]
&\operatorname{Tor}_1^{P}(k,k) \ar[rr]^-{\operatorname{Tor}_1^{\kappa}(k,k)}
&&\operatorname{Tor}_1^{Q}(k,k)\ar[rr]
&&0{\qquad\qquad}
}$$ of $k$-vector spaces. Since $\operatorname{Tor}_*^{\kappa}(k,k)$ is a homomorphism of algebras, it gives $$\operatorname{Tor}^Q_1(k,k)^2=(\operatorname{Im}(\operatorname{Tor}_1^{\kappa}(k,k))^2
\subseteq\operatorname{Im}(\operatorname{Tor}_2^{\kappa}(k,k))=\operatorname{Ker}(\operatorname{Hom}_k(\delta,k))\,.$$ Thus, for each $\epsilon\in\operatorname{Ext}^1_P(Q,k)$ one gets $\delta(\epsilon)(\operatorname{Tor}^Q_1(k,k)^2)=0$, as desired.
The exact sequence in the hypothesis of the lemma induces an isomorphism $$\label{eq:eth}
\eth\colon \operatorname{Hom}_P(V,k){\ensuremath{\xrightarrow}}{\ \cong\ }\operatorname{Ext}^1_P(Q,k)\,.$$ of $k$-vector spaces. Setting ${\widetilde}\iota=\delta\eth$, and noting that one has $\pi^1(P)=\operatorname{Ext}^1_P(k,k)$ and $\pi^1(Q)=\operatorname{Ext}^1_Q(k,k)$, one gets the desired exact sequence from that for $\operatorname{Ext}$’s.
The following definition uses [@Av:Golod 4.6]; see \[Gol\] for the standard definition.
\[ex:Gol\] A surjective homomorphism $\kappa\colon P\to Q$ is said to be *Golod* if the induced map $\pi^*(\kappa)\colon\pi^*(Q)\to\pi^*(P)$ is surjective and its kernel is a free Lie algebra.
When $\kappa$ is Golod $\operatorname{Ker}(\pi^*(\kappa))$ is the free Lie algebra on a graded $k$-vector space $W$, with $W^i=0$ for $i\le1$ and $\operatorname{rank}_k W^i=\operatorname{rank}_k\operatorname{Ext}^{i-2}_P(Q,k)$ for all $i\ge2$.
\[TensorAlgebra\] Let ${\widetilde}V$ denote the graded vector space with ${\widetilde}V{}^i=0$ for $i\ne2$ and ${\widetilde}V{}^2=\operatorname{Hom}_P(V,k)$, let ${\ensuremath{\mathsf T}}({\widetilde}V)$ be the tensor algebra of ${\widetilde}V$, and let $$\iota^*\colon {\ensuremath{\mathsf T}}({\widetilde}V)\to\operatorname{Ext}^*_Q(k,k)$$ be the unique homomorphism of graded $k$-algebras with $\iota^2={\widetilde}\iota$; see Lemma *\[iota\]*.
If $\beta$, $\gamma$, $\kappa$, and $\kappa'$ are surjective homomorphisms of rings, the diagram $$\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{
0 \ar[r]
& V \ar[r]^{\iota} \ar[d]_-{\alpha}
& P \ar[r]^{\kappa} \ar[d]_-{\beta}
& Q \ar[r] \ar[d]_{\gamma}
& 0
\\
0 \ar[r]
& V' \ar[r]^{\iota'}
& P' \ar[r]^{\kappa'}
& Q' \ar[r]
& 0
}$$ commutes, and its rows are exact, then the following maps are equal: $$\iota^*\circ{\ensuremath{\mathsf T}}(\operatorname{Hom}_{\beta}(\alpha,k))=\operatorname{Ext}^*_{\gamma}(k,k)\circ{\widetilde}\iota'^*
\colon {\ensuremath{\mathsf T}}({\widetilde}{V}{}')\to \operatorname{Ext}^*_{Q}(k,k)\,.$$
If $V$ is cyclic and $\iota(V)$ is contained in ${\ensuremath{\mathfrak p}}^2$, or if the homomorphism $\kappa$ is Golod, then $\iota^*$ is injective, and $\operatorname{Ext}^*_Q(k,k)$ is free as a left and as a right ${\ensuremath{\mathsf T}}({\widetilde}V)$-module.
The maps ${\widetilde}\iota$ and ${\widetilde}\iota{}'$ are the compositions of the rows in the following diagram, which commutes by the naturality of the maps $\eth$ from and $\delta$ from : $$\xymatrixrowsep{2.5pc}
\xymatrixcolsep{3.5pc}
\xymatrix{
{\operatorname{Hom}_P(V,k)}\ar[r]^-{\eth}
&\operatorname{Ext}^1_P(Q,k) \ar[r]^{\delta}
&\operatorname{Ext}^2_Q(k,k)
\\
{\operatorname{Hom}_{P'}(V',k)}\ar[r]^-{\eth'} \ar[u]^{\operatorname{Hom}_{\beta}(\alpha,k)}
&\operatorname{Ext}^1_{P'}(Q',k) \ar@{->}[u]_{\operatorname{Ext}^1_{\pi}(\beta,k)}
\ar[r]^{\delta'}
&\operatorname{Ext}^2_{Q'}(k,k) \ar@{->}[u]_-{\operatorname{Ext}^2_{\gamma}(k,k)}
}$$
Set $W^2=\iota^2({\widetilde}V^2)$. The subalgebra $E=\iota^*({\ensuremath{\mathsf T}}({\widetilde}V))$ of $\operatorname{Ext}^*_{Q}(k,k)$ is generated by $W^2$. Lemma \[iota\] shows that $W^2$ is contained in $\pi^2(Q)$, so $E$ is the universal enveloping algebra of the Lie subalgebra $\omega^*$ of $\pi^*(Q)$, generated by $W^2$.
The Poincaré-Birkhoff-Witt Theorem (e.g., [@Av:barca 10.1.3.4]) implies that the universal enveloping algebra $U$ of $\pi^*(Q)$ is free as a left and as a right $E$-module. Recall, from \[pi\], that $U$ equals $\operatorname{Ext}^*_Q(k,k)$. Thus, it suffices to show that $\iota^*$ is injective. This is equivalent to injectivity of $\iota^2$ plus freeness of the associative $k$-algebra $E$; the latter condition can be replaced by freeness of the Lie algebra $\omega^*$.
If $V$ is contained in ${\ensuremath{\mathfrak p}}^2$, then $\operatorname{Ext}^1_{\kappa}(k,k)$ is surjective, so $\iota^2$ is injective by Lemma \[iota\]. If $V$ is, in addition, cyclic, then $W^2$ is a $k$-subspace of $\pi^*(Q)$, generated by a non-zero element of even degree. Any such subspace is a free Lie subalgebra.
When $\kappa$ is Golod, $\pi^1(\kappa)$ is surjective by \[ex:Gol\], so $\iota^2$ is injective by Lemma \[iota\]. Now $\operatorname{Ker}\pi^*(\kappa)$ is a free Lie algebra, again by \[ex:Gol\], hence so is its subalgebra $\omega^*$.
Cohomology of fiber products {#sec:Cohomology of fiber products}
============================
The cohomology algebra of fiber products is known, and its structure is used in the next section. To describe it, we recall a construction of coproduct of algebras.
\[coproduct\] Let $B$ and $C$ be graded $k$-algebras, with $B^0=k=C^0$ and $B^n=0=C^n$ for all $n<0$. Thus, there exist isomorphisms $B\cong{\ensuremath{\mathsf T}}(X)/K$ and $C\cong{\ensuremath{\mathsf T}}(Y)/L$, where $X$ and $Y$ are graded $k$-vector spaces, and $K$ and $L$ are ideals in the respective tensor algebras, satisfying $K\subseteq X\otimes_kX$ and $L\subseteq Y\otimes_kY$. The algebra $B\sqcup C={\ensuremath{\mathsf T}}(X\oplus Y)/(K,L)$ is a *coproduct* of $B$ and $C$ in the category of graded $k$-algebras.
Before proceeding we fix some notation.
\[convention\] When $(R,{\ensuremath{\mathfrak r}},k)$ and $(S,{\ensuremath{\mathfrak s}},k)$ are local rings, we let ${\ensuremath{\varepsilon_R}}\colon R\to k$ and ${\ensuremath{\varepsilon_S}}\colon S\to k$ denote the canonical surjections, and form the commutative diagram $$\begin{aligned}
\xi=&\qquad
\begin{gathered}
\xymatrixrowsep{1pc}
\xymatrixcolsep{2.5pc}
\xymatrix{
&R
\ar[dr]^{{\ensuremath{\varepsilon_R}}}
\\
R\times_kS{\ }
\ar[dr]_{\sigma}
\ar[ur]^{\rho}
&&{\ }k{\qquad}
\\
&S
\ar[ur]_{{\ensuremath{\varepsilon_S}}}
}
\end{gathered}
\\
\intertext{of local rings. The induced commutative diagram of graded $k$-algebras}
&\qquad \begin{gathered}
\xymatrixrowsep{1pc}
\xymatrixcolsep{2.2pc}
\xymatrix{
&{\quad}\operatorname{Ext}^*_{R}(k,k)
\ar[dr]^{\operatorname{Ext}^*_{\rho}(k,k)}
\\
k
\ar[ur]
\ar[dr]
&&{\quad}\operatorname{Ext}^*_{R\times_kS}(k,k){\quad}
\\
&{\quad}\operatorname{Ext}^*_{S}(k,k)
\ar[ur]_{\operatorname{Ext}^*_{\sigma}(k,k)}
}
\end{gathered}
\\
\intertext{see \eqref{pi}, determines a homomorphism of graded $k$-algebras }
\label{eq:xi}
\xi^*&\colon\operatorname{Ext}^*_{R}(k,k)\sqcup\operatorname{Ext}^*_{S}(k,k){\ensuremath{\longrightarrow}}\operatorname{Ext}^*_{R\times_kS}(k,k)\,.
\end{aligned}$$
The following result is [@Mo 3.4]; for $k$-algebras, see also [@PP Ch.3, 1.1].
\[coproduct cohomology\] The map $\xi^*$ in is an isomorphism of graded $k$-algebras.
To describe some invariants of modules over fiber products, we recall that the *Poincaré series* of a finite module $M$ over a local ring $(Q,{\ensuremath{\mathfrak q}},k)$ is defined by $${\ensuremath{P^{Q}_{M}}} = \sum_i \operatorname{rank}_k \operatorname{Ext}_Q^i(M,k)\,z^i\in{\ensuremath{\mathbb{Z}}}[\![z]\!]\,.$$
\[DrKr\] Dress and Krämer [@DK Thm.1] proved that each finite $R$-module $M$ satisfies $${\ensuremath{P^{R\times_kS}_{M}}}
= {\ensuremath{P^{R}_{M}}}\cdot\frac{{\ensuremath{P^{S}_{k}}}}{{\ensuremath{P^{R}_{k}}}
+ {\ensuremath{P^{S}_{k}}}
- {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}\,.$$ Formulas for Poincaré series of $S$-modules are obtained by interchanging $R$ and $S$.
\[thm:FibProdGolHom\] Let $(R,{\ensuremath{\mathfrak r}},k)$ and $(S,{\ensuremath{\mathfrak s}},k)$ be local rings and let $\varphi\colon R \to R'$ and $\psi\colon S \to S'$ be surjective homomorphisms of rings.
For the induced map $\varphi\times_k\psi\colon R\times_kS
\to R'\times_kS'$ one has an equality $${\ensuremath{P^{R\times_kS}_{R'\times_kS'}}}=
\frac{{\ensuremath{P^{R}_{R'}}}{\ensuremath{P^{S}_{k}}}+{\ensuremath{P^{S}_{S'}}}{\ensuremath{P^{R}_{k}}}-{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
{{\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}.$$
Set $I=\operatorname{Ker}(\varphi)$ and $J=\operatorname{Ker}(\psi)$. The first equality below holds because one has $\operatorname{Ker}(\varphi\times_k\psi)=I\oplus J$ as ideals; the second one comes from \[DrKr\]: $$\begin{aligned}
{\ensuremath{P^{R\times_kS}_{R'\times_kS'}}}
&=1+z\cdot({\ensuremath{P^{R\times_kS}_{I}}}+{\ensuremath{P^{R\times_kS}_{J}}})
\\
&=1+z\cdot\left(\frac{{\ensuremath{P^{R}_{I}}}{\ensuremath{P^{S}_{k}}}}
{{\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}+
\frac{{\ensuremath{P^{S}_{J}}}{\ensuremath{P^{R}_{k}}}}
{{\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}\right)
\\
&=1+\frac{z}{{\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}\cdot
\left(\frac{{\ensuremath{P^{R}_{R'}}}-1}{z}\cdot{\ensuremath{P^{S}_{k}}}+
\frac{{\ensuremath{P^{S}_{S'}}}-1}{z}\cdot{\ensuremath{P^{R}_{k}}}\right)
\\
&=\frac{{\ensuremath{P^{R}_{R'}}}{\ensuremath{P^{S}_{k}}}+{\ensuremath{P^{S}_{S'}}}{\ensuremath{P^{R}_{k}}}-{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
{{\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}.
\qedhere
\end{aligned}$$
We recall Levin’s [@Le:Gol] original definition of Golod homomorphism in terms of Poincaré series. The symbol $\preccurlyeq$ stands for termwise inequality of power series.
\[Gol\] Every surjective ring homomorphism $R\to R'$ with $(R,{\ensuremath{\mathfrak r}},k)$ local satisfies $${\ensuremath{P^{R'}_{k}}}\preccurlyeq \frac{{\ensuremath{P^{R}_{k}}}}{1+z-z{\ensuremath{P^{R}_{R'}}}}\,,$$ see, for instance, [@Av:barca 3.3.2]. Equality holds if and only if $R\to R'$ is *Golod*.
The following result is due to Lescot [@Ls 4.1].
\[cor:FibProdGolHom\] If $\varphi$ and $\psi$ are Golod, then so is $\varphi\times_k\psi$.
When the homomorphisms $\varphi$ and $\psi$ are Golod the following equalities hold: $$\begin{aligned}
\frac{1}{{\ensuremath{P^{R'\times_kS'}_{k}}}}
& = \frac{1}{{\ensuremath{P^{R'}_{k}}}} + \frac{1}{{\ensuremath{P^{S'}_{k}}}} - 1
\\
&= \frac{(1+z-z{\ensuremath{P^{R}_{R'}}})}{{\ensuremath{P^{R}_{k}}}}+ \frac{(1+z-z{\ensuremath{P^{S}_{S'}}})}{{\ensuremath{P^{S}_{k}}}} - 1
\\
& = \frac{(1+z-z{\ensuremath{P^{R}_{R'}}} ){\ensuremath{P^{S}_{k}}} + (1+z-z{\ensuremath{P^{S}_{S'}}}){\ensuremath{P^{R}_{k}}} -
{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
{{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
\\
& = \frac{(1+z)({\ensuremath{P^{R}_{k}}}+{\ensuremath{P^{S}_{k}}}-{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}})
- z({\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{S'}}} + {\ensuremath{P^{S}_{k}}}{\ensuremath{P^{R}_{R'}}}-{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}})}{{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
\\
& = \frac{(1+z-z{\ensuremath{P^{R\times_kS}_{R'\times_kS'}}})\big({\ensuremath{P^{R}_{k}}} + {\ensuremath{P^{S}_{k}}} - {\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}\big)}
{{\ensuremath{P^{R}_{k}}}{\ensuremath{P^{S}_{k}}}}
\\
&=\frac{1+z-z{\ensuremath{P^{R\times_kS}_{R'\times_kS'}}}}{{\ensuremath{P^{R\times_kS}_{k}}}}\,.\end{aligned}$$ The first and last come from \[DrKr\], the second from the definition, the penultimate one from the proposition. Stringing them together, we see that $\varphi\times_k\psi$ is Golod.
Cohomology of connected sums {#sec:Cohomology of connected sums}
============================
We compute the cohomology algebra of a connected sum of local rings over certain Golod homomorphisms, using amalgams of graded $k$-algebras.
\[amalgam\] Let $\beta\colon B\gets A\to C\,{:}\,\gamma$ be homomorphisms of graded $k$-algebras.
Let $B\sqcup_AC$ denote the quotient of the coproduct $B\sqcup C$, see \[coproduct\], by the two-sided ideal generated by the set $\{\beta(a)-\gamma(a)\mid a\in A\}$. It comes equipped with canonical homomorphisms of graded $k$-algebras $\gamma'\colon B\to B\sqcup_AC\gets C\,{:}{\hskip1.5pt}\beta'$, satisfying $\gamma'\beta=\beta'\gamma$. The universal property of coproducts implies that $B\sqcup_AC$ is an *amalgam* of $\beta$ and $\gamma$ in the category of graded $k$-algebras.
If $B$ and $C$ are free as left graded $A$-modules and as right graded $A$-modules, then Lemaire [@Lm 5.1.5 and 5.1.10] shows that the maps $\gamma'$ and $\beta'$ are injective, and $$\label{eq:amalgam2}
\frac1{H_{B\sqcup_AC}}=\frac1{H_B}+\frac1{H_C}-\frac1{H_A}\,.$$
\[setup\] Given a connected sum diagram with local rings $(R,{\ensuremath{\mathfrak r}},k)$ and $(S,{\ensuremath{\mathfrak s}},k)$, $T=k$, and canonical surjection ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$, set $R'=R/{\ensuremath{\iota_R}}(V)$ and $S'=S/{\ensuremath{\iota_S}}(V)$.
We refine to a commutative diagram $$\Xi=\qquad \begin{gathered}
\xymatrixrowsep{2pc}
\xymatrixcolsep{0.9pc}
\xymatrix{
&&& R
\ar@{->>}[rr]_(.3){\varphi}
\ar@/^2.8pc/[drrrrr]^{{\ensuremath{\varepsilon_R}}}
&& R'
\ar@/^1pc/[drrr]
\\
V
\ar@/^1pc/[urrr]^-{{\ensuremath{\iota_R}}}
\ar@{->}[rr]^-{\iota}
\ar@/_1pc/[drrr]_-{{\ensuremath{\iota_S}}}
&& R\times_kS
\ar@{->>}[ur]^-{\rho}
\ar@{->>}[rr]^-{\kappa}
\ar@{->>}[dr]_-{\sigma}
&& R\#_kS
\ar@{->>}[ur]^-{\rho'}
\ar@{->>}[rr]^-{\varkappa}
\ar@{->>}[dr]_-{\sigma'}
&& R'\times_kS'
\ar@{->>}[rr]
\ar@{->>}[ul]
\ar@{->>}[dl]
&& k
\\
&&& S
\ar@/_2.8pc/[urrrrr]_{{\ensuremath{\varepsilon_S}}}
\ar@{->>}[rr]^(.3){\psi}
&& S'
\ar@/_1pc/[urrr]
}
\end{gathered}$$ where $\iota(v)=({\ensuremath{\iota_R}}(v),{\ensuremath{\iota_S}}(v))$ and two-headed arrows denote canonical surjections.
Proposition \[TensorAlgebra\] now gives a commutative diagram of graded $k$-algebras: $$\label{eq:hopfDiag}
\xymatrixrowsep{2.5pc}
\xymatrixcolsep{.2pc}
\begin{gathered}
\xymatrix{
&&&&& \operatorname{Ext}^*_{R'}(k,k)
\ar[dr]
\ar[dl]_(.7){\operatorname{Ext}^*_{\rho'}(k,k)}
\\
{\ensuremath{\mathsf T}}({\widetilde}V)
\ar@/^1.5pc/[urrrrr]^{{\ensuremath{\iota_R}}^*}
\ar[rrrr]^-{\iota^*}
\ar@/_1.5pc/[drrrrr]_{{\ensuremath{\iota_S}}^*}
&&&&\operatorname{Ext}^*_{R\#_kS}(k,k)
&&\operatorname{Ext}^*_{R'\times_kS'}(k,k)
\ar[ll]_{\operatorname{Ext}^*_{\varkappa}(k,k)}
\\
&&&&& \operatorname{Ext}^*_{S'}(k,k)
\ar[ur]
\ar[ul]^(.7){\operatorname{Ext}^*_{\sigma'}(k,k)}
}
\end{gathered}$$
By \[amalgam\], the preceding diagram defines a homomorphism of graded $k$-algebras $$\label{eq:induced}
\Xi^*\colon
\operatorname{Ext}^*_{R'}(k,k)\sqcup_{{\ensuremath{\mathsf T}}({\widetilde}V)}\operatorname{Ext}^*_{S'}(k,k)
\longrightarrow
\operatorname{Ext}^*_{R\#_kS}(k,k)\,.$$
\[connected\] Assume that ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ in *\[setup\]* are injective and non-zero.
If the homomorphism $\varkappa\colon R\#_kS\to R'\times_kS'$ is Golod, in particular, if
1. the rings $R$ and $S$ are Gorenstein of length at least $3$, or
2. the homomorphisms $\varphi$ and $\psi$ are Golod,
then $\Xi^*$ in is an isomorphism, and the canonical maps below are injective: $$\operatorname{Ext}^*_{R'}(k,k)
{\ensuremath{\xrightarrow}}{\operatorname{Ext}^*_{\rho'}(k,k)}
\operatorname{Ext}^*_{R\#_kS}(k,k)
{\ensuremath{\xleftarrow}}{\operatorname{Ext}^*_{\sigma'}(k,k)}
\operatorname{Ext}^*_{S'}(k,k)\,.$$
\[cor:connected\] When $\varkappa$ is Golod, for every $R'$-module $N$ one has $${\ensuremath{P^{R\#_kS}_{N}}}=
{\ensuremath{P^{R'}_{N}}}\cdot\frac{{\ensuremath{P^{S'}_{k}}}}
{{\ensuremath{P^{R'}_{k}}}+{\ensuremath{P^{S'}_{k}}}-(1-rz^2)\cdot {\ensuremath{P^{R'}_{k}}} {\ensuremath{P^{S'}_{k}}}}\,,$$ where $r=\operatorname{rank}_kV$ (and thus, $r=1$ under condition *(a)*). Formulas for Poincaré series of $S'$-modules are obtained by interchanging $R'$ and $S'$.
In preparation for the proofs, we review a few items.
\[specialGol\] When $(P,{\ensuremath{\mathfrak p}},k)$ is a local ring and $\kappa\colon P\to Q$ a surjective homomorphism with ${\ensuremath{\mathfrak p}}\operatorname{Ker}(\kappa)=0$, the following inequality holds, with equality if and only if $\kappa$ is Golod: $${\ensuremath{P^{Q}_{k}}}\preccurlyeq
\frac{{\ensuremath{P^{P}_{k}}}}{1-\operatorname{rank}_k(\operatorname{Ker}(\kappa))\cdot z^2\cdot {\ensuremath{P^{P}_{k}}}}\,.$$
Indeed, the short exact sequence of $P$-modules $0\to\operatorname{Ker}(\kappa)\to P\to Q\to0$ yields ${\ensuremath{P^{P}_{Q}}} = 1 + \operatorname{rank}_k(\operatorname{Ker}(\kappa))\cdot z\cdot{\ensuremath{P^{P}_{k}}}$, so the assertion follows from \[Gol\].
The Golod property may be lost under composition or decomposition, but:
\[socSeqLemma\] Let $P{\ensuremath{\xrightarrow}}{\kappa}Q{\ensuremath{\xrightarrow}}{\varkappa}P'$ be surjective homomorphisms of rings.
When ${\ensuremath{\mathfrak p}}\operatorname{Ker}(\varkappa\kappa)=0$ holds, the map $\varkappa\kappa$ is Golod if and only if $\varkappa$ and $\kappa$ are.
Set $\operatorname{rank}_k\operatorname{Ker}(\kappa) = r$ and $\operatorname{rank}_k\operatorname{Ker}(\varkappa) = r'$. [From]{} \[specialGol\] one gets $$\begin{aligned}
{\ensuremath{P^{P'}_{k}}} \preccurlyeq \frac{{\ensuremath{P^{Q}_{k}}}}{1-r'z^2\cdot{\ensuremath{P^{Q}_{k}}}}
\preccurlyeq
\frac{{\displaystyle\frac{{\ensuremath{P^{P}_{k}}}}{1 - rz^2\cdot{\ensuremath{P^{P}_{k}}}}}}
{1 - r'z^2\cdot{\displaystyle\frac{{\ensuremath{P^{P}_{k}}}}{1-rz^2\cdot{\ensuremath{P^{P}_{k}}}}}}
= \frac{{\ensuremath{P^{P}_{k}}}}{1 - (r+r')z^2\cdot{\ensuremath{P^{P}_{k}}}}\,.\end{aligned}$$ One has $\operatorname{rank}_k\operatorname{Ker}(\varkappa\kappa) = r+r'$, so the desired assertion follows from \[specialGol\].
\[GolSocle\] When $(Q,{\ensuremath{\mathfrak q}},k)$ is an artinian Gorenstein ring with $\operatorname{edim}Q\ge2$, the canonical surjection $Q\to Q/\operatorname{Soc}Q$ is a Golod homomorphism; see [@LA Theorem 2].
For $Q=R\#_kS$ and $P'=R'\times_kS'$, we have a commutative diagram, with $\theta$ the canonical surjection, see \[amalgam\], and $\xi^*$ the bijection from \[coproduct cohomology\]: $$\xymatrixrowsep{2pc}
\xymatrixcolsep{4pc}
\xymatrix{
\operatorname{Ext}^*_{R'}(k,k)\sqcup_{{\ensuremath{\mathsf T}}({\widetilde}V)}\operatorname{Ext}^*_{S'}(k,k)
\ar[r]^-{\Xi^*}
&\operatorname{Ext}^*_{Q}(k,k)
\\
\operatorname{Ext}^*_{R'}(k,k)\sqcup\operatorname{Ext}^*_{S'}(k,k)
\ar[r]^-{\cong}_-{\xi^*}\ar[u]^-{\theta}
&\operatorname{Ext}^*_{P'}(k,k)
\ar[u]_-{\operatorname{Ext}^*_{\varkappa}(k,k)}
}$$ The map $\operatorname{Ext}^*_{\varkappa}(k,k)$ is surjective because $\varkappa$ is Golod, see \[ex:Gol\], so $\Xi^*$ is surjective.
Set $D=\operatorname{Ext}^*_{R'}(k,k)\sqcup_{{\ensuremath{\mathsf T}}({\widetilde}V)}\operatorname{Ext}^*_{R'}(k,k)$. By Proposition \[TensorAlgebra\], ${\ensuremath{\iota_R}}^*$, $\iota^*$, and ${\ensuremath{\iota_S}}^*$ turn their targets into free graded ${\ensuremath{\mathsf T}}({\widetilde}V)$-modules, left and right, so gives: $$\frac1{H_D}
=\frac1{{\ensuremath{P^{R'}_{k}}}}+\frac1{{\ensuremath{P^{S'}_{k}}}}-\frac1{H_{{\ensuremath{\mathsf T}}({\widetilde}V)}}
=\frac1{{\ensuremath{P^{R'}_{k}}}}+\frac1{{\ensuremath{P^{S'}_{k}}}}-(1-rz^2)\,.$$ On the other hand, from \[specialGol\] and \[DrKr\] we obtain $$\label{seriesQ}
\frac1{{\ensuremath{P^{Q}_{k}}}}
=\frac1{{\ensuremath{P^{P'}_{k}}}}+rz^2
=\left(\frac1{{\ensuremath{P^{R'}_{k}}}}+\frac1{{\ensuremath{P^{S'}_{k}}}}-1\right)+rz^2\,.$$ Thus, one has $H_D={\ensuremath{P^{Q}_{k}}}$. This implies that the surjection $\Xi^*$ is an isomorphism.
The injectivity of $\operatorname{Ext}^*_{\rho'}(k,k)$ and $\operatorname{Ext}^*_{\sigma'}(k,k)$ now results from Proposition \[TensorAlgebra\].
It remains to show that condition (a) or (b) implies that $\varkappa$ is Golod.
\(a) Let $R$ and $S$ be artinian Gorenstein of length at least $3$. The socle of $R$ is equal to the maximal non-zero power of ${\ensuremath{\mathfrak r}}$, and ${\ensuremath{\mathfrak r}}^2=0$ would imply $\operatorname{\ell}(R)=2$, so we have $\operatorname{Soc}R\subseteq{\ensuremath{\mathfrak r}}^2$. By symmetry, we also have $\operatorname{Soc}S\subseteq{\ensuremath{\mathfrak s}}^2$.
Set $P=R\times_kS$. By definition, $Q$ equals $P/pP$, where $p$ is a non-zero element in $\operatorname{Soc}P$. The maximal ideal ${\ensuremath{\mathfrak p}}$ of $P$ is equal to ${\ensuremath{\mathfrak r}}\oplus{\ensuremath{\mathfrak s}}$, so $\operatorname{Soc}P=\operatorname{Soc}R\oplus\operatorname{Soc}S$ is in ${\ensuremath{\mathfrak r}}^2\oplus{\ensuremath{\mathfrak s}}^2$. This gives the equality below, and the first inequality: $$\operatorname{edim}Q=\operatorname{edim}P\ge\operatorname{edim}R+\operatorname{edim}S -\operatorname{edim}k\ge2\,.$$ Since the ring $Q$ is artinian Gorenstein by Theorem \[gorenstein\], and the kernel of the map $Q\to P'$ is non-zero and is in $\operatorname{Soc}Q$, this homomorphism is a Golod by \[GolSocle\].
\(b) If $\varphi$ and $\psi$ are Golod, then so is $\varphi\times_k\psi$ by Corollary \[cor:FibProdGolHom\]. [From]{} the equality $\varphi\times_k\psi=
\varkappa\kappa$ and Lemma \[socSeqLemma\], one concludes that $\varkappa$ is Golod.
As $\operatorname{Ext}_{\rho'}(k,k)$ is injective, the first equality in the string $${{\ensuremath{P^{Q}_{N}}}}={{\ensuremath{P^{R'}_{N}}}}\cdot\frac{{\ensuremath{P^{Q}_{k}}}}{{\ensuremath{P^{R'}_{k}}}}
={{\ensuremath{P^{R'}_{N}}}}\cdot\frac{{\ensuremath{P^{S'}_{k}}}}{{\ensuremath{P^{R'}_{k}}}+
{\ensuremath{P^{S'}_{k}}}-(1-rz^2){\ensuremath{P^{R'}_{k}}}{\ensuremath{P^{S'}_{k}}}}$$ follows from a result of Levin; see [@Le:large 1.1]. The second one comes from .
Indecomposable Gorenstein rings {#sec:ciSum}
===============================
In this section we approach the problem of identifying Gorenstein rings that cannot be decomposed in a non-trivial way as a connected sum of Gorenstein local rings. Specifically, we prove that complete intersection rings have no such decomposition over regular rings, except in a single, well understood special case.
Recall that, by Cohen’s Structure Theorem, the ${\ensuremath{\mathfrak r}}$-adic completion $\widehat R$ of a local ring$(R,{\ensuremath{\mathfrak r}},k)$ is isomorphic to ${\widetilde}R/K$, with $({\widetilde}R,{\widetilde}{\ensuremath{\mathfrak r}},k)$ regular local and $K\subseteq{{\widetilde}{\ensuremath{\mathfrak r}}}^2$. One says that $R$ is *complete intersection* (*of codimension $c$*) if $K$ can be generated by a ${\widetilde}R$-regular sequence (of length $c$). A *hypersurface* ring is a complete intersection ring of codimension $1$; it is *quadratic* in case $K$ is generated by an element in ${{\widetilde}{\ensuremath{\mathfrak r}}}^2\smallsetminus{{\widetilde}{\ensuremath{\mathfrak r}}}^3$.
We also need homological characterizations of complete intersection rings:
\[ciLie\] A local ring $(R,{\ensuremath{\mathfrak r}},k)$ is complete intersection if and only if $\pi^3(R)=0$, if and only if ${\ensuremath{P^{R}_{k}}}(z)=(1+z)^b/(1-z)^c$ with $b,c\in{\ensuremath{\mathbb{Z}}}$, see [@GL 3.5.1].
If $R$ is complete intersection, then $\operatorname{codim}R=\operatorname{rank}_k\pi^2(R)=c$; see [@GL 3.4.3].
Now we return to the setup and notation of Section \[sec:ConnSum\], which we recall:
\[setupSum2\] The rings in the diagram are local: $(R,{\ensuremath{\mathfrak r}},k)$, $(S,{\ensuremath{\mathfrak s}},k)$ and $(T,{\ensuremath{\mathfrak t}},k)$.
The maps ${\ensuremath{\varepsilon_R}}$ and ${\ensuremath{\varepsilon_S}}$ are surjective; set $I=\operatorname{Ker}({\ensuremath{\varepsilon_R}})$ and $J=\operatorname{Ker}({\ensuremath{\varepsilon_S}})$.
The maps ${\ensuremath{\iota_R}}$ and ${\ensuremath{\iota_S}}$ are injective.
\[regular\] When $R$ and $S$ are Gorenstein of dimension $d$, $T$ is regular of dimension $d$, and ${\ensuremath{\iota_R}}(V)=(0:I)$ and ${\ensuremath{\iota_S}}(V)=(0:J)$, the ring $R\#_TS$ is a local complete intersection if and only if one of the following conditions holds:
1. $R$ is a quadratic hypersurface ring and $S$ is a complete intersection ring.
In this case, $R\#_TS\cong S$.
2. $S$ is a quadratic hypersurface ring and $R$ is a complete intersection ring.
In this case, $R\#_TS\cong R$.
3. $R$ and $S$ are non-quadratic hypersurface rings.
In this case, $\operatorname{codim}(R\#_TS)=2$.
Let $(P,{\ensuremath{\mathfrak p}},k)$ denote the local ring $R\times_TS$, see Lemma \[local1\], and $Q=R\#_TS$.
If $e(R)=1$ or $e(S)=1$, then $R\#_TS=0$ by Proposition \[multiplicity\] and Theorem \[gorenstein\]. Else, the ring $Q$ is local, see \[trivial\]; let ${\ensuremath{\mathfrak q}}$ denote its maximal ideal.
If $e(R)=2$, then $Q\cong S$ by Proposition \[multiplicity\], so $Q$ and $S$ are complete intersection simultaneously. The case $e(S)=2$ is similar, so we assume $e(R)\ge3$ and $e(S)\ge3$.
The $P$-modules $P$, $Q$, $R$, $S$, and $T$ are Cohen-Macaulay of dimension $d$; see Proposition \[cmProd\] and Theorem \[gorenstein\]. Tensoring the diagram with $P[y]_{{\ensuremath{\mathfrak p}}[y]}$ over $P$, we may assume that $k$ is infinite. Choose a sequence ${\ensuremath{\boldsymbol{x}}}$ in $P$ that is regular on $P$ and $T$ and satisfies $\operatorname{\ell}(T/{\ensuremath{\boldsymbol{x}}}T)=e(T)$; see \[lem:multiplicity\]. Since $T$ is a regular ring, we have $e(T)=1$, hence $T/{\ensuremath{\boldsymbol{x}}}T=k$, so the image of ${\ensuremath{\boldsymbol{x}}}$ in $T$ is a minimal set of generators of ${\ensuremath{\mathfrak t}}$. The surjective homomorphism $Q\to T$ induces a surjective $k$-linear map ${\ensuremath{\mathfrak q}}/{\ensuremath{\mathfrak q}}^2\to{\ensuremath{\mathfrak t}}/{\ensuremath{\mathfrak t}}^2$, so the image of ${\ensuremath{\boldsymbol{x}}}$ in $Q$ extends to a minimal generating set of ${\ensuremath{\mathfrak q}}$.
Since ${\ensuremath{\boldsymbol{x}}}$ is a system of parameters for $P$, and $Q$, $R$, and $S$ are $d$-dimensional Cohen-Macaulay $P$-modules, ${\ensuremath{\boldsymbol{x}}}$ is also a system of parameters for each one of them. Thus, ${\ensuremath{\boldsymbol{x}}}$ is a regular sequence on $Q$, $R$, and $S$. Since ${\ensuremath{\boldsymbol{x}}}$ is part of a minimal set of generators of ${\ensuremath{\mathfrak q}}$, the ring $Q$ is complete intersection of codimension $c$ if and only if so is $Q/{\ensuremath{\boldsymbol{x}}}Q$. Also, $R$ and $S$ are Gorenstein if and only so are $R/{\ensuremath{\boldsymbol{x}}}R$ and $S/{\ensuremath{\boldsymbol{x}}}S$, and they satisfy $\operatorname{\ell}(R)\ge e(R)\ge3$ and $\operatorname{\ell}(S)\ge e(S)\ge3$; see \[lem:multiplicity\]. Lemma \[regularSum\] gives an isomorphism of rings $Q/{\ensuremath{\boldsymbol{x}}}Q\cong (R/{\ensuremath{\boldsymbol{x}}}R)\#_k(S/{\ensuremath{\boldsymbol{x}}}S)$. Thus, after changing notation, for the rest of the proof we may assume $Q=R\#_kS$, where $R$ and $S$ are artinian Gorenstein rings that are not quadratic hypersurfaces.
Let $Q$ be complete intersection and assume $\operatorname{edim}R\ge 2$. Set $R'=R/\operatorname{Soc}R$. Theorem \[connected\] shows that the homomorphism $Q\to R'$ induces an injective homomorphism of cohomology algebras, and hence one of homotopy Lie algebras; see \[pi\]. This gives the second inequality in the following string, where the first inequality comes from \[ex:Gol\] (because $R\to R'$ is Golod by \[GolSocle\]), and the equality from \[ciLie\]: $$\operatorname{rank}_k\operatorname{Ext}^1_R(R',k)\le\operatorname{rank}_k\pi^3(R')\le\operatorname{rank}_k\pi^3(Q)=0\,.$$ It follows that $R'$ is free as an $R$-module. On the other hand, it is annihilated by $\operatorname{Soc}R$, and this ideal is non-zero because $R$ is artinian. This contradiction implies $\operatorname{edim}R=1$, so $R$ is a hypersurface ring. By symmetry, so is $S$.
Conversely, if $R$ and $S$ are hypersurface rings, then Corollary \[cor:connected\] gives $${\ensuremath{P^{Q}_{k}}}
=\frac1{1-z}\cdot\frac{\displaystyle\frac1{1-z}}
{\displaystyle\frac1{1-z}+\frac1{1-z}-(1-z^2)\cdot\frac1{1-z}\cdot\frac1{1-z}}
=\frac1{(1-z)^2}\,.$$ This implies that $Q$ is a complete intersection ring of codimension $2$; see \[ciLie\].
Acknowledgement {#acknowledgement .unnumbered}
===============
We thank Craig Huneke for several useful discussions.
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[^1]: L.L.A. was partly supported by NSF grant DMS 0803082.
|
---
abstract: 'We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0 < \alpha <\frac {4} {N-2}$, for every $\mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= \mu |x|^{-\frac {2} {\alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.'
address:
- '$^1$Université Pierre et Marie Curie & CNRS, Laboratoire Jacques-Louis Lions, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France'
- '$^2$Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21944–970 Rio de Janeiro, R.J., Brazil'
- '$^3$Laboratoire J.A. Dieudonné, UMR CNRS 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France'
- '$^4$Université Paris 13, Sorbonne Paris Cité, LAGA CNRS UMR 7539, 99 Avenue J.-B. Clément, F-93430 Villetaneuse, France'
author:
- Thierry Cazenave$^1$
- 'Flávio Dickstein$^{1,2}$'
- Ivan Naumkin$^3$
- 'Fred B. Weissler$^4$'
title: 'Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value'
---
[^1] [^2] [^3]
Introduction {#Intro}
============
In this paper we prove the existence of radially symmetric self-similar solutions of the nonlinear heat equation on ${{\mathbb R}}^N $ $$\label{NLHE}
u_t = \Delta u + |u|^\alpha u$$ with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$, where $0 < \alpha <\frac {4} {N-2}$ ($0< \alpha < \infty $ if $N=1,2$) and $ {\mu }\in {{\mathbb R}}$, $ {\mu }\neq 0$. These solutions are classical solutions, in $C((0,\infty); {{C_0({{\mathbb R}}^N )}})$, and the initial value is realized in the sense of $L^1({{\mathbb R}}^N) + L^\infty({{\mathbb R}}^N)$ if $\frac{2}{N} < \alpha < \frac {4} {N-2}$, and in the sense of $\mathcal{D}'({{\mathbb R}}^N\setminus\{0\})$ if $0 < \alpha \le \frac {2} {N}$. In all these cases, as $t \to 0$, the solution approaches the initial value uniformly on the exterior of any ball around $0$. In fact, for every $\alpha$ in the range $0 < \alpha < \frac {4} {N-2}$, and every $ {\mu }\in {{\mathbb R}}$, $ {\mu }\not = 0$, we prove the existence of [*infinitely*]{} many self-similar solutions of with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$.
This result is significant for several reasons. First, while it is already known that for every $\alpha > 0$, the equation has at least one self-similar solution with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$ if $ {\mu }\in {{\mathbb R}}$ is sufficiently small, including ${\mu }=0$, we establish this fact for [*all*]{} $ {\mu }\in {{\mathbb R}}$. Our restriction to the range $0 < \alpha < \frac {4} {N-2}$ is sharp; for $\alpha \ge \frac {4} {N-2}$, does not admit a radially symmetric self-similar solution with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$ if $| {\mu }|$ is large. See Remark \[zRem2\] , and , as well as [@CW Theorem 6.1].
Second, in the range $0 < \alpha < \frac {4} {N-2}$, it is known (Remark \[zRem2\] and ) that there exists an arbitrarily large (finite) number of self-similar solutions with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$ if $| {\mu }|$ is sufficiently small, and only if $ {\mu }= 0$ is the existence of infinitely many such solutions known (Remark \[zRem2\] , and ). Here we establish the existence of infinitely many self-similar solutions with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$, for all $ {\mu }\in {{\mathbb R}}$.
Third, for all $ {\mu }\in {{\mathbb R}}$ in the case $0 < \alpha \le \frac {2} {N}$, and for all $ {\mu }\in {{\mathbb R}}$ with $| {\mu }|$ sufficiently large in the case $\frac{2}{N} < \alpha < \frac {4} {N-2}$, the self-similar solutions we construct are all sign-changing. In other words, the positive initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$, with $ {\mu }> 0$, gives rise to solutions which assume both signs for every $t > 0$. This is particularly striking since equation does not have [*any nonnegative*]{} local in time solution with initial value $ {\mu }|x|^{-\frac {2} {\alpha }}$ with $ {\mu }> 0$ sufficiently large in the case $\alpha >\frac {2} {N} $. The same is true for all $ {\mu }> 0$ in the case $0 < \alpha \le \frac {2} {N}$ since $| \cdot |^{-\frac {2} {\alpha }} \not \in L^1_{{\mathrm{loc}}}({{\mathbb R}}^N ) $. (See Proposition \[eNEX1\] in the Appendix for details on these nonexistence properties.) Thus, we construct sign-changing solutions of with a positive initial value for which there is no local in time nonnegative solution.
This last point merits further explanation. Consider the initial value problem $$\label{Inlh}
\begin{cases}
u_t = \Delta u + |u|^\alpha u \\
u ( 0, \cdot ) = {u_0 }(\cdot )
\end{cases}$$ where $u= u(t,x)$, $t\ge 0$, $x \in \Omega $ where $\Omega $ is a domain in ${{\mathbb R}}^N $ (possibly $\Omega ={{\mathbb R}}^N $), and $\alpha >0$. In the case where $\Omega \not = {{\mathbb R}}^N $, we impose Dirichlet boundary conditions. It is well-known that this problem is locally well-posed in ${{C_0(\Omega )}}$, and also in $L^p (\Omega ) $ for $p\ge 1$, $p>\frac {N\alpha } {2}$. See [@Weissler1; @Weissler2; @BrezisC]. Moreover, if ${u_0 }\ge 0$ in $\Omega $, then the resulting solution satisfies $u\ge 0$ in $\Omega $. This is a consequence of the iterative method used to construct solutions, based on Duhamel’s formula.
On the other hand, this problem is not well-posed in $L^p (\Omega ) $ if $\alpha >\frac {2} {N}$ and $1\le p < \frac {N\alpha } {2}$. For example, regular initial values can yield multiple solutions which are continuous into these spaces. See [@HarauxW; @Baras; @QuittnerS]. Also, it was observed in [@Weissler2; @Weissler4] (see also [@QuittnerS; @LaisterRSVL; @FujishimaI]) that there are nonnegative ${u_0 }\in L^p (\Omega )$ for which there is no local-in-time nonnegative solution in the weakest possible sense. This last fact has often been considered as a proof of the non-existence of solutions of associated to those initial values. However, the possibility remains that positive initial data can give rise to local solutions which assume both positive and negative values. This of course would seemingly violate the maximum principle. Such a possibility is not completely unknown. Indeed, in [@HarauxW] both positive and negative solutions were constructed with initial value $0$. Even though these solutions are regular for $t>0$, they are too singular as $(t,x)\to (0,0)$ for any maximum principle to apply.
The present paper gives the first result, to our knowledge, of the existence of a sign-changing solution of with an initial value ${u_0 }\in L^1_{{\mathrm{loc}}}({{\mathbb R}}^N ) $, ${u_0 }\ge 0$ for which [*no*]{} local in time nonnegative solution exists. While the initial value ${u_0 }(x) = {\mu }|x|^{-\frac {2} {\alpha }}$ is not in any $L^p$ space, we show in a subsequent article [@CDNW2] that the solutions constructed here can be perturbed to give solutions to with nonnegative initial value in $L^p$, $1\le p < \frac {N\alpha } {2}$, with the same property.
In order to state our results more precisely, we recall some known facts about self-similar solutions of . A self-similar solution of is a solution of the form $$\label{fpr1}
u(t, x) = t^{-\frac {1} {\alpha }} f \Bigl( \frac {x} {\sqrt t} \Bigr),$$ where $f : {{\mathbb R}}^N \to {{\mathbb R}}$ is the profile of the self-similar solution $u$ given by . In order for $u$ given by to be a classical solution of for $t > 0$, the profile $f$ must be of class $C^2$ and satisfy the elliptic equation $$\label{fpr2}
\Delta f + \frac {1} {2} x\cdot \nabla f + \frac {1} {\alpha } f + |f|^\alpha f = 0.$$ In our investigations, we look only for radially symmetric profiles (except if $N = 1$), and so we write, by abuse of notation, $f(r) = f(x)$ where $r= |x|$, so that $f : [0, \infty) \to {{\mathbb R}}$ is of class $C^2$, and satisfies the following initial value ODE problem, $$\begin{gathered}
\displaystyle f''(r) + \Big(\frac{N-1}{r} + \frac{r}{2}\Big)f'(r) + \frac{1}{\alpha}f(r) + |f(r)|^{\alpha}f(r) = 0 \label{fpr3}\\
f(0) = a, \quad f'(0) = 0 \label{fpr3:1}\end{gathered}$$ for some $a\in {{\mathbb R}}$. Of course, if $N = 1$, it is not necessary that $f'(0) = 0$ in order to obtain a regular profile on ${{\mathbb R}}$. See Theorem \[eMain3\] below.
It is known [@HarauxW] that the problem - is well-posed. More precisely, given $a \in {{\mathbb R}}$, there exists a unique solution $f_a \in C^2([0,\infty);{{\mathbb R}})$. Furthermore the limit $$\label{prlm}
L(a) = \lim_{r\to\infty}r^{\frac{2}{\alpha}}f_a(r) \in {{\mathbb R}}$$ exists, and is a locally Lipschitz function of $a \in {{\mathbb R}}$. If $L(a) = 0$, then $f _a $ decays exponentially. See [@PTW] for more precise information on the asymptotic behavior of solutions to -. Moreover, if $a \neq 0$, then $f_a$ has at most finitely many zeros and we set $$\label{nz}
N(a) = \text{the number of zeros of the function $f_a$} .$$ Finally, given a radially symmetric self-similar solution $u$ of the form , with $f = f_a$, its initial value can be easily determined. Indeed, if $x \neq 0 $, $$\label{ssiv}
\lim_{t \to 0}u(t,x) = \lim_{t \to 0}t^{-\frac {1} {\alpha }} f_a \Bigl( \frac {|x|} {\sqrt t} \Bigr)
= |x|^{-\frac {2} {\alpha }}\lim_{r\to\infty}r^{\frac{2}{\alpha}}f_a(r)
= L(a)|x|^{-\frac {2} {\alpha }}.$$ It follows that every radially symmetric regular self-similar solution of with initial value ${u_0 }= {\mu }|x|^{-\frac {2} {\alpha }}$ is given, [*via*]{} the formula , by a profile $f = f_a$ which is a solution of - such that $L(a) = {\mu }$. Moreover, and imply that $$\label{fEP1}
|u(t, x)| \le C (t+ |x|^2 )^{- \frac {1} {\alpha }} .$$ One deduces easily from and that $$\label{fEP2}
u(t) {\mathop{\longrightarrow}}_{ t\downarrow 0 } {u_0 }\text{ in } L^q ( \{ |x|>\varepsilon \} ) \text{ for all } \varepsilon >0 \text{ and } q \ge 1, q >\frac {N\alpha } {2} ,$$ and that if $\alpha > \frac {2} {N}$, then $$\label{fEP3}
u(t) {\mathop{\longrightarrow}}_{ t\downarrow 0 } {u_0 }\text{ in } L^p ({{\mathbb R}}^N ) +L^q ({{\mathbb R}}^N ) \text{ for all } 1\le p < \frac {N\alpha } {2} <q .$$
We are now able to state our main result. It concerns the case $N\ge 2$. In dimension $1$, we have a similar result (see Theorem \[eMain3\] below), whose proof is somewhat different.
\[eMain1\] Assume $$\label{fNZ1}
N\ge 2, \quad \alpha >0,\quad \text{and}\quad (N-2) \alpha < 4 ,$$ and let ${\mu }\in {{\mathbb R}}$, $ {\mu }\not = 0$. There exists $m_0\ge 0$ such that for all $m \ge m_0$ there exist at least two different, radially symmetric regular self-similar solutions $u$ of with initial value ${u_0 }= {\mu }|x|^{-\frac {2} {\alpha }}$ in the sense , and also if $\alpha >\frac {2} {N}$, and whose profiles have exactly $m$ zeros. These solutions are such that $u\in C^1((0,\infty ), L^r ({{\mathbb R}}^N ) )$ and $\Delta u, |u|^\alpha u\in C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r\ge 1$, $r> \frac {N\alpha } {2}$.
Furthermore, if $\alpha >\frac {2} {N}$, the solutions satisfy the integral equation $$\label{fNZ2}
u(t) = e^{t\Delta } {u_0 }+ \int _0^t e^{ (t-s) \Delta } |u(s)|^\alpha u(s)\, ds$$ where each term is in $C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r> \frac {N\alpha } {2} > 1$. Moreover, the map $t \mapsto u (t) - e^{t \Delta } {u_0 }$ is in $ C([0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r \ge 1$ such that $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$.
To prove this theorem, we need to show, among other things, that the function $L$ takes on every value $ {\mu }\in{{\mathbb R}}$ infinitely often. More precisely, given $ {\mu }\in {{\mathbb R}}$, $ {\mu }\neq 0$, we need to show that for all sufficiently large integers $m$, there exist at least two values of $a$ such that $L(a) = {\mu }$ and $N(a) = m$. Note that since $L(-a) = -L(a)$, it suffices to consider $ {\mu }> 0$. To put these assertions in the appropriate historical context, and describe our approach to the proof, we recall some of the known results about the solutions $f_a$ of - and the function $L$ defined in . We let $$\label{bta}
\beta = \frac {2} {\alpha } \Bigl( N -2 - \frac {2} {\alpha } \Bigr).$$ If $\beta > 0$, i.e. if $\alpha > \frac{2}{N-2}$, then $$\label{statsingsol}
u(x) = \beta^{\frac{1}{\alpha}}|x|^{-\frac{2}{\alpha}}$$ is a singular stationary solution of . It is also self-similar, with singular profile $f(x) = \beta^{\frac{1}{\alpha}}|x|^{-\frac{2}{\alpha}}$. We will see later (see Theorem \[eMain2\]) that has other singular self-similar solutions when $\beta \ge 0$. Hence the need to specify that the self-similar solutions in Theorem \[eMain1\] are regular.
The detailed study of the profiles $f_a$ is based on the numbers $$\label{am}
a_m = \inf\{a > 0; N(a) \ge m+1\}$$ for $m\ge 0$, first defined in [@Weissler6]. The following remark recalls some of the important properties of these numbers.
\[zRem2\]
1. \[eSE1\] If $0 < a_m < \infty$, then $L(a_m) = 0$ and $N(a_m) = m$. Furthermore, if $0 < a_m < \infty$ for all large $m$, then $a_m \to \infty$ as $m\to \infty $. See [@Weissler6 Theorem 1].
2. \[eSE2\] If $\frac {2} {N} < \alpha < \frac{4}{N-2}$, then $0 < a_m < \infty$ for all $m \ge 0$. In particular, there are infinitely many radially symmetric self-similar solutions of with initial value $0$, including one which is positive. See [@HarauxW] and [@Weissler6 Theorem 1].
3. \[eSE3\] If $0 < \alpha \le \frac {2} {N}$, then there exists $m_0 \ge 0$ such that $0 < a_m < \infty$ for all $m \ge m_0$. In particular, there are infinitely many radially symmetric self-similar solutions of with initial value $0$. See [@Weissler6 Theorems 1 and 2] and [@Yanagida Proposition 2]. (In this case, $a_0 =0$ by [@HarauxW Theorem 5 (a)].)
4. \[eSE3:1\] Parts and above show that Theorem \[eMain1\] is already known in the case ${\mu }=0$. It suffices to consider the profiles $f_a$ with $a= \pm a_m$.
5. \[eSE4\] If $0 < \alpha < \frac{2}{N-2}$, then $N(a)$ is a nondecreasing function of $a > 0$ and the numbers $a_m$ are uniquely determined by the property that $L(a_m) = 0$ and $N(a_m) = m$. Furthermore, if $0 < a_m < a < a_{m+1}$, then $N(a) = m+1$ and $L(a) > 0$. See [@Yanagida].
6. \[eSE5\] If $\frac{2}{N-2} \le \alpha < \frac{4}{N-2}$, then $a_0$ is uniquely determined by the property $L(a_0) = 0$ and $N(a_0) = 0$ (i.e. $f_{a_0}(r) > 0$ for all $r > 0$). Moreover, if $0 < a < a_0$, then $f_a(r) > 0$ for all $r > 0$ and $L(a) > 0$. If $a > a_0$, then $N(a) \ge 1$. (See [@DohmenH].) In this range of $\alpha $, it is also true that for every $m \ge 0$, there exists $a_m < a < a_{m+1}$ such that $N(a) = m+ 1$ and $L(a) > 0$. This is not explicitly proved anywhere, as far as we know. It does follow from the results in [@Weissler6] along with a slight improvement of Proposition 3.7 in that paper which is straightforward to prove.
7. \[eSE6\] If $\alpha \ge \frac{4}{N-2}$ and $a > 0$, then $f_a(r) > 0$ for all $r > 0$ and $L(a) > 0$ (see [@HarauxW]). Moreover, $L:{{\mathbb R}}\to {{\mathbb R}}$ is bounded. Indeed, there is no local in time positive solution of with the initial value ${\mu }|x|^{-\frac {2} {\alpha }}$ for ${\mu }$ large, see [@Weissler4 Theorem 1] (and Lemma \[eQSol\]).
In light of these properties and in view of , to prove Theorem \[eMain1\], it would suffice to show that the successive maxima of $ | L (a)| $ on the intervals $[ a_m , a _{ m+1 } ]$ tend to infinity. Our approach to proving Theorem \[eMain1\], however, is not based directly on a study of $L(a)$ using equation -. Instead, we begin by specifying an arbitrary value of $L(a)$ and constructing the solution of - starting at infinity. This idea was previously used in [@SoupletW], which introduced the [*inverted*]{} profile equation for this purpose.
If $f$ is the profile of a radially symmetric self-similar solution, we set $$\label{fIP1}
w(s) = s^{-\frac{1} {\alpha}}f \Bigl( \frac{1} {\sqrt{s}} \Bigr),$$ for $0 < s < \infty$. The profile equation for $f$, i.e. , is equivalent to the following equation for $w$, $$\label{IPE}
4s^2w''(s) + 4\gamma sw'(s) -w'(s) - \beta w(s) + |w(s)|^\alpha w(s) = 0,$$ where $\beta$ is given by and $$\label{gma}
\gamma = \frac{2}{\alpha} - \frac{N-4}{2}.$$ If $f = f_a$, we see that $$\label{fILA}
\begin{cases}
\displaystyle a = \lim_{r \to 0}f_a(r) = \lim_{s\to\infty}s^{\frac{1}{\alpha}}w(s),\\
\displaystyle L(a) = \lim_{r\to\infty}r^{\frac{2}{\alpha}}f_a(r) = \lim_{s\to 0}w(s) = w(0).
\end{cases}$$ For an arbitrary solution $w: (0,\infty) \to {{\mathbb R}}$ of , it is not clear that $\lim_{s\to 0}w(s) = w(0)$ exists. In spite of the highly singular nature of equation at $s = 0$, global regular solutions $w: [0,\infty) \to {{\mathbb R}}$ can be constructed given any fixed values of $w(0)$ and $w'(s_0)$ for sufficiently small $s_0 > 0$. See [@SoupletW] and Section \[sIPE\] below for the details.
Theorem \[eMain1\] therefore reduces to showing that for every $ {\mu }> 0$ and for every $m > m_0$, there exist at least $2$ solutions of with $w(0) = {\mu }$, which have precisely $m$ zeros, and such that $\lim_{s\to\infty}s^{\frac{1}{\alpha}}w(s)$ exists and is finite.
It is therefore important to understand the asymptotic behavior as $s \to \infty$ of solutions to . It was shown in [@SoupletW] that if $\alpha < \frac{4}{N-2}$, then $\lim_{s\to \infty}w(s)$ always exists and is a stationary solution of . If $\beta > 0$, which corresponds to $\frac{2}{N-2} < \alpha < \frac{4}{N-2}$, then $w(s)$ must tend to either $0$ or $\pm\beta^{\frac{1}{\alpha}}$. In the case $\beta \le 0$, i.e. $0 < \alpha \le \frac{2}{N-2}$, all solutions must satisfy $w(s) \to 0$ as $s \to \infty$.
The possible asymptotic behaviors of $w(s)$ can be determined from the following partially formal argument. Setting $w(s) = z(t)$ where $t = \log s$, we obtain $$4z''(t) + 4(\gamma - 1)z'(t) - e^{-t}z'(t) - \beta z(t) + |z(t)|^\alpha z(t) = 0.$$ Deleting the term $e^{-t}z'(t)$ on the (intuitive) basis that this will be negligible as $t \to \infty$ yields the autonomous differential equation $$\label{autoode}
4z''(t) + 4(\gamma - 1)z'(t) - \beta z(t) + |z(t)|^\alpha z(t) = 0.$$ Note that $\gamma -1>0$ if $(N-2) \alpha < 4$. In the case $\beta < 0$, a standard phase plane stability analysis shows that all solutions of have one of the following two asymptotic behaviors for large $t > 0$: $$\label{fastslow1}
\begin{cases}
z(t) \sim z_1(t) = e^{-\frac{t}{\alpha}}\\
z(t) \sim z_2(t) =
\begin{cases}
e^{-t\big(\frac{1}{\alpha}-\frac{N-2}{2}\big)} & N \ge 3\\
te^{-\frac{t}{\alpha}} & N = 2 .
\end{cases}
\end{cases}$$ This translates, for solutions of , into the following two possible asymptotic behaviors: $$\label{fastslow2}
\begin{cases}
w(s) \sim \phi _1(s) = s^{-\frac{1}{\alpha}}\\
w(s) \sim \phi _2(s) =
\begin{cases}
s^{-\frac{1}{\alpha}+\frac{N-2}{2}} & N \ge 3\\
s^{-\frac{1}{\alpha}}\log s & N = 2 .
\end{cases}
\end{cases}$$ If $\beta =0$, $w$ exhibits analogous behaviors with $\phi _1 (s) = s^{-\frac {1} {\alpha }}$ and $\phi _2 (s) = (\log s)^{-\frac {1} {\alpha }}$. If $\beta >0$, the corresponding asymptotic behaviors of $w$ are given by $\phi _1(s) = s^{-\frac{1}{\alpha}}$ and $\phi _2(s) = \beta^{\frac{1}{\alpha}}$.
It is now relatively clear what must be done. These asymptotic behaviors must be proved for solutions $w(s)$ of , and the existence of solutions with asymptotic behavior like $\phi _1(s)$ with arbitrarily large number of zeros must be established. It turns out that for $N\ge 2$ the long time behavior determined by $\phi _2$, which represents slower decay than the one determined by $\phi _1$, is stable, so that the asymptotic behavior we are looking for is unstable. Our approach to proving the existence of solutions with the behavior determined by $\phi _1$ is a shooting argument. More precisely, we show the existence of solutions with the asymptotic behavior given by $\phi _2$ with arbitrarily large number of zeros. Solutions with long time behavior given by $\phi _1$ are found in the transitional regions where the number of zeros increases.
Solutions with the asymptotic behavior $\phi _2$ give rise formally to self-similar solutions of with singular profiles. It turns out that these are genuine weak solutions of when $\beta \ge 0$.
\[eMain2\] Assume $N\ge 3$ and let $ \frac {2} {N-2} \le \alpha < \frac {4} {N-2}$. There exist an integer $ \overline{m} \ge 0$ and an increasing sequence $( \overline{{\mu }} _m) _{ m \ge \overline{m} } \subset (0, \infty )$ such that for all $m\ge \overline{m} $ there exists a radially symmetric profile $ h _m$ which is a solution of in the sense of distributions, has exactly $m$ zeros, is regular for $r>0$, has the singularity $$\label{feMain2}
\begin{cases}
r^{\frac {2} {\alpha }} h _m (r) \to (-1)^m \beta^{\frac {1} {\alpha }} & \frac {2} {N-2} < \alpha < \frac {4} {N-2} \\
|\log r|^{\frac {1} {\alpha }}r^{\frac {2} {\alpha }} h _m (r) \to (-1)^m ( {\frac {2} {\alpha }} ) ^{\frac {2} {\alpha }} & \alpha = \frac {2} {N-2}
\end{cases}$$ as $r\downarrow 0$, and the asymptotic behavior $r^\frac {2} {\alpha } h_m (r) \to \overline{{\mu }} _m$ as $r \to \infty $.
If $u$ is given by with $f= h_m$, and ${u_0 }= \overline{{\mu }} _m |x|^{-\frac {2} {\alpha }}$, then $u$ is a solution of , and also of , in $C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) ) $ where $p, q$ satisfy $$\label{eSP1:5}
\begin{cases}
1\le p< \frac {N\alpha } {2(\alpha +1)} < \frac {N\alpha } {2} <q & \text{if } \alpha >\frac {2} {N-2} \\
1= p < \frac {N\alpha } {2} < q & \text{if } \alpha =\frac {2} {N-2} . \\
\end{cases}$$ Moreover, $u$ satisfies and . In addition, the map $t \mapsto u (t) - e^{t \Delta } {u_0 }$ is in $ C([0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$ if $\alpha > \frac {2} {N-2}$, and in $ C([0,\infty ), L^1 ({{\mathbb R}}^N ) + L^r ({{\mathbb R}}^N ) )$ for all $r>1$ if $\alpha = \frac {2} {N-2}$.
If $\alpha <\frac {2} {N-2}$, the singular profiles do not give rise to weak solutions of , see Remark \[eRemz\].
In the case $N=1$, the conclusion of Theorem \[eMain1\] holds. However, the asymptotic analysis of $w(s) $ is much simpler, since the profile equation has no singularity at $r=0$. This last property also allows us to construct odd profiles. Our result in this case is the following.
\[eMain3\] Assume $N =1$ and let $\alpha >0$ and $ {\mu }\not = 0$.
1. \[eMain3:1\] There exists $m_0\ge 0$ such that for all $m \ge m_0$ there exist at least two different, even, regular self-similar solutions of with initial value ${u_0 }= {\mu }|x|^{-\frac {2} {\alpha }}$ in the sense , and also if $\alpha > 2$, and whose profiles have exactly $2 m$ zeros.
2. \[eMain3:2\] There exists $m_0\ge 0$ such that for all $m \ge m_0$ there exist at least two different, odd, regular self-similar solutions of with initial value ${u_0 }= {\mu }|x|^{-\frac {2} {\alpha } -1} x$ in the sense , and also if $\alpha >2$, and whose profiles have exactly $2m +1$ zeros.
These solutions satisfy $u\in C^1((0,\infty ), L^r ({{\mathbb R}}) )$ and $u _{ xx }, |u|^\alpha u\in C((0,\infty ), L^r ({{\mathbb R}}) )$ for all $r\ge 1$, $r> \frac {\alpha } {2}$. Furthermore, if $\alpha >2$, they are solutions of the integral equation , where each term is in $C((0,\infty ), L^r ({{\mathbb R}}) )$ for all $r> \frac {\alpha } {2} > 1$. Moreover, the map $t \mapsto u (t) - e^{t \Delta } {u_0 }$ is in $ C([0,\infty ), L^r ({{\mathbb R}}) )$ for all $r \ge 1$ such that $\frac {\alpha } {2(\alpha +1) } < r < \frac {\alpha } {2}$.
The rest of the paper is organized as follows. The proof of Theorem \[eMain1\] is given in Section \[sProof\]. This proof depends on several intermediate results, which are stated in Section \[sProof\], but whose proofs are deferred until Sections \[sIPE\] to \[sFNL\]. Theorems \[eMain2\] and \[eMain3\] are proved in Sections \[sSinPro\] and \[sDim1\], respectively.
Proof of Theorem $\ref{eMain1}$ {#sProof}
===============================
In this section, we prove Theorem \[eMain1\]. The bulk of the work is the construction of the appropriate profiles, as stated in the following theorem.
\[eMT2\] Suppose . For every ${\mu }>0$, there exist an integer $m_{\mu }$ and four sequences $(a _{ {\mu }, m }^\pm ) _{ m \ge m _ {\mu }}\subset (0,\infty )$ and $(b_{ {\mu }, m }^\pm ) _{ m \ge m _{\mu }}\subset (-\infty , 0)$ such that the following properties hold, with the notation -.
1. \[eMT2:1\] $ a_{ {\mu }, m }^\pm \to \infty $ and $ b_{ {\mu }, m }^\pm \to - \infty $ as $m\to \infty $.
2. \[eMT2:2\] $L (a_{ {\mu }, m }^\pm ) =L (b_{ {\mu }, m }^\pm ) ={\mu }$ for all $m\ge m _ {\mu }$.
3. \[eMT2:3\] For all $m\ge m _ {\mu }$, $a_{ {\mu }, m }^- < a_{ {\mu }, m }^+$ and $ N (a_{ {\mu }, m }^\pm ) =2m$.
4. \[eMT2:4\] For all $m\ge m _ {\mu }$, $b_{ {\mu }, m }^+ < b_{ {\mu }, m }^-$ and $ N (b_{ {\mu }, m }^\pm ) =2m +1$.
The proof of Theorem \[eMT2\] depends on a series of propositions, which are stated below and proved in the subsequent sections. In the last part of this section, we give the proof of Theorem \[eMT2\] assuming these propositions and, at the very end of this section, obtain Theorem \[eMain1\] as a consequence.
As mentioned in Section \[Intro\], our approach is based on the study of the inverted profile equation . It is convenient to set $$\label{fNot1}
g(s) = -\beta s+ |s|^\alpha s, \quad s\in {{\mathbb R}}$$ and $$\label{fNot2}
G(s) = -\frac {\beta } {2} s^2+ \frac {1} {\alpha +2} |s|^{\alpha + 2}, \quad s\in {{\mathbb R}}$$ We collect in the following proposition an existence result for solutions of with appropriate initial conditions, as well as several properties of these solutions. These results are in part taken from [@SoupletW], and the detailed proof is given in Section \[sIPE\].
\[eEX1:b1\] Assume $N\ge 1$ and $\alpha >0$. There exists ${{\eta}}>0$ such that for all ${{\mu }}\ge {1 }$, there exists a unique solution $w_{{\mu }} \in C^1([0,\infty )) \cap C^2(0,\infty )$ of satisfying $$\label{fTT1:b1}
w_{{\mu }} (0)= {\mu }, \quad {\textstyle{ w_{{\mu }}'( {{\eta}}{\mu }^{- \alpha } ) }} =0 , \quad \| w_{\mu }\| _{ L^\infty (0, 2 {{\eta}}{\mu }^{- \alpha } ) } \le 10 {\mu }.$$ Moreover, $w _{\mu }$ satisfies the following properties.
1. \[eEX1:6:b1\] Given any $T>0$, $w_{{\mu }}$ depends continuously on ${{\mu }}$ in $C^1([0,T])$.
2. \[eEX1:2:b1\] If $$\label{fENE1:b2}
H_{{\mu }} (s)= 2s^2 w_{{\mu }} '(s)^2 + G (w_{\mu }(s))$$ with the notation -, then $$\label{fENE3:b1}
H_{{\mu }} '(s) = w_{{\mu }} '(s)^2 [1 - 4(\gamma -1) s]$$ for all $s\ge 0$.
3. \[eEX1:1:b1\] If $(N-2) \alpha < 4$, then $w_{{\mu }}(s)$ and $sw_{{\mu }}'(s)$ are bounded as $s\to \infty $.
4. \[eEX1:2:b2\] If $(N-2) \alpha < 4$, then $H_{{\mu }} (s) $ defined by has a finite limit as $s\to \infty $.
5. \[eEX1:3:b1\] If $(N-2) \alpha < 4$, then $s w_{{\mu }}' (s) \to 0$ as $s\to \infty $. Moreover, if $\beta \le 0$, then $w_{{\mu }}(s) \to 0$ as $s\to \infty $; and if $\beta >0$, then $w_{{\mu }}(s)$ converges to either $0$ or $\pm \beta ^{\frac {1} {\alpha }}$, for $\beta >0$.
6. \[eEX1:2:6\] $w _{\mu }$ has a finite number of zeros.
7. \[eEX1:2:7\] If $w_{\mu }(s) \to 0$ as $s\to \infty $, then $w _{\mu }' $ has a finite number of zeros, and $w_{\mu }w_{\mu }' <0$ for $s$ large.
There is an arbitrary choice in Proposition \[eEX1:b1\]. Indeed, as the proof of the proposition shows, ${{\eta}}$ can be any sufficiently small positive number. In fact, all that is needed in the subsequent arguments is a collection of solutions $(w_{{\mu }} ) _{ {\mu }\ge 1 }\subset C^1([0,\infty )) \cap C^2(0,\infty )$ of such that $w_{\mu }(0) ={\mu }$ and Property of Proposition \[eEX1:b1\] is true.
The key ingredient in the proof of Theorem \[eMT2\] is that the number of zeros of $w_{\mu }$ is arbitrarily large for large ${\mu }$. This is stated in the following proposition.
\[fFIn1\] Assume $N\ge 1$, $\alpha >0$, and let $( w_{\mu }) _{ {\mu }\ge 1 }$ be the collection of solutions of given by Proposition $\ref{eEX1:b1}$. Given any $T>0$ and $m\in {{\mathbb N}}$, there exists $ \overline{{\mu }} \ge {1 }$ such that if ${\mu }\ge \overline{{\mu }} $, then $w_{\mu }$ has at least $m$ zeros on $[0,T]$. In particular, if $$\label{fFIn1:1}
\text{$ {{\widetilde {N}}}({\mu }) $ is the number of zeros of $w_{\mu }$ on $(0,\infty )$, }$$ then $ {{\widetilde {N}}}({\mu }) \to \infty $ as ${\mu }\to \infty $.
The proof of Proposition \[fFIn1\] is given in Section \[sAB\]. The second ingredient we use is a classification of the possible asymptotic behaviors of $w_{\mu }(s)$ as $s\to \infty $. We let $$\label{fPR1}
\lambda _1 = \frac {1} {\alpha } , \quad \lambda _2 = \frac {1} {\alpha } - \frac {N-2} {2} .$$ Note that $\lambda _1> \lambda _2$ if $N\ge 3$ and $\lambda _1= \lambda _2$ if $N=2$. Moreover, $$\label{fNZE3}
\beta = - 4 \lambda _1 \lambda _2,\quad \gamma = 1 + \lambda _1 + \lambda _2$$ where $\beta $ and $\gamma $ are defined by and . We also define $$\label{fPR2}
\phi _1 (s) = s^{- \lambda _1}$$ and $$\label{fPR3}
\phi _2 (s) =
\begin{cases}
1 & N\ge 3 \text{ and }\alpha > \frac {2} {N-2} \\
(\log s )^{-\frac {1} {\alpha }} & N\ge 3 \text{ and }\alpha = \frac {2} {N-2} \\
s^{- \lambda _2} & N\ge 3 \text{ and }\alpha < \frac {2} {N-2} \\
s^{- \lambda _1} \log s & N=2.
\end{cases}$$ The following result shows that as $s\to \infty $, the solutions $w_{\mu }(s)$ of given by Proposition \[eEX1:b1\] decay either as $\phi _2$ (slow decay), or else as $\phi _1$ (fast decay).
\[eAB1\] Assume , and let $( w_{\mu }) _{ {\mu }\ge 1 }$ be the collection of solutions of given by Proposition $\ref{eEX1:b1}$. If $\phi _1$ and $\phi _2$ are defined by and , respectively, then the following properties hold.
1. \[eAB1:1\] The limit $$\label{eAB2}
{{\mathcal L}_2}({\mu })=\lim_{s\to \infty} \frac {w_{\mu }(s) } {\phi _2(s)}$$ exists and is finite.
2. \[eAB1:2\] If ${{\mathcal L}_2}( {\mu })=0$, then $$\label{eAB4}
{{\mathcal L}_1}( {\mu })=\lim_{s\to \infty} \frac {w_ {\mu }(s) } {\phi _1(s)}$$ exists and is finite, and ${{\mathcal L}_1}( {\mu })\ne 0$.
Proposition \[eAB1\] in the case $\beta >0$ follows from the results in [@SoupletW]. For $\beta \le 0$, the proof requires the separate study of the various cases $N\ge 3$ and $\beta <0$; $N\ge 3$ and $\beta =0$; $N=2$. The proof is given in Section \[sec:auxiliary\].
Proposition \[eAB1\] is relevant for the following reason. To prove Theorem \[eMT2\] we need to find $a>0$ such that $L (a) = {\mu }$. For the solution $w$ of the inverted profile equation defined by in terms of the profile $f_a$, this means, by , that $w (0)= {\mu }$. The existence of solutions $w$ of such that $w (0)= {\mu }$ follows from Proposition \[eEX1:b1\] (at least if ${\mu }$ is sufficiently large). However, in order that these solutions correspond to a profile $f_a$, we must ensure that $f$ defined by in terms of $w$ has a finite limit as $r\to 0$. This last condition is equivalent to the fact that $ \frac {w(s)} {\phi _1(s)}=s^{\frac {1} {\alpha }} w(s)$ has a finite limit as $s\to \infty $. By Proposition \[eAB1\], we see that we should look for solutions $w_{\mu }(s)$ of such that ${{\mathcal L}_2}( {\mu })=0$.
The last ingredient needed in the proof of Theorem \[eMT2\] concerns the local behavior of $ {{\widetilde {N}}}( {\mu }) $, the number of zeros of $w_{\mu }$.
\[eEX2\] Assume . Let ${{\mathcal L}_2}$ be defined by and ${{\widetilde {N}}}$ by , and consider $ \overline{{\mu }} \ge {1 }$.
1. \[eEX2:1\] If ${{\mathcal L}_2}( \overline{{\mu }} )\ne 0$, then there exists $\delta >0$ such that if ${\mu }\ge {1 }$ and $ | {\mu }- \overline{{\mu }} | \le \delta $, then $ {{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} ) $.
2. \[eEX2:2\] If ${{\mathcal L}_2}( \overline{{\mu }} )= 0$, then there exists $\delta >0$ such that if ${\mu }\ge {1 }$ and $ | {\mu }- \overline{{\mu }} | \le \delta $, then either $ {{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} ) $ or else $ {{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} ) +1 $ and ${{\mathcal L}_2}( {\mu }) \not = 0$.
The proof of Proposition \[eEX2\] is carried out in Section \[sFNL\].
We are now in a position to complete the proof of Theorem \[eMT2\], assuming Propositions \[eEX1:b1\], \[fFIn1\], \[eAB1\] and \[eEX2\]. We first give a lemma, which will also be used in the proofs of Theorems \[eMain2\] and \[eMain3\].
\[eTPA15\] Assume . Let ${{\mathcal L}_2}$ be defined by and ${{\widetilde {N}}}$ by , and set $ \overline{m} = {{\widetilde {N}}}(2) +1$. It follows that there exists an increasing sequence $( {\mu }_m)_{m\ge \overline{m}} \subset [ {1 }, \infty )$ with the following properties.
1. \[eTPA15:1\] $ {\mu }_m \to \infty $ as $m\to \infty $, ${{\mathcal L}_2}( {\mu }_m)=0$, $ {{\widetilde {N}}}( {\mu }_m) =m$.
2. \[eTPA15:2\] For every $m\ge \overline{m} $, there exists ${\mu }\in ( {\mu }_m, {\mu }_{ m+1 } )$ such that $ {{\widetilde {N}}}( {\mu }) =m +1$ and ${{\mathcal L}_2}( {\mu }) \not = 0$.
Given $m\ge \overline{m} $, let $$\label{eNZ1:9}
E_m= \{ {\mu }\ge 2 ;\, {{\widetilde {N}}}( {\mu })\ge m+1 \}.$$ It follows from Proposition \[fFIn1\] that $E_m \not = \emptyset$, and we define $$\label{eNZ1:10}
{\mu }_m = \inf E _m \in [2, \infty ).$$ By Proposition \[eEX2\], ${{\widetilde {N}}}( {\mu }) \le {{\widetilde {N}}}(2)+1=\overline{m} \le m$ for ${\mu }\ge 2$ close to $2$. Thus we see that ${\mu }_m > 2$. Suppose that $ {{\widetilde {N}}}( {\mu }_m ) \le m-1$. It follows from Proposition \[eEX2\] that ${{\widetilde {N}}}({\mu }) \le m$ for ${\mu }$ close to ${\mu }_m$, contradicting . Suppose now that ${{\widetilde {N}}}( {\mu }_m ) \ge m+1$. Applying again Proposition \[eEX2\], we deduce that ${{\widetilde {N}}}( {\mu }) \ge m+1$ for ${\mu }$ close to ${\mu }_m$, contradicting again . So we must have $ {{\widetilde {N}}}({\mu }_m) =m$. Moreover, if ${{\mathcal L}_2}( {\mu }_m)\not = 0$, then ${{\widetilde {N}}}( {\mu }) = m$ for ${\mu }$ close to ${\mu }_m$, contradicting once more ; and so, ${{\mathcal L}_2}( {\mu }_m) = 0$. We finally prove that $ {\mu }_m \to \infty $ as $m\to \infty $. Otherwise, there exist $ \overline{{\mu }}\in [{1 }, \infty )$ and a sequence $m_k \to \infty $ such that ${\mu }_{m_k} \to \overline{{\mu }} $ as $k \to \infty $ and $ {{\widetilde {N}}}( {\mu }_{m_k}) =m_k $. It follows from Proposition \[eEX2\] that $ {{\widetilde {N}}}({\mu }_{m_k}) \le {{\widetilde {N}}}( \overline{{\mu }} ) +1$ for all sufficiently large $k$, which is absurd. This proves Property .
Given $m\ge \overline{m} $, let $$\label{fTPA66}
J_m = \{{\mu }\ge {\mu }_m ; \, {{\widetilde {N}}}( \nu ) \le m \text{ for all } \nu\in [ {\mu }_m ,{\mu }] \}.$$ We see that ${\mu }_m \in J_m $, so that $J_m \not = \emptyset$. Moreover, $ {{\widetilde {N}}}( {\mu }_{ m+1 }) =m +1 $, so that ${\mu }_{ m+1 } \not \in J_m $. In particular, $J_m $ is bounded and we set $$\label{fTPA67}
\overline{{\mu }} _m = \sup J_m .$$ We claim that $$\label{fTPA68}
{{\widetilde {N}}}( \overline{{\mu }} _m) =m.$$ Indeed, if ${{\widetilde {N}}}( \overline{{\mu }} _m) \le m-1$, then $ {{\widetilde {N}}}( {\mu }) \le m$ for ${\mu }$ close to $ \overline{{\mu }} _m$ by Proposition \[eEX2\], which contradicts . Furthermore, if ${{\widetilde {N}}}( \overline{{\mu }} _m) \ge m + 1$ then, again by Proposition \[eEX2\], we have $ {{\widetilde {N}}}({\mu }) \ge m+1$ for ${\mu }$ close to $\overline{{\mu }} _m$. This contradicts once more , thus proving . In addition, , , and Proposition \[eEX2\] imply that $ {{\mathcal L}_2}(\overline{{\mu }} _m ) =0 $. Consequently, it follows from Proposition \[eEX2\] that for ${\mu }$ close to $ \overline{{\mu }} _m$, we have either ${{\widetilde {N}}}( {\mu }) =m$ or else ${{\widetilde {N}}}( {\mu }) = m+1$ and $ {{\mathcal L}_2}( {\mu }) \not = 0$. In particular, $ \overline{{\mu }}_m < {\mu }_{ m+1 } $ by . Since, by , there exists ${\mu }$ arbitrarily close to $ \overline{{\mu }}_m $ such that $ {{\widetilde {N}}}( {\mu }) =m+1$, Property follows.
\[Proof of Theorem $\ref{eMT2}$\] We first prove that there exists a sequence $(\alpha _m) _{ m\ge \overline{m} } \subset (0,\infty ) $ satisfying (with the notation -) $$\begin{gathered}
\alpha _m {\mathop{\longrightarrow}}_{ m\to \infty }\infty \text{ and } (-1)^m L (\alpha _m ) {\mathop{\longrightarrow}}_{ m\to \infty }\infty \label{eNZ3:1} \\
N (\alpha _m ) = m . \label{eNZ3:3} \end{gathered}$$ To see this, we consider the collection $( w_{\mu }) _{ {\mu }\ge 1 }$ of solutions of given by Proposition \[eEX1:b1\], and the sequence $({\mu }_m) _{ m\ge \overline{m} }$ of Lemma \[eTPA15\]. Since ${{\mathcal L}_2}( {\mu }_m )=0$, it follows from Proposition \[eAB1\] that ${{\mathcal L}_1}( {\mu }_m )$ is well defined and ${{\mathcal L}_1}( {\mu }_m )\not = 0$. On the other hand, $ {{\widetilde {N}}}({\mu }_m) =m$ and $w _{ {\mu }_m } (0)= {\mu }_m >0$, so that $(-1)^m w _{ {\mu }_m } (s) >0$ for $s$ large. Thus we see that that $ (-1)^m {{\mathcal L}_1}( {\mu }_m ) > 0$. Setting $\alpha _m= (-1)^m {{\mathcal L}_1}( {\mu }_{ m } )$, it follows from and that $$\label{fTPA15}
f _{ \alpha _m }(r)= (-1)^m r^{-\frac {2} {\alpha }} w _{ {\mu }_m } ( r^{-2} )$$ for all $r>0$. Moreover $N (\alpha _m) = {{\widetilde {N}}}( {\mu }_m ) =m $, $L (\alpha _m) = (-1)^m w _{ {\mu }_m } (0)= (-1)^m {\mu }_m $. In particular, $(-1)^m L (\alpha _m) \to \infty $ as $m\to \infty $, and since $L$ is continuous $[0, \infty ) \to {{\mathbb R}}$, we see that $\alpha _m \to \infty $ as $m\to \infty $. This proves and .
Fix ${\mu }>0$. We construct the sequences $(a _{ {\mu }, m }^\pm ) _{ m \ge \overline{m} } $. Consider the sequence $\alpha _m \to \infty $ constructed above and satisfying -. By , we may choose an integer $m_ {\mu }$ sufficiently large so that $$\label{fMT2:1}
L ( \alpha _{2m}) > {\mu }\text{ and } L ( \alpha _{2m -1}) <- {\mu }\text{ for all } m\ge m_ {\mu }.$$ By continuity of $L$, there exist $$\label{fMT2:2}
0 < \alpha _{2m -1} < a_m^- < \alpha _{2m} < a_m ^+ < \alpha _{2m +1}$$ such that $$\label{fMT2:3}
L ( a_m^- )= L (a_m ^+ ) =0 \text{ and } L(a) >0 \text{ for all } a_m^- < a < a_m^+ .$$ Next, since $N (\alpha _{2m}) =2m $ (by ), it follows from , and [@Weissler6 Proposition 3.7] (this is in fact an immediate consequence of [@HarauxW Proposition 3.4]) that $$\label{fMT2:4}
N ( a) = 2m \text{ for all } a_m^- < a < a_m^+ .$$ From and , we deduce that there exist $$\label{fMT2:6}
a_m^- < a _{ {\mu }, m }^- < \alpha _{2m} < a _{ {\mu },m }^+ < a_m^+$$ such that $L ( a _{ {\mu }, m }^\pm ) = {\mu }$. Moreover, it follows from and that $N ( a _{ {\mu },m }^\pm ) = 2 m$, and from , and that $a _{ {\mu },m }^\pm \to \infty $ as $m\to \infty $. Thus we see that the sequences $(a _{ {\mu },m }^\pm ) _{ m \ge m_ {\mu }}$ satisfy all the conclusions of the theorem.
By considering $\alpha _{ 2m+1 } \to \infty $ (instead of $\alpha _{2m}$), one constructs as above a sequence $( \widetilde{b} _{ {\mu }, m }^\pm ) _{ m \ge m_ {\mu }}\subset (0, \infty )$ such that $ \widetilde{b} _{ {\mu }, m }^\pm \to \infty $ as $m\to \infty $, $L ( \widetilde{b} _{ {\mu }, m }^\pm ) = - {\mu }$, $ \widetilde{b} _{ {\mu }, m }^- < \widetilde{b} _{ {\mu }, m }^+$ and $ N ( \widetilde{b} _{ {\mu }, m }^\pm ) =2m +1$. The result follows by letting $b _{ {\mu }, m }^\pm = - \widetilde{b} _{ {\mu }, m }^\pm$.
With the completion of the proof of Theorem \[eMT2\], the existence of the appropriate profiles has been established. It remains to show that the resulting solutions given by formula have the properties required by Theorem \[eMain1\]. To this end, we prove the following lemma, which includes the case of non-radially symmetric profiles.
\[eQSol\] Let $\alpha >0$ and let $f\in C^2 ({{\mathbb R}}^N )$ be a solution of such that $ |f(x)| + |x\cdot \nabla f(x)|\le C (1+ |x|^2)^{-\frac {1} {\alpha }}$ and $ |x|^\frac {2} {\alpha } f(x)- \omega (x) \to 0$ as $ |x| \to \infty $, for some $\omega \in C ({{\mathbb R}}^N \setminus \{0\})$ homogeneous of degree $0$. If $u (t, x) $ is defined by for all $t>0$ and $x\in {{\mathbb R}}^N $, then $u\in C^1((0,\infty ), L^r ({{\mathbb R}}^N ) )$ and $\Delta u, |u|^\alpha u\in C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r\ge 1$, $r> \frac {N\alpha } {2}$, and $u$ satisfies with ${u_0 }= \omega (x) |x|^{-\frac {2} {\alpha }}$. If in addition $\alpha >\frac {2} {N}$, then $u$ also satisfies , and $u $ is a solution of where each term is in $C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for $r> \frac {N\alpha } {2}$. Moreover, the map $t \mapsto u (t) - e^{t \Delta } {u_0 }$ is in $ C([0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r \ge 1$ such that $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$.
If $\alpha \le \frac {2} {N}$, then ${u_0 }\not \in L^1 _{{\mathrm{loc}}}({{\mathbb R}}^N ) $ (except if $\omega \equiv 0$), so that the first term on the right hand side of does not make sense, hence altogether, does not make sense.
\[Proof of Lemma $\ref{eQSol}$\] Since $f\in C^2 ({{\mathbb R}}^N )$ is a solution of , it follows from formula and elementary calculations that $u\in C^2((0,\infty )\times {{\mathbb R}}^N )$ is a solution of on $(0,\infty )\times {{\mathbb R}}^N$. Moreover, it is not difficult to show by dominated convergence that $u\in C ((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r\ge 1$, $r>\frac {N\alpha } {2}$. Differentiating with respect to $t$, we see that $$\label{feQSol4}
u_t= - \frac {1} {\alpha t}u - \frac {1} {2t} x\cdot \nabla u ,$$ and since $ |f(x)| + |x\cdot \nabla f(x)|\le C (1+ |x|^2)^{-\frac {1} {\alpha }}$, we have $u\in C^1 ((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r\ge 1$, $r>\frac {N\alpha } {2}$. Moreover, $ |f|^{\alpha +1}\le C (1+ |x|^2)^{-\frac {\alpha +1} {\alpha }}\le C (1+ |x|^2)^{-\frac {1 } {\alpha }}$, from which it follows easily that $ |u|^\alpha u \in C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r\ge 1$, $r> \frac {N\alpha } {2}$. The regularity of $\Delta u$ now follows from equation . Next, $ |f(x)|\le C (1+ |x|^2)^{-\frac {1 } {\alpha }}\le C |x|^{-\frac {2 } {\alpha }}$, so that $ |u(t,x) |\le C |x|^{-\frac {2} {\alpha }}$. Moreover given $x\not = 0$, $$\label{fTR1}
\lim_{t \to 0}u(t,x) = \lim_{t \to 0}t^{-\frac {1} {\alpha }} f \Bigl( \frac { x } {\sqrt t} \Bigr)
= |x|^{-\frac {2} {\alpha }}\lim_{r\to\infty}r^{\frac{2}{\alpha}} f \Bigl( \frac {x} { |x|} r \Bigr)
= \omega (x) |x|^{-\frac {2} {\alpha }},$$ and property follows by dominated convergence.
If $\alpha >\frac {2} {N}$, then $ |\cdot |^{-\frac {2} {\alpha }}\in L^p ({{\mathbb R}}^N ) +L^q ({{\mathbb R}}^N ) $ for all $1\le p<\frac {N\alpha } {2} < q $, which implies property . Moreover, the regularity of $u$ on $(0,\infty )$ ensures that $$\label{feQSol1}
u(t) = e^{(t- \varepsilon )\Delta } u(\varepsilon ) + \int _\varepsilon ^t e^{ (t-s) \Delta } |u(s)|^\alpha u(s)\, ds$$ for all $0< \varepsilon < t$. Consider now $r \ge 1$ such that $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$. In particular $f\in L^{(\alpha +1) r} ({{\mathbb R}}^N ) $, therefore, $$\label{feQSol3}
\begin{split}
\| e^{(t-s) \Delta } |u(s)|^\alpha u(s) \| _{ L^r } & \le \| |u(s)|^\alpha u(s) \| _{ L^r } = \| u(s)\| _{ L^{ ( \alpha +1) r } }^{\alpha +1}
\\ &= s^{- \frac {\alpha +1} {\alpha } + \frac {N} {2 r }} \| f \| _{ L^{(\alpha +1) r } }^{\alpha +1} .
\end{split}$$ Note that $- \frac {\alpha +1} {\alpha } + \frac {N} {2 r } >-1$ because $r < \frac {N\alpha } {2}$. Applying , one easily passes to the limit in as $\varepsilon \downarrow 0$ and obtain equation . Since the first two terms in are in $C((0,\infty ), L^r ({{\mathbb R}}^N ) )$ for $r> \frac {N\alpha } {2}$, so is the integral term. Finally, that $ u (t) - e^{t \Delta } {u_0 }\in C([0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $r \ge 1$ such that $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$ easily follows from and .
We finally complete the proof of Theorem \[eMain1\]. We may assume ${\mu }>0$ without loss of generality, and we apply Theorem \[eMT2\]. The profiles $f= f_{a^\pm }$ with $a^\pm = a^\pm _{ {\mu }, \frac {m} {2} }$ if $m\ge m_{\mu }$ is even and $a^\pm = b^\pm _{ {\mu }, \frac {m-1} {2} }$ if $m\ge m_{\mu }$ is odd, are two different radially symmetric solutions of with $m$ zeros. Moreover, $r^{\frac {2} {\alpha }} f_{a^\pm} (r) \to {\mu }$ as $r\to \infty $, and it follows from [@HarauxW Proposition 3.1] that $ |f_{a^\pm} (r)| + r |f'_{a^\pm} (r)|\le C( 1+ r^2)^{-\frac {1} {\alpha }}$. Theorem \[eMain1\] is now an immediate consequence of Lemma \[eQSol\], where $f (x) =f_{a^\pm} ( |x|)$ and $\omega (x) \equiv {\mu }$.
The inverted profile equation {#sIPE}
=============================
This section is devoted to the proof of Proposition \[eEX1:b1\].
We use the notation of [@SoupletW Theorem 2.5]. In particular, we consider the functions $K_1$ and $K_2$, both continuous on $[0,\infty )$, with $K_1(0)= 4$. We fix $t_0 >0$ sufficiently small so that $$\label{fLDC0}
\sup _{ 0\le s\le t_0 } K_1(s) \le 5$$ and we set $$\label{fLDC0:1}
\nu _0 = \sup _{ 0\le s\le t_0 } K_2(s) .$$ Note that for all $M>0$ $$\label{fLDC00}
\sup _{ |y|\le M } |g'(y)| \le |\beta |+ (\alpha +1) M^\alpha ,$$ where $g$ is given by . We fix ${{\eta}}>0$ sufficiently small so that $$\begin{gathered}
2 {{\eta}}\le t_0 \label{fTPA12} \\
5 {{\eta}}\nu_0 ( |\beta | + (\alpha +1) 10^\alpha )
\le 1 . \label{fLDC1} \end{gathered}$$ It follows from – that if ${{\mu }} \ge {1 }$ and $$R = {\mu }, \quad M= 10 {\mu }, \quad T= 2 {{\eta}}{\mu }^{- \alpha }$$ then $$\label{fLDC2}
\frac {R} {M} (1+ K_1 (T) ) + T K_2(T) \sup _{ |{{\mu }} |\le M } |g'({{\mu }} )| \le 1 .$$ Applying , we deduce from [@SoupletW Theorem 2.5 and Proposition 2.4] that for all ${{\mu }} \ge {1 }$, there exists a solution $w_{{\mu }} \in C^1([0,2 {{\eta}}{\mu }^{- \alpha } ]) \cap C^2((0, 2 {{\eta}}{\mu }^{- \alpha } ])$ of satisfying the conditions . Moreover, [@SoupletW Proposition 3.1] implies that the solution $w_{{\mu }} $ can be extended to $[0, \infty )$. This proves the first part of the statement.
The last statement of [@SoupletW Theorem 2.5] shows that given $ \overline{{{\mu }} } \ge {1 }$, $w_{{\mu }} $ depends continuously on ${{\mu }} $ at $ \overline{{{\mu }} } $ in $C^1([0, {{\eta}}{\mu }^{- \alpha }])$. Since equation is not degenerate on $[ {{\eta}}{\mu }^{- \alpha }, T]$ for $T> {{\eta}}{\mu }^{- \alpha }$, continuous dependence in $C^1([0,T])$ follows easily.
Identity follows from elementary calculations.
Moreover, [@SoupletW Proposition 3.1 (i)] implies that the solution $w_{{\mu }} $ satisfies .
If $(N-2) \alpha \le 4$, then $\gamma >1$, see . It follows that $H_{{\mu }} (s)$ defined by is nonincreasing for $s$ large, and bounded by , so it has a finite limit as $s\to \infty $. This proves Property .
The second statement of Property is a consequence of [@SoupletW Propositions 3.3 and 3.2]. In particular, $G (w_{\mu }(s))$ defined by has a limit as $s\to \infty $. Since $H_{{\mu }} (s)$ also has a limit, we see that $s^2 w_{{\mu }} '(s)^2$ must have a limit, which is necessarily $0$. This proves the first statement of Property .
We now turn to the proof of Property . If $\beta >0$ and $w_{\mu }(s)\to \pm \beta ^{\frac {1} {\alpha }}$ then the result is clearly true. Otherwise, if $\beta >0$ and $w_{\mu }(s)\not\to \pm \beta ^{\frac {1} {\alpha }}$, or if $\beta \le 0$, then $w_{\mu }(s)\to 0$, by Property . We first consider the case $N \ge 3$, so that $\lambda _1 > \lambda _2$ by . Given $\sigma \in {{\mathbb R}}$, let $$\label{fBN5a0}
z(s)=s^\sigma w_{\mu }(s).$$ It follows from and that $$\label{fBN5a}
4s^2z''+ 4 (1 + \lambda _1 + \lambda _2 -2\sigma )sz'-z'+ 4 (\sigma -\lambda _1) (\sigma -\lambda _2) z+\frac {\sigma } {s}z+ \frac {1} {s^{ \alpha \sigma }} |z|^\alpha z=0.$$ If we fix $\sigma >0$, $\sigma \in ( \lambda _2, \lambda _1)$, then $ 4 (\sigma -\lambda _1) (\sigma -\lambda _2) < 0$. Since $s^{-\alpha \sigma }|z|^\alpha=|w|^\alpha \to 0$ as $s\to \infty$, we deduce from that if $s_0>0$ is sufficiently large, then for $s\ge s_0$, $z(s) z''(s)>0$ whenever $z' (s) =0$. In other words, at any point $s\ge s_0$ where $z'(s)=0$, $z^2$ has a local minimum. Thus $z$ can vanish at most once for $s\ge s_0$.
If $N=1$, so that $\lambda _1< \lambda _2$ by , the above argument works with $\sigma \in ( \lambda _1, \lambda _2)$.
The case $N=2$, where $\lambda _1 = \lambda _2$, requires a more delicate argument. We let $\sigma =\lambda _1$, so that equation becomes $$\label{fBN5a1}
4s^2z''+ 4 sz'-z' + \frac {\lambda _1 } {s}z+ \frac {1} {s } |z|^\alpha z=0.$$ Multiplying by $z' $ yields the energy identity $$\Bigl( 2 s^2 z'\null ^2 + \frac {\lambda _1} {2s} z^2 + \frac {1} {(\alpha +2) s} |z|^{\alpha +2} \Bigr) '= z' \null ^2 - \frac {\lambda _1} {2s^2} z^2- \frac {1} {(\alpha +2) s^2} |z|^{\alpha +2} \le z' \null ^2$$ so that $$\Bigl( 2 s^2 z'\null ^2 + \frac {\lambda _1} {2s} z^2 + \frac {1} {(\alpha +2) s} |z|^{\alpha +2} \Bigr) ' \le \frac {1} {2s^2} \Bigl( 2 s^2 z'\null ^2 + \frac {\lambda _1} {2s} z^2 + \frac {1} {(\alpha +2) s} |z|^{\alpha +2} \Bigr) .$$ Integrating the above inequality yields $$\label{fBN5a3}
2 s^2 z' (s) ^2 \le e^{\frac {1} {2}- \frac {1} {2s}} \Bigl( 2 z'(1) ^2 + \frac {\lambda _1} {2} z(1) ^2 + \frac {1} {\alpha +2} |z (1) |^{\alpha +2} \Bigr)$$ for $s\ge 1$. Therefore, $ | z' (s)| \le C (1+s)^{-1}$, so that $$\label{fBN5a4}
|z (s)| \le C \log (2 +s)$$ for $s\ge 1$. We now define for $s> 1$ $$\label{fBN5a5}
t^{\frac {1} {2}} v (t)=z (s), \quad t = \log s .$$ It follows in particular that $$\label{fbw1}
\frac {dt} {ds}= \frac {1} {s}.$$ We deduce from and that $$\label{fbw2}
z'(s)= \frac {1} {s} \Bigl( t^{\frac {1} {2}} v'(t) + \frac {1} {2} t^{- \frac {1} {2}} v(t) \Bigr)$$ so that $$\label{fbw3}
s z'(s)= t^{\frac {1} {2}} v'(t) + \frac {1} {2} t^{- \frac {1} {2}} v(t) .$$ Next, differentiating with respect to $s$ and applying again , we obtain $$\label{fbw4}
s^2 z''(s) = - \Bigl( t^{\frac {1} {2}} v'(t) + \frac {1} {2} t^{- \frac {1} {2}} v(t) \Bigr) + \Bigl( t^{\frac {1} {2}} v''(t) + t^{- \frac {1} {2}} v '(t) - \frac {1} {4} t^{- \frac {3} {2}} v (t) \Bigr) .$$ It follows from , , and that $$\label{eNZE3:3}
4v ''+\Bigl ( \frac { 4 } {t}-e^{-t}\Bigr ) v '-\Bigl (\frac { 1 } {t^2}+e^{-t}\Bigl (\frac {1} {2 t}-\frac {1} {\alpha } - ( t^{\frac {1} {2}} |v |)^\alpha \Bigr)\Bigr ) v =0.$$ Note that by and , $$t^{\frac {1} {2}} |v | = |z (s)| \le C (\log (2 +s)) \le C t$$ for $t$ large, so that $$\label{fTPA19}
t^2 e^{-t}\Bigl (\frac {1} {2 t}-\frac {1} {\alpha } - ( t^{\frac {1} {2}} |v |)^\alpha \Bigr) {\mathop{\longrightarrow}}_{ t\to \infty } 0 .$$ It follows from and that for $t$ large, if $v' $ vanishes, then $v '' $ has the sign of $v$. Arguing as in the case $N\ge 3$, we conclude that $v$ has a finite number of zeros.
We finally prove Property , so we suppose $w_{\mu }(s) \to 0$ as $s\to \infty $. By Property , $w_{\mu }(s) \not = 0$ for $s$ large, and we deduce from that if $w_{\mu }' (s)= 0$, then $$4s^2 w_{\mu }'' (s)= (\beta - |w_{\mu }(s)|^\alpha ) w_{\mu }(s) .$$ It easily follows that if $s$ is large, then either $w_{\mu }'' (s) $ has the sign of $w_{\mu }(s)$ (if $\beta >0$), or else $w_{\mu }'' (s) >0$ has the opposite sign of $w_{\mu }(s)$ (if $\beta <0$ or if $\beta =0$) whenever $w_{\mu }' (s)=0$. This shows that $w_{\mu }' $ cannot vanish for $s$ large. Since $w_{\mu }(s) \to 0$ as $s\to \infty $, we conclude that $w_{\mu }w_{\mu }' <0$ for $s$ large.
Arbitrarily many zeros {#sAB}
======================
This section, in its entirety, constitutes the proof of Proposition \[fFIn1\]. We set $$\label{frd}
b = \frac {1} {4\gamma +8}.$$ It will be shown that there exists a function $\iota :(1, \infty) \to (0,\infty)$ such that $\iota ({\mu }) \to 0$ as ${\mu }\to \infty$ with the property that if $I \subset (0, b )$ is an interval on which $w_{\mu }(s) \neq 0$, then $|I| \le \iota({\mu })$. This implies the proposition. Indeed, if $\iota({\mu }) < \frac {T} {m}$, where $T < b$, then $w_{\mu }$ has at least $m$ zeros on $[0,T]$.
Thus we fix an interval $I \subset (0, b )$ for which $w_{\mu }(s) > 0$ for all $s \in I$. (The case $w_{\mu }(s) < 0$ can be handled analogously.) We need to estimate $|I|$ as a function of ${\mu }$, and it suffices to do so for ${\mu }> 1$ sufficiently large.
The crucial observation is that $H_{\mu }$ given by is increasing on $[0,b]$, by , so that $$\label{fTPA01}
2s^2w_{\mu }'(s)^2 + G(w_{\mu }(s)) = H _{\mu }(s) > H _{\mu }(0) = G( {\mu })$$ for $ s\in (0, b ] $. The first largeness condition we impose on ${\mu }$ is that $$\label{frd1}
{\mu }> 2\big(2(\alpha + 2)|\beta|\big)^{\frac{1}{\alpha}}.$$ This condition guarantees, in particular, that there is at most one value of $s \in I$ where $w_{\mu }'(s) = 0$. To see this, suppose $s \in I$ and $w_{\mu }'(s) = 0$. It follows by and that $$G(w_{\mu }(s)) > G( {\mu }) > 0,$$ and so it must be that $w_{\mu }(s) > {\mu }$. This implies, again by , that $g(w_{\mu }(s)) > 0$, which in turn implies, by equation , that $w_{\mu }''(s) < 0$. In other words, at any point in $I$ where $w_{\mu }'(s) = 0$ the solution $w_{\mu }$ must have a local maximum. Hence there is at most one such point in $I$.
It follows that the interval $I$ can be partitioned into four pairwise disjoint subintervals, which depend on ${\mu }$, $$\label{frd3}
I = I_{\mu }^1 \cup I_{\mu }^2 \cup I_{\mu }^3 \cup I_{\mu }^4,$$ given by $$\begin{gathered}
I_{\mu }^1 = \{s \in I: w_{\mu }(s) \le \frac{{\mu }}{2}, w_{\mu }'(s) > 0\}, \quad
I_{\mu }^2 = \{s \in I: w_{\mu }(s) > \frac{{\mu }}{2}, w_{\mu }'(s) > 0\}, \\
I_{\mu }^3 = \{s \in I: w_{\mu }(s) > \frac{{\mu }}{2}, w_{\mu }'(s) \le 0\}, \quad
I_{\mu }^4 = \{s \in I: w_{\mu }(s) \le \frac{{\mu }}{2}, w_{\mu }'(s) < 0\} .\end{gathered}$$ Note that one or more of these intervals might be empty or contain just a single point. We now proceed to estimate the lengths of these four intervals.
Consider first the interval $I_{\mu }^1$. If $s \in I_{\mu }^1$, then by and $$w_{\mu }'(s)^2 \ge \frac{1}{2s^2}(G({\mu }) - G(w_{\mu }(s))
\ge \frac{1}{2b^2}(G({\mu }) - G({{\mu }} / {2}))
\ge \frac{{\mu }^{\alpha+2}}{8b^2(\alpha + 2)}.$$ Consequently, if $s < t$ and $s,t \in I_{\mu }^1$, then $$|w_{\mu }(t) - w_{\mu }(s)|
\ge \frac {t-s } {2b\sqrt{2(\alpha + 2)}} {\mu }^{1 + \frac {\alpha} {2}}.$$ Since $|w_{\mu }(t) - w_{\mu }(s)| \le \frac{{\mu }}{2}$ if $s,t \in I_{\mu }^1$, this implies $$\label{frd4}
|I_{\mu }^1| \le b\sqrt{2(\alpha + 2)}{\mu }^{- \frac{\alpha}{2}}.$$ In exactly the same way, we also have $$\label{frd5}
|I_{\mu }^4| \le b\sqrt{2(\alpha + 2)}{\mu }^{- \frac{\alpha}{2}}.$$
We next turn our attention to $I_{\mu }^2$. To estimate its length, we partition this interval into further subintervals. Differentiating we get $$4s^2 w _ {\mu }'''-(1-(4\gamma +8)s)w _ {\mu }''+(4\gamma -\beta +(\alpha +1)|w _ {\mu }|^\alpha )w _ {\mu }'=0.$$ Therefore, if $w _ {\mu }''(s)=0$ for some $s \in I_{\mu }^2$, we have from that $w _ {\mu }'''(s)<0$, and so $w_{\mu }'' $ has a local maximum at that point. It follows that $w _ {\mu }''$ can have at most one zero in $I_{\mu }^2$. We may therefore define the intervals $$J_{\mu }^1 = \{s \in I_{\mu }^2: w_{\mu }''(s) \ge 0\}, \quad J_{\mu }^2 = \{s \in I_{\mu }^2: w_{\mu }''(s) < 0\}$$ so that $I_{\mu }^2 = J_{\mu }^1 \cup J_{\mu }^2$.
If $s \in J_{\mu }^1$, then $$w_{\mu }'(s) \ge \frac{g(w_{\mu }(s))}{1 - 4\gamma s}
\ge \frac{w_{\mu }(s)^{\alpha+1}}{2(1 - 4\gamma s)}
\ge \frac{1}{2}w_{\mu }(s)^{\alpha+1},$$ where again we use and . In other words, $(w_{\mu }(s)^{-\alpha})' \le -\frac{\alpha}{2}$. Integrating, we see that if $s < t$ and $s,t \in J_{\mu }^1$, then $$t - s \le \frac{2}{\alpha}w_{\mu }(s)^{-\alpha}
\le \frac{2^{\alpha + 1}}{\alpha}{\mu }^{-\alpha}.$$ This shows that $$\label{frd6}
|J_{\mu }^1| \le \frac{2^{\alpha + 1}}{\alpha}{\mu }^{-\alpha}.$$
If $s \in J_{\mu }^2$, then $$(w_{\mu }'(s) - w_{\mu }(s)^{\frac{\alpha}{2}+1})'
= w_{\mu }''(s) - ({\frac{\alpha}{2}+1})w_{\mu }'(s) < 0.$$ Therefore, $J_{\mu }^2$ is the union of two intervals, $J_{\mu }^{2+}$ and $J_{\mu }^{2-}$, where, respectively, $w_{\mu }'(s) \ge w_{\mu }(s)^{\frac{\alpha}{2}+1}$, and $w_{\mu }'(s) < w_{\mu }(s)^{\frac{\alpha}{2}+1}$. On $J_{\mu }^{2+}$, we have that $(w_{\mu }(s)^{-\frac{\alpha}{2}})' \le -\frac{\alpha}{2}$, which integrates to yield, if $s < t$ and $s,t \in J_{\mu }^{2+}$, $$t - s \le \frac{2}{\alpha}w_{\mu }(s)^{-\frac{\alpha}{2}}
\le \frac{2^{1+\frac{\alpha}{2}}}{\alpha}{\mu }^{-\frac{\alpha}{2}}.$$ This implies that $$\label{frd7}
|J_{\mu }^{2+}| \le \frac{2^{1+\frac{\alpha}{2}}}{\alpha}{\mu }^{-\frac{\alpha}{2}}.$$ On the other hand, if $s \in J_{\mu }^{2-}$, then $$\begin{split}
4b^2 w_{\mu }'' (s) & \le 4s^2 w_{\mu }'' (s) = (1-4\gamma s)w _ {\mu }'(s)-g(w_ {\mu }(s)) \\
& \le w _ {\mu }'(s)-g(w_ {\mu }(s)) \le w _{\mu }(s)^{1+ \frac {\alpha } {2}} - \frac {1} {2} w_ {\mu }(s)^{\alpha +1} \\
& \le - \frac {1} {4} w_ {\mu }(s)^{\alpha +1},
\end{split}$$ where we need to impose an additional largeness condition on ${\mu }$, [ i.e.]{} $$\label{frd8}
{\mu }\ge 2^{1 + \frac{4}{\alpha}}.$$ Consequently, if $s<t$ and $s, t \in J_{\mu }^{2-}$, then $$w_{\mu }' (s) \ge w_{\mu }' (s) - w_{\mu }' (t) \ge \frac {1} {16 b^2} \int _s^t w_ {\mu }(\sigma )^{\alpha +1} d\sigma \ge \frac {t-s} {16 b^2}
w_ {\mu }(s )^{\alpha +1} .$$ Since $w_{\mu }' (s) \le w _{\mu }(s)^{1+ \frac {\alpha } {2}} $ and $w_{\mu }(s) \ge \frac {{\mu }} {2}$, we deduce that $$1 \ge \frac {t-s} {16 b^2} w_ {\mu }(s )^{\frac {\alpha } {2}} \ge \frac {t-s} {2^{4+ \frac {\alpha } {2}} b^2} {\mu }^{\frac {\alpha } {2}} .$$ Therefore, $$\label{frd9}
|J_{\mu }^{2-}| \le 2^{ 4 + \frac {\alpha } {2}}b^2 {\mu }^{ - \frac {\alpha } {2} }.$$ For future reference, we recall that $$\label{frd10}
I_{\mu }^2 = J_{\mu }^1 \cup J_{\mu }^{2+}\cup J_{\mu }^{2-},$$ and so the length of $I_{\mu }^2$ is estimated by , and .
Finally, we estimate the length of the interval $I_{\mu }^3$. Observe first that since $g(w_{\mu }(s)) > 0$ on $I_{\mu }^3$ (by ) and $w_{\mu }'(s) \le 0$ on $I_{\mu }^3$, it follows from that $w_{\mu }''(s) < 0$ on $I_{\mu }^3$. Hence, for $s\in I_{\mu }^3$, $$4b ^2 w _{\mu }'' (s) +g(w_{\mu }(s) )\le 4s^2 w _{\mu }'' (s) +g(w_{\mu }(s) ) = (1 - 4\gamma s) w_{\mu }' (s) \le 0$$ so that $$(2 b ^2 w _{\mu }'(s)^2+ G( w _{\mu }(s)))'=(4b^2 w _{\mu }''(s)+ g(w _{\mu }(s)))w _{\mu }'(s) \ge 0.$$ Therefore, if $s < t$ and $s,t \in I_{\mu }^3$, $$\label{fTPA08:3}
2b ^2 w _{\mu }'(t)^2 + G(w _{\mu }(t)) \ge 2b^2w _{\mu }' (s )^2 +
G ( w _{\mu }(s)) \ge G( w _{\mu }(s )) .$$ Recall that $w_{\mu }'' <0$ and $w_{\mu }' \le 0$ on $I_{\mu }^3$, so that $w_{\mu }(s) > w_{\mu }(t)$. Since $G(w _{\mu }(t)) > 0$ by , we deduce that $G(w _{\mu }(s)) > G(w _{\mu }(t)) >0$, and it follows from that $$\frac {-\sqrt{2} b w _{\mu }'(t)} {(G(w _{\mu }(s))-G(w _{\mu }(t)))^{\frac {1} {2}}} \ge 1.$$ Integrating, we obtain $$\label{fNP10}
\begin{split}
t - s & \le \int_s^t \frac {-\sqrt{2} b w _{\mu }'(\tau)} {(G(w _{\mu }(s))-G(w _{\mu }(\tau)))^{\frac {1} {2}}}d\tau\\
& = \int_{\frac {w _{\mu }(t)} {w _{\mu }(s)}}^{1} \frac {\sqrt 2 b w _{\mu }(s)} {(G(w _{\mu }(s))- G( z w _{\mu }(s) ))^{\frac {1} {2}}}\, dz,
\end{split}$$ where we have set $z = \frac{w _{\mu }(\tau)}{w _{\mu }(s)}$, for $s \le \tau \le t$. Furthermore, once again using , $$\begin{split}
G(w _{\mu }(s)) - G(zw _{\mu }( s) ) & = \frac {w _{\mu }(s) ^{\alpha +2} } {\alpha +2} (1-z^{\alpha +2}) - \frac {\beta w _{\mu }(s) ^2 } {2} (1-z^2) \\
&\ge \Bigl( \frac {w _{\mu }( s) ^{\alpha +2}} {\alpha +2}- \frac { |\beta| w _{\mu }( s) ^2} {2}\Bigr) (1-z^{\alpha +2}) \\ & \ge \frac {w _{\mu }(s) ^{\alpha +2}} {2(\alpha +2)} (1-z^{\alpha +2}).
\end{split}$$ Putting this into , we obtain that $$t - s \le \frac {2 b \sqrt{\alpha +2} } { w _{\mu }(s) ^{\frac {\alpha } {2}} } \int_0 ^{1} \frac { dz} { (1-z^{\alpha +2}) ^{\frac {1} {2}}} \le \frac {2^{1+\frac {\alpha } {2}} b \sqrt{\alpha +2} } { {\mu }^{\frac {\alpha } {2}} } \int_0 ^{1} \frac { dz} { (1-z^{\alpha +2}) ^{\frac {1} {2}}},$$ so that $$\label{frd11}
|I_{\mu }^3| \le 2^{1+\frac {\alpha } {2}} b \sqrt{\alpha +2} \Bigl( \int_0 ^{1} \frac { dz} { (1-z^{\alpha +2}) ^{\frac {1} {2}}} \Bigr) \, {\mu }^{-\frac {\alpha } {2}}.$$
The proof is now complete. Indeed, by and , the function $\iota({\mu })$, for ${\mu }$ satisfying and , can be taken as the sum of the right hand sides of the estimates , , , , and , where $b$ is given by
Asymptotic behavior of the solutions of {#sec:auxiliary}
========================================
In this section, we give the proof of Proposition \[eAB1\], which concerns the asymptotic behavior as $s\to \infty $ of the solutions $( w_{\mu }) _{ {\mu }\ge 1 }$ of given by Proposition $\ref{eEX1:b1}$. This behavior must be studied separately in each of the four different cases
1. \[ic1\] $N\ge 3$ and $\beta >0$;
2. \[ic2\] $N\ge 3$ and $\beta <0$;
3. \[ic3\] $N\ge 3$ and $\beta =0$;
4. \[ic4\] $N=2$.
Although this section is somewhat technical, the results are not surprising and the methods used are standard techniques. We note that some of the results in this section will also be used in the proof of Proposition \[eEX2\].
Before we begin the detailed analysis, we observe that in all cases, if ${{\mathcal L}_1}( {\mu })$ given by exists and is finite., then $f(r)=r^{-\frac {2} {\alpha }}w_{\mu }(r^{-2})$ solves - with $a ={{\mathcal L}_1}({\mu })$. Since $w_{\mu }\not \equiv 0$, we deduce that ${{\mathcal L}_1}({\mu })\ne 0$.
Proof of Proposition $\ref{eAB1}$ in the case $N\ge 3$ and $\beta > 0$ {#sec:beta_positive}
----------------------------------------------------------------------
\[eEX1:c1\] Assume $N\ge 3$ and $\beta >0$ and let ${\mu }\ge {1 }$. If $w_{\mu }(s) \to 0$ as $s\to \infty $, then the limit exists and is finite.
It follows from Proposition \[eEX1:b1\] that $w_{{\mu }} $ has a constant sign for $s$ large. The result is now a consequence of [@SoupletW Proposition 3.5]. (Note that this last proposition assumes that $w _{{\mu }} $ has a constant sign on $(0,\infty )$, but the argument uses only the fact that $w _{{\mu }} $ has a constant sign for $s$ large.)
Proposition \[eAB1\] follows from Proposition \[eEX1:b1\] and Lemma \[eEX1:c1\].
Proof of Proposition $\ref{eAB1}$ in the case $N\ge 3$ and $\beta < 0$ {#sec:beta_negative}
----------------------------------------------------------------------
Throughout this subsection, we assume that $N\ge 3$ and $\beta <0$, so that $\lambda _1 > \lambda _2 >0$ with the notation . We define $$\label{fBNb2}
Lw=4s^2w''+4\gamma sw'-\beta w$$ so that $Lw_{\mu }={W}_{\mu }$, where $$\label{fTPA55}
{W}_ {\mu }=w_ {\mu }'-|w_ {\mu }|^\alpha w_ {\mu }.$$ Let $\phi _1$ and $\phi _2$ be defined by and , respectively. A simple calculation shows that $L\phi _1=L\phi _2=0$. It then straightforward to check that $w_{\mu }$ satisfies the following variation of the parameter formula $$\label{fBN3}
\begin{split}
w_{\mu }= & \left (\frac {\lambda _2 w_{\mu }(1)+w_{\mu }'(1)} {\lambda _2-\lambda _1}-\frac {1} {2N-4}\int_1^s \tau^{\lambda _1-1} {W}_{\mu }\, d\tau \right )\phi _1 \\
& + \left (\frac {\lambda _1 w_{\mu }(1)+ w_{\mu }'(1)} {\lambda _1-\lambda _2}+\frac {1} {2N-4}\int_1^s \tau^{\lambda _2-1} {W}_{\mu }\, d\tau \right )\phi _2
\end{split}$$ for all $s>0$.
\[eBN5\] Given any ${\mu }\ge 1$, the limit exists and is finite, the map $s\mapsto s^{\lambda _2-1} {W}_ {\mu }(s)$ belongs to $L^1(1,\infty)$, and $$\label{eBN6}
{{\mathcal L}_2}( {\mu })=\frac {\lambda _1 w_ {\mu }(1)+w_ {\mu }'(1)} {\lambda _1-\lambda _2}+\frac {1} {2N-4}\int_1^\infty \tau^{\lambda _2-1}{W}_ {\mu }(\tau)\, d\tau .$$
We first prove that $$\label{fBN10}
\sup_{s \ge 1} \, [s^{\lambda _2} | w_ {\mu }(s)|+ s^{\lambda _2+1}|w_ {\mu }'(s)| ] ( \log s )^{-1} < \infty .$$ Indeed, $z(s)=s^{\lambda _2 }w_ {\mu }(s)$ satisfies with $\sigma =\lambda _2$, i.e. (using ) $$4s^2z''+ 2 N sz'-z' +\frac {\lambda _2 } {s}z+ \frac {1} {s^{ \alpha \lambda _2 }} |z|^\alpha z=0.$$ Therefore, if $$\label{fBN6}
\widetilde{H} (s)=2s^2|z'|^2+\frac {\lambda _2 } {2s} z^2 +\frac {1} {(\alpha +2) s^{\alpha \lambda _2}}|z|^{\alpha +2}$$ then $$\label{fBN6b}
\widetilde{H} '(s)=-\frac {\alpha \lambda _2 |z|^{\alpha +2}} {(\alpha +2)s^{\alpha \lambda _2 +1}} - \frac {\lambda _2 |z|^2} {2s^2}+(1 - 2 (N - 2 ) s ) | z ' | ^ 2$$ It follows that the right hand side of is negative for $ s>s_0=\frac {1} { 2 (N - 2 ) }$; and so $ \widetilde{H} (s) $ is bounded. In particular, $ | z ' (s)| \le \frac {C} {s}$, so that $ | z (s)| \le C \log s$. Estimate easily follows.
Next, we deduce from and the property $\lambda _2\alpha <1$ that $$\label{eBN7}
|{W}_{\mu }(s)|\le |w_ {\mu }'(s)|+|w_ {\mu }(s)|^{\alpha +1}\le Cs^{-\lambda_2(\alpha +1)} (\log s)^{\alpha +1}$$ for $s$ large. Therefore, $s^{\lambda _2-1} {W}_ {\mu }(s)\in L^1(1,\infty)$ and $$\label{eBN7u}
\begin{split}
\int_1^s \tau^{\lambda _1-1} |{W}_ {\mu }(\tau)|\, d\tau & \le \int_1^s \tau^{\lambda _1 - \lambda _2 (\alpha +1) -1} (\log \tau )^{\alpha +1} \, d\tau \\ &\le C\max \{s^{\lambda _1-\lambda_2(\alpha +1)},1 \} (\log s)^{\alpha +1} .
\end{split}$$ Hence, from we see that ${{\mathcal L}_2}( {\mu })$ is well defined and that holds.
\[eBN8\] If ${{\mathcal L}_2}( {\mu })=0 $, then the limit exists and is finite.
If ${{\mathcal L}_2}( {\mu })=0 $ then and yield $$\label{eBN9}
\begin{split}
w_ {\mu }= & \left (\frac {\lambda _2 w_ {\mu }(1)+w_ {\mu }'(1)} {\lambda _2-\lambda _1}-\frac {1} {2N-4}\int_1^s \tau^{\lambda _1-1}{W}_ {\mu }(\tau)\, d\tau \right )\phi _1 \\
& - \left (\frac {1} {2N-4}\int_s^\infty \tau^{\lambda _2-1}{W}_ {\mu }(\tau)\, d\tau \right )\phi _2
\end{split}$$ and $$\label{eBN9:b1}
\begin{split}
w_ {\mu }'= & \left (\frac {\lambda _2 w_ {\mu }(1)+w_ {\mu }'(1)} {\lambda _2-\lambda _1}-\frac {1} {2N-4}\int_1^s \tau^{\lambda _1-1}{W}_ {\mu }(\tau)\, d\tau \right )\phi '_1 \\
& - \left (\frac {1} {2N-4}\int_s^\infty \tau^{\lambda _2-1}{W}_ {\mu }(\tau)\, d\tau \right )\phi ' _2 .\end{split}$$ It follows that $$\sup_{s >0} \, [ (1+s) ^{\lambda _2} |w_ {\mu }(s)|+ (1+s) ^{\lambda _2 +1}|w_ {\mu }'(s)| ] <\infty .$$ Indeed, $(1+s) ^{\lambda _2} |w_ {\mu }(s)|$ is bounded by Lemma \[eBN5\]. Moreover, since $s^{\lambda _2-1} |{W}_ {\mu }(s)|$ is integrable at infinity, again by Lemma \[eBN5\], we deduce from formula that $(1+s) ^{\lambda _2 +1}|w_ {\mu }'(s)| $ is bounded. We now set $$\label{eBN4}
R= \Bigl\{ \rho >0,\ \sup_{s >0}\ \{ (1+s) ^\rho |w_ {\mu }(s)|+ (1+s) ^{\rho +1}|w_ {\mu }'(s)|\}<\infty \Bigr\}$$ so that $R$ is an interval and $\lambda _2\in R$. We claim that $\lambda _1\in R$. We prove this by contradiction, so we assume $\lambda _1 \not \in R $. Therefore may choose $\rho >0$ such that $$\label{fTPA77}
\rho \in R, \quad \rho \ge \lambda _2, \quad \rho < \lambda _1, \quad (\alpha +1) \rho > \sup R, \quad (\alpha +1) \rho \not = \lambda _1.$$ Note that, since $\rho \in R$, we have $$\label{fTPA99}
|{W}_{\mu }(s) | \le C ( s ^{-\rho -1} + s ^{ -(\alpha +1) \rho } )$$ for $s\ge 1$. Moreover, $\alpha \rho = \frac {\rho } {\lambda _1}< 1$, so that $ | {W}_{\mu }(s) | \le C s ^{ -(\alpha +1) \rho } $; and so $$\label{eBN10}
\int_1^s \tau^{\lambda _1-1}|{W}_ {\mu }(\tau) | \, d\tau \le C \theta (s)$$ where $$\label{eBN10b}
\theta (s)=
\begin{cases}
s^{\lambda _1-(\alpha +1)\rho} & (\alpha +1)\rho< \lambda _1 \\
\log (1+s) & (\alpha +1)\rho = \lambda _1 \\
1 & (\alpha +1)\rho > \lambda _1
\end{cases}$$ and $$\label{eBN11}
\int_s^\infty \tau^{\lambda _2-1}|{W}_ {\mu }(\tau) | \, d\tau \le Cs^{\lambda _2-(\alpha +1)\rho} .$$ Using , and, respectively, and we obtain $$\label{eBN11b}
|w_ {\mu }(s)| + s |w_ {\mu }'(s)| \le C \max\{ s^{- \lambda _1} \theta (s), s^{-(\alpha +1) \rho } \} .$$ Since $(\alpha +1) \rho \not \in R$ by , we deduce from that $$\max\{ s^{-\lambda _1} \theta (s) , s^{-(\alpha +1) \rho } \} = s^{-\lambda _1} \theta (s)$$ so that $$\label{fTPA78}
|w_ {\mu }(s)| + s |w_ {\mu }'(s)| \le s^{-\lambda _1} \theta (s) .$$ Since $\lambda _1\not \in R$ by assumption and $(\alpha +1) \rho \not \in R$ (again by ), it follows from and that $\theta (s) =\log (1+s)$. This means that $\lambda _1= (\alpha +1) \rho $, which contradicts the last condition in . This contradiction establishes that $\lambda _1 \in R$.
To show that $ {{\mathcal L}_1}( {\mu })$ is well defined, we let $\rho =\lambda _1$ in , so that $|{W}_{\mu }(s)|\le Cs^{-(\lambda _1+1)}$. Thus, $s^{\lambda _1-1}{W}_ {\mu }(s)\in L^1(1,\infty)$. Moreover, $$\int_s^\infty \tau^{\lambda _2-1}{W}_ {\mu }(\tau)\, d\tau \le C \int_s^\infty \tau^{- (\lambda _1- \lambda _2)-2}\, d\tau \le C s^{-( \lambda _1- \lambda _2)-1}.$$ Thus we deduce from that the limit $ {{\mathcal L}_1}( {\mu })$ exists and is finite.
Proposition \[eAB1\] follows from Lemmas \[eBN5\] and \[eBN8\].
Proof of Proposition $\ref{eAB1}$ in the case $N\ge 3$ and $\beta = 0$ {#proof-of-propositionrefeab1-in-the-case-nge-3-and-beta-0}
----------------------------------------------------------------------
Throughout this subsection, we assume that $N\ge 3$ and $\beta =0$, so that $\lambda _1 =\frac {1} {\alpha } > \lambda _2 =0$ and $\gamma = \frac {1} {\alpha } +1$. Moreover, $\phi _2(s)=(\log s)^{-\frac {1} {\alpha }}$. We first study the limit .
\[eBN2b\] Given any ${\mu }\ge {1 }$, the limit ${{\mathcal L}_2}( {\mu })$ given by exists and is finite, and either ${{\mathcal L}_2}( {\mu })= 0$ or else ${{\mathcal L}_2}( {\mu })= \pm ( \frac {2} {\alpha } )^{\frac {2} {\alpha }} $. Moreover, $$\label{fTPA44}
\frac {s w_{\mu }' (s)} {w_{\mu }(s)} {\mathop{\longrightarrow}}_{ s\to \infty } 0$$ if and only if ${{\mathcal L}_2}( {\mu })= \pm ( \frac {2} {\alpha } )^{\frac {2} {\alpha }} $.
Since $w_{\mu }$ has a finite number of zeros, we may suppose $w_{\mu }(t)>0$ for $t$ large. Set $$v(t)=t^{\frac {1} {\alpha }} w _{\mu }(s) = \frac { w_{\mu }(s)} {\phi _2(s)}$$ with $t=\log s$ and $s>1$, so that $v (t) >0$ for $t$ large and ${{\mathcal L}_2}( {\mu })= \lim _{ t\to \infty } v(t)$ if this last limit exists. Note that $$\label{fTPA94}
\frac {sw _{\mu }'(s)} {w _{\mu }(s)}=\frac {v'(t)} {v(t)}-\frac {1} {\alpha t}.$$ Since $w_{\mu }$ is a solution of , a straightforward calculation shows that $v$ satisfies $$\label{fFH18}
4t v''+\Bigl (\frac {4t} {\alpha }-\frac {8} {\alpha }-te^{-t}\Bigr )v'-\Bigr (\frac {4} {\alpha ^2}-\frac {4(\alpha +1)} {\alpha ^2 t}-\frac {e^{-t}} {\alpha }\Bigl )v+|v|^\alpha v=0 .$$ Set $$\label{fFH19}
\Psi (v)=\frac {1} {\alpha +2}|v|^{\alpha +2}-\frac {2} {\alpha ^2}|v|^2$$ and $$\label{fFH19:1}
V (t) =2t v' (t) ^2+\Bigl (\frac {2(\alpha +1)} { \alpha ^ 2 t} +\frac {1 } {2\alpha }e^{-t}\Bigr ) v (t) ^2 + \Psi (v (t) )$$ so that in particular $V (t)$ is bounded below. It follows using that $$\label{fTTPPAA}
V' (t) =-\Bigl (\frac {4t} {\alpha }-\frac {8} {\alpha }-2-te^{-t}\Bigr ) v' (t) ^2-\Bigl (\frac {2(\alpha +1)} {\alpha ^2 t^2} + \frac {1} {2\alpha }e^{-t}\Bigr )v (t)^2.$$ Therefore, $V' (t) <0$ for $t$ large, and $V$ is decreasing; and so $V(t)$ has a finite limit $V^\infty $ as $t\to \infty $. It follows that $v (t) $ and $ t |v'(t) |^2 $ are bounded as $t\to \infty$, hence $v'(t) \to 0$ as $t\to \infty $.
We next show that $v(t) $ has a limit as $t\to \infty $, i.e. that ${{\mathcal L}_2}({\mu })$ exists. For this, we set $$\label{fTTPPAA1}
v^+ = \limsup _{ t\to \infty } v (t), \quad v^- = \liminf _{ t\to \infty } v (t) .$$ Assuming $v^- < v^+$, it follows that $v$ oscillates asymptotically between $v^-$ and $v^+$, so that there exist $t_n^\pm \to \infty $ such that $v' (t_n^\pm ) =0$, $v (t_n^\pm ) \to v^\pm$. Letting $t= t_n ^\pm$ in , then $n\to \infty $, we deduce that $$\label{fTTPPAA2}
V^\infty = \Psi ( v^- ) = \Psi ( v^+) .$$ Since $\Psi(v^+)=\Psi(v^-)$, there exists $ \underline{v}\in (v^- ,v^+ )$ such that $\Psi'(\underline{v})=0$. From we obtain $ \underline{v}=(\frac {2} {\alpha } )^{\frac {2} {\alpha }}$. Consider now $ \underline{v}<a<b<v^+$ and $\tau _n \uparrow \infty$ such that $v(\tau_{2n})= a$, $v(\tau_{2n+1})= b$, with $v(t)\in [a,b]$ for $t\in [\tau _{2n}, \tau_{2n+1}]$. Thus $v(t) ^\alpha \ge a^\alpha > \underline{v} ^\alpha =\frac {4} {\alpha ^2}$ for $t\in [\tau _{2n}, \tau_{2n+1}]$. On the other hand, we can write in the form $$\label{fFH18:1}
\Bigl (v' + \Bigl( \frac {1} {\alpha } - \frac {2} {\alpha t} - \frac {e^{-t}} {4} \Bigr)v \Bigr )' = \frac {1} {4t} \Bigr ( \underline{v}^\alpha - |v| ^\alpha + \frac {4(\alpha -1)} {\alpha ^2 t }-\frac {1} {\alpha }e^{-t} + t e^{-t} \Bigl ) v .$$ We observe that the right-hand side of is negative on $ [\tau _{2n}, \tau_{2n+1}] $ for $n$ sufficiently large. Integrating on $ ( \tau _{2n}, \tau_{2n+1} ) $, we obtain $$v' ( \tau_{2n+1} ) + \Bigl( \frac {1} {\alpha } - \frac {2} {\alpha \tau_{2n+1}} - \frac {e^{-\tau_{2n+1}}} {4} \Bigr) b \le v' ( \tau_{2n } ) + \Bigl( \frac {1} {\alpha } - \frac {2} {\alpha \tau_{2n }} - \frac {e^{-\tau_{2n }}} {4} \Bigr) a.$$ We now let $n\to \infty $. Since $v'( \tau _n) \to 0$, we obtain $b\le a$, which is absurd. It follows that there exists $v_{\infty}\ge 0$ such that $v(t)\to v_{\infty}$ as $t\to \infty$.
We now claim that either $v_\infty =0$ or else $v_\infty = \underline{v} $. Indeed, suppose first $v_\infty \in ( 0, \underline{v} )$, so that $(\underline{v}^\alpha - v_\infty ^\alpha) v_\infty <0$. It follows from that there exists $\delta >0$ such that $$\Bigl (v' + \Bigl( \frac {1} {\alpha } - \frac {2} {\alpha t} - \frac {e^{-t}} {4} \Bigr)v \Bigr )' \le - \frac {\delta } {t}$$ for $t $ large. Integrating the above inequality yields a contradiction with the fact that $v$ and $v'$ are bounded. We obtain in the same way a contradiction if we assume that $v_\infty > \underline{v} $. Thus either $v(t) \to 0$, or else $v(t) \to (\frac {2} {\alpha })^{\frac {2} {\alpha }}$ as $t\to \infty $. This proves the first part of the statement.
Suppose now ${{\mathcal L}_2}( {\mu }) = (\frac {2} {\alpha })^{\frac {2} {\alpha }}$, i.e. $v(t) \to (\frac {2} {\alpha })^{\frac {2} {\alpha }}$. Since $v'(t) \to 0$, it follows that $$\label{fTPA82}
\frac {v'(t)} {v(t)} {\mathop{\longrightarrow}}_{ t\to \infty } 0.$$ Applying , we deduce that holds. Conversely, assume , so holds. We prove that $v(t) \to (\frac {2} {\alpha })^{\frac {2} {\alpha }}$ by contradiction. Otherwise, $v(t) \to 0$, and it follows from that $v' + ( \frac {1} {\alpha } - \frac {2} {\alpha t} - \frac {e^{-t}} {4} )v$ is nondecreasing for $t$ large. Since both $v(t)$ and $v'(t)$ converge to $0$ as $t\to \infty $, we deduce that $v' + ( \frac {1} {\alpha } - \frac {2} {\alpha t} - \frac {e^{-t}} {4} )v\le 0$ for $t$ large. It follows that $\frac {v'} {v} \le - \frac {1} {2\alpha }$ for $t$ large, contradicting . This completes the proof.
\[eFH1\] Let ${\mu }\ge {1 }$ and let $s_{\mu }>0$ be sufficiently large so that $$w_{\mu }(s) w_{\mu }' (s) < 0, \quad s>s_{\mu }.$$ (See Proposition $\ref{eEX1:b1}$ .) If $$\label{fFH1}
h_{1, {\mu }} (s) = \frac {sw _{\mu }' (s)} {w_{\mu }(s)} + \frac {1} {\alpha }, \quad h_{2, {\mu }} (s) = \frac {sw _{\mu }' (s)} {w_{\mu }(s)}$$ for $s>s_{\mu }$, then $$\begin{aligned}
\frac {d} {ds} ( s^{- \frac {N-2} {2} } e^{\frac {1} {4s }} h_{1, {\mu }} (s) ) &= - s^{- \frac {N} {2} } e^{\frac {1} {4s }} \Bigl( \frac {1 } {4\alpha s} + \frac { |w_{\mu }|^\alpha } {4} + h_{1, {\mu }} ^2 (s) \Bigr) \label{fFH2} \\
\frac {d} {ds} ( s^{\frac {N-2} {2}} e^{\frac {1} {4s }} h_{2, {\mu }} (s) ) &= - s^{\frac {N-4} {2} } e^{\frac {1} {4s }} \Bigl( \frac { |w _{\mu }|^\alpha } {4} + h_{2, {\mu }}^2 (s) \Bigr) \label{fFH2b}\end{aligned}$$ for all $s>s_ {\mu }$ and $j=1, 2$.
Identity is formula (3.1) in [@SoupletW], and is straightforward to verify. Formula can be obtained from and the identity $h_{2, {\mu }} (s) = h_{1, {\mu }} (s) -\frac {1} {\alpha }=h_{1, {\mu }} (s) - \frac {N-2} {2}$.
\[eFH2\] Let ${\mu }\ge {1 }$. With the notation , we have $$\label{fFH3}
h_{2, {\mu }} (s) {\mathop{\longrightarrow}}_{ s\to \infty } \ell$$ with either $\ell =0$ or else $\ell = - \frac {N-2} {2}$.
We first show that holds for some $\ell \in [-\infty , 0]$. Note that $h_{ 2, {\mu }} (s) <0$ for all $s>s_{\mu }$. Suppose there exists $\sigma _{\mu }>s_ {\mu }$ such that $h_{ 2, {\mu }} ( \sigma _{\mu }) \le - \frac {N-2} {2}$. This means that $h_{ 1, {\mu }} ( \sigma _{\mu }) \le 0$, then by , we deduce that $h_{ 1, {\mu }} (s) <0$ for all $s> \sigma _{\mu }$. Furthermore, since the right-hand side of is negative, $$s^{- \frac {N-2} {2}} e^{\frac {1} {4s }} h_{ 1, {\mu }} ' (s) \le - h_{ 1, {\mu }} (s) \frac {d} {ds} ( s^{- \frac {N-2} {2}} e^{\frac {1} {4s }}) = h_{ 1, {\mu }} (s) \Bigl( \frac {N-2} {2} + \frac {1} {4s} \Bigr) s^{- \frac {N } {2}} e^{\frac {1} {4s }} <0$$ for $s> \sigma _{\mu }$. Thus $h_{ 1, {\mu }}$ is decreasing, and so is $h_{ 2, {\mu }}$. The desired conclusion then follows.
It remains to consider the case $- \frac {N-2} {2} < h_{ 2, {\mu }} (s) < 0$ for $s>s_ {\mu }$. Suppose by contradiction that $h_{ 2, {\mu }}$ does not have a limit. Let $$-\frac {N-2} {2}\le \underline{ \ell } = \liminf _{ s\to \infty } h_{ 2, {\mu }} (s) < \overline{ \ell } = \limsup _{ s\to \infty } h_{ 2, {\mu }} (s) \le 0$$ and let $\ell \in ( \underline{ \ell} , \overline{ \ell} )$. Consider an increasing sequence $s _n \to \infty $ such that $$\label{fFH4}
h_{ 2, {\mu }} (s _n) = \ell \text{ and } h_{ 2, {\mu }}( s) \ge \ell \text{ for }s _{ 2n }\le s\le s _{ 2n+1 }.$$ Integrating on $(s _{ 2n }, s _{ 2n+1 })$ and applying we obtain $$\begin{split}
( s _{ 2n+1 }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n+1 } }} - s _{ 2n }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n} }} ) \ell & = - \int _{ s _{ 2n } }^{s _{ 2n+1 }} s^{\frac {N-4} {2} } e^{\frac {1} {4s }} \Bigl( \frac { |w _{\mu }|^\alpha } {4} + h_{ 2, {\mu }} (s) ^2 \Bigr)\, ds \\ & \ge - \int _{ s _{ 2n } }^{s _{ 2n+1 }} s^{\frac {N-4} {2} } e^{\frac {1} {4s }} \Bigl( \frac { |w _{\mu }|^\alpha } {4} + \ell ^2 \Bigr)\, ds .
\end{split}$$ Since $w _{\mu }(s) \to 0$ as $s \to \infty $, given any $0<\delta < \ell ^2$, we have $ \frac { |w _{\mu }(s)|^\alpha } {4}< \delta $ for $s$ large, and we deduce that $$( s _{ 2n+1 }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n+1 } }} - s _{ 2n }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n} }} ) \ell \ge - \int _{ s _{ 2n } }^{s _{ 2n+1 }} s^{\frac {N-4} {2} } e^{\frac {1} {4s }} ( \ell ^2 +\delta )\, ds .$$ Since $\ell <0$, we deduce that $$\frac { s _{ 2n+1 }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n+1 } }} - s _{ 2n }^{\frac {N-2} {2}} e^{\frac {1} {4s _{ 2n} }} } {\int _{ s _{ 2n } }^{s _{ 2n+1 }} s^{\frac {N-4} {2} } e^{\frac {1} {4s }} ds } \le \frac { \ell ^2 +\delta } {- \ell} .$$ Letting $n\to \infty $ (and since $e^{\frac {1} {4s }} \to 1$ as $s\to \infty $), we conclude that $$\frac {N-2} {2} \le \frac { \ell ^2 +\delta } {- \ell} .$$ Letting now $\delta \downarrow 0$, we obtain $\ell \le - \frac {N-2} {2}$, which is absurd since $\ell > \underline{\ell} \ge - \frac {N-2} {2}$.
Thus we have shown that holds for some $\ell \in [-\infty , 0]$, and we finally prove that $\ell =0 $ or $\ell = - \frac {N-2} {2}$. Integrating on $(1, s)$, we obtain $$e^{\frac {1} {4s }} h_{ 2, {\mu }} (s) = s^{- \frac {N-2} {2}} e^{\frac {1} {4 }} h_{ 2, {\mu }} (1) - s^{ - \frac {N-2} {2}} \int _1 ^s \tau ^{\frac {N-4} {2} } e^{\frac {1} {4\tau }} \Bigl( \frac { |w _{\mu }|^\alpha } {4} + h_{ 2, {\mu }} (\tau ) ^2 \Bigr) \, d\tau ,$$ and so, letting $s\to \infty $ and applying , $$\label{fFH5}
\ell = - \lim _{ s\to \infty } \frac { \int _1 ^s \tau ^{\frac {N-4} {2} } e^{\frac {1} {4\tau }} \Bigl( \frac { |w _{\mu }|^\alpha } {4} + h_{ 2, {\mu }} (\tau ) ^2 \Bigr) \, d\tau } { s^{ \frac {N-2} {2}}} .$$ Both the numerator and the denominator in the right-hand side of go to $\infty $ with $s$. By l’Hôpital’s rule $$\ell = - \lim _{ s\to \infty } \frac {s ^{\frac {N-4} {2} } e^{\frac {1} {4s }} ( \frac { |w _{\mu }|^\alpha } {4} + h_{ 2, {\mu }} (s ) ^2 ) } {\frac {N-2} {2} s^{ \frac {N-4} {2}}} = - \frac {2} {N-2} \ell ^2.$$ Therefore, either $\ell = - \frac {N-2} {2}$ or else $\ell=0$.
\[eFH3\] Given ${\mu }\ge 1$, if ${{\mathcal L}_2}( {\mu })=0 $, then the limit exists and is finite.
We follow the argument of [@SoupletW proof of Proposition 3.5]. Suppose ${{\mathcal L}_2}( {\mu })=0 $. It follows from Lemma \[eBN2b\] that does not hold. Therefore, holds with $\ell = - \frac {N-2} {2}$, so that $h_{ 1, {\mu }}(s) \to 0$ as $s\to \infty $. Since the right-hand side of is negative, we deduce that $h_{ 1, {\mu }}(s) >0$ for $s> s_{\mu }$. Moreover, integrating on $(s, \infty )$, we obtain $$\begin{split}
s^{- \frac {N-2} {2} } e^{\frac {1} {4s }} h_{ 1, {\mu }} (s) & = \int _{ s }^\infty \tau ^{- \frac {N} {2} } e^{\frac {1} {4\tau }} \Bigl( \frac { 1 } {4 \alpha \tau } + \frac { |w _{\mu }|^\alpha } {4} + h_{ 1, {\mu }} (\tau ) ^2 \Bigr) \, d\tau \\
& \le e^{\frac {1} {4s }} \Bigl( \frac { 1 } {4 \alpha s } + \frac { |w _{\mu }(s) |^\alpha } {4} + \sup _{ \tau \ge s } h_{ 1, {\mu }} (\tau ) ^2 \Bigr) \int _s^\infty \tau ^{- \frac {N} {2} } d\tau \\
& = \frac {2} {N-2} s^{- \frac {N-2} {2} } e^{\frac {1} {4s }} \Bigl( \frac {1 } {4 \alpha s } + \frac { |w _{\mu }(s) |^\alpha } {4} + \sup _{ \tau \ge s } h_{ 1, {\mu }} (\tau ) ^2 \Bigr) .
\end{split}$$ This implies $$\sup _{ \tau \ge s } h_{ 1, {\mu }} (\tau ) \le \frac {2} {N-2} \Bigl( \frac { 1 } {4 \alpha s } + \frac { |w _{\mu }(s) |^\alpha } {4} + \sup _{ \tau \ge s } h_{ 1, {\mu }} (\tau ) ^2 \Bigr)$$ so that (since $h_{ 1, {\mu }}(s) \to 0$) $$\label{fFH6}
\sup _{ \tau \ge s } h_{ 1, {\mu }} (\tau ) \le \frac {4} {N-2} \Bigl( \frac { 1 } {4 \alpha s } + \frac { |w _{\mu }(s) |^\alpha } {4} \Bigr)$$ for $s$ large. Since $h_{ 1, {\mu }}(s) \to 0$, we have $\frac {w_{\mu }'} {w _{\mu }} + \frac {1} {2\alpha s}\le 0$ for $s$ large, by . It follows that $w _{\mu }(s) \le C s^{- \frac {1 } {2\alpha } }$, so that by , $h_{ 1, {\mu }} (s) \le C s^{ -\frac {1} {2} }$. Therefore, $\frac {w _{\mu }'(s)} {w _{\mu }(s)} + \frac {1} {\alpha s} = O( s^{-\frac {3} {2} })$, from which the existence of the limit easily follows.
Proposition \[eAB1\] follows from Lemmas \[eBN2b\] and \[eFH3\].
Proof of Proposition $\ref{eAB1}$ in the case $N=2$ {#NeqD}
---------------------------------------------------
We set $\lambda =\frac {1} {\alpha }$, so that $\phi _1(s)=s^{-\lambda }$ and $\phi _2 (s) = s^{-\lambda }\log s$. It is straightforward to verify the following variation of the parameter formula $$\label{fNd1}
w_{\mu }(s) =c_{1,{\mu }}(s)\phi _1 (s) +c_{2,{\mu }}(s)\phi _2 (s),$$ where $$\label{fNd1bis}
\begin{split}
&c_{1,{\mu }}(s)=w_{{\mu }}(1)-\frac {1} {4}\int_1^s \tau^{\lambda -1} (\log \tau ) {W}_{\mu }(\tau)\, d\tau,\\
&c_{2,{\mu }}(s)=\lambda w_{{\mu }}(1)+w'_{{\mu }}(1) +\frac {1} {4}\int_1^s \tau^{\lambda -1} {W}_{\mu }(\tau)\, d\tau
\end{split}$$ and ${W}_{\mu }$ is given by .
\[eNd10\] Given $ \overline{{\mu }} \ge {1 }$, there exists $C>0$ such that $$\label{fNd2d}
\sup_{{1 }\le {\mu }\le \overline{{\mu }} }\, \sup_{s \ge 0} \, \frac {(1+ s)^{\lambda }} {\log (2+s)} \, ( (1 + s) | w_{\mu }'(s) |+| w_{\mu }(s)|)\le C$$ and $$\label{fNd2e}
\sup_{{1 }\le {\mu }\le \overline{{\mu }} }\, \sup_{s \ge 0} \, \frac {(1+ s)^{\lambda +1}} { (\log (2+s))^{\alpha +1}} |{W}_{\mu }(s)|\le C.$$
$z(s)=s^\lambda w_{\mu }(s)$ is a solution of equation and it follows from and that $$\sup _{ s\ge 1 } \Bigl( s | z' (s)| + \frac {1} {\log (2+s)} |z (s)| \Bigr)\le C$$ where $C$ depends only on $ |z(1)|+ |z'(1)|$. This implies that $$\sup_{s \ge 1} \, \frac {(1+ s)^{\lambda }} {\log (2+s)} \, ( (1 + s) | w_{\mu }'(s) |+| w_{\mu }(s)|)\le C$$ where $C$ depends only on $ |w_{\mu }(1)|+ |w_{\mu }'(1)|$. Estimate now follows from the continuous dependence of $w_ {\mu }$ and $w'_ {\mu }$ on $ {\mu }$ given by Proposition \[eEX1:b1\] . Finally, is a direct consequence of .
\[eTPA2\] For all ${\mu }\ge {1 }$, the map $\tau \mapsto \tau^{\lambda -1} (\log \tau ) {W}_{\mu }(\tau)$ is integrable at $\infty $. Moreover, the functions $c _{ j, {\mu }} (s)$ defined by have finite limits $c _{ j, {\mu }}^\infty $ as $s\to \infty $, given by $$\label{fNd1t}
\begin{split}
&c_{1,{\mu }}^\infty =w_{{\mu }}(1)-\frac {1} {4}\int_1^\infty \tau^{\lambda -1} (\log \tau ) {W}_{\mu }(\tau)\, d\tau,\\
&c_{2,{\mu }}^\infty =\lambda w_{{\mu }}(1)+w'_{{\mu }}(1) +\frac {1} {4}\int_1^\infty \tau^{\lambda -1} {W}_{\mu }(\tau)\, d\tau .
\end{split}$$ In addition, the maps ${\mu }\mapsto c_{j,{\mu }}^\infty $ are continuous $[{1 }, \infty ) \to {{\mathbb R}}$ and, given $ \overline{{\mu }} \ge {1 }$, there exists a constant $C$ such that $$\label{fTPA35}
| c _{ j, {\mu }} ^\infty - c _{ j, {\mu }} (s)| \le C s^{- \frac {1} {2}}$$ for $j=1,2$, $s\ge 1$ and ${1 }\le {\mu }\le \overline{{\mu }} $.
Fix $ \overline{{\mu }} > {1 }$. It follows from that there exists a constant $C$ such that $$\label{fTPA36}
\tau^{\lambda -1} (\log \tau ) |{W}_{\mu }(\tau) | \le C \tau ^{-2} ( \log (2 +\tau )^{\alpha +2}$$ for all $\tau \ge 1$ and all ${1 }\le {\mu }\le \overline{{\mu }} $. This shows the integrability property, the existence of limits $c _{ j, {\mu }}^\infty $, formulas and estimates . We now prove the continuity. Let ${\mu }_1, {\mu }_2 \in [ {1 }, \overline{{\mu }} )$. Given $s\ge 1$, we deduce from that $$|c _{ j, {\mu }_1 }^\infty - c _{ j, {\mu }_2 }^\infty | \le |c _{ j, {\mu }_1 } (s) - c _{ j, {\mu }_2 } (s) | + C s^{- \frac {1} {2}} .$$ Given $\varepsilon >0$, we first fix $s_0$ sufficiently large so that $C s_0^{- \frac {1} {2}}\le \frac {\varepsilon } {2}$. By continuous dependence (Proposition \[eEX1:b1\] ), if $ | {\mu }_1 - {\mu }_2 |$ is sufficiently small, then $ |c _{ j, {\mu }_1 } (s_0) - c _{ j, {\mu }_2 } (s_0) | \le \frac {\varepsilon } {2}$. Therefore, $|c _{ j, {\mu }_1 }^\infty - c _{ j, {\mu }_2 }^\infty | \le \varepsilon $, which proves continuity on $[{1 }, \overline{{\mu }} )$. Since $ \overline{{\mu }} $ is arbitrary, this completes the proof.
\[eNd11\] Let ${\mu }\ge {1 }$, and let $c _{ 1, {\mu }}^\infty $ and $c _{ 2, {\mu }}^\infty $ be given by . The limit exists and ${{\mathcal L}_2}( {\mu }) = c _{ 2, {\mu }}^\infty $. Moreover, if ${{\mathcal L}_2}({\mu })=0$ then the limit exists and ${{\mathcal L}_1}({\mu })= c _{ 1, {\mu }}^\infty $.
By formula , $$\frac {w_{\mu }(s)} {\phi _2 (s)} = \frac {c _{ 1, {\mu }} (s) } {\log s} + c _{ 2, {\mu }}(s).$$ It follows from Lemma \[eTPA2\] that the limit is well defined and ${{\mathcal L}_2}( {\mu }) = c _{ 2, {\mu }}^\infty $. Moreover, it follows from formula again that $$\frac {w_{\mu }(s)} {\phi _1 (s)} = c _{ 1, {\mu }} (s) + c _{ 2, {\mu }}(s) \log s.$$ Therefore, if ${{\mathcal L}_2}( {\mu }) =0$ (hence $c _{ 2, {\mu }}^\infty =0$), then by $$\Bigl| \frac {w_{\mu }(s)} {\phi _1 (s)} - c _{ 1, {\mu }}^\infty \Bigr| \le | c _{ 1, {\mu }} (s) - c _{ 1, {\mu }}^\infty | + | c _{ 2, {\mu }} (s) - c _{ 2, {\mu }}^\infty | \log s {\mathop{\longrightarrow}}_{ s\to \infty }0.$$ Thus we see that ${{\mathcal L}_1}({\mu })$ is well defined and $ {{\mathcal L}_1}({\mu })=c _{ 1, {\mu }}^\infty$.
Proposition \[eAB1\] follows from Lemma \[eNd11\].
Local behavior of the number of zeros {#sFNL}
=====================================
This section is devoted to the proof of Proposition \[eEX2\]. We consider separately the cases $N\ge 3$ and $N=2$.
The case $N\ge 3$
-----------------
The proof is inspired by the proof of [@McLeodTW Lemma 4]. Since $\lambda _1 > \frac {\lambda _1 + \lambda _2} {2} > \max \{ 0, \lambda _2 \}$ we may fix $$\label{fNZE1}
\quad \sigma \in (\max \{ 0, \lambda _2 \}, \textstyle { \frac {\lambda _1 + \lambda _2} {2} } ).$$ We set $$\label{fNZE2}
a_1= 1 + \lambda _1 + \lambda _2 -2 \sigma , \quad a_2 =-2 (\sigma - \lambda _1) (\sigma - \lambda _2) , \quad a_3 = 4 (\lambda _1 + \lambda _1 -2\sigma )$$ so that by $$\label{fNZE4}
a_1 >1, \quad a_2 >0, \quad a_3 >0.$$ Given ${\mu }\ge {1 }$ we set $$\label{fNZE5}
z _{\mu }(s) = s^\sigma w_{\mu }(s)$$ so that (see ) $$\label{fNZE6}
4s^2 z _{\mu }''+ 4 a_1 s z _{\mu }' - z ' - 2 a_2 z _{\mu }+ \frac {\sigma } {s} z _{\mu }+ s^{-\alpha \sigma }| z _{\mu }|^\alpha z _{\mu }=0.$$ Moreover, we set $$\label{fNZE7}
\widetilde{H} _{\mu }(s) = 2 s^2 | z _{\mu }' |^2 - a_2 z _{\mu }^2 + \frac {\sigma } {2 s} z _{\mu }^2 + \frac {s^{-\sigma \alpha }} {\alpha +2} | z _{\mu }|^{\alpha +2} .$$ Elementary calculations show that $$\label{fNZE8}
\widetilde{H} _{\mu }'(s) = -\frac {\alpha \sigma s^{- \alpha \sigma -1} } { \alpha +2} | z _{\mu }|^{\alpha +2} -\frac {\sigma } {2s^2} z _{\mu }^2 +(1 - a_3 s)| z _{\mu }'|^2 .$$ In particular, $$\label{fNZE8b1}
\widetilde{H} _{\mu }'(s) \le 0, \quad s\ge \frac {1} {a_3}.$$ Given $\delta >0$, it follows from that $$\begin{split}
s^{ - \delta} \frac {d} {ds} [ s^\delta \widetilde{H} _{\mu }(s) ] & = \widetilde{H} _{\mu }' (s) +\frac {\delta } {s} \widetilde{H} _{\mu }(s) \\
= - [ ( a_3 - 2\delta ) s -1] | z _{\mu }' |^2 & - \frac { z _{\mu }^2} { 2 s^2} [ (1- \delta ) \sigma + 2 \delta a_2 s ] - \frac {s^{- \alpha \sigma -1} | z _{\mu }|^{\alpha +2}} {\alpha +2} [ \alpha \sigma -\delta ] .
\end{split}$$ Therefore, if we fix $$\label{fNZE10}
0 < \delta < \min \Bigl\{ \frac {a_3} {4}, 1, \alpha \sigma \Bigr\}$$ then $$\label{fNZE11}
\frac {d} {ds} [ s^\delta \widetilde{H} _{\mu }(s) ] \le - \delta a_2 s^{ \delta -1 } z _{\mu }(s)^2, \quad s\ge \frac {2} {a_3}.$$ Suppose first ${{\mathcal L}_2}( \overline{{\mu }}) \not = 0 $, for instance ${{\mathcal L}_2}( \overline{{\mu }}) > 0 $. Since $\sigma > \max\{0, \lambda _2 \}$, we see that $ z _{ \overline{{\mu }} } (s) \to \infty $, and it follows from that $ \frac {d} {ds} [ s^\delta \widetilde{H} _{ \overline{{\mu }} } (s) ] \le - c s^{\delta -1}$ for $s$ large, with $c>0$. Since $s^{\delta -1}$ is not integrable at $\infty $, there exists $s_0\ge \frac {2} {a_3}$ such that $\widetilde{H} _{ \overline{{\mu }} } (s_0) <0$. By continuous dependence, we have $\widetilde{H} _{\mu }( s_0 ) <0 $ for ${\mu }$ close to $ \overline{{\mu }} $. Therefore, by , $\widetilde{H} _{\mu }( s ) <0 $ for $s\ge s_0$ and ${\mu }$ close to $ \overline{{\mu }} $, and it follows from that $w_{\mu }$ does not vanish for $s\ge s_0$ and ${\mu }$ close to $ \overline{{\mu }} $. Again by continuous dependence of $w_{\mu }$ on ${\mu }$ in $C^1([0, s_0 ]) $, we conclude that $w_{\mu }$ has the same number of zeros as $w _{ \overline{{\mu }} }$ provided ${\mu }$ is sufficiently close to $ \overline{{\mu }} $. This proves Property of Proposition \[eEX2\].
We now assume ${{\mathcal L}_2}( \overline{{\mu }} )=0$. It follows from Proposition \[eAB1\] that ${{\mathcal L}_1}( \overline{{\mu }} )$ exists and is finite, therefore by $$\label{fNZE12}
z _{ \overline{{\mu }} } (s) {\mathop{\longrightarrow}}_{ s\to \infty }0.$$ Moreover, we deduce from equation that for large $s$, if $ z _{ \overline{{\mu }} } '$ vanishes, then $z _{ \overline{{\mu }} } '' $ has the sign of $z _{ \overline{{\mu }} }$. It easily follows that $ z _{ \overline{{\mu }} } '$ has constant sign for $s$ large. Therefore, we may assume without loss of generality that there exists $$\label{fNZE13b2}
\overline{s} \ge \frac {2} {a_3}$$ such that $$\label{fNZE13}
z _{ \overline{{\mu }} } (s) >0 \text{ and } z _{ \overline{{\mu }} } '(s) <0 \text{ for } s\ge \overline{s} .$$ We now let $\varepsilon >0$ to be specified later, and we consider $ {\mu }\ge {1 }$ such that $$\label{fNZE13b1}
| {\mu }- \overline{ {\mu }} | \le \varepsilon .$$ It follows easily from continuous dependence (Proposition \[eEX1:b1\] ), and the fact that if $w_{\mu }(s)= 0$ then $w_{\mu }'(s) \not = 0$, that $w_{\mu }$ has ${{\widetilde {N}}}( \overline{ {\mu }} )$ zeros on $(0, \overline{s} ) $ if $\varepsilon $ is sufficiently small. This means that $z_{\mu }$ has ${{\widetilde {N}}}( \overline{ {\mu }} ) $ zeros on $(0, \overline{s}) $. Moreover, by choosing $\varepsilon $ possibly smaller, we have $$\label{fNZE14}
z _{\mu }( \overline{s} ) >0 \text{ and } z _{\mu }' ( \overline{s} ) <0 .$$ If $z _{\mu }>0$ on $( \overline{s}, \infty ) $, then $w _{\mu }$ has ${{\widetilde {N}}}( \overline{{\mu }} ) $ zeros on $(0,\infty )$.
Assume now that $z _{\mu }$ has a zero on $( \overline{s} , \infty )$, and let $ s_ {\mu }> \overline{s} $ be the smallest such zero. By and continuous dependence, $$\label{fNZE15}
s_ {\mu }{\mathop{\longrightarrow}}_{ \varepsilon \to 0 } \infty .$$ In particular, if $\varepsilon $ is sufficiently small, then $$\label{fNZE16}
s_ {\mu }\ge \frac {\sigma } {a_2} .$$ Moreover, $ z _{\mu }' ( s_ {\mu }) <0$; and so we may define $ s_ {\mu }< \tau _{\mu }\le \infty $ by $$\label{fNZE17}
\tau _{\mu }= \sup \{ s> s_ {\mu }; \, z _{\mu }' <0 \text{ on } ( s_ {\mu }, s ) \} .$$ If $\tau _{\mu }=\infty $, then $z _{\mu }$ has ${{\widetilde {N}}}( \overline{{\mu }} ) +1$ zeros on $(0, \infty )$ and $\limsup z _{\mu }(s) <0$ as $s\to \infty $. Therefore, $ \limsup \frac {w _{\mu }(s)} {\phi _1(s)}=\infty$, showing that ${{\mathcal L}_2}( {\mu })\ne 0$. If $\tau _{\mu }<\infty $, then $z _{\mu }(\tau _{\mu }) <0$, $z _{\mu }' (\tau _{\mu })= 0$ and $z _{\mu }'' (\tau ) \ge 0$, so that by $$|z _{\mu }(\tau _{\mu })|^\alpha \ge \tau _{\mu }^{\alpha \sigma } \Bigl( 2a_2 - \frac {\sigma } {\tau _{\mu }} \Bigr).$$ Applying , we deduce that $ |z _{\mu }( \tau _{\mu })| \ge a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma } $. Thus we see that there exist $ s _{\mu }< t_1 < t_2\le \tau _{\mu }$ such that $$\label{fNZE19}
z _{\mu }( t_1 ) = - \frac {1} {2} a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma } , \quad z _{\mu }( t_2 ) = - a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma }$$ and $$\label{fNZE20}
- \frac {1} {2} a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma }\ge z _{\mu }(s) \ge - a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma }, \quad t_1 \le s\le t_2 .$$ It follows from , and that $$\begin{split}
t_2 ^\delta \widetilde{H} _{\mu }(t_2 ) - \overline{s} ^\delta \widetilde{H} _{\mu }( \overline{s} ) & \le - \delta a_2 \int _{ \overline{s} }^{t_2} s^{ \delta -1 } z _{\mu }(s)^2 \le - \delta a_2 \int _{t_1} ^{t_2} s^{ \delta -1 } z _{\mu }(s)^2 \\ & \le - \frac {\delta } {4} a_2^{\frac {\alpha +2} {\alpha }} \tau ^{2\sigma } \int _{t_1} ^{t_2} s^{ \delta -1 } = - \frac {1 } {4} a_2^{\frac {\alpha +2} {\alpha }} \tau ^{2\sigma } (t_2 ^\delta - t_1^\delta ) .
\end{split}$$ Since $s ^\delta \widetilde{H} _{\mu }(s)$ is nonincreasing on $( \overline{s} , \infty )$ by and , and since by continuous dependence, $ \overline{s} ^\delta \widetilde{H} _{\mu }( \overline{s} ) $ is bounded uniformly on $ {\mu }$ satisfying , we have $$\label{fNZE21}
\begin{split}
\tau _{\mu }^\delta \widetilde{H} _{\mu }(\tau _{\mu }) & \le t_2 ^\delta \widetilde{H} _{\mu }(t_2 ) \le \overline{s} ^\delta \widetilde{H} _{\mu }( \overline{s} ) - \frac {1 } {4} a_2^{\frac {\alpha +2} {\alpha }} \tau ^{2\sigma } (t_2 ^\delta - t_1^\delta ) \\ & \le C - \frac {1 } {4} a_2^{\frac {\alpha +2} {\alpha }} \tau ^{2\sigma } (t_2 ^\delta - t_1^\delta )
\end{split}$$ for some constant $C$ independent of $ {\mu }$. Moreover, it follows from that $$\label{fNZE22}
\frac {1} {2} a_2^{\frac {1} {\alpha }} \tau _{\mu }^{ \sigma } = \Bigl| \int _{ t_1 }^{t_2} z _{\mu }' \Bigr| .$$ Furthermore, $ \widetilde{H} _{\mu }$ is nonincreasing on $( \overline{s} , \infty )$ by and , from which it follows (using and ) that $$\label{fNZE23}
s^2 |z _{\mu }' (s)|^2 \le \frac {1} {2} \widetilde{H} _{\mu }( \overline{s} ) + \frac {a_2} {2} z _{\mu }(s)^2 \le C + \frac {a_2^{\frac {\alpha +2} {\alpha }}} {8} \tau _{\mu }^{2\sigma } , \quad t_1\le s\le t_1.$$ In particular, $$\label{fNZE24}
\int _{ t_1 }^{t_2} |z _{\mu }'| \le C \tau _{\mu }^\sigma \log \frac {t_2 } {t_1}.$$ Formulas and imply that $\log \frac {t_2 } {t_1}$ is bounded below, so that there exists $\nu >0$ independent of $ {\mu }$ satisfying such that $t_2 \ge (1+ \nu ) t_1$. Since $t_1 \ge s_{\mu }\to \infty $ as $\varepsilon \to 0$ by , we see that if $\varepsilon $ is sufficiently small, then $t_2 ^\delta - t_1^\delta\ge 1$. Therefore, we deduce from and the fact that $\tau _{\mu }\ge s_{\mu }\to \infty $ as $\varepsilon \to 0$, that $ \widetilde{H} _{\mu }(\tau _{\mu }) <0$ if $\varepsilon $ is sufficiently small. Applying and , we conclude that $ \widetilde{H} _{\mu }(s ) \le \widetilde{H} _{\mu }(\tau _{\mu }) <0$ for $s\ge \tau _{\mu }$. It now follows from that there exists $\rho >0$ such that $z _{\mu }(s) \le -\rho $ for $s\ge \tau _{\mu }$. Therefore $z _{\mu }$ has ${{\widetilde {N}}}( \overline{{\mu }} ) +1$ zeros on $(0, \infty )$ and $ \frac {w _{\mu }(s)} {\phi _1(s)}\to \infty$. By Proposition \[eAB1\] we deduce that ${{\mathcal L}_2}( {\mu })\ne 0$. This proves Property of Proposition \[eEX2\], and completes the proof in the case $N\ge 3$.
The case $N=2$
--------------
We continue with the notation established in Subsection \[NeqD\]. We first consider $ \overline{{\mu }} \ge {1 }$ such that ${{\mathcal L}_2}( \overline{{\mu }} )\not = 0$. It follows that there exists $ \overline{s} >0$ such that $$\label{fTPAC}
| w_{ \overline{{\mu }} } (s)|\ge \frac {1} {2} |{{\mathcal L}_2}( \overline{{\mu }} )| \phi _2 (s)$$ for $s\ge \overline{s} $. We deduce from formula that $$w_{\mu }(s) - c _{ 2, {\mu }}^\infty \phi _2 (s) = c_{1,{\mu }}(s)\phi _1 (s) + (c_{2,{\mu }}(s)- c _{ 2, {\mu }}^\infty )\phi _2 (s)$$ so that, by continuity of the map ${\mu }\mapsto c_{1,{\mu }}^\infty $ (Lemma \[eTPA2\]) and estimate $$| w_{\mu }(s) - c _{ 2, {\mu }}^\infty \phi _2 (s) | \le C \phi _1 (s) + C s^{-\frac {1} {2}}\phi _2 (s)$$ for some constant $C$ independent of $s\ge 1$ and ${\mu }$ with $1\le {\mu }\le \overline{{\mu }} +1 $. Thus we see that there exists $ \widetilde{s} \ge 1$ such that $$| w_{\mu }(s) - c _{ 2, {\mu }}^\infty \phi _2 (s) | \le \frac {1} {10} |{{\mathcal L}_2}( \overline{{\mu }} )| \phi _2 (s)$$ for $s\ge \widetilde{s} $ and ${1 }\le {\mu }\le \overline{{\mu }} +1$. Therefore, $$\begin{split}
|w_{\mu }(s) - w _{ \overline{{\mu }} } (s)|& \le | w_{\mu }(s) - c _{ 2, {\mu }}^\infty \phi _2 (s) |+| c _{ 2, \overline{{\mu }} }^\infty- c _{ 2, {\mu }}^\infty | \phi _2 (s) +| w_{ \overline{{\mu }} } (s) - c _{ 2, \overline{{\mu }} }^\infty \phi _2 (s) |
\\& \le \frac {1} {4} |{{\mathcal L}_2}( \overline{{\mu }} )| \phi _2 (s)
\end{split}$$ for $s\ge \widetilde{s} $ and ${\mu }$ close to $ \overline{{\mu }} $. Since $ |w_{\mu }(s)|\ge |w _{ \overline{{\mu }} } (s)| - |w_{\mu }(s) - w _{ \overline{{\mu }} } (s)|$, we deduce by applying that if ${\mu }$ is close to $ \overline{{\mu }} $, then $w_{\mu }$ has no zero on $[ \max \{ \overline{s}, \widetilde{s} \}, \infty )$. By possibly assuming that ${\mu }$ is closer to $ \overline{{\mu }} $, it follows from continuous dependence (Proposition \[eEX1:b1\] ) that ${{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} )$, which proves Property of Proposition \[eEX2\].
We next assume ${{\mathcal L}_2}( \overline{{\mu }} )=0$. Given ${\mu }\ge {1 }$, define $v_{\mu }(t)$ for $t >0$ by $$\label{eNZE3:0}
t^{\frac {1} {2}} v_{\mu }(t)=s^{\lambda }w_{\mu }(s), \quad t= \log s$$ so that $v_ {\mu }$ is a solution of . We deduce from and Proposition \[eAB1\] that $$\label{fTPAC1}
t^{ \frac {1} {2}} v_{\mu }(t) {\mathop{\longrightarrow}}_{ t\to \infty } {{\mathcal L}_1}( {\mu }) .$$ Moreover, it follows from and that $$\label{fTPAC7}
M: = \sup_{{1 }\le {\mu }\le \overline{{\mu }}+1 }\, \sup_{t \ge 1} \, t^{ - \frac {1} {2}} |v_{\mu }(t)| <\infty .$$ We deduce from and that for $t$ large, if $ v _{ \overline{{\mu }} } '(t)=0$ then $ v _{ \overline{{\mu }} } ''(t)$ and $v _{ \overline{{\mu }} } (t)$ have the same sign. Since $v _{ \overline{{\mu }} } (t) \to 0$ as $t\to \infty $ (by ), it easily follows that there exists $ \overline{t}_{ \overline{{\mu }} } > 1 $ such that $ v_{ \overline{{\mu }} } (t) v_{ \overline{{\mu }} }'(t)<0$ for $ t\ge \overline{t} _{ \overline{{\mu }} }$. Thus we may assume without loss of generality that $$\label{eNZE3:9}
v _{ \overline{{\mu }} }( t )>0 \quad \text{and} \quad v _{ \overline{{\mu }} } '( t)<0 \quad \text{for} \quad t\ge \overline{t} _{ \overline{{\mu }} }.$$ In particular, $w _{ \overline{{\mu }} }$ has ${{\widetilde {N}}}( \overline{{\mu }} )$ zeros on $(0, \log \overline{t} _{ \overline{{\mu }} }) $, so that by continuous dependence, $w_{\mu }$ also has ${{\widetilde {N}}}( \overline{{\mu }} )$ zeros on $(0, \log \overline{t} _{ \overline{{\mu }} }) $ provided $ |{\mu }- \overline{{\mu }} |$ is sufficiently small. Therefore, we need only show that, by possibly assuming $ |{\mu }- \overline{{\mu }} |$ smaller, either $$\label{fTPAC2}
v_{\mu }(t) >0, \quad t\ge \overline{t} _{ \overline{{\mu }} }$$ or else $$\label{fTPAC3}
\text{$v_{\mu }$ has one zero on $[ \overline{t}_{ \overline{{\mu }} }, \infty ) $ and ${{\mathcal L}_2}( {\mu }) \not = 0$. }$$ By , we have $v_{\mu }( \overline{t} _{ \overline{{\mu }} } )>0 $ if $ | {\mu }- \overline{{\mu }} |$ is sufficiently small, and we define $t_{\mu }> \overline{t} _{ \overline{{\mu }} } $ by $$t_{\mu }= \sup\{ t> \overline{t} _{ \overline{{\mu }} } ;\, v_{\mu }> 0 \text{ on } ( \overline{t} _{ \overline{{\mu }} }, t)\}.$$ If $t_{\mu }=\infty $, then we are in case . Assume now $t_{\mu }<\infty $, so that $v_{\mu }(t_{\mu }) =0$ and $v_{\mu }' (t_{\mu }) < 0$, and set $$\widetilde{t}_{\mu }=\sup\{ t> t_{\mu };\, v_{\mu }'<0 \text{ on } (t_{\mu }, t)\}.$$ We claim that $ \widetilde{t} _{\mu }=\infty$. Assuming this claim, we see that $v_{\mu }$ has one zero on $[ \overline{t}, \infty ) $ and that $\limsup v_{\mu }(t) <0$ as $t\to \infty $. Therefore, $t^{\frac {1} {2}}v_{\mu }(t) \to -\infty $ as $t\to \infty $. By , this means that $s ^\lambda w_{\mu }(s) \to -\infty $ as $s\to \infty $, and Proposition \[eAB1\] implies that ${{\mathcal L}_2}( {\mu })\not = 0$. Thus we see that we are in case , and this completes the proof. We finally prove that $ \widetilde{t} _{\mu }=\infty$. Otherwise, $v_{\mu }' ( \widetilde{t} _{\mu }) =0 \text{ and } v_{\mu }'' ( \widetilde{t} _{\mu }) \ge 0$. Since $v_{\mu }( \widetilde{t} _{\mu }) <0$, equation yields $$\frac { 1 } { \widetilde{t}_{\mu }^2}+e^{- \widetilde{t}_{\mu }}\Bigl (\frac {1} { 2 \widetilde{t}_{\mu }}-\frac {1} {\alpha } - \widetilde{t}_{\mu }^{\frac {\alpha } {2} } |v_{\mu }|^\alpha \Bigr) \le 0$$ so that $$\label{fTPAC4}
\widetilde{t}_{\mu }^{\frac {\alpha } {2} } |v_{\mu }( \widetilde{t}_{\mu }) |^\alpha \ge e^{ \widetilde{t}_{\mu }}\frac { 1} { \widetilde{t}_{\mu }^2}+ \frac {1} { 2 \widetilde{t}_{\mu }}-\frac {1} {\alpha }$$ By and continuous dependence, $t_{\mu }$, hence $ \widetilde{t}_{\mu }$ can be made arbitrarily large by assuming $ |{\mu }- \overline{{\mu }} |$ sufficiently small. In particular, with $M>0$ defined by , we have $$e^{ \widetilde{t}_{\mu }} \frac { 1} { \widetilde{t}_{\mu }^2}+ \frac {1} { 2 \widetilde{t}_{\mu }}-\frac {1} {\alpha } \ge (2 M \widetilde{t}_{\mu })^\alpha$$ if $ |{\mu }- \overline{{\mu }} |$ is sufficiently small. Applying , we obtain $ \widetilde{t}_{\mu }^{ - \frac {1 } {2} } |v_{\mu }( \widetilde{t}_{\mu }) | \ge 2M$, which contradicts . Thus $ \widetilde{t} _{\mu }=\infty$, which completes the proof for $N=2$.
Singular profiles {#sSinPro}
=================
In what follows, we prove Theorem \[eMain2\]. We suppose $N\ge 3$ and $\frac {2} {N-2} \le \alpha < \frac {4} {N-2}$, and we consider the collection $( w_{\mu }) _{ {\mu }\ge 1 }$ of solutions of given by Proposition $\ref{eEX1:b1}$. It follows from Proposition \[eAB1\] and that the corresponding profile $$\label{fTPA42}
\widetilde{f}_{\mu }(r)=r^{-\frac {2} {\alpha }}w_{\mu }(r^{-2})$$ behaves like $ {{\mathcal L}_2}({\mu }) r^{- \frac {2} {\alpha }} \phi _2 ( r ^{-2}) $ as $r\to 0$. Thus we see that if $ {{\mathcal L}_2}({\mu }) \not = 0$ (i.e. if $w_{\mu }$ has slow decay), then $ \widetilde{f}_{\mu }(r)$ is singular at $r = 0$. On the other hand, one verifies easily that $ \widetilde{f}_{\mu }\in L^{\alpha +1}_{{\mathrm{loc}}}({{\mathbb R}}^N )$. Furthermore, it follows from that $$\label{fTPA42b2}
r^{1+ \frac {2} {\alpha }} \widetilde{f}_{\mu }' (r) = - \frac {2} {\alpha } w_{\mu }(r^{-2}) - 2 r^{-2} w_{\mu }' (r^{-2});$$ and so $$\label{fTPA42b1}
r^{ \frac {2} {\alpha }} ( |\widetilde{f}_{\mu }(r)| + r |\widetilde{f}_{\mu }' (r)| ) \le C \sup _{ s\ge 0 } \, ( |w_{\mu }(s)| + s |w_{\mu }' (s)| ) \le C$$ where we apply Proposition \[eEX1:b1\] in the last inequality.
We claim that $ \widetilde{f}_{\mu }$ is a solution of in the sense of distributions, i.e. $$\label{fTPA61}
\int_{{{\mathbb R}}^N} \widetilde{f} _{\mu }\Bigl( \Delta \varphi - \frac {1} {2} \nabla \cdot (x \varphi ) + \frac {1} {\alpha } \varphi + | \widetilde{f} _{\mu }|^\alpha \varphi \Bigr) \, dx =0$$ for all $\varphi \in C^\infty _{{\mathrm{c}}}( {{\mathbb R}}^N)$. To see this, we let $\varepsilon >0$. Since $ \widetilde{f}_{\mu }\in L^{\alpha +1}_{{\mathrm{loc}}}({{\mathbb R}}^N )$, we see that $$\label{fTPA62}
\begin{split}
\Bigl| \int_{ \{ |x|<\varepsilon \}} \widetilde{f} _{\mu }\Bigl( \Delta \varphi - \frac {1} {2} \nabla \cdot (x \varphi ) + \frac {1} {\alpha } \varphi &+ | \widetilde{f} _{\mu }|^\alpha \varphi \Bigr) \Bigr| \\& \le C \int_{ \{ |x|<\varepsilon \}} ( |\widetilde{f}_{\mu }| + | \widetilde{f} _{\mu }|^{ \alpha +1}){\mathop{\longrightarrow}}_{ \varepsilon \downarrow 0 } 0 .
\end{split}$$ On the other hand, $\widetilde{f}_{\mu }$ is a classical solution of on $ {{\mathbb R}}^N \setminus \{ 0\} $, so that integration by parts yields $$\label{fTPA63}
\begin{split}
\int_{ \{ |x| >\varepsilon \}} \widetilde{f} _{\mu }\Bigl( \Delta \varphi - \frac {1} {2} \nabla \cdot (x \varphi ) + \frac {1} {\alpha } \varphi &+ | \widetilde{f} _{\mu }|^\alpha \varphi \Bigr) \\& = \int_{ \{ |x|=\varepsilon \}} \Bigl( \widetilde{f} _{\mu }\frac {\partial \varphi } {\partial r}- \varphi \frac {\partial \widetilde{f} _{\mu }} {\partial r} -\frac {\varepsilon } {2} \widetilde{f}_{\mu }\varphi \Bigr) .
\end{split}$$ Therefore, follows from and provided we show that $$\int_{ \{ |x|=\varepsilon \}} \Bigl( |\widetilde{f} _{\mu }| + \Bigl| \frac {\partial \widetilde{f} _{\mu }} {\partial r} \Bigr| \Bigr) {\mathop{\longrightarrow}}_{ \varepsilon \downarrow 0 } 0 ,$$ i.e. $r^{N-1} ( | \widetilde{f}_{\mu }(r) | + | \widetilde{f}_{\mu }' (r)|) \to 0$ as $r \downarrow 0$. If $\alpha > \frac {2} {N-2}$, this is a consequence of . If $\alpha = \frac {2} {N-2}$, this follows from , , and the fact that $ |w_{\mu }( s ) | + s |w_{\mu }' ( s ) | \to 0$ as $s\to \infty $ by Proposition \[eEX1:b1\] .
We now conclude the proof of Theorem \[eMain2\] as follows. By Lemma \[eTPA15\] , there exist an integer $ \overline{m} $ and an increasing sequence $( \overline{{\mu }}_m ) _{ m\ge \overline{m} } \subset [ {1 }, \infty )$ such that ${{\widetilde {N}}}( \overline{{\mu }} _m ) = m$ and $ {{\mathcal L}_2}( \overline{{\mu }} _m )\not = 0$. Next, we note that by Proposition \[eEX1:b1\] (for $\frac {2} {N-2} < \alpha < \frac {4} {N-2}$) and Lemma \[eBN2b\] (for $\alpha = \frac {2} {N-2}$), $ {{\mathcal L}_2}( \overline{{\mu }} _m ) = \pm \nu$ where $$\nu =
\begin{cases}
\beta ^{\frac {1} {\alpha }} & \frac {2} {N-2} < \alpha < \frac {4} {N-2} \\
(\frac {2} {\alpha } )^{\frac {2} {\alpha }} & \alpha = \frac {2} {N-2} .
\end{cases}$$ Since $ w _{ \overline{{\mu }}_m }$ has $m$ zeros, we see that $ {{\mathcal L}_2}( \overline{{\mu }} _m ) = (-1)^m \nu $. Consequently, the profiles $h_m = \widetilde{f} _{ \overline{{\mu }} _m }$ satisfy the first part of Theorem \[eMain2\].
Since $ |h_m (r)| + r | h_m '(r) | \le C r^{-\frac {2} {\alpha }}$ by and, as noted previously, $ h_m = \widetilde{f} _{ \overline{{\mu }} _m } \in L^{\alpha +1}_{{\mathrm{loc}}}({{\mathbb R}}^N )$, the second part of Theorem \[eMain2\] is an immediate consequence of the following lemma, with $f= h_m$ and $\omega (x) \equiv \overline{{\mu }}_m $.
\[eSP1\] Let $\alpha \ge \frac {2} {N-2}$, and let $f\in C^2( {{\mathbb R}}^N \setminus \{0\} ) \cap L^{\alpha +1}_{{\mathrm{loc}}}({{\mathbb R}}^N )$ be a solution of in the sense of distributions and satisfy $$\label{eSP1:1}
|f(x)| + | x \cdot \nabla f(x)|\le C |x|^{-\frac {2} {\alpha }} \quad x\not = 0 ,$$ and $ |x|^\frac {2} {\alpha } f(x)- \omega (x) \to 0$ as $ |x| \to \infty $, for some $\omega \in C ({{\mathbb R}}^N \setminus \{0\})$ homogeneous of degree $0$. If $u \in C^2( (0,\infty )\times ({{\mathbb R}}^N \setminus \{0\}) $ is defined by for $t>0$ and $x\not = 0 $, then $$\label{eSP1:3:1}
| u | + t |u _t | + |x\cdot \nabla u| \le C |x|^{- \frac {2} {\alpha }}$$ and $$\label{eSP1:3:2}
u, u_t , x\cdot \nabla u\in C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) ) \text{ for } 1\le p< \frac {N\alpha } {2}< q .$$ Furthermore, $$\label{eSP1:4}
\Delta u, |u|^\alpha u\in C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) )$$ and $u$ is a solution of , and also of with ${u_0 }(x)= \omega (x) |x|^{-\frac {2} {\alpha }}$, in $C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) ) $ where $p, q$ satisfy . Moreover, $u$ satisfies and . In addition, the map $t \mapsto u (t) - e^{t \Delta } {u_0 }$ is in $ C([0,\infty ), L^r ({{\mathbb R}}^N ) )$ for all $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$ if $\alpha > \frac {2} {N-2}$, and in $ C([0,\infty ), L^1 ({{\mathbb R}}^N ) + L^r ({{\mathbb R}}^N ) )$ for all $r>1$ if $\alpha = \frac {2} {N-2}$.
The proof is similar to that of Lemma \[eQSol\], but some extra care is needed because of the possible singularity of $f$ at $x=0$. Differentiating with respect to $t$, we see that $u$ satisfies equation for $t>0$ and $x\not = 0$. Property follows from and , and property is then a consequence of the dominated convergence theorem. Next, we deduce from estimate if $\alpha >\frac {2} {N-2}$, and the assumption $f \in L^{\alpha +1}_{{\mathrm{loc}}}({{\mathbb R}}^N )$ if $\alpha =\frac {2} {N-2}$ that $ |u|^\alpha u\in C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) ) $ whenever $p,q$ satisfy . In addition, it follows from that for every $t>0$, $$\label{fEId2}
\Delta u + \frac {1} {2t} x\cdot \nabla u + \frac {1} {\alpha t} u + |u|^\alpha u=0$$ in the sense of distributions. In particular, we see that $ \Delta u\in C((0,\infty ), L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) ) $ whenever $p,q$ satisfy . Moreover, and imply that $u$ is a solution of . Properties and follow from the fact that $u (t, x) \to {u_0 }(x) $ for all $x\not = 0$ by , estimate , and dominated convergence.
We now show that $u$ satisfies for all $0< \varepsilon <t $. To see this, we consider $\varphi \in C^\infty _{{\mathrm{c}}}({{\mathbb R}}^N )$ such that $\varphi (x)=1$ for $ |x| \le 1$. Note that $ \Delta (\varphi u) = \varphi \Delta u + 2\nabla u\cdot \nabla \varphi + u\Delta \varphi $, so that $$\label{fEId3}
(\varphi u)_t = \Delta (\varphi u) - 2\nabla u\cdot \nabla \varphi - u\Delta \varphi + \varphi |u|^\alpha u.$$ Therefore, it follows from that $\varphi u\in C^1 ((0,\infty ), L^p ({{\mathbb R}}^N ) )$; and from and that $\Delta (\varphi u), \nabla u\cdot \nabla \varphi , \varphi |u|^\alpha u \in C ((0,\infty ), L^p ({{\mathbb R}}^N ) )$, for $1< p< \frac {N\alpha } {2(\alpha +1)}$ if $\alpha >\frac {2} {N-2}$ and $p=1$ if $\alpha =\frac {2} {N-2}$. In the case $\alpha =\frac {2} {N-2}$, note also that $\nabla (\varphi u)\in C((0,\infty ), L^1 ({{\mathbb R}}^N ) )$ by and dominated convergence. Therefore $\varphi u$ has the required regularity so that implies $$\label{fEId4}
\varphi u(t)= e^{ (t- \varepsilon ) \Delta } \varphi u(\varepsilon ) + \int _\varepsilon ^t e^{t-s) \Delta } [ - 2\nabla u\cdot \nabla \varphi - u\Delta \varphi + \varphi |u|^\alpha u ]$$ in $L^p ({{\mathbb R}}^N ) $ for all $0<\varepsilon <t$. Next, we have $$\label{fEId5}
((1- \varphi) u)_t = \Delta ((1- \varphi ) u) + 2\nabla u\cdot \nabla \varphi + u\Delta \varphi +(1- \varphi ) |u|^\alpha u.$$ Arguing as above, one sees that $(1-\varphi )u$ has the required regularity so that $$\label{fEId6}
(1-\varphi ) u(t)= e^{(t- \varepsilon ) \Delta } (1-\varphi ) u(\varepsilon ) + \int _\varepsilon ^t e^{t-s) \Delta } [ 2\nabla u\cdot \nabla \varphi +u\Delta \varphi + (1- \varphi ) |u|^\alpha u ]$$ in $L^q ({{\mathbb R}}^N ) $ for all $q >\frac {N \alpha } {2} $ and $0<\varepsilon <t$. Summing up and , we see that $u$ satisfies in $ L^p ({{\mathbb R}}^N ) + L^q ({{\mathbb R}}^N ) $ for all $p, q$ satisfying and all $0< \varepsilon <t $.
We now prove the last statement in the lemma, which will imply equation . For this, we consider separately the cases $\alpha >\frac {2} {N-2}$ and $\alpha =\frac {2} {N-2}$. Suppose first $\alpha >\frac {2} {N-2}$ and let $\frac {N\alpha } {2(\alpha +1) } < r < \frac {N\alpha } {2}$. Since $ | \cdot |^{- \frac {2(\alpha +1)} {\alpha }} \in L^1 ({{\mathbb R}}^N ) +L^r ({{\mathbb R}}^N ) $, we have $e^{ \Delta } | \cdot |^{- \frac {2(\alpha +1)} {\alpha }} \in L^r ({{\mathbb R}}^N ) $, so that by scaling $$\| e^{(t-s) \Delta } | \cdot |^{- \frac {2(\alpha +1)} {\alpha }} \| _{ L^r } = (t-s) ^{- \frac {\alpha +1} {\alpha } + \frac {N} {2r} } \| e^{ \Delta } | \cdot |^{- \frac {2(\alpha +1)} {\alpha }} \| _{ L^r }.$$ Therefore, since $ |f(x)|\le C |x|^{-\frac {2} {\alpha }}$, we have $$\| e^{(t-s) \Delta } |u(s)|^\alpha u(s) \| _{ L^r }\le C \| e^{(t-s) \Delta } | \cdot |^{- \frac {2(\alpha +1)} {\alpha }} \| _{ L^r } = C (t-s) ^{- \frac {\alpha +1} {\alpha } + \frac {N} {2r} }.$$ Since $ - \frac {2(\alpha +1)} {\alpha } >-1$, we see that the integral term in is continuous $[0, \infty ) \to L^r ({{\mathbb R}}^N ) $, and that we can let $\varepsilon \downarrow 0$ in the integral term in . Since $e^{(t-\varepsilon ) \Delta } u(\varepsilon ) \to e^{t\Delta } {u_0 }$ as $\varepsilon \downarrow 0$ in $L^q ({{\mathbb R}}^N )$ for all $q>\frac {N\alpha } {2}$, this completes the proof in the case $\alpha >\frac {2} {N-2}$.
Finally, in the case $\alpha = \frac {2} {N-2}$, recall that $f\in L^{ \alpha +1} _{{\mathrm{loc}}}({{\mathbb R}}^N ) $. We fix $T>0$ and write for $0< t <T$ $$|u|^\alpha u = 1 _{ \{ |x|<\sqrt t \} } |u|^\alpha u + 1 _{ \{ \sqrt t < |x| < \sqrt T\} } |u|^\alpha u + 1 _{ \{ |x|>\sqrt T \} } |u|^\alpha u = : F_1 + F_2 + F_3.$$ We have by $$\begin{split}
\| e^{(t-s) \Delta } F_1 (s) \| _{ L^1 } & \le \| F_1 (s) \| _{ L^1 }= \int _{ \{ |x|<\sqrt s \} } s^{-\frac {\alpha +1} {\alpha }} \Bigl| f \Bigl( \frac {x} {\sqrt s} \Bigr) \Bigr|^{\alpha +1} dx \\ & = \int _{ \{ |x|< 1 \} } |f(x)|^{\alpha +1}dx
\end{split}$$ since $ \frac {\alpha +1} {\alpha } = \frac {N} {2}$. Moreover, since $ |u ( t, x)|^{\alpha +1} \le C |x|^{-\frac {2(\alpha +1)} {\alpha }} = C |x|^{-N }$, $$\| e^{(t-s) \Delta } F_2 (s) \| _{ L^1 } \le \| F_2 (s) \| _{ L^1 } \le \int _{ \{ \sqrt s < |x| < \sqrt T \} } |x| ^{ - N } dx \le C \Bigl| \log \frac {s} {T} \Bigr| .$$ Furthermore, given $r>1$, $$\begin{split}
\| e^{(t-s) \Delta } F_3 (s) \| _{ L^r } & \le \| F_2 (s) \| _{ L^r } \\ & \le \Bigl( \int _{ \{ |x| > \sqrt T \} } |x| ^{ - rN } dx \Bigr)^{\frac {1} {p}} \le C<\infty .
\end{split}$$ because $r> \frac {N\alpha } {2(\alpha +1)}$. Therefore, $$\| e^{(t-s) \Delta } |u(s)|^\alpha u(s) \| _{ L^1 + L^r } \le C (1 + |\log s| )$$ and one can easily complete the proof as in the case $\alpha >\frac {2} {N-2}$.
In the case $\alpha > \frac {2} {N-2}$, the singular stationary solution $u$ of given by is in particular a self-similar solution. Its profile (which is $u$ itself) satisfies the assumptions of Lemma \[eSP1\], so that $u$ is a solution of . Note that $u$ is time-independent, but the other two terms in do depend on time.
\[eRemz\] If $\alpha <\frac {2} {N-2}$ and ${{\mathcal L}_2}( {\mu })\not = 0$, then the singularity of $ \widetilde{f}_{\mu }(r)$ at $0$ is of order $r^{-(N-2)}$. Therefore $\Delta \widetilde{f} _{\mu }$ has a Dirac mass at the origin, so that $ \widetilde{f} _{\mu }$ is not a solution of in the sense of distributions.
The case of dimension 1 {#sDim1}
=======================
The purpose of this section is to prove Theorem \[eMain3\]. Its proof is similar to that of Theorem \[eMain1\], with two major differences. The first one is that the proofs of the results analogous to Propositions \[eAB1\] and \[eEX2\] are completely elementary. This is due to the fact that equation is not singular at $r=0$. (It seems that there is no simplification in the proof of Proposition \[fFIn1\] when $N=1$.) The second major difference is that Theorem \[eMain3\] concerns both radially symmetric (i.e. even) profiles, and odd profiles.
Throughout this section, we suppose $N=1$. We first observe that if $f\in C^2([0,\infty ))$ is a solution of on $(0,\infty )$ and if $f'(0)=0$, then extending $f$ by setting $f(x)= f(-x)$ for $x<0$ yields a solution of on ${{\mathbb R}}$. Similarly, if $f (0)=0$, then extending $f$ by setting $f(x)= -f(-x)$ for $x<0$ also yields a solution of . Therefore, we need only construct solutions of on $(0,\infty )$ that satisfy either $f '(0)= 0$ (corresponding to an even profile) or $f (0)= 0$ (corresponding to an odd profile).
We consider the collection $( w_{\mu }) _{ {\mu }\ge 1 }$ of solutions of given by Proposition $\ref{eEX1:b1}$. We recall that, by Proposition \[eEX1:b1\] , $$\label{fTPA27b1}
\text{the map ${\mu }\mapsto w_{\mu }$ is continuous $[{1 }, \infty ) \to C^1([0, 1])$.}$$ Next, we define $$\label{fTPA25}
\widetilde{f}_{\mu }(r)= r^{-\frac {2} {\alpha }} w _{ {\mu }} (r^{-2})$$ for $r>0$. It follows that $ \widetilde{f}_{\mu }$ is a solution of the profile equation and that $$\label{fTPA26}
\widetilde{f}_{\mu }(1)= w _{ {\mu }} (1), \quad \widetilde{f}_{\mu }' (1)= -2 w _{ {\mu }} ' (1) - \frac {2} {\alpha } w _{ {\mu }} (1) .$$ Moreover, since $N=1$, equation is not singular at $r=0$, so that $ \widetilde{f}_{\mu }$ can be extended to a solution of for all $r\in {{\mathbb R}}$. (This follows from an obvious energy argument.) One deduces easily by using and that $$\label{fTPA27}
\text{the map ${\mu }\mapsto \widetilde{f}_{\mu }$ is continuous $[{1 }, \infty ) \to C^1([0, 1])$.}$$ Thus we see that the study of $w_{\mu }$ on $[0,\infty )$ is equivalent to the study of $w_{\mu }$ on $[0,1]$ and the study of $ \widetilde{f}_{\mu }$ on $[0, 1]$, The problem is therefore reduced to compact intervals, which considerably simplifies the analysis.
Since $N=1$, we have $\lambda _1=\frac {1} {\alpha }$, $\lambda _2=\frac {1} {\alpha }+\frac {1} {2}$, $\beta<0$, and $\lambda _2>\lambda _1 >0$. In particular, unlike in the case $N\ge 3$, $s^{-\lambda _1}$ decays more slowly than $s^{-\lambda _2}$, and also represents the generic behavior as $s\to \infty $ of solutions of . Consequently, in this section we define $$\label{fL2N1}
{{\mathcal L}_2}( {\mu }) = \lim _{ s\to \infty } s^{\lambda _1} w_{\mu }(s) = \lim _{ s\to \infty } s^{\frac {1} {\alpha }} w_{\mu }(s)$$ and $$\label{fL1N1}
{{\mathcal L}_1}( {\mu }) = \lim _{ s\to \infty } s^{\lambda _2} w_{\mu }(s) = \lim _{ s\to \infty } s^{\frac {1} {\alpha }+ \frac {1} {2}} w_{\mu }(s)$$ whenever these limits exist.
\[eTPA3\] The following properties are simple consequences of the above observation.
1. \[eTPA3:1\] Since both $w_{\mu }$ and $ \widetilde{f}_{\mu }$ have only a finite number of zeros on the compact interval $[0,1]$, it follows from that $w_{\mu }$ has at most a finite number of zeros on $[0,\infty )$. This gives a quick proof of Proposition \[eEX1:b1\] in the case $N=1$. Recall that the possible zero of $ \widetilde{f} _{\mu }$ at $r=0$ is not counted in ${{\widetilde {N}}}( {\mu })$, where ${{\widetilde {N}}}( {\mu })$ is given by .
2. \[eTPA3:2\] Formula implies $$\widetilde{f}_{\mu }(0) = \lim _{ r\downarrow 0 } \widetilde{f}_{\mu }(r) = \lim _{ s\to \infty } s^{\frac {1} {\alpha }} w_{\mu }(s) .$$ Therefore, the limit exists and $$\label{fTPA27:10}
{{\mathcal L}_2}( {\mu }) = \widetilde{f}_{\mu }(0) .$$ Since $ \widetilde{f}_{\mu }(0) $ depends continuously on ${\mu }$, ${{\mathcal L}_2}$ is continuous $[{1 }, \infty ) \to {{\mathbb R}}$. In addition, $$\widetilde{f}_{\mu }' (0) = \lim _{ r\downarrow 0 } \frac {\widetilde{f}_{\mu }(r) - \widetilde{f}_{\mu }(0)} {r} = \lim _{ s\to \infty } s^{\frac {1} {2}} (s^{\frac {1} {\alpha }} w_{\mu }(s) -{{\mathcal L}_2}({\mu }) ).$$ Thus we see that if ${{\mathcal L}_2}({\mu }) =0$, then the limit exists and $$\label{fTPA27:20}
{{\mathcal L}_1}( {\mu }) = \widetilde{f}_{\mu }' (0) .$$ In particular, ${{\mathcal L}_1}( {\mu }) \not = 0$ for otherwise we would have $\widetilde{f}_{\mu }(0) = \widetilde{f}_{\mu }' (0) =0$, hence $ \widetilde{f} _{\mu }\equiv 0$.
3. \[eTPA3:3\] It follows from that $$\lim _{ r\to \infty } r^{\frac {2} {\alpha }} \widetilde{f}_{\mu }(r) = w _{\mu }(0) = {\mu }\ge {1 }$$ so that no zero of $ \widetilde{f}_{\mu }$ can appear or disappear at infinity by varying ${\mu }$. Moreover if $r>0$, then $\widetilde{f}_{\mu }' (r)\not = 0$ whenever $\widetilde{f}_{\mu }(r) =0$, so that no zero of $ \widetilde{f}_{\mu }$ on $( 0, \infty ) $ can appear or disappear by varying ${\mu }$. Thus we see that ${{\widetilde {N}}}( {\mu })$ can only change at a ${\mu }$ for which $ \widetilde{f}_{\mu }(0) = 0$, i.e. ${{\mathcal L}_2}( {\mu }) = 0$. In particular, if ${1 }\le {\mu }_1 < {\mu }_2$, ${{\widetilde {N}}}( {\mu }_1) \not = {{\widetilde {N}}}({\mu }_2)$ and $ \widetilde{f} _{ {\mu }_1 } (0) \widetilde{f} _{ {\mu }_2 } (0) \not = 0 $, then there exists ${\mu }\in ({\mu }_1, {\mu }_2)$ such that $ \widetilde{f}_{\mu }(0) =0 $.
4. \[eTPA3:4\] It follows from Property above that if ${{\mathcal L}_2}( \overline{{\mu }} ) \not = 0$, then ${{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} )$ for ${\mu }$ close to $\ \overline{ {\mu }} $.
5. \[eTPA3:5\] Suppose ${{\mathcal L}_2}( \overline{{\mu }} ) = 0$ (i.e. $ \widetilde{f} _{ \overline{{\mu }} } (0)= 0$), so that $ \widetilde{f}_{\overline{{\mu }}} ' (0) \not = 0$. Without loss of generality we may suppose $ \widetilde{f}_{\overline{{\mu }}} ' (0) > 0$. It follows from that there exist $\delta , \rho >0$ such that if $ |{\mu }- \overline{{\mu }} |\le \delta $, then $$\begin{gathered}
\frac {1} {2} \widetilde{f}_{ \overline{{\mu }} } ' (0) \le \widetilde{f} _{\mu }'(r) \le 2 \widetilde{f}_{ \overline{{\mu }} } ' (0) , \quad 0\le r\le \rho , \label{eTPA3:5:1} \\
| \widetilde{f} _{\mu }(0) |\le \frac {\rho } {4} \widetilde{f}_{ \overline{{\mu }} } ' (0) . \label{eTPA3:5:2}\end{gathered}$$ Therefore, $ \widetilde{f}_ {\mu }$ is increasing in $[0,\rho ]$ and $$\label{eTPA3:5:3}
\widetilde{f} _{\mu }(r) \ge \widetilde{f} _{\mu }(0) + \frac {r} {2} \widetilde{f}_{ \overline{{\mu }} } ' (0) \quad 0\le r\le \rho .$$ It follows in particular from and that $$\label{eTPA3:5:4}
\widetilde{f} _{\mu }(\rho ) \ge \frac {\rho } {4} \widetilde{f}_{ \overline{{\mu }} } ' (0) >0.$$ If $ \widetilde{f} _{\mu }(0) \ge 0$, then $ \widetilde{f} _{\mu }(r) > 0$ on $(0, \rho ]$ by , so that ${{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} ) $. If $ \widetilde{f} _{\mu }(0) < 0$, then $ \widetilde{f} _{\mu }$ has exactly one zero on $(0, \rho ]$ by and the fact that $ \widetilde{f}_{\mu }$ is increasing, so that ${{\widetilde {N}}}( {\mu }) = {{\widetilde {N}}}( \overline{{\mu }} ) +1$.
6. \[eTPA3:6\] It follows from Properties and above that Proposition \[eEX2\] holds as well in the case $N=1$, where ${{\mathcal L}_2}( {\mu })$ and ${{\mathcal L}_1}( {\mu })$ are now given by and .
In order to prepare the proof of Theorem \[eMain3\], we make the following observation.
\[eLFN1\] Let ${{\widetilde {N}}}( {\mu }) $ be as defined by . Set $ \overline{m} = {{\widetilde {N}}}(2) +1$, $$\label{eNZn1:9}
E_m= \{ {\mu }\ge 2 ;\, {{\widetilde {N}}}( {\mu })\ge m+1 \}$$ and $$\label{eNZn1:10}
{\mu }_m = \inf E _m \in [2, \infty )$$ for $m\ge \overline{m} $. It follows that $E_m \not = \emptyset$ and $2\le {\mu }_m <\infty $ for all $m\ge \overline{m} $. Moreover, the sequence $({\mu }_m) _{ m\ge \overline{m} }$ is increasing and satisfies $$\label{fTPA90}
{\mu }_m {\mathop{\longrightarrow}}_{ m\to \infty } \infty, \quad {{\widetilde {N}}}( {\mu }_m) =m, \quad {{\mathcal L}_2}( {\mu }_m)=0 .$$
It follows from Proposition \[fFIn1\] that $E_m \not = \emptyset$, so that $2\le {\mu }_m <\infty $. The remaining properties follow from the argument used in the proof of Lemma \[eTPA15\] (using Remark \[eTPA3\] where the proof of Lemma \[eTPA15\] uses Proposition \[eEX2\]).
\[Proof of Theorem $\ref{eMain3}$ \] We consider the sequence $({\mu }_m) _{ m\ge \overline{m} }$ given by Lemma \[eLFN1\]. We construct an increasing sequence $ (\widetilde{ {\mu }}_m) _{ m\ge \overline{m} +1 } \subset ( {1 }, \infty ) $ such that, with the notation $$\label{fTPA90:1}
\widetilde{{\mu }} _m {\mathop{\longrightarrow}}_{ m\to \infty } \infty, \quad
{{\widetilde {N}}}( \widetilde{{\mu }} _m) =m , \quad
\widetilde{f} _{ \widetilde{{\mu }} _m } ( 0) \not = 0 , \quad
\widetilde{f} _{ \widetilde{{\mu }} _m } ' ( 0) = 0.$$ Since $w _{ {\mu }_m } (0) = {\mu }_m >0$ and $ {{\widetilde {N}}}({\mu }_m) = m$, we deduce that $(-1)^m w _{ {\mu }_m }(s) >0$ for $s$ large. Since ${{\mathcal L}_1}( {\mu }_m )\not = 0$ is well defined (because ${{\mathcal L}_2}( {\mu }_m )=0$), we conclude that $(-1)^m {{\mathcal L}_1}( {\mu }_m ) > 0$. Therefore, it follows from and that $$\begin{gathered}
\widetilde{f} _{ {\mu }_m } ( 0) =0 , \label{fTPA17} \\
(-1)^m \widetilde{f} _{ {\mu }_m } ' ( 0)> 0 . \label{fTPA27:2} \end{gathered}$$ We now consider $m\ge \overline{m} +1$. Note that for $1\le {\mu }< {\mu }_m$, we have ${{\widetilde {N}}}({\mu })\le m$ by -. Since $\widetilde{f} _{ {\mu }_{m-1} } ' ( 0) \widetilde{f} _{ {\mu }_m } ' ( 0)<0$ by , the map ${\mu }\mapsto \widetilde{f}_{\mu }'(0) $ has at least one zero on $({\mu }_{ m-1 }, {\mu }_m)$. We denote by $ \widetilde{{\mu }} _m$ the largest such zero, and it follows that $$\begin{gathered}
{{\widetilde {N}}}( {\mu }) \le m,\quad \widetilde{{\mu }}_m \le {\mu }\le {\mu }_m \label{ffC1} \\
(-1)^m \widetilde{f} _{ {\mu }} ' ( 0)> 0, \quad \widetilde{{\mu }}_m < {\mu }\le {\mu }_m \label{ffC2} \\
\widetilde{f} _{ \widetilde{{\mu }}_m } ' ( 0)= 0 . \label{ffC3} \end{gathered}$$ We claim that $$\label{ffC4}
{{\widetilde {N}}}( \widetilde{{\mu }}_m ) = m .$$ Assuming , the conclusion easily follows. Indeed, ${\mu }_{ m-1 }< \widetilde{{\mu }}_m < {\mu }_m $, so that $ (\widetilde{ {\mu }}_m) _{ m\ge \overline{m} +1 }$ is increasing and $ \widetilde{{\mu }} _m \to \infty $. Moreover, since $ \widetilde{f} _{ \widetilde{{\mu }}_m } \not \equiv 0$, implies that $ \widetilde{f} _{ \widetilde{{\mu }}_m } ( 0)\not = 0$. Together with , this shows the the sequence $ (\widetilde{ {\mu }}_m) _{ m\ge \overline{m} +1 } $ has the desired properties.
We now prove . It follows from Remark \[eTPA3\] that there exists $\varepsilon >0$ such that for every ${\mu }_m- \varepsilon \le {\mu }\le {\mu }_m$, we have either $(-1)^m \widetilde{f}_{\mu }(0) < 0 $ and ${{\widetilde {N}}}({\mu }) =m+1$ or $(-1)^m \widetilde{f}_{\mu }(0) \ge 0 $ and ${{\widetilde {N}}}({\mu }) =m$. Applying , we deduce that $(-1)^m \widetilde{f}_{\mu }(0) \ge 0 $ and ${{\widetilde {N}}}({\mu }) =m$ for ${\mu }_m- \varepsilon \le {\mu }\le {\mu }_m$. We now decrease ${\mu }$, and we note that, as long as $(-1)^m \widetilde{f}_{\mu }' (0) > 0 $, we may keep applying Remark \[eTPA3\] (if $ \widetilde{f}_{\mu }(0) = 0 $) or Remark \[eTPA3\] (if $(-1)^m \widetilde{f}_{\mu }(0) > 0 $), so that $(-1)^m \widetilde{f}_{\mu }(0) \ge 0 $ and ${{\widetilde {N}}}({\mu })= m$. It follows that $(-1)^m \widetilde{f}_{\mu }' (0) > 0 $, $(-1)^m \widetilde{f}_{\mu }(0) \ge 0 $ and ${{\widetilde {N}}}({\mu })= m$ for all $ \widetilde{{\mu }}_m < {\mu }\le {\mu }_m $. Property now follows from Remark \[eTPA3\] .
We next show that there exists a map $m : (0,\infty ) \to {{\mathbb N}}$ such that, given any ${\mu }>0$, there exist four sequences $(a _{ {\mu }, m }^\pm ) _{ m \ge m _ {\mu }}\subset (0,\infty )$ and $(b_{ {\mu }, m }^\pm ) _{ m \ge m_{\mu }}\subset (-\infty , 0)$ satisfying Properties – of Theorem \[eMT2\]. This is done exactly as in the proof of Theorem \[eMT2\], starting with the sequence $( \widetilde{{\mu }}_m) _{ m\ge \overline{m} +1 } $ defined above instead of the sequence $( {\mu }_m) _{ m\ge \overline{m} }$ of Lemma \[eTPA15\].
Finally, we may assume ${\mu }>0$ without loss of generality, and we see that the profiles $f= f_{a^\pm} $ with $a^\pm = a^\pm _{ {\mu }, \frac {m} {2} }$ if $m\ge m_{\mu }$ is even and $a^\pm = b^\pm _{ {\mu }, \frac {m-1} {2} }$ if $m\ge m_{\mu }$ is odd, are two different radially symmetric solutions of with $m$ zeros on $[ 0, \infty )$. Moreover, $r^{\frac {2} {\alpha }} f_{a^\pm} (r) \to {\mu }$ as $r\to \infty $, and it follows from [@HarauxW Proposition 3.1] that $ |f_{a^\pm} (r)| + r |f'_{a^\pm} (r)|\le C( 1+ r^2)^{-\frac {1} {\alpha }}$. Theorem \[eMain3\] is now an immediate consequence of Lemma \[eQSol\], where $f (x) =f_{a^\pm} ( |x|)$ and $\omega (x) \equiv {\mu }$.
Part of Theorem \[eMain3\] concerns odd solutions of . Therefore, given any $b\in {{\mathbb R}}$, we consider the solution $g_b$ of $$\label{fTPA60}
\begin{cases}
\displaystyle g_b '' + \frac {r} {2} g_b ' + \frac {1} {\alpha } g_b + |g_b |^\alpha g_b =0 \quad r\ge 0\\
g_b (0)= 0, \quad g'_b (0)= b .
\end{cases}$$ The solutions $g_b$ of have properties similar to the solutions $f_a$ of – (with $N=1$). We summarize some of these properties in the following proposition.
\[eTPA4\] Problem is globally well posed, $$\label{fTPA1b61}
\sup _{ r\ge 0 } \, (1+ r^2) ^{\frac {1} {\alpha }} ( | g_b (r) | + r | g_b ' (r) | ) < \infty ,$$ and the limit $$\label{fTPA161}
L _1 (b) = \lim_{r\to\infty}r^{\frac{2}{\alpha}}g_b(r) \in {{\mathbb R}}$$ exists, and is a continuous function of $b \in {{\mathbb R}}$. Moreover, if $b \neq 0$, then $g_b$ has at most finitely many zeros on $(0,\infty )$, and we set $$\label{fTPA162}
N_1(b ) = \text{the number of zeros of the function $g_b$ on $(0,\infty )$} .$$ If $L_1 (b) \not = 0$ for some $b\not = 0$, then $N_1 $ is constant in some neighborhood of $b$.
Let $b\in {{\mathbb R}}$ and let $g_b$ be the solution of . It follows from standard energy arguments that $g_b$ exists globally. We set $$\label{fPLE1}
z_b (s) = s^{-\frac{1} {\alpha}} g_b \Bigl( \frac{1} {\sqrt{s}} \Bigr),\quad 0<s< \infty$$ so that $z_b$ is a solution of on $(0,\infty )$. It follows from [@SoupletW Proposition 2.4] that $z_b \in C^1([0,\infty ))$; and so $z_b (0) = \lim _{ s\to 0 } z_b (s)$ exists and is finite. Applying , we obtain the existence and finiteness of the limit with $L _1 (b) = z_b (0)$. Moreover, $ |z_b (s) | + s |z_b ' (s) |$ is bounded on $[0,1]$, so that by , $(1+ r^2) ^{\frac {1} {\alpha }} ( | g_b (r) | + r | g_b ' (r) | )$ is bounded on $[1,\infty )$. Since $(1+ r^2) ^{\frac {1} {\alpha }} ( | g_b (r) | + r | g_b ' (r) | )$ is clearly bounded on $[0, 1]$, we obtain .
The continuous dependence of $L _1 (b) = z_b (0)$ on $b$ follows from arguments in [@SoupletW]. More precisely, let $$\label{fPLE0}
\sigma = \frac {1} {4(\gamma -1)},\quad \tau = \frac {1} {\sqrt \sigma }= 2 \sqrt{\gamma -1}$$ and note that, given any $\tau <T<\infty $, the map $ b\mapsto g_b $ is continuous $ {{\mathbb R}}\to C^1([\tau , T])$. Applying , this implies that, given any $0<\varepsilon <\sigma $, $$\label{fPLE1b}
\text{the map }b\mapsto z_b \text{ is continuous } {{\mathbb R}}\to C^1( [ \varepsilon , \sigma ]).$$ Moreover, the map $s\mapsto 2s^2 z_b '(s)^2 + G (z_b (s))$ is nondecreasing on $(0,\sigma )$ by and , so that $z_b$ is bounded on $(0,\sigma )$ in terms of $z_b(\sigma )$ and $z_b ' (\sigma )$. Therefore, by , $$\label{fPLE2}
\sup _{ 0<s<\sigma } |z_b (s)| \le A ( g_b( \tau ), g_b ' (\tau ))$$ where $A$ is a continuous function of its arguments. Let $b\in {{\mathbb R}}$. It follows from [@SoupletW formula (2.3)] that for all $0\le s \le \sigma $ $$\label{fPLE2b}
\begin{split}
z_b (s) = & z_b (\sigma ) - \sigma ^\gamma e^{\frac {1} {4\sigma }} \Bigl( \int _s^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \, dt \Bigr) z_b'(\sigma ) \\ & - \frac {1} {4} \int _s^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} g (z _{ b } (r) ) \, dr dt
\end{split}$$ Given $b_0, b \in {{\mathbb R}}$ and applying , we have for all $0\le s \le \sigma $ $$\begin{split}
|z _{ b_0 } (s)- z _{ b } (s) | \le & |z _{ b_0 } (\sigma )- z _{ b } (\sigma ) | \\ & + \sigma ^\gamma e^{\frac {1} {4\sigma }} \Bigl( \int _0^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \, dt \Bigr) |z _{ b_0 } ' (\sigma ) - z _{ b } ' (\sigma ) | \\ & + \frac {1} {4} \int _0^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | \, dr dt
\end{split}$$ Setting $$C_1 = \sigma ^\gamma e^{\frac {1} {4\sigma }} \int _0^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \, dt <\infty ,$$ we deduce that, given any $0< \nu <\sigma $ $$\begin{split}
\sup _{ 0\le s\le \sigma } & |z _{ b_0 } (s)- z _{ b } (s) | \le |z _{ b_0 } (\sigma )- z _{ b } (\sigma ) | + C_1 |z _{ b_0 } ' (\sigma ) - z _{ b } ' (\sigma ) | \\ & + \Bigl( \frac {1} {4} \int _0^\nu t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} \, dr dt \Bigr) \sup _{ 0\le r\le \nu } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | \\ & + \Bigl( \frac {1} {4} \int _\nu ^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} \, dr dt \Bigr) \sup _{ \nu \le r\le \sigma } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | .
\end{split}$$ We observe that (see [@SoupletW Lemma 2.1]) $$C_2= \sup _{ 0<s<\sigma } \frac {1} {s}\int _0^s t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} \, dr dt <\infty$$ and $$C_3 = \int _0 ^\sigma t ^{-\gamma } e^{-\frac {1} {4t }} \int _t ^\sigma r^{\gamma -2} e^{\frac {1} {4r}} \, dr dt <\infty$$ so that $$\label{fPLE3}
\begin{split}
\sup _{ 0\le s\le \sigma } & |z _{ b_0 } (s)- z _{ b } (s) | \le |z _{ b_0 } (\sigma )- z _{ b } (\sigma ) | + C_1 |z _{ b_0 } ' (\sigma ) - z _{ b } ' (\sigma ) | \\ & + C_2 \nu \sup _{ 0< r < \nu } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | + C_3 \sup _{ \nu < r < \sigma } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | .
\end{split}$$ We now fix $b_0\in {{\mathbb R}}$ and $\delta >0$. We deduce from that if $\eta >0$ is sufficiently small, then $$\label{fPLE6}
\sup _{ |b- b_0| \le \eta } \{ |z _{ b_0 } (\sigma )- z _{ b } (\sigma ) | + C_1 |z _{ b_0 } ' (\sigma ) - z _{ b } ' (\sigma ) |\} \le \frac {\delta } {3} .$$ Moreover, it follows from that there exists $0 < C_4< \infty $ such that $$\label{fPLE4}
\sup _{ |b- b_0| \le \eta } \sup _{ 0< r < \sigma } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | \le C_4,$$ and we fix $0< \nu < \sigma $ sufficiently small so that $$\label{fPLE5}
C_2 C_4 \nu \le \frac {\delta } {3}.$$ For this fixed value of $\nu$, it follows from that, assuming $\eta >0$ possibly smaller $$\label{fPLE7}
C_3 \sup _{ \nu < r < \sigma } | g (z _{ b_0 } (r) ) - g (z _{ b } (r) ) | \le \frac {\delta } {3}.$$ Estimates , , , and yield $$\sup _{ |b- b_0| \le \eta } \sup _{ 0\le s\le \sigma } |z _{ b_0 } (s)- z _{ b } (s) | \le \delta .$$ Therefore, $$\label{fPLE1b2}
\text{the map }b\mapsto z_b \text{ is continuous } {{\mathbb R}}\to C( [ 0 ,\sigma ]).$$ In particular, $L_1 (b)= z_b (0)$ depends continuously on $b$.
We next show that if $b\not = 0$, then $g_b$ has finitely many zeroes on $(0,\infty )$. It is clear that $g_b$ has finitely many zeros on $[0,\tau ]$, where $\tau $ is defined by . Applying , it remains to show that $z_b$ has finitely many zeros on $[0,\sigma ]$. This is clear if $z_b (0) \not = 0$. Thus we now assume $z_b (0)=0$, and it follows from [@SoupletW Proposition 2.7 (i)] that there exists $0<\varepsilon \le \sigma $ such that $z_b '$ has at most one zero on $(0, \varepsilon )$, so that $z_b$ has at most two zeros on $[0,\varepsilon ]$. Since equation is not singular on $[\varepsilon ,\sigma ]$, $z_b$ has finitely many zeros on $[\varepsilon ,\sigma ]$, which proves the desired property.
Finally, if $L ( \overline{b} ) \not = 0$, i.e. $z_ {\overline{b} } (0) \not = 0$, then if follows from that if $ |b - b_0|$ is sufficiently small, then there exists $\varepsilon >0$ such that $ | z_b (s) |\ge \varepsilon $ for all $0\le s\le \sigma $. By , this means that $ |g_b (r) |\ge \varepsilon r^{-\frac {2} {\alpha }} $ for $r\ge \tau $, with $\tau $ defined by . Since $g_b$ has the same number of zeros as $g _{ b_0 }$ on $[0, \tau )$ for $ |b-b_0|$ sufficiently small, we deduce that $N_1 (b)= N_1 (b_0)$ provided $ |b-b_0|$ sufficiently small.
\[Proof of Theorem $\ref{eMain3}$ \] We first claim that the sequence $({\mu }_m) _{ m\ge \overline{m} }$ of Lemma \[eLFN1\] gives rise to a sequence $(\beta _m) _{ m\ge \overline{m} } \subset (0,\infty ) $ satisfying (with the notation -) $$\begin{gathered}
\beta _m {\mathop{\longrightarrow}}_{ m\to \infty }\infty \label{eNZ3n1:1} \\
(-1)^m L_1 (\beta _m ) {\mathop{\longrightarrow}}_{ m\to \infty }\infty \label{eNZ3n1:1b1} \\
N_1 (\beta _m ) = m . \label{eNZ3n1:3} \end{gathered}$$ Indeed, we have $w _{ {\mu }_m } (0) = {\mu }_m >0$. Moreover, ${{\mathcal L}_2}( {\mu }_m )=0$, so that by Remark \[eTPA3\] the limit exists and ${{\mathcal L}_1}( {\mu }_m )\not = 0$. On the other hand, $ {{\widetilde {N}}}({\mu }_m) =m$, so that $(-1)^m w _{ {\mu }_m } (s) >0$ for $s$ large. Thus we see that that $ (-1)^m {{\mathcal L}_1}( {\mu }_m ) > 0$. We set $$\beta _m= (-1)^m {{\mathcal L}_1}( {\mu }_{ m } ),$$ so that $$g _{ \beta _m }(r)= (-1)^m r^{-\frac {2} {\alpha }} w _{ {\mu }_m } ( r^{-2} )$$ for all $r>0$. Moreover $N_1 (\beta _m) = {{\widetilde {N}}}( {\mu }_m ) =m $, $L_1 (\beta _m) = (-1)^m w _{ {\mu }_m } (0)= (-1)^m {\mu }_m $. In particular, $(-1)^m L _1 (\beta _m) \to \infty $ as $m\to \infty $, and since $L_1$ is continuous $[0, \infty ) \to {{\mathbb R}}$, we see that $\beta _m \to \infty $ as $m\to \infty $. This establishes the claim.
Let ${\mu }>0$. By , we may choose an integer $m_ {\mu }$ sufficiently large so that $$\label{fMT2n1:1}
L_1 (\beta _{2m}) > {\mu }\text{ and } L_1 (\beta _{2m-1 }) < - {\mu }\text{ for all } m\ge m_ {\mu }.$$ It follows from Proposition \[eTPA4\] (continuity of $L_1$) that there exist $$\label{fMT2n1:2}
0 < \beta _{ 2m-1 }< c_m^- < \beta _{2m} < c_m ^+ < \beta _{ 2m+1 }$$ such that $$\label{fMT2n1:3}
L_1 ( c_m^- )= L_1 (c_m ^+ ) =0 \text{ and } L_1( c ) >0 \text{ for all } c_m^- < c < c_m^+ .$$ Next, since $N_1 (\beta _{2m}) =2m $ (by ), we deduce from , and Proposition \[eTPA4\] that $$\label{fMT2n1:4}
N_1 ( c) = 2m \text{ for all } c_m^- < c < c_m^+ .$$ From and , it follows that there exist $$\label{fMT2n1:6}
c_m^- < c _{ {\mu }, m }^- < \beta _{2m} < c _{ {\mu },m }^+ < c_m^+$$ such that $L_1 (c _{ {\mu }, m }^\pm ) = {\mu }$. We deduce from and that $N_1 ( c _{ {\mu },m }^\pm ) = 2 m$, and from , and that $c _{ {\mu },m }^\pm \to \infty $ as $m\to \infty $. Thus we see that the sequences $(c _{ {\mu },m }^\pm ) _{ m \ge m_ {\mu }}$ satisfy
1. $c_{ {\mu }, m }^- <c _{ {\mu }, m }^+$ and $ c_{ {\mu }, m }^\pm \to \infty $ as $m\to \infty $;
2. $L_1 (c_{ {\mu }, m }^\pm ) ={\mu }$ for all $m\ge m _ {\mu }$;
3. $ N_1 (c_{ {\mu }, m }^\pm ) =2m$ for all $m\ge m _ {\mu }$.
By considering $\beta _{ 2m+1 } \to \infty $ (instead of $\beta _{2m}$), one constructs as above a sequence $( d _{ {\mu }, m }^\pm ) _{ m \ge m_ {\mu }} $ such that
1. $d _{ {\mu }, m }^+ < d_{ {\mu }, m }^-$ and $ d_{ {\mu }, m }^\pm \to -\infty $ as $m\to \infty $;
2. $L_1 (d_{ {\mu }, m }^\pm ) ={\mu }$ for all $m\ge m _{\mu }$;
3. $ N_1 (d_{ {\mu }, m }^\pm ) =2m +1$ for all $m\ge m _{\mu }$.
Finally, we may assume ${\mu }>0$ without loss of generality, and we define $ \widetilde{g} _{b ^\pm }$ by $$\widetilde{g}_{b ^\pm } (x)=
\begin{cases}
g_{b ^\pm } (x) & x\ge 0 \\ - g_{b ^\pm } (-x) & x <0
\end{cases}$$ with ${b ^\pm }= c^\pm _{ {\mu }, \frac {m} {2} }$ if $m\ge m_{\mu }$ is even and ${b ^\pm }= d^\pm _{ {\mu }, \frac {m-1} {2} }$ if $m\ge m_{\mu }$ is odd. We observe that $ \widetilde{g}_{b ^\pm } $ are two different solutions of the profile equation , which are odd and have $m$ zeros on $(0,\infty )$, hence $2m+1$ zeros on ${{\mathbb R}}$. Moreover, $ |x|^{\frac {2} {\alpha }} \widetilde{g}_{b ^\pm } (x) \to {\mu }$ as $x\to \infty $ and $ |x|^{\frac {2} {\alpha }} \widetilde{g}_{b ^\pm } (x) \to -{\mu }$ as $x\to -\infty $. Since $(1 + |x|^2)^{\frac {1} {\alpha }} ( |\widetilde{g}_{b ^\pm } (x)| + | x \widetilde{g}_{b ^\pm } '(x)|)$ is bounded on ${{\mathbb R}}$ by , Theorem \[eMain3\] now follows from Lemma \[eQSol\], where $f= \widetilde{ g } _{b^\pm }$ and $\omega (x)= 1$ for $x>0$ and $\omega (x)=-1$ for $x<0$.
Nonexistence of local, nonnegative solutions {#sNEX}
============================================
As explained in the introduction, the main achievement of this paper is the construction of solutions of equation with positive initial values, i.e. ${\mu }|x|^{-\frac {2} {\alpha }}$, for which no local in time nonnegative solution exists. This last assertion is a consequence of [@Weissler4 Theorem 1]. However, the result in [@Weissler4] only concerns nonnegative solutions of the integral equation . On the other hand, if $0< \alpha \le \frac {2} {N}$, the solutions constructed in Theorems \[eMain1\] and \[eMain3\] do not satisfy the integral equation. For completeness, we state and prove below a proposition which establishes nonexistence of nonnegative solutions for both classical solutions of , and solutions of the integral equation .
\[eNEX1\] Let ${\mu }>0 $, $\alpha >0$ and let ${u_0 }(x)= {\mu }|x|^{-\frac {2} {\alpha }}$.
1. \[eNEX1:1\] Suppose either $0<\alpha \le \frac {2} {N}$, or else $\alpha >\frac {2} {N}$ and ${\mu }>{\mu }_0$ where $$\label{fNEX1}
\frac {1} {{\mu }_0}= \alpha ^{\frac {1} {\alpha } } 2^{- \frac {2} {\alpha } } \pi ^{ - \frac {N} {2}} \int _{ {{\mathbb R}}^N } e^{- |y|^2} | y |^{-\frac {2} {\alpha }} dy.$$ It follows that for all $T>0$, there is no solution $u \in L^\infty _{{\mathrm{loc}}}((0,T), L^\infty ({{\mathbb R}}^N ) )$ of , $u\ge 0$, which is a classical solution on $(0,T)$, and satisfies the initial condition in the sense that $u(t) \to {u_0 }$ in $L^1_{{\mathrm{loc}}}({{\mathbb R}}^N \setminus \{0\}) $.
2. \[eNEX1:2\] If $\alpha > \frac {2} {N}$ and ${\mu }>{\mu }_0$ with ${\mu }_0$ defined by , then for all $T>0$, there is no measurable, almost everywhere finite, function $u:(0,T)\times {{\mathbb R}}^N \to {{\mathbb R}}$, $u\ge 0$, which satisfies the integral equation . (Note that all terms in are integrals of nonnegative, measurable functions, which are well defined, possibly infinite.)
Suppose $\alpha >\frac {2} {N}$, so that ${u_0 }\in L^1 ({{\mathbb R}}^N ) +L^\infty ({{\mathbb R}}^N ) $. It follows, since ${u_0 }$ is radially symmetric, radially decreasing, and homogeneous, that $$\label{fNEX2}
(\alpha t)^{\frac {1} {\alpha } } \| e^{t \Delta } {u_0 }\| _{ L^\infty } = (\alpha t)^{\frac {1} {\alpha } } e^{t \Delta } {u_0 }(0) = \alpha ^{\frac {1} {\alpha } } e^{ \Delta } {u_0 }(0) = \frac {{\mu }} {{\mu }_0}$$ where ${\mu }_0 $ is defined by . Therefore, if ${\mu }>{\mu }_0$, then $ (\alpha t)^{\frac {1} {\alpha } } \| e^{t \Delta } {u_0 }\| _{ L^\infty } >1 $ for all $t>0$, and Property follows from [@Weissler4 Theorem 1].
We next prove Property . Suppose there exist $T>0$ and a solution $u$ on $(0,T)$ in the sense of . Since $u$ is a classical solution on $(0 ,T)$, it is also a solution of the integral equation starting at $u(\tau )$ for $0<\tau <T$, i.e. $$\label{feFBW2:3}
u(t)= e^{(t- \tau ) \Delta } u( \tau ) + \int _\tau ^t e^{(t- s ) \Delta } |u(s)|^\alpha u(s) \,ds$$ for all $0< \tau <t<T$. Suppose first $\alpha >\frac {2} {N}$ and ${\mu }>{\mu }_0$. We deduce from [@Weissler4 Theorem 1] that for all $0<\tau <T$, $$\sup _{ 0<t<T-\tau } ( \alpha t )^{\frac {1} {\alpha }} \| e^{t\Delta } u(\tau ) \| _{ L^\infty } \le 1.$$ Given a compact subset $K$ of ${{\mathbb R}}^N \setminus \{0\}$, it follows that $$\sup _{ 0<t<T-\tau } ( \alpha t )^{\frac {1} {\alpha }} \| e^{t\Delta } {\mathbbm 1}_K u(\tau ) \| _{ L^\infty } \le 1.$$ Since ${\mathbbm 1}_K u(\tau )\to {\mathbbm 1}_K {u_0 }$ as $\tau \downarrow 0$ in $L^1 ({{\mathbb R}}^N ) $, we obtain $$\sup _{ 0<t<T } ( \alpha t )^{\frac {1} {\alpha }} \| e^{t\Delta } {\mathbbm 1}_K {u_0 }\| _{ L^\infty } \le 1.$$ Since $K\subset \subset {{\mathbb R}}^N \setminus \{0\}$ is arbitrary, we conclude that $$\sup _{ 0<t<T } ( \alpha t )^{\frac {1} {\alpha }} \| e^{t\Delta } {u_0 }\| _{ L^\infty } \le 1$$ which is in contradiction with . Suppose now $0<\alpha \le \frac {2} {N}$. It follows from that $$\label{fSL1}
u(t)\ge e^{(t- \tau ) \Delta } u( \tau )$$ for all $0< \tau <t<T$. We fix $0<t<T$ and $M>0$. Since ${u_0 }$ is not integrable at $x=0$, there exists $0<\varepsilon <1$ such that if $K= \{ \varepsilon < |x| < 1\}$, then $e^{t \Delta }({\mathbbm 1}_K {u_0 }) (0) \ge M$. Since $u(\tau )\ge {\mathbbm 1}_K u(\tau )$, we have $u(t, 0)\ge e^{(t- \tau ) \Delta } ({\mathbbm 1}_K u( \tau )) (0)$ by , and we deduce by letting $\tau \to 0$ that $u(t, 0) \ge M$. Since $M>0$ is arbitrary, we deduce that $u(t, 0) =\infty $, which is absurd.
[99]{}
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[^1]: Research supported by the “Brazilian-French Network in Mathematics"
[^2]: Flávio Dickstein was partially supported by CNPq (Brasil).
[^3]: Ivan Naumkin thanks the project ERC-2014-CdG 646.650 SingWave for its financial support, and the Laboratoire J.A. Dieudonné of the Université de Nice Sophia-Antipolis for its kind hospitality.
|
---
author:
- 'Arnaud Gocsei[^1], Fouad Sahel[^2]'
date: 5 May 2010
title: |
Analysis of the sensitivity to discrete dividends :\
A new approach for pricing vanillas[^3]
---
**Abstract**
The incorporation of a dividend yield in the classical option pricing model of Black-Scholes results in a minor modification of the Black-Scholes formula, since the lognormal dynamic of the underlying asset is preserved. However, market makers prefer to work with cash dividends with fixed value instead of a dividend yield. Since there is no closed-form solution for the price of a European Call in this case, many methods have been proposed in the literature to approximate it. Here, we present a new approach. We derive an exact analytic formula for the sensitivity to dividends of an European option. We use this result to elaborate a proxy which possesses the same Taylor expansion around 0 with respect to the dividends as the exact price. The obtained approximation is very fast to compute (the same complexity than the usual Black-Scholes formula) and numerical tests show the extreme accuracy of the method for all practical cases.\
\
\
**Key words:** Equity options, discrete dividends.
Introduction {#intro}
============
In the classical Black-Scholes framework, we can find in the literature three main ways of inserting cash dividends into the model [^4] :
1. **Escrowed model.** Assume that the asset price minus the present value of all dividends to be paid before the maturity of the option follows a Geometric Brownian Motion.
2. **Forward model.** Assume that the asset price plus the forward value of all dividends from past dividend dates to today, follows a Geometric Brownian Motion.
3. **Piecewise lognormal model.** Assume that the asset price shows a jump downward at each dividend date (equal to the cash dividend payment at that date) and follows a Geometric Brownian Motion between those dates.
Although the first two models lead to a closed-form solution, they are not satisfactory. Indeed, the option price obtained in these models is not continuous at dividend dates. Moreover, if one considers two options with different maturities $T_{1}<T_{2}$, the first two models lead to different asset price process dynamics for $t\leq T_{1}$, since the dividends paid between $T_{1}$ and $T_{2}$ are taken into account in one case but not in the other.
Therefore, it is the piecewise lognormal model which is prefered from a theoritical point of view. This paper is dedicated to find a robust pricing proxy for this model. We consider an underlying following a Black-Scholes dynamic between dividend detachement dates and paying cash dividends at discrete times $0<T_{1}<\ldots <T_{n}<T$, i.e. :
- for $T_i\leq t<T_{i+1}$ : $$dS_{t} = r S_{t}dt + \sigma S_{t} dW_{t},$$
- at time $T_i$ : $$S_{T_i^+} = S_{T_i^-} - D_i (S_{T_i^-}),$$
where $r$ is the interest rate, assumed constant, $W$ is a standard Brownian motion and $D_{i}$ is the dividend policy defined by : $$D_{i}(S)=\left\{
\begin{array}{lll}
C_{i} & \text{if} & S>C_{i}, \\
S & \text{if} & S\leq C_{i},\end{array}\right.$$The cash amounts $C_{1},\ldots ,C_{n}$ are *known* at the initial date 0 and each $C_i$ represents the dividend cash amount eventually paid at time $T_{i}$.
The dividend policy $D_{i}$ is a *liquidator* policy as the stock price is absorbed at zero at time $T_{i}$ if $S_{T_{i}}<C_{i}$. Consequently, the stock price remains positive. Note that as a practical matter, for most applications, the definition of $D_{i}(S)$ when $S\leq C_{i}$ has negligible financial effects[^5], as the probability that a stock price drops below a declared dividend at a fixed time is typically small. It just ensures the positivity of the price.
In this paper, we are interested in computing the fair price of the European Call $\operatorname*{Call}(S_{0},K)$ with strike $K$ and maturity $T$. Since there is no closed-formula, one should recover the price via PDE methods using a finite difference scheme, with boundary conditions at each $T_{i}$ ensuring the continuity of the price of the Call. This procedure can be time-consuming if one considers a maturity $T=20$ years and an underlying paying as much as one dividend a week. Therefore, when computation speed is at stake, one would prefer a fast and accurate proxy for the price.
We review in the following section three of the existing methods in the literature and discuss their limitations.
Existing Methods
================
1. **Method of moments matching.** We approximate the stock price process $S$ by a process $\tilde{S}$ with a shifted log-normal dynamic under the risk-neutral pricing measure : $$\tilde{S}_{t}=\lambda +M\exp \left( -\frac{1}{2}\sigma ^{^{\prime
}2}t+\sigma ^{^{\prime }}W_{t}\right) .$$The three parameters $\lambda ,M$ and $\sigma ^{^{\prime }}$ are calibrated so that the first three moments of $\tilde{S}_{T}$ match the first three moments of $S_{T}$. This method reduces to the pricing of a European Call on a modified underlying $\tilde{S}$, which can be done using the usual Black-Scholes formula. This proxy does not work well if the stock pays dividends frequently, the maturity is greater than 5 years or the option is deep in-the-money.
2. In [@bos2002], Bos and Vandermark define a mixture of the Escrowed and Forward models, using linear pertubations of first order. They derive a proxy resulting in spot/strike adjustment : $$\operatorname*{Call}(S_{0},K)\approx \operatorname*{Call^{BS}}(S^{\ast },K^{\ast }),$$where $\operatorname*{Call^{BS}}$ is the usual Black-Scholes function and: $$\begin{aligned}
S^{\ast }=& S_{0}-\sum_{i=1}^{n}\left( 1-\frac{T_{i}}{T}\right)
C_{i}e^{-rT_{i}}, \label{Setoile} \\
K^{\ast }=& K+\sum_{i=1}^{n}\frac{T_{i}}{T}C_{i}e^{r(T-T_{i})}.
\label{Ketoile}\end{aligned}$$This proxy works better for at-the-money options and small maturities but results in serious mis-pricing for in-and out-of-the-money options and large maturities.
3. In [@bos2003], Bos, Gairat and Shepeleva derive a more accurate proxy than the previous one by considering a volatility adjustment : $$\operatorname*{Call}(S_{0},K)\approx \operatorname*{Call^{BS}}(S^{\ast },K,\sigma (S^{\ast },K,T)),$$with $S^{\ast }$ given by (\[Setoile\]): $$\begin{aligned}
\sigma (S^{\ast },K,T)^{2}= & \sigma ^{2}+\sigma \sqrt{\frac{\pi }{2T}}\Bigg\{\frac{e^{\frac{a^{2}}{2}}}{S^{\ast }}\sum_{i=1}^{n}C_{i}e^{-rT_{i}}\left[ N(a)-N\left( a-\sigma \frac{T_{i}}{\sqrt{T}}\right) \right] \\
+& \frac{e^{\frac{b^{2}}{2}}}{S^{\ast 2}} \sum_{i,j=1}^{n}
C_{i}C_{j}e^{-r(T_{i}+T_{j})}\left[ N(b)-N\left( b-2\sigma \frac{min(T_{i},T_{j})}{\sqrt{T}}\right) \right],\end{aligned}$$where $N(x)$ is the normal distribution function and: $$a=\frac{1}{\sigma \sqrt{T}}\left( \log \left( \frac{S^{\ast }}{K}\right)
+(r-\sigma ^{2}/2)T\right) ,\ b=a+\frac{1}{2}\sigma \sqrt{T}.$$ This proxy will be a good benchmark to test the accuracy of our method presented in the following section.
The method
==========
Motivations and notations
-------------------------
Consider $\operatorname*{Call}(S_0,K)$ as a function of the dividends $C_{1},\ldots ,C_{n}$: $$\operatorname*{Call}(S_{0},K)=\operatorname*{Call}(C_{1},\ldots ,C_{n}).$$ Although there is no closed-form formula for $\operatorname*{Call}(C_1,\ldots,C_n)$, we prove in annex \[annex1\] that we can still compute explicitely its sensitivities to dividends at the origin. More precisely, we have for all $k\in \mathbb{N}$ and $1\leq i_{1},\ldots,i_{k}\leq n$:
$$\label{derivative}
\frac{\partial^k \operatorname*{Call}}{\partial C_{i_1}\ldots\partial C_{i_k}}(0)=(-1)^k\frac{\partial^k \operatorname*{Call^{BS}}}{\partial S^k}\left(S_0e^{-\sigma^2\sum_{q=1}^kT_{i_q}},K,T\right)e^{-r\sum_{q=1}^kT_{i_q}-\sigma^2\sum_{q=2}^k(q-1)T_{i_q}}.$$
We use this result to derive an accurate approximation of $\operatorname*{Call}(C_{1},\ldots ,C_{n})$. Before explaining our method, we need first to introduce some notations. For all functions $f$ of $n$ variables $x_{1},\ldots ,x_{n}$ and $\forall \alpha \in \mathbb{N}$, we note $T_{\alpha
}f$ the $\alpha^{th}$ order Taylor series at 0 of $f$: $$T_{\alpha }f(x_{1},\ldots ,x_{n}):=\sum_{k=0}^{\alpha }\sum_{i_{1},\ldots
,i_{k}=1}^{n}\frac{x_{i_{1}}\ldots x_{i_{k}}}{i_{1}!\ldots i_{k}!}\frac{\partial ^{k}f}{\partial x_{i_{1}}\ldots \partial x_{i_{k}}}(0).$$We introduce the space $\mathcal{A}_{\alpha }$ of functions having the same $\alpha^{th}$ Taylor series at 0 as the function $\operatorname*{Call}(C_{1},\ldots
,C_{n})$: $$\mathcal{A}_{\alpha }:=\{f,T_{\alpha }f=T_{\alpha }\operatorname*{Call}\}.$$The order $\alpha $ quantifies how near is $f(C_{1},\ldots ,C_{n})$ from $\operatorname*{Call}(C_{1},\ldots ,C_{n})$ when the dividends are small. This precision increase with $\alpha $.
Functions $f$ in $\mathcal{A}_{\alpha }$ are naturally good candidates to approximate $\operatorname*{Call}$. However, the difference $\operatorname*{Call}(C_{1},\ldots
,C_{n})-f(C_{1},\ldots ,C_{n})$ can be quite big if the dividends are not small enough. For instance, for $\alpha=2$ and 3, we test the accuracy of the natural choice consisting of taking $$f(C_{1},\ldots ,C_{n}):=T_{\alpha}\operatorname*{Call}(C_{1},\ldots ,C_{n}).$$ Figure \[toto\] shows the relative error of the price of a European Call when using this approximation. We assume that the stock pays a fixed dividend $C$ every year and we analyse how the relative error varies when we increase $C$. We can see that both the second and third order Taylor series give accurate results when the dividends are small but when the dividends increase, they both lead to serious mis-pricing.
\[toto\] {width="12cm"}
Therefore, the approximations $T_{\alpha }\operatorname*{Call}$, $\alpha=2, 3$, are not satisfying. Thus, one need to find for $\alpha \geq 2$, a function $\operatorname*{Proxy}\in \mathcal{A}_{\alpha }$, different from $T_{\alpha }\operatorname*{Call}$, which gives an accurate approximation of $\operatorname*{Call}$ for all practical values of $C_{1},\ldots ,C_{n}$ (not necessarily very small). We explain in the next subsection how we determine $\operatorname*{Proxy}\in\mathcal{A}_{\alpha}$.
Spot/Strike adjustment
----------------------
Like Bos and Vandermark in [@bos2002], we search our function $\operatorname*{Proxy}$ under the form: $$\label{approxCall}
\operatorname*{Proxy}(C_{1},\ldots ,C_{n}):= \operatorname*{Call^{BS}}(S^{\ast}(C_1,\ldots,C_n),K^{\ast}(C_1,\ldots,C_n)),$$ with: $$\begin{aligned}
S^{*}(C_1,\ldots,C_n)=& S_0+\sum_{k=0}^{\alpha}\sum_{i_{1},\ldots,
i_{k}=1}^n a_{i_1,\ldots,i_k}C_{i_1}\ldots C_{i_k}, \label{spotadj} \\
K^{*}(C_1,\ldots,C_n)=& K+\sum_{k=0}^{\alpha}\sum_{i_{1},\ldots, i_{k}=1}^n
b_{i_1,\ldots,i_k}C_{i_1}\ldots C_{i_k}. \label{strikeadj}\end{aligned}$$ The reason why we perform a spot/strike adjustment is that it allows to recover the exact price when the dividends are paid spot or at maturity.
The coefficients $a_{i_1,\ldots,i_k}$ and $b_{i_1,\ldots,i_k}$ are calculated recursively. They are entirely determined by the two following conditions:
1. $\frac{\partial^k \operatorname*{Proxy}}{\partial C_{i_1}\ldots\partial C_{i_k}}(0)=\frac{\partial^k \operatorname*{Call}}{\partial C_{i_1}\ldots\partial C_{i_k}}(0), \forall
k\leq \alpha$,
2. We impose our proxy to satisfy the Call-Put parity[^6]: $$\begin{aligned}
\operatorname*{Call^{BS}}(S^{\ast },K^{\ast })-\operatorname*{Put^{BS}}(S^{*},K^{*})=&
S_{0}-Ke^{-rT}-\sum_{i=1}^{n}C_{i}e^{-rT_{i}}, \label{callput} \\
i.e.\hspace{3.5cm} S^{\ast }-K^{\ast }e^{-rT}=&
S_{0}-Ke^{-rT}-\sum_{i=1}^{n}C_{i}e^{-rT_{i}}. \label{eq2}\end{aligned}$$
Let’s detail the calculus:
- **Computation of $a_i$ and $b_i$:** the equality $\frac{\partial \operatorname*{Proxy}}{\partial C_i}(0)=\frac{\partial \operatorname*{Call}}{\partial C_i}(0)$ reads: $$N(d_{1})a_i-e^{-rT}N(d_{2})b_i=-e^{-rT_{i}}N(d(T_{i})), \label{Eq1}$$ where : $$\begin{aligned}
d_{1}=& \frac{1}{\sigma \sqrt{T}}(\ln (S_{0}/K)+(r+\sigma ^{2}/2)T), \\
d_{2}=& d_{1}-\sigma \sqrt{T}, \\
d(t)=& d_{1}-\frac{\sigma }{\sqrt{T}}t,\hspace{0.5cm}0\leq t\leq T.\end{aligned}$$ The differentiation of (\[eq2\]) writes: $$a_i-e^{-rT}b_i=-e^{-rT_{i}}. \label{Eq2}$$ Solving the linear system (\[Eq1\])-(\[Eq2\]) gives: $$\begin{aligned}
a_i=& -e^{-rT_{i}}\frac{N(d(T_{i}))-N(d_{2})}{N(d_{1})-N(d_{2})}, \\
b_i=& e^{r(T-T_{i})}\frac{N(d_{1})-N(d(T_{i}))}{N(d_{1})-N(d_{2})}.\end{aligned}$$
- **Computation of $a_{i,j}$ and $b_{i,j}$:** the equality $\frac{\partial^2 \operatorname*{Proxy}}{\partial C_i\partial C_j}(0)=\frac{\partial^2 \operatorname*{Call}}{\partial C_i\partial C_j}(0)$ and two succesive differentiations in ([eq2]{}) give the following linear system: $$\begin{aligned}
N(d_{1})a_{i,j}-e^{-rT}N(d_{2})b_{i,j}=& \beta , \label{caca} \\
a_{i,j}-e^{-rT}b_{i,j}=& 0, \label{pipi}\end{aligned}$$ where: $$\begin{aligned}
\beta = \frac{\partial ^{2}\operatorname*{Call}}{\partial C_{i}\partial C_{j}}(0)- a_ia_j\frac{\partial ^{2}\operatorname*{Call^{BS}}}{\partial S^{2}}(S_{0},K) - (a_ib_j+a_jb_i)\frac{\partial ^{2}\operatorname*{Call^{BS}}}{\partial S\partial K}(S_{0},K) - b_ib_j\frac{\partial ^{2}\operatorname*{Call^{BS}}}{\partial K^{2}}(S_{0},K).\end{aligned}$$ After some direct computations, we obtain: $$\begin{aligned}
a_{i,j}=& \frac{1}{\gamma} e^{-r(T_i+T_j)}\Bigg[a+b\Big(N(d(T_i))+N(d(T_j))\Big)+cN(d(T_i))N(d(T_j))+de^{\sigma^2T_i}N^{^{\prime }}(d(T_i+T_j))\Bigg],
\\
b_{i,j}=& e^{rT}a_{i,j},\end{aligned}$$ with: $$\begin{aligned}
\gamma=& \sigma S\sqrt{T}N^{^{\prime }}(d_1)\Big(N(d_1)-N(d_2)\Big)^3 \\
a=& -\Big(N(d_2)N^{^{\prime }}(d_1)-N(d_1)N^{^{\prime }}(d_2)\Big)^2 \\
b=& \Big(N^{^{\prime }}(d_1)-N^{^{\prime }}(d_2)\Big)\Big(N(d_2)N^{^{\prime
}}(d_1)-N(d_1)N^{^{\prime }}(d_2)\Big) \\
c=& -\Big(N^{^{\prime }}(d_1)-N^{^{\prime }}(d_2)\Big)^2 \\
d=& N^{^{\prime }}(d_1)\Big(N(d_1)-N(d_2)\Big)^2 \\\end{aligned}$$
- **Computation of $a_{i_1,\ldots,i_k}$ and $b_{i_1,\ldots,i_k}$, $k\geq 3$:** the previous method can be reproduced recursively. Knowing all the values $a_{j_1,\ldots,j_m}$ and $b_{j_1,\ldots,j_m}$, $m\leq k-1$, we obtain $a_{i_1,\ldots,i_k}$ and $b_{i_1,\ldots,i_k}$ by solving a linear system of the form :
$$\begin{aligned}
\frac{\partial^{k}\operatorname*{Call^{BS}}}{\partial S^k}(S_{0},K) a_{i_1,\ldots,i_k}+\frac{\partial^{k}\operatorname*{Call^{BS}}}{\partial K^k}(S_{0},K)b_{i_1,\ldots,i_k}=& u , \\
a_{i_1,\ldots,i_k}-e^{-rT}b_{i_1,\ldots,i_k}=& 0.\end{aligned}$$
We have presented a simple and general method to derive a function $\operatorname*{Proxy}$ in $\mathcal{A}_{\alpha}$ for any $\alpha\in\mathbb{N}$. As for the order $\alpha$ that we choose effectively for our tests, the second order computation is a good choice for performance and accuracy. Before presenting the numerical results, we recall some desirable properties of our second order proxy (\[approxCall\]):
1. fast computation, even when one considers a large number $n$ of dividends.
2. recovery of exact price when all dividends are paid spot or at maturity.
3. arbitrage free with the Call-Put parity.
4. guarantee of the continuity of the Call price at dividend detachement dates.
5. accuracy for all practical configurations, even for the extreme cases (deep in-the-money-option with large maturity and high frequency of dividends) for which the already existing methods of the financial literature might lead to serious mis-pricing.
Numerical tests
===============
Test on an underlying paying dividends with low frequency
---------------------------------------------------------
We test the accuracy of our proxy on a stock with the following parameters: $S_0=100$, $r=3\%$, $\sigma=30\%$. We suppose that the stock pays a dividend of 3 in the middle of every year. We compute the Call price with strike $K\in\{50,75,100,125,150,175,200\}$ and maturity $T\in\{5,10,15,20\}$ using four methods:
1. the finite difference method,
2. the method of moments matching.
3. the spot/vol adjustment of Bos, Gairat and Shepeleva[@bos2003],
4. our proxy with spot/strike adjustment given by (\[spotadj\])-([strikeadj]{}),
Remember that no approximation is made in the finite difference method. The results are given in the following tables.
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 47.14 & 33.85 & 24.42 & 17.79 & 13.12 & 9.79 & 7.39\
Method of moments & 47.17 & 33.87 & 24.42 & 17.78 & 13.10 & 9.77 & 7.38\
Proxy BGS & 47.11 & 33.84 & 24.42 & 17.80 & 13.13 & 9.81 & 7.41\
Proxy GS & 47.14 & 33.85 & 24.42 & 17.79 & 13.12 & 9.79 & 7.39\
**Relative error (in%):** & & & & & & &\
Method of moments & 0.07 & 0.06 & 0.01 & -0.05 & -0.11 & -0.16 & -0.20\
Proxy BGS & -0.05 & -0.05 & -0.02 & 0.03 & 0.10 & 0.17 & 0.24\
Proxy GS & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 46.85 & 38.21 & 31.66 & 26.58 & 22.56 & 19.34 & 16.71\
Method of moments & 47.07 & 38.38 & 31.77 & 26.64 & 22.59 & 19.34 & 16.69\
Proxy BGS & 46.65 & 38.08 & 31.59 & 26.57 & 22.60 & 19.41 & 16.81\
Proxy GS & 46.85 & 38.21 & 31.66 & 26.58 & 22.56 & 19.34 & 16.71\
**Relative error (in%):** & & & & & & &\
Method of moments & 0.49 & 0.45 & 0.35 & 0.23 & 0.11 & -0.01 & -0.13\
Proxy BGS & -0.43 & -0.35 & -0.21 & -0.04 & 0.15 & 0.35 & 0.55\
Proxy GS & 0.00 & -0.01 & -0.01 & -0.01 & -0.01 & -0.01 & -0.02\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 46.47 & 40.48 & 35.73 & 31.85 & 28.63 & 25.91 & 23.59\
Method of moments & 47.12 & 41.04 & 36.18 & 32.21 & 28.91 & 26.13 & 23.75\
Proxy BGS & 45.79 & 40.01 & 35.43 & 31.70 & 28.60 & 25.98 & 23.74\
Proxy GS & 46.49 & 40.49 & 35.73 & 31.85 & 28.63 & 25.91 & 23.59\
**Relative error (in%):** & & & & & & &\
Method of moments & 1.41 & 1.38 & 1.27 & 1.13 & 0.98 & 0.82 & 0.66\
Proxy BGS & -1.47 & -1.19 & -0.85 & -0.49 & -0.12 & 0.24 & 0.60\
Proxy GS & 0.02 & -0.01 & -0.02 & -0.03 & -0.03 & -0.04 & -0.04\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 46.02 & 41.74 & 38.22 & 35.26 & 32.72 & 30.51 & 28.57\
Method of moments & 47.35 & 42.95 & 39.30 & 36.21 & 33.55 & 31.24 & 29.20\
Proxy BGS & 44.33 & 40.47 & 37.30 & 34.63 & 32.33 & 30.32 & 28.55\
Proxy GS & 46.10 & 41.76 & 38.23 & 35.26 & 32.71 & 30.50 & 28.56\
**Relative error (in%):** & & & & & & &\
Method of moments & 2.89 & 2.90 & 2.82 & 2.69 & 2.54 & 2.38 & 2.21\
Proxy BGS & -3.71 & -3.07 & -2.44 & -1.82 & -1.23 & -0.65 & -0.09\
Proxy GS & 0.14 & 0.03 & -0.02 & -0.04 & -0.05 & -0.06 & -0.07\
{width="12cm"}
Test on an underlying paying dividends with high frequency
----------------------------------------------------------
We now check the accuracy of our proxy on an underlying paying dividends every week. This situation occurs when considering an index like S&P 500 or Eurostox 50. We take the following parameters: $S_0=3000$, $r=3\%$, $\sigma=30\%$. We suppose that the stock pays a dividend of 2 every week. We compute the Call price with strike $K$ such as $K/S_0\in\{0.5,0.75,1,1.25,1.5,1.75,2\}$ and maturity $T\in\{5,10,15,20\}$.
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 1359.87 & 972.67 & 699.65 & 508.71 & 374.45 & 279.07 & 210.47\
Method of moments & 1361.05 & 973.29 & 699.70 & 508.39 & 373.97 & 278.54 & 209.98\
Proxy BGS & 1358.88 & 972.02 & 699.52 & 508.99 & 375.00 & 279.75 & 211.21\
Proxy GS & 1359.87 & 972.69 & 699.68 & 508.73 & 374.47 & 279.07 & 210.47\
**Relative error (in%):** & & & & & & &\
Method of moments & 0.09 & 0.06 & 0.01 & -0.06 & -0.13 & -0.19 & -0.23\
Proxy BGS & -0.07 & -0.07 & -0.02 & 0.05 & 0.15 & 0.25 & 0.35\
Proxy GS & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 1319.62 & 1075.07 & 890.03 & 746.82 & 633.81 & 543.14 & 469.40\
Method of moments & 1327.05 & 1080.52 & 893.49 & 748.67 & 634.44 & 542.91 & 468.55\
Proxy BGS & 1311.37 & 1069.84 & 887.68 & 746.80 & 635.57 & 546.23 & 473.42\
Proxy GS & 1319.68 & 1075.04 & 889.96 & 746.72 & 633.69 & 543.02 & 469.25\
**Relative error (in%):** & & & & & & &\
Method of moments & 0.56 & 0.51 & 0.39 & 0.25 & 0.10 & -0.04 & -0.18\
Proxy BGS & -0.63 & -0.49 & -0.26 & 0.00 & 0.28 & 0.57 & 0.86\
Proxy GS & 0.00 & 0.00 & -0.01 & -0.01 & -0.02 & -0.02 & -0.03\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 1287.50 & 1122.13 & 990.74 & 883.63 & 794.68 & 719.49 & 655.20\
Method of moments & 1308.65 & 1140.08 & 1005.32 & 895.16 & 803.52 & 726.17 & 660.09\
Proxy BGS & 1258.42 & 1102.31 & 978.74 & 877.99 & 794.07 & 723.01 & 662.03\
Proxy GS & 1288.47 & 1122.33 & 990.66 & 883.42 & 794.31 & 719.10 & 654.79\
**Relative error (in%):** & & & & & & &\
Method of moments & 1.64 & 1.60 & 1.47 & 1.31 & 1.11 & 0.93 & 0.75\
Proxy BGS & -2.26 & -1.77 & -1.21 & -0.64 & -0.08 & 0.49 & 1.04\
Proxy GS & 0.08 & 0.02 & -0.01 & -0.02 & -0.05 & -0.06 & -0.06\
{width="12cm"}
[|c|ccccccc|]{}\
\
\
$K/S_0$ & 0.5 & 0.75 & 1 & 1.25 & 1.50 & 1.75 & 2\
**Price:** & & & & & & &\
FD (exact price) & 1260.33 & 1144.53 & 1049.11 & 968.59 & 899.43 & 839.22 & 786.23\
Method of moments & 1303.36 & 1183.64 & 1083.92 & 999.27 & 926.36 & 862.79 & 806.82\
Proxy BGS & 1184.43 & 1088.21 & 1008.55 & 940.91 & 882.41 & 831.09 & 785.57\
Proxy GS & 1264.53 & 1145.94 & 1049.44 & 968.43 & 899.04 & 838.71 & 785.66\
**Relative error (in%):** & & & & & & &\
Method of moments & 3.41 & 3.42 & 3.32 & 3.17 & 2.99 & 2.81 & 2.62\
Proxy BGS & -6.02 & -4.92 & -3.87 & -2.86 & -1.89 & -0.97 & -0.08\
Proxy GS & 0.33 & 0.12 & 0.03 & -0.02 & -0.04 & -0.06 & -0.07\
{width="12cm"}
Conclusion
==========
We have presented a new approach to deal with cash dividends in equity option pricing in a piecewise lognormal model for the underlying. Our method relies on the derivation of an analytic formula for the sensitivity to dividends of a European option. We obtain a closed-form formula for a European Call which gives both very accurate results for all practical cases.
Computation of the dividend sensitivities {#annex1}
=========================================
Consider a European option of maturity $T$ with payoff $h(S_{T})$, with $S$ the stock price following the piecewise lognormal dynamic presented in the introduction. We note its fair price at time 0 $$\Pi (S_{0},T,C_{1},\ldots ,C_{n})$$We denote: $$\Pi ^{BS}(S_{0},T)$$the fair price of the option if $S$ does not pay dividends. The partial derivatives $$\frac{\partial ^{k}\Pi }{\partial C_{i_{1}}\ldots \partial C_{i_{k}}}(S_{0},T,0,\ldots ,0)$$are related to the usual Black-Scholes greeks by the following formula:
\[prop\] For $k\in\mathbb{N}$ and $1\leq i_1\leq\ldots\leq i_k\leq n$, we have: $$\begin{aligned}
\frac{\partial^k \Pi}{\partial C_{i_1}\ldots\partial C_{i_k}}(S_0,T,0,\ldots,0)=(-1)^k\frac{\partial^k \Pi^{BS}}{\partial S^k}\left(S_0e^{-\sigma^2\sum_{q=1}^kT_{i_q}},T\right)e^{-r\sum_{q=1}^kT_{i_q}-\sigma^2\sum_{q=2}^k(q-1)T_{i_q}}.\end{aligned}$$
Follows a proof of this formula.
First step: a recursive formula
-------------------------------
We introduce some notations:
- We define the natural filtration $(\mathcal{F}_{t})_{t\geq 0}$ associated with the brownian motion $W$. We suppose the filtration right continuous.
- We define for all $0\leq t_1\leq t_2$: $$X_{t_1\to t_2}:=e^{(r-\sigma^2/2)(t_2-t_1)+\sigma (W_{t_2}-W_{t_1})},$$
- We denote $\phi (S_{0},S,t)$ the log-normal density associated with the variable $S_{0}X_{0\rightarrow t}$
- We define the functions of $n+1$ variables $(h_i)_{0\leq i \leq n}$ such as: $$h_i(S_{T_i},C_1,\ldots,C_n):=e^{-r(T-T_i)}E[h(S_T)|\mathcal{F}_{T_i}].$$
For the sake of simplicity, when there is no confusion, we will simply denote $h_{i}(S)$ instead of $h_{i}(S,C_{1},\ldots ,C_{n})$. Note that we have $\Pi (S_{0},T,C_{1},\ldots ,C_{n})=h_{0}(S_{0})$. We can compute the functions $h_{i}$ recursively beginning with $h_{n}$: $$h_{n}(S)=\Pi ^{BS}(S,T-T_{n}),$$and by conditioning, $\forall i\leq n-1$: $$\begin{aligned}
h_{i}(S)=& e^{-r(T_{i+1}-T_{i})}E[h_{i+1}((SX_{t_{i}\rightarrow
t_{i+1}}-C_{i+1})_{+})|\mathcal{F}_{T_{i}}] \notag \\
=& e^{-r(T_{i+1}-T_{i})}\int_{C_{i+1}}^{\infty }h_{i+1}(S_{i+1}-C_{i+1})\phi
(S,S_{i+1},T_{i+1}-T_{i})dS_{i+1}, \label{intergal}\end{aligned}$$
Now, we show how these relations allow us to compute recursively the partial derivatives: $$\frac{\partial ^{k}\Pi }{\partial C_{i_{1}}\ldots \partial C_{i_{k}}}(S_{0},0,\ldots ,0),$$for $1\leq i\leq n$, $k\in \mathbb{N}^{\ast }$ and $1\leq i_{1}\leq \ldots
\leq i_{k}\leq n$. First, note that a direct application of the theorem of differentiation under the integral sign in the relation (\[intergal\]) proves that the functions $h_{i}$, $0\leq i\leq n$ are infinitely differentiable. Then, using the markov property of the log-normal densities: $$\int_{0}^{\infty }\phi (S_{i},S_{i+1},t_{i})\phi
(S_{i+1},S_{i+2},t_{i+1})dS_{i+1}=\phi (S_{i},S_{i+2},t_{i}+t_{i+1}),$$we obtain: $$\begin{aligned}
\frac{\partial ^{k}\Pi }{\partial C_{i_{1}}\ldots \partial C_{i_{k}}}(S_{0},0,\ldots ,0)=& -e^{-rT_{i_{1}}}E\left[ \frac{\partial ^{k}h_{i_{1}}}{\partial S\partial C_{i_{2}}\ldots \partial C_{i_{k}}}(S_{0}X_{0\rightarrow
T_{i_{1}}},0,\ldots ,0)\right] , \notag \\
=& -e^{-rT_{i_{1}}}\int_{0}^{\infty }\frac{\partial ^{k}h_{i_{1}}}{\partial
S\partial C_{i_{2}}\ldots \partial C_{i_{k}}}(S_{i_{1}},0,\ldots ,0)\phi
(S_{0},S_{i_{1}},T_{i_{1}})dS_{i_{1}}. \label{fifi}\end{aligned}$$This relation will be very useful for a recursive proof of proposition [prop]{} since it reduces by one the number of differentiations with respect to the dividends.
Second step: a martingale argument
----------------------------------
The proof of proposition \[prop\] relies heavily on this simple but crucial lemma:
\[lemma\] Consider a process following a Black-Scholes dynamic $S_t=S_0e^{(r-\sigma^2/2)t+\sigma W_t}$, $0\leq t\leq T$. Then, for all integer $k\geq 0$ and for all real number $a\geq 0$, the process: $$Z_t:=\frac{\partial^k \Pi^{BS}}{\partial S^k}\left(S_te^{k\sigma^2(t-a)},T-t\right)e^{(k-1)(r+k\sigma^2/2)t}$$ is a martingale.
The following corollary is easy to derive.
\[cor\] For $k\in\mathbb{N}^*$, $0\leq t\leq T$ and $a\geq 0$, we have: $$\label{bob}
E\left[\frac{\partial^k \Pi^{BS}}{\partial S^k}\left(S_te^{k\sigma^2(t-a)},T-t\right)\right]= \frac{\partial^k \Pi^{BS}}{\partial S^k}\left(S_0e^{-k\sigma^2a},T\right)e^{-(k-1)(r+k\sigma^2/2)t}$$
*Proof of lemma \[lemma\]:* The drift of $Z_t$ is: $$\begin{aligned}
&e^{(k-1)(r+k\sigma^2/2)t}\Bigg[(k-1)(r+k\sigma^2/2)\frac{\partial^k \Pi^{BS}}{\partial S^k}-\frac{\partial^{k+1}\Pi^{BS}}{\partial t\partial S^k} \notag
\\
&+(r+k\sigma^2)S_te^{k\sigma^2(t-a)}\frac{\partial^{k+1} \Pi^{BS}}{\partial
S^{k+1}} +\frac{1}{2}\sigma^2S_t^2e^{2k\sigma^2(t-a)}\frac{\partial^{k+2}
\Pi^{BS}}{\partial S^{k+2}}\Bigg], \label{plop}\end{aligned}$$ where all the derivatives in the last formula are evaluated in $(S_te^{k\sigma^2(t-a)},T-t)$. Remember that $\Pi^{BS}$ satisfies the Black-Scholes PDE: $$-\frac{\partial\Pi^{BS}}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2
\Pi^{BS}}{\partial S^2} +rS\frac{\partial \Pi^{BS}}{\partial S}-r\Pi^{BS}=0.$$ Now, differentiate $k$ times this equation with respect to $S$: $$\label{BSn}
-\frac{\partial^{k+1}\Pi^{BS}}{\partial t\partial S^k} +\frac{1}{2}\sigma^2S^2\frac{\partial^{k+2} \Pi^{BS}}{\partial S^{k+2}} +(r+k\sigma^2)S\frac{\partial^{k+1} \Pi^{BS}}{\partial S^{k+1}} +(k-1)(r+k\sigma^2/2)\frac{\partial^k \Pi^{BS}}{\partial S^k}=0.$$ One immediately checks that the term in bracket in (\[plop\]) is equal to the left term in (\[BSn\]) evaluated in $(S_te^{k\sigma^2(t-a)},T-t)$, and therefore is equal to 0.
$\Box$
Third step: Proof of proposition \[prop\]
-----------------------------------------
We argue by recurrence on the number $k$ of differentiations with respect to the dividends:
- If $k=0$, the proposition is trivially true as it simply says: $$\Pi(S_0,T,0,\ldots,0)=\Pi^{BS}(S,T).$$
- Now, suppose that the property is true at rank $k$. We want to prove that it remains true at rank $k+1$. We have by relation (\[fifi\]):
$$\begin{aligned}
\label{alice}
\frac{\partial^{k+1} \Pi}{\partial C_{i_1} \ldots \partial C_{i_{k+1}}}(S_0,0,\ldots,0)=& -e^{-rT_{i_1}}E\left[\frac{\partial^{k+1} h_{i_1}}{\partial S\partial C_{i_2} \ldots \partial C_{i_{k+1}}}(S_0X_{0\to
T_{i_1}},0,\ldots,0)\right].\end{aligned}$$
By hypothesis of recurrence, we have:
$$\begin{aligned}
& \frac{\partial ^{k}h_{i_{1}}}{\partial C_{i_{2}}\ldots \partial C_{i_{k+1}}}(S,0,\ldots ,0) \\
& =(-1)^k \frac{\partial ^{k}\Pi ^{BS}}{\partial S^{k}}\left( Se^{-\sigma
^{2}\sum_{q=2}^{k+1}(T_{i_{q}}-T_{i_{1}})},T-T_{i_{1}}\right) \times
e^{-r\sum_{q=2}^{k+1}(T_{i_{q}}-T_{i_{1}})-\sigma
^{2}\sum_{q=3}^{k+1}(q-2)(T_{i_{q}}-T_{i_{1}})}, \\
& =(-1)^k \frac{\partial ^{k}\Pi ^{BS}}{\partial S^{k}}\left( Se^{(k+1)\sigma
^{2}(T_{i_{1}}-a )},T-T_{i_{1}}\right) \times e^{\left[ (k+1)r+\frac{1}{2}k(k-1)\sigma ^{2}\right] T_{i_{1}}-(k+1)ra -\sigma
^{2}\sum_{q=3}^{k+1}(q-2)T_{i_{q}}},\end{aligned}$$
where we set: $$a =\frac{1}{k+1}\sum_{q=1}^{k+1}T_{i_{q}}.$$We differentiate with respect to $S$: $$\begin{aligned}
& \frac{\partial ^{k+1}h_{i_{1}}}{\partial S\partial C_{i_{2}}\ldots
\partial C_{i_{k+1}}}(S,0,\ldots ,0) \\
& =(-1)^k \frac{\partial ^{k+1}\Pi ^{BS}}{\partial S^{k+1}}\left( Se^{(k+1)\sigma
^{2}(T_{i_{1}}-a )},T-T_{i_{1}}\right) \times e^{(k+1)\left( r+\frac{1}{2}k\sigma ^{2}\right) T_{i_{1}}-(k+1)(r+\sigma ^{2})a -\sigma
^{2}\sum_{q=3}^{k+1}(q-2)T_{i_{q}}}.\end{aligned}$$
Inserting this formula into (\[alice\]) and using corollary \[cor\], we get: $$\begin{aligned}
& \frac{\partial ^{k+1}\Pi }{\partial C_{i_{1}}\ldots \partial C_{i_{k+1}}}(0,\ldots ,0) \\
=& (-1)^{k+1} \frac{\partial ^{k+1}\Pi ^{BS}}{\partial S^{k+1}}\left( Se^{-(k+1)\sigma
^{2}a },T\right) \exp \left( -(k+1)(r+\sigma ^{2})a -\sigma
^{2}\sum_{q=3}^{k+1}(q-2)T_{i_{q}}\right) , \\
=& (-1)^{k+1} \frac{\partial ^{k+1}\Pi ^{BS}}{\partial S^{k+1}}\left( Se^{-\sigma
^{2}\sum_{q=1}^{k+1}T_{i}},T\right) e^{-r\sum_{q=1}^{k+1}T_{i_{q}}-\sigma
^{2}\sum_{q=2}^{k+1}(q-1)T_{i_{q}}}.\end{aligned}$$
$\Box$
[9]{} M. Bos, S. Vandermark. *Finessing fixed dividends*, Risk, September 2002.
R. Bos, A. Gairat, A.Shepeleva. *Dealing with discrete dividends*, Risk, January 2002.
E.G. Haug et al. *Back to basics: a new approach to the discrete dividend problem*, Wilmott magazine, pp. 37-47, 2003.
M.H Vellekoop, J.W. Nieuwenhuis. *Efficient Pricing of Derivatives On Assets with Discrete Dividends*, Applied Mathematical Finance, Vol. 13, No. 3,265-284, September 2006.
[^1]: Arnaud Gocsei is a quantitative analyst in the Equity model validation team at Société Générale. E-mail: arnaud.gocsei@sgcib.com.
[^2]: Fouad Sahel is head of the Equity model validation team at Société Générale. E-mail: mohammed-fouad.sahel@sgcib.com.
[^3]: The opinions expressed in this article are those of the authors alone, and do not reflect the views of Société Générale, its subsidiaries or affiliates.
[^4]: We take here the terminology used in [@vellekoop].
[^5]: This becomes less true when considering large maturities and dividends.
[^6]: The right term in equation (\[callput\]) is not rigorously exact since $e^{-rT}E[S_{T}]$ is not equal to $S_{0}-\sum_{i=1}^{n}C_{i}e^{-rT_{i}}$, but the two quantities are very close.
|
---
abstract: 'To describe the tunneling dynamics of a stack of two-dimensional fermionic superfluids in an optical potential, we derive an effective action functional from a path integral treatment. This effective action leads, in the saddle point approximation, to equations of motion for the density and the phase of the superfluid Fermi gas in each layer. In the strong coupling limit (where bosonic molecules are formed) these equations reduce to a discrete nonlinear Schrödinger equation, where the molecular tunneling amplitude is reduced for large binding energies. In the weak coupling (BCS) regime, we study the evolution of the stacked superfluids and derive an approximate analytical expression for the Josephson oscillation frequency in an external harmonic potential. Both in the weak and intermediate coupling regimes the detection of the Josephson oscillations described by our path integral treatment constitutes experimental evidence for the fermionic superfluid regime.'
address: 'TFVS, Universiteit Antwerpen, Universiteitsplein 1, B2610 Antwerpen, Belgium.'
author:
- 'M. Wouters, J. Tempere$^{\ast }$, J. T. Devreese$^{\ast \ast }$'
date: 'December 5th, 2003.'
title: Path integral formulation of the tunneling dynamics of a superfluid Fermi gas in an optical potential
---
\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
Introduction
============
Recent experiments have demonstrated that both degenerate Fermi gases and Bose-Einstein condensates (BECs) can be loaded in one-dimensional optical lattices created by standing laser waves [AndersonScience282,CataliottiScience293,ArimondoPRL87,PhillipsJPB35,ModugnoPRA68]{}. The atoms are trapped in the valleys of the periodic potential, and form a stack of ‘pancake’ shaped clouds weakly coupled to each other. When the laser power is large enough, the gases in the separate valleys become quasi two-dimensional (quasi-2D). An additional parabolic potential, provided by external magnetic fields, and with corresponding oscillator length much larger than the period of the periodic potential, can be applied. In the ground state of this system, the fermionic and/or bosonic atoms are distributed in the lattice sites near the bottom of the additional parabolic trap.
The possibility to load a BEC in a periodic potential has led to the observation of the Mott-Insulator phase transition [@GreinerNAT415] and the detection of Bloch oscillations and Josephson currents through the potential barriers separating the layers of superfluid [CataliottiScience293,ArimondoPRL87,PhillipsJPB35]{}. To observe the Josephson effect, Cataliotti [*et al.*]{} suddenly displaced the additional parabolic potential, placing the stack of quasi-2D BECs out of equilibrium [CataliottiScience293]{}. Josephson currents allow the superfluid to tunnel between different layers and perform pendulum-like oscillations around the equilibrium position, driven by the external harmonic potential.
In Ref. [@CataliottiScience293], a critical Josephson current was found when the BEC was moved too far out of equilibrium, indicating the breakdown of superfluidity across the layers [@MenottiNJP5]. The existence of the Josephson currents is a direct manifestation of phase coherence across layers. For a one-dimensional array of dilute Fermi gases the superfluid regime is predicted to be accessible [@HofstetterPRL89], and also in this case the Josephson effect will be a signature of superfluidity.
In this paper, we derive an effective action, starting from the path integral representation of the partition function of the coupled quasi-2D layers filled with two different species of fermions at temperature zero. From this effective action, we derive equations of motion that allow us to study the dynamics of the phase and the density of the fermionic superfluid. Based on our results for the equations of motions of a fermionic superfluid in a one-dimensional potential, we describe what would happen if a fermionic superfluid instead of a BEC would be subject to the experiment of Ref. [CataliottiScience293]{} as illustrated in Fig. 1. We show that when the interatomic interactions are weak there are loosely bound BCS-pairs that can tunnel coherently through the barriers that separate the potential valleys. When the interatomic interaction becomes stronger, confinement induced quasi-2D bosonic molecules with high binding energy are formed and the superfluid becomes a Bose-Einstein condensate of these molecules and we calculate their tunneling energy.
The effective action
====================
In the derivation of the effective action, we follow rather closely the approach suggested by S. De Palo [*et al.* ]{}in Ref. [@DePaloPRB60] and start from the path integral representation of the partition function for a system consisting of layers of 2D fermions$$Z=\int {\cal D}\psi _{j,\sigma }^{\dag }(x){\cal D}\psi _{j,\sigma }(x)\exp
\left\{ -S\left[ \psi _{j,\sigma }^{\dag }(x),\psi _{j,\sigma }(x)\right]
\right\} ,$$where the action is given by$$\begin{aligned}
S\left[ \psi _{j,\sigma }^{\dag }(x),\psi _{j,\sigma }(x)\right] &=&\sum_{j}%
%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
\limits_{0}^{\beta }d\tau
%TCIMACRO{\dint }%
%BeginExpansion
\displaystyle\int
%EndExpansion
d^{2}{\bf x}\left[ \sum_{\sigma }\psi _{j,\sigma }^{\dag }\left( x\right)
\left( \partial _{\tau }-\frac{\nabla ^{2}}{2m}+V_{ext}\left( j\right) -\mu
\right) \psi _{j,\sigma }\left( x\right) \right. \nonumber \\
&&-U\psi _{j,\uparrow }^{\dag }\left( x\right) \psi _{j,\downarrow }^{\dag
}\left( x\right) \psi _{j,\downarrow }\left( x\right) \psi _{j,\uparrow
}\left( x\right) \nonumber \\
&&\left. +t_{1}\sum_{\sigma }\left( \psi _{j,\sigma }^{\dag }\left( x\right)
\psi _{j+1,\sigma }\left( x\right) +\psi _{j+1,\sigma }^{\dag }\left(
x\right) \psi _{j,\sigma }\left( x\right) \right) \right] . \label{S}\end{aligned}$$Here, $\beta =1/(k_{B}T)$ where $T$ denotes the temperature and $k_{B}$ the Boltzmann constant. The three-vector notation $x=\left( {\bf x,}\tau \right)
$ is used. The field $\psi _{j,\sigma }\left( x\right) $ belongs to a fermion of mass $m$ in layer $j$ and spin $\sigma $ ($\uparrow $ or $%
\downarrow $). The potential $V_{ext}\left( j\right) $ is an additional external potential that the fermions are subjected to. The attraction strength between the fermions is determined by $U$. The interlayer tunneling energy for a fermion is denoted by the real number $t_{1}$ that can be calculated with the approximate formula from Ref. [martikainencondmat0304]{} for the tunneling energy in a optical potential $%
V\left( z\right) =V_{0}\sin ^{2}\left( 2\pi z/\lambda \right) $ with wave length $\lambda $ and depth $V_{0}$:$$t_{1}=\frac{m\omega _{L}^{2}\lambda ^{2}}{8\pi ^{2}}\left[ \frac{\pi ^{2}}{4}%
-1\right] e^{-\left( \lambda /4\ell _{L}\right) ^{2}}, \label{t1}$$where $\omega _{L}=\sqrt{8\pi ^{2}V_{0}/\left( m\lambda ^{2}\right) }$ and $%
\ell _{L}=\sqrt{1/m\omega _{L}}$ are respectively the trapping frequency and the oscillator length that an atom feels in in de $z$-direction.
In appendix A, the reduction of (\[S\]) to an effective action is given and here we only give a summary. In order to grasp the most important part of the path integral in the superfluid state, the interaction between the fermions is decoupled by the Hubbard-Stratonovich (HS) transformation with the complex HS-field $\Delta _{j}^{HS}(x)$. After integration over the fermion fields, one is left with an effective action in terms of the HS-fields.
However, no information about the physical density of the system can be read off from such an effective action. In order have access to this quantity in the effective action, we introduce it by multiplying the partition function with the constant$$\begin{aligned}
C &=&\int {\cal D}\zeta _{j}^{HS}(x){\cal D}\rho _{j}(x)\exp \left\{
-\sum_{j}%
%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
\limits_{0}^{\beta }d\tau \int d^{2}{\bf x}\;i\zeta _{j}^{HS}\left( x\right)
\right. \nonumber \\
&&\left. \times \left[ \rho _{j}\left( x\right) -\psi _{j,\uparrow }^{\dag
}\left( x\right) \psi _{j,\uparrow }\left( x\right) -\psi _{j,\downarrow
}^{\dag }\left( x\right) \psi _{j,\downarrow }\left( x\right) \right]
\right\} . \label{eqC}\end{aligned}$$Carrying out the functional integral over $\zeta _{j}^{HS}\left( x\right) $ alone gives $\delta \lbrack \rho _{j}\left( x\right) -\psi _{j,\uparrow
}^{\dag }\left( x\right) \psi _{j,\uparrow }\left( x\right) -\psi
_{j,\downarrow }^{\dag }\left( x\right) \psi _{j,\downarrow }\left( x\right)
]$ and thus $\rho _{j}\left( x\right) $ corresponds to the physical density of the system along any path. Next, the complex field $\Delta _{j}^{HS}(x)$ is separated in a modulus and a phase. This phase is important for the low energy dynamics and therefore it is advantageous to introduce it explicitly $$\Delta _{j}^{HS}\left( x\right) =\left| \Delta _{j}^{HS}\left( x\right)
\right| e^{i\theta _{j}\left( x\right) }. \label{deltaexplicit}$$We then arrive at the following expression for the partition function $$Z\propto \int {\cal D}\left| \Delta _{j}^{HS}\left( x\right) \right| {\cal D}%
\zeta _{j}^{HS}(x){\cal D}\theta _{j}(x){\cal D}\rho _{j}(x)\exp \left[ -S_{%
\text{eff}}\right] ,$$where $S_{\text{eff}}$ is given in appendix A, expression (\[Seff\]).
To describe the low-energy dynamics of the density and the phase of the superfluid, the paths along which $\theta _{j}(x)$ and $\rho _{j}(x)$ vary slowly in comparison to the fermionic frequencies (Fermi energy and binding energy) will be of importance. Along these paths, we make a saddle point approximation for the remaining fields $\left| \Delta _{j}^{HS}(x)\right| $ and $\zeta _{j}^{HS}(x)$ in appendix B. The fluctuations $\delta \left|
\Delta _{j}^{HS}(x)\right| $ and $\delta \zeta _{j}^{HS}(x)$ around the saddle point values $\left| \Delta _{j}^{(0)}(x)\right| $ and $\zeta
_{j}^{(0)}(x)$ can be treated perturbatively. The saddle point value for the effective action is calculated in the appendix, and given by expression ([Seffsp]{}). The saddle point equations are $$\begin{aligned}
\frac{1}{U} &=&\int \frac{d^{2}{\bf k}}{\left( 2\pi \right) ^{2}}\frac{%
1-2n_{F}\left[ E_{j}(k)\right] }{2E_{j}\left( k\right) }, \label{egap0} \\
\rho _{j}(x) &=&\int \frac{d^{2}{\bf k}}{\left( 2\pi \right) ^{2}}\left(
\frac{k^{2}}{2m}-i\zeta _{j}^{(0)}(x)\right) \left\{ \frac{2n_{F}\left[
E_{j}(k)\right] -1}{E_{j}\left( k\right) }+1\right\} , \label{edens}\end{aligned}$$with $n_{F}(E)=1/(e^{\beta E}+1)$ the Fermi-Dirac distribution function and $%
E_{j}\left( k\right) $ the local BCS energy defined by$$E_{j}\left( k\right) =\sqrt{\left( \frac{k^{2}}{2m}-i\zeta
_{j}^{(0)}(x)\right) +\left| \Delta _{j}^{(0)}(x)\right| ^{2}}.
\label{BCSenergy}$$Equations (\[edens\]) and (\[BCSenergy\]) show that the saddle point value $\zeta _{j}^{(0)}(x)$ can be interpreted as a chemical potential $%
z_{j}(x)=i\zeta _{j}^{(0)}(x)$. The first saddle point equation (\[egap0\]) corresponds to the BCS gap equation, whereas the second saddle point equation leads to the BCS equation fixing the chemical potential $z_{j}(x)$ in layer $j$ as a function of the density $\rho _{j}(x)$ in layer $j$.
As we have introduced a momentum independent contact-interaction, equation (\[egap0\]) has to be regularized as described in [@houbiersPRA56], after which it becomes$$\frac{-1}{T_{00}\left( E\right) }=\int \frac{d^{2}{\bf k}}{\left( 2\pi
\right) ^{2}}\left[ \frac{2n_{F}\left[ E_{j}(k)\right] -1}{2E_{j}\left(
k\right) }-\frac{1}{k^{2}/m-E+i\varepsilon }\right] , \label{egap}$$where $T_{00}\left( E\right) $ is the low-momentum limit of the $T$-matrix. This equation has no ultraviolet divergences. In two dimensions, at low energy, $T_{00}\left( E\right) $ is given by [RanderiaPRB41,AdhikariAmJPhys54]{} $$\frac{1}{T_{00}\left( E\right) }=\frac{m}{4}\left[ \frac{-1}{\pi }\ln \left[
E/E_{b}\right] +i\right] , \label{Tmatrix}$$where $E_{b}$ is the energy of the 2D bound state that always exists in two dimensions. For the optical potential $V_{0}\sin ^{2}\left( 2\pi z/\lambda
\right) $ described earlier, the binding energy of the quasi-2D bound state is given by [@petrovPRA64]$$E_{b}=\frac{C\hbar \omega _{L}}{\pi }\exp \left( \sqrt{2\pi }\frac{\ell _{L}%
}{a}\right) , \label{Ebind}$$with $a$ the scattering length of the fermionic atoms and $C\approx 0.915.$
Josephson current at $T=0$
==========================
Equations of motion for density and phase
-----------------------------------------
We now proceed with an analysis at $T=0$ and with the assumption that the energy $t_{1}$ is small compared to the other energies, such that a perturbational expansion with $t_{1}$ as a small parameter is possible. In this case, equations (\[edens\]) and (\[egap\]) can be solved analytically for $\left\vert \Delta _{j}^{(0)}(x)\right\vert $ and $z_{j}(x)$ (see also [@RanderiaPRB41]) :$$\begin{aligned}
\left\vert \Delta _{j}^{(0)}(x)\right\vert &=&\sqrt{\frac{2\pi \rho _{j}(x)}{%
m}E_{b}}, \label{sp1} \\
z_{j}(x) &=&\frac{\pi \rho _{j}(x)}{m}-\frac{E_{b}}{2}. \label{sp2}\end{aligned}$$
As we want to study the current perpendicular to the layers in which the atoms are confined, we have to calculate the terms in the effective action that couple the different layers. In our perturbational expansion of the effective action (\[Seffsp\]), the lowest order contribution comes from the term in the self energy (\[Sigma\]) proportional to $t_{1}$. We find that the contribution equals$$\begin{aligned}
&&-2t_{1}^{2}\sum_{j}\frac{1}{\beta }\sum_{\omega _{n}=(2n+1)\pi /\beta
}\int \frac{d^{2}{\bf k}}{\left( 2\pi \right) ^{2}}\frac{1}{\left[ \omega
_{n}^{2}+E_{j+1}^{2}\left( k\right) \right] \left[ \omega
_{n}^{2}+E_{j}^{2}\left( k\right) \right] } \\
&&\times \left\{ -\omega _{n}^{2}+\xi _{j+1}\left( k\right) \xi _{j}\left(
k\right) +\left\vert \Delta _{j+1}^{(0)}(x)\right\vert \left\vert \Delta
_{j}^{(0)}(x)\right\vert \cos \left[ \theta _{j+1}(x)-\theta _{j}(x)\right]
\right\} ,\end{aligned}$$with $\xi _{j}\left( k\right) =k^{2}/\left( 2m\right) -z_{j}(x)$ the free fermion dispersion and $E_{j}$ the BCS dispersion relation (\[BCSenergy\]). In this expression, the part proportional to $\cos \left[ \theta
_{j+1}(x)-\theta _{j}(x)\right] $ is responsible for the Josephson tunneling between the layers and we write it symbolically as $$S_{\text{tunnel}}=-\sum_{j}\text{ }\int_{0}^{\beta }d\tau \int d{\bf x}\text{
}T_{j+1,j}\cos \left[ \theta _{j+1}(x)-\theta _{j}(x)\right] .
\label{Stunnel}$$Evaluating this term with the supposition that the gap $\left\vert \Delta
_{j}^{(0)}(x)\right\vert $ and chemical potential $z_{j}(x)$ vary slowly with $j$ leads to$$T_{j+1,j}=\frac{t_{1}^{2}m}{4\pi }\left( 1+\frac{z_{j}(x)}{\sqrt{\left\vert
\Delta _{j}^{(0)}(x)\right\vert ^{2}+z_{j}^{2}(x)}}\right) =\frac{%
t_{1}^{2}\rho _{j}(x)}{2\pi \rho _{j}(x)/m+E_{b}}.$$A similar contribution to the energy can be obtained in a BCS-approach, following Ref. [@TanakaphysicaC219].
We now take as an approximation for the effective action $$S_{\text{J}}\left[ \theta _{j}(x),\rho _{j}(x)\right] =S_{\text{eff}}^{\text{%
sp}}+S_{\text{tunnel}},$$with $S_{\text{eff}}^{\text{sp}}$ given by (\[Seffsp\]) from which $%
\left| \Delta _{j}^{0}(x)\right| $ and $\zeta _{j}^{0}(x)$ have been eliminated using equations (\[sp1\]),(\[sp2\]). Having obtained an expression for the action that only depends on $\theta _{j}(x)$ and $\rho
_{j}(x)$ we can finally proceed with deriving equations of motion by extremizing $S_{\text{J}}$ with respect to these fields. Because we want to give a dynamical interpretation to these equations, we write them in real time ($i\partial _{\tau }\rightarrow \partial _{t}$). Extremizing with respect to the phase field $\theta _{j}(x)$ results in $$\begin{aligned}
\partial _{t}\frac{\rho _{j}\left( x\right) }{2} &=&-\frac{{\bf \nabla }%
\theta _{j}\left( x\right) \cdot {\bf \nabla }\rho _{j}\left( x\right) }{4m}
\nonumber \\
&&+T_{j,j-1}\sin \left[ \theta _{j}(x)-\theta _{j-1}(x)\right] \nonumber \\
&&-T_{j+1,j}\sin \left[ \theta _{j+1}(x)-\theta _{j}(x)\right] ,
\label{eqcont}\end{aligned}$$and the derivative with respect to $\rho _{j}\left( x\right) $ yields$$\begin{aligned}
-\partial _{t}\frac{\theta _{j}\left( x\right) }{2} &=&\frac{\left[ \nabla
\theta _{j}\left( x\right) \right] ^{2}}{8m}+V_{ext}\left( j\right)
+z_{j}-\mu \nonumber \\
&&-\frac{\partial T_{j+1,j}}{\partial \rho _{j}\left( x\right) }\cos \left[
\theta _{j+1}(x)-\theta _{j}(x)\right] \nonumber \\
&&-\frac{\partial T_{j,j-1}}{\partial \rho _{j}\left( x\right) }\cos \left[
\theta _{j}(x)-\theta _{j-1}(x)\right] . \label{eEuler}\end{aligned}$$We have calculated the derivative $\partial T_{j+1,j}/\partial \rho _{j}$ if the density varies smoothly with the layer index ($\rho _{j+1}\approx \rho
_{j}\approx \rho _{j-1}$) and is constant in the plane:$$\frac{\partial T_{j+1,j}}{\partial \rho _{j}(x)}=\frac{t_{1}^{2}}{2}\frac{%
E_{b}}{\left[ 2\pi \rho _{j}(x)/m+E_{b}\right] ^{2}}.$$
Oscillations of the superfluid in an optical lattice
----------------------------------------------------
We now introduce the wave function $$\psi _{j}(t)=\sqrt{\frac{\rho _{j}(t)}{2}}e^{i\theta _{j}(t)},
\label{wavfie}$$with $\rho _{j}$ and $\theta _{j}$ only depending on time and layer index $j$. That is, we assume that within a layer, the density and the phase are homogeneous, but that they can still vary over time and over layers. The wave function (\[wavfie\]) obeys the Schrödinger equation$$\begin{aligned}
i\frac{d}{dt}\psi _{j} &=&\left( 2V_{ext}\left( j\right) +2z_{j}-2\mu
\right) \psi _{j}-\frac{t_{1}^{2}}{2\pi \rho _{j}/m+E_{b}}\psi _{j}\left(
e^{i\left( \theta _{j-1}-\theta _{j}\right) }+e^{i\left( \theta
_{j+1}-\theta _{j}\right) }\right) \nonumber \\
&&-\psi _{j}t_{1}^{2}\frac{4\pi \rho _{j}/m}{\left( E_{b}+2\pi \rho
_{j}/m\right) ^{2}}\left[ \cos \left( \theta _{j+1}-\theta _{j}\right) +\cos
\left( \theta _{j}-\theta _{j-1}\right) \right] , \label{eqpsitot}\end{aligned}$$where we can take in the approximation of slowly varying density that we used before, namely $\psi _{j-1}\approx \psi _{j}e^{i\left( \theta
_{j-1}-\theta _{j}\right) }$ and $\psi _{j+1}\approx \psi _{j}e^{i\left(
\theta _{j+1}-\theta _{j}\right) }$, so that we have in the limit $\pi \rho
_{j}/m\ll E_{b}$ that $$i\frac{d}{dt}\psi _{j}=\left( 2V_{ext}\left( j\right) +2z_{j}-2\mu \right)
\psi _{j}-\frac{t_{1}^{2}}{2\pi \rho _{j}/m+E_{b}}\left( \psi _{j-1}+\psi
_{j+1}\right) , \label{eqpsi}$$which is the nonlinear discrete Schrödinger equation for an array of Bose-Einstein condensates [@trombettoniPRL86], where the boson tunneling matrix element is given by $t_{1}^{2}/\left( 2\pi \rho _{j}/m+E_{b}\right) $. This is a decreasing function of $E_{b}$, which has the physical consequence that for increasing molecular binding energy the coupling between the layers becomes weaker and that in the Bose-Einstein limit ($%
E_{b}\rightarrow \infty $), no tunneling of molecules occurs. This may be related to the fact that the binding together of atoms in molecules is strongly influenced by the the confinement potential and the quasi-2D nature of the gas (as can be seen from eq. \[Ebind\]). In the intermediate states in the calculation of the amplitude of the tunneling process, the molecular state is unlikely to survive as a bound state. Hence, the molecular binding energy is to be added to the energy barrier for tunneling.
We know from [@CataliottiScience293], that [*in the bosonic limit*]{} equation (\[eqpsi\]) allows for oscillations where the phase difference $%
\theta _{j+1}-\theta _{j}$ is locked to a constant value $\Delta \theta $ and the differential equation governing the dynamics of the center of mass coordinate and this phase difference $\Delta \theta $ is of the pendulum type. Our numerical simulations of (\[eqpsitot\]) suggest that [*in the BCS-regime*]{} there is a similar current through the array.
>From equations (\[eqcont\]) and (\[eEuler\]), we can derive a simplified analytical formula to estimate the oscillation frequency in an external harmonic potential $V_{ext}\left( j\right) =\Omega j^{2}$. Assuming that the phase difference between neighboring layers is constant, this becomes$$\partial _{t}R=2\left\langle T_{j,j-1}\right\rangle \sin \left( \theta
_{j}-\theta _{j-1}\right) , \label{esimp1}$$where $$\left\langle T_{j,j-1}\right\rangle =\frac{1}{\sum_{j}\rho _{j}\left(
x\right) }\sum_{j}T_{j,j-1}$$is the average of $T_{j,j-1}$ over the lattice sites. The initial density profile $\rho _{j}(t=0)$ is calculated from the chemical potentials $z_{j}$ through (\[sp2\]) and these chemical potentials are derived from ([eEuler]{}), $z_{j}=\mu -V_{ext}\left( j\right) $, for a particular choice of $%
\mu $. The last two terms from equation (\[eqpsitot\]) seem to have only a minor influence on the frequency so that we omit them in the analytic calculation. Equation (\[eEuler\]) then becomes$$\partial _{t}\left( \theta _{j}-\theta _{j-1}\right) =-4\Omega R.
\label{esimp2}$$For small oscillations, (\[esimp1\]) and (\[esimp2\]) lead to the frequency $$\omega =\sqrt{8\Omega \left\langle T_{j,j-1}\right\rangle }. \label{wanal}$$
Taking $^{40}$K atoms with a central density of $n=10^{9}$ cm$^{-2},$ an optical wavelength $\lambda =754$ nm, and an axial frequency $\omega
_{a}=2\pi \times 24$ Hz, we plot in Fig. 2 the analytical frequency (curves) obtained from equation (\[wanal\]) and compare it with a numerical calculation (symbols) based on equation (\[eqpsitot\]). The oscillation frequency is plotted as a function of the inverse scattering length which appears in expression (\[Ebind\]) of the binding energy. Fig. 2 shows that the estimation (\[wanal\]) agrees almost perfectly with the numerical results in the bosonic limit, and that also in the BCS regime there is reasonable quantitative agreement. The inset of Fig. 2 shows that far in the bosonic regime ($a>0$), the oscillation frequency decreases rapidly as an exponential function of $1/a$. Nevertheless in the cross-over regime, the oscillation frequencies are high enough for the observation of Josephson currents to be useful as a tool to investigate the superfluidity of an ultracold Fermi gas, analogous to the experiments of Ref. [CataliottiScience293]{} in the bosonic case.
Conclusions
===========
We have derived an effective action and the resulting equations of motion to describe the dynamics of a fermionic superfluid in a layered system and applied this formalism to study the center of mass motion of an atomic Fermi gas in the potential formed by an optical standing wave. We find that a Fermi gas [*in the BCS regime* ]{}can perform superfluid oscillations through the optical lattice, similar to those that have been observed for condensates of bosonic atoms [CataliottiScience293,ArimondoPRL87,PhillipsJPB35]{}, when the gas is not in equilibrium in the harmonic trapping potential superimposed on the optical lattice. An analytical approximate expression (\[wanal\]) for the oscillation frequency is derived and the predictions of this expression are tested with numerical simulations of the full equations of motion ([eqpsitot]{}). For a superfluid Fermi gas [*in the BEC regime*]{}, we find that the tunneling is suppressed when the molecular binding energy becomes large. We conclude that superfluidity in Fermi gases can be revealed through Josephson currents in optical lattices if the Fermi gas is either in the BCS regime, or in the weakly-bound molecular BEC regime.
Acknowledgments
===============
We thank G. Modugno for useful discussions. Two of the authors (M. W. and J. T.) are supported financially by the Fund for Scientific Research - Flanders (Fonds voor Wetenschappelijk Onderzoek – Vlaanderen). This research has been supported financially by the FWO-V projects Nos. G.0435.03, G.0306.00, the W.O.G. project WO.025.99N.and the GOA BOF UA 2000, IUAP.
Also at: Lyman Laboratory of Physics, Harvard University, Cambridge, Massachusetts 02138.
Also at: Technische Universiteit Eindhoven, P. B. 513, 5600 MB Eindhoven, The Netherlands.
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Derivation of the effective action
==================================
In order to grasp the most important part of the path integral in the paired-fermion state, the interaction between the fermions is decoupled by the Hubbard-Stratonovich transformation, which transforms (\[S\]) into $$Z=\int {\cal D}\psi _{j,\sigma }^{\dag }(x){\cal D}\psi _{j,\sigma }(x){\cal %
D}\Delta _{j}^{HS}(x){\cal D}\Delta _{j}^{HS,\dagger }(x)\exp \left\{
-S^{\left( 1\right) }\right\} ,$$with$$\begin{aligned}
S^{\left( 1\right) } &=&\sum_{j}\int_{0}^{\beta }d\tau \int d^{2}{\bf x}%
\left[ \frac{\left\vert \Delta _{j}^{HS}\left( x\right) \right\vert ^{2}}{U}%
\right. \nonumber \\
&&+\sum_{\sigma =\pm 1}\psi _{j,\sigma }^{\dag }\left( x\right) \left(
\partial _{\tau }-\frac{\nabla ^{2}}{2m}+V_{ext}\left( j\right) -\mu \right)
\psi _{j,\sigma }\left( x\right) \nonumber \\
&&-\Delta _{j}^{HS}\left( x\right) \psi _{j,\uparrow }^{\dag }\left(
x\right) \psi _{j,\downarrow }^{\dag }\left( x\right) -\Delta _{j}^{HS,\dag
}\left( x\right) \psi _{j,\downarrow }\left( x\right) \psi _{j,\uparrow
}\left( x\right) \nonumber \\
&&+\left. t_{1}\sum_{\sigma }\left( \psi _{j,\sigma }^{\dag }\left( x\right)
\psi _{j+1,\sigma }\left( x\right) +\psi _{j+1,\sigma }^{\dag }\left(
x\right) \psi _{j,\sigma }\left( x\right) \right) \right] .\end{aligned}$$In order to keep track of the total density, we introduce the constant $C$ (see expression (\[eqC\])). In order to investigate the BCS gap and the phase we separate the complex field $\Delta _{j}^{HS}(x)$ in a modulus and a phase (expression (\[deltaexplicit\])) and also transform the fermion fields as $\psi _{j,\sigma }\left( x\right) \rightarrow \psi _{j,\sigma
}\left( x\right) e^{i\theta _{j}\left( x\right) /2}$. Additionally, we shift the field $i\zeta _{j}^{HS}\left( x\right) $ according to $$i\zeta _{j}^{HS}\left( x\right) \rightarrow i\zeta _{j}^{HS}\left( x\right)
+i\partial _{\tau }\frac{\theta _{j}\left( x\right) }{2}+\frac{\left( \nabla
\theta _{j}\left( x\right) \right) ^{2}}{8m}+V_{ext}\left( j\right) -\mu$$and use the Nambu spinor notation $\eta _{j}\left( x\right) =\left( \psi
_{j,\uparrow }\left( x\right) ,\psi _{j,\downarrow }^{\dag }\left( x\right)
\right) ^{T}$. After this procedure, the partition function can be written as $$Z\propto \int \left\vert \Delta _{j}^{HS}(x)\right\vert {\cal D}\eta
_{j}^{+}(x){\cal D}\eta _{j}(x){\cal D}\left\vert \Delta
_{j}^{HS}(x)\right\vert {\cal D}\theta _{j}(x){\cal D}\zeta _{j}^{HS}(x)%
{\cal D}\rho _{j}(x)\exp \left\{ -S^{\left( 2\right) }\right\} ,$$with the action $S^{\left( 2\right) }=S_{0}+S^{(3)}[\eta _{j}^{+}(x),\eta
_{j}(x)]$ where$$\begin{aligned}
S_{0} &=&\sum_{j}\int_{0}^{\beta }d\tau \int d^{2}{\bf x}\left\{ \frac{%
\left\vert \Delta _{j}^{HS}(x)\right\vert ^{2}}{U}\right. \nonumber \\
&&\left. +\left[ i\zeta _{j}^{HS}(x)+i\partial _{\tau }\frac{\theta
_{j}\left( x\right) }{2}+\frac{\left( \nabla \theta _{j}\left( x\right)
\right) ^{2}}{8m}+V_{ext}\left( j\right) -\mu \right] \rho _{j}\left(
x\right) \right\} \label{Snul}\end{aligned}$$does not contain the fermion fields any more and $$\begin{aligned}
S^{(3)} &=&\sum_{j}\int_{0}^{\beta }d\tau \int d^{2}{\bf x}\left\{ \eta
_{j}^{\dag }\left( x\right) \left[ \left( \partial _{\tau }-i\frac{\nabla
\theta _{j}\left( x\right) }{2m}\nabla -i\frac{\nabla ^{2}\theta _{j}\left(
x\right) }{4m}\right) \sigma _{0}\right. \right. \nonumber \\
&&+\left. \left( -\frac{\nabla ^{2}}{2m}-i\zeta _{j}^{HS}\left( x\right)
\right) \sigma _{3}-\left\vert \Delta _{j}^{HS}\left( x\right) \right\vert
\sigma _{1}\right] \eta _{j}\left( x\right) \nonumber \\
&&\left. +\left[ t_{1}\eta _{j}^{\dag }\left( x\right) e^{i\left( \theta
_{j+1}-\theta _{j}\right) \sigma _{3}/2}\sigma _{3}\eta _{j+1}\left(
x\right) +\text{h.c.}\right] \right\}\end{aligned}$$is the part of the action that still depends on them. The Pauli matrices are denoted by $\sigma _{i}$. Since $S^{(3)}$ is quadratic in the fermion fields, the path integral over these fields can be performed, resulting in $$\int {\cal D}\eta _{j}^{+}(x){\cal D}\eta _{j}(x)\exp \{-S^{(3)}\}=\det
[-G^{-1}],$$where the Green’s function $G$ is a matrix in coordinate space as in layer space and is given by$$\begin{aligned}
-G^{-1}\left( x,j;x^{\prime },j^{\prime }\right) &=&\delta \left(
x-x^{\prime }\right) \left\{ \delta _{jj^{\prime }}\left[ \left( \partial
_{\tau }-i\frac{\nabla \theta _{j}\left( x\right) }{2m}\nabla -i\frac{\nabla
^{2}\theta _{j}\left( x\right) }{4m}\right) \sigma _{0}\right. \right.
\nonumber \\
&&\left. +\left( -\frac{\nabla ^{2}}{2m}-i\zeta _{j}^{HS}(x)\right) \sigma
_{3}-\left\vert \Delta _{j}^{HS}(x)\right\vert \sigma _{1}\right] \nonumber
\\
&&\left. +\delta _{j+1,j^{\prime }}t_{1}e^{i\left( \theta _{j+1}-\theta
_{j}\right) \sigma _{3}/2}\sigma _{3}+\delta _{j-1,j^{\prime
}}t_{1}e^{-i\left( \theta _{j+1}-\theta _{j}\right) \sigma _{3}/2}\sigma
_{3}\right\} . \label{Greens}\end{aligned}$$The partition sum can then be written as $$Z\propto \int {\cal D}\left\vert \Delta _{j}^{HS}(x)\right\vert {\cal D}%
\theta _{j}(x){\cal D}\zeta _{j}^{HS}(x){\cal D}\rho _{j}(x)\exp \left\{ -S_{%
\text{eff}}\right\} \label{Z4}$$where $$S_{\text{eff}}=S_{0}+\text{Tr}\left[ \ln (-G^{-1})\right] \label{Seff}$$with $S_{0}$ given by (\[Snul\]) and the Green’s function given by ([Greens]{}).
Saddle point expansion for the effective action
===============================================
>From the path integrations in (\[Z4\]), we ultimately want to extract equations of motion for $\theta _{j}(x),$ $\rho _{j}(x)$. To describe the low-energy dynamics, the contributions with $\theta _{j}(x),$ $\rho _{j}(x)$ varying slow in comparison with the fermionic frequencies (Fermi energy and binding energy) will be important. Along the paths of slowly varying $\theta
_{j}(x),$ $\rho _{j}(x)$, we can make the saddle point expansion in the fields $\left| \Delta _{j}^{HS}\right| ,\zeta _{j}^{HS}$, setting$$\begin{aligned}
\left| \Delta _{j}^{HS}(x)\right| &=&\left| \Delta _{j}^{(0)}\left( x\right)
\right| +\delta \left| \Delta _{j}^{HS}(x)\right| , \\
\zeta _{j}^{HS}(x) &=&\zeta _{j}^{(0)}\left( x\right) +\delta \zeta
_{j}^{HS}(x),\end{aligned}$$where also $\left| \Delta _{j}^{(0)}\left( x\right) \right| $ and $\zeta
_{j}^{(0)}\left( x\right) $ will vary slowly in comparison to the fermion frequencies.
Expanding the Green’s function (\[Greens\]) around the saddle point values $\left| \Delta _{j}^{(0)}\right| $ and $\zeta _{j}^{(0)}=-iz_{j}$ leads to $$G^{-1}=G_{0}^{-1}+\Sigma ,$$where the saddle point contribution is given by $$-G_{0}^{-1}\left( x,j;x^{\prime },j^{\prime }\right) =\delta \left(
x-x^{\prime }\right) \delta _{jj^{\prime }}\left[ \partial _{\tau }\sigma
_{0}+\left( -\frac{\nabla ^{2}}{2m}-z_{j}\right) \sigma _{3}-\left| \Delta
_{j}^{(0)}\right| \sigma _{1}\right] .$$This saddle point contribution can be diagonalized by going to Fourier space : $$G_{0}^{-1}\left( {\bf k,}\omega ,j;{\bf k}^{\prime },\omega ^{\prime
},j^{\prime }\right) =\delta _{jj^{\prime }}\delta _{\omega \omega ^{\prime
}}\delta \left( {\bf k}-{\bf k}^{\prime }\right) \left[ i\omega \sigma
_{0}-\left( \frac{k^{2}}{2m}-z_{j}\right) \sigma _{3}+\left| \Delta
_{j}^{(0)}\right| \sigma _{1}\right] .$$and thus$$\begin{aligned}
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \ln \left[ -G_{0}^{-1}\right] \right] &=&\sum_{j}\frac{1}{\beta }%
\sum_{\omega _{n}=(2n+1)\pi /\beta }\int \frac{d^{2}{\bf k}}{(2\pi )^{2}}%
\text{ } \nonumber \\
&&\times \ln \left[ -\omega _{n}^{2}-\left[ k^{2}/(2m)-z_{j}(x)\right]
^{2}+\left| \Delta _{j}^{(0)}(x)\right| ^{2}\right] . \label{G0}\end{aligned}$$In this expression, the $x$ dependence of $z_{j}(x)$ and $\left| \Delta
_{j}^{(0)}(x)\right| $ has been reintroduced, still assuming that these fields vary slowly in comparison to the relevant fermion frequencies. The self energy $\Sigma $ equals$$\begin{aligned}
-\Sigma \left( x,j;x^{\prime },j^{\prime }\right) &=&\delta \left(
x-x^{\prime }\right) \delta _{jj^{\prime }}\left[ -i\sigma _{3}\delta \zeta
_{j}^{HS}(x)-(\delta \left| \Delta _{j}^{HS}(x)\right| )\sigma _{1}\right.
\nonumber \\
&&\left. -\left( i\frac{\nabla \theta _{j}}{2m}\nabla +i\frac{\nabla
^{2}\theta _{j}}{4m}\right) \sigma _{0}\right] \nonumber \\
&&+\delta \left( x-x^{\prime }\right) \left[ \delta _{j+1,j^{\prime
}}t_{1}e^{i\left[ \theta _{j+1}(x)-\theta _{j}(x)\right] \sigma
_{3}/2}\sigma _{3}\right. \nonumber \\
&&\left. +\delta _{j-1,j^{\prime }}t_{1}e^{-i\left[ \theta _{j+1}(x)-\theta
_{j}(x)\right] \sigma _{3}/2}\sigma _{3}\right] , \label{Sigma}\end{aligned}$$so that$$\begin{aligned}
\det \left[ -G^{-1}\right] &=&\exp \left\{
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \ln \left( -G^{-1}\right) \right] \right\} =\exp \left\{
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \ln \left( -G_{0}^{-1}-\Sigma \right) \right] \right\} \nonumber \\
&=&\exp \left\{
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \ln \left( -G_{0}^{-1}\right) \right] -\sum_{n=1}^{\infty }\frac{%
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \left( -G_{0}^{-1}\Sigma \right) ^{n}\right] }{n}\right\} , \label{G}\end{aligned}$$where $%
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
$ denotes the trace over all the variables (coordinates, imaginary time, layer index and Nambu space). Expanding the action, $S_{\text{eff}}$, expression (\[Seff\]), around the saddle point values $\left| \Delta
_{j}^{(0)}(x)\right| $ and $z_{j}(x)=i\zeta _{j}^{(0)}(x)$, and using the result (\[G\]) to lowest order, we find $$\begin{aligned}
S_{\text{eff}}^{\text{sp}} &=&%
%TCIMACRO{\limfunc{Tr}}%
%BeginExpansion
\mathop{\rm Tr}%
%EndExpansion
\left[ \ln \left( -G_{0}^{-1}\right) \right] +\sum_{j}\int_{0}^{\beta }d\tau
\int d^{2}{\bf x}\left\{ \frac{\left| \Delta _{j}^{(0)}(x)\right| ^{2}}{U}%
\right. \nonumber \\
&&\left. +\left[ z_{j}(x)+i\partial _{\tau }\frac{\theta _{j}\left( x\right)
}{2}+\frac{\left( \nabla \theta _{j}\left( x\right) \right) ^{2}}{8m}%
+V_{ext}\left( j\right) -\mu \right] \rho _{j}\left( x\right) \right\}
\label{Seffsp}\end{aligned}$$The saddle point values $\left| \Delta _{j}^{\left( 0\right) }(x)\right| $ and $z_{j}(x)=i\zeta _{j}^{(0)}(x)$ are now equal to the extremum of ([Seffsp]{}) and they fulfill the equations (\[egap0\]) and (\[edens\]). The lowest order Green’s function $G_{0}$ is given by (\[G0\]).
|
---
---
[**Additive and product properties of Drazin inverses of elements in a ring**]{}\
[ **Abstract:**]{} 0truemm0truemm We study the Drazin inverses of the sum and product of two elements in a ring. For Drazin invertible elements $a$ and $b$ such that $a^2b=aba$ and $b^2a=bab$, it is shown that $ab$ is Drazin invertible and that $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. Moreover, the formulae of $(ab)^D$ and $(a+b)^D$ are presented. Thus, a generalization of the main result of Zhuang, Chen et al. (Linear Multilinear Algebra 60 (2012) 903-910) is given.\
[ **Keywords:**]{} Drazin inverse, group inverse, spectral idempotent, ring\
[ **AMS Subject Classifications:**]{} 15A09, 16U80
**Introduction**
=================
Throughout this paper, $R$ is an associative ring with unity $1$. The symbols $R^D$, $R^{\rm nil}$ denote, the sets of Drazin invertible elements, nilpotent elements of $R$, respectively. The commutant of an element $a\in R$ is defined as ${\rm comm}(a)=\{x\in R: ax=xa\}$. An element $a\in R$ is Drazin invertible if there exists $b\in R$ such that $$b\in {\rm comm}(a),~ bab=b, ~a^k=a^{k+1}b \eqno(1.1)$$ for some positive integer $k$. The such least $k$ is called the Drazin index of $a$, denoted by $k={\rm ind}(a)$. If ${\rm ind}(a)=1$, then $b$ is called the group inverse of $a$ and denoted by $a^\#$.
The conditions in (1.1) are equivalent to $$b\in {\rm comm}(a), ~bab=b, ~a-a^2b\in R^{\rm nil}.\eqno(1.2)$$ Drazin \[7\] proved that if $a$ is Drazin invertible and $ab=ba$, then $a^Db=ba^D$. By $a^\pi=1-aa^D$ we mean the spectral idempotent of $a$.
The problem of Drazin inverse of the sum of two Drazin invertible elements was first considered by Drazin in his celebrated paper \[7\]. It was proved that $(a+b)^D=a^D+b^D$ under the condition that $ab=ba=0$ in associative rings. It is well known that the product $ab$ of two commutative Drazin invertible elements $a$, $b$ is Drazin invertible and $(ab)^D=a^Db^D=b^Da^D$ in a ring. In 2011, Wei and Deng \[12\] considered the relations between the Drazin inverses of $A+B$ and $1+A^DB$ for two commutative complex matrices $A$ and $B$. For two commutative Drazin invertible elements $a,b\in R$,Zhuang, Chen et al. \[16\] proved that $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. Moreover, the representation of $(a+b)^D$ was obtained. More results on Drazin inverse can be found in \[1-6,8,11-16\].
For any elements $a, b\in R$, $ab=ba$ implies that $a^2b=aba$ and $b^2a=bab$. However, the converse need not be true. For example, take
$a=\left(
\begin{array}{cc}
1 & 0 \\
0 & 0\\
\end{array}
\right)
$, $b=\left(
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}
\right)
$.
It is easy to get $a^2b=aba$ and $b^2a=bab$. However, $ab\neq ba$. Under the conditions $P^2Q=PQP$ and $Q^2P=QPQ$, Liu, Wu and Yu \[9\] characterized the relations between the Drazin inverses of $P+Q$ and $1+P^DQ$ for complex matrices $P$ and $Q$ by using the methods of splitting complex matrices into blocks. In this paper, we extend the results in \[9\] to a ring $R$. For $a, b\in R^D$, it is shown that $ab\in R^D$ and that $a+b\in R^D$ if and only if $1+a^Db\in R^D$ under the conditions $a^2b=aba$ and $b^2a=bab$. Moreover, the expressions of $(ab)^D$ and $(a+b)^D$ are presented. Consequently, some results in \[9,12,16\] can be deduced from our results.
**Some lemmas**
===============
In this section, we start with some useful lemmas.
Let $a,b\in R$ with $a^2b=aba$. Then for any positive integer $i$, the following hold:\
$(1)$ $ a^{i+1}b =a^iba=aba^i,$ $a^{2i}b=a^iba^i,$$(2.1)$\
$(2)$ $(ab)^i=a^ib^i.$$(2.2)$
\(1) Since $a^2b=aba$, we have $a^{i+1}b=a^{i-1}a^2b=a^{i-1}aba=a^iba$. This shows that $a^ib\in {\rm comm}(a)$ and $a^ib\in {\rm comm}(a^i)$.
From $ab\in {\rm comm}(a)$, it follows that $ab\in {\rm comm}(a^i)$, i.e., $aba^i=a^{i+1}b$. Thus, $a^{i+1}b =a^iba=aba^i$.
Therefore, we also obtain $a^{2i}b=a^iba^i$ by $a^ib\in {\rm comm}(a^i)$.
\(2) follows by induction.
Let $a,b\in R$ with $a^2b=aba$ and $b^2a=bab$.\
$(1)$ If $a$ or $b$ is nilpotent, then $ab$ and $ba$ are nilpotent.\
$(2)$ If $a$ and $b$ are nilpotent, then $a+b$ is nilpotent.
\(1) By Lemma 2.1(2).
\(2) Note that $(a+b)^k=\displaystyle{\sum_{i=0}^{k-1}}C_{k-1}^i(a^{k-i}b^i+b^{k-i}a^i)$ in \[9\]. We have $(a+b)^k=0$ by taking $k={\rm ind}(a)+{\rm ind}(b)$.
Let $a,b \in R$ with $a^2b=aba$ and $b^2a=bab$. If $a \in R^D$, then\
$(1)$ $(a^D)^2b=a^Dba^D,$$(2.3)$\
$(2)$ $b^2a^D=ba^Db.$$(2.4)$
\(1) Since $a^2b=aba$, we have $ab\in {\rm comm}(a^D)$ by \[7, Theorem 1\]. Hence,
$(a^D)^2b=(a^D)^2a^Dab=(a^D)^2aba^D=a^Dba^D$.
\(2) Note that $ab\in {\rm comm}(a^D)$. It follows that
$b^2a^D=b^2a(a^D)^2=bab(a^D)^2=b(a^D)^2ab=ba^Db$.
Let $a,b \in R^D$ with $a^2b=aba$ and $b^2a=bab$. Then\
$(1)$ $\{ab, a^Db, ab^D, a^Db^D\} \subseteq {\rm comm}(a),$ $(2.5)$\
$(2)$ $ \{ba,b^Da,ba^D,b^Da^D\} \subseteq {\rm comm}(b)$. $(2.6)$
It is enough to prove (1) since we can obtain (2) by the symmetry of $a$ and $b$.
\(1) By hypothesis, we obtain $ab\in {\rm comm}(a)$ and
$aa^Db=(a^D)^2a^2b=(a^D)^2aba=a^Dba$.
Since $ba\in {\rm comm}(b)$ implies that $ba\in {\rm comm}(b^D)$, it follows that
$ab^Da=a(b^D)^2ba=aba(b^D)^2=a^2b(b^D)^2=a^2b^D$.
Note that $ab^Da=a^2b^D$. We get
$aa^Db^D=(a^D)^2a^2b^D=(a^D)^2ab^Da=a^Db^Da$.
Let $a, b\in R^D$ and $\xi=1+a^Db$. If $a^2b=aba$ and $b^2a=bab$, then $\{a, a^D, ab, a^Db, ab^D, a^Db^D\} \subseteq {\rm comm}(\xi).\hfill(2.7)$
Since $a^Db\in {\rm comm}(a)$, we have $a\in {\rm comm}(\xi)$. By \[7, Theorem 1\], $a^D\in {\rm comm}(\xi)$.
Observing that $(a^Db)ab \stackrel{(2.5)}{=}a(a^Db)b=a^D(ab)b=ab(a^Db)$, it follows that $ab\in {\rm comm}(\xi)$.
Similarly, $a^Db, ab^D, a^Db^D\in {\rm comm}(\xi)$.
Hence, $\{a, a^D, ab, a^Db, ab^D, a^Db^D\} \subseteq {\rm comm}(\xi)$.
Let $a, b\in R^D$ with $a^2b=aba$ and $b^2a=bab$. Then for any positive integer $i$, the following hold:\
$(1)$ $ab^Db^Da=(ab^D)^2=a^2(b^D)^2$, $(2.8)$\
$(2)$ $(ab^D)^{i+1}=ab^D(b^Da)^i=a^{i+1}(b^D)^{i+1}$, $(2.9)$\
$(3)$ $(a^Db)^{i+1}=a^Db(ba^D)^i=(a^D)^{i+1}b^{i+1}$,$(2.10)$\
$(4)$ $ba^Da^Db=(ba^D)^2=b^2(a^D)^2$, $(2.11)$\
$(5)$ $(ba^D)^{i+1}=ba^D(a^Db)^i=b^{i+1}(a^D)^{i+1}$, $(2.12)$\
$(6)$ $(b^Da)^{i+1}=b^Da(ab^D)^i=(b^D)^{i+1}a^{i+1}$.$(2.13)$
It is sufficient to prove (1)-(3).
\(1) We have $$ab^Db^Da=ab^D(b^Da)\stackrel{(2.6)}{=}a(b^Da)b^D=(ab^D)^2.$$
According to Lemma 2.4, we get $$ab^Db^Da=ab^D(ab^D)\stackrel{(2.5)}{=}a(ab^D)b^D=a^2(b^D)^2.$$
\(2) It is just (1) for $i=1$. Assume that the equality holds for $i=k$, i.e., $$(ab^D)^{k+1}=ab^D(b^Da)^k=a^{k+1}(b^D)^{k+1}.$$ For $i=k+1$, $$\begin{aligned}
(ab^D)^{k+2}=ab^D(ab^D)^{k+1}=ab^Dab^D(b^Da)^k \stackrel{(2.8)}{=}ab^Db^Da (b^Da)^k=ab^D(b^Da)^{k+1}\end{aligned}$$ and
$(ab^D)^{k+2}=ab^D(ab^D)^{k+1}=ab^Da^{k+1}(b^D)^{k+1}\stackrel{(2.5)}{=}a^{k+1}ab^D(b^D)^{k+1}= a^{k+2}(b^D)^{k+2}$.
Thus, (2) holds for any positive integer $i$.
\(3) Its proof is similar to (2).
Let $a, b\in R^D$ with $a^2b=aba$ and $b^2a=bab$. If $a_1=b^\pi a^\pi b$ and $a_2=bb^Daa^\pi$, then $a_1-a_2$ is nilpotent.
Firstly, we prove $a_1=b^\pi a^\pi b$ is nilpotent. According to Lemma 2.6, we have the following equalities $$b^\pi ba^\pi b=b^2b^\pi a^\pi \eqno(2.14)$$ and $$b a^\pi b^\pi =bb^\pi a^\pi.\eqno(2.15)$$
Hence, we obtain $$\begin{aligned}
(b^\pi a^\pi b)^3&=& b^\pi a^\pi( b b^\pi a^\pi b )b^\pi a^\pi b \stackrel{(2.14)}{=} b^\pi a^\pi( b^\pi b^2 a^\pi )b^\pi a^\pi b \\
&=& b^\pi a^\pi b^\pi (b^2 a^\pi b^\pi) a^\pi b \stackrel{(2.15)}{=}b^\pi a^\pi b^\pi (b^2 b^\pi a^\pi ) a^\pi b \\
&=& b^\pi a^\pi (b b^\pi)^2 a^\pi a^\pi b= b^\pi a^\pi (b b^\pi)^2 a^\pi b.\end{aligned}$$
By induction, $(b^\pi a^\pi b)^{n+1}=b^\pi a^\pi (bb^\pi)^n a^\pi b$. Since $bb^\pi$ is nilpotent, $b^\pi a^\pi b=a_1$ is nilpotent.
Secondly, we show that $a_2=bb^Daa^\pi$ is nilpotent. As $$\begin{aligned}
(bb^D aa^\pi)^2 &=& (bb^D aa^\pi)(bb^D aa^\pi)= bb^Da(1-aa^D)bb^Da(1-aa^D)\\
&=& bb^D(1-aa^D)abb^Da(1-aa^D)\\
&\stackrel{(2.5)}{=}& bb^Dab(1-aa^D)b^Da(1-aa^D)\\
&\stackrel{(2.6)}{=}& bbb^Da(1-aa^D)b^Da(1-aa^D)\\
&=& bbb^D(1-aa^D)ab^Da(1-aa^D)\\
&\stackrel{(2.5)}{=}& bbb^Dab^D(1-aa^D)a(1-aa^D)\\
&\stackrel{(2.6)}{=}& bbb^Db^Da(1-aa^D)a(1-aa^D)\\
&=& bb^Da(1-aa^D)a(1-aa^D)\\
&=& bb^D (aa^\pi)^2,\end{aligned}$$ by induction, $(bb^D aa^\pi)^n=bb^D (aa^\pi)^n$. Since $aa^\pi$ is nilpotent, $bb^D aa^\pi=a_2$ is nilpotent.
Finally, we prove that $a_1-a_2$ is nilpotent.
Since
$a_1^2a_2= b^\pi a^\pi b b^\pi a^\pi b bb^Daa^\pi\stackrel{(2.6)}{=} b^\pi a^\pi bb^\pi b^2b^Daa^\pi=0$
and $$\begin{aligned}
a_2a_1&=& bb^Daa^\pi b^\pi a^\pi b=bb^Da^\pi (a b^\pi) a^\pi b\\
&\stackrel{(2.5)}{=}& bb^D(a b^\pi) a^\pi a^\pi b=bb^D(a b^\pi) a^\pi b\\
&=& b(b^Da) b^\pi a^\pi b\stackrel{(2.6)}{=}b b^\pi (b^Da) a^\pi b\\
&=&0,\end{aligned}$$
we have $a_1^2a_2=a_1a_2a_1=0$ and $a_2^2a_1=a_2a_1a_2=0$.
As $a_1$ and $a_2$ are nilpotent, $b^\pi a^\pi b-bb^Daa^\pi=a_1-a_2$ is nilpotent by Lemma 2.2(2).
Let $a, b\in R^D$ with $a^2b=aba$ and $b^2a=bab$ and $\xi=1+a^Db\in R^D$. Suppose $b_1=a\xi\xi^\pi+\xi^Daa^\pi$ and $b_2=b^\pi a^\pi b-bb^Daa^\pi$. Then $b_1+b_2$ is nilpotent.
Since $a\xi\xi^\pi$, $\xi^Daa^\pi$ are nilpotent and $a\xi \xi^\pi$ commutes with $\xi^Daa^\pi$, it follows that $b_1=a\xi\xi^\pi+\xi^Daa^\pi$ is nilpotent. By Lemma 2.7, $a_1-a_2=b_2$ is nilpotent.
First we give two useful equalities $$aa^\pi bb^\pi a=a^\pi ab^\pi (ba) \stackrel{(2.6)}{=}a^\pi (ab)ab^\pi \stackrel{(2.5)}{=}a^2a^\pi bb^\pi\eqno(2.16)$$ and $$bab^\pi a^\pi b a \stackrel{(2.5)}{=}ba^\pi ab^\pi ba=ba^\pi (abb^\pi)a\stackrel{(2.5)}{=}b(abb^\pi)a^\pi a=(ba)bb^\pi aa^\pi\stackrel{(2.6)}{=}b^2b^\pi a^2a^\pi. \eqno(2.17)$$
According to Lemma 2.5, we have $$\begin{aligned}
b_1^2b_2&=&(a\xi \xi^\pi+\xi^Daa^\pi)^2(b^\pi a^\pi b -bb^Daa^\pi)\\
&\stackrel{(2.7)}{=}& (a^2\xi^2\xi^\pi+(\xi^D)^2a^2a^\pi)(b^\pi a^\pi b-bb^Daa^\pi)\\
&=& a^2\xi^2\xi^\pi b^\pi a^\pi b -a^2\xi^2\xi^\pi bb^Daa^\pi+(\xi^D)^2a^2a^\pi b^\pi a^\pi b-(\xi^D)^2a^2a^\pi bb^Daa^\pi\\
&\stackrel{(2.7)}{=}& a^2a^\pi\xi^2\xi^\pi bb^\pi -a^3a^\pi\xi^2\xi^\pi bb^D+(\xi^D)^2a^2a^\pi bb^\pi-(\xi^D)^2a^3a^\pi bb^D.\end{aligned}$$
By Lemma 2.4 and Lemma 2.5, we obtain $$\begin{aligned}
&&b_1b_2b_1=(a\xi\xi^\pi+\xi^Daa^\pi)(b^\pi a^\pi b -bb^Daa^\pi)(a\xi\xi^\pi+\xi^Daa^\pi)\\
&=&(a\xi\xi^\pi b^\pi a^\pi b -a\xi\xi^\pi bb^Daa^\pi +\xi^Daa^\pi b^\pi a^\pi b-\xi^Da^2a^\pi bb^D)(a\xi\xi^\pi+\xi^Daa^\pi)\\
&=& a\xi\xi^\pi b^\pi a^\pi b a\xi\xi^\pi-a\xi\xi^\pi bb^Daa^\pi a\xi\xi^\pi+\xi^Daa^\pi b^\pi a^\pi b a\xi\xi^\pi-\xi^Da^2a^\pi bb^Da\xi\xi^\pi \\
&&+a\xi\xi^\pi b^\pi a^\pi b \xi^Daa^\pi -a\xi\xi^\pi bb^Daa^\pi \xi^Daa^\pi+\xi^Daa^\pi b^\pi a^\pi b\xi^Daa^\pi- \xi^Da^2a^\pi bb^D\xi^D aa^\pi \\
&\stackrel{(2.7)}{=}& a\xi\xi^\pi a\xi\xi^\pi b^\pi a^\pi b -a\xi\xi^\pi aa^\pi a\xi\xi^\pi bb^D+\xi^Daa^\pi a^\pi a\xi\xi^\pi b^\pi b -\xi^Da^2a^\pi a\xi\xi^\pi bb^D \\
&&+a\xi\xi^\pi a^\pi \xi^Daa^\pi b^\pi b -a\xi\xi^\pi aa^\pi \xi^Daa^\pi bb^D+\xi^Daa^\pi a^\pi \xi^Daa^\pi b^\pi b- \xi^Da^2a^\pi \xi^D aa^\pi bb^D \\
&\stackrel{(2.16)}{=}& a^2a^\pi\xi^2\xi^\pi bb^\pi -a^3a^\pi\xi^2\xi^\pi bb^D+(\xi^D)^2a^2a^\pi bb^\pi-(\xi^D)^2a^3a^\pi bb^D.\end{aligned}$$
Hence, $b_1^2b_2=b_1b_2b_1$.
In view of Lemma 2.7 and $b^Db^\pi=0$, we have $$\begin{aligned}
b_2^2b_1&=&(b^\pi a^\pi b -bb^Daa^\pi)^2(a\xi \xi^\pi +\xi^Daa^\pi)\\
&\stackrel{(2.6)}{=}&(b^\pi a^\pi b^2b^\pi a^\pi-b^\pi a^\pi b^2b^Daa^\pi +bb^Da^2a^\pi)(a\xi \xi^\pi +\xi^Daa^\pi)\\
&=&b^\pi a^\pi b^2b^\pi aa^\pi\xi\xi^\pi-b^\pi a^\pi b^2b^Daa^\pi a\xi\xi^\pi+b^2b^Da^2a^\pi a\xi\xi^\pi\\
&&+b^\pi a^\pi b^2b^\pi a^\pi\xi^Daa^\pi-b^\pi a^\pi b^2b^Daa^\pi\xi^Daa^\pi+bb^Da^2a^\pi\xi^Daa^\pi\\
&\stackrel{(2.7)}{=}& b^\pi a^\pi b^2b^\pi aa^\pi\xi\xi^\pi-b^\pi a^\pi b^2b^Da^2a^\pi\xi\xi^\pi+b^2b^Da^3a^\pi \xi\xi^\pi\\
&&+b^\pi a^\pi b^2b^\pi aa^\pi\xi^D-b^\pi a^\pi b^2b^Da^2a^\pi\xi^D+bb^Da^3a^\pi\xi^D,\end{aligned}$$ and $$\begin{aligned}
b_2b_1b_2&=&(b^\pi a^\pi b -bb^Daa^\pi)(a\xi \xi^\pi +\xi^Daa^\pi)(b^\pi a^\pi b -bb^Daa^\pi)\\
&\stackrel{(2.7)}{=}&(b^\pi a^\pi b a\xi \xi^\pi+b^\pi a^\pi b aa^\pi\xi^D -bb^Da^2a^\pi \xi \xi^\pi-bb^Da^2a^\pi\xi^D)(b^\pi a^\pi b -bb^Daa^\pi)\\
&=&b^\pi a^\pi b a\xi \xi^\pi b^\pi a^\pi b +b^\pi a^\pi b aa^\pi\xi^D b^\pi a^\pi b -bb^Da^2a^\pi \xi \xi^\pi b^\pi a^\pi b\\
&&-bb^Da^2a^\pi\xi^D b^\pi a^\pi b-b^\pi a^\pi b a\xi \xi^\pi bb^Daa^\pi -b^\pi a^\pi ba a^\pi\xi^D bb^Daa^\pi \\
&&+bb^Da^2a^\pi \xi\xi^\pi bb^Daa^\pi +bb^Da^2a^\pi \xi^D bb^Daa^\pi\\
&\stackrel{(2.6)}{=}&b^\pi a^\pi b a\xi \xi^\pi b^\pi a^\pi b +b^\pi a^\pi b aa^\pi\xi^D b^\pi a^\pi b -bb^Db^\pi a^2a^\pi \xi \xi^\pi a^\pi b \\
&&-bb^D b^\pi a^2a^\pi\xi^D a^\pi b-b^\pi a^\pi b a\xi \xi^\pi bb^Daa^\pi -b^\pi a^\pi ba a^\pi\xi^D bb^Daa^\pi \\
&&+bb^Da^2a^\pi \xi\xi^\pi bb^Daa^\pi +bb^Da^2a^\pi \xi^D bb^Daa^\pi\\
&=& b^\pi a^\pi b a\xi \xi^\pi b^\pi a^\pi b +b^\pi a^\pi b aa^\pi\xi^D b^\pi a^\pi b -b^\pi a^\pi b a\xi \xi^\pi bb^Daa^\pi \\
&&-b^\pi a^\pi ba a^\pi\xi^D bb^Daa^\pi +bb^Da^2a^\pi \xi\xi^\pi bb^Daa^\pi +bb^Da^2a^\pi \xi^D bb^Daa^\pi\\
&\stackrel{(2.17)}{=}& b^\pi a^\pi b^2b^\pi aa^\pi\xi\xi^\pi-b^\pi a^\pi b^2b^Da^2a^\pi\xi\xi^\pi+b^2b^Da^3a^\pi \xi\xi^\pi\\
&&+b^\pi a^\pi b^2b^\pi aa^\pi\xi^D-b^\pi a^\pi b^2b^Da^2a^\pi\xi^D+bb^Da^3a^\pi\xi^D.\end{aligned}$$
Therefore, $b_2^2b_1=b_2b_1b_2$.
By Lemma 2.2(2), it follows that $b_1+b_2$ is nilpotent.
**Main results**
==================
In this section, we consider the formulae on the Drazin inverses of the product and sum of two elements of $R$.
Let $a,b \in R^D$ with $a^2b=aba$ and $b^2a=bab$. Then $ab\in R^D$ and
$(ab)^D=a^Db^D$.
Let $x=a^Db^D$. We prove that $x$ is the Drazin inverse of $ab$ by showing the following results: (1) $(ab)x=x(ab)$; (2) $ x(ab)x=x$; (3) $(ab)^k=(ab)^{k+1}x$ for some positive integer $k$.
\(1) Note that $a(ab)=(ab)a$ implies $(ab)a^D=a^D(ab)$. It follows that $$\begin{aligned}
(ab)x &=& (ab)a^Db^D\stackrel{(2.5)}{=}a^Dabb^D \\
&=& a(a^Db^D)b\stackrel{(2.5)}{=} a^Db^Dab \\
&=& x(ab).\end{aligned}$$
$(2)$ We calculate directly that $$\begin{aligned}
x(ab)x &=& (a^Db^D)aba^Db^D\stackrel{(2.5)}{=}a(a^Db^D)ba^Db^D=aa^Db(b^Da^D)b^D \\
&=& aa^D(b^Da^D)bb^D=a(a^Db^D)a^Dbb^D \stackrel{(2.5)}{=}aa^D(a^Db^D)bb^D\\
&=& x.\end{aligned}$$
$(3)$ Take $k={\rm max}\{{\rm ind}(a),{\rm ind}(b)\}$. Then $$\begin{aligned}
(ab)^{k+1}x&=& (ab)^{k+1}a^Db^D=a^D(ab)^{k+1}b^D \\
&\stackrel{(2.2)}{=}& a^Da^{k+1}b^{k+1}b^D=a^kb^k\\
&\stackrel{(2.2)}{=}& (ab)^k.\end{aligned}$$
Hence, $(ab)^D=a^Db^D$.
[In \[16, Lemma 2\], for $a,b \in R^D$ with $ab=ba$, it is proved that $(ab)^D=a^Db^D=b^Da^D$. However, in Theorem 3.1, $(ab)^D=a^Db^D \neq b^Da^D$. Such as, take $a=\left(
\begin{array}{cc}
1 & 0 \\
1 & 0 \\
\end{array}
\right)
$ and $b=\left(
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}
\right)
$. By calculations, we obtain $a^2b=aba$ and $b^2=bab$, but $a^Db^D \neq b^Da^D$]{}.
Let $a,b \in R^D$. If $a^2b=aba$, $b^2a=bab$ and ${\rm ind}(a)=s$, then $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. Moreover, we have $$\begin{aligned}
(a+b)^D &=& a^D(1+a^Db)^D+a^\pi b[a^D(1+a^Db)^D]^2+\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \\
&& +b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi,\end{aligned}$$ and $(1+a^Db)^D=a^\pi +a^2a^D(a+b)^D$.
Suppose that $a+b$ is Drazin invertible. We prove that $1+a^Db$ is Drazin invertible. Write $1+a^Db=a_1+b_1$, where $a_1=a^\pi$, $b_1=a^D(a+b)$.
Note that $(a^D)^2(a+b)=a^D(a+b)a^D$ and $(a+b)^2a^D=(a+b)a^D(a+b)$. By Theorem 3.1, it follows that $a^D(a+b)=b_1$ is Drazin invertible and
$(b_1)^D=[a^D(a+b)]^D=(a^D)^D(a+b)^D=a^2a^D(a+b)^D$.
From Lemma 2.4, we obtain that $a^Db$ commutes with $aa^D$. Hence, $a^D(a+b)\in {\rm comm}(a^\pi)$ and $a_1b_1=b_1a_1=0$. By \[7, Corollary 1\], it follows that $(1+a^Db)^D=a^\pi +a^2a^D(a+b)^D$.
Conversely, let $\xi=1+a^Db$ be Drazin invertible and $$\begin{aligned}
x &=& a^D\xi^D+a^\pi b(a^D\xi^D)^2+\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi+b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi\\
&=& x_1+x_2,\end{aligned}$$ where $x_1= a^D\xi^D+a^\pi b(a^D\xi^D)^2$, $x_2=\displaystyle{\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi}$.
According to $a^D\in {\rm comm}(\xi^D)$ and $(ba^D)^i=b^i(a^D)^i$, we have $$\begin{aligned}
(a+b)a^\pi b (a^D)^2&=&(a+b)(1-aa^D)b(a^D)^2=(a-a^2a^D+b-baa^D)b(a^D)^2\\
&=& ab(a^D)^2-a^2a^Db(a^D)^2+b^2(a^D)^2-baa^Db(a^D)^2\\
&=& (a^D)^2ab-a^2(a^D)^2a^Db+b^2(a^D)^2-ba(a^D)^2a^Db\\
&=& b^2(a^D)^2-b(a^D)^2b\\
&\stackrel{(2.6)}{=}& b^2(a^D)^2-(ba^D)^2\\
&\stackrel{(2.11)}{=}&0.\end{aligned}$$
Thus, $(a+b)a^\pi b(a^D\xi^D)^2=0$.
Similarly, $(a+b)b^\pi a(b^D)^2=0$. Hence, $(a+b)b^\pi a \displaystyle{\sum_{i=0}^{s-2}(i+1)}(b^D)^{i+2}(-a)^ia^\pi =0$.
Next, we show that $x$ is the Drazin inverse of $(a+b)$ in 3 steps.
Step 1. First we prove that $x(a+b)=(a+b)x$. Put $y_1=(a+b)a^D\xi^D$ and $y_2=(a+b)\displaystyle{\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi}$. Then we have $$\begin{aligned}
(a+b)x &=&(a+b) [a^D \xi^D+a^\pi b(a^D \xi^D)^2+\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \\
&& +b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi]\\
&=&(a+b)[a^D \xi^D + \sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi]\\
&=& y_1+y_2.\end{aligned}$$
Second we show that $x_1(a+b)=y_1$ and $x_2(a+b)=y_2$.
Since $a^D\xi^D=\xi^Da^D$, we get $$\begin{aligned}
x_1(a+b) &=& [a^D\xi^D+a^\pi b(a^D(\xi^D)^2](a+b)\\
&\stackrel{(2.5)}{=}& a^D(a+b)\xi^D+ a^\pi b(a^D)^2(a+b)(\xi^D)^2\\
&=& a^D(a+b)\xi^D+(a^\pi ba^D+a^\pi ba^Da^Db)(\xi^D)^2\\
&=& a^D(a+b)\xi^D+a^\pi ba^D\xi(\xi^D)^2\\
&=& a^D(a+b)\xi^D +a^\pi ba^D\xi^D\\
&=& (a^Da+a^Db+ba^D-aa^Dba^D)\xi^D\\
&=& (a+b)a^D \xi^D\\
&=& y_1.\end{aligned}$$
By induction, $$\begin{aligned}
bb^D(ab^D)^i&=&(bb^Dab^D)^i=(b^Da)^i.\end{aligned}$$
Note that $a^sa^\pi=0$ and $$\begin{aligned}
\nonumber
b^Da^\pi b &=& b^D(1-aa^D)b=bb^D-(b^Da^D)ba\stackrel{(2.6)}{=}bb^D-bb^Da^Da \\
&=& bb^Da^\pi.\end{aligned}$$
We have $$\begin{aligned}
&&x_2(a+b)-y_2=[\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi](a+b)\\
&&-(a+b)\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&-\sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^{i+1}a^\pi+\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi b-b^\pi a\sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^{i+1}a^\pi\\
&&+b^\pi a\sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi b-(a+b)\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&-\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +\sum_{i=0}^{s-1}b^D(-b^Da)^ia^\pi b-b^\pi ab^D\sum_{i=0}^{s-2}(i+1)(-b^Da)^ib^Da^\pi b\\
&&+b^\pi a(b^D)^2\sum_{i=0}^{s-2}(i+1)(-b^Da)^ia^\pi b-a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi-b\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&\stackrel{(3.2)}{=}&-\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi+\sum_{i=0}^{s-1}(-b^Da)^ibb^Da^\pi+b^\pi \sum_{i=0}^{s-2}(i+1)(-ab^D)^{i+2}a^\pi\\
&& -b^\pi\sum_{i=0}^{s-2}(i+1)(-ab^D)^{i+1}bb^Da^\pi +\sum_{i=0}^{s-1}(-ab^D)^{i+1}a^\pi -\sum_{i=0}^{s-1}(-b^Da)^ibb^Da^\pi\\
&=& -\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +\sum_{i=0}^{s-1}(-ab^D)^{i+1}a^\pi +b^\pi[\sum_{i=0}^{s-2}(i+1)(-ab^D)^{i+2}a^\pi\\
&&-\sum_{i=0}^{s-2}(i+1)(-ab^D)^{i+1}a^\pi]\\
&=& -\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +\sum_{i=0}^{s-1}(-ab^D)^{i+1}a^\pi -b^\pi \sum_{i=1}^{s-1}(-ab^D)^ia^\pi\\
&=& -\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +bb^D\sum_{i=1}^{s-1}(-ab^D)^ia^\pi\\
&\stackrel{(3.1)}{=}& -\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +\sum_{i=1}^{s-1}(-bb^Dab^D)^ia^\pi\\
&=& -\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi +\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\\
&=& -\sum_{i=1}^s(-b^Da)^ia^\pi +\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\\
&=& -\sum_{i=1}^{s-1}(-b^Da)^ia^\pi -(-b^Da)^sa^\pi +\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\\
&\stackrel{(2.13)}{=}&-\sum_{i=1}^{s-1}(-b^Da)^ia^\pi -(-b^D)^sa^sa^\pi +\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\\
&=&0.\end{aligned}$$
Hence, $x_2(a+b)=y_2$. It follows that $x(a+b)=(a+b)x$.
Step 2. We show that $x(a+b)x=x$. Note that $a+b=a\xi+a^\pi b$. We have $$\begin{aligned}
&&x(a+b)x = x(a+b)[a^D\xi^D+\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi]\\
&=& (a+b)[a^D\xi^D+\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi][a^D\xi^D+\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi]\\
&=&(a+b)(a^D\xi^D)^2+(a+b)a^D\xi^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&&+(a+b)\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&z_1+z_2+z_3,\end{aligned}$$ where $z_1=(a+b)(a^D\xi^D)^2$, $z_2=(a+b)a^D\xi^D\displaystyle{\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi}$ and $$z_3=(a+b)\displaystyle{\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi}.$$
Now we prove that $z_1+z_2+z_3=x$.
Firstly, we have $$\begin{aligned}
z_1&=& (a+b)(a^D\xi^D)^2=(a\xi+a^\pi b)(a^D\xi^D)^2\\
&=& a\xi(a^D\xi^D)^2+a^\pi b(a^D\xi^D)^2\\
&\stackrel{(2.5)}{=}&a^D\xi^D+ a^\pi b(a^D\xi^D)^2,\end{aligned}$$ $$\begin{aligned}
z_2&=& (a+b)a^D\xi^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& \xi^D aa^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +b\xi^Da^D \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&\xi^D \sum_{i=0}^{s-1}a^Da b^D(-b^Da)^ia^\pi +b\xi^D \sum_{i=0}^{s-1}(a^D)^2ab^D(-b^Da)^ia^\pi\\
&\stackrel{(2.9)}{=}&-\xi^D\sum_{i=0}^{s-1}a^D(-ab^D)^{i+1}a^\pi -b\xi^D\sum_{i=0}^{s-1}(a^D)^2(-ab^D)^{i+1}a^\pi\\
&\stackrel{(2.5)}{=}& -\xi^D\sum_{i=0}^{s-1}(-ab^D)^{i+1}a^Da^\pi -b\xi^D\sum_{i=0}^{s-1}(-ab^D)^{i+1}(a^D)^2a^\pi\\
&=& 0.\end{aligned}$$
Secondly, we show that $$\begin{aligned}
z_3&=& \sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi+b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi.\end{aligned}$$
Indeed, we have $$\begin{aligned}
&&z_3= (a+b)\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& b\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&[bb^Da^\pi +\sum_{i=1}^{s-1}(-b^Da)^ia^\pi]\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=&bb^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi-bb^Daa^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi
+\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&&+ a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi-bb^Daa^D\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi
+\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&&+ a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi+bb^D\sum_{i=0}^{s-1}(-ab^D)^{i+1}a^Da^\pi
+\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&&+ a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi+\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&&+a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&=& \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi +m_1+m_2,\end{aligned}$$ where $$m_1=\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi,$$ $$m_2=a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi.$$
In view of the equality $(3.3)$, it is enough to prove that $$\begin{aligned}
m_1+m_2&=&b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi.\end{aligned}$$
Since $$\begin{aligned}
&&b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi = (1-bb^D)a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi\\
&=&a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi-bb^Da \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi,\end{aligned}$$ we only need to show $$m_1=-bb^Da \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi,$$ $$m_2=a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi.$$
Since $a^sa^\pi=0$ and $bb^D$ commutes with $b^Da$, we get $$\begin{aligned}
m_1&=&\sum_{i=1}^{s-1}(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi= \sum_{i=1}^s(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi\\
&\stackrel{(2.6)}{=}& \sum_{i=1}^s(-bb^Db^Da)^ia^\pi\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi=bb^D\sum_{i=1}^s(-b^Da)^ia^\pi\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi\\
&=& -bb^D\sum_{i=1}^s(-b^Da)^{i-1}b^Daa^\pi\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi\\
&\stackrel{(2.6)}{=}&-bb^D\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^i(b^Daa^\pi)b^Da^\pi\\
&=&-bb^D\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^ib^D(b^Daa^\pi)a^\pi\\
&=&-bb^D\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^i(b^D)^2aa^\pi\\
&\stackrel{(2.6)}{=}& -bb^D(b^D)^2a\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^ia^\pi\\
&\stackrel{(2.6)}{=} &-b(b^D)^2ab^D\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^ia^\pi\\
&\stackrel{(2.6)}{=}&-bb^Da(b^D)^2\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^ia^\pi \\
&=&-bb^Da(b^D)^2\sum_{i=0}^{s-2}(i+1)(-b^Da)^ia^\pi\\
&\stackrel{(2.13)}{=}& -bb^Da\sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi.\end{aligned}$$
Similarly, $$\begin{aligned}
m_2&=&a\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi \stackrel{(2.6)}{=} a\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi \sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi\\
&\stackrel{(2.6)}{=}&a\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi b^Da^\pi\stackrel{(2.6)}{=}a\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^i b^Db^Da^\pi a^\pi\\
&=&a\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^i (b^D)^2 a^\pi\stackrel{(2.6)}{=}a(b^D)^2\sum_{i=0}^{s-1}(-b^Da)^i\sum_{i=0}^{s-1}(-b^Da)^i a^\pi\\
&=& a(b^D)^2\sum_{i=0}^{s-2}(i+1)(-b^Da)^i a^\pi\stackrel{(2.13)}{=} a(b^D)^2\sum_{i=0}^{s-2}(i+1)(b^D)^i(-a)^i a^\pi\\
&=& a\sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^i a^\pi.\end{aligned}$$
Thus, $$\begin{aligned}
z_3&=& \sum _{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi+b^\pi a \sum_{i=0}^{s-2}(i+1)(b^D)^{i+2}(-a)^ia^\pi.\end{aligned}$$
Therefore, $x(a+b)x=x$.
Step 3. We prove that $(a+b)-(a+b)^2x$ is nilpotent.
Note that the proof of step 1. We have $$\begin{aligned}
&&(a+b)-(a+b)^2x = (a+b)-[a^D\xi^D+\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi](a+b)^2\\
&\stackrel{(2.5)}{=}&(a+b)-\xi^Da^D(a+b)^2-\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi(a+b)^2 \\
&=&(a+b)-\xi^Da(a^D(a+b))^2-\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi(a^2+ab+ba+b^2)\\
&=& a\xi+a^\pi b-\xi^Da(\xi-a^\pi)^2-\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi a^2\\
&&-\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi ab-\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi ba\\
&&- \sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^i a^\pi b^2\\
&\stackrel{(2.13)}{=}& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)+\sum_{i=0}^{s-1}(-b^Da)^{i+1}a a^\pi+\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&-\sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi ba - \sum_{i=0}^{s-1}b^D(-b^Da)^ia^\pi b^2\\
&\stackrel{(3.2)}{=}& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)+\sum_{i=0}^{s-1}(-b^Da)^{i+1}a a^\pi+\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&-\sum_{i=0}^{s-1}(-b^Da)^ibb^Daa^\pi - \sum_{i=0}^{s-1}b^D(-b^Da)^ia^\pi b^2\\
&=& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)-bb^Daa^\pi +\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&- \sum_{i=0}^{s-1}b^D(-b^Da)^ia^\pi b^2\\
&\stackrel{(2.6)}=& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)-bb^Daa^\pi +\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&- \sum_{i=0}^{s-1}(-b^Da)^ib^Da^\pi b^2\\
&\stackrel{(3.2)}{=}& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)-bb^Daa^\pi +\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&- \sum_{i=0}^{s-1}(-b^Da)^ibb^Da^\pi b\\
&\stackrel{(2.6)}{=}& a\xi+a^\pi b-\xi^Da(\xi^2-a^\pi)-bb^Daa^\pi +\sum_{i=0}^{s-1}(-b^Da)^{i+1}a^\pi b \\
&&- \sum_{i=0}^{s-1}bb^D(-b^Da)^ia^\pi b\\
&=&a\xi +a^\pi b-\xi^Da\xi^2+\xi^Daa^\pi -bb^Daa^\pi-bb^Da^\pi b\\
&=&(a\xi-\xi^Da\xi^2)+\xi^Daa^\pi+(a^\pi b-bb^Da^\pi b)-bb^Daa^\pi\\
&=& a\xi \xi^\pi+\xi^Daa^\pi +b^\pi a^\pi b -bb^Daa^\pi\\
&=& b_1+b_2.\end{aligned}$$
By Lemma 2.8, $(a+b)-(a+b)^2x=b_1+b_2$ is nilpotent.
The proof is completed.
$[16, {\rm Theorem}~3]$ Let $a,b \in R^D$ with $ab=ba$. Then $a+b$ is Drazin invertible if and only if $1+a^Db$ is Drazin invertible. In this case, we have $$\begin{aligned}
(a+b)^D &=& (1+a^Db)^D a^D+b^D(1+aa^\pi b^D)^{-1}a^\pi\\
&=& a^D(1+a^Db)^Dbb^D+b^\pi(1+bb^\pi a^D)^{-1}+b^D(1+aa^\pi b^D)^{-1}a^\pi,\end{aligned}$$ and
$(1+a^Db)^D=a^\pi +a^2a^D(a+b)^D$.
Since $ab=ba$, we have $a$, $b$, $a^D$ and $b^D$ commute with each other. Hence, it follows that $a^\pi b[a^D(1+a^Db)^D]^2=0$ and $b^\pi a \displaystyle{\sum_{i=0}^{s-2}(i+1)}(b^D)^{i+2}(-a)^ia^\pi=0$, where $s={\rm ind}(a)$.
Since $aa^\pi b^D$ is nilpotent, $1+aa^\pi b^D$ is invertible and $$\begin{aligned}
(1+aa^\pi b^D)^{-1} &=& 1+(-aa^\pi b^D)+(-aa^\pi b^D)^2+\cdots +(-a a^\pi b^D)^{s-1}\\
&=& \sum _{i=0}^{s-1}(-b^Daa^\pi)^i.\end{aligned}$$ We have $$\begin{aligned}
b^D(1+aa^\pi b^D)^{-1}a^\pi &=& b^D \sum _{i=0}^{s-1}(-b^Daa^\pi)^i a^\pi= b^D \sum_{i=0}^{s-1}(b^D)^i(-a)^ia^\pi\\
&=&\sum_{i=0}^{s-1}(b^D)^{i+1}(-a)^ia^\pi.\end{aligned}$$
Therefore, $(a+b)^D = (1+a^Db)^D a^D+b^D(1+aa^\pi b^D)^{-1}a^\pi$.
**ACKNOWLEDGMENTS**
This research is supported by the National Natural Science Foundation of China (10971024), the Specialized Research Fund for the Doctoral Program of Higher Education (20120092110020), the Natural Science Foundation of Jiangsu Province (BK2010393) and the Foundation of Graduate Innovation Program of Jiangsu Province(CXLX13-072).
[s2]{}
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---
abstract: 'We give combinatorial descriptions of the restrictions to $T$-fixed points of the classes of structure sheaves of Schubert varieties in the $T$-equivariant $K$-theory of Grassmannians and of maximal isotropic Grassmannians of orthogonal and symplectic types. We also give formulas, based on these descriptions, for the Hilbert series and Hilbert polynomials at $T$-fixed points of the corresponding Schubert varieties. These descriptions and formulas are given in terms of two equivalent combinatorial models: excited Young diagrams and set-valued tableaux. The restriction fomulas are positive, in that for a Schubert variety of codimension $d$, the formula equals $(-1)^d$ times a sum, with nonnegative coefficients, of monomials in the expressions $(e^{-{\alpha}} -1)$, as $\alpha$ runs over the positive roots. In types $A_n$ and $C_n$ the restriction formulas had been proved earlier by [@Kre:05], [@Kre:06] by a different method. In type $A_n$, the formula for the Hilbert series had been proved earlier by [@LiYo:12]. The method of this paper, which relies on a restriction formula of Graham [@Gra:02] and Willems [@Wil:06], is based on the method used by Ikeda and Naruse [@IkNa:09] to obtain the analogous formulas in equivariant cohomology. The formulas we give differ from the $K$-theoretic restriction formulas given by Ikeda and Naruse [@IkNa:11], which use different versions of excited Young diagrams and set-valued tableaux. We also give Hilbert series and Hilbert polynomial formulas which are valid for Schubert varieties in any cominuscule flag variety, in terms of the $0$-Hecke algebra.'
address:
- ' Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, GA 30602 '
- ' Department of Mathematics, University of Wisconsin - Parkside, Kenosha, WI 53140 '
author:
- William Graham
- Victor Kreiman
bibliography:
- 'refs\_eyd.bib'
title: 'Excited Young diagrams, equivariant K-theory, and Schubert varieties'
---
Introduction
============
In this paper we use equivariant $K$-theory to obtain information about the local structure of Schubert varieties in Grassmannians or maximal isotropic Grassmannians of orthogonal or symplectic type. Such a Grassmannian is a generalized flag variety of the form $X = G/P$, where $G$ is one of the groups $G = {\operatorname{SL}}_n({\mathbb{C}})$, $G = {\operatorname{SO}}_n({\mathbb{C}})$ or $G = {\operatorname{Sp}}_{2n}({\mathbb{C}})$, and $P$ is a parabolic subgroup. These varieties have long attracted attention because of their connections with combinatorics and representation theory. Let $B \supset T$ denote a Borel subgroup and maximal torus of $G$, and let $B^-$ denote the opposite Borel subgroup to $B$. If $Y$ is a Schubert variety in $X$ (that is, the closure of a $B^-$-orbit in $X$), then the structure sheaf ${\mathcal{O}}_Y$ defines an element $[{\mathcal{O}}_Y]$ in $K_T(X)$, the Grothendieck group of $T$-equivariant coherent sheaves on $X$. If $i: \{ x \} \hookrightarrow X$ is the inclusion of a $T$-fixed point, then the class $i_x^*[{\mathcal{O}}_Y]$ is an element in the representation ring $R(T)$ of $T$. The restriction $i_x^*[{\mathcal{O}}_Y]$ enables one to describe the ring of functions on the tangent cone of $Y$ at $x$ as a representation of $T$. We will refer to the class $i_x^*[{\mathcal{O}}_Y]$ as the restriction or pullback of the class $[{\mathcal{O}}_Y]$ to the fixed point $x$. The main results of this paper are combinatorial formulas for these pullback classes. Such formulas have particular interest because in all of these cases (except for the odd orthogonal case) the generalized flag variety is cominuscule, which means that the restriction formulas yield formulas for the Hilbert series and Hilbert functions of the local rings ${\mathcal{O}}_{Y,x}$.[^1] As a consequence of our formulas we deduce that in these examples (and in fact for Schubert varieties in any cominuscule flag variety), the Hilbert function coincides with the Hilbert polynomial. Note that the restriction fomulas in this paper are positive, in that for a Schubert variety of codimension $d$, the formula is $(-1)^d$ times a sum, with nonnegative coefficients, of monomials in the expressions $(e^{-{\alpha}} -1)$, as $\alpha$ runs over the positive roots. Similar positivity results occur in the structure constants for the equivariant $K$-theory of flag varieties; see [@GrRa:04], [@GrKu:08]. One consequence of the positivity in our restriction formulas is that the constants defining the Hilbert series and Hilbert polynomial are given by positive termed enumerative formulas, i.e., one obtains them by counting well defined algebraic or combinatorial objects.
The combinatorial formulas for the $i_x^*[{\mathcal{O}}_Y]$ were obtained earlier, in the cases of the Grassmannian and the Lagrangian Grassmannian, by Kreiman [@Kre:05], [@Kre:06]. The formulas were derived there by using equivariant Gröbner degenerations of Schubert varieties in the neighborhood of a $T$-fixed point; these degenerations were obtained in [@KoRa:03], [@Kre:03], [@KrLa:04], and [@GhRa:06]. This method is discussed in more detail in Section \[s.Grassmannian\]. The approach taken in this paper is different, and modeled on the approach taken by Ikeda and Naruse [@IkNa:09], who obtained restriction formulas in equivariant cohomology. The main tool of Ikeda and Naruse is a formula of Andersen-Jantzen-Soergel [@AJS:94] and Billey [@Bil:99]. This gives the restriction of a Schubert class to a $T$-fixed point in terms of expressions in what is called the nil-Coxeter (or nil-Hecke) algebra. The nil-Coxeter formula works for the full flag variety (and hence for any generalized flag variety). In the classical cominuscule cases, however, it can be used to give formulas in terms of combinatorics related to Young diagrams. To obtain the formulas in equivariant $K$-theory, we replace the cohomology formula by an analogous $K$-theory formula in the $0$-Hecke algebra, obtained by Graham and Willems. We use this to obtain general formulas for the Hilbert series and Hilbert polynomials for Schubert varieties of cominuscule flag varieties at $T$-fixed points. In the Grassmanian cases, we again relate these formulas to Young diagrams. Our formulas are in terms of excited Young diagrams (the term is due to Ikeda and Naruse [@IkNa:09]; these were called subsets of Young diagrams in [@Kre:05], [@Kre:06]). We have generalized the definition of excited Young diagrams for the $K$-theory formulas; the earlier definitions, which were used for the equivariant cohomology formulas, are what we call reduced excited Young diagrams. Reduced excited Young diagrams were discovered independently by Kreiman [@Kre:05], [@Kre:06] and Ikeda and Naruse [@IkNa:09]. In type $A_n$, excited Young diagrams are the same combinatorial objects as the pipe dreams of [@WoYo:12] (see also [@LiYo:12]). We also give formulas in terms of an alternative, but equivalent, combinatorial model, namely set-valued tableaux. The set-valued tableaux which we use were introduced in [@Kre:05], [@Kre:06]. In type $A_n$, they also appeared in [@WoYo:12], where they were identified as special types of flagged set-valued tableaux. Flagged set-valued tableaux, which were introduced in [@KMY:09], generalize both set-valued tableaux [@Buc:02] and flagged tableaux [@Wac:85]. Ikeda and Naruse [@IkNa:11] used somewhat different versions of excited Young diagrams and set-valued tableaux to obtain combinatorial formulas for functions $G_{\lambda}(x|b)$, $GB^{(n)}_{\lambda}(x|b)$, $GC^{(n)}_{\lambda}(x|b)$, and $GD^{(n)}_{\lambda}(x|b)$. These are functions in variables $x_1,x_2, \ldots, x_n$ and $b_1,b_2, \ldots$, which depend on a parameter ${\beta}$. The function $G_{\lambda}(x|b)$ is equal to the factorial Grothendieck polynomial of Macnamara [@Mcn:06]. These functions represent the classes of structure sheaves of Schubert varieties in equivariant $K$-theory, in types $A$, $B$, $C$, and $D$, respectively. In particular, Ikeda and Naruse prove that if $G$ is of type $A$, $B$, $C$, or $D$, then if one takes a function of the appropriate type, sets ${\beta}= -1$, and chooses an appropriate specialization of the variables (depending on $\mu$), the result is the restriction of $[{\mathcal{O}}_{X^{\lambda}}]$ to the point corresponding to $\mu$. Thus, their results lead to combinatorial formulas for the pullbacks of the structure sheaves of Schubert varieties. These formulas are different from the formulas given in this paper. See Section \[ss.previous\] for examples comparing the formulas in this paper with the results of [@IkNa:11].
Since the $0$-Hecke restriction formula is valid for any generalized flag variety, it is natural to ask why we focus on the Grassmannians and maximal isotropic Grassmannians. There are are two important properties which are relevant to these cases. First, given any $T$-fixed point in a cominuscule flag variety, there exists a vector $\xi$ in the Lie algebra of $T$, such that for any weight ${\alpha}$ of $T$ on the tangent space of the flag variety at that point, ${\alpha}(\xi) = -1$ (see Proposition \[p.cominuscule\]). This property implies that in the cominuscule cases, restrictions in equivariant $K$-theory give information about the Hilbert series (see Proposition \[p.fixedhilbert\]). Second, in the cases we consider—that is, in all cominuscule flag varieties, as well as the maximal isotropic Grassmannians in the odd orthogonal case—the Schubert varieties and $T$-fixed points are parametrized by elements of the Weyl group which are fully commutative in the sense of Stembridge [@Ste:96]. This property is used when we connect the $0$-Hecke formula to Young diagrams.
The contents of the paper are as follows. Section \[s.background\] describes the relation between restriction formulas in equivariant $K$-theory and the tangent cone at $T$-fixed points, and explains how, in the case of cominuscule flag varieties, this is connected with multiplicities and Hilbert series. This connection has been known for some time; we learned about it from Michel Brion, who pointed out that in the cominuscule case, equivariant multiplicities can be used to compute multiplicities. Section \[ss.mult\] contains Proposition \[p.fixedchar\], which is a version of a result in an unpublished paper of Bressler [@Bre], who used it to give a proof of a formula of Kumar [@Kum:96 Theorem 2.2] describing multiplicities in the tangent cone of Schubert varieties in terms of the $0$-Hecke algebra. This is is also related to work of Rossmann [@Ros:89]. The connection to Hilbert series is given in Proposition \[p.fixedhilbert\]. This connection to the Hilbert series is known—see for example [@IkNa:09 Section 9] or [@LiYo:12]; we have given some details not explained in these references. Section \[ss.backgroundHecke\] recalls some definitions about the equivariant $K$-theory of the flag variety, and states the $0$-Hecke pullback formula (Theorem \[t.pullback\]); related formulas were given by Graham [@Gra:02] and Willems [@Wil:06]. Section \[ss.cominuscule\] contains Proposition \[p.cominuscule\], which shows that cominuscule flag varieties have the geometric property needed to apply Proposition \[p.fixedhilbert\]. We originally learned this result from Brion; it is also used in [@IkNa:09 Section 9]. Combining the above results yields Hilbert series and Hilbert polynomial formulas (Theorem \[t.cominusculeformula\]) and a formula for the multiplicity of a $T$-fixed point (Corollary \[c.cominusculemult\]), which are valid for arbitrary (not necessarily classical) cominuscule flag varieties; these formulas are given in Section \[ss.Hilbertcominuscule\].
Section \[s.nilHecke\] proves some results about fully commutative elements and the $0$-Hecke algebra which we need to obtain the connection with Young diagrams.
Section \[s.Grassmannian\] concerns the case where $G = {\operatorname{SL}}(n,{\mathbb{C}})$, so $X = {\operatorname{Grass }}(d,n)$, the Grassmannian variety of all $d$-dimensional subspaces of ${\mathbb{C}}^n$. This case is the foundation of all the classical cases. Section \[ss.partitions\_An\] gives background about Grassmannian permutations and partitions, which index the Schubert varieties and the $T$-fixed points. Section \[ss.eyd\] defines excited Young diagrams. The first main result is Theorem \[t.ktheory\_eyd\], the restriction formula in terms of excited Young diagrams. This theorem is proved by finding a reduced expression for a Grassmannian permutation which is related to the Young diagram of the corresponding partition (Section \[ss.eyd\]), and then interpreting the terms of the $0$-Hecke restriction formula in terms of excited Young diagrams (Proposition \[p.subsequences\]). Section \[ss.setvalued\_tableaux\] defines set-valued tableaux, which are in some ways easier to work with than excited Young diagrams; Theorem \[t.ktheory\_svt\] gives the restriction formula in terms of set-valued tableaux. This theorem is proved by establishing a bijection between appropriate collections of set-valued tableaux and excited Young diagrams.
Section \[s.orthosymplectic\] deals with the remaining classical cases, the maximal isotropic Grassmannians of orthogonal or symplectic types. In these cases, the Schubert varieties and $T$-fixed points are indexed by shifted Young diagrams. The restriction formulas, which are obtained by adapting the methods of the previous section, are in terms of excited shifted Young diagrams (Theorem \[t.ktheory\_eyd\_bcd\]) and set-valued shifted tableaux (Theorem \[t.ktheory\_syt\_bcd\]).
Appendix \[s.appendix\_roots\] reviews some facts about root systems and Weyl groups. Appendix \[s.appendix\_restriction-opp\] explains the relationship between different versions of the $0$-Hecke restriction formula.
Equivariant $K$-theory of the flag variety, Hilbert series and multiplicities {#s.background}
=============================================================================
Equivariant $K$-theory, Hilbert series and multiplicities {#ss.mult}
---------------------------------------------------------
In this section we review some results relating equivariant $K$-theory to the local rings of functions at $T$-fixed points. We have included some proofs for the convenience of the reader. Let $T = ({\mathbb{C}}^*)^m$ denote a complex torus. Let $R(T)$ denote the representation ring of $T$; this is the set of all ${\mathbb{ Z}}$-linear combinations of $e^{{\lambda}}$, where ${\lambda}$ is a weight of $T$.
If $M$ is a scheme with a $T$-action, let $K_T(M)$ denote the Grothendieck group of coherent sheaves on $X$. If $M$ is smooth, then $K_T(M)$ can be identified with the Grothendieck group of vector bundles on $M$. A $T$-equivariant coherent sheaf ${\mathcal{F}}$ on $M$ defines a class $[{\mathcal{F}}] \in K_T(M)$. In particular, a closed $T$-invariant subscheme $Z$ of $M$ defines a class $[{\mathcal{O}}_Z] \in K_T(M)$. If $M$ is a point then $K_T(M)$ is identified with $R(T)$.
Let $X$ be a smooth $T$-variety and $x \in X^T$ be an isolated fixed point. Let $\Phi(T_x)$ denote the set of weights of $T$ on the tangent space $T_x X$. The fixed point $x$ is said to be attractive if there is a half-space in ${{\mathfrak t}}^*$ containing $\Phi(T_x)$. This implies that $x$ has a $T$-stable neighborhood in $X$ which is $T$-equivariantly isomorphic to $T_x X$, and such that $x$ corresponds to the origin in $T_x X$. Using this fact we can prove results about the pullbacks of classes in $K_T(X)$ to $x$ by reducing to the case where $X$ is a vector space with a linear $T$-action and $x$ is the origin (see Proposition \[p.fixedchar\]).
Let $Y \subset X$ be a $T$-stable subscheme containing $x$, and let $ {\mathcal{O}}_{Y,x}$ denote the local ring of $Y$ at $x$ with maximal ideal ${{\mathfrak m}}= {{\mathfrak m}}_{Y,x}$. Let $ {\operatorname{Gr }}{\mathcal{O}}_{Y,x} = \oplus_{i=0}^{\infty} {{\mathfrak m}}^i / {{\mathfrak m}}^{i+1}$. By definition, the tangent cone of $Y$ at $x$ is ${\operatorname{Spec}}( {\operatorname{Gr }}{\mathcal{O}}_{Y,x})$ (see [@Kum:96 Section 2]). Let $\hat{R}$ denote the set of expressions of the form $\sum_{\mu \in \hat{T}} c_{\mu} e^{\mu}$. The group $T$ acts on ${\operatorname{Gr }}{\mathcal{O}}_{Y,x}$ with finite multiplicities, so we can define ${\operatorname{char}}( {\operatorname{Gr }}{\mathcal{O}}_{Y,x} ) \in \hat{R}$ as ${\operatorname{char}}( {\operatorname{Gr }}{\mathcal{O}}_{Y,x} ) = \sum m_{\mu} e^{\mu}$, where $m_{\mu}$ is the multiplicity of the weight $\mu$ in $Gr {\mathcal{O}}_{Y,x} $.
Let $f$ be an element of the quotient field of $R(T)$ of the form $$f = \frac{r}{\prod_{\mu \in \hat{T}} (1 - e^{\mu})^{n_{\mu}}},$$ where $r \in R(T)$, $n_{\mu} \in {\mathbb{ Z}}_{\ge 0}$, and such that there is a half-space in ${{\mathfrak t}}^*$ containing all the $\mu$ with $n_{\mu} \neq 0$. Define $F(f) \in \hat{R}$ to be the series $$r \prod_{\mu \in \hat{T}} ( \sum 1 + e^{\mu} + e^{2 \mu} + \cdots )^{n_{\mu}}).$$
The following proposition is a version of a result in an unpublished paper of Bressler, and is also related to [@Ros:89 Lemma 1.1]. Bressler [@Bre] used this result to give a proof of a formula of Kumar [@Kum:96 Theorem 2.2] describing the multiplicities in the ring of functions on the tangent cone to a Schubert variety at a $T$-fixed point in terms of the $0$-Hecke algebra (see [@Kum:96 Remark 2.13]).
\[p.fixedchar\] Let $x$ be an attractive fixed point in the smooth $T$-variety $X$, and let $Y \subset X$ be a $T$-stable subscheme containing $X$. Let $i: \{ x \} \hookrightarrow X$ denote the inclusion, and $[{\mathcal{O}}_Y] \in K_T(X)$ the class of the structure sheaf of $Y$. Then $${\operatorname{char}}({\operatorname{Gr }}{\mathcal{O}}_{Y,x}) = F\left(\frac{i^*[{\mathcal{O}}_Y]}{\prod_{\mu \in \Phi(T_x)}(1 - e^{-\mu})}\right).$$
There is a $T$-stable affine open neighborhood of $x$ in $X$ which is $T$-equivariantly isomorphic to $V = T_x X$ (see [@Bia:73 Corollary 2]). We can replace $X$ by this neighborhood and therefore assume $X = V = {\operatorname{Spec}}A$, where $A = S(V^*)$. Let $I$ denote the ideal of $Y$ in $A$ and $B = A/I$, so $Y = {\operatorname{Spec}}B$. Let ${{\mathfrak n}}$ denote the maximal ideal of $x$ in $B$. Then ${\mathcal{O}}_{Y,x} = B_{{{\mathfrak n}}}$, and ${{\mathfrak m}}= {{\mathfrak n}}B_{{{\mathfrak n}}} \subset B_{{{\mathfrak n}}}$. Define ${\operatorname{Gr }}B = \oplus_{i=0}^{\infty} {{\mathfrak n}}^i / {{\mathfrak n}}^{i+1}$. The natural map $B / {{\mathfrak n}}^i \to B_{{{\mathfrak n}}} / {{\mathfrak m}}^i$ is an isomorphism for all $i$. This implies that the natural map ${{\mathfrak n}}^i / {{\mathfrak n}}^{i+1} \to {{\mathfrak m}}^i / {{\mathfrak m}}^{i+1}$ is an isomorphism for all $i$, so we obtain a $T$-equivariant isomorphism ${\operatorname{Gr }}{\mathcal{O}}_{Y,x} \to {\operatorname{Gr }}B$. Therefore ${\operatorname{char}}( {\operatorname{Gr }}{\mathcal{O}}_{Y,x} ) = {\operatorname{char}}({\operatorname{Gr }}B)$, which in turn is equal to ${\operatorname{char}}(B)$.
There exists a $T$-equivariant resolution of $B$ by finite free $A$-modules $$0 \to F_d \to F_{d-1} \to \cdots \to F_0 \to B \to 0$$ where each $F_j$ is isomorphic to $\oplus_i A \otimes {\mathbb{C}}_{\lambda_{i,j}}$. Here $A \otimes {\mathbb{C}}_{\lambda_{i,j}}$ denotes the $A$-module $A$ with $T$-action twisted by $\lambda_{i,j} \in \hat{T}$. (See [@Ros:89 Lemma 1.1].) This resolution corresponds to the resolution of ${\mathcal{O}}_Y$ over ${\mathcal{O}}_X$: $$0 \to {\mathcal{F}}_d \to {\mathcal{F}}_{d-1} \to \cdots \to {\mathcal{F}}_0 \to {\mathcal{O}}_Y \to 0$$ where ${\mathcal{F}}_j$ is isomorphic to $\oplus_i {\mathcal{O}}_X \otimes {\mathbb{C}}_{\lambda_{i,j}}$ In $R(T)$, $i^*[{\mathcal{O}}_X] = 1$, and therefore $$i^*[{\mathcal{O}}_Y] = \sum_{i,j} (-1)^j e^{\lambda_{i,j}} i^*[{\mathcal{O}}_X] = \sum_{i,j} (-1)^j e^{\lambda_{i,j}}.$$ On the other hand, $$\begin{aligned}
{\operatorname{char}}(B) & = & \sum_j (-1)^j {\operatorname{char}}(F_j) = \sum_{i,j} (-1)^j e^{\lambda_{i,j}} {\operatorname{char}}(A) \\
& = & \sum_{i,j} (-1)^j e^{\lambda_{i,j}} F\left(\frac{1}{\prod_{\mu \in \Phi(T_x)}(1 - e^{-\mu})}\right) \\
& = & F\left(\frac{i^*[{\mathcal{O}}_Y]}{\prod_{\mu \in \Phi(T_x)}(1 - e^{-\mu})}\right),\end{aligned}$$ as desired.
The Hilbert function of ${\operatorname{Gr }}{\mathcal{O}}_{Y,x}$ is by definition the function $n \mapsto \dim ({{\mathfrak m}}^n / {{\mathfrak m}}^{n+1})$. For $n >> 0$, this function is a polynomial, which we denote by $h(Y,x)(n)$. Let $r$ denote the degree of $h(Y,x)(n)$. The multiplicity of $Y$ at $x$, which we denote by ${\operatorname{mult}}(Y,x)$, is $r!$ times the leading coefficient of $h(Y,x)(n)$. The Hilbert series $H(Y,x)(t)$ is the generating function associated to the Hilbert function of ${\operatorname{Gr }}{\mathcal{O}}_{Y,x}$. By definition, $$H(Y,x)(t)= \sum_{i = 0}^{\infty} \dim ({{\mathfrak m}}^i / {{\mathfrak m}}^{i+1}) t^i.$$
Let $S$ denote a formal sum $S = \sum_{\mu \in \hat{T}} c_{\mu} e^{\mu}$. Suppose that there exists an element $\xi \in {{\mathfrak t}}$ such that $\mu(\xi)$ is a nonnegative integer for each $\mu$ with $c_{\mu} \neq 0$, and such that for each $n \in {\mathbb{ Z}}_{\ge 0}$, there exist only finitely many $\mu$ with $c_{\mu} \neq 0$ and $\mu(\xi) = n$. Then define $ev_{\xi} S$ to be the power series $ev_{\xi} S = \sum_{\mu \in \hat{T}} t^{\mu(\xi)}$.
\[p.fixedhilbert\] Keep the hypotheses of Proposition \[p.fixedchar\], and assume in addition that there exists $\xi \in {{\mathfrak t}}$ such that ${\alpha}(\xi) = -1$ for each ${\alpha}\in \Phi(T_x)$. Then $$H(Y,x)(t) = {\operatorname{ev}}_{\xi} {\operatorname{char}}({\operatorname{Gr }}{\mathcal{O}}_{Y,x}) = \frac{{\operatorname{ev}}_{\xi} i^* [{\mathcal{O}}_Y]}{(1-t)^d},$$ where $d = \dim X$.
Let $z_1,\ldots,z_q$ be a basis for ${{\mathfrak m}}/{{\mathfrak m}}^2\subseteq (T_xX)^*$, with weights $-{\alpha}_1,\ldots,-{\alpha}_q$, ${\alpha}_k\in \Phi(T_x)$. For ${\mathbf{j}}=(j_1,\ldots,j_i)$ a sequence of integers between 1 and $q$, define $z_{\mathbf{j}}=z_{j_1}\cdots z_{j_i}\in {{\mathfrak m}}^i/{{\mathfrak m}}^{i+1}$, and let ${\mu}_{\mathbf{j}}=-{\alpha}_{j_1}-\cdots-{\alpha}_{j_i}$, the weight of $z_{\mathbf{j}}$. Let ${\mathcal{B}}_i$ be a collection of $z_{\mathbf{j}}$’s which forms a basis for ${{\mathfrak m}}^i/{{\mathfrak m}}^{i+1}$. Then ${\operatorname{char}}({\operatorname{Gr }}{\mathcal{O}}_{Y,x})=\sum_{i=0}^{\infty}\sum_{z_{\mathbf{j}}\in {\mathcal{B}}_i}e^{{\mu}_{\mathbf{j}}}$, and $${\operatorname{ev}}_{\xi} {\operatorname{char}}({\operatorname{Gr }}{\mathcal{O}}_{Y,x})=\sum_{i=0}^{\infty}\sum_{z_{\mathbf{j}}\in {\mathcal{B}}_i} t^{{\mu}_{\mathbf{j}}(\xi)}=\sum_{i=0}^{\infty}\sum_{z_{\mathbf{j}}\in {\mathcal{B}}_i}t^i=H(Y,x)(t).$$
Equivariant K-theory of flag varieties {#ss.backgroundHecke}
--------------------------------------
In this section we recall some background about equivariant $K$-theory and flag varieties. Let $G$ be a complex semisimple algebraic group, $B$ a Borel subgroup of $G$, and $T$ a maximal torus in $B$. Let $P \supset B$ be a parabolic subgroup of $G$. Let ${{\mathfrak g}}, {{\mathfrak b}}, {{\mathfrak t}}$ and ${{\mathfrak p}}$ denote the Lie algebras of these groups. Given a representation $V$ of $T$, let $\Phi(V) \subset {{\mathfrak t}}^*$ denote the set of weights of $V$. Let $\Phi = \Phi({{\mathfrak g}})$ denote the set of roots of ${{\mathfrak g}}$ with respect to ${{\mathfrak t}}$, and let $\Phi^+$ denote the set of positive roots, chosen so that the positive root spaces are in ${{\mathfrak b}}$, i.e. , so that $\Phi^+ = \Phi({{\mathfrak b}})$. The set of negative roots is $\Phi^- = -\Phi^+$. Let $L$ be a Levi subgroup of $P$ containing $T$, and let $\Phi_{{{\mathfrak l}}} = \Phi({{\mathfrak l}})$, and $\Phi^+_{{{\mathfrak l}}} = \Phi_{{{\mathfrak l}}} \cap \Phi^+$, $\Phi^-_{{{\mathfrak l}}} = \Phi_{{{\mathfrak l}}} \cap \Phi^-$. Let $B^-$ denote the opposite Borel subgroup to $B$ and ${{\mathfrak b}}^-$ its Lie algebra. Let $W = N_G(T)/T$ denote the Weyl group; we will often use the same letter to denote an element of $W$ and a representative in $N_G(T)$. Let $S$ denote the set of simple reflections in $W$, so $(W,S)$ is a Coxeter system. Let $W_P = W_L$ denote the Weyl group of $L$. Then $W_P$ is a subgroup of $W$. Each coset $wW_P$ in $W$ contains a unique minimal length element and we let $W^P$ denote the set of minimal length coset representatives in $W$. The element $w$ is in $W^P$ if and only if $w(\Phi^+_{{{\mathfrak l}}}) \subset \Phi^+$ (cf. [@BiLa:00 2.5.3]).
Let $Y = G/B$, $X = G/P$, and let $\pi: Y \to X$ denote the projection. We will need a formula for the pullback of the class in $K_T(X)$ of the structure sheaf of a Schubert variety to a fixed point. We explain how to obtain this formula from the known formula for the corresponding problem on $Y$. If $w \in W$, we define the Schubert varieties $X^w = \overline{B^- \cdot wP} \subset X$ and $Y^w = \overline{B^- \cdot wB} \subset Y$. The variety $X^w$ only depends on the coset $wW_P$, and if we take $w \in W^P$, then ${\operatorname{codim}}X^w = l(w)$. Since $\pi$ is a flat map, it induces a map $\pi^*: K_T(X) \to K_T(Y)$ satisfying $\pi^*[{\mathcal{O}}_{X^w}] = [{\mathcal{O}}_{\pi^{-1}(X^w)}] = [{\mathcal{O}}_{Y^w}]$. If $v \in W^P$, let $i_v: \{ pt \} \to X$ (resp. $j_v: \{ pt \} \to Y$) denote the map taking the point to $vP$ (resp. $vB$). Because $X$ and $Y$ are smooth, there are induced pullback maps $i_v^*: K_T(X) \to R(T)$ and $j_v^*: K_T(Y) \to R(T)$. As $i_v = \pi \circ j_v$, we have $i_v^* = j_v^* \circ \pi^*$, and therefore $$i_v^*[{\mathcal{O}}_{X^w}] = j_v^* \pi^* [{\mathcal{O}}_{X^w}] = j_v^* [{\mathcal{O}}_{Y^w}].$$
There is a general formula for $i_v^*[{\mathcal{O}}_{X^w}] $. To state it we need to define the $0$-Hecke algebra associated to the Coxeter system $(W,S)$, over the ring $R$. This algebra is a free $R$-algebra with basis $H_w$, for $w \in W$, and the multiplication is characterized by the following relations: $H_1$ is the identity element (here $1$ denotes the identity element of $W$); for $s \in S$, $w \in W$, we have $H_s H_w = H_{sw}$ if $l(sw)> l(w)$, $H_s H_w = H_{w}$ if $l(sw)< l(w)$, and $H_s^2 = H_s$.
\[r.nilHecke\] The term $0$-Hecke algebra has been used (see, for example, [@Car:86], [@Fay:05]) for the algebra with basis $J_w$ (for $w \in W$), characterized by the properties $J_s J_w = J_{sw}$ if $l(sw)> l(w)$, $J_s J_w = J_{w}$ if $l(sw)< l(w)$, and $J_s^2 = -J_s$ (and the formula in [@Gra:02] is given in terms of this algebra). If we set $H_s = - J_s$, we see that this algebra is the same as the algebra defined above, and we can translate between the two presentations, since $$J_{s_1} J_{s_2} \ldots J_{s_k} = (-1)^{l(u) - k} J_u
\Leftrightarrow H_{s_1} H_{s_2} \ldots H_{s_k} = H_u.$$ In the case of equivariant cohomology, the appropriate algebra is one in which the relations $J_s^2 = - J_s$ are replaced by relations of the form $T_s^2 = 0$. This algebra has been called the nil-Coxeter algebra or the nil-Hecke ring or algebra (see for example [@FoSt:94], [@KoKu:86], [@Gin:97]).
\[d.tvj\] Let ${\mathbf{s}}= (s_1, s_2, \ldots, s_l)$ be a sequence of simple reflections. Define $T(w, {\mathbf{s}})$ to be the set of subsequences ${\mathbf{t}}= (s_{i_1}, \ldots, s_{i_m})$, $1\leq i_1<\cdots<i_m\leq l$ such that $H_{s_{i_1}} H_{s_{i_2}} \cdots H_{s_{i_m}} = H_w$. Define $l({\mathbf{t}}) = m$, and define $e({\mathbf{t}}) = l({\mathbf{t}}) - l(w)$.
In the above definition of $T(w,{\mathbf{s}})$, if $(i_1,\ldots,i_m)$ and $(j_1,\ldots,j_m)$ are different subsequences of $(1,\ldots,l)$, then we regard $(s_{i_1}, \ldots, s_{i_m})$ and $(s_{j_1}, \ldots, s_{j_m})$ as different subsequences of ${\mathbf{s}}$, even if they have the same entries.
We can now state the restriction formula.
\[t.pullback\] Let $v,w \in W^P$. Fix a reduced expression ${\mathbf{s}}= (s_{1}, \ldots s_{l})$ for $v$, and for $c = 1, \ldots, l$, let $r(c) = s_{1} s_{2} \cdots s_{c-1}({\alpha}_c)$. Then $$\begin{aligned}
\label{e.pullback}
i_v^*[{\mathcal{O}}_{X^w}] & = & (-1)^{l(w)} \sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}} }(e^{-r(i_1)}-1) (e^{-r(i_2)} - 1) \cdots (e^{-r(i_m)} - 1) \\
& = & \sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}} (-1)^{e({\mathbf{t}})} (1 - e^{-r(i_1)}) (1 - e^{-r(i_2)}) \cdots (1 - e^{-r(i_m)})
\end{aligned}$$ where $T_{w,{\mathbf{s}}}$ and $e({\mathbf{t}})$ are as in Definition \[d.tvj\].
This formula can be deduced from formulas for restrictions to fixed points given by Graham and Willems (see [@Gra:02], [@Wil:06]). (The paper [@Wil:06] gives formulas for restrictions of a different basis than the basis of structure sheaves of Schubert varieties, but the relationship between bases given in [@GrKu:08 Proposition 2.2] allows one to deduce formulas for the restrictions of one basis from the formulas for another basis.) See also [@Knu:08].
\[r.tangent\] Observe that $\{ r(1), r(2), \ldots, r(l) \} \subset \Phi(T_v(G/P))$ (in the notation of Theorem \[t.pullback\]). Indeed, the set $\{ r(1), r(2), \ldots, r(l) \}$ of Theorem \[t.pullback\] is equal to $v(\Phi^-) \cap \Phi^+$, and the set $\Phi(T_v(G/P))$ of weights of $T_v(G/P)$ is $v(\Phi({{\mathfrak g}}/{{\mathfrak p}})) = v(\Phi^- \setminus \Phi^-_{{{\mathfrak l}}})$. Because $v \in W^P$, $v(\Phi^-_{{{\mathfrak l}}}) \subset \Phi^-$. Therefore $v(\Phi^-) \cap \Phi^+ \subset v(\Phi^- \setminus \Phi^-_{{{\mathfrak l}}})$, which implies the assertion.
Cominuscule flag varieties {#ss.cominuscule}
--------------------------
Let $P \supset B$ be a standard parabolic subgroup of $G$; let ${{\mathfrak l}}$ be a Levi subalgebra of ${{\mathfrak p}}$ containing ${{\mathfrak t}}$, and let ${{\mathfrak u}}$ denote the nilradical of ${{\mathfrak p}}$, so we have a Levi decomposition ${{\mathfrak p}}= {{\mathfrak l}}+ {{\mathfrak u}}$. Let ${{\mathfrak u}}^-$ denote the nilradical of the opposite parabolic subalgebra to ${{\mathfrak p}}$. If $P$ is maximal, then $P$ corresponds to some simple root $\alpha$, in the sense that the simple roots of ${{\mathfrak l}}$ are the simple roots of ${{\mathfrak g}}$ other than $\alpha$.
\[l.rootl\] Let ${\alpha}_1, \ldots, {\alpha}_r$ denote the simple roots for ${{\mathfrak g}}$ and let $P$ denote the maximal parabolic subgroup corresponding to ${\alpha}_i$. If ${\alpha}= \sum_k n_k {\alpha}_k$ is a root of ${{\mathfrak g}}$, and $n_i = 0$, then ${\alpha}$ is a root of ${{\mathfrak l}}$.
By replacing ${\alpha}$ by $-{\alpha}$ if necessary we may assume each $n_k \ge 0$ (i.e. ${\alpha}$ is a positive root). The proof is by induction on $\sum n_k$. If $\sum n_k = 1$ then ${\alpha}$ is simple and, as noted above, ${\alpha}$ is a root of ${{\mathfrak l}}$. Suppose now that the statement of the lemma holds for all roots $\beta = \sum m_k {\alpha}_k$ with $\sum m_k < \sum n_k$. Since $({\alpha}, {\alpha}) = \sum n_k ({\alpha}, {\alpha}_k) > 0$, there exists some $j \neq i$ with $({\alpha}, {\alpha}_j)>0$. Then $\beta = s_{{\alpha}_j}({\alpha}) = \sum m_k {\alpha}_k$ is a root, necessarily positive (since the only root which changes sign under $s_{{\alpha}_j}$ is ${\alpha}_j \neq \beta$). Moreover, $\sum m_k = \sum n_k - 2\frac{({\alpha},{\alpha}_j)}{({\alpha}_j,{\alpha}_j)} < \sum n_k$. Our inductive hypothesis implies that $\beta$ is a root of ${{\mathfrak l}}$; since the set of roots of ${{\mathfrak l}}$ is preserved by $s_{{\alpha}_j}$, we conclude that $s_{{\alpha}_j} \beta = \alpha$ is a root of ${{\mathfrak l}}$.
If $G$ is simple, a maximal parabolic subgroup $P$ is called cominuscule if the corresponding simple root $\alpha$ appears with coefficient equal to $1$ when the highest root of $G$ is written as a sum of simple roots. The corresponding generalized flag variety $G/P$ is also called cominuscule. Cominuscule flag varieties have the following useful property (which we learned from Michel Brion).
\[p.cominuscule\] Let $G/P$ be a cominuscule generalized flag variety. For any $v \in W^P$, there exists an element $\xi \in {{\mathfrak t}}$ (depending on $v$) such that for any weight $\alpha$ of $T$ on $T_{vP}(G/P)$, we have $\alpha(\xi) = -1$.
Let ${\alpha}_1, \ldots, {\alpha}_r$ denote the simple roots of ${{\mathfrak g}}$; these form a basis of ${{\mathfrak t}}^*$, and we denote the dual basis of ${{\mathfrak t}}$ by $\xi_1, \ldots, \xi_r$. Assume that $P$ corresponds to the simple root ${\alpha}_i$. First suppose that $v= e$ is the identity. In this case, we can take $\xi = \xi_i$. The reason is that as $T$-representations, $$T_{eP} (G/P) \cong {{\mathfrak g}}/{{\mathfrak p}}\cong \oplus {\mathbb{C}}_{-\alpha},$$ where the sum is over the positive roots of ${{\mathfrak g}}$ which are not roots of ${{\mathfrak l}}$. If ${\alpha}= \sum n_k {\alpha}_k$ is such a root, then $n_i > 0$ by Lemma \[l.rootl\]; $n_i \le 1$ since ${\alpha}_i$ occurs with coefficient $1$ in the highest root of ${{\mathfrak g}}$, so $n_i = 1$, and then $-{\alpha}(\xi_i) = - n_i = -1$ as asserted. For general $v \in W^P$, we can take $\xi = v \xi_i$, since the set of weights of $T_{vP} (G/P)$ is $v$ applied to the set of weights of $T_{eP} (G/P)$.
Hilbert series and Hilbert polynomials in cominuscule flag varieties {#ss.Hilbertcominuscule}
--------------------------------------------------------------------
We can now describe the Hilbert series and Hilbert polynomial of a Schubert variety in a cominuscule generalized flag variety at a $T$-fixed point. We will write $H(X^w, v)(t)$ for the Hilbert series $H(X^w, vP)(t)$ and $h(X^w,v)(n)$ for the Hilbert polynomial $h(X^w,vP)(n)$.
\[t.cominusculeformula\] Let $G/P$ be a cominuscule generalized flag variety and $v, w \in W^P$. Fix a reduced expression ${\mathbf{s}}= (s_{1}, \ldots s_{l})$ for $v$. Let $d = \dim G/P$. The Hilbert series $H(X_w,v)$ is given by $$H(X^w,v)(t) = \sum_{{\mathbf{t}}\in T_{w, {\mathbf{s}}}} \frac{(-1)^{e({\mathbf{t}})}}{(1-t)^{d - l({\mathbf{t}})}}.$$ The Hilbert function is equal to the Hilbert polynomial $h(X^w,v)(n)$ for all $n$, and is given by the formula $$h(X^w,v)(n) = \sum_{{\mathbf{t}}\in T_{w, {\mathbf{s}}} }(-1)^{e({\mathbf{t}})}
\begin{pmatrix}
n + d - l({\mathbf{t}}) - 1 \\
d - l({\mathbf{t}}) -1.
\end{pmatrix}$$
By Proposition \[p.cominuscule\], there exists $\xi$ in ${{\mathfrak t}}$ so that ${\alpha}(\xi) = -1$ for each weight ${\alpha}$ of $T_v(G/P)$. By Remark \[r.tangent\], each $r(i)$ is a weight of $T_v(G/P)$. Hence ${\operatorname{ev}}_{\xi}(e^{-r(i)}) = t$. Therefore, Proposition \[p.fixedhilbert\] and Theorem \[t.pullback\] imply $$\begin{split}
H(X^w,v)(t)=\frac{{\operatorname{ev}}_\xi i_v^* [{\mathcal{O}}_{X^w}]}{(1-t)^d}
&= \frac{{\operatorname{ev}}_\xi \sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}} (-1)^{e({\mathbf{t}})} (1 - e^{-r(i_1)}) (1 - e^{-r(i_2)}) \cdots (1 - e^{-r(i_m)})}{(1-t)^d}\\
&= \frac{\sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}} (-1)^{e({\mathbf{t}})} (1-t)^{m}}{(1-t)^d}
= \sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}} \frac{(-1)^{e({\mathbf{t}})} }{(1-t)^{d-l({\mathbf{t}})}}.
\end{split}$$ Recalling the identity $\displaystyle
\frac{1}{(1-t)^k}=\sum_{n=0}^{\infty} {n+k-1\choose k-1}t^n,
$ we obtain $$\begin{split}
H(X^w,v)(t)&=\sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}}(-1)^{e({\mathbf{t}})}\sum_{n=0}^{\infty}
{n+d-l({\mathbf{t}})-1\choose d-l({\mathbf{t}})-1}t^n\\
&=\sum_{n=0}^{\infty}\left(\sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}}(-1)^{e({\mathbf{t}})}
{n+d-l({\mathbf{t}})-1\choose d-l({\mathbf{t}})-1}\right)t^n.
\end{split}$$ Thus $$h(X^w,v)(n)=\sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}}(-1)^{e({\mathbf{t}})}
{n+d-l({\mathbf{t}})-1\choose d-l({\mathbf{t}})-1}.$$
We may alternatively index the summations for the Hilbert series and Hilbert polynomial by the nonnegative integers ${\mathbb{ N}}$.
\[c.cominusculemult\] Let $G/P$ be a cominuscule generalized flag variety and $v, w \in W^P$, $v\geq w$. Fix a reduced expression ${\mathbf{s}}= (s_{1}, \ldots s_{l})$ for $v$. Let $d_w=\dim X^w=\dim G/P-l(w)$. For $k\in{\mathbb{ N}}$, define $m_k=\#\{{\mathbf{t}}\in T_{w,{\mathbf{s}}}\mid e({\mathbf{t}})=k\}$. Then $$\begin{aligned}
{3}
&H(X^w,v)(t)& &= &\ \ &\sum_{k\in{\mathbb{ N}}}\frac{(-1)^k m_k}{(1-t)^{d_w-k}}\\
&h(X^w,v)(n)& &=& &\sum_{k\in{\mathbb{ N}}} (-1)^k m_k {n+d_w-k-1\choose d_w-k-1}\\
&{\operatorname{mult}}(X^w,v)& &=& &m_0\end{aligned}$$
Let $d=\dim G/P$. Note that $e({\mathbf{t}})=l({\mathbf{t}})-l(w)$, so $d-l({\mathbf{t}})=(d-l(w))-e({\mathbf{t}})=d_w-e({\mathbf{t}})$. Thus $$H(X^w,v)(t) = \sum_{{\mathbf{t}}\in T_{w, {\mathbf{s}}}} \frac{(-1)^{e({\mathbf{t}})}}{(1-t)^{d - l({\mathbf{t}})}}
=\sum_{{\mathbf{t}}\in T_{w,{\mathbf{s}}}}\frac{(-1)^{e({\mathbf{t}})}}{(1-t)^{d_w-e({\mathbf{t}})}}
=\sum_{k\in{\mathbb{ N}}}\frac{(-1)^k m_k}{(1-t)^{d_w-k}}.$$ The formula for the Hilbert polynomial follows similarly. The highest degree term of $h(X^w,v)(n)$ is $m_0 n^{d_w-1}/(d_w-1)!$, implying that ${\operatorname{mult}}(X^w,v)=m_0$.
The formula for multiplicity in the above corollary can be restated as follows. Let $G/P$ be a cominuscule generalized flag variety and $v, w \in W^P$. Fix a reduced expression $(s_{1}, \ldots s_{l})$ for $v$. Recall that $v\geq w$ in the Bruhat order if and only if there exists a subsequence $(i_1,\ldots,i_m)$ of $({1}, \ldots, {l})$ such that $(s_{i_1},\ldots, s_{i_m})$ is a reduced expression for $w$; in this case, ${\operatorname{mult}}(X^w,v)$ is equal to the number of such subsequences.
The $0$-Hecke algebra and fully commutative elements {#s.nilHecke}
====================================================
This section contains some results about fully commutative elements and the $0$-Hecke algebra which we need to connect the pullback formula of Theorem \[t.pullback\] with the combinatorics of Young diagrams.
Given any $q$-tuple ${\mathbf{s}}= (s_1, \ldots, s_q)$ of elements of $S$, let $H_{{\mathbf{s}}} = H_{s_1} H_{s_2} \cdots H_{s_q}$. If $q = 0$ we define $H_{{\mathbf{s}}} = H_1$. Let $({\mathbf{s}}, s)$ denote the $q+1$-tuple $(s_1, \ldots, s_q,s)$, and given a $q'$-tuple ${\mathbf{s}}= (s'_1, \ldots, s'_{q'})$, let $({\mathbf{s}}, {\mathbf{s}}') = (s_1, \ldots, s_q, s'_1, \ldots, s'_{q'})$. The length of a $q$-tuple ${\mathbf{s}}$ is $l({\mathbf{s}}) = q$. If $$\label{e.reduced}
w = s_1 s_2 \cdots s_q$$ and $l(w) = l({\mathbf{s}})$ then we will say ${\mathbf{s}}$ is a reduced expression for $w$. We will also use the term reduced expression to refer to the equation .
We begin with some preliminary results.
\[l.minlength\] If $H_{{\mathbf{s}}} = H_w$, then $l({\mathbf{s}}) \ge l(w)$. If $l({\mathbf{s}}) = l(w)$, then ${\mathbf{s}}$ is a reduced expression for $w$.
We proceed by induction on $l({\mathbf{s}})$. If $l({\mathbf{s}}) = 0$ or $l({\mathbf{s}}) = 1$ then the lemma is trivial. Suppose the lemma is true for tuples of length $q$, and ${\mathbf{s}}= ({\mathbf{t}}, s_{q+1})$, where ${\mathbf{t}}= (s_1, \ldots, s_q)$. Then $H_{{\mathbf{t}}} = H_u$ for $u \in W$ with $l({\mathbf{t}}) \ge l(u)$. Then $w$ equals either $u$ or $u s_{q+1}$, so $$l({\mathbf{s}}) = l({\mathbf{t}})+1 \ge l(u)+1 \ge l(w).$$
Now assume that $l({\mathbf{s}}) = l(w)$, so $l(w)=q+1$. Then since $l(u)\leq l({\mathbf{t}})=q$ and $l(w)$ equals either $l(u)$ or $l(u)+1$, we must have that $l(u)=q$ and $l(w)=l(u)+1$. Thus $w$ must equal $u s_{q+1}$. By the inductive hypothesis, $u=s_1\cdots s_q$. Therefore $w$ equals $s_1\cdots s_{q+1}$.
The right (resp. left) weak order on $W$ is the transitive closure of the relation $u <_R us$ (resp. $u<_L su$) for $u \in W$, $s \in S$ with $l(u) < l(us)$ (resp. $l(u)< l(su)$). Given a reduced expression $w = s_1 \cdots s_q$, for any $k<q$, we have $s_1 \ldots s_k <_R w$, and for any $k>1$, we have $s_k s_{k+1} \cdots s_q <_L w$. We can extend these results to the Hecke algebra; we only state the version using $<_R$.
\[p.weakorder\] Suppose that ${\mathbf{s}}= (s_1, \ldots, s_q)$ and $H_{{\mathbf{s}}} = H_w$. Suppose that $k \le q$ and ${\mathbf{t}}= (s_1, \ldots, s_k)$ is a reduced expression for $u = s_1 \cdots s_k$. Then $u \le_R w$.
Let ${\mathbf{r}}= (s_{k+1}, \ldots, s_q)$, so ${\mathbf{s}}= ({\mathbf{t}}, {\mathbf{r}})$. It suffices to show that there is a subsequence ${\mathbf{q}}$ of ${\mathbf{r}}$ such that $({\mathbf{t}}, {\mathbf{q}})$ is a reduced expression for $w$. We proceed by induction on $l({\mathbf{s}})$. Lemma \[l.minlength\] implies that $l({\mathbf{s}}) \ge l(w)$. If $l({\mathbf{s}})= l(w)$ we are done. Otherwise, there is some $j \ge k$ such that $$u < u s_{k+1} < \cdots u s_{k+1} \cdots s_j > u s_{k+1} \cdots s_j s_{j+1}.$$ Let ${\mathbf{r}}'$ denote the sequence ${\mathbf{r}}$ with $s_{j+1}$ deleted, and let ${\mathbf{s}}' = ({\mathbf{t}}, {\mathbf{r}}')$. Since $H_{s_1} \ldots H_{s_j} = H_{s_1} \ldots H_{s_j} H_{s_{j+1}}$, we have $H_{{\mathbf{s}}'} = H_{{\mathbf{s}}}$. Our inductive hypothesis to ${\mathbf{s}}'$ implies that there is a subsequence ${\mathbf{q}}$ of ${\mathbf{r}}'$ such that $({\mathbf{t}}, {\mathbf{r}}')$ is a reduced expression for $w$. Since ${\mathbf{q}}$ is also a subsequence of ${\mathbf{r}}$, the result follows.
Given two elements $s,t \in S$, let $m(s,t)$ denote the order of $st$ in $W$. Given any $q$-tuple ${\mathbf{s}}= (s_1, \ldots, s_q)$ of elements of $S$, let ${\mathbf{s}}(s,t)$ denote the sub-tuple of ${\mathbf{s}}$ formed by the occurrences of $s$ and $t$. For example, if ${\mathbf{s}}= (s,t,u,s,u,s,v,s,t)$ then ${\mathbf{s}}(s,t) = (s,t,s,s,s,t)$.
Given $w \in W$ of length $q$, let $V_w$ denote the set of all $q$-tuples ${\mathbf{s}}$ which are reduced expressions for $w$. Form a graph $G_w$ with vertex set $V_w$, such that ${\mathbf{s}}= (s_1, \ldots, s_q), {\mathbf{t}}= (t_1, \ldots, t_q) \in V_w$ are joined by an edge if there are elements $s,t \in S$ and a sequence of indices $i, i+1, i+2, \ldots, i+ m(s,t)$ such that $(s_{i+1}, s_{i+2}, \ldots, s_{i+m(s,t)}) = (s,t, \ldots )$ and $(t_{i+1}, t_{i+2}, \ldots, t_{i+m(s,t)}) = (t,s, \ldots )$ (we will say that this edge corresponds to the braid relation between $s$ and $t$).
An element $w \in W$ is called fully commutative if any reduced expression for $w$ can be obtained from any other by using only the relations $st = ts$ where $s$ and $t$ are commuting elements of $S$. Suppose that $w$ is fully commutative and $m(s,t) \ge 3$ (that is, $s$ and $t$ do not commute). Then as observed by Stembridge [@Ste:96], there is no edge in the graph corresponding to the braid relation between $s$ and $t$. Stembridge also observed that this implies that if ${\mathbf{s}}$ and ${\mathbf{t}}$ are joined by an edge, then ${\mathbf{s}}(s,t) = {\mathbf{t}}(s,t)$, so since the graph is connected, ${\mathbf{s}}(s,t) = {\mathbf{t}}(s,t)$ for any two elements ${\mathbf{s}}$ and ${\mathbf{t}}$ of $V_w$. Write $w(s,t) = {\mathbf{s}}(s,t)$ where ${\mathbf{s}}$ is any element of $V_w$.
Observe that $w(s,t)$ can have repeated elements. For example, in type $B_3$, if $w=s_2 s_3 s_2 s_1$ (which is fully commutative), then $w(s_1,s_2) = (s_2, s_2, s_1)$.
\[l.reflectionsametimes\] Let $w$ be fully commutative, and let ${\mathbf{s}}$, ${\mathbf{t}}\in V_w$. Then any $s\in S$ occurs the same number of times in ${\mathbf{s}}$ as in ${\mathbf{t}}$.
If two elements of $V_w$ are connected by an edge, then they differ only by the interchange of two elements of $S$. Therefore $s$ must occur the same number of times in both elements. Since the graph $G_w$ is connected, the result follows.
\[p.statistic2\] Suppose ${\mathbf{s}}= (s_1, \ldots, s_q)$ satisfies $H_{{\mathbf{s}}} = H_w$, where $w$ is fully commutative, and suppose that $q > l = l(w)$. Then there exist $i<j$ such that $s_i=s_j$ and $s_i$ commutes with $s_k$ for every $k$, $i<k<j$.
Since $q>l$, there is some index $j$ such that $$\label{e.statistic}
s_1 < s_1 s_2 < \cdots < s_1 s_2 \ldots s_{j-1}>s_1 s_2 \ldots s_{j-1} s_j.$$ Let $s = s_j$ and $u = s_1 s_2 \cdots s_{j-1}$; then ${\mathbf{s}}' = (s_1, \ldots, s_{j-1})$ is a reduced expression for $u$. By Proposition \[p.weakorder\], $u \le_R w$, so $u$ is fully commutative (see [@Ste:96]). Because $us<s$, there is a reduced expression for $u$ which ends in $s$. Therefore any reduced expression for $u$ must have at least one term equal to $s$. In particular this holds for the reduced expression ${\mathbf{s}}'$. Let $i$ be the largest integer with $1 \le i \le j-1$ satisfying $s_i=s$. It suffices to show that $s_k$ commutes with $s$ for all $k$ with $i<k<j$. Suppose this fails; then $t = s_k$ does not commute with $s$ for some $k$ with $i<k<j$. We have chosen $i$ so that $s$ is not an element of the set $\{s_{k+1} , s_{k+2}, \ldots,s_{ j-1} \}$. Therefore $u(s,t) = {\mathbf{s}}'(s,t)$ ends in $t$. On the other hand, $u(s,t) = s$ since $u$ has a reduced expression ending in $s$. This is a contradiction. We conclude that $s_k$ commutes with $s$ for all $k$ with $i<k<j$, as desired.
Applications to the Grassmannian {#s.Grassmannian}
================================
Let $G$ be ${\operatorname{SL}}_n({\mathbb{C}})$, $P_d$ the maximal parabolic subgroup of $G$ corresponding to the simple root ${\alpha}_d$, $B^-$ the Borel subgroup of lower triangular matrices in $G$, and $T$ the group of diagonal matrices in $G$. The Weyl group $W=N_G(T)/T$ is isomorphic to $S_n$, the permutation group on $n$ elements. The Weyl group $W_{P_d}$ of $P_d$ is isomorphic to $S_d\times S_{n-d}$, and the set $W^{P_d}$ of minimal length coset representatives of $W/W_{P_d}$ consists of the permutations $w=(w_1,\ldots,w_n)$ such that $w_1<\cdots<w_d$ and $w_{d+1}<\cdots<w_n$. The coset space $G/P_d$ is identified with ${\operatorname{Grass }}(d,n)$, the Grassmannian variety of $d$-dimensional complex subspaces of ${\mathbb{C}}^n$. It is an irreducible projective variety of complex dimension $d(n-d)$. The cosets $wP_d$, $w\in W^{P_d}$, are precisely the $T$-fixed points of $G/P_d$. By abuse of notation, we often denote $wP_d$ by $w$. The Schubert variety $X^w$ is by definition ${\overline{B^-wP_d}}\subseteq G/P_d$. It is an irreducible projective variety of codimension $l(w)$. For $v,w\in W^{P_d}$, $v\in X^w$ if and only if $v\geq w$ in the Bruhat order.
In this section, we give formulas for $i_v^*[{\mathcal{O}}_{X^w}]$ (Theorems \[t.ktheory\_eyd\] and \[t.ktheory\_svt\]), as well as the Hilbert series, Hilbert polynomial, and multiplicity of $X^w$ at $v$ (Section \[ss.hilbseries\_An\]). These are reformulations of Theorem \[t.pullback\] and Corollary \[c.cominusculemult\], expressed in terms of indexing sets which are specific to the combinatorics of the the symmetric group and the Grassmannian. Namely, our indexing sets are [*excited Young diagrams*]{} and [*set-valued tableaux*]{}. The term excited Young diagram is due to Ikeda and Naruse [@IkNa:09]; in [@Kre:05] this is called a subset of a Young diagram. In this paper we have modified the definition of excited Young diagram for our applications to $K$-theory; the earlier definition corresponds to our [*reduced*]{} excited Young diagram. Reduced excited Young diagrams were discovered independently by Kreiman [@Kre:05] and Ikeda and Naruse [@IkNa:09]. A related version of excited Young diagram introduced in [@IkNa:11] is discussed in Section \[ss.previous\].
Our formula for $i_v^*[{\mathcal{O}}_{X^w}]$, expressed in terms of set-valued tableaux, was obtained in [@Kre:05]. The derivation of the formula there relies on an equivariant Gröbner degeneration of $X^w$ in a neighborhood of $v$ to a union of coordinate subspaces; this degeneration is due to [@KoRa:03], [@Kre:03], [@KrLa:04], and [@Kre:08]. In [@Kre:05], computing $i_v^*[{\mathcal{O}}_{X^w}]$ involves cataloging the weights of the intersections of these coordinate subspaces. Each coordinate subspace is expressed as $V(m_T)$, where $m_T$ is a monomial indexed by the entries of a Young tableau $T$. The intersection $V(m_{T_1})\cap\cdots \cap V(m_{T_k})$ is then equal to $V(m_{T_1},\ldots,m_{T_k})=V(m_{T_1\cup\cdots\cup T_k})$, where $T_1\cup\cdots\cup T_k$ is the set-valued tableau whose entry in each box is equal to the union of the entries of $T_1,\ldots,T_k$ in the same box. In this way set-valued tableaux arise naturally from this approach.
Our approach is different. In both this section and the next one, our methods are modeled on those of Ikeda and Naruse [@IkNa:09]. We generalize their arguments from reduced excited Young diagrams to excited Young diagrams, and correspondingly from nil-Coxeter algebras to 0-Hecke algebras. In several places, we use their results directly. Whereas set-valued tableaux are more suitable for the methods used in [@Kre:05], excited Young diagrams are more suitable for the methods used here.
Permutations, partitions, and Young diagrams {#ss.partitions_An}
--------------------------------------------
A **partition** is a sequence of integers ${\lambda}=({\lambda}_1,\ldots,{\lambda}_d)$ such that ${\lambda}_1\geq\cdots\geq{\lambda}_d\geq 0$. Let ${\mathcal{P}}_{d,n-d}$ denote the set of partitions such that ${\lambda}_1\leq n-d$. A **Young diagram** is a set of boxes arranged in a left justified array, such that the row lengths weakly decrease from top to bottom. To any partition ${\lambda}$ we associate the Young diagram $D_{\lambda}$ whose $i$-th row contains ${\lambda}_i$ boxes.
Let $d=5$, $n=11$, and ${\lambda}=(4,4,2,1)\in{\mathcal{P}}_{d,n-d}$. The Young diagram $D_{\lambda}$ fits inside a $d\times (n-d)$ rectangle.
(.5,.5) – ++(6,0) – ++(0,5) – ++(-6,0) – cycle; in [ [(1,5)]{},[(2,5)]{},[(3,5)]{},[(4,5)]{}, [(1,4)]{},[(2,4)]{},[(3,4)]{},[(4,4)]{}, [(1,3)]{}, [(2,3)]{}, [(1,2)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{}
The map $W^{P_d}\to {\mathcal{P}}_{d,n}$ given by $v\mapsto {\lambda}_v$, where $$\label{e.lambdav}
({\lambda}_v)_i=v_{d+1-i}-(d+1-i), i=1,\ldots,d,$$ is a bijection. Thus $W^{P_d}$, ${\mathcal{P}}_{d,n}$, and the set of Young diagrams which fit inside a $d\times (n-d)$ rectangle can all be identified; each parametrizes the $T$-fixed points of $G/P_d$. We now record several properties of these sets and relationships among them.
An **inversion of a permutation** $v=\{v_1,\ldots,v_n\}\in S_n$ is a pair $(i,j)$ for which $i<j$ and $v_i>v_j$. The number of inversions of $v$ equals $l(v)$.
Assume that $v\in W^{P_d}$. If $(i,j)$ is an inversion, then $i\in \{1,\ldots,d\}$ and $j\in\{d+1,\ldots,n\}$. For $i\in\{1,\ldots,d\}$ and $j\in\{d+1,\ldots,n\}$, define $l_i(v)=\#\{k\in \{d+1,\ldots,n\}\mid v_i>v_k\}$ and $l^j(v)=\#\{k\in \{1,\ldots,d\}\mid v_k>v_j\}$. Then $$\label{e.lenghtcomps}
l_i(v)=v_i-i, \qquad
l^j(v)=j-v_j,$$ and $$\label{e.lengthfromcomps}
l(v)=\sum\limits_{i=1}^d l_i(v)=\sum\limits_{j=d+1}^n l^j(v).$$ Equations and imply $$\label{e.partitionfromcomps}
({\lambda}_v)_i=l_{d+1-i}(v).$$ The length $|{\lambda}|$ of a partition ${\lambda}$ is defined to be ${\lambda}_1+\cdots+{\lambda}_d$.
\[l.lengthofperm\] For $v\in W^{P_d}$, $|{\lambda}_v|=l(v)$.
By and , $|{\lambda}_v|=\sum_{i=1}^d ({\lambda}_v)_i=\sum_{i=1}^d l_{d+1-i}(v)=\sum_{i=1}^d l_i(v)=l(v)$.
For ${\lambda}\in {\mathcal{P}}_{d,n}$, define the partition ${\lambda}^t=({\lambda}^t_1,\ldots,{\lambda}^t_{n-d})\in P_{n-d,n }$ by ${\lambda}^t_j=\#\{i\in \{1,\ldots,d\}\mid {\lambda}_i\geq j\}$, $j=1,\ldots,n-d$. Then ${\lambda}^t_j$ is the number of boxes in the $j$-th column of $D_{\lambda}$. We call ${\lambda}^t$ the **transpose** of ${\lambda}$.
\[l.trans\_partition\] For $v\in W^{P_d}$, $({\lambda}_v)^t_j=(d+j)-v_{d+j}$, $j=1,\ldots,n-d$.
$({\lambda}_v)^t_j
=\#\{i\in \{1,\ldots,d\}\mid ({\lambda}_v)_i\geq j\}
=\#\{i\in \{1,\ldots,d\}\mid l_{d+1-i}(v)\geq j\}
=\#\{i\in \{1,\ldots,d\}\mid l_i(v)\geq j\}
=\#\{i\in \{1,\ldots,d\}\mid v_i>v_{d+j}\}
=l^{d+j}(v)
=(d+j)-v_{d+j}.
$
The following lemma will be needed in Section \[s.orthosymplectic\]. We say that a partition ${\lambda}$ is **symmetric** if ${\lambda}^t={\lambda}$, and that a Young diagram is symmetric if the length of column $j$ equals the length of row $j$ for all $j$. Clearly ${\lambda}$ is symmetric if and only if $D_{\lambda}$ is symmetric. We say that a permutation $v=(v_1,\ldots,v_{2n})$ is **symmetric** if $v_{{\overline{\imath}}}={\overline{v_i}}$, $i=1,\ldots,n$, where ${\overline{x}}=2n+1-x$.
\[l.symmsymm\] The permutation $v=(v_1,\ldots,v_{2n})\in W^{P_{n}}$ is symmetric if and only if ${\lambda}_{v}$ is symmetric.
We have [$$\begin{aligned}
v_{{\overline{\imath}}}={\overline{v_i}}, i=1,\ldots, n
&\iff v_{{\overline{\imath}}}+v_i=2n+1,i=1,\ldots,n\\
&\iff v_{n+j}+v_{n+1-j}=2n+1, j=1,\ldots, n\\
&\iff (n+j)-v_{n+j}=v_{n+1-j}-(n+1-j), j=1,\ldots, n\\
&\iff ({\lambda}_v)^t_j=({\lambda}_v)_j, j=1,\ldots, n\end{aligned}$$ ]{}
Restriction formula in terms of excited Young diagrams {#ss.eyd}
------------------------------------------------------
Our convention is to number the rows of a Young diagram from top to bottom and the columns from left to right. The box in row $i$ and column $j$ is denoted by $(i,j)$.
Suppose that $C$ is any subset of $D_{\lambda}$, $(i,j)\in C$, and $(i+1,j),(i,j+1),(i+1,j+1)\in D_{\lambda}\setminus C$. Define an **excitation of type 1** to be an operation which replaces $C$ by $C'=C\setminus (i,j)\cup (i+1,j+1)$, and denote such an operation by $C{{\ \tikz[baseline={([yshift=-.7ex]current bounding box.center)}]{\draw[>=latex,->] (0,0) -- (.85,0);}\ }}C'$. Define an **excitation of type 2** to be an operation which replaces $C$ by $C''=C\cup (i+1,j+1)$, and denote such an operation by $C{{\ \tikz[baseline={([yshift=-.7ex]current bounding box.center)}]{\draw[double distance = 1.5pt,>=latex,->] (0,0) -- (.85,0);}\ }}C''$.
Let ${\lambda},{\mu}\in I_{d,n}$. If ${\lambda}_i\leq {\mu}_i$, $i=1,\ldots,d$, then the map $(i,j)\mapsto (i,j)$ embeds $D_{\lambda}$ as a subset of $D_{\mu}$. An **excited Young diagram** of $D_{\lambda}$ in $D_{\mu}$ is defined to be a subset of $D_{\mu}$ which can be obtained by applying a sequence of excitations to the subset $D_{\lambda}$. An excited Young diagram is said to be **reduced** if it can be obtained from $D_{\lambda}$ by applying only type 1 excitations. Denote the set of excited Young diagrams of $D_{\lambda}$ in $D_{\mu}$ by ${\mathcal{E}}_{\lambda}({\mu})$, and the set of reduced excited Young diagrams by ${\mathcal{E}}^{\text{red}}_{\lambda}({\mu})$ (see Figure \[f.eyd\_all\]).
\[t.ktheory\_eyd\] Let $w\leq v\in W^{P_d}$, and let ${\lambda}={\lambda}_w$, ${\mu}={\lambda}_v$ be the corresponding partitions. Then $\displaystyle
i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{C\in{\mathcal{E}}_{{\lambda}}({\mu})}\prod_{(i,j)\in C}\left(e^{{\epsilon}_{v_{d+1-i}}-{\epsilon}_{v_{d+j}}}-1\right)
$
\[ex.restriction\_A\] Let $n=7$, $d=3$. For $v=\{4,6,7,1,2,3,5\},w=\{1,3,5,2,4,6,7\}\in W^{P_d}$, ${\mu}={\lambda}_v=(4,4,3)$ and ${\lambda}={\lambda}_w=(2,1)$. Thus $l(w)=|{\lambda}_w|=3$. The set of excited Young diagrams ${\mathcal{E}}_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_all\]. For any $C\in {\mathcal{E}}_{{\lambda}}({\mu})$ and $(i,j)\in C$, a simple method of finding the indices $v_{d+j}$ and $v_{d+1-i}$ of Theorem \[t.ktheory\_eyd\] is to label the rows and columns of $C$ with the entries of $v$ as indicated below. Then $v_{d+1-i}$ and $v_{d+j}$ are the row $i$ and column $j$ labels respectively. For example, for $$C\ =\
\begin{tikzpicture}[scale=.6,every node/.style={scale=1},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos in
{{(1,3)},{(2,3)},{(2,1)}}
{\draw[fill=blue!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,3)},{(2,3)},{(3,3)},{(4,3)},
{(1,2)},{(2,2)},{(3,2)},{(4,2)},
{(1,1)}, {(2,1)},{(3,1)}
}
{\draw \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0,1) node{$4$}; \draw (0,2) node{$6$}; \draw (0,3) node{$7$};
\draw (1,4) node{$1$}; \draw (2,4) node{$2$}; \draw (3,4) node{$3$};
\draw (4,4) node{$5$};
\end{tikzpicture}\ \in{\mathcal{E}}_{{\lambda}_w}({\lambda}_v),$$
we have $\prod_{(i,j)\in C}\left(e^{{\epsilon}_{v_{d+1-i}}-{\epsilon}_{v_{d+j}}}-1\right)
=\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right).
$
By Theorem \[t.ktheory\_eyd\], [$$\begin{aligned}
i_v^*[{\mathcal{O}}_{X^w}] &=
-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)
-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)\\
&- \left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)
-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\\
&-\left(e^{{\epsilon}_6-{\epsilon}_2}-1\right)\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)\\
&-\left(e^{{\epsilon}_7-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_7-{\epsilon}_2}-1\right)
\left(e^{{\epsilon}_6-{\epsilon}_1}-1\right)\left(e^{{\epsilon}_6-{\epsilon}_3}-1\right)
\left(e^{{\epsilon}_4-{\epsilon}_2}-1\right).\end{aligned}$$ ]{}
Theorem \[t.ktheory\_eyd\] is a reformulation of Theorem \[t.pullback\], in which the indexing set $T(w,{\mathbf{s}})$ and integer $r(c)$ of the latter theorem are replaced by expressions involving excited Young diagrams. These replacements are given in Proposition \[p.subsequences\].
Let $w,v\in W^{P_d}$, and let ${\lambda}={\lambda}_w$ and ${\mu}={\lambda}_v$ be the corresponding partitions. Fill in each box $(i,j)$ of $D_{{\mu}}$ with the simple reflection $s_{d+j-i}$, thus obtaining a reflection-valued tableau denoted by $T_{{\mu}}$. Then $v=s_{i_1}s_{i_2}\cdots s_{i_l}$, where $s_{i_1},s_{i_2},\ldots,s_{i_l}$ are the entries of $T_{{\mu}}$ read from right to left, beginning with the bottom row, then the next row up, etc. Since $l=|D_{{\mu}}|=|{\mu}|=l(v)$, this decomposition is reduced. To any subset $C\subseteq D_{{\mu}}$, form the subsequence ${\mathbf{s}}_C=(s_{j_1},\ldots,s_{j_q})$ of $(s_{i_1},\ldots,s_{i_l})$ whose entries lie in the set $C$ of boxes of $T_{{\mu}}$. If $C$ and $D$ are different subsets of $D_{\mu}$, then we regard ${\mathbf{s}}_C$ and ${\mathbf{s}}_D$ as different subsequences of $(s_{i_1},\ldots,s_{i_l})$, even if they have the same entries.
\[ex.reduceddecomp\] Let $n=8$, $d=4$. For $v=\{3,5,6,8,1,2,4,7\}\in W^{P_d}$, ${\mu}={\lambda}_v=(4,3,3,2)$, $$\begin{aligned}
T_{{\mu}}&=
\begin{tikzpicture}[
scale=.4,
every node/.style={scale=.8},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos / \label in
{ {(1,4)}/{s_4},{(2,4)}/{s_5},{(3,4)}/{s_6},{(4,4)}/{s_7},
{(1,3)}/{s_3},{(2,3)}/{s_4},{(3,3)}/{s_5},
{(1,2)}/{s_2},{(2,2)}/{s_3},{(3,2)}/{s_4},
{(1,1)}/{s_1},{(2,1)}/{s_2}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\end{tikzpicture}
\intertext{and $s_2s_1s_4s_3s_2s_5s_4s_3s_7s_6s_5s_4$ is a reduced decomposition for $v$. For}
C&=
\begin{tikzpicture}[scale=.4,every node/.style={scale=.8},baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos in
{{(1,2)},{(1,3)},{(2,3)},{(3,2)},{(4,4)}}
{\draw[fill=green!90] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,4)},{(2,4)},{(3,4)},{(4,4)},
{(1,3)},{(2,3)},{(3,3)},
{(1,2)},{(2,2)},{(3,2)},
{(1,1)},{(2,1)}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
}
\end{tikzpicture}
\subset D_{{\mu}},\end{aligned}$$ we have ${\mathbf{s}}_C=(s_4,s_2,s_4,s_3,s_7)$.
\[p.subsequences\] Denote the reduced decomposition $(s_{i_1},\cdots, s_{i_l})$ for $v$ obtained above by ${\mathbf{s}}_v$. By definition, $T(w,{\mathbf{s}}_v)=\{{\mathbf{s}}_C\mid C\subseteq D_{\mu}, H_{{\mathbf{s}}_C}=H_w\}$. We have
- $T(w,{\mathbf{s}}_v)=\{{\mathbf{s}}_C\mid C\in{\mathcal{E}}_{{\lambda}}({\mu})\}$.
- Let $(i,j)$ be the box of $T_{{\mu}}$ containing $s_{i_c}$. Then $r(c)={\epsilon}_{v_{d+j}}-{\epsilon}_{v_{d+1-i}}$.
The reduced decomposition ${\mathbf{s}}_v$ can be deduced from a more general method of producing reduced decompositions of arbitrary permutations (see [@Man:01 Remark 2.1.9]). The decomposition here also appears in Ikeda and Naruse [@IkNa:09]. Proposition \[p.subsequences\](ii) is due to Ikeda and Naruse [@IkNa:09]. A version of Proposition \[p.subsequences\](i) which involves the nil-Coxeter alegra and (what we call) reduced excited Young diagrams is also proved in [@IkNa:09].
For (co)Grassmannian permutations $w$ and $v$, the notion of a [*pipe dream for $w$ on $D(v)$*]{}, introduced by Woo and Yong [@WoYo:12] (see also [@LiYo:12]), is closely related to that of an excited Young diagram of $D_{\lambda}$ in $D_{\mu}$. In our language, a [*pipe dream for $w$ on $D(v)$*]{} is equal to a subset $C$ of $D_{\mu}$ such that $H_{{\mathbf{s}}_C}=H_w$. By Proposition \[p.subsequences\](i), the set of all such pipe dreams is equal to ${\mathcal{E}}_{\lambda}({\mu})$. General pipe dreams have been studied by [@BeBi:93], [@FoKi:96], [@KnMi:04], [@KnMi:05]. The excited Young diagrams introduced in [@IkNa:11] are also related to those of this paper (see Section \[ss.previous\]).
### Proof of Proposition \[p.subsequences\](i) {#ss.proof_p.subsequences}
Let $C$ be a subset of $D_{\mu}$. Proposition \[p.subsequences\](i) is equivalent to $$\label{e.subsequences}
H_{{\mathbf{s}}_C}=H_w\text{ if and only if }C\in{\mathcal{E}}_{\lambda}({\mu}).$$ The proof of the reverse implication is fairly straightforward (see Lemma \[l.containmentone\]). The proof of the forward implication (Lemma \[l.containmenttwo\]) is by induction on $e_1(C)$ and $e_2(C)$, defined below. Lemma \[l.hecketoeyd\] helps us to translate from excited Young diagrams to 0-Hecke algebras.
\[d.energy\] If $H_{{\mathbf{s}}_C}=H_w$, then define
1. $|C|=$ number of boxes of $C$
2. $e_1(C)=(1/2)(\sum_{(i,j)\in C}(i+j)-\sum_{(i,j)\in D_{\lambda}}(i+j))$
3. $e_2(C)=|C|-|D_{{\lambda}}|=|C|-|{\lambda}|$
We call $e_1(C)$ and $e_2(C)$ the **type 1 energy** and **type 2 energy** of $C$ respectively.
\[l.hecketoeyd\] If $H_{{\mathbf{s}}_C}=H_w$, then
1. $|C|=l({\mathbf{s}}_C)$.
2. $e_2(C)=e({\mathbf{s}}_C)$.
3. $e_2(C)\geq 0$, and $e_2(C)=0$ if and only if ${\mathbf{s}}_C$ is a reduced expression for $w$.
\(i) Clear from the definition of ${\mathbf{s}}_C$.
\(ii) $e_2(C)=|C|-|{\lambda}|=l({\mathbf{s}}_C)-l(w)=e({\mathbf{s}}_C)$, where the last equality is the definition of $e({\mathbf{s}}_C)$. (iii) By (ii), this can be restated: If $H_{{\mathbf{s}}_C}=H_w$, then $l({\mathbf{s}}_C)-l(w)\geq 0$, and $l({\mathbf{s}}_C)-l(w)=0$ if and only if ${\mathbf{s}}_C$ is a reduced expression for $w$. This follows immediately from Lemma \[l.minlength\].
If $C\in{\mathcal{E}}_{\lambda}({\mu})$, then one can give interpretations of $e_1(C)$ and $e_2(C)$ in terms of excitations; these interpretations are not required for the sequel. An excitation of type 1 applied to $C$ does not alter $|C|$, whereas an excitation of type 2 increases it by 1. Thus $e_2(C)$ is the number of excitations of type 2 which must be applied in order to obtain $C$ from $D_{{\lambda}}$. In particular, $e_2(C)\geq 0$, and $e_2(C)=0$ if and only if $C$ is reduced. In general, the number of type 1 excitations which must be applied in order to obtain $C$ from $D_{\lambda}$ is not defined, since it may be possible to obtain $C$ by two different sequences of excitations which have a different number of type 1 excitations. However, if $C$ is reduced, then this number is defined and is given by $e_1(C)$.
**Diagonal $k$** of $D_{\lambda}$ is defined to be the set of boxes $(i,j)$ such that $j-i=k$. We say that box $(i,j)$ **lies on diagonal $k$** if $j-i=k$. Impose the following total order on the boxes of $D_{\mu}$: $(i,j)<(k,l)$ if $i>k$ or if $i=k$ and $j>l$.
\[l.modifysubsequence\] Let ${\alpha}'<{\alpha}$ be two boxes of $D_{\mu}$ which lie on the same diagonal $k$. Suppose that ${\alpha}\in C$, ${\alpha}'\not\in C$, and that $C$ contains no box ${\beta}$, ${\alpha}'<{\beta}<{\alpha}$, which lies on diagonal $k-1$, $k$, or $k+1$. Then $H_{{\mathbf{s}}_{C\,\setminus {\alpha}\,\cup\,{\alpha}'}}=H_{{\mathbf{s}}_{C \,\cup\,{\alpha}'}}=H_{{\mathbf{s}}_C}$.
$$C=\ \
\begin{tikzpicture}[xscale=.4,yscale=.4,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.6}]
\foreach \pos in
{{(4,8)},{(3,7)},{(4,7)},{(5,7)},{(4,6)},{(5,6)},(6,6),(5,5),(6,5)}
{\draw[fill=red!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw[fill=green!90] (3,8) +(-.5,-.5) rectangle ++(.5,.5);
\draw (0.5,0.5) -- ++(2,0) -- ++(0,1) -- ++(1,0) -- ++(0,2) -- ++(4,0) -- ++(0,3) -- ++(3,0) -- ++(0,3) -- ++(-10,0)
-- cycle;
\node[inner sep=1pt] (src_one) at (1,9) {${\alpha}$};
\node (dst_one) at (3,8) {};
\draw[->] (src_one) to [out=-40,in=180] (dst_one);
\node[inner sep=3pt] (src_two) at (8.5,4) {${\alpha}'$};
\node (dst_two) at (6,5) {};
\draw[->] (src_two.west) to [out=170,in=-90] (dst_two);
\end{tikzpicture}$$
Boxes ${\alpha}'$ and ${\alpha}$ of $T_{\mu}$ contain the same reflection, namely $s_{k+d}$; let $r=k+d$. To create ${\mathbf{s}}_{C\,\setminus {\alpha}\,\cup\,{\alpha}'}$ from ${\mathbf{s}}_C$, one moves the reflection $s_r$ of ${\mathbf{s}}_C$ which lies in box ${\alpha}$ of $C$ past all of the reflections of ${\mathbf{s}}_C$ which lie in boxes ${\beta}$ of $C$, ${\alpha}'<{\beta}<{\alpha}$. Since all such reflections lie outside of diagonals $k-1$, $k$, and $k+1$, $s_r$ commutes with them. Thus $H_{{\mathbf{s}}_{C\,\setminus {\alpha}\,\cup\,{\alpha}'}}=H_{{\mathbf{s}}_C}$. One proves $H_{{\mathbf{s}}_{C \,\cup\,{\alpha}'}}=H_{{\mathbf{s}}_C}$ similarly.
\[l.containmentone\] If $C\in{\mathcal{E}}_{{\lambda}}({\mu})$, then $H_{{\mathbf{s}}_C}=H_w$.
Since $C$ is obtained by applying a sequence of excitations to $D_{{\lambda}}$, and $H_w=H_{{\mathbf{s}}_{D_{{\lambda}}}}$, we need only prove that applying a single excitation to a subset $D$ of $D_{\mu}$ does not alter $H_{{\mathbf{s}}_D}$. This is a special case of Lemma \[l.modifysubsequence\].
\[l.containmenttwo\] If $H_{{\mathbf{s}}_C}=H_w$, then $C\in {\mathcal{E}}_{{\lambda}}({\mu})$.
We prove this lemma for the following three cases, which increase in generality: first when $e_2(C)=0$ and $e_1(C)=0$, then when $e_2(C)=0$ and $e_1(C)$ is arbitrary, and finally with no restrictions on $C$. Each serves as the base case of an inductive proof for the subsequent more general case.
**Case 1**. $e_2(C)=0$ and $e_1(C)=0$.
Since $e_2(C)=0$, ${\mathbf{s}}_C$ is a reduced expression for $w$ by Lemma \[l.hecketoeyd\](iii). Thus, by Lemma \[l.reflectionsametimes\], any reflection must occur the same number of times in ${\mathbf{s}}_C$ as in ${\mathbf{s}}_{D_{{\lambda}}}$. Thus on each diagonal, $C$ must have the same number of boxes as $D_{{\lambda}}$ does. Since $e_1(C)=0$, $C=D_{{\lambda}}$.
**Case 2**. $e_2(C)=0$ and $e_1(C)$ arbitrary.
The proof is by induction on $e_1(C)$. If $e_1(C)=0$, then we are done, by Case 1. Assume that $e_1(C)>0$. Then $C\neq D_{{\lambda}}$. Since $e_2(C)=0$, ${\mathbf{s}}_C$ is a reduced expression for $w$ and therefore $C$ must have the same number of boxes as $D_{\lambda}$ does. Thus there must be some box which is contained in $D_{{\lambda}}$ but not in $C$. Let ${\alpha}$ be the maximal such, and let $k$ be the diagonal of ${\alpha}$. As in Case 1, on each diagonal, $C$ must have the same number of boxes as $D_{{\lambda}}$ does. This implies that $C$ must contain a box lying on diagonal $k$ and in a lower row than ${\alpha}$; let ${\alpha}'$ be the maximal such. By maximality of ${\alpha}'$, $C$ contains no box ${\beta}$, ${\alpha}'<{\beta}<{\alpha}$, which lies on diagonal $k$. Since ${\mathbf{s}}_{C}(s_k,s_{k-1})={\mathbf{s}}_{D_{{\lambda}}}(s_k,s_{k-1})$ (see Section \[s.nilHecke\]), $C$ contains no box ${\beta}$, ${\alpha}'<{\beta}<{\alpha}$, which lies on diagonal $k-1$. Since ${\mathbf{s}}_{C}(s_k,s_{k+1})={\mathbf{s}}_{D_{{\lambda}}}(s_k,s_{k+1})$, $C$ contains no box ${\beta}$, ${\alpha}'<{\beta}<{\alpha}$, which lies on diagonal $k+1$.
$$D_{{\lambda}}=\ \
\begin{tikzpicture}[xscale=.4,yscale=.4,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.6}]
\foreach \pos in
{{(1,9)},{(2,9)},{(3,9)},{(4,9)},
(1,8),(2,8),(3,8)}
{\draw[fill=green!90] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0.5,0.5) -- ++(2,0) -- ++(0,1) -- ++(1,0) -- ++(0,2) -- ++(4,0) -- ++(0,3) -- ++(3,0) -- ++(0,3) -- ++(-10,0)
-- cycle;
\node[inner sep=1pt] (src_one) at (5,7) {${\alpha}$};
\node (dst_one) at (3,8) {};
\draw[->] (src_one) to [out=170,in=-90] (dst_one);
\end{tikzpicture}
\qquad\qquad
C=\ \
\begin{tikzpicture}[xscale=.4,yscale=.4,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.6}]
\foreach \pos in
{{(1,9)},{(2,9)},{(3,9)},{(4,9)},
(1,8),(2,8),(6,5)}
{\draw[fill=green!90] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{{(3,8)},{(4,8)},{(3,7)},{(4,7)},{(5,7)},{(4,6)},{(5,6)},(6,6),(5,5)}
{\draw[fill=red!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0.5,0.5) -- ++(2,0) -- ++(0,1) -- ++(1,0) -- ++(0,2) -- ++(4,0) -- ++(0,3) -- ++(3,0) -- ++(0,3) -- ++(-10,0)
-- cycle;
\node[inner sep=3pt] (src_two) at (8.5,4) {${\alpha}'$};
\node (dst_two) at (6,5) {};
\draw[->] (src_two.west) to [out=170,in=-90] (dst_two);
\end{tikzpicture}$$
Let $A=C\setminus {\alpha}'\cup {\alpha}$. By Lemma \[l.modifysubsequence\], $H_{{\mathbf{s}}_A}=H_{{\mathbf{s}}_C}=H_w$. Furthermore, $e_1(A)<e_1(C)$. By induction, $A\in{\mathcal{E}}_{{\lambda}}({\mu})$. Since $C$ is obtained by applying (type 1) excitations to $A$, $C\in{\mathcal{E}}_{{\lambda}}({\mu})$.
**Case 3**. No restrictions on $C$.
The proof is by induction on $e_2(C)$. If $e_2(C)=0$, then we are done, by Case 2. Assume that $e_2(C)> 0$. Write ${\mathbf{s}}_C=(s_{j_1},\ldots,s_{j_q})$. Since $e_2(C)>0$, $q>l(w)$ by Lemma \[l.hecketoeyd\](iii). By assumption $H_{{\mathbf{s}}_C}=H_w$. Proposition \[p.statistic2\] implies that there exist $a<b$ such that $j_a=j_b$ and $s_{j_a}$ commutes with $s_{j_c}$ for every $c$, $a<c<b$. Let ${\alpha}'$ and ${\alpha}$ be the boxes of $C$ containing $s_{j_a}$ and $s_{j_b}$ respectively, and let $k$ be the diagonal of ${\alpha}$ and ${\alpha}'$. Then $C$ contains no box ${\beta}$, ${\alpha}'<{\beta}<{\alpha}$, which lies on diagonal $k-1$, $k$, or $k+1$.
$$C=\ \
\begin{tikzpicture}[xscale=.4,yscale=.4,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.6}]
\foreach \pos in
{(2,8),(5,5)}
{\draw[fill=green!90] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{{(3,8)},{(2,7)},{(3,7)},{(4,7)},{(5,6)},(3,6),{(4,6)},(5,6),(4,5)}
{\draw[fill=red!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0.5,0.5) -- ++(2,0) -- ++(0,1) -- ++(1,0) -- ++(0,2) -- ++(4,0) -- ++(0,3) -- ++(3,0) -- ++(0,3) -- ++(-10,0)
-- cycle;
\node[inner sep=1pt] (src_one) at (-.5,9) {${\alpha}$};
\node (dst_one) at (2,8) {};
\draw[->] (src_one) to [out=-40,in=180] (dst_one);
\node[inner sep=3pt] (src_two) at (8.5,4) {${\alpha}'$};
\node (dst_two) at (5,5) {};
\draw[->] (src_two.west) to [out=170,in=-90] (dst_two);
\end{tikzpicture}$$
Let $A=C\setminus {\alpha}'$. By Lemma \[l.modifysubsequence\], $H_{{\mathbf{s}}_A}=H_{{\mathbf{s}}_C}=H_w$. Furthermore, $e_2(A)<e_2(C)$. By induction, $A\in{\mathcal{E}}_{{\lambda}}({\mu})$. Since $C$ can be obtained by applying (type 2) excitations to $A$, $C\in{\mathcal{E}}_{{\lambda}}({\mu})$.
### Proof of Proposition \[p.subsequences\](ii)
By and Lemma \[l.trans\_partition\], there are $v_{d+1-i}-(d+1-i)$ boxes in row $i$ and $(d+j)-v_{d+j}$ boxes in column $j$ of $D_{{\lambda}_v}$. Recall that the entry of box $(l,m)$ of $T_{{\mu}}$ is $s_{d+m-l}$. Thus the entry of the rightmost box of row $i$ of $T_{{\mu}}$ is $s_{d+(v_{d+1-i}-(d+1-i))-i}=s_{v_{d+1-i}-1}$, and the entry of the bottom box of column $j$ is $s_{d+j-((d+j)-v_{d+j})}=s_{v_{d+j}}$. Figure \[f.r(ic)\_proof\] shows some of the entries of $T_{{\mu}}$: $s_x$ and $s_y$ are the reflections in the rightmost box of row $i$ and lowest box of column $j$ respectively. Thus $x=v_{d+1-i}-1$, $y=v_{d+j}$. Also, $p=d+j-i$.
$$T_{{\mu}}=
\,
\begin{tikzpicture}[xscale=.6,yscale=.6,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.7}]
\draw[gray!7,fill=gray!7] (0.5,0.5) -- ++(9,0) -- ++(0,2) -- ++(1,0) -- ++(0,2) -- ++(2,0) -- ++(0,2) -- ++(-9,0) -- ++(0,-1) -- ++(-3,0) -- cycle;
\draw[gray!30,fill=gray!30] (2.5,0.5) -- ++(0,6) -- ++(10,0) -- ++(0,-1) -- ++(-9,0) -- ++(0,-5) -- cycle;
\foreach \pos / \label in
{ {(3,6)}/{},{(4,6)}/{s_{p+1}},{(5,6)}/{s_{p+2}},{(6,6)}/{s_{p+3}},{(12,6)}/{s_{x}},
{(2,5)}/{s_{p-2}},{(3,5)}/{s_{p-1}},{(4,5)}/{s_{p}},{(5,5)}/{s_{p+1}},{(6,5)}/{s_{p+2}},{(12,5)}/{s_{x-1}},
{(2,4)}/{s_{p-3}},{(3,4)}/{s_{p-2}},{(4,4)}/{s_{p-1}},{(5,4)}/{s_{p}},{(6,4)}/{s_{p+1}},
{(2,1)}/{s_{y-1}},{(3,1)}/{s_{y}},{(4,1)}/{s_{y+1}},{(5,1)}/{s_{y+2}}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\draw (3,9) node{$j$};
\draw (0,6) node{$i$};
\draw (9,6) node{$\cdots$};
\draw (3,2.5) node{$\vdots$};
\draw (0.5,0.5) -- ++(9,0) -- ++(0,2) -- ++(1,0) -- ++(0,2) -- ++(2,0) -- ++(0,4) -- ++(-12,0) -- cycle;
\end{tikzpicture}$$
In the expression $r(c)=s_{i_1}s_{i_2}\cdots s_{i_{c-1}}({\alpha}_{i_c})= s_{i_1}s_{i_2}\cdots s_{p+1}({\epsilon}_p-{\epsilon}_{p+1})$, the reflections $s_{i_j}$ which lie outside darkly shaded boxes in Figure \[f.r(ic)\_proof\] can be removed. Thus [$$\begin{aligned}
r(c)
&= s_y \cdots s_{p-2}s_{p-1}s_x\cdots s_{p+2} s_{p+1} ({\epsilon}_p-{\epsilon}_{p+1})\\
&= s_y\cdots s_{p-2}s_{p-1}s_x\cdots s_{p+2}({\epsilon}_p-{\epsilon}_{p+2})\\
&= s_y\cdots s_{p-2}s_{p-1}s_x\cdots ({\epsilon}_p-{\epsilon}_{p+3})\\
&= s_y\cdots s_{p-2}s_{p-1}({\epsilon}_p-{\epsilon}_{x+1})\\
&= s_y\cdots s_{p-2}({\epsilon}_{p-1}-{\epsilon}_{x+1})\\
&= s_y\cdots ({\epsilon}_{p-2}-{\epsilon}_{x+1})\\
&= {\epsilon}_{y}-{\epsilon}_{x+1}
={\epsilon}_{v_{d+j}}-{\epsilon}_{v_{d+1-i}}.
$$ ]{}
Restriction formula in terms of set-valued tableaux {#ss.setvalued_tableaux}
---------------------------------------------------
Let ${\lambda}$ be a partition. A **set-valued filling** of $D_{\lambda}$ is a function $T$ which assigns to each box $(i,j)$ of $D_{{\lambda}}$ a nonempty subset $T(i,j)$ of $\{1,\ldots,d\}$. We call ${\lambda}$ the **shape** of $T$. We call $(i,j)$ a box of $T$, and refer to an element of $T(i,j)$ as an entry of box $(i,j)$ of $T$, or just an entry of $T$. A set-valued filling $T$ in which each entry of box $(i,j)$ of $T$ is less than or equal to each entry of box $(i,j+1)$ and strictly less than each entry of box $(i+1,j)$ is said to be **semistandard**. A **set-valued Young tableau**, or just **set-valued tableau**, is defined to be a semistandard set-valued filling of $D_{\lambda}$. A **Young tableau**, or just **tableau**, is a set-valued tableau in which each box contains a single entry.
Let ${\mu}$ be a partition. We say that a set-valued tableau is **restricted by ${\mu}$** if, for any box $(i,j)\in T$ and any entry $x$ of $(i,j)$, $$\label{e.restrictedbyv}
x+j-i\leq {\mu}(x).$$ Denote by ${\mathcal{T}}_{{\lambda}}({\mu})$ (resp. ${\mathcal{T}}^{\text{red}}_{{\lambda}}({\mu})$) the set of set-valued tableaux (resp. tableaux) of shape ${\lambda}$ which are restricted by ${\mu}$ (see Figure \[f.syt\_all\]).
\[t.ktheory\_svt\] Let $w\leq v\in W^{P_d}$, and let ${\lambda}={\lambda}_w$, ${\mu}={\lambda}_v$ be the corresponding partitions. Then $$i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{T\in{\mathcal{T}}_{{\lambda}}({\mu})}\prod_{(i,j)\in T}\prod_{x\in T(i,j)}\left(e^{{\epsilon}_{v_{d+1-x}}-{\epsilon}_{v_{d+x+j-i}}}-1\right)$$
Theorem \[t.ktheory\_svt\], which appeared in [@Kre:05], is essentially the same statement as Theorem \[t.ktheory\_eyd\], except that the indexing set ${\mathcal{E}}_{{\lambda}}({\mu})$ has been replaced by ${\mathcal{T}}_{{\lambda}}({\mu})$. Equation below defines a map $f$ between these indexing sets, and Proposition \[p.fbijective\] establishes that $f$ is a bijection. A similar and related bijection appears in [@KMY:09]. The map $f$ restricts to a bijection between ${\mathcal{E}}^{\text{red}}_{{\lambda}}({\mu})$ and ${\mathcal{T}}^{\text{red}}_{{\lambda}}({\mu})$, which was given in [@Kre:05] and [@WoYo:12], and is closely related to a bijection in [@Kog:00]. A bijection between ${\mathcal{E}}^{\text{red}}_{{\lambda}}({\mu})$ and the nonintersecting lattice paths of [@Kra:01], [@Kra:05], [@KoRa:03], [@Kre:03], [@KrLa:04], [@Kre:08] was given by [@Kre:05].
Set-valued tableaux of shape ${\lambda}$ restricted by ${\mu}$ were introduced in [@Kre:05]. They also appeared in [@WoYo:12], where they were identified as special types of flagged set-valued tableaux. General flagged set-valued tableaux, which were introduced in [@KMY:09], are set-valued tableaux whose entries in row $i$ are less than or equal to the $i$-th coordinate of a fixed vector ${{\mathfrak b}}$, which is called the flag. The specific flags utilized in [@KMY:09] are similar to ours. However, whereas their flags depend on one vexillary permutation, ours depend on two Grassmannian permutations, namely $w$ and $v$. We point out that [@WoYo:12] applies more generally to covexillary permutations.
One difference between set-valued tableaux and excited Young diagrams is that the former are defined locally, whereas the latter are not. One can determine whether a set-valued tableau $T$ of shape ${\lambda}$ lies in ${\mathcal{T}}_{{\lambda}}({\mu})$ by checking whether its entries satisfy the inequalities . In particular, one need only look only at $T$. On the other hand, according to the definition, in order to determine whether a subset $C$ of $D_{{\lambda}}$ lies in ${\mathcal{E}}_{{\lambda}}({\mu})$, one must search for a sequence of excitations which when applied to $D_{{\lambda}}$ produces $C$. One can give a local criterion for membership in ${\mathcal{E}}_{{\lambda}}({\mu})$ based on Proposition \[p.subsequences\](i): $C$ lies in ${\mathcal{E}}_{{\lambda}}({\mu})$ precisely when the product of the reflections of ${\mathbf{s}}_C$ equals $w$. Of course, checking this requires calculations in the 0-Hecke algebra.
Let $n=7$, $d=3$. For $v=\{4,6,7,1,2,3,5\}$ and $w=\{1,3,5,2,4,6,7\}$, ${\mu}={\lambda}_v=(4,4,3)$ and ${\lambda}={\lambda}_w=(2,1)$. The set-valued tableaux ${\mathcal{T}}_{{\lambda}}({\mu})$ appear in Figure \[f.syt\_all\]. The expression for $i_v^*[{\mathcal{O}}_{X^w}]$ computed using Theorem \[t.ktheory\_svt\] is the same as the expression computed using Theorem \[t.ktheory\_eyd\] in Example \[ex.restriction\_A\].
### Proof of Theorem \[t.ktheory\_svt\]
Define $f:{\mathcal{T}}_{{\lambda}}({\mu})\to\{\text{subsets of }D_{{\mu}}\}$ by $$\label{e.f}
\begin{split}
f(T)&= \{(x,x+j-i)\mid (i,j)\in T, x\in T(i,j)\}\\
&=\{(x,k\backslash\mid (i,k\backslash \in T, x\in T(i,k\backslash\}
\end{split}$$ for $T\in {\mathcal{T}}_{{\lambda}}({\mu})$, where $(i,k\backslash$ denotes the box in row $i$, diagonal $k$ of $T$. To see that every box of $f(T)$ lies in $D_{{\mu}}$, and thus $f$ is well defined, observe that for every box $(i,j)\in T$ and entry $x\in T(i,j)$, $1\leq x\leq d$ and $1\leq x+j-i\leq {\mu}(x)$. Indeed, $1\leq x\leq d$ by definition. Semistandardness of $T$ forces $i\leq x$; hence $1\leq x+j-i$. The final inequality, namely $x+j-i\leq {\mu}(x)$, is . As stated above, we prove Theorem \[t.ktheory\_svt\] by showing that $f$ is in fact a bijection from ${\mathcal{T}}_{\lambda}({\mu})$ to ${\mathcal{E}}_{\lambda}({\mu})$ (Proposition \[p.fbijective\]).
We emphasize that $T$ and $f(T)$ are associated to Young diagrams of different shapes: $T$ has shape ${\lambda}$, and $f(T)$ is a subset of $D_{\mu}$, which has shape ${\mu}$. The subset $f(T)$ of $D_{{\mu}}$ contains, for each integer $x$ of box $(i,j)$ of $T$, box $(x, x+j-i)$ of $D_{{\mu}}$. We write $f(T|_x)=(x, x+j-i)=(x,j-i\backslash$. Observe that $f(T|_x)$ lies in the same diagonal as $x$, namely $j-i$, but in row $x$ instead of $i$. This suggests a more qualitative description of $f$. Suppose that the entries of some box ${\alpha}$ of $T$ are $x_1,\ldots,x_k$. Corresponding to these entries, $f(T)$ will have boxes in $D_{{\mu}}$ in the same diagonal as ${\alpha}$, and rows $x_1,\ldots,x_k$; thus the entries of $T$ record the rows of the boxes of $f(T)$. The inequality merely ensures that the boxes of $f(T)$ actually lie in $D_{\mu}$.
In Section \[ss.eyd\] we defined excitations on subsets of Young diagrams. Here we need an analogous operation on set-valued tableaux. Let $T\in{\mathcal{T}}_{{\lambda}}({\mu})$, and suppose that $x\in T(i,j)$, $x\not\in T(i,j+1)$, $x+1\not\in T(i,j)$, $x+1\not\in T(i+1,j)$, $(x+1)+j-i\leq {\mu}(x+1)$. An excitation of type 1 replaces entry $x$ with $x+1$ in box $(i,j)$ of $T$, and an excitation of type 2 adds entry $x+1$ to box $(i,j)$ of $T$. Both types of excitations preserve semistandardness and the property of being restricted by ${\mu}$. Let $T^{\text{top}}\in {\mathcal{T}}_{{\lambda}}({\mu})$ be defined by $T^{\text{top}}(i,j)=\{i\}$, i.e., each box in row $i$ contains the single entry $i$. Then $f(T^{\text{top}})=D_{{\lambda}}$.
Any element $T\in {\mathcal{T}}_{{\lambda}}({\mu})$ can be obtained by applying a sequence of excitations to $T^{\text{top}}$.
For any set-valued tableau $T$ of shape ${\lambda}$, define $s(T)$ to be the sum of the entries of $T$. Then $s(T)\geq s(T^{\text{top}})$, and $s(T)=s(T^{\text{top}})$ if and only if $T=T^{\text{top}}$. We proceed by induction on $s(T)$. Assume that $s(T)>s(T^{\text{top}})$, and thus $T\neq T^{\text{top}}$.
At least one box of $T$ contains more than one entry. Choose any such box. Define $T'$ to be the set-valued tableau obtained by removing the second smallest entry from this box of $T$.
Every box of $T$ contains exactly one entry. Let $(i,j)$ be the largest box for which the entries of $T$ and $T^{\text{top}}$ do not agree (where we use the order on boxes introduced in Section \[ss.proof\_p.subsequences\]). Let $T'$ be the tableau obtained from $T$ by subtracting one from its entry in box $(i,j)$.
In both cases, $T'\in {\mathcal{T}}_{{\lambda}}({\mu})$ and $s(T')<s(T)$. By the induction hypothesis, $T'$ can be obtained by applying a sequence of excitations to $T^{\text{top}}$. Furthermore, $T$ can be obtained by applying a sequence of excitations to $T'$. The result follows.
Excitations commute with $f$, as made precise by the following lemma, which follows from the definitions.
\[l.fcommutesexcitation\] Let $T\in{\mathcal{T}}_{{\lambda}}({\mu})$, and let $C=f(T)$. Let $x$, $x<d$, be an entry of box $(i,j)$ of $T$, and let $(a,b)=f(T|_x)\in C$. Then $x+1\not\in T(i+1,j)$, $x\not\in T(i,j+1)$, $x+1\not\in T(i,j)$, $(x+1)+j-i\leq {\mu}(x+1)$ if and only $(a+1,b)\not\in C, (a,b+1)\not\in C,(a+1,b+1)\not\in C$, $(a+1,b+1)\in D_{{\mu}}$ respectively. If all of these conditions are satisfied, let ${\nu}$ be an excitation of $T$ modifying $x$, and let ${\mu}$ be the excitation of $C$ of the same type as ${\nu}$ modifying $(a,b)$. Then $f({\nu}(T))={\mu}(f(T))$. We say that ${\mu}$ corresponds to ${\nu}$ under $f$.
The image of $f$ lies in ${\mathcal{E}}_{{\lambda}}({\mu})$.
Let $T\in {\mathcal{T}}_{{\lambda}}({\mu})$. Then $T={\nu}_k\cdots{\nu}_1(T^{\text{top}})$ for some excitations ${\nu}_1,\ldots,{\nu}_k$. By Lemma \[l.fcommutesexcitation\], $f(T)={\mu}_k\cdots{\mu}_1(D_{{\lambda}})\in {\mathcal{E}}_{{\lambda}}({\mu})$, where ${\mu}_1,\ldots{\mu}_k$ correspond to ${\nu}_1,\ldots{\nu}_k$ under $f$.
\[l.fsurjective\] The image of $f$ equals ${\mathcal{E}}_{{\lambda}}({\mu})$.
Let $C\in {\mathcal{E}}_{{\lambda}}({\mu})$. Then $C={\mu}_k\cdots{\mu}_1(D_{{\lambda}})$ for some excitations ${\mu}_1,\ldots,{\mu}_k$. We prove that $C\in f({\mathcal{T}}_{{\mu}}({\lambda}))$ by induction on $k$. For the base case, we use the fact that ${\lambda}=f(T^{\text{top}})$. Let $C'={\mu}_{k-1}\cdots{\mu}_1(D_{{\lambda}})$. The excitation ${\mu}_k$ modifies some box $(a,b)$ of $C'$. Thus $(a+1,b)\not\in C, (a,b+1)\not\in C,(a+1,b+1)\not\in C$, $(a+1,b+1)\in D_{{\mu}}$. By the induction hypothesis, $C'=f(T')$ for some $T'\in{\mathcal{T}}_{{\lambda}}({\mu})$. Let $x\in T'(i,j)$ be such that $f(T|_x)=(a,b)$, and let ${\nu}$ be the excitation of the same type as ${\mu}_k$ modifying $x$. By Lemma \[l.fcommutesexcitation\], $C={\mu}_k C'={\mu}_k f(T')=f({\nu}T')$.
\[l.finjective\] The map $f$ is injective.
Let $C\in{\mathcal{E}}_{{\lambda}}({\mu})$. By Lemma \[l.fsurjective\], there exists $T\in{\mathcal{T}}_{{\lambda}}({\mu})$ such that $f(T)=C$. We give a constructive proof of the uniqueness of $T$ by filling in the boxes of $T$ one diagonal at a time, beginning with the largest diagonal. As we shall see, there is only one way to fill in the boxes so that $T$ is semistandard and $f(T)=C$.
The largest diagonal $r$ of $D_{{\lambda}}$ contains a single box, namely $(1,r\backslash$. For each box $(x,r\backslash$ of $C$, place an $x$ in box $(1,r\backslash$ of $T$. Now assume that we have filled in each box in diagonals $q+1,\ldots,r$ of $T$ with a nonempty set of positive integers. Let $(x,q\backslash\in C$. In order to satisfy $f(T)=C$, we must place an $x$ in some box of diagonal $q$ of $T$. In order for $T$ to be semistandard, we must place this $x$ in the unique box $(i,q\backslash$ of $T$ such that $x$ is strictly greater than all entries of box $(i-1,q+1\backslash$ and weakly less than all entries of box $(i,q+1\backslash$. Surjectivity of $f$ guarantees the existence of such a box $(i,q\backslash$. Surjectivity of $f$ also ensures that if this procedure is carried out for every box of diagonal $q$ of $C$, every box of diagonal $q$ of $T$ will have at least one number placed inside of it.
From Lemmas \[l.fsurjective\] and \[l.finjective\] we have
\[p.fbijective\] The map $f$ is a bijection from ${\mathcal{T}}_{{\lambda}}({\mu})$ to ${\mathcal{E}}_{{\lambda}}({\mu})$.
Hilbert series and Hilbert polynomials of points on Schubert varieties {#ss.hilbseries_An}
----------------------------------------------------------------------
In type $A_n$, all of the maximal parabolic subgroups $P_d$ are cominuscule (cf. [@BiLa:00 9.0.14]). Thus Corollary \[c.cominusculemult\] may be used to compute the Hilbert series, Hilbert polynomial, and multiplicity at $v$ of a Schubert variety $X^w$ in the Grassmannian ${\operatorname{Grass }}(d,n)$. Let ${\lambda}$ and ${\mu}$ be the partitions corresponding to $w$ and $v$ respectively. In the present setting, the constant $m_k$ of Corollary \[c.cominusculemult\] is equal to the number of excited Young diagrams $C\in{\mathcal{E}}_{{\lambda}}({\mu})$ such that the number of boxes of $C$ is $k+|{\lambda}|$ (due to Proposition \[p.subsequences\](i), Definition \[d.energy\](iii), and Lemma \[l.hecketoeyd\](ii)). In terms of set-valued tableaux, $m_k$ is equal to the number of $T\in{\mathcal{T}}_{{\lambda}}({\mu})$ with $k+|{\lambda}|$ entries.
Let $n=7$, $d=3$, $w=\{1,3,5,2,4,6,7\}$, and $v=\{4,6,7,1,2,3,5\}$, as in Example \[ex.restriction\_A\]. Then ${\lambda}={\lambda}_w=(2,1)$, ${\mu}={\lambda}_v=(4,4,3)$, $l(w)=|{\lambda}|=3$, and $d_w=d(n-d)-l(w)=9$. The set of excited Young diagrams ${\mathcal{E}}_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_all\], and the set of set-valued tableaux ${\mathcal{T}}_{{\lambda}}({\mu})$ appears in Figure \[f.syt\_all\]. From either of these figures, one reads off $m_0=5$, $m_1=5$, and $m_2=1$. Hence [$$\begin{aligned}
H(X^w,v)(t)&=\frac{5}{(1-t)^9}-\frac{5}{(1-t)^8}+\frac{1}{(1-t)^7},\\
h(X^w,v)(i)&=5{i +8\choose 8}-5{i+7\choose 7}+{i+6\choose 6},\\
{\operatorname{mult}}(X^w,v)&=5.\end{aligned}$$ ]{}
Several other multiplicity formulas have appeared in the literature: inductive [@LaWe:90], determinantal [@RoZe:01], [@WoYo:12], [@LiYo:12], and enumerative [@Kra:01], [@KoRa:03] [@Kre:03], [@KrLa:04], [@Kra:05], [@Kre:08]. The inductive formula of Lakshmibai and Weyman, which holds more generally for minuscule $G/P$, was used by Rosenthal and Zelevinsky to prove the determinantal formula, which in turn was used by Krattenthaler to prove the enumerative formula, which counts nonintersecting lattice paths. The formula given above appeared earlier in [@IkNa:09] and [@WoYo:12], and it can also be deduced from [@Kre:05] together with the multiplicity formulas of [@KoRa:03], [@Kre:03], [@KrLa:04].
Formulas for the Hilbert series and Hilbert polynomial of $X^w$ at $v$ were obtained by [@Kra:05]. They were derived using an expression for the Hilbert function of $X^w$ at $v$ given in [@KoRa:03], [@Kre:03], [@KrLa:04], and [@Kre:08]. The formulas given here appeared earlier in [@LiYo:12]. We remark that the formulas of [@LiYo:12] apply more generally to covexillary permutations.
Applications to the orthogonal and Lagrangian Grassmannians {#s.orthosymplectic}
===========================================================
In types $B_n$, $C_n$, and $D_n$, let $G$ be ${\operatorname{SO}}_{2n+1}({\mathbb{C}})$, ${\operatorname{Sp}}_{2n}({\mathbb{C}})$, and ${\operatorname{SO}}_{2n}({\mathbb{C}})$ respectively. Each of these groups $G$ is defined to be the subgroup of a general linear group preserving a specified nondegenerate symmetric or skew symmetric inner product (see Appendix \[s.appendix\_roots\]). Let $P_n$ be the maximal parabolic subgroup of $G$ corresponding to simple root ${\alpha}_n$.
- The coset space $G/P_n$ is identified with the **odd orthogonal Grassmannian** $OG(n,2n+1)$, which parametrizes the maximal ($n$ dimensional) isotropic subspaces of ${\mathbb{C}}^{2n+1}$. It has the structure of an irreducible projective variety of dimension $n(n+1)/2$.
- The coset space $G/P_n$ is identified with the **Lagrangian Grassmannian** $LG(n,2n)$, which parametrizes the maximal ($n$ dimensional) isotropic subspaces of ${\mathbb{C}}^{2n}$. It has the structure of an irreducible projective variety of dimension $n(n+1)/2$.
- The coset space $G/P_n$ is identified with the **even orthogonal Grassmannian** $OG(n,2n)$, which parametrizes one of the two components of the maximal ($n$-dimensional) isotropic subspaces of ${\mathbb{C}}^{2n}$. The even orthogonal Grassmannian has the structure of an irreducible projective variety of dimension $n(n-1)/2$.
Let $B^-$ be the Borel subgroup of lower triangular matrices in $G$, and $T$ the subgroup of diagonal matrices in $G$. The Weyl group $W=N_G(T)/T$ embeds into $\{w=(w_1,\ldots,w_{2n})\in S_{2n}\mid w_{{\overline{\imath}}}={\overline{w_i}}\}$, where, for $x\in\{1,\ldots,2n\}$, ${\overline{x}}=2n+1-x$. In particular, $w\in W$ is uniquely determined by $w_1,\ldots,w_n$. The Weyl group $W_{P_n}$ is isomorphic to $S_n$, and the set of **minimal length coset representatives** of $W/W_{P_n}$ is given by $$\label{e.WP}
W^{{P_n}} = \left\{(w_1,\ldots,w_{2n})\in W\mid w_1<\cdots<w_n \right\}.$$ The cosets $w{P_n}$, $w\in W^{P_n}$, are precisely the $T$-fixed points of $G/{P_n}$. By abuse of notation, we sometimes denote $w{P_n}$ by just $w$. The Schubert variety $X^w$ is by definition ${\overline{B^-wP_n}}\subseteq G/{P_n}$. It is an irreducible projective variety of codimension $l(w)$. For $v,w\in W^{P_n}$, $v\in X^w$ if and only if $v\geq w$ in the Bruhat order.
In this section, we give formulas for $i_v^*[{\mathcal{O}}_{X^w}]$ (Theorems \[t.ktheory\_eyd\_bcd\] and \[t.ktheory\_syt\_bcd\]), as well as the Hilbert series, Hilbert polynomial, and multiplicity of $X^w$ at $v$ (Section \[ss.hilbseries\_BCD\]). These are reformulations of Theorem \[t.pullback\] and Corollary \[c.cominusculemult\], expressed in terms of [*excited shifted Young diagrams*]{} ([@Kre:06], [@IkNa:09]) and [*set-valued shifted tableaux*]{}. The term excited shifted Young diagram is due to Ikeda and Naruse [@IkNa:09]; in [@Kre:06], in which only the case $G={\operatorname{Sp}}_{2n}({\mathbb{C}})$ is studied, this is called a subset of a Young diagram. Also, in this paper we have modified the definition of excited shifted Young diagram for our applications to $K$-theory; the earlier definitions correspond to our [*reduced*]{} excited shifted Young diagrams. Reduced excited Young diagrams were discovered independently by Kreiman [@Kre:06] and Ikeda and Naruse [@IkNa:09]. A related version of excited shifted Young diagram was introduced in [@IkNa:11] (see Section \[ss.previous\]).
In type $C_n$, our formula for $i_v^*[{\mathcal{O}}_{X^w}]$, expressed in terms of set-valued shifted tableaux, was obtained earlier in [@Kre:06]. The derivation of the formula there relies on an equivariant Gröbner degeneration of $X^w$ in a neighborhood of $v$; this degeneration is due to [@GhRa:06]. Our approach is different. As in Section \[s.Grassmannian\], our methods in this section are modeled on those of Ikeda and Naruse [@IkNa:09], and in several places, we use their results directly. Formulas for $i_v^*[{\mathcal{O}}_{X^w}]$ which were obtained in [@IkNa:11] are discussed in Section \[ss.previous\].
Strict partitions and shifted Young diagrams {#ss.strict_partitions_bcd}
--------------------------------------------
A partition ${\lambda}=({\lambda}_1,\ldots,{\lambda}_n)\in {\mathcal{P}}_{n,n}$ is said to be **strict** if ${\lambda}_i={\lambda}_{i+1}$ implies ${\lambda}_{i}={\lambda}_{i+1}=0$. Let ${\mathcal{S}}{\mathcal{P}}_{n}$ denote the set of such partitions. A **shifted Young diagram** is an array of boxes arranged such that the row lengths strictly decrease from top to bottom and the leftmost box of row $i$ lies in column $i$. To a strict partition ${\lambda}$ we associate the shifted Young diagram $D'_{\lambda}$ whose $i$-th row contains ${\lambda}_i$ boxes. Then $D'_{{\lambda}}$ fits in the upper triangular boxes of an $n\times n$ square.
Let $n=6$, and ${\lambda}=(5,4,2,1)\in{\mathcal{S}}{\mathcal{P}}_{n}$. The shifted Young diagram $D'_{\lambda}$ fits in the upper triangular boxes of an $n\times n$ square.
(.5,.5) – ++(6,0) – ++(0,6) – ++(-6,0) – cycle; in [ [(1,6)]{},[(2,6)]{},[(3,6)]{},[(4,6)]{},[(5,6)]{}, [(2,5)]{},[(3,5)]{},[(4,5)]{},[(5,5)]{}, [(3,4)]{},[(4,4)]{}, [(4,3)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{}
Define a map $W^{P_n}\to {\mathcal{S}}{\mathcal{P}}_{n}$, $w\mapsto {\lambda}'_w$, as follows. Given $w\in W^{P_n}$, form the Young diagram $D_{{\lambda}_w}$. By Lemma \[l.symmsymm\], $D_{{\lambda}_w}$ is symmetric. In types $B_n$ and $C_n$, remove all boxes $(i,j)$ of $D_{{\lambda}_w}$ such that $i> j$; in type $D_n$, remove all boxes $(i,j)$ of $D_{{\lambda}_w}$ such that $i\geq j$. The resulting shifted Young diagram corresponds to ${\lambda}'_w$. More explicitly, in types $B_n$ and $C_n$, $$\label{e.glprimewbc}
({\lambda}'_w)_i=\max\{({\lambda}_w)_i-(i-1),0\}, i=1,\ldots,n,$$ and in type $D_n$, $$\label{e.glprimewd}
({\lambda}'_w)_i=\max\{({\lambda}_w)_i-i,0\}, i=1,\ldots,n-1.$$ As in type $A_n$, $l(w)=|{\lambda}'_w|$.
\[ex.strict\_partitions\] Let $n=6$, and $w=(1,4,{\overline{6}},{\overline{5}},{\overline{3}},{\overline{2}},2,3,5,6,{\overline{4}},{\overline{1}})$\
$=(1,4,7,8,10,11,2,3,5,6,9,12)\in W^{P_n}$. Then ${\lambda}_w=(5,5,4,4,2)\in {\mathcal{P}}_{n,n}$. $$\begin{tikzpicture}[scale=.4,
baseline={([yshift=-.5ex]current bounding box.center)}]
\draw (.5,.5) -- ++(6,0) -- ++(0,6) -- ++(-6,0) -- cycle;
\foreach \pos in
{ {(1,6)},{(2,6)},{(3,6)},{(4,6)},{(5,6)},
{(2,5)},{(3,5)},{(4,5)},{(5,5)},
{(3,4)},{(4,4)},
{(4,3)}
}
{\draw[fill=gray!20] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,5)},
{(1,4)}, {(2,4)},
{(1,3)}, {(2,3)},{(3,3)},
{(1,2)}, {(2,2)}
}
{\draw[dashed] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (3,7) node[anchor=south] {$B_n, C_n$};
\end{tikzpicture}
\qquad\qquad\qquad
\begin{tikzpicture}[scale=.4,
baseline={([yshift=-.5ex]current bounding box.center)}]
\draw (.5,.5) -- ++(6,0) -- ++(0,6) -- ++(-6,0) -- cycle;
\foreach \pos in
{{(2,6)},{(3,6)},{(4,6)},{(5,6)},
{(3,5)},{(4,5)},{(5,5)},
{(4,4)}
}
{\draw[fill=gray!20] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,6)},
{(1,5)},{(2,5)},
{(1,4)}, {(2,4)},{(3,4)},
{(1,3)}, {(2,3)}, {(3,3)}, {(4,3)},
{(1,2)}, {(2,2)}
}
{\draw[dashed] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (3,7) node[anchor=south] {$D_n$};
\end{tikzpicture}$$
In types $B_n$ and $C_n$, ${\lambda}'_w=(5,4,2,1)$, and in type $D_n$, ${\lambda}'_w=(4,3,1)$. These are the strict partitions obtained by removing the unshaded boxes from $D_{{\lambda}_w}$ in the diagrams above.
In types $B_n$ and $C_n$, the map $W^{P_n}\to {\mathcal{S}}{\mathcal{P}}_{n}$ is a bijection, and in type $D_n$, the map $W^{P_n}\to {\mathcal{S}}{\mathcal{P}}_{n-1}$ is a bijection. The assertion in type $D_n$ follows from the fact that both $W^{P_n}\to Q$ and $Q\to {\mathcal{S}}{\mathcal{P}}_{n-1}$ are bijections, where $Q$ is the set of symmetric partitions with at most $n$ rows and an even number of boxes of the form $(i,i)$ (i.e., lying on diagonal 0). The map $W^{P_n}\to Q$, $v\mapsto {\lambda}_v$, is well defined and bijective by Lemma \[l.symmsymm\] and . The map $Q\to {\mathcal{S}}{\mathcal{P}}_{n-1}$, ${\lambda}_v\mapsto {\lambda}'_v$, is bijective by straightforward combinatorial arguments. In types $B_n$ and $C_n$, the proof is similar but simpler.
Thus $W^{P_n}$ in type $D_n$ and $W^{P_{n-1}}$ in type $B_{n-1}$ are both in bijection with ${\mathcal{S}}{\mathcal{P}}_{n-1}$. We identify the element of $W^{P_{n}}$ in type $D_{n}$ with the element of $W^{P_{n-1}}$ in type $B_{n-1}$ which corresponds to the same strict partition of ${\mathcal{S}}{\mathcal{P}}_{n-1}$. This in turn gives an identification between the fixed points of ${\operatorname{OG}}(n,2n)$ and of ${\operatorname{OG}}(n-1,2n-1)$.
\[ex.fixedpts\_bijection\_BD\] In type $D_n$, with $n=6$, let $w=(1,4,{\overline{6}},{\overline{5}},{\overline{3}},{\overline{2}},2,3,5,6,{\overline{4}},{\overline{1}})$ as in Example \[ex.strict\_partitions\]. Then ${\lambda}_w=(5,5,4,4,2)\in {\mathcal{P}}_{n,n}$ and ${\lambda}'_w=(4,3,1)\in{\mathcal{S}}{\mathcal{P}}_{n-1}$. Let ${\lambda}={\lambda}'_w$. In type $B_{n-1}$, ${\lambda}\in{\mathcal{S}}{\mathcal{P}}_{n-1}$ corresponds to $(4,4,3,2)\in {\mathcal{P}}_{n-1,n-1}$, which corresponds to $u=(1,4,{\overline{5}},{\overline{3}},{\overline{2}},2,3,5,{\overline{4}},{\overline{1}})\in W^{P_{n-1}}$.
(.5,.5) – ++(5,0) – ++(0,5) – ++(-5,0) – cycle; in [ [(1,5)]{},[(2,5)]{},[(3,5)]{},[(4,5)]{}, [(2,4)]{},[(3,4)]{},[(4,4)]{}, [(3,3)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} in [ [(1,4)]{}, [(1,3)]{}, [(2,3)]{}, [(1,2)]{}, [(2,2)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} (3,6) node\[anchor=south\] [$B_{n-1}$]{};
Thus we identify the Weyl group element $w$ in type $D_n$ with $u$ in type $B_{n-1}$, and we identify the point $w$ in ${\operatorname{OG}}(n,2n)$ with $u$ in ${\operatorname{OG}}(n-1,2n-1)$.
Another identification between $W^{P_n}$ in type $D_n$ and $W^{P_{n-1}}$ in type $B_{n-1}$ appears in [@RaUp:10 1.3]. Given $w=(w_1,\ldots,w_{2n})\in W^{P_n}$ in type $D_n$, merely remove $n$ and ${\overline{n}}$ from the entries $w_1,\ldots,w_{2n}$ in order to produce element $u$ of $B_{n-1}$. (In Example \[ex.fixedpts\_bijection\_BD\], $u$ can be obtained from $w$ by removing $6$ and ${\overline{6}}$ from the entries $w_1,\ldots,w_{12}$.) In the other direction, suppose we are given $u=(u_1,\ldots,u_{2n-2})\in W^{P_{n-1}}$ in type $B_{n-1}$. Add $n$ and ${\overline{n}}$ to the entries of $u$ in such a way that the new permutation is symmetric, and the first $n$ of its entries have an even number of barred entries and are increasing. For the remainder of this section, we prove that this identification is the same as ours. Let ${\lambda}=({\lambda}_1,\ldots,{\lambda}_{n-1})\in{\mathcal{S}}{\mathcal{P}}_{n-1}$. Let ${\lambda}={\lambda}'_w$ for $w=(w_1,\ldots,w_{2n})\in W^{P_n}$ in type $D_n$, and let ${\lambda}={\lambda}'_u$ for $u=(u_1,\ldots,u_{2n-2})\in W^{P_{n-1}}$ in type $B_{n-1}$. We wish to show that $u$ is obtained by removing $n$ and ${\overline{n}}$ from the entries of $w$. This is equivalent to the assertion $\{w_1,\ldots,w_n\}\setminus \{n,{\overline{n}}\}=\{u_1,\ldots,u_{n-1}\}$.
\[l.partsbd\] For $1\leq i\leq n-1$, $$\begin{aligned}
& B_{n-1}:\quad u_{n-i}\geq {\overline{n-1}}\text{ if and only if }{\lambda}_i>0,\text{ in which case }u_{n-i}={\overline{n-{\lambda}_i}} \\
& D_n:\quad w_{n+1-i}\geq {\overline{n-1}}\text{ if and only if }{\lambda}_i>0,\text{ in which case }w_{n+1-i}={\overline{n-{\lambda}_i}}\end{aligned}$$
Note that in type $B_{n-1}$, ${\overline{x}}=2n-1-x$, whereas in type $D_{n}$, ${\overline{x}}=2n+1-x$.
We prove this for type $D_n$. The proof for type $B_{n-1}$ is similar. By , for $i=1,\ldots,n$, $({\lambda}_w)_i=w_{n+1-i}-(n+1-i)$. Thus $$\label{e.lem_technical}
({\lambda}_w)_i-i=w_{n+1-i}-(n+1)=w_{n+1-i}-{\overline{n}}.$$ Thus $w_{n+1-i}\geq {\overline{n-1}}$ if and only if $w_{n+1-i}>{\overline{n}}$ if and only if $({\lambda}_w)_i>i$ if and only if ${\lambda}_i=({\lambda}'_w)_i>0$, by . In this case $({\lambda}_w)_i-i=({\lambda}'_w)_i={\lambda}_i$, so implies $w_{n+1-i}={\lambda}_i+{\overline{n}}={\overline{n-{\lambda}_i}}$.
Recall for $x=1,\ldots,n$, exactly one of $x$ or ${\overline{x}}$ occurs among $w_1,\ldots,w_n$, and for $y=1,\ldots,n-1$, exactly one of $y$ or ${\overline{y}}$ occurs among $u_1,\ldots,u_{n-1}$. In particular exactly one of $n$ or ${\overline{n}}$ occurs among $w_1,\ldots,w_n$. If one excludes this entry, then Lemma \[l.partsbd\] implies that the barred entries of $w_1,\ldots,w_n$ and of $u_1,\ldots,u_{n-1}$ are the same, and hence the unbarred entries must be the same as well. This completes the proof.
The orthogonal Grassmannians {#ss.orthog_grassmannians}
----------------------------
In this section we show that the restriction formula in type $B_n$ can be obtained from the restriction formula in type $D_{n+1}$. Let $S$ and $T$ be the set of diagonal matrices in ${\operatorname{SO}}_{2n+2}({\mathbb{C}})$ and ${\operatorname{SO}}_{2n+1}({\mathbb{C}})$ respectively. The $(n+1)$-st diagonal entry of any element of $T$ must be 1, and the map ${\operatorname{diag}}(t_1,\ldots,t_{2n+1})\mapsto {\operatorname{diag}}(t_1,\ldots,t_{n},1,1,t_{n+2},\ldots,t_{2n+1})$ embeds $T$ as a closed subgroup of $S$. Let $Y={\operatorname{OG}}(n+1,2n+2)$ and $X={\operatorname{OG}}(n,2n+1)$. There is a $T$-equivariant isomorphism $\pi:Y\to X$ (cf. [@RaUp:10 1.3]). Hence $\pi^*:K_T(Y)\to K_T(X)$ is an isomorphism. The sets $\{T$-fixed points of $Y\}$, $\{T$-fixed points of $X\}$, and $\{S$-fixed points of $Y\}$ can be identified. The identification of the first two sets follows from the $T$-equivariance of $\pi$, and the identification of the last two sets is described in Section \[ss.strict\_partitions\_bcd\]. In fact, these two identifications are the same (cf. [@RaUp:10 1.3]).
If $S$ acts on any scheme $M$, then there is a natural restriction homomorphism\
${\operatorname{res}}:K_S(M)\to K_T(M)$ taking the class of an $S$-equivariant sheaf ${\mathcal{F}}$ to the class of the same sheaf, but viewed as equivariant with respect to the $T$ action. If $w$ is any $S$-fixed point (equivalently, $T$-fixed point) of $Y$, then the square and triangle in the diagram
\(m) \[matrix of math nodes, row sep=3.5em, column sep=3.5em, text height=1.5ex, text depth=0.25ex\] [ K\_S(Y) & K\_T(Y) & K\_T(X)\
K\_S(w)& K\_T(w) &\
]{}; (m-1-1) edge node\[auto\] [${\operatorname{res}}$]{} (m-1-2) (m-1-3) edge node\[auto,swap\] [$\pi^*$]{} node\[auto\] [$\sim$]{} (m-1-2) (m-2-1) edge node\[auto,swap\] [${\operatorname{res}}$]{} (m-2-2) (m-1-1) edge node\[auto,swap\] [$i_w^*$]{} (m-2-1) (m-1-2) edge node\[auto,swap\] [$i_w^*$]{} (m-2-2) (m-1-3) edge node\[auto\] [$i_w^*$]{} (m-2-2);
commute. Let ${{\mathfrak s}}^*$ and ${{\mathfrak t}}^*$ be the duals of the Lie algebras of $S$ and $T$ respectively, with bases as given in Appendix \[s.appendix\_roots\]. Define $f:{{\mathfrak s}}^*\to {{\mathfrak t}}^*$ by ${\epsilon}_i\mapsto {\epsilon}_i$, $i=1,\ldots,n$, ${\epsilon}_{n+1}\mapsto 0$. Then ${\operatorname{res}}:K_S(w)\to K_T(w)$ is the homomorphism defined by $e^{{\mu}}\mapsto e^{f({\mu})}$.
The isomorphism $\pi$ identifies Schubert varieties of $Y$ with Schubert varieties of $X$ (cf. [@RaUp:10 1.3]). We shall denote a Schubert variety in $Y$ as $Y^w$ and $\pi(Y^w)$ as $X^w$.
\[p.BfromD\_restriction\] Let $w$ denote both a $T$-fixed point of $X$ and the corresponding $S$-fixed point of $Y$. Then $i_w^*[X^w]={\operatorname{res}}(i_w^*[Y^w])$.
We have ${\operatorname{res}}[Y_w]=[Y_w]$, where the first $[Y_w]$ lies in $K_S(Y)$ and the second in $K_T(Y)$. Further, $\pi^*[X^w]=[\pi^{-1}(X^w)]=[Y^w]$. Hence $
{\operatorname{res}}(i_w^*[Y^w])=(i_w^*\circ (\pi^*)^{-1}\circ {\operatorname{res}})[Y^w]=i_w^*[X^w].
$
Restriction formula in terms of excited shifted Young diagrams {#ss.eyd_bcd}
--------------------------------------------------------------
Suppose that ${\lambda}$ is a strict partition and $S$ is a subset of $D'_{\lambda}$. As in type $A_n$, an excitation moves or adds a box to $S$. Let $(i,j)\in S$. If $i<j$, then there are two possible excitations based on $(i,j)$, and these are the same as in type $A_n$: if $(i+1,j),(i,j+1),(i+1,j+1)\in D_{\lambda}\setminus S$, then excitation of type 1 replaces $S$ by $S\setminus (i,j)\cup (i+1,j+1)$, and an excitation of type 2 replaces $S$ by $S\cup (i+1,j+1)$. If $i=j$, then the excitations are described as follows:
- If $(i,i+1), (i+1,i+1)\in D'_{\lambda}\setminus S$, then an excitation of type 1 replaces $S$ by $S\setminus (i,i)\cup (i+1,i+1)$, and an excitation of type 2 replaces $S$ by $S\cup (i+1,i+1)$.
- If $(i,i+1),(i+1,i+1), (i+1,i+2), (i+2,i+2)\in D'_{\lambda}\setminus S$, then an excitation of type 1 replaces $S$ by $S\setminus (i,i)\cup (i+2,i+2)$, and an excitation of type 2 replaces $S$ by $S\cup (i+2,i+2)$.
Let ${\lambda},{\mu}\in {\mathcal{S}}{\mathcal{P}}_{n}$. If ${\lambda}_i\leq {\mu}_i$, $i=1,\ldots,n$, the map $(i,j)\mapsto (i,j)$ embeds $D'_{{\lambda}}$ as a subset of $D'_{{\mu}}$. An **excited shifted Young diagram** of $D'_{\lambda}$ in $D'_{\mu}$ is a subset of $D'_{\mu}$ which can be obtained by applying a sequence of excitations to the subset $D'_{\lambda}$. An excited shifted Young diagram is said to be **reduced** if it can be obtained by applying only type 1 excitations. Denote the set of excited shifted Young diagrams of $D'_{\lambda}$ in $D'_{\mu}$ by ${\mathcal{E}}'_{\lambda}({\mu})$, and the set of reduced excited shifted Young diagrams by ${\mathcal{E}}'^{\text{red}}_{\lambda}({\mu})$.
\[t.ktheory\_eyd\_bcd\] Let $w\leq v\in W^{P_n}$, and let ${\lambda}$, ${\mu}$ be the corresponding strict partitions. [$$\begin{aligned}
& B_n:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{C\in{\mathcal{E}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in C}\left(e^{-2^{-\delta_{ij}}({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}})}-1\right)\\
& C_n:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{C\in{\mathcal{E}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in C}\left(e^{-({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}})}-1\right)\\
& D_{n}:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{C\in{\mathcal{E}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in C}\left(e^{-({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j+1}})}-1\right)\end{aligned}$$ ]{} where for $1\leq m\leq n$, ${\epsilon}_{{\overline{m}}}$ is defined to equal $-{\epsilon}_m$.
We now have two methods for computing $i_v^*[{\mathcal{O}}_{X^w}]$ in type $B_n$. We can either employ the above formula for type $B_n$, or we can invoke Proposition \[p.BfromD\_restriction\]: first use the formula for type $D_{n+1}$ and then set every ${\epsilon}_{n+1}$ to 0. In general, the two methods produce different expressions for $i_v^*[{\mathcal{O}}_{X^w}]$. From the combinatorics alone it is not clear that these two expressions result in the same value. In general, the latter method is computationally simpler because it involves fewer excited Young diagrams. We illustrate both methods in the examples.
\[ex.restriction\_BC\] Let $n=4$. Let $v=\{2,{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,{\overline{2}}\}$, $w=\{1,2,{\overline{4}},{\overline{3}},3,4,{\overline{2}},{\overline{1}}\}$ in type $C_n$ or $B_n$. Then ${\lambda}_v=(4,3,3,1)$, ${\mu}={\lambda}'_v=(4,2,1)$, ${\lambda}_w=(2,2)$, and ${\lambda}={\lambda}'_w=(2,1)$. Thus $l(w)=|{\lambda}'_w|=3$. The set of excited shifted Young diagrams ${\mathcal{E}}'_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_allbc\]. For any $C\in{\mathcal{E}}'_{{\lambda}}({\mu})$ and $(i,j)\in C$, a simple method for finding the indices $v_{n+i}$ and $v_{n+j}$ appearing in Theorem \[t.ktheory\_eyd\_bcd\] is to label the rows of $C$, from top to bottom, as well as the columns, from left to right, with the entries $v_{n+1},v_{n+2},\ldots$ of $v$ (where ${\overline{x}}$ is replaced by $-x$). Then $v_{n+i}$ and $v_{n+j}$ are the row $i$ and column $j$ labels respectively. For example, for $$C=\
\begin{tikzpicture}[scale=.6,every node/.style={scale=1},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos in
{{(1,3)},{(2,3)},{(3,1)}}
{\draw[fill=blue!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,3)},{(2,3)},{(3,3)},{(4,3)},
{(2,2)},{(3,2)},
{(3,1)}
}
{\draw \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0,1) node{$4$}; \draw (0,2) node{$3$}; \draw (0,3) node{$1$};
\draw (1,4) node{$1$}; \draw (2,4) node{$3$}; \draw (3,4) node{$4$};
\draw (4,4) node{$-2$};
\end{tikzpicture}
\in{\mathcal{E}}'_{{\lambda}}({\mu}),$$ in type $B_n$, $$\begin{aligned}
&\prod_{(i,j)\in C}\Big(e^{-2^{-{\delta}_{ij}}({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}})}-1\Big)
=\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-{\epsilon}_4}-1\Big),
\intertext{and in type $C_n$,}
&\prod_{(i,j)\in C}\Big(e^{-({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}})}-1\Big)
=\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-2{\epsilon}_4}-1\Big).\end{aligned}$$ Invoking Theorem \[t.ktheory\_eyd\_bcd\], we obtain, in type $B_n$, [$$\begin{aligned}
i_v^*[{\mathcal{O}}_{X^w}]=&
-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-{\epsilon}_3}-1\Big)
-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-{\epsilon}_4}-1\Big)\\
&-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)
\Big(e^{-{\epsilon}_4}-1\Big)
-\Big(e^{-{\epsilon}_3}-1\Big)\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)
\Big(e^{-{\epsilon}_4}-1\Big)\\
&-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-{\epsilon}_3}-1\Big)\Big(e^{-{\epsilon}_4}-1\Big)\\
&-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)\Big(e^{-{\epsilon}_4}-1\Big)\\
&-\Big(e^{-{\epsilon}_1}-1\Big)\Big(e^{-{\epsilon}_3}-1\Big)
\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)\Big(e^{-{\epsilon}_4}-1\Big),
\intertext{and in type $C_n$,}
i_v^*[{\mathcal{O}}_{X^w}]=&
-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-2{\epsilon}_3}-1\Big)
-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-2{\epsilon}_4}-1\Big)\\
&-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)
\Big(e^{-2{\epsilon}_4}-1\Big)
-\Big(e^{-2{\epsilon}_3}-1\Big)\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)
\Big(e^{-2{\epsilon}_4}-1\Big)\\
&-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-2{\epsilon}_3}-1\Big)\Big(e^{-2{\epsilon}_4}-1\Big)\\
&-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-({\epsilon}_1+{\epsilon}_3)}-1\Big)
\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)\Big(e^{-2{\epsilon}_4}-1\Big)\\
&-\Big(e^{-2{\epsilon}_1}-1\Big)\Big(e^{-2{\epsilon}_3}-1\Big)
\Big(e^{-({\epsilon}_3+{\epsilon}_4)}-1\Big)\Big(e^{-2{\epsilon}_4}-1\Big).
\end{aligned}$$ ]{}
\[ex.restriction\_D\] Let $n=6$. In type $D_n$, let $v=\{2,6,{\overline{5}},{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,5,{\overline{6}},{\overline{2}}\}$ and $w=\{1,2,4,6,{\overline{5}},{\overline{3}},3,5,{\overline{6}},{\overline{4}},{\overline{2}},{\overline{1}}\}$. Then ${\lambda}_v=(6,5,5,5,4,1)$, ${\mu}={\lambda}'_v=(5,3,2,1)$, ${\lambda}_w=(4,3,2,1)$, and ${\lambda}={\lambda}'_w=(3,1)$. Thus $l(w)=|{\lambda}'_w|=4$. The set of excited shifted Young diagrams ${\mathcal{E}}'_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_alld\]. For any $C\in{\mathcal{E}}'_{{\lambda}}({\mu})$ and $(i,j)\in C$, a simple method of finding the indices $v_{n+i}$ and $v_{n+j+1}$ of Theorem \[t.ktheory\_eyd\_bcd\] is to label the rows of $C$, from top to bottom, with the numbers $v_{n+1},v_{n+2},\ldots$, and the columns, from left to right, with the numbers $v_{n+2},v_{n+3},\ldots$. For both row and column labels, ${\overline{x}}$ is replaced by $-x$. Then $v_{n+i}$ and $v_{n+j+1}$ are the row $i$ and column $j$ labels respectively. For example, for $$C=\
\begin{tikzpicture}[scale=.6,every node/.style={scale=1},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos in
{{(1,4)},{(2,4)},{(4,3)},{(4,1)}}
{\draw[fill=blue!30] \pos +(-.5,-.5) rectangle ++(.5,.5);}
\foreach \pos in
{ {(1,4)},{(2,4)},{(3,4)},{(4,4)},{(5,4)},
{(2,3)},{(3,3)},{(4,3)},
{(3,2)},{(4,2)},
{(4,1)}
}
{\draw \pos +(-.5,-.5) rectangle ++(.5,.5);}
\draw (0,1) node{$5$}; \draw (0,2) node{$4$}; \draw (0,3) node{$3$}; \draw (0,4) node{$1$};
\draw (1,5) node{$3$}; \draw (2,5) node{$4$}; \draw (3,5) node{$5$};
\draw (4,5) node{$-6$}; \draw (5,5) node{$-2$};
\end{tikzpicture}
\in{\mathcal{E}}'_{{\lambda}}({\mu}),$$ we have $$\prod_{(i,j)\in C}\left(e^{-({\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j+1}})}-1\right)
=\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right).$$ By Theorem \[t.ktheory\_eyd\_bcd\], [$$\begin{aligned}
i_v^*[{\mathcal{O}}_{X^w}]&
=
\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right)\left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)\left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_3+{\epsilon}_5)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right)\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)
\left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right) \left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right)\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)
\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right) \left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_3+{\epsilon}_5)}-1\right)
\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\\
&+\left(e^{-({\epsilon}_1+{\epsilon}_3)}-1\right)\left(e^{-({\epsilon}_1+{\epsilon}_4)}-1\right)
\left(e^{-({\epsilon}_1+{\epsilon}_5)}-1\right)\left(e^{-({\epsilon}_3-{\epsilon}_6)}-1\right)
\left(e^{-({\epsilon}_3+{\epsilon}_4)}-1\right)\left(e^{-({\epsilon}_5-{\epsilon}_6)}-1\right)\end{aligned}$$ ]{}
Let $n=5$. In type $B_n$, let $v=\{2,{\overline{5}},{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,5,{\overline{2}}\}$ and $w=\{1,2,4,{\overline{5}},{\overline{3}},3,5,{\overline{4}},{\overline{2}},{\overline{1}}\}$. Then $v$ and $w$ are identified with the corresponding elements of $W^{P_{n+1}}$ in Example \[ex.restriction\_D\] (see Section \[ss.strict\_partitions\_bcd\]). By Proposition \[p.BfromD\_restriction\], $i_v^*[{\mathcal{O}}_{X^w}]$ can be obtained from the same expression in Example \[ex.restriction\_D\] by replacing each ${\epsilon}_6$ by 0.
Theorem \[t.ktheory\_eyd\_bcd\] is a reformulation of Theorem \[t.pullback\], in which the indexing set $T(w,{\mathbf{s}})$ and integer $r(c)$ of the latter theorem are expressed in terms of excited shifted Young diagrams. These replacements are described explicitly in Proposition \[p.subsequences\_bcd\].
Let $w,v\in W^{P_n}$, and let ${\lambda}={\lambda}'_w$ and ${\mu}={\lambda}'_v$ be the corresponding strict partitions. Form a reflection-valued shifted tableau $T'_{{\mu}}$ as follows:
$B_n$, $C_n$
: Fill each box $(i,j)$ of $D'_{{\mu}}$ with the reflection $s_{n+i-j}$.
$D_{n}$
: Fill each box $(i,i)$ of $D'_{{\mu}}$ with $s_{n}$ if $i$ is odd or $s_{n-1}$ if $i$ is even; fill each box $(i,j)$, $i<j$, with $s_{n+i-(j+1)}$.
Then $v=s_{i_1}\cdots s_{i_l}$, where $s_{i_1},\ldots, s_{i_l}$ are the entries of $T'_{{\mu}}$ read from right to left, beginning with the bottom row, then the next row up, etc. This decomposition is reduced. To any subset $C$ of $D'_{{\mu}}$, form the subsequence ${\mathbf{s}}_C=(s_{j_1},\ldots,s_{j_q})$ of $(s_{i_1},\ldots,s_{i_l})$ whose entries lie in the set $C$ of boxes of $T'_{{\mu}}$. If $C$ and $D$ are different subsets of $D'_{\mu}$, then we regard ${\mathbf{s}}_C$ and ${\mathbf{s}}_D$ as different subsequences of $(s_{i_1},\ldots,s_{i_l})$, even if they have the same entries.
\[ex.reduceddecompbc\] Let $n=6$. For $v=\{3,6,{\overline{5}},{\overline{4}},{\overline{2}},{\overline{1}},1,2,4,5,{\overline{6}},{\overline{3}}\}\in W^{P_n}$ in $C_n$ and\
$v=\{3,6,{\overline{5}},{\overline{4}},{\overline{2}},{\overline{1}},7,1,2,4,5,{\overline{6}},{\overline{3}}\}\in W^{P_n}$ in $B_n$, ${\lambda}_v=(6,6,5,5,4,2)$, ${\mu}={\lambda}'_v=(6,5,3,2)$, $$T'_{{\mu}}=
\begin{tikzpicture}[
scale=.5,
every node/.style={scale=.8},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos / \label in
{ {(1,4)}/{s_6},{(2,4)}/{s_5},{(3,4)}/{s_4},{(4,4)}/{s_3},{(5,4)}/{s_2},{(6,4)}/{s_1},
{(2,3)}/{s_6},{(3,3)}/{s_5},{(4,3)}/{s_4},{(5,3)}/{s_3},{(6,3)}/{s_2},
{(3,2)}/{s_6},{(4,2)}/{s_5},{(5,2)}/{s_4},
{(4,1)}/{s_6},{(5,1)}/{s_5}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\end{tikzpicture}$$ and $s_5s_6s_4s_5s_6s_2s_3s_4s_5s_6s_1s_2s_3s_4s_5s_6$ is a reduced decomposition for $v$.
\[ex.reduceddecompd\] Let $n=7$. For $v=\{3,6,7,{\overline{5}},{\overline{4}},{\overline{2}},{\overline{1}},1,2,4,5,{\overline{7}},{\overline{6}},{\overline{3}}\}\in W^{P_n}$ in $D_n$,\
${\lambda}_v=(7,7,6,6,4,4,2)$ and ${\mu}={\lambda}'_v=(6,5,3,2)$, $$T'_{{\mu}}=
\begin{tikzpicture}[
scale=.5,
every node/.style={scale=.8},
baseline={([yshift=-.5ex]current bounding box.center)}]
\foreach \pos / \label in
{ {(1,4)}/{s_7},{(2,4)}/{s_5},{(3,4)}/{s_4},{(4,4)}/{s_3},{(5,4)}/{s_2},{(6,4)}/{s_1},
{(2,3)}/{s_6},{(3,3)}/{s_5},{(4,3)}/{s_4},{(5,3)}/{s_3},{(6,3)}/{s_2},
{(3,2)}/{s_7},{(4,2)}/{s_5},{(5,2)}/{s_4},
{(4,1)}/{s_6},{(5,1)}/{s_5}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\end{tikzpicture}$$ and $s_5s_6s_4s_5s_7s_2s_3s_4s_5s_6s_1s_2s_3s_4s_5s_7$ is a reduced decomposition for $v$.
\[p.subsequences\_bcd\] Denote the reduced decomposition $(s_{i_1},\cdots, s_{i_l})$ for $v$ obtained above by ${\mathbf{s}}_v$. By definition, $T(w,{\mathbf{s}}_v)=\{{\mathbf{s}}_C\mid C\subseteq D'_{\mu}, H_{{\mathbf{s}}_C}=H_w\}$. We have
- $T(w,{\mathbf{s}}_v)=\{{\mathbf{s}}_C\mid C\in{\mathcal{E}}'_{{\lambda}}({\mu})\}$.
- Let $(i,j)$ be the box of $T'_{{\mu}}$ containing $s_{i_c}$. Define ${\epsilon}_{{\overline{m}}}=-{\epsilon}_m$ for $1\leq m\leq n$. Then [ $$\begin{split}
& B_n:\quad r(c)=2^{-{\delta}_{ij}}({\epsilon}_{v_{n+i+1}} +{\epsilon}_{v_{n+j+1}})\\
& C_n:\quad r(c)={\epsilon}_{v_{n+i}} +{\epsilon}_{v_{n+j}}\\
& D_n:\quad r(c)={\epsilon}_{v_{n+i}} +{\epsilon}_{v_{n+j+1}}
\end{split}$$ ]{}
The reduced decomposition ${\mathbf{s}}_v$ is due to Ikeda and Naruse [@IkNa:09]. Proposition \[p.subsequences\_bcd\](ii) is as well, although our expressions for the constants $r(c)$ are different than theirs. A version of Proposition \[p.subsequences\_bcd\](i) which involves the nil-Coxeter algebra and (what we call) reduced excited Young diagrams is also proved in [@IkNa:09].
Proposition \[p.subsequences\_bcd\] is the counterpart for types $B_n$, $C_n$, and $D_n$ of Proposition \[p.subsequences\]. The proof of part (i) carries over with very minor modifications. We omit the details. We prove part (ii) below.
\[d.energy\_bcd\] Let $C$ be a subset of $D_{\mu}$ such that $H_{{\mathbf{s}}_C}=H_w$. Define
1. $|C|=$ number of boxes of $C$
2. $B_n$, $C_n$: $e_1(C)=(1/2)(\sum_{(i,j)\in C}(i+j)-\sum_{(i,j)\in D_{\lambda}}(i+j))$\
$D_n$: $e_1(C)=(1/2)(\sum_{(i,j)\in C, i<j}(i+j)+\sum_{(i,i)\in C}i-\sum_{(i,j)\in D_{\lambda}, i<j}(i+j)-\sum_{(i,i)\in D_{\lambda}}i)$
3. $e_2(C)=|C|-|D'_{{\lambda}}|=|C|-|{\lambda}|$
We call $e_1(C)$ and $e_2(C)$ the **type 1 energy** and **type 2 energy** of $C$ respectively.
### Proof of Proposition \[p.subsequences\_bcd\](ii)
**1. Type $C_n$**. Recall that entry of box $(l,m)$ of $T'_{{\lambda}_v}$ is $s_{n+l-m}$. By , the rightmost box of row $i$ lies in column $v_{n+1-i}-(n+1-i)$; by Lemma \[l.trans\_partition\], the lowest box of column $j$, assuming that this box lies to the right of the ‘descending staircase’, lies in row $n+j-v_{n+j}$. The entries of these two boxes in $T'_{{\mu}}$ are $s_a$ and $s_b$ respectively, where $$\begin{aligned}
a&=n+i-(v_{n+1-i}-(n+1-i))=2n+1-v_{n+1-i}={\overline{v_{n+1-i}}}=v_{n+i}.\label{e.rightboxc}\\
b&= n+((n+j)-v_{n+j})-j=(2n+1)-v_{n+j}-1={\overline{v_{n+j}}}-1.\label{e.bottomboxc}\end{aligned}$$ We consider three cases: $i<j$ and $(j,j)\in D'_{{\mu}}$, $i<j$ and $(j,j)\not\in D'_{{\mu}}$, and $i=j$.
[*Case 1. $i<j$ and $(j,j)\in D'_{{\mu}}$*]{}. Let $s_x, s_y$ denote the reflections which lie in the rightmost box of rows $i$ and $j$ respectively. Figure \[f.Case1\_bc\] shows some of the entries of $T'_{{\mu}}$.
$$T'_{{\mu}}=\,
\begin{tikzpicture}[xscale=.6,yscale=.6,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.7}]
\draw (0,8) node{$i$};
\draw (7,10) node{$j$};
\draw[gray!30,fill=gray!30] (14.5,8.5) -- ++(-8,0) -- ++(0,-6) -- ++(7,0) -- ++(0,1) -- ++(-6,0) -- ++(0,4) -- ++(7,0) -- cycle;
\foreach \pos / \label in
{ {(7,8)}/{{\alpha}_p},{(8,8)}/{s_{p-1}},{(9,8)}/{s_{p-2}},{(14,8)}/{s_{x}},
{(6,7)}/{s_{p+2}},{(7,7)}/{s_{p+1}},{(8,7)}/{s_{p}},{(9,7)}/{s_{p-1}},
{(6,6)}/{s_{p+3}},{(7,6)}/{s_{p+2}},{(8,6)}/{s_{p+1}},{(9,6)}/{s_{p}},
{(7,3)}/{s_n},{(8,3)}/{s_{n-1}},{(9,3)}/{s_{n-2}},{(13,3)}/{s_y}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\draw (0.5,9.5)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(1,0) -- ++(0,1)
-- ++(2,0) -- ++(0,1)
-- ++(1,0) -- ++(0,3)
-- ++(1,0) -- ++ (0,4)
-- cycle;
\draw (7,4.5) node {$\vdots$};
\draw (11,3) node {$\cdots$};
\draw (11,8) node {$\cdots$};
\end{tikzpicture}$$
In the expression $r(c)=s_{i_1}s_{i_2}\cdots s_{i_{c-1}}({\alpha}_{i_c})=s_{i_1}s_{i_2}\cdots s_{p-2}s_{p-1}({\epsilon}_p-{\epsilon}_{p+1})$, the reflections $s_{i_j}$ which lie outside of the shaded boxes can be removed. We have [$$\begin{aligned}
r(c)
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots s_{p+2} s_{p+1} s_x\cdots s_{p-2} s_{p-1} ({\epsilon}_p-{\epsilon}_{p+1})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots s_{p+2} s_{p+1} s_x\cdots s_{p-2} ({\epsilon}_{p-1}-{\epsilon}_{p+1})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots s_{p+2} s_{p+1} s_x\cdots ({\epsilon}_{p-2}-{\epsilon}_{p+1})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots s_{p+2} s_{p+1} ({\epsilon}_{x}-{\epsilon}_{p+1})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots s_{p+2} ({\epsilon}_{x}-{\epsilon}_{p+2})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n\cdots ({\epsilon}_{x}-{\epsilon}_{p+3})\\
&= s_y\cdots s_{n-2}s_{n-1}s_n({\epsilon}_{x}-{\epsilon}_{n})\\
&= s_y\cdots s_{n-2}s_{n-1}({\epsilon}_{x}+{\epsilon}_{n})\\
&= s_y\cdots s_{n-2}({\epsilon}_{x}+{\epsilon}_{n-1})\\
&= s_y\cdots ({\epsilon}_{x}+{\epsilon}_{n-2})
= {\epsilon}_{x}+{\epsilon}_{y}={\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}},\end{aligned}$$ ]{} where the last equality is due to .
[*Case 2. $i<j$ and $(j,j)\not\in D'_{{\mu}}$*]{}.
$$T'_{{\mu}}=\,
\begin{tikzpicture}[xscale=.6,yscale=.6,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.7}]
\draw (0,8) node{$i$};
\draw (11,10) node{$j$};
\draw[gray!30,fill=gray!30] (14.5,8.5) -- ++(-4,0) -- ++(0,-7) -- ++(1,0) -- ++(0,6) -- ++(3,0) -- cycle;
\foreach \pos / \label in
{ {(14,8)}/{s_{x}}, {(11,2)}/{s_z}, {(11,8)}/{{\alpha}_p}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\draw (0.5,9.5)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(1,0) -- ++(0,1)
-- ++(2,0) -- ++(0,1)
-- ++(1,0) -- ++(0,3)
-- ++(1,0) -- ++ (0,4)
-- cycle;
\draw (11,5) node {$\vdots$};
\draw (12.5,8) node {$\cdots$};
\end{tikzpicture}$$
As in the above calculation, reflections lying outside of the shaded region can be removed from the expression $r(c)=s_{i_1}s_{i_2}\cdots s_{i_{c-1}}({\alpha}_{i_c})$, where ${\alpha}_{i_c}={\alpha}_{p}={\epsilon}_p-{\epsilon}_{p+1}$. Let $s_x, s_z$ be the reflections which lie in the rightmost box of row $i$ and bottom box of column $j$ respectively. One checks that $r(c)={\epsilon}_{x}-{\epsilon}_{z+1}={\epsilon}_{v_{n+i}}-{\epsilon}_{{\overline{v_{n+j}}}}={\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j}}$, using and .
[*Case 3: $i=j$*]{}. In this case, $r(c)
= \cdots s_{x}\cdots s_{n-2}s_{n-1}(2{\epsilon}_n)=2{\epsilon}_x={\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+i}}$.
**2. Type $B_n$**. The Weyl group in type $B_n$ is the same as in type $C_n$, and the simple roots other than ${\alpha}_n$ are identical as well. It follows that if $i<j$, then $r(c)$ is the same value computed in type $C_n$. If $i=j$, then $ r(c) = \cdots s_{x}\cdots s_{n-2}s_{n-1}({\epsilon}_n)={\epsilon}_x=2^{-1}({\epsilon}_x+{\epsilon}_x)$.
**3. Type $D_n$**.
Recall that for each element $v\in W^{P_n}$, $D'_{{\mu}}=D'_{{\lambda}'_v}$ is formed by removing all boxes $(i,j)$ of $D_{{\lambda}_v}$ such that $i\geq j$. The column number of any box in $D'_{{\lambda}'_v}$ is one less than the column number of the corresponding box of $D_{{\lambda}_v}$. Thus in order to ‘place’ a box in $D'_{{\lambda}'_v}$ into its appropriate box in $D_{{\lambda}_v}$, one must add one to its column index.
Recall also that for $l<m$ and for $l=m$, $l$ even, the entry of box $(l,m)$ of $T'_{{\lambda}_v}$ is $s_{n+l-(m+1)}$. By , the rightmost box of row $i$ lies in column $v_{n+1-i}-(n+1-i)$ of $D_{{\lambda}_v}$, and thus in column $v_{n+1-i}-(n+1-i)-1$ of $D'_{{\lambda}'_v}$; by Lemma \[l.trans\_partition\], the lowest box of column $j$, assuming that this box lies to the right of the ‘descending staircase’, lies in row $n+j+1-v_{n+j+1}$. The entries of these two boxes in $T'_{\mu}$ are $s_a$ and $s_b$ respectively, where $$\begin{aligned}
a&=n+i-(v_{n+1-i}-(n+1-i)-1+1)=2n+1-v_{n+1-i}={\overline{v_{n+1-i}}}=v_{n+i}\label{e.rightboxd}\\
b&=n+((n+j+1)-v_{n+j+1})-(j+1)=(2n+1)-v_{n+j+1}-1={\overline{v_{n+j+1}}}-1\label{e.bottomboxd}\end{aligned}$$ We consider four possibilities for $(i,j)\in D'_{{\lambda}'_v}$: $i<j$, $(j+1,j+1)\in D'_{{\lambda}'_v}$, $j$ odd; $i<j$, $(j+1,j+1)\in D'_{{\lambda}'_v}$, $j$ even; $i<j$, $(j+1,j+1)\not\in D'_{{\lambda}'_v}$; and $i=j$.
[*Cases 1 and 2: $i<j$ and $(j+1,j+1)\in D'_{{\mu}}$*]{}. Let $s_x, s_y$ denote the reflections which lie in the rightmost box of row $i$ and row $j+1$ respectively. Reflections $s_{i_k}$ not lying in the shaded boxes of Figure \[f.Case1\_d\] can be removed from the expression $r(c)=s_{i_1}s_{i_2}\cdots s_{i_{c-1}}({\alpha}_{i_c})$, where ${\alpha}_{i_c}={\alpha}_{p}={\epsilon}_p-{\epsilon}_{p+1}$.
$$T'_{{\mu}}=\,
\begin{tikzpicture}[xscale=.6,yscale=.6,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.7}]
\draw (0,7) node{$i$};
\draw (6,9) node{$j$};
\draw[gray!30,fill=gray!30] (14.5,7.5) -- ++(-9,0) -- ++(0,-5) -- ++(1,0) -- ++(0,4) -- ++(8,0) -- cycle;
\draw[gray!30,fill=gray!30] (6.5,2.5) -- ++(7,0) -- ++(0,-1) -- ++(-7,0) -- cycle;
\foreach \pos / \label in
{ {(6,7)}/{{\alpha}_p},{(7,7)}/{s_{p-1}},{(8,7)}/{s_{p-2}},{(14,7)}/{s_{x}},
{(6,6)}/{s_{p+1}},{(7,6)}/{s_{p}},{(8,6)}/{s_{p-1}},
{(6,4)}/{s_{n-2}},{(7,4)}/{s_{n-3}},{(8,4)}/{s_{n-4}},
{(6,3)}/{s_{a}},
{(7,2)}/{s_b},{(8,2)}/{s_{n-2}},{(9,2)}/{s_{n-3}},{(13,2)}/{s_y}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\draw (6,5) node{\vdots};
\draw (11,7) node{$\cdots$};
\draw (11,2) node{$\cdots$};
\draw (0.5,8.5)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(4,0) -- ++(0,1)
-- ++(1,0) -- ++(0,3)
-- ++(1,0) -- ++ (0,4)
-- cycle;
\end{tikzpicture}$$
If $j$ is odd then $a=n$ and $b=n-1$; otherwise $a=n-1$ and $b=n$. In either case, $r(c)={\epsilon}_{x}+{\epsilon}_{y}={\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j+1}}$, where the last equality is due to .
[*Case 3: $i<j$ and $(j+1,j+1)\not\in D'_{{\mu}}$*]{}. Let $s_x, s_z$ be the reflections which lie in the rightmost box of row $i$ and bottom box or column $j$ respectively. Reflections $s_{i_k}$ not in the shaded boxes of Figure \[f.Case3\_d\] can be removed from the expression $r(c)=s_{i_1}s_{i_2}\cdots s_{i_{c-1}}({\alpha}_{i_c})$, where ${\alpha}_{i_c}={\alpha}_{p}={\epsilon}_p-{\epsilon}_{p+1}$. One checks that $r(c)={\epsilon}_x-{\epsilon}_{z+1}={\epsilon}_{v_{n+i}}-{\epsilon}_{{\overline{v_{n+j+1}}}}={\epsilon}_{v_{n+i}}+{\epsilon}_{v_{n+j+1}}$, where the last equality is due to and .
$$T'_{{\mu}}=\,
\begin{tikzpicture}[xscale=.6,yscale=.6,
baseline={([yshift=-.5ex]current bounding box.center)},
every node/.style={scale=.7}]
\draw (0,7) node{$i$};
\draw (9,9) node{$j$};
\draw[gray!30,fill=gray!30] (14.5,7.5) -- ++(-6,0) -- ++(0,-7) -- ++(1,0) -- ++(0,6) -- ++(5,0) -- cycle;
\foreach \pos / \label in
{ {(14,7)}/{s_{x}},{(9,7)}/{{\alpha}_p},{(9,1)}/{s_z}
}
{
\draw \pos +(-.5,-.5) rectangle ++(.5,.5);
\draw \pos node{$\label$};
}
\draw (9,4) node{\vdots};
\draw (12,7) node{$\cdots$};
\draw (0.5,8.5)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(0,-1) -- ++(1,0)
-- ++(4,0) -- ++(0,1)
-- ++(1,0) -- ++(0,3)
-- ++(1,0) -- ++ (0,4)
-- cycle;
\end{tikzpicture}$$
[*Case 4: $i=j$*]{}. The analysis is similar to the other cases.
Relationship with previously obtained restriction formulas {#ss.previous}
----------------------------------------------------------
The formulas in this paper for $i_v^*[{\mathcal{O}}_{X^w}]$ for types $A_n$ and $C_n$, expressed in terms of set-valued tableaux, appeared earlier in [@Kre:05] and [@Kre:06]. A restriction formula in type $A_n$ can also be obtained by specializing the factorial Grothendieck polynomials of [@Mcn:06]. This restriction formula is generalized to types $B_n$, $C_n$, and $D_n$ in [@IkNa:11]. The main difference in the restriction formulas of this paper is that they are positive, meaning that they result in $(-1)^{l(w)}$ times sums of monomials in $e^{-{\alpha}}-1$, where ${\alpha}$ is a positive root (see Theorem \[t.pullback\] and Remark \[r.tangent\]). In the two examples below, we compare the restriction formulas of this paper and of [@IkNa:11].
The formulas of [@IkNa:11], whose notation and definitions we adopt in this section, use the binary operators $\oplus$ and $\ominus$ (see [@FoKi:94], [@FoKi:96]): $$a\oplus b=a+b+{\beta}ab\ \text{ and }\ a\ominus b=\frac{a-b}{1+{\beta}b},$$ where ${\beta}$ is a parameter. Note that $\ominus$ is the inverse of $\oplus$. Define $\oplus x= 0\oplus x=x$, $\ominus x=0\ominus x$, and set the parameter ${\beta}$ equal to $-1$. One checks that $(1-e^x)\oplus (1-e^y)=1-e^{x+y}$, $(1-e^x)\ominus (1-e^y)=1-e^{x-y}$, and $\ominus (1-e^{x})=1-e^{-x}$.
We work in type $C_n$, $n=2$. Let $v=w=\{2,4,1,3\}=\{2,{\overline{1}},1,{\overline{2}}\}\in W^{P_n}$. Then ${\lambda}_v=(2,1)$, ${\mu}={\lambda}'_v=(2,0)$, and ${\lambda}={\lambda}'_w=(2,0)$.
We first compute $i_v^*[{\mathcal{O}}_{X^w}]$ using Theorem \[t.ktheory\_eyd\_bcd\]. The set ${\mathcal{E}}'_{{\lambda}}({\mu})$ consists of the single element
in [[(1,1)]{},[(2,1)]{}]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} in [ [(1,1)]{},[(2,1)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} (0,1) node[$1$]{}; (1,2) node[$1$]{}; (2,2) node[$-2$]{};
Thus $$\label{e.pull_smallex}
i_v^*[{\mathcal{O}}_{X^w}]=(e^{-2{\epsilon}_1}-1)(e^{-({\epsilon}_1-{\epsilon}_2)}-1).$$
We next compute $i_v^*[{\mathcal{O}}_{X^w}]$ using the methods of [@IkNa:11]. The elements of ${\mathcal{E}}_n^I({\lambda})$ [@IkNa:11 9.2] are
By [@IkNa:11 (9.6)], $$\begin{aligned}
{\operatorname{GC}}_{\lambda}^{(n)}(x\,|\,b)= & (x_1\oplus x_1)(x_1\oplus x_2)+(x_1\oplus x_1)(x_2\oplus b_1)(1-x_1\oplus x_2)\\
&+(x_2\oplus x_2)(x_2\oplus b_1)(1-x_1\oplus x_1).
\intertext{Substituting $b_{\mu}=(\ominus b_{{\mu}_1},\ldots,\ominus b_{{\mu}_r},0,0,0,\ldots)=(\ominus b_{2},0,0,\ldots)$ for $x$,}
{\operatorname{GC}}_{\lambda}^{(n)}(b_{\mu}\,|\,b)= & (\ominus b_{2}\ominus b_{2})(\ominus b_{2}\oplus 0)+(\ominus b_{2}\ominus b_{2})(0\oplus b_1)(1-\ominus b_{2}\oplus 0)\\
&+(0\oplus 0)(0\oplus b_1)(1-\ominus b_{2}\ominus b_{2})\\
= & (\ominus b_{2}\ominus b_{2})(\ominus b_{2})+(\ominus b_{2}\ominus b_{2})(b_1)(1-\ominus b_{2}).\end{aligned}$$ Substituting $1-e^{t_i}$ for $b_i$, $i=1,2$, and then replacing $t_1,t_2$ by ${\epsilon}_2,{\epsilon}_1$ respectively in order to account for the ordering of the roots of the Dynkin diagram in [@IkNa:11 4.6], we obtain $$\label{e.pull_smallex_in}
i_v^*[{\mathcal{O}}_{X^w}] = (1-e^{-2{\epsilon}_1})(1-e^{-{\epsilon}_1})+(1-e^{-2{\epsilon}_1})(1-e^{{\epsilon}_2})(1-(1-e^{-{\epsilon}_1})).$$ One checks that this agrees with .
\[ex.in\_d5\] Consider type $D_n$, $n=5$. Let $v=\{2,4,5,8,10,1,3,6,7,9\}=$\
$\{2,4,5,{\overline{3}},{\overline{1}},1,3,{\overline{5}},{\overline{4}},{\overline{2}}\}$, $w=\{1,2,5,7,8,3,4,6,9,10\}=\{1,2,5,{\overline{4}},{\overline{3}},3,4,{\overline{5}},{\overline{2}},{\overline{1}}\}\in W^{P_n}$. Then ${\lambda}_v=(5,4,2,2,1)$, ${\mu}={\lambda}'_v=(4,2)$, ${\lambda}_w=(3,3,2,0,0)$, ${\lambda}={\lambda}'_w=(2,1)$.
We first compute $i_v^*[{\mathcal{O}}_{X^w}]$ using Theorem \[t.ktheory\_eyd\_bcd\]. The set ${\mathcal{E}}'_{{\lambda}}({\mu})$ consists of the single element
in [[(1,2)]{},[(2,2)]{},[(2,1)]{}]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} in [ [(1,2)]{},[(2,2)]{},[(3,2)]{},[(4,2)]{}, [(2,1)]{},[(3,1)]{} ]{} [+(-.5,-.5) rectangle ++(.5,.5);]{} (0,1) node[$3$]{}; (0,2) node[$1$]{}; (1,3) node[$3$]{}; (2,3) node[-$5$]{}; (3,3) node[-$4$]{}; (4,3) node[-$2$]{};
Thus $$\label{e.pull_bigger}
i_v^*[{\mathcal{O}}_{X^w}]=-(e^{-({\epsilon}_1+{\epsilon}_3)}-1)(e^{-({\epsilon}_1-{\epsilon}_5)}-1)
(e^{-({\epsilon}_3-{\epsilon}_5)}-1).$$
We next compute $i_v^*[{\mathcal{O}}_{X^w}]$ using the methods of [@IkNa:11]. The elements of ${\mathcal{E}}_n^I({\lambda})$ [@IkNa:11 9.2] are
By [@IkNa:11 (9.6)], $$\begin{aligned}
{\operatorname{GD}}_{\lambda}^{(n)}(x\,|\,b)= & (x_1\oplus x_2)(x_1\oplus x_3)(x_2\oplus x_3)\\
& + (x_1\oplus x_2)(x_1\oplus x_3)(x_4\oplus b_1)(1-x_2\oplus x_3)\\
& + (x_1\oplus x_2)(x_2\oplus x_4)(x_4\oplus b_1)(1-x_1\oplus x_3)\\
& + (x_1\oplus x_2)(x_3\oplus b_1)(x_4\oplus b_1)(1-x_1\oplus x_3)(1-x_2\oplus x_4)\\
& + (x_3\oplus x_4)(x_3\oplus b_1)(x_4\oplus b_1)(1-x_1\oplus x_2).
\intertext{Substituting $b_{\mu}=(\ominus b_{{\mu}_1+1},\ldots,\ominus b_{{\mu}_r+1},0,0,0,\ldots)=(\ominus b_{5},\ominus b_{3},0,0,0,\ldots)$ for $x$,}
{\operatorname{GD}}_{\lambda}^{(n)}(b_{\mu}\,|\,b)= & (\ominus b_{5}\ominus b_3)(\ominus b_{5})(\ominus b_3)\\
& + (\ominus b_{5}\ominus b_3)(\ominus b_{5})(b_1)(1-\ominus b_3)\\
& + (\ominus b_{5}\ominus b_3)(\ominus b_3)(b_1)(1-\ominus b_{5})\\
& + (\ominus b_{5}\ominus b_3)(b_1)(b_1)(1-\ominus b_{5})(1-\ominus b_3).\end{aligned}$$ Substituting $1-e^{t_i}$ for $b_i$, $i=1,\ldots,5$, and then replacing $t_1,t_2,t_3,t_4,t_5$ by ${\epsilon}_5,{\epsilon}_4,{\epsilon}_3,{\epsilon}_2,{\epsilon}_1$ respectively in order to account for the ordering of the roots of the Dynkin diagram in [@IkNa:11 4.6], we obtain $$\label{e.pull_bigger_in}
\begin{split}
i_v^*[{\mathcal{O}}_{X^w}] = & (1-e^{-({\epsilon}_1-{\epsilon}_3)})(1-e^{-{\epsilon}_1})(1-e^{-{\epsilon}_3})\\
& + (1-e^{-({\epsilon}_1-{\epsilon}_3)})(1-e^{-{\epsilon}_1})(1-e^{{\epsilon}_5})(1-(1-e^{-{\epsilon}_3}))\\
& + (1-e^{-({\epsilon}_1-{\epsilon}_3)})(1-e^{-{\epsilon}_3})(1-e^{{\epsilon}_5})(1-(1-e^{-{\epsilon}_1}))\\
& + (1-e^{-({\epsilon}_1-{\epsilon}_3)})(1-e^{{\epsilon}_5})(1-e^{{\epsilon}_5})(1-(1-e^{-{\epsilon}_1}))(1-(1-e^{-{\epsilon}_3})).
\end{split}$$ One checks that this agrees with .
Although the excited Young diagrams introduced in [@IkNa:11] and in this paper are both related to the reduced excited Young diagrams of [@IkNa:09], [@Kre:05], [@Kre:06], they are different combinatorial objects. While the excited Young diagrams of this paper reside in Young diagrams, those of [@IkNa:11] reside in a grid which is unbounded on the right. This differs from the reduced excited Young diagrams of [@IkNa:09], [@Kre:05], [@Kre:06], as well as the excited Young diagrams in this paper, since these all reside in Young diagrams. Another difference between the excited Young diagrams of [@IkNa:11] and of here is that the former are produced by modifying reduced excited Young diagrams by adding ‘$\times$’ symbols; an excited Young diagram of [@IkNa:11] with $k$ of these symbols encodes $2^k$ of what we would call excited Young diagrams.
Restriction formula in terms of set-valued shifted tableaux {#s.setvalued_shiftedtableaux}
-----------------------------------------------------------
Let ${\lambda}$ be a strict partition. A **set-valued filling** of $D'_{\lambda}$ is a function $T$ which assigns to each box $(i,j)$ of $D'_{{\lambda}}$ a nonempty subset $T(i,j)$ of $\{1,\ldots,n\}$. We call ${\lambda}$ the **shape** of $T$. We often call $(i,j)$ a box of $T$, and refer to an element of $T(i,j)$ as an entry of box $(i,j)$ of $T$, or just an entry of $T$. A set-valued filling $T$ in which each entry of box $(i,j)$ of $T$ is less than or equal to each entry of box $(i,j+1)$ and strictly less than each entry of box $(i+1,j)$ is said to be **semistandard**. A **set-valued shifted Young tableau**, or just **set-valued shifted tableau**, is a semistandard set-valued filling of $D'_{\lambda}$. A **shifted Young tableau** or just **shifted tableau** is a set-valued tableau in which each box contains a single entry.
Let ${\mu}$ be a strict partition. We say that a set-valued shifted tableau is **restricted by ${\mu}$** if, for any box $(i,j)\in T$ and any entry $x$ of $(i,j)$, $$\label{e.restrictedbyv_bcd}
j-i\leq {\mu}(x)-1.$$ Denote by ${\mathcal{T}}'_{{\lambda}}({\mu})$ (resp. ${\mathcal{T}}'^{\text{red}}_{{\lambda}}({\mu})$) the set of set-valued shifted tableaux (resp. shifted tableaux) of shape ${\lambda}$ which are restricted by ${\mu}$ (see Figures \[f.syt\_allbc\] and \[f.syt\_alld\]). The following theorem in type $C_n$ appeared in [@Kre:06].
\[t.ktheory\_syt\_bcd\] Let $w\leq v\in W^{P_n}$, and let ${\lambda}={\lambda}'_w$ and ${\mu}={\lambda}'_v$ be the corresponding strict partitions. Then [$$\begin{aligned}
& B_n:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{T\in{\mathcal{T}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in T}\prod_{x\in T(i,j)}\left(e^{-2^{-\delta_{ij}}({\epsilon}_{v_{n+x+1}}+{\epsilon}_{v_{n+x+j-i+1}})}-1\right)\\
& C_n:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{T\in{\mathcal{T}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in T}\prod_{x\in T(i,j)}\left(e^{-({\epsilon}_{v_{n+x}}+{\epsilon}_{v_{n+x+j-i}})}-1\right)\\
& D_{n}:\quad i_v^*[{\mathcal{O}}_{X^w}]=(-1)^{l(w)}\sum_{T\in{\mathcal{T}}'_{{\lambda}}({\mu})}\prod_{(i,j)\in T}\prod_{x\in T(i,j)}\left(e^{-({\epsilon}_{v_{n+x}}+{\epsilon}_{v_{n+x+j-i+1}})}-1\right),\end{aligned}$$ ]{} where for $1\leq m\leq n$, ${\epsilon}_{{\overline{m}}}$ is defined to equal $-{\epsilon}_m$.
Theorem \[t.ktheory\_syt\_bcd\] is essentially the same statement as Theorem \[t.ktheory\_eyd\_bcd\], except that the indexing set ${\mathcal{E}}'_{{\lambda}}({\mu})$ has been replaced by ${\mathcal{T}}'_{{\lambda}}({\mu})$. This replacement is given by the map $f:{\mathcal{T}}'_{{\lambda}}({\mu})\to\{\text{subsets of }D'_{{\mu}}\}$, $$\label{e.f_bcd}
\begin{split}
f(T)= \{(x,x+j-i)\mid (i,j)\in{\lambda}, x\in T(i,j)\}.
\end{split}$$
\[p.fbijective\_bcd\] The map $f$ is a bijection from ${\mathcal{T}}'_{{\lambda}}({\mu})$ to ${\mathcal{E}}'_{{\lambda}}({\mu})$.
The proof of this proposition is so similar to that of Proposition \[p.fbijective\] that we omit the details.
Hilbert series and Hilbert polynomials of points on Schubert varieties {#ss.hilbseries_BCD}
----------------------------------------------------------------------
In types $C_n$ and $D_n$, the parabolic subgroup ${P_n}$ is cominuscule (cf. [@BiLa:00 9.0.14]). Thus Corollary \[c.cominusculemult\] may be used to compute the Hilbert series, Hilbert polynomial, and multiplicity of $X^w$ at $v$. In the present setting, the constant $m_k$ of Corollary \[c.cominusculemult\] is equal to the number of excited shifted Young diagrams $C\in{\mathcal{E}}'_{{\lambda}}({\mu})$ such that the number of boxes of $C$ is $k+|{\mu}|$. In terms of set-valued shifted Young tableaux, $m_k$ is equal to the number of $T\in{\mathcal{T}}'_{{\lambda}}({\mu})$ with $k+|{\mu}|$ entries.
In type $C_n$, $n=4$, let $w=\{1,2,{\overline{4}},{\overline{3}},3,4,{\overline{2}},{\overline{1}}\}$, $v=\{2,{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,{\overline{2}}\}$, as in Example \[ex.restriction\_BC\]. Then ${\lambda}={\lambda}'_w=(2,1)$, ${\mu}={\lambda}'_v=(4,2,1)$, $l(w)=|{\lambda}'_w|=3$, and $d_w=n(n+1)/2-l(w)=7$. The set of excited shifted Young diagrams ${\mathcal{E}}'_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_allbc\], and the set of set-valued shifted tableaux ${\mathcal{T}}'_{{\lambda}}({\mu})$ appears in Figure \[f.syt\_allbc\]. From either of these figures, one sees that $m_0=4$ and $m_1=3$. Hence [$$\begin{aligned}
H(X^w,v)(t)&=\frac{4}{(1-t)^7}-\frac{3}{(1-t)^6},\\
h(X^w,v)(i)&=4{i +6\choose 6}-3{i+5\choose 5},\\
{\operatorname{mult}}(X^w,v)&=4.\end{aligned}$$ ]{}
\[ex.Hilbert\_Dn\] In type $D_n$, $n=6$, let $w=\{1,2,4,6,{\overline{5}},{\overline{3}},3,5,{\overline{6}},{\overline{4}},{\overline{2}},{\overline{1}}\}$ and $v=\{2,6,{\overline{5}},{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,5,{\overline{6}},{\overline{2}}\}$, as in Example \[ex.restriction\_D\]. Then ${\lambda}={\lambda}'_w=(3,1)$, ${\mu}={\lambda}'_v=(5,3,2,1)$, $l(w)=|{\lambda}'_w|=4$, and $d_w=n(n-1)/2 - l(w)=11$. The set of excited shifted Young diagrams ${\mathcal{E}}'_{{\lambda}}({\mu})$ appears in Figure \[f.eyd\_alld\], and the set of set-valued shifted tableaux ${\mathcal{T}}'_{{\lambda}}({\mu})$ appears in Figure \[f.syt\_alld\]. From either of these figures one reads off $m_0=5$, $m_1=5$, and $m_2=1$. Hence $$\begin{aligned}
H(X^w,v)(t)&=\frac{5}{(1-t)^{11}}-\frac{5}{(1-t)^{10}}+\frac{1}{(1-t)^{9}},\\
h(X^w,v)(i)&=5{i +10\choose 10}-5{i+9\choose 9}+{i+8\choose 8},\\
{\operatorname{mult}}(X^w,v)&=5.\end{aligned}$$
In type $B_n$, ${P_n}$ is not cominuscule, so Corollary 1.14 may not be applied directly. However, since the isomorphism $\pi:{\operatorname{OG}}(n+1,2n+2)\to {\operatorname{OG}}(n,2n+1)$ identifies $T$-fixed points and Schubert varieties (see Section \[ss.orthog\_grassmannians\] or [@RaUp:10 1.3]), properties of singularities of Schubert varieties in ${\operatorname{OG}}(n,2n+1)$ can be obtained from those of Schubert varieties in ${\operatorname{OG}}(n+1,2n+2)$.
In type $B_n$, $n=5$, let $w=(1,2,4,{\overline{5}},{\overline{3}},3,5,{\overline{4}},{\overline{2}},{\overline{1}})$ and\
$v=(2,5,{\overline{4}},{\overline{3}},{\overline{1}},1,3,4,{\overline{5}},{\overline{2}})$. Then $v$ and $w$ are identified with the corresponding elements of $W^{P_{n+1}}$ of Example \[ex.Hilbert\_Dn\] (see Section \[ss.strict\_partitions\_bcd\]). Therefore $H(X^w,v)(t)$, $h(X^w,v)(i)$, and ${\operatorname{mult}}(X^w,v)$ are the same as in that example.
Other multiplicity formulas appear in [@LaWe:90], [@GhRa:06], and [@RaUp:10]. The above formula for the multiplicity of $X^w$ at $v$, expressed in terms of reduced excited shifted Young diagrams, appeared earlier in [@IkNa:09]. In type $C_n$, the formula can be deduced from [@Kre:06] and [@GhRa:06]. Formulas for the Hilbert function of $X^w$ at $v$ appear in [@GhRa:06], [@RaUp:10], and [@Upa:09].
Root systems and Weyl groups in types $A_n$, $B_n$, $C_n$, and $D_n$ {#s.appendix_roots}
====================================================================
We review some facts about the classical root systems and Weyl groups.
**Type $A_{n-1}$**. The special linear group ${\operatorname{SL}}_{n}({\mathbb{C}})$ is equal to $G=\{g\in {\operatorname{GL}}_{n}({\mathbb{C}})\mid \det(g)=1\}$. The Lie algebra ${{\mathfrak g}}={{\mathfrak s}}{{\mathfrak l}}_{n}({\mathbb{C}})=\{a\in {{\mathfrak g}}{{\mathfrak l}}_{n}({\mathbb{C}})\mid {\operatorname{trace}}(a)=0\}$. The set of diagonal matrices in ${{\mathfrak g}}$ forms a Cartan subalgebra $${{\mathfrak h}}=\{{\operatorname{diag}}(a_1,\ldots,a_{n})\mid a_1,\ldots,a_{n}\in{\mathbb{C}}, \sum a_i = 0\}.$$ For $1\leq i\leq n$, let ${\epsilon}_i\in{{\mathfrak h}}^*$ be the linear functional $${\epsilon}_i({\operatorname{diag}}(a_1,\ldots,a_{n}))=a_i.$$ Then $\{{\epsilon}_1,\ldots,{\epsilon}_{n}\}$ span ${{\mathfrak h}}^*$. The set of roots ${\Phi}$ of ${{\mathfrak g}}$ relative to ${{\mathfrak h}}$ is $\{{\epsilon}_i- {\epsilon}_j, 1\leq i\neq j\leq n\}$. The set $${\Delta}=\{{\alpha}_1={\epsilon}_1-{\epsilon}_2,\ldots,{\alpha}_{n-1}={\epsilon}_{n-1}-{\epsilon}_{n}\}$$ forms a base of ${\Phi}$, with respect to which the set of positive roots is $${\Phi}^+=\{{\epsilon}_i-{\epsilon}_j, 1\leq i<j\leq n\}.$$ For $i\in\{1,\ldots,n-1\}$, the reflection $s_i$ along ${\alpha}_i$ is given by: $$s_i:
\begin{cases}
&{\epsilon}_i\mapsto {\epsilon}_{i+1}\\
&{\epsilon}_{i+1}\mapsto{\epsilon}_{i}
\end{cases}.$$ The Weyl group $W=\langle s_1,\ldots,s_n\rangle\subseteq {\operatorname{GL}}({{\mathfrak h}}^*)$ is the group of permutations of $\{{\epsilon}_1,\ldots,{\epsilon}_{n}\}$, which is isomorphic to $S_n$. Denote a permutation $w\in S_n$ by its one-line notation $w=(w_1,\ldots,w_n)$, where $w(1)=w_1$ , $w(2)=w_2$, etc.
**Type $C_n$**. Define the inner product $\langle x,y\rangle=x^tJy$, $x,y\in{\mathbb{C}}^{2n}$, where $J$ is the antidiagonal $2n\times 2n$ matrix whose top $n$ antidiagonal entries are 1’s and whose bottom $n$ antidiagonal entries are -1’s. The symplectic group ${\operatorname{Sp}}_{2n}({\mathbb{C}})$ is equal to $G=\{g\in {\operatorname{GL}}_{2n}({\mathbb{C}})\mid \langle gu,gv\rangle = \langle u,v\rangle, u,v\in{\mathbb{C}}^{2n}\}$. The Lie algebra ${{\mathfrak g}}={{\mathfrak s}}{{\mathfrak p}}_{2n}({\mathbb{C}})=\{a\in {{\mathfrak g}}{{\mathfrak l}}_{2n}({\mathbb{C}})\mid \langle au,v\rangle +\langle u,av\rangle=0, u,v\in{\mathbb{C}}^{2n}\}$. The set of diagonal matrices in ${{\mathfrak g}}$ forms a Cartan subalgebra $${{\mathfrak h}}=\{{\operatorname{diag}}(a_1,\ldots,a_n,-a_n,\ldots,-a_1)\mid a_1,\ldots,a_n\in{\mathbb{C}}\}.$$ For $1\leq i\leq n$, let ${\epsilon}_i\in{{\mathfrak h}}^*$ be the linear functional $${\epsilon}_i({\operatorname{diag}}(a_1,\ldots,a_n,-a_n,\ldots,-a_1))=a_i.$$ Then $\{{\epsilon}_1,\ldots,{\epsilon}_n\}$ forms a basis for ${{\mathfrak h}}^*$. The set of roots ${\Phi}$ of ${{\mathfrak g}}$ relative to ${{\mathfrak h}}$ is $\{\pm{\epsilon}_i\pm {\epsilon}_j, 1\leq i\neq j\leq n\}\cup\{\pm 2{\epsilon}_i,i=1,\ldots,n\}$. The set $${\Delta}=\{{\alpha}_1={\epsilon}_1-{\epsilon}_2,\ldots,{\alpha}_{n-1}={\epsilon}_{n-1}-{\epsilon}_n\}\cup \{{\alpha}_n=2{\epsilon}_n\}$$ forms a base of ${\Phi}$, with respect to which the set of positive roots is $${\Phi}^+=\{{\epsilon}_i\pm{\epsilon}_j, 1\leq i<j\leq n\}\cup \{2{\epsilon}_i,i=1,\ldots,n\}.$$ The reflection $s_i$ along ${\alpha}_i$ is given by: $$\begin{split}
&s_i:
\begin{cases}
&{\epsilon}_i\mapsto {\epsilon}_{i+1}\\
&{\epsilon}_{i+1}\mapsto{\epsilon}_{i}\\
&{\epsilon}_j\mapsto{\epsilon}_j, j\neq i,i+1
\end{cases},
\quad
\text{ if }i\in\{1,\ldots,n-1\}
\\
&s_n:
\begin{cases}
&{\epsilon}_n\mapsto -{\epsilon}_{n}\\
&{\epsilon}_j\mapsto{\epsilon}_j, j\neq n
\end{cases}
\end{split}$$ The Weyl group $W=\langle s_1,\ldots,s_n\rangle\subseteq {\operatorname{GL}}({{\mathfrak h}}^*)$ is the group of permutations and sign changes of $\{{\epsilon}_1,\ldots,{\epsilon}_n\}$. More precisely, $W\cong S_n\ltimes{\mathbb{ Z}}_2^{n}$.
The map $W\to S_{2n}$ given by $s_i\mapsto (i,i+1)({\overline{i+1}},{\overline{i}})$, $i\neq n$, $s_n\mapsto (i,{\overline{i}})$, is a monomorphism, identifying $W$ with $$\label{e.Weyl_Cn}
W\cong \{(w_1,\ldots,w_{2n})\in S_{2n}\mid w_{{\overline{\imath}}}={\overline{w_i}}, 1\leq i\leq n\}.$$
**Type $B_n$**. Define the inner product $\langle x,y\rangle=x^tJy$, $x,y\in{\mathbb{C}}^{2n+1}$, where $J$ is the antidiagonal $(2n+1)\times (2n+1)$ matrix all of whose antidiagonal entries are 1’s, except for the entry in row and column $n+1$, which is 2. The odd orthogonal group ${\operatorname{SO}}_{2n+1}({\mathbb{C}})$ is equal to $G=\{g\in {\operatorname{GL}}_{2n+1}({\mathbb{C}})\mid \langle gu,gv\rangle = \langle u,v\rangle, u,v\in{\mathbb{C}}^{2n+1}\}$.The Lie algebra ${{\mathfrak g}}={{\mathfrak s}}{{\mathfrak o}}_{2n+1}({\mathbb{C}})=\{a\in {{\mathfrak g}}{{\mathfrak l}}_{2n+1}({\mathbb{C}})\mid \langle au,v\rangle +\langle u,av\rangle=0, u,v\in{\mathbb{C}}^{2n+1}\}$. The set of diagonal matrices in ${{\mathfrak g}}$ forms a Cartan subalgebra $${{\mathfrak h}}=\{{\operatorname{diag}}(a_1,\ldots,a_n,0,-a_n,\ldots,-a_1)\mid a_1,\ldots,a_n\in{\mathbb{C}}\}.$$ For $1\leq i\leq n$, let ${\epsilon}_i\in{{\mathfrak h}}^*$ be the linear functional $${\epsilon}_i({\operatorname{diag}}(a_1,\ldots,a_n,0,-a_n,\ldots,-a_1))=a_i.$$ Then $\{{\epsilon}_1,\ldots,{\epsilon}_n\}$ forms a basis for ${{\mathfrak h}}^*$. The set of roots ${\Phi}$ of ${{\mathfrak g}}$ relative to ${{\mathfrak h}}$ is $\{\pm{\epsilon}_i\pm {\epsilon}_j, 1\leq i\neq j\leq n\}\cup\{\pm{\epsilon}_i,i=1,\ldots,n\}$. The set $${\Delta}=\{{\alpha}_1={\epsilon}_1-{\epsilon}_2,\ldots,{\alpha}_{n-1}={\epsilon}_{n-1}-{\epsilon}_n\}\cup \{{\alpha}_n={\epsilon}_n\}$$ forms a base of ${\Phi}$, with respect to which the set of positive roots is $${\Phi}^+=\{{\epsilon}_i\pm{\epsilon}_j, 1\leq i<j\leq n\}\cup \{{\epsilon}_i,i=1,\ldots,n\}.$$
Since the roots in types $B_n$ and $C_n$ agree up to scalar multiples, they have the same Weyl group. Thus also gives an identification of the Weyl group in type $B_n$.
**Type $D_n$**. Define the inner product $\langle x,y\rangle=x^tJy$, $x,y\in{\mathbb{C}}^{2n}$, where $J$ is the antidiagonal $2n\times 2n$ matrix all of whose antidiagonal entries are 1’s. The even orthogonal group ${\operatorname{SO}}_{2n}({\mathbb{C}})$ is equal to $G=\{g\in {\operatorname{GL}}_{2n}({\mathbb{C}})\mid \langle gu,gv\rangle = \langle u,v\rangle, u,v\in{\mathbb{C}}^{2n}\}$. The Lie algebra ${{\mathfrak g}}={{\mathfrak s}}{{\mathfrak o}}_{2n}({\mathbb{C}})=\{a\in {{\mathfrak g}}{{\mathfrak l}}_{2n}({\mathbb{C}})\mid \langle au,v\rangle +\langle u,av\rangle=0, u,v\in{\mathbb{C}}^{2n}\}$, where $\langle x,y\rangle=x^tJy$, $x,y\in{\mathbb{C}}^{2n}$. The set of diagonal matrices in ${{\mathfrak g}}$ forms a Cartan subalgebra $${{\mathfrak h}}=\{{\operatorname{diag}}(a_1,\ldots,a_n,-a_n,\ldots,-a_1)\mid a_1,\ldots,a_n\in{\mathbb{C}}\}.$$ For $1\leq i\leq n$, let ${\epsilon}_i\in{{\mathfrak h}}^*$ be the linear functional $${\epsilon}_i({\operatorname{diag}}(a_1,\ldots,a_n,-a_n,\ldots,-a_1))=a_i.$$ Then $\{{\epsilon}_1,\ldots,{\epsilon}_n\}$ forms a basis for ${{\mathfrak h}}^*$. The set of roots ${\Phi}$ of ${{\mathfrak g}}$ relative to ${{\mathfrak h}}$ is $\{\pm{\epsilon}_i\pm {\epsilon}_j, 1\leq i\neq j\leq n\}$. The set $${\Delta}=\{{\alpha}_1={\epsilon}_1-{\epsilon}_2,\ldots,{\alpha}_{n-1}={\epsilon}_{n-1}-{\epsilon}_n,{\alpha}_n={\epsilon}_{n-1}+{\epsilon}_n\}$$ forms a base of ${\Phi}$, with respect to which the set of positive roots is $${\Phi}^+=\{{\epsilon}_i\pm{\epsilon}_j, 1\leq i<j\leq n\}.$$ The reflection $s_i$ along ${\alpha}_i$ is given by: $$\begin{split}
&s_i:
\begin{cases}
&{\epsilon}_i\mapsto {\epsilon}_{i+1}\\
&{\epsilon}_{i+1}\mapsto{\epsilon}_{i}\\
&{\epsilon}_j\mapsto{\epsilon}_j, j\neq i,i+1
\end{cases},
\quad
\text{ if }i\in\{1,\ldots,n-1\}
\\
&s_n:
\begin{cases}
&{\epsilon}_{n-1}\mapsto -{\epsilon}_{n}\\
&{\epsilon}_n\mapsto -{\epsilon}_{n-1}\\
&{\epsilon}_j\mapsto{\epsilon}_j, j\neq n-1,n
\end{cases}
\end{split}$$ The Weyl group $W=\langle s_1,\ldots,s_n\rangle\subseteq {\operatorname{GL}}({{\mathfrak h}}^*)$ is the group of permutations and even number of sign changes of $\{{\epsilon}_1,\ldots,{\epsilon}_n\}$. More precisely, $W\cong S_n\ltimes{\mathbb{ Z}}_2^{n-1}$.
The map $W\to S_{2n}$ given by $s_i\mapsto (i,i+1)({\overline{i+1}},{\overline{i}})$, $i\neq n$, $s_n\mapsto (n,n+1)(n-1,n)(n+1,n+2)(n,n+1)$, is a monomorphism, identifying $W$ with $$\label{e.Weyl_Dn}
W\cong\left\{(w_1,\ldots,w_{2n})\in S_{2n}\mid
w_{{\overline{\imath}}}={\overline{w_i}}, 1\leq i\leq n,
\#\{i<n\mid w_i>n\}\text{ is even}.
\right\}$$
Restriction formulas and opposite Schubert varieties {#s.appendix_restriction-opp}
====================================================
In this section we explain the relation between the restriction formulas for Schubert varieties and for opposite Schubert varieties. We have included this because some references use the opposite Schubert varieties— indeed, the formulas in [@Gra:02] are for $ i_x^*[{\mathcal{O}}_{X_w}] $, where $X_w = \overline{B \cdot wB}$ is the opposite Schubert variety to $X^w$. However, the formula of Theorem \[t.pullback\] can be obtained from the formula for opposite Schubert varieties by using Proposition \[p.translate\] below. To prove this proposition we need two lemmas. Let $*$ denote the involution of $R(T)$ defined by $*(e^{\lambda}) = e^{- \lambda}$. If $T$ acts on any scheme $M$, we can define a new action $\odot$ of $T$ by the rule $t \odot m = t^{-1}m$. Write $K_T(M,\odot)$ to denote the equivariant $K$-theory of $M$ with the $\odot$ action. Any coherent sheaf on $M$ which is equivariant with respect to the original $T$-action is equivariant with respect to the $\odot$ action. There is a map $K_T(M) \to K_T(M,\odot)$, $\xi \mapsto \xi_{\odot}$, taking the class of a $T$-equivariant sheaf ${\mathcal{F}}$ to the class of the same sheaf, but viewed as equivariant with respect to the $\odot$ action. Observe that if $M$ is a point, then $K_T(M) = R(T)$ and $\xi_{\odot} = *\xi$.
\[l.dual\] Suppose $T$ acts on a smooth scheme $X$. Let $x \in X^T$ and let $i: \{ x \} \to X$ denote the inclusion. Then for any $\xi \in K_T(M)$, $$i^*(\xi_{\odot}) = *( i^*\xi).$$
If we write $\xi = \sum_j a_j [V^j]$, where each $V^j$ is a $T$-equivariant vector bundle, then for each j, the fiber $V^j_x$ is a representation of $T$. We have $i^* \xi = \sum a_j [V^j_x]$ and $$i^*(\xi_{\odot}) = \sum a_j ([V^j_x]_{\odot}) = * ( \sum a_j ([V^j_x])) = *( i^*\xi).$$
\[l.involution\] There exists an involution $\Psi: G \to G$ such that $\Psi(t) = t^{-1}$ for $t \in T$, $\Psi(B) = B^-$, and $\psi(nT) = nT$ for $n \in N_G(T)$.
Given a root ${\beta}$ of ${{\mathfrak g}}$, let ${{\mathfrak g}}_{\beta}$ denote the corresponding root space. For each simple root ${\alpha}$ of ${{\mathfrak g}}$ we can find elements $x_{{\alpha}} \in {{\mathfrak g}}_{{\alpha}}$, $x_{-{\alpha}} \in {{\mathfrak g}}_{-{\alpha}}$, and $h_{{\alpha}} \in {{\mathfrak t}}$ such that $x_{{\alpha}}, \{h_{{\alpha}}, x_{-{\alpha}} \}$ is an ${\mathfrak sl}_2$-triple (see [@Sam:90 Section 2.4]. There is an involution $\psi$ of ${{\mathfrak g}}$ which acts by multiplication by $-1$ on ${{\mathfrak t}}$ such that if ${\alpha}$ is any simple root, then $\psi(x_{{\alpha}}) = x_{-{\alpha}}$ (see [@Hum:72 Proposition 14.3]). It follows that $\psi({{\mathfrak b}}) = {{\mathfrak b}}^-$. Let ${\widetilde}{G}$ denote the simply connected algebraic group with Lie algebra ${{\mathfrak g}}$, and let ${\widetilde}{T}$ denote the subgroup of ${\widetilde}{G}$ with Lie algebra ${{\mathfrak t}}$. Because ${\widetilde}{G}$ is simply connected, $\psi$ lifts to an automorphism $\Psi$ of ${\widetilde}{G}$. Moreover, since $\psi$ acts by multiplication by $-1$ on ${{\mathfrak t}}$, $\Phi$ takes any element of ${\widetilde}{T}$ to its inverse. The group $G$ is isomorphic to $G/Z_1$, where $Z_1$ is a subgroup of the center $Z$ of ${\widetilde}{G}$. Since $Z_1 \subset {\widetilde}{T}$, and $Z_1$ is closed under inverses, $\Psi(Z_1) = Z_1$. Therefore $\Psi$ descends to an automorphism (also denoted $\Psi$) of $G$. The assertions $\Psi(t) = t^{-1}$ for $t \in T$, $\Psi(B) = B^-$ follow from the corresponding properties of $\psi$. Finally, let $J_{{\alpha}} = \frac{\pi}{2}(x_{{\alpha}} - x_{-{\alpha}})$. The simple reflection $s_{{\alpha}}$ in $W$ is represented by the element $n_{{\alpha}} = \exp(J_{{\alpha}}) \in N_G(T)$ (see [@Sam:90 Section 2.15]). The argument in Samelson shows that $s_{{\alpha}}$ is also represented by the element $ \exp(-J_{{\alpha}}) = \Psi(n_{{\alpha}})$. Hence $\Psi(n_{{\alpha}} T) = n_{{\alpha}} T$. Given any $n \in N_G(T)$, $nT = n_{{\alpha}_{1}} n_{{\alpha}_2} \cdots n_{{\alpha}_k} T$ for some simple roots ${\alpha}_1, \ldots, {\alpha}_k$. It follows that $\Psi(nT) = nT$, as claimed.
Let ${\widetilde}{X} = G/B^-$, ${\widetilde}{X}_w= \overline{B^- \cdot w B^-} \subset {\widetilde}{X}$. Let ${\widetilde}{i}_x: \{ pt \} \to {\widetilde}{X}$ be the map ${\widetilde}{i}_x(pt) = xB^-$.
\[p.translate\] $$i_x^*[{\mathcal{O}}_{X^w}] = *(i_{xw_0}^*[{\mathcal{O}}_{X_{ww_0}}]).$$
The map $\phi: G/B \to G/B^-$ defined by $\phi(gB) = g w_0 B^-$ is a $G$-equivariant isomorphism. Since $\phi(xB) = x w_0 B^-$ and $\phi(X^w) = {\widetilde}{X}_{ww_0}$, we have $$\label{e.translate1}
i_x^*[{\mathcal{O}}_{X^w}] = {\widetilde}{i}_{xw_0}[{\mathcal{O}}_{{\widetilde}{X}_{ww_0}}].$$ Let $\Psi$ denote the involution of $G$ from Lemma \[l.involution\]. Since $\Psi(B) = B^-$, there is an induced map (which we also denote by $\Psi$) $G/B \to G/B^-$, $gB \mapsto \psi(g)B^-$. Since $\Psi(nT) = nT$ and $\Psi(B) = B^-$, for any $u \in W$, $\Psi(X_u) = {\widetilde}{X}_u$. The map $\Psi$ is $T$-equivariant if $T$ acts by left multiplication on $G/B^-$, and by $t \odot gB = t^{-1} gB$ on $G/B$. Therefore, $\Psi^*[{\mathcal{O}}_{{\widetilde}{X}_u}] = [{\mathcal{O}}_{X_u}]_{\odot}$, where the subscript $\odot$ indicates that we are using the $\odot$ action of $T$. Therefore, $$\label{e.translate2}
{\widetilde}{}i^*_x[{\mathcal{O}}_{{\widetilde}{X}_u}] = i^*( [{\mathcal{O}}_{X_u}]_{\odot}) = *(i^* [{\mathcal{O}}_{X_u}]),$$ where the second equality follows from Lemma \[l.dual\]. The proposition follows from and , taking $u = w w_0$.
[^1]: The maximal isotropic Grassmannians in the $SO(2n+1)$ and $SO(2n+2)$ cases are isomorphic, so one can obtain Hilbert series and multiplicity formulas in the odd orthogonal case as well. See Section \[ss.orthog\_grassmannians\].
|
---
abstract: 'The two leading twist, quark helicity conserving generalized parton distributions (GPDs) of $^3$He, accessible, for example, in coherent deeply virtual Compton scattering (DVCS), are calculated in impulse approximation (IA). Their sum, at low momentum transfer, is found to be largely dominated by the neutron contribution, so that $^3$He is very promising for the extraction of the neutron information. Anyway, such an extraction could be not trivial. A technique, able to take into account the nuclear effects included in the IA analysis in the extraction procedure, even at moderate values of the momentum transfer, is proposed. Coherent DVCS arises therefore as a crucial experiment to access, for the first time, the neutron GPDs and the orbital angular momentum of the partons in the neutron.'
author:
- 'M. Rinaldi'
- 'S. Scopetta'
date: 'Received: date / Accepted: date'
title: Generalized parton distributions of $^3$He and the neutron orbital structure
---
Generalized Parton Distributions (GPDs) [@uno] parameterize the non-perturbative hadron structure in hard exclusive processes, allowing to access unique information such as, for example, the parton total angular momentum [@rassegne]. By subtracting from the latter the helicity quark contribution, measured in other hard processes, the parton orbital angular momentum (OAM), contributing to the nucleon spin, could be then estimated, a crucial step towards the solution of the so called “Spin Crisis”.
-9.3cm [Fig.1: (a): The magnetic ff of $^3$He, $G_M^3(\Delta^2)$, with $\Delta^{\mu} = \sqrt{-\Delta^2}$. Full line: the present IA calculation, obtained as the x-integral of $\sum_q \tilde{G}^{3,q}_M$ (see text). Dashed line: experimental data [@dataff]. (b): The quantity $x_3 \tilde{G}^3_M(x,\Delta^2,\xi)$, where $x_3 = M_3/M \ x$ and $\xi_3 = M_3/M \ \xi$, shown at $\Delta^2 = -0.1
\ \mbox{GeV}^2$ and $\xi_3=0.1$, together with the neutron (dashed) and the proton (dot-dashed) contribution.]{}
The cleanest process to access GPDs is Deeply Virtual Compton Scattering (DVCS), i.e. $eH \longmapsto e'H' \gamma$ when $Q^2\gg M^2$ ($Q^2=-q \cdot q$ is the momentum transfer between the leptons $e$ and $e'$, $\Delta^2$ the one between hadrons $H$ and $H'$ with momenta $P$ and $P'$, and $M$ is the nucleon mass. Another relevant kinematical variable is the so called skewedness, $\xi = - \Delta^+/(P^+ + P^{'+})$ [^1]). Despite severe difficulties to extract GPDs from experiments, data for proton and nuclear targets are being analyzed, see, i.e., Refs. [@data1; @data2]. The measurement of GPDs for nuclei could be crucial to distinguish between different models of nuclear medium modifications of the nucleon structure, an impossible task in the analysis of DIS experiments only. Moreover, the neutron measurement, which requires nuclear targets, is a very relevant information because it permits, together with the proton one, a flavor decomposition of GPDs. In studies of the neutron polarization, $^3$He plays a special role, due its spin structure (see, e.g., Ref. [@3He]). This is true in particular for GPDs. In fact, among the latters, the ones of interest here are $H_q(x,\Delta^2,\xi)$ and $E_q(x,\Delta^2,\xi)$. $^3$He, among the light nuclei, is the only one for which the combination $\tilde{G}_M^{3,q}(x,\Delta^2,\xi) = H^{3}_q(x,\Delta^2,\xi)+
E^{3}_q(x,\Delta^2,\xi)$ of its GPDs could be dominated by the neutron, being $^2$H and $^4$He not suitable to this aim, as discussed in Ref. [@noiold]. To what extent this fact can be used to extract the neutron information, is shown in Refs. [@noiold; @noiarxive], and summarized here.
The formal treatment of $^3$He GPDs in Impulse Approximation (IA) can be found in Refs. [@scopetta], where, for the GPD $H$ of $^3$He, $H_q^3$, a convolution-like equation in terms of the corresponding nucleon quantity is found. Very recently, the treatment has been extended to $\tilde{G}_M^{3,q}$ (see Refs. [@noiold; @noiarxive] for details), yielding
-1cm
-2mm {width="57mm"}
-4.7cm
6.7cm {width="57mm"}
Fig.2: (a): The quantity $x_3 \tilde{G}^{n,q}_M(x,\Delta^2,\xi)$ for the neutron at $\Delta^2=-0.1 \ \mbox{GeV}^2$ and $\xi_3=0.1$ with $u$, $d$ and $u+d$ contributions (full lines), compared with the approximation $x_3 \tilde{G}^{n,q,extr}_M(x,\Delta^2,\xi)$, Eq. (6), (dashed). (b): The ratio $r_n(x, \Delta^2,\xi)= \tilde{G}_M^{n,extr}(x, \Delta^2,\xi)
/ \tilde{G}_M^{n}(x, \Delta^2,\xi)$, in the forward limit (full), at $\Delta^2=-0.1 \ \mbox{GeV}^2$ and $\xi_3=0$ (dashed) and at $\Delta^2=-0.1 \ \mbox{GeV}^2$ and $\xi_3=0.1$ (dot-dashed).
-.55cm $$\begin{aligned}
\tilde G_M^{3,q}(x,\Delta^2,\xi) =
\sum_N
\int dE
\int d\vec{p}~
\tilde{P}^3_N(\vec{p}, \vec{p'},E)
{\xi' \over \xi}
\tilde G_M^{N,q}(x',\Delta^2,\xi'),\end{aligned}$$ -3mm where $x'$ and $\xi'$ are the variables for the bound nucleon GPDs and $p \, (p'= p + \Delta)$ is its 4-momentum in the initial (final) state. Besides, $\tilde{P}^3_N(\vec{p}, \vec{p'},E)$ is a proper combination of components of the spin dependent, one body off diagonal spectral function:
-3cm 5.5cm
[Fig.3: $r_n(x, \Delta^2,\xi)= \tilde{G}_M^{n,extr}(x, \Delta^2,\xi)
/ \tilde{G}_M^{n}(x, \Delta^2,\xi)$, at $\Delta^2$ = 0.1 GeV$^2$ and $\xi_3 = 0$, using the model of Ref. [@Rad1] for the nucleon GPDs (dashed) and the one of Ref. [@sv] (full).]{}
1.3cm $$\begin{aligned}
\label{spectral1}
P^N_{SS',ss'}(\vec{p},\vec{p}\,',E)
=
\dfrac{1}{(2 \pi)^6}
\dfrac{M\sqrt{ME}}{2}
\int d\Omega _t
%\\
% \times
\sum_{\substack{s_t}} \langle\vec{P'}S' |
\vec{p}\,' s',\vec{t}s_t\rangle_N
\langle \vec{p}s,\vec{t}s_t|\vec{P}S\rangle_N~,
%\nonumber \end{aligned}$$ where $S,S'(s,s')$ are the nuclear (nucleon) spin projections in the initial (final) state, respectively, and $E= E_{min} +E_R^*$, being $E^*_R$ the excitation energy of the two-body recoiling system. The main quantity appearing in the definition Eq. (\[spectral1\]) is the intrinsic overlap integral $$\langle \vec{p} ~s,\vec{t} ~s_t|\vec{P}S\rangle_N
=
\int d \vec{y} \, e^{i \vec{p} \cdot \vec{y}}
\langle \chi^{s}_N,
\Psi_t^{s_t}(\vec{x}) | \Psi_3^S(\vec{x}, \vec{y})
\rangle~
\label{trueover}$$ between the wave function of $^3$He, $\Psi_3^S$, with the final state, described by two wave functions: [*i)*]{} the eigenfunction $\Psi_t^{s_t}$, with eigenvalue $E = E_{min}+E_R^*$, of the state $s_t$ of the intrinsic Hamiltonian pertaining to the system of two [*interacting*]{} nucleons with relative momentum $\vec{t}$, which can be either a bound or a scattering state, and [*ii)*]{} the plane wave representing the nucleon $N$ in IA. For a numerical evaluation of Eq. (1), the overlaps, Eq. (3), appearing in Eq. (2) and corresponding to the analysis of Ref. [@overlap] in terms of Av18 [@pot] wave functions [@AV18], have been used, together with a simple nucleonic model for $\tilde G_M^{N,q}$ [@Rad1] (see Ref. [@noiarxive] for details). Since there are no $^3$He data available, it is possible to verify only a few general GPDs properties, i.e., the forward limit and the first moments. In particular the calculation of $H^{3}_q(x,\Delta^2,\xi)$ fulfills these constraints [@scopetta]. In the $\tilde G_M^{3,q}(x,\Delta^2,\xi)$ case, since there is no observable forward limit for $E^{3}_q(x,\Delta^2,\xi)$, the only possible check is the first moment: $
\sum_q \int dx \, \tilde G_M^{3,q}(x,\Delta^2,\xi) = G_M^3(\Delta^2);
$ where $G_M^3(\Delta^2)$ is the magnetic form factor (ff) of $^3$He. The result obtained is in perfect agreement with the one-body part of the AV18 calculation presented in Ref. [@schiavilla] (see Fig.1a). Moreover, for the values of $\Delta^2$ which are relevant for the coherent process under investigation here, i.e., $-\Delta^2 \ll 0.15$ GeV$^2$, our results compare well also with the data [@dataff]. With the comfort of this succesfull check, results for GPDs of $^3$He are now discussed. In the forward limit, necessary to measure OAM, the neutron contribution strongly dominates the $^3$He quantity, but increasing $\Delta^2$ the proton contribution grows up (see Fig.1b), in particular for the $u$ flavor [@noiold; @noiarxive]. It is therefore necessary to introduce a procedure to safely extract the neutron information from $^3$He data. This can be done by observing that Eq. (1) can be written as $$\begin{aligned}
\tilde G_M^{3,q}(x,\Delta^2,\xi) =
\sum_N \int_{x_3}^{M_A \over M} { dz \over z}
g_N^3(z, \Delta^2, \xi )
\tilde G_M^{N,q} \left( {x \over z},
\Delta^2,
{\xi \over z},
\right)~,
%\nonumber\end{aligned}$$ where $g_N^3(z, \Delta^2, \xi )$ is a “light cone off-forward momentum distribution” which, close to the forward limit, is strongly peaked around $z=1$. Therefore, for $x_3 = (M_A/M) x < 1$:
2.5cm -5.5mm $$\begin{aligned}
\tilde G_M^{3,q}(x,\Delta^2,\xi)
& \simeq & {low \,\Delta^2} \simeq
\sum_N
\tilde G_M^{N,q} \left( x, \Delta^2, {\xi } \right)
\int_0^{M_A \over M} { dz }
g_N^3(z, \Delta^2, \xi )
\nonumber
\\
& = &
G^{3,p,point}_M(\Delta^2)
\tilde G_M^p(x, \Delta^2,\xi)
+
G^{3,n,point}_M(\Delta^2)
\tilde G_M^n(x,\Delta^2,\xi)~.
%\nonumber\end{aligned}$$ -1mm Here, the magnetic point like ff, $G_M^{3,N,point}(\Delta^2)=\int_0^{M_A \over M} dz \, g_N^3(z,\Delta^2,\xi), $ which would give the nucleus ff if the proton and the neutron were point-like particles with their physical magnetic moments, are introduced. These quantities are very well known theoretically and depend weakly on the potential used in the calculation [@noiarxive].
Eq. (5) can now be used to extract the neutron contribution: -5mm $$\begin{aligned}
\label{extr}
\tilde G_M^{n,extr}(x, \Delta^2,\xi) \simeq
{1 \over G^{3,n,point}_M(\Delta^2)}
\left\{ \tilde G_M^3(x, \Delta^2,\xi)
-
G^{3,p,point}_{M}(\Delta^2)
\tilde G_M^p(x, \Delta^2,\xi) \right\}~.\end{aligned}$$
In Fig. 2a, the comparison between the free neutron GPDs, used as input in the calculation, and the ones extracted using our calculation for $\tilde G_M^3$ and the proton model for $\tilde G_M^p$, shows that the procedure works nicely even beyond the forward limit. The only theoretical ingredients are the magnetic point like ffs, which are completely under control. This is even clearer in Fig. 2b, where the ratio $
r_n (x,\Delta^2,\xi) =
{\tilde G_M^{n,extr}(x, \Delta^2,\xi)
\over
\tilde G_M^{n}(x, \Delta^2,\xi)}
$ is shown in a few kinematical regions. The procedure works for $x < 0.7$, where data are expected from JLab. Moreover, the extraction procedure depends weakly on the used nucleonic model (see Fig. 3 and Ref. [@noiarxive]).
In closing, we have shown that coherent DVCS off $^3$He at low momentum transfer $\Delta^2$ is an ideal process to access the neutron GPDs; if data were taken at higher $\Delta^2$, a relativistic treatment [@lussino] and/or the inclusion of many body currents, beyond the present IA scheme, should be implemented.
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Friar, J. L. [*et al.*]{}, Phys. Rev. C [**42**]{} 2310 (1990); Ciofi degli Atti, C. [*et al.*]{} , Phys. Rev. C [**48**]{} 968 (1993).
Rinaldi, M. and Scopetta, S. Phys. Rev. C 85, 062201(R) (2012). Rinaldi, M. and Scopetta, S. arXiv:1208.2831 \[nucl-th\]. Scopetta, S. Phys. Rev. C [**70**]{} 015205 (2004); Phys. Rev. C [**79**]{} 025207 (2009). Kievsky, A., Pace, E., Salmè, G. and Viviani, M. Phys. Rev. C [**56**]{}, 64 (1997). Wiringa, R. B., Stoks V. G. J. and Schiavilla, R. Phys. Rev. C [**51**]{} 38 (1995). Kievsky, A., Viviani, M. and Rosati, S. Nucl. Phys. A [**577**]{} 511 (1994). Musatov, I. V. and Radyushkin, A. V. Phys. Rev. D [**61**]{} 074027 (2000). Marcucci, L. E., Riska, D. O. and Schiavilla, R. Phys. Rev. C [**58**]{}, 3069 (1998). Amroun, A. [*et al.*]{}, Nucl. Phys. A [**579**]{}, 596 (1994). Scopetta, S. and Vento, V. Eur. Phys. J. A [**16**]{}, 527 (2003). Scopetta, S. Del Dotto, A., Pace, E. and Salmè, G. Il Nuovo Cimento C, 35, 101 (2012) arXiv:1110.6757 \[nucl-th\].
[^1]: In this paper, $a^{\pm}=(a^0 \pm a^3)/\sqrt{2}$
|
---
abstract: |
Recently, George Andrews has given a Glaisher style proof of a finite version of Euler’s partition identity. We generalise this result by giving a finite version of Glaisher’s partition identity. Both the generating function and bijective proofs are presented.
2010 Mathematics Subject Classification: 05A15, 05A17, 05A19, 05A30, 11P81.
Keywords: partition, identity, bijection, generating function.
author:
- Darlison Nyirenda
- |
Darlison Nyirenda\
\
[School of Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa.]{}\
[Darlison.Nyirenda@wits.ac.za]{}
date:
-
-
title: 'Generalising a finite version of Euler’s partition identity'
---
Introduction
============
The so-called Euler’s theorem has been widely studied. In the language of integer partitions, the theorem implies that the number of partitions of $n$ into odd parts is equal to the number of partitions of $n$ into distinct parts. This identity, called Euler’s partition identity, has been refined (see [@eric]).\
J. W. L. Glaisher gave a bijective proof of the identity (see [@andrewsg]). Furthermore, its finite version was given together with bijective proofs (see [@corteel], [@bradford] ). We recall this version below.
\[inde\] The number of partitions of $n$ into odd parts each $\leq 2N$ equals the number of partitions of $n$ into parts each $\leq 2N$ in which the parts $\leq N$ are distinct.
For example, if $n = 10$, $N = 3$, then the seven partitions of $n$ into odd parts each $\leq 6$ are\
$$5 + 5,\, 5 + 3 + 1 + 1,\, 5 + 1 + 1 + 1 + 1 + 1,\, 3 + 3 + 3 + 1,$$ $$3 + 3 + 1 + 1 + 1 + 1,\, 3 + 1 + 1 + 1 + 1 + 1 + 1 + 1,\, 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1.$$ And the seven partitions with parts $\leq 6$ such that parts $\leq 3$ are distinct are\
$$6 + 1,\, 6 + 3 + 1,\, 5 + 5, 5 + 4 + 1,\, 5 + 3 + 2,\, 4 + 4 + 2,\, 4 + 3 + 2 + 1.$$ However, the bijections for Theorem \[inde\] given in [@bradford] are complicated, and motivated by their complexity, George Andrews gave a much simpler proof that is Glaisher style (see [@andrewsg]).\
It is clear that Euler’s partition identity is a specific case of Glaisher’s partition identity (see Theorem \[glais\]) when $s = 2$.
\[glais\] The number of partitions of $n$ into parts not divisible by $s$ is equal to the number of partitions of $n$ into parts not repeated more than $s - 1$ times.
We are then naturally led to ask as to whether a finite version of Glaisher’s partition identity that generalises Theorem \[inde\] is possible. If so, can we find a bijective proof thereof reminiscent of George Andrews Glaisher style proof?\
The goal of this paper is to fully address the questions above.\
Our main result is as follows:
\[indemain\] Let $s$ be a positive integer. The number of partitions of $n$ into parts not divisible by $s$ each $\leq sN$ equals the number of partitions of $n$ into parts each $\leq sN$ in which the parts $\leq N$ occur at most $s - 1$ times.
Observe that Theorem \[inde\] is obtained by setting $s = 2$.\
In the subsequent section, we give a generating function proof of the result, and in the section thereafter, a bijective proof that is Glaisher style.
First Proof of Theorem \[indemain\]
===================================
Let $\mathcal{O}_{s,N}(n)$ denote the number of partitions of $n$ in which each part is not divisible by $s$ and $\leq sN$, and $\mathcal{D}_{s,N}(n)$ denote the number of partitions of $n$ in which each part is $\leq sN$ and all parts $\leq N$ occur at most $s - 1$ times. Thus $$\sum_{n = 0}^{\infty}\mathcal{O}_{s,N}(n)q^{n} = \prod_{n = 1}^{N}\frac{1}{(1 - q^{sn - 1})(1 - q^{sn - 2})\ldots (1 - q^{sn - s + 1})}$$ and $$\sum_{n = 0}^{\infty}\mathcal{D}_{s,N}(n)q^{n} = \frac{\prod_{n = 1}^{N} (1 + q^{n} + q^{2n} + \ldots + q^{(s - 1)n})}{\prod_{n = 1}^{(s - 1)N}(1 - q^{n + N})}.$$ Observe that $$\begin{aligned}
\sum_{n = 0}^{\infty}\mathcal{D}_{s,N}(n)q^{n} & = \frac{\prod_{n = 1}^{N} (1 + q^{n} + q^{2n} + \ldots + q^{(s - 1)n})}{\prod_{n = 1}^{(s - 1)N}(1 - q^{n + N})}\\
& = \frac{\prod_{n = 1}^{N}(1 - q^{n}) (1 + q^{n} + q^{2n} + \ldots + q^{(s - 1)n})}{\prod_{n = 1}^{N}(1 - q^{n})\prod_{n = 1}^{(s - 1)N}(1 - q^{n + N})}\\
& = \frac{\prod_{n = 1}^{N}(1 - q^{sn})}{\prod_{n = 1}^{sN}(1 - q^{n})}\\
& = \prod_{n = 1}^{N}\frac{1}{(1 - q^{sn - 1})(1 - q^{sn - 2})\ldots (1 - q^{sn - s + 1})}\\
& = \sum_{n = 0}^{\infty}\mathcal{O}_{s,N}(n)q^{n}.
\end{aligned}$$
Second Proof of Theorem \[indemain\]
====================================
We give a simple Glaisher style extension of the bijection given by George Andrews [@andrewsg].\
Consider a partition $\lambda$ enumerated by $\mathcal{O}_{s,N}(n)$. Each part is of the form $sm - t$ for some $t = 1, 2, \ldots, s - 1$. We can rewrite the partition as $$\sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}} f_{i,t}(sm_{i} - t)$$ where $f_{i,t}$ is the multiplicity of the part $sm_{i} - t$, $r_{t}$ is the number of parts that are $\equiv -t \pmod{s}$.\
Note that there exists a unique $\alpha_{i,t}$ such that $N < (sm_{i} - t)s^{\alpha_{i,t}} \leq sN$. Rather than taking a complete $s$-ary expansion of $f_{i,t}$, we instead do the following: we find the aforementioned $\alpha_{i,t}$ and use division algorithm to compute $\beta_{i,t}$ and $e_{i,t}$ from the equation $$f_{i,t} = \beta_{i,t}s^{\alpha_{i,t}} + e_{i,t}\,\,\,\text{where}\,\,\,0\leq e_{i,t} \leq s^{\alpha_{i,t}} - 1.$$ Then write the $s$-ary expansion of $e_{i,t}$, i.e., $$e_{i,t} = \sum_{j = 0}^{b_i}a_{j,t}s^{j}\,\,\,\,\text{where}\,\,\,\, 0\leq a_{j,t} \leq s - 1.$$ So $f_{i,t} = \sum_{j = 0}^{b_i}a_{j,t}s^{j} + \beta_{i,t}s^{\alpha_{i,t}}$ and thus $$\begin{aligned}
\sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}} f_{i,t}(sm_{i} - t) & = \sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}}( \sum_{j = 0}^{b_i}a_{j,t}s^{j} + \beta_{i,t}s^{\alpha_{i,t}})(sm_{i} - t)\\
& = \sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}}\sum_{j = 0}^{b_i}a_{j,t}(sm_{i} - t)s^{j} + \sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}}\beta_{i,t}(sm_{i} - t)s^{\alpha_{i,t}}\\
& = \sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}}(a_{0,t}(sm_{i} - t) + a_{1,t}(sm_{i} - t)s + \ldots \\
& \hspace{5mm} \,\ldots + a_{b_i,t}(sm_{i} - t)s^{b_i}) + \, \sum_{t = 1}^{s - 1}\sum_{i = 1}^{r_{t}}\beta_{i,t}(sm_{i} - t)s^{\alpha_{i,t}}\end{aligned}$$
which is the image of $\lambda$ with parts $(sm_{i} - t)s^{j} \leq N$ that have multiplicity $a_{j,t} \leq s - 1$, and parts $(sm_{i} - t)s^{\alpha_{i,t}} \in [N + 1, sN]$ that have multiplicity $ 0 \leq \beta_{i,t} < \infty$. This image is a partition enumerated by $\mathcal{D}_{s,N}(n)$.\
The inverse of the bijection is not difficult to contruct.\
We demontrate the bijection using $n = 177, s = 3, N = 4$ and\
$\lambda = (11^{6}, 7^{5}, 5^{7},4^{5}, 2^{2}, 1^{17})$.\
Note that $11.3^{0} = 11.1 \in [5,12]$, $7.3^{0} = 7.1 \in [5,12]$, $5.3^{0} = 5.1 \in [5,12]$, $4.3^{1} = 4.3 \in [5,12]$, $2.3^{1} = 2.3 \in [5,12]$, and $1.3^{2} = 1.9 \in [5,12]$.\
$11^{6}$ is mapped to $11(6.1 + 0) = 6.11$, which is interpreted as $11^{6}$. Similarly, $7^{5}$ to $7^{5}$, and $5^{7}$ to $5^{7}$.\
$4^{5}$ goes to $4(1.3 + 2)$, using division algorithm $5 = 1.3 + 2$, and now taking the 3-ary expansion of 2; $4(1.3 + 2) = 4(1.3 + 2.3^{0}) = 1.12 + 2.4$, which is $(12, 4^{2})$. Continuing in this manner, we have the mapping $2^{2} \mapsto 2^{2}$ and $1^{17} \mapsto (9,3^{2},1^{2})$. Thus $$\lambda \mapsto (12,11^{6},9,7^{5},5^{7},4^{2},3^{2},1^{2}).$$
[1]{} G. E. Andrews, *Euler’s partition identity-finite version*, preprint. G. E. Andrews, K. Eriksson, Integer partitions, Cambridge University Press, 2004. L. Bradford, M. Harris, B. Jones, A. Komarinski, C. Matson, E. O’shea, *The refined lecture hall theorem via abacus diagrams*, Ramanujan J., **34** (2014), 163 – 176. S. Corteel, C. D. Savage, *Lecture hall theorems, $q$-series and truncated objects*, J. Combin. Theory Ser. A, 108(**2**)(2004), 217 – 245. J. W. L. Glaisher, *A theorem in partitions*, Messenger of Math., **12** (1883), 158 – 170.
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---
abstract: 'We compute the posterior distributions of the initial population and parameter of binary branching processes, in the limit of a large number of generations. We compare this Bayesian procedure with a more naïve one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances.'
title: Asymptotics of posteriors for binary branching processes
---
Introduction {#s.intro}
============
This paper is devoted to some estimation procedures of binary branching processes in a Bayesian setting. To be more specific, let $(X_n)_{n\ge0}$ denote a Galton-Watson process which starts from the initial population $X_0\ge1$ and whose offspring is ruled by the distribution $$(1-U)\,\delta_1+U\delta_2\quad\mbox{with}\ 0<U<1,$$ where $\delta_x$ denotes the Dirac mass at $x$. This means that, at every generation, each individual dies and is replaced by $1$ or $2$ individuals, with probability $1-U$ and $U$ respectively, independently of the fate of the other individuals, and that $X_n$ counts generation $n$.
In a Bayesian framework, the initial population $X_0$ and the offspring parameter $U$ are both random and unknown. To keep things simple, we also assume that $X_0$ and $U$ are independent, and we wish to estimate them from the observation of a finite path $x_{1:n}=(x_k)_{1\le k\le n}$ of the process $X_{1:n}=(X_k)_{1\le k\le n}$ up to a given time $n\ge1$.
Well known motivations for such a study are various biological settings where one observes $X_{1:n}$ but $X_0$ and $U$ are unknown. One example is the modeling of polymerase chain reaction. Probabilistic models of polymerase chain reactions were proposed and studied by Sun (1995), Weiss and von Haeseler (1995) and (1997), Peccoud and Jacob (1996), Piau (2002), (2004), (2005), and Jagers and Klebaner (2003). Recently, Lalam and Jacob (2007) introduced and studied the Bayesian setting above, see also Lalam (2007). For other Bayesian approaches of branching processes, see Scott (1987), Prakasa Rao (1992), Mendoza and Gutiérrez-Peña (2000), and, for the interesting model of bisexual branching process, Molina, González and Mota (1998) for example. Finally, the idea of studying a branching process backwards, but to estimate its age rather than its initial population, is in Klebaner and Sagitov (2002).
In models of polymerase chain reactions and in similar contexts, the initial population $X_0$ is the size of a small sample, extracted at random from a much larger population. This suggests that the initial population $X_0$ should be Poisson distributed, say with parameter $\Lambda$. We assume that $\Lambda$ is random as well. Jeffreys’ principle, see Kass and Wasserman (1996), then indicates that the prior distributions of $\Lambda$ and $U$ should be proportional to measures which we compute below. To sum up the result of these computations, the prior of $\Lambda$ is easy to write down but improper and the prior of $U$ is awkward but proper. However, the posterior of $(X_0,U)$ conditionally on $X_{1:n}$ is a proper distribution, which can be computed explicitly. In particular, this posterior distribution depends only on $X_1$, $X_n$ and $S_n=X_1+\cdots+X_n$. Unfortunately, it is also rather unwieldy.
In such situations, one may rely on numerical algorithms, based on MCMC for example, to simulate the posterior distributions with any prescribed degree of accuracy. Rather, we look for simple asymptotics in realistic regimes. Namely, we assume that $n$ is large and we are interested in the asymptotic posterior distribution of $(X_0,U)$ assuming that $X_n$ is large and that the ratio $S_n/X_n$ converges to a finite limit. This assumption is almost surely fulfilled by the paths of binary branching processes since these are supercritical. In this setting, we show that the posterior distributions indeed converge and we compute explicitly their limit.
Results
=======
To describe our results, we introduce some notations. Let $x_{0:\infty}=(x_n)_{n\ge0}$ denote a sequence of positive integers. We say that such a sequence is admissible if, for every nonnegative $n$, $x_n\le x_{n+1}\le2x_n.$ We say that an admissible sequence is regular if furthermore, $x_n/s_n$ converges to a positive limit when $n$ goes to infinity, where $s_n=x_1+\cdots+x_n$. The binary index $B(x_{0:\infty})$ of a regular admissible sequence $x_{0:\infty}$ is the real number in $]0,1]$ defined by $$B(x_{0:\infty})=\lim_{n\to\infty}\frac{x_{n+1}}{s_n}.$$ The renormalized index $R(x_{0:\infty})$ of a regular admissible sequence $x_{0:\infty}$ is the real number in $[0,+\infty[$ defined by $$R(x_{0:\infty})=\lim_{n\to\infty}\frac{(s_n-x_{n+1})^2}{4x_{n+1}s_n}.$$ Almost every (sequence which can be realized as a) path of a binary branching process is admissible and regular. The renormalized index is a function of the binary index, namely $R(x_{0:\infty})=\varrho(B(x_{0:\infty}))$ where, for every $u$ in $]0,1]$, $$\varrho(u)=\frac{(1-u)^2}{4u}.$$ The binary index and the normalized index are asymptotic quantities, in the sense that, for every nonnegative integer $n$, the indexes of a regular admissible sequence $x_{0:\infty}$ do not depend on the first values $x_{0:n}$.
From now on, letters $k$ and $n$ are used to enumerate generations of the process (that is, the time) and symbols $x$, $x_k$, $x_n$ and $y$ are used to measure population sizes.
\[d.munu\] For every positive real number $r$ and every positive integer $x$, the finite discrete measure $\nu(r,x)$ and the discrete probability measure $\mu(r,x)$, both on the positive integers, are defined by $$\nu(r,x)=\sum_{y=h(x)}^x\binom{2y}{y}\,\binom{y}{x-y}\,r^y\,\delta_y,
\qquad
\mu(r,x)=\frac{\nu(r,x)}{|\nu(r,x)|}.$$ For every positive integer $x$, the integer $h(x)$ in the formula above is the upper half of $x$, that is, the smallest integer such that $2h(x)\ge x$. In other words, $h(2x)=h(2x-1)=x$ for every positive integer $x$.
Our main result is as follows.
\[ta\] (1) The path $X_{0:\infty}$ of a binary branching process with parameter $U$ is almost surely regular admissible and its binary index is almost surely $B(X_{0:\infty})=U$.
\(2) Assume that the prior distribution of $(X_0,U)$ satisfies Jeffreys’ principle. Then, for every regular admissible sequence $x_{1:\infty}$ with binary index $u=B(x_{1:\infty})$ in $]0,1[$, the posterior distribution of $(X_0,U)$ conditionally on $X_{1:n}=x_{1:n}$ converges when $n$ goes to infinity to the distribution $
\mu(\varrho(u),x_1)\otimes\delta_{u}.
$
Theorem \[ta\] shows that the limit posterior distribution of $X_0$ when $n$ goes to infinity is almost surely $\mu(r,x)$ with $r=\varrho(U)$ and $x=X_1$. Unless $r=0$, $r=1$ or $x=1$, $\mu(r,x)$ is not degenerate, hence the value of $X_0$ can be determined only with some uncertainty, even from an infinite trajectory $X_{1:\infty}$. On the contrary, $U$ is a function of the infinite trajectory $X_{1:\infty}$.
The limit distribution $\mu(\varrho(u),x_1)\otimes\delta_{u}$ in theorem \[ta\] converges to the Dirac distribution at $(x_1,0)$ when $u$ converges to $0$ and to the Dirac distribution at $(h(x_1),1)$ when $u$ converges to $1$. Our next result describes the intuitively obvious variations of $\mu(r,x)$ with respect to $r$ and $x$. First, since $r=\varrho(u)$ is a decreasing function of $u$ and the offspring distribution of the branching process is stochastically increasing with $u$, one should expect $\mu(r,x)$ to increase stochastically when $r$ increases. Likewise, since $x$ represents the population at time $1$, one should expect $\mu(r,x)$, which represents the population at time $0$, to increase stochastically when $x$ increases.
We recall that a measure $\mu_1$ is stochastically larger than a measure $\mu_2$ if and only if $\mu_1([z,+\infty))\ge\mu_2([z,+\infty))$ for every real number $z$.
\[lo\] For every positive integer $x$, the family $(\mu(r,x))_{r\ge0}$ is stochastically increasing. For every positive real number $r$, the family $(\mu(r,x))_{x\ge1}$ is stochastically increasing.
We now characterize the limit of $\mu(r,x)$ for every fixed value of $r$, when $x$ converges to infinity.
\[tb\] Fix $u$ in $]0,1[$. For every positive integer $x$, let $\xi_x$ denote a random variable with distribution $\mu(\varrho(u),x)$. When $x$ converges to infinity, the expectation and the mode of $\xi_x/x$ both converge to $$m_u=1/(1+u),$$ and the random variables $\left(\xi_x-m_ux\right)/\sqrt{x}$ converge in distribution to a centered Gaussian distribution with variance $$\sigma^2_u=u(1-u)/(1+u)^3.$$
For the sake of comparison, we turn to another natural way to estimate initial populations of branching processes with known offspring distributions, based on hitting times. To describe this in the setting of binary branching processes, we first introduce some notations.
Fix a real number $u$ in $]0,1[$, and let $(\varepsilon_x)_{x\ge1}$ denote a sequence of independent Bernoulli random variables with distribution $(1-u)\,\delta_1+u\,\delta_2$. For every positive integer $x$, let $\sigma_x:=\varepsilon_1+\cdots+\varepsilon_x$. Define the distribution of the hitting time $\eta_x$ by the relation $$\mathbb{P}(\eta_x=y)=\mathbb{P}(\sigma_y=x\,|\,H_x),\qquad
\mbox{where}\ H_x=\{\exists z\ge1\,;\,\sigma_z=x\}.$$
When the value of $u$ is known, an estimation procedure of $X_0$ based on $X_1=x$ is to propose the value $y$ for $X_0$ with probability $\mathbb{P}(\eta_x=y)$, thus an estimator of $X_0$ when $X_1=x$ is the distribution of $\eta_x$.
Recall that $m_u=1/(1+u)$ and $\sigma^2_u=u(1-u)/(1+u)^3$.
\[tc\] Fix a real number $u$ in $]0,1[$. For every positive integer $x$, $$\left|\mathbb{E}(\eta_x)-m_ux\right|\le2u/(1+u)^2\le1/2.$$ Furthermore, when $x$ converges to infinity, $(\eta_x-m_ux)/\sqrt{x}$ converges in distribution to a centered Gaussian variable with variance $\sigma^2_u$.
The rest of the paper is organized as follows. We prove theorem \[ta\] and proposition \[lo\] in section \[s.ta\] and theorem \[tb\] in section \[s.tb\]. Finally, the proof of theorem \[tc\], sharper bounds on $\mathbb{E}(\eta_x)$ and a brief comparison with another, non Bayesian, estimation procedure are in section \[s.tc\].
Posterior distributions {#s.ta}
=======================
Preliminaries
-------------
Jeffreys’ principle, see Kass and Wasserman (1996), indicates that the prior measure for a parameter $\theta$ governing the distribution $\nu_\theta$ of a random variable $Z$ should have a density proportional to $J(\theta)^{1/2}$, where $$J(\theta)=-\mathbb{E}_{\theta}\left(\frac{\partial^2}{\partial\theta^2}\log\nu_\theta(Z)\right).$$ We apply this to the parameter $(\Lambda,U)$. Parts of lemma \[l.prior\] are in Lalam and Jacob (2007).
\[l.prior\] For every positive integer $n$, the prior measure for $(\Lambda,U)$ according to Jeffreys’ principle and based on $X_{0:n}$ is the product of the prior measures for $\Lambda$ and $U$. The prior measures for $\Lambda$ and for $U$ are respectively proportional to the measures $\mathrm{d}\lambda/\sqrt\lambda$ on $\lambda>0$ and $\pi_n(u)\,\mathrm{d} u$ on $0<u<1$, where $$\pi_n(u)=\sqrt{\frac{(1+u)^n-1}{u^2(1-u)}}.$$ In particular, the prior of $U$ is proper.
Assume that $X_0$ is Poisson distributed with parameter $\Lambda$ and that $X_{0:n}$ is a binary branching process with parameter $U$. Then the distribution $\nu_{\Lambda,U}$ of $X_{0:n}$ is such that $$\nu_{\Lambda,U}(x_{0:n})=\mathrm{e}^{-\Lambda}\frac{\Lambda^{x_0}}{x_0!}\prod_{k=1}^n\binom{x_{k-1}}{x_k-x_{k-1}}
U^{x_k-x_{k-1}}(1-U)^{2x_{k-1}-x_k}.$$ Up to a factor $C(x_{0:n})$ which does not depend on $(\Lambda,U)$, $\log\nu_{\Lambda,U}(x_{0:n})$ is $$-\Lambda+x_0\log\Lambda+
(x_n-x_0)\log U+(s_n-2x_n+2x_0)\log(1-U)+C(x_{0:n}).$$ This is the sum of a function of $\Lambda$ and a function of $U$, hence the prior measures are product measures. As regards the prior for $\Lambda$, $$\frac{\partial^2}{\partial\Lambda^2}\log\nu_\Lambda(x_0)
=
-\frac{x_0}{\Lambda^2},
\qquad\mbox{hence}\
J(\Lambda)=\frac{\mathbb{E}_\Lambda(X_0)}{\Lambda^2}=\frac1{\Lambda}.$$ As regards the prior for $U$, $$\frac{\partial^2}{\partial U^2}\log\nu_U(x_{0:n})
=
-(x_n-x_0)/U^2-(s_n-2x_n+2x_0)/(1-U)^2,$$ hence $$J_n(U)=\frac{\mathbb{E}_U(X_n-X_0)}{U^2}+\frac{\mathbb{E}_U(S_n-2X_n+2X_0)}{(1-U)^2},$$ where $S_n=X_1+\cdots+X_n$. Since $\mathbb{E}_U(X_k)=(1+U)^k\mathbb{E}(X_0)$ for every nonnegative integer $k$, one finds that $J_n(U)=\mathbb{E}(X_0)\pi_n(U)^2$ with the notations of the lemma.
Finally, up to multiplicative constants, $\pi_n(u)$ behaves like $1/\sqrt{u}$ when $u$ converges to $0$ and like $1/\sqrt{1-u}$ when $u$ converges to $1$. Hence, $\pi_n$ is integrable and there exists a (proper) prior distribution for $U$. This concludes the proof of lemma \[l.prior\].
From now on, we fix a positive integer $n$, we assume that the observations are $X_{1:n}=x_{1:n}$ with $x_{1:n}=(x_k)_{1\le k\le n}$ and we recall that $s_n=x_1+\cdots+x_n$. The posterior distribution in lemma \[l5\] is similar, but not equal, to a posterior distribution computed in Lalam and Jacob (2007).
\[l5\] The posterior distribution of $(X_0,U)$ conditionally on $X_{1:n}=x_{1:n}$ depends only on $x_1$, $x_n$ and $s_n$, and is proportional to the measure $$\sum_{x=h(x_1)}^{x_1}2^{-2x}\,\binom{2x}{x}\,\binom{x}{x_1-x}\,
u^{x_n-x}\,(1-u)^{s_n-2x_n+2x}\,\pi_n(u)\,\delta_x\otimes\mathrm{d} u.$$
Fix $u$, $x_{1:n}$ and $x$ such that $h(x_1)\le x\le x_1$. Then, the conditional probability $\mathbb{P}(U\in\mathrm{d}
u,X_0=x\,\vert\,X_{1:n}=x_{1:n})$ is proportional to $$\nu_U(\mathrm{d} u)\int\nu_\Lambda(\mathrm{d}\lambda)
\mathbb{P}_\lambda(X_0=x)\mathbb{P}_u(X_{1:n}=x_{1:n}\,\vert\,X_0=x),$$ where $\nu_U(\mathrm{d} u)=\pi_n(u)\mathrm{d} u$ and $\nu_\Lambda(\mathrm{d}\lambda)=\mathrm{d}\lambda/\sqrt\lambda$. Hence, $$\int\nu_\Lambda(\mathrm{d}\lambda)\mathbb{P}_\lambda(X_0=x)
=
\frac{\Gamma(x+1/2)}{\Gamma(x+1)}
=
\sqrt\pi\,2^{-2x}\,\binom{2x}{x}.$$ Likewise, using the computations in the proof of lemma \[l.prior\], one gets $$\mathbb{P}_u(X_{1:n}=x_{1:n}\,\vert\,X_0=x)=C(x_{1:n})\,\binom{x}{x_1-x}\,
u^{x_n-x}\,(1-u)^{s_n-2x_n+2x},$$ where $C(x_{1:n})$ does not depend on $(x,u)$. This concludes the proof of lemma \[l5\].
Proof of theorem \[ta\]
-----------------------
Part (1) follows from the fact that, when $n$ converges to infinity, $X_n/(1+U)^n$ converges almost surely to a random positive and finite limit.
A sketch of the proof of part (2) is as follows. Consider the distribution in lemma \[l5\] and assume that $x_n$ converges to infinity and that $x_n/(s_n-x_n)$ converges to $v$. Then $s_n-x_n$ is equivalent to $x_n/v$, hence $$u^{x_n}(1-u)^{s_n-2x_n}=\left(u^v(1-u)^{1-v}\right)^{s_n-x_n+o(x_n)}.$$ The inner parenthesis is maximal when $u=v$, and the exponent converges to infinity, hence this contribution becomes concentrated around the value $u=v$. The remaining factor involving $u$ in the distribution described in lemma \[l5\] is $\varrho(u)^x$, and the convergence to $\mu(\varrho(v),x_1)$ follows.
For a detailed proof of part (2), we consider a sequence $x_{1:\infty}$ such that $x_n$ converges to infinity and $x_n/(s_n-x_n)$ converges to $v$. For every positive integer $n$, we introduce random variables $(T_n,U_n)$ distributed as $(X_0,U)$ conditionally on $X_{1:n}=x_{1:n}$. We first show the convergence in probability of $U_n$, then the convergence in distribution of $(T_n,U_n)$.
\[l.pru\] With the notations above, $U_n$ converges to $v$ in probability.
Lemma \[l5\] yields $$\mathbb{P}(T_n=x,U_n\in\mathrm{d} u)=c_np_x\varrho(u)^xb_n(u)q_n(u)\,\mathrm{d} u,$$ where $c_n$ denotes a normalizing constant which is independent on $x$ and $u$, $p_x$ depends only on $x$ and $x_1$, $b_n(u)$ depends only on $u$, $x_n$ and $s_n$, and $q_n(u)$ depends only on $u$ and $n$. More precisely, for every integer $x$ such that $x_1\le2x\le2x_1$ and every real number $u$ in $]0,1[$, $$\begin{aligned}
p_x & = & \binom{2x}{x}\binom{x}{x_1-x},
\\
b_n(u) & = & u^{x_n-1/2}(1-u)^{s_n-2x_n-1/2},
\\
q_n(u) & = &
%\sqrt{u(1-u)}\,\pi_n(u)=
\sqrt{\frac{(1+u)^n-1}{u}}.\end{aligned}$$ We aim to show that, for every integer $x$ such that $p_x$ is positive and every positive real number $z$, when $n$ converges to infinity, $$\mathbb{P}(T_n=x,|U_n-v|\ge z)\ll \mathbb{P}(T_n=x).$$ Since the function $q_n$ is nondecreasing, $$\mathbb{P}(T_n=x,|U_n-v|\ge z)\le c_np_xq_n(1)\int_{|u-v|\ge
z}\varrho(u)^xb_n(u)\,\mathbf{1}_{[0,1]}(u)\,\mathrm{d} u,$$ and $$\mathbb{P}(T_n=x)\ge c_np_xq_n(0)\int_0^1\varrho(u)^xb_n(u)\,\mathrm{d} u.$$ The ratio of the two integrals written above is $\mathbb{P}(|B_n-v|\ge z)$, where $B_n$ is a beta random variable of parameters $(\alpha_n,\beta_n)$, with $$\alpha_n=x_n-x+1/2,\quad
\beta_n=s_n-2x_n+2x+1/2.$$ Since $\alpha_n$ and $\beta_n$ both converge to infinity and $\alpha_n/(\alpha_n+\beta_n)$ converges to $v$, it is an easy matter to show that $B_n$ converges in probability to $v$. However, we need a stronger statement, namely the fact that $\mathbb{P}(|B_n-v|\ge z)\ll q_n(0)/q_n(1)$. Note that $q_n(0)=\sqrt{n}$ and $q_n(1)\sim2^{n/2}$, hence $q_n(0)/q_n(1)\ll1$.
One can write an elementary proof of this, based on the representation of beta random variables with integer parameters as ratios of sums of i.i.d. exponential random variables and on large deviations properties of these sums. Instead, we rely on approximations of beta distributions by normal distributions provided by Alfers and Dinges (1984). A rephrasing of corollary 1 on page 405 of this paper is as follows. Let $(Y_k)_k$ denote a sequence of beta random variables of parameters $(ka_k,k(1-a_k))$. Assume that $k$ converges to infinity and that $a_k$ converges to a limit $0<a<1$. Then, for every fixed $y$ such that $a<y<1$, the ratio $$\frac{\mathbb{P}(Y_k\ge y)}{\mathbb{P}\left(Z\ge\sqrt{2k\ell(a_k,y)}\right)}$$ converges to a finite and positive limit, which depends on $a$ and $y$ only, where $Z$ denotes a standard Gaussian random variable, and $\ell$ denotes the function defined by $$\ell(\alpha,y)=\alpha\log\left(\frac{\alpha}{y}\right)
+
(1-\alpha)\log\left(\frac{1-\alpha}{1-y}\right).$$ Since $a_k$ converges to $a$ and $\ell(\alpha,y)$ is a continuous function of $\alpha$, standard estimates of Gaussian tails and the result by Alfers and Dinges show that there exists a positive constant $C<1$, independent on $k$, such that for every $k$ large enough, $$\mathbb{P}(Y_k\ge y)\le C^k.$$ Applying this to our setting, first to the random variables $B_n$ and to $y=v+z$, then to the random variables $1-B_n$ and to $y=1-v+z$, one gets the existence of a constant $C<1$ such that, for every $n$ large enough, $$\mathbb{P}(|B_n-v|\ge z)\le 2C^{\alpha_n+\beta_n}.$$ Since $\alpha_n+\beta_n=s_{n-1}+x+1\ge s_{n-1}\gg n$, $2C^{\alpha_n+\beta_n}\ll q_n(0)/q_n(1)$, and the proof of lemma \[l.pru\] is complete.
We now apply lemma \[l.pru\] to the proof of part (2). Introduce the finite sums $$p(u)=\sum_xp_x\varrho(u)^x.$$ For every $u$ in $]0,1[$, the distribution of $T_n$ conditionally on $U_n=u$ is independent on $n$ and such that $$\mathbb{P}(T_n=x\,|\,U_n=u)=p(u)^{-1}p_x\varrho(u)^x.$$ Hence, for every measurable subset $B$ of $]0,1[$, $$\mathbb{P}(T_n=x,U_n\in B)=
\mathbb{E}\left(p(U_n)^{-1}p_x\mathbf{1}_B(U_n)\varrho(U_n)^x\right).$$ The function $u\mapsto p(u)^{-1}p_x\mathbf{1}_B(u)\varrho(u)^x$ is bounded by $1$ on $]0,1[$ and, as soon as $v$ is not in the boundary of $B$, continuous at $u=v$. Since $U_n$ converges in distribution to $v$, this implies that $\mathbb{P}(T_n=x,U_n\in B)$ converges to $p(v)^{-1}p_x\mathbf{1}_B(v)\varrho(v)^x$, for instance for every interval $B=[0,u]$ with $u\ne v$. This is equivalent to the desired convergence in distribution.
Remarks
-------
For every positive integer $n$ and every admissible sample, $s_n\ge2x_n(1-1/2^n)$ since $x_k\ge x_{k+1}/2$ for every nonnegative integer $k$, hence $s_n-x_n\ge
x_n+o(x_n)$ and $u\le1$ in the asymptotics that we consider. Furthermore, the function $\varrho$ decreases from $\varrho(0^+)=+\infty$ to $\varrho(1^-)=0$.
The measures $\mu(r,x)$ for the first values of $x$ are as follows: $\mu(r,1)=\delta_1$, $$\mu(r,2)=\frac{\delta_1+3r\delta_2}{1+3r},\quad
\mu(r,3)=\frac{3\delta_2+5r\delta_3}{3+5r},\quad
\mu(r,4)=\frac{3\delta_2+30r\delta_3+35r^2\delta_4}{3+30r+35r^2},$$ and $$\mu(r,5)=\frac{15\delta_3+70r\delta_4+63r^2\delta_5}{15+70r+63r^2}.$$
Proof of proposition \[lo\]
---------------------------
The monotonicity with respect to $r$ is valid in a wider setting, described in proposition \[pw\] below, but the monotonicity with respect to $x$ is more specific.
\[pw\] Let $\mu$ denote a nonzero bounded measure with exponential moments. For every real number $a$, introduce the measures $\nu_a$ and $\mu_a$ defined by the relations $\nu_a(\mathrm{d}
x)=\mathrm{e}^{ax}\mu(\mathrm{d} x)$ and $\mu_a=\nu_a/|\nu_a|$. Then the family $(\mu_a)_{a}$ is stochastically nondecreasing.
Fix $x$. The derivative of $\mu_a([x,+\infty))$ with respect to $a$ has the sign of $D(x)$, with $$D(x)=\int_{y\ge x}y\mathrm{e}^{ay}\mu(\mathrm{d} y)\,\int\mathrm{e}^{az}\mu(\mathrm{d} z)
-\int_{y\ge x}\mathrm{e}^{ay}\mu(\mathrm{d} y)\,\int z\mathrm{e}^{az}\mu(\mathrm{d} z).$$ The variations of $D(x)$ with respect to $x$ are given by $$\mathrm{d} D(x)=\mathrm{e}^{ax}\mu(\mathrm{d} x)\,\int(y-x)\mathrm{e}^{ay}\mu(\mathrm{d} y).$$ The integral in the right hand side is a nonincreasing function of $x$. Since $D(0)=D(\infty)=0$, the function $x\mapsto D(x)$ is nondecreasing for $x\le x_a$ and nonincreasing for $x\ge x_a$, where $x_a$ solves the equation $$\int y\mathrm{e}^{ay}\mu(\mathrm{d} y)
=
x_a\int\mathrm{e}^{ay}\mu(\mathrm{d} y).$$ This proves that $D(x)\ge0$ for every $x$, hence $\mu_a([x,+\infty))\le\mu_b([x,+\infty))$ for every $a\le b$. This concludes the proof of proposition \[pw\].
We turn to the monotonicity of $\mu(r,x)$ with respect to $x$. We fix a value of $r$ and write every $\nu(r,x)$ as $$\nu(r,x)=\sum_ya_y^x\delta_y.$$ We want to prove that for every $x$, $G(z)\ge0$ for every $z$, with $$G(z)=\sum_ya_y^x \sum_{y\ge z}a_y^{x+1}- \sum_{y\ge z}a_y^x \sum_{y}a_y^{x+1}.$$ One sees that $G(0)=G(\infty)=0$, and simple computations show that $$F(z)=\frac1{a_z^{x}}(G(z+1)-G(z))
=\sum_{y}a_y^{x+1}-\frac{a_z^{x+1}}{a_z^x}\sum_ya_y^x.$$ At this point, we use the specific form of the coefficients $a_z^x$, which yields $$\frac{a_z^{x+1}}{a_z^x}=2\frac{(2x+1)(2z-x)}{(x+1)(x+1-z)}.$$ This shows that $(F(z))_z$ is a nonincreasing sequence, hence $G(z+1)-G(z)\ge0$ if $z<z_*$ and $G(z+1)-G(z)\le0$ if $z\ge z_*$, for a given $z_*$. Hence the sequence $(G(z))_z$ is nondecreasing on $z\le z_*$ and nonincreasing on $z\ge z_*$. Since $G(0)=G(\infty)=0$, this implies that $G(z)\ge0$ for every positive $z$. This concludes the proof of proposition \[lo\].
Limit posterior distributions of initial populations {#s.tb}
====================================================
Expectations
------------
Let $u$ in $]0,1[$ and $r=\varrho(u)$. We are interested in the limit as $x\to\infty$ of the sequence $$\frac1{x}\,\mathbb{E}(\xi_x)=\frac{A(r,x)}{x\,B(r,x)},$$ with the notations $$A(r,x)=\sum_yy\,\nu(r,x)(y)=\sum_yy\,\binom{2y}{y}\,\binom{y}{x-y}\,r^y,$$ and $$B(r,x)=|\nu(r,x)|=\sum_y\binom{2y}{y}\,\binom{y}{x-y}\,r^y.$$
For every positive $\lambda$ and $r$, introduce $$C_\lambda(r,z)=(1-4rz(1+z))^{-\lambda}=\sum_{x\ge0}c_\lambda(r,x)\,z^x.$$
Starting from the expansion $$(1-4z)^{-1/2}=\sum_{x\ge0}\binom{2x}{x}\,z^x,$$ one can write $B(r,x)$ as the coefficient of $z^x$ in the expansion of $C_{1/2}(r,x)$ along the powers of $z$, namely, $B(r,x)=c_{1/2}(r,x).$ Likewise, $A(r,x)$ is $r$ times the derivative of $B(r,x)$ with respect to $r$, hence $A(r,x)$ is the coefficient of $z^x$ in the expansion of $2rz(1+z)C_{3/2}(r,x)$ along the powers of $z$. This yields $$A(r,x)=2r\,\left(c_{3/2}(r,x-1)+c_{3/2}(r,x-2)\right),$$ and $$x\,B(r,x)=2r\,\left(c_{3/2}(r,x-1)+2c_{3/2}(r,x-2)\right).$$
\[d.gammam\] For every positive $r$, introduce $$\gamma(r)=\frac12\left(\sqrt{\frac{1+r}{r}}-1\right),
\qquad
m(r)=\frac{1+\gamma(r)}{1+2\gamma(r)}
=
\frac12\left(1+\sqrt{\frac{r}{1+r}}\right).$$
Note that, for every $u$ in $]0,1[$, $$\gamma(\varrho(u))=\frac{u}{1-u},\qquad
m(\varrho(u))=\frac1{1+u}=m_u.$$
\[l.cl\] For every positive $\lambda$ and $r$, when $x$ converges to infinity, $$c_\lambda(r,x)\sim c_\lambda(r)\,x^{\lambda-1}\,\gamma(r)^{-x},
\qquad
c_\lambda(r)=m(r)^\lambda/\Gamma(\lambda).$$
This is a consequence of known expansions of powers of $1/(1-z)$. First, recall that $$(1-z)^{-\lambda}
=
\sum_{x\ge0}d_\lambda(x)\,z^x,\quad
d_\lambda(x)
=
\frac{\Gamma(x+\lambda)}{\Gamma(x+1)\Gamma(\lambda)}
\sim
\frac{x^{\lambda-1}}{\Gamma(\lambda)}.$$ We use this and the decomposition $$1-4rz(1+z)=\left(1-\frac{z}{\gamma(r)}\right)\,\left(1+\frac{z}{\gamma(r)+1}\right),$$ to get the expansion $$C_\lambda(r,z)=
\sum_xd_\lambda(x)\,\left(\frac{z}{\gamma(r)}\right)^x\,
\sum_xd_\lambda(x)\,\left(\frac{-z}{1+\gamma(r)}\right)^x,$$ which implies $$c_\lambda(r,x)=d_\lambda(x)\,\gamma(r)^{-x}\,
\sum_{y=0}^x\left(\frac{-\gamma(r)}{\gamma(r)+1}\right)^y\,d_\lambda(y)\,
\frac{d_\lambda(x-y)}{d_\lambda(x)}.$$ When $x$ converges to infinity, the ratios $d_\lambda(x-y)/d_\lambda(x)$ converge to $1$, hence, by dominated convergence, $$c_\lambda(r,x)\sim
d_\lambda(x)\,\gamma(r)^{-x}\,
\sum_{y\ge0}\left(\frac{-\gamma(r)}{\gamma(r)+1}\right)^y\,d_\lambda(y)
=
d_\lambda(x)\,\gamma(r)^{-x}\,\left(1+\frac{\gamma(r)}{1+\gamma(r)}\right)^{-\lambda},$$ where the equality stems from the definition of the coefficients $d_\lambda(\cdot)$. Plugging the equivalent of $d_\lambda(x)$ into this and using the fact that $1+\gamma(r)/(1+\gamma(r))=1/m(r)$, one deduces lemma \[l.cl\].
Lemma \[l.cl\] for $\lambda=\frac32$ yields that, when $x$ converges to infinity, there exists a constant $\alpha$, whose value is irrelevant, such that $$A(r,x)\sim2r\alpha\,x^{1/2}\,\gamma(r)^{-x}\,\gamma(r)\,(1+\gamma(r)),$$ and $$x\,B(r,x)-A(r,x)\sim2r\alpha\,x^{1/2}\,\gamma(r)^{-x}\,\gamma(r)^2.$$ Hence $(x\,B(r,x)-A(r,x))/A(r,x)$ converges to $\gamma(r)/(1+\gamma(r))$, and $$\frac{A(r,x)}{x\,B(r,x)}
\quad\mbox{converges to}\quad
\frac{1+\gamma(r)}{1+2\gamma(r)}=m(r).$$ This is the desired convergence of the expectations because, as mentioned above, the relation $r=\varrho(u)$ means that $m(r)=m_u$.
Modes
-----
To study the mode of $\xi_x$, one compares $\nu(r,x)(y+1)$ to $\nu(r,x)(y)$. The ratios $$\frac{\nu(r,x)(y+1)}{\nu(r,x)(y)}
=
\frac{(y+1/2)\,(x-y)\,r}{(y+1-x/2)\,(y+1/2-x/2)}$$ are the terms of a nonincreasing sequence indexed by $y$. Writing $y$ as $y=x\,(1+s)/(2s)$ with $s\ge 1$, when $x$ is large, one gets $$\frac{\nu(r,x)(y+1)}{\nu(r,x)(y)}\sim\,r\,(s^2-1).$$ This implies that the sequence $(\nu(r,x)(y))_{y}$ is increasing on $y\le y_*$ and decreasing on $y\ge y_*$, for a value of $y_*$ such that $y_*=x\,(1+s_*)/(2s_*)+o(x)$ with $s_*^2=1+1/r.$ Finally, this shows that, when $r=\varrho(u)$, the mode of $\mu(r,x)$ is at $x/(1+u)+o(x)$.
Distributions
-------------
Our next computation is based on characteristic functions. Fix $u$ in $]0,1[$ and let $r=\varrho(u)$. For every positive integer $x$, introduce $$F_x(t)=\mathbb{E}\left(\exp\left(t\,\frac{\xi_x-x\,m}{\sqrt{x}}\right)\right).$$ Recall that $$m(r)=\frac{1+\gamma(r)}{1+2\gamma(r)},\qquad
\frac1{1+2\gamma(r)}=\sqrt{\frac{r}{1+r}},\qquad
r=\frac{(1-u)^2}{4u}.$$ Since $\mathbb{E}(\exp(t\xi_x))
=
B(r\mathrm{e}^t,x)/B(r,x)$, $$F_x(t)=\mathrm{e}^{-t\,\sqrt{x}\,m}\,B(r\mathrm{e}^{t/\sqrt{x}},x)/B(r,x).$$ We turn to the study of the sequence of functions $(B(\cdot,x))_{x\ge1}$.
Since $B(r,x)=c_{1/2}(r,x)$, a consequence of lemma \[l.cl\] is that, when $x$ converges to infinity, $$\frac{B(r\mathrm{e}^{t/\sqrt{x}},x)}{B(r,x)}
\sim
\left(\frac{\gamma(r)}{\gamma(r\mathrm{e}^{t/\sqrt{x}})}\right)^{x}
\frac{S(x,t/\sqrt x)}{m(r)^{1/2}},$$ where, for every $s$, $$S(x,s)=
\sum_{y=0}^x\left(\frac{-\gamma(r\mathrm{e}^{s})}
{\gamma(r\mathrm{e}^{s})+1}\right)^y\,d_{1/2}(y)\,
\frac{d_{1/2}(x-y)}{d_{1/2}(x)}.$$ We get rid of the fraction involving $S(x,t/\sqrt x)$ through lemmas \[ld\] and \[ls\].
\[ld\] For every nonnegative $x$ and $y$, $d_{1/2}(y)\,d_{1/2}(x)\le d_{1/2}(x+y)$.
A probabilistic proof is as follows. For every nonnegative $x$, $d_{1/2}(x)=2^{-2x}\binom{2x}{x}$ is the probability that a simple symmetric random walk on the integer line is at its starting point after $2x$ steps. Hence $d_{1/2}(x+y)$ is the probability that the random walk is at its starting point after $2x+2y$ points and $d_{1/2}(y)\,d_{1/2}(x)$ is the probability that the random walk is at its starting point after $2x$ steps and also after $2x+2y$ points. The latter event being included in the former, this shows the desired inequality.
\[ls\] When $x$ converges to infinity, $S(x,t/\sqrt{x})$ converges to $m(r)^{1/2}$.
Since $S(x,0)$ converges to $S(\infty,0)=m(r)^{1/2}$ when $x$ converges to infinity, we show that $S(x,t/\sqrt{x})-S(x,0)$ converges to $0$. By lemma \[ld\], the ratios of coefficients $d_{1/2}$ involved in $S(x,t/\sqrt{x})$ and $S(x,0)$ are bounded by $1$. Adding terms such that $y\ge x+1$, one gets $|S(x,t/\sqrt{x})-S(x,0)|\le T(t/\sqrt{x})$, where $$T(t/\sqrt{x})=\sum_{y=0}^{+\infty}\left|\left(\frac{\gamma(r\mathrm{e}^{t/\sqrt{x}})}
{\gamma(r\mathrm{e}^{t/\sqrt{x}})+1}\right)^y
-
\left(\frac{\gamma(r)}
{\gamma(r)+1}\right)^y\right|.$$ All the terms in the sum have the same sign, hence $$T(t/\sqrt{x})=
\left|\sum_{y=0}^{+\infty}\left(\frac{\gamma(r\mathrm{e}^{t/\sqrt{x}})}
{\gamma(r\mathrm{e}^{t/\sqrt{x}})+1}\right)^y
-
\left(\frac{\gamma(r)}
{\gamma(r)+1}\right)^y\right|.$$ One can compute the sum of each geometric series. This yields $$|S(x,t/\sqrt{x})-S(x,0)|\le T(t/\sqrt{x})
=
\left|\gamma(r\mathrm{e}^{t/\sqrt{x}})-\gamma(r)\right|,$$ which proves the lemma since $\gamma(\cdot)$ is a continuous function.
Lemma \[ls\] shows that $$\frac{B(r\mathrm{e}^{t/\sqrt{x}},x)}{B(r,x)}\sim
\left(\frac{\gamma(r\mathrm{e}^{t/\sqrt{x}})}{\gamma(r)}\right)^{-x}.$$ The rest of the proof is standard. A Taylor expansion of $\gamma(\cdot)$ around $r$ yields $$\gamma(r\mathrm{e}^{t/\sqrt{x}})=
\gamma(r)+(\mathrm{e}^{t/\sqrt{x}}-1)\,\gamma'(r)+
(\mathrm{e}^{t/\sqrt{x}}-1)^2\,\gamma''(r)/2+o((\mathrm{e}^{t/\sqrt{x}}-1)^2).$$ Using the expansion of $\mathrm{e}^{t/\sqrt{x}}$ along powers of $1/\sqrt{x}$ and dividing everything by $\gamma(r)$, one gets $$\frac{\gamma(r\mathrm{e}^{t/\sqrt{x}})}{\gamma(r)}=
1+\left(r\,\frac{\gamma'(r)}{\gamma(r)}\right)\,\frac{t}{\sqrt{x}}+
\left(r\,\frac{\gamma'(r)}{\gamma(r)}+
r^2\frac{\gamma''(r)}{\gamma(r)}\right)\,\frac{t^2}{2x}
+o\left(\frac1{x}\right).$$ Note that $$r\frac{\gamma'(r)}{\gamma(r)}=-m(r).$$ Taking logarithms, writing the ratio of functions $\gamma$ as $$(\gamma(r\mathrm{e}^{t/\sqrt{x}})/\gamma(r))^{-x}=
\exp(-x\,\log(\gamma(r\mathrm{e}^{t/\sqrt{x}})/\gamma(r))),$$ and using the expansion $\log(1+z)=z-z^2/2+o(z^2)$ when $z=o(1)$, one gets that $F_x(t)$ is equivalent to the exponential of $$-tm(r)\sqrt{x}-x\,
\left(-m(r)\,t/\sqrt{x}+(m_2(r)-m(r))\,t^2/(2x)-m(r)^2\,t^2/(2x)+o(1/x)\right),$$ where $$m_2(r)=r^2\frac{\gamma''(r)}{\gamma(r)}.$$ Finally, $F_x(t)$ converges to $\mathrm{e}^{\sigma^2(r)\,t^2/2}$, with $$\sigma^2(r)=m(r)^2+m(r)-m_2(r).$$ Using the definitions of $m(r)$ and $m_2(r)$ as functions of $\gamma(r)$ and its derivatives, one gets $$\sigma^2(r)=r\,\left(-r\,\frac{\gamma'(r)}{\gamma(r)}\right)'=r\,m'(r).$$ Using the formula for $m(r)$ at the beginning of this section, one gets finally $$\sigma^2(r)=\frac14\sqrt{\frac{r}{(1+r)^3}}=\frac{u\,(1-u)}{(1+u)^3}
=\sigma^2_u.$$ The proof is complete.
Conditional hitting times {#s.tc}
=========================
Proof of theorem \[tc\]
-----------------------
We introduce the renewal process $(\zeta_x)_{x\ge1}$ with increments $(\varepsilon_x)_{x\ge1}$, that is $$\zeta_x=\inf\{y\ge1\,;\,\sigma_y\ge x\}.$$ The usual central limit theorem for renewal processes states that $(\zeta_x-mx)/\sqrt{x}$ converges in distribution to a centered Gaussian variable whose variance is the variance $u(1-u)$ of every $\varepsilon_x$ divided by the cube of the mean $1+u$ of every $\varepsilon_x$, that is $u(1-u)/(1+u)^3=\sigma^2_u$.
Our next lemma expresses the distribution of $\eta_x$ for every positive $x$ in terms of the distributions of the random variables $(\zeta_z)_{1\le z\le x+1}$.
\[l.eta\] For every positive $x$ and $y$, $$\mathbb{P}(\eta_x=y)
=
\frac{1+u}{1-(-u)^{x+1}}\,\sum_{z=0}^{x-1}(-u)^z\mathbb{P}(\zeta_{x+1-z}=y+1).$$
Let $x$ and $y$ denote positive integers. We begin with the fact that $$\{\zeta_{x+1}=y+1\}=\{\sigma_{y}=x\}\cup\{\sigma_{y}=x-1,\varepsilon_{y+1}=2\},$$ hence $$\mathbb{P}(\sigma_{y}=x)=\mathbb{P}(\zeta_{x+1}=y+1)-u\,\mathbb{P}(\sigma_{y}=x-1).$$ Iterating this recursion, one gets $$\mathbb{P}(H_x)\,\mathbb{P}(\eta_x=y)=\mathbb{P}(\sigma_{y}=x)
=
\sum_{z=0}^{x-1}(-u)^z\,\mathbb{P}(\zeta_{x+1-z}=y+1).$$ Summing over every positive value of $y$ and using the facts that $\mathbb{P}(\zeta_{z}=1)=0$ if $z\ge3$ and that $\mathbb{P}(\zeta_{2}=1)=u$, one gets $$\mathbb{P}(H_x)=(-u)^{x-1}(1-u)+\sum_{z=0}^{x-2}(-u)^z=\frac{1-(-u)^{x+1}}{1+u}.$$ This concludes the proof.
Lemma \[l.eta\], the fact that $|u|<1$ and the convergence of the distribution of $(\zeta_x-mx)/\sqrt{x}$, imply the same convergence for the distribution of $(\eta_x-mx)/\sqrt{x}$.
Finally, $(\xi_x-mx)/\sqrt{x}$, $(\zeta_x-mx)/\sqrt{x}$ and $(\eta_x-mx)/\sqrt{x}$ all converge in distribution to the same limit, which is the centered Gaussian distribution with variance $\sigma^2_u$.
Sharp bounds
------------
\[l.ex\] For every positive $x$, $$\mathbb{E}(\eta_x)=\frac{x+1}{1+u}\,
\frac{1+(-u)^{x+2}}{1-(-u)^{x+1}}-\frac{1+u^2}{(1+u)^2}.$$
For instance, $$\mathbb{E}(\eta_1)=1,\quad\mathbb{E}(\eta_2)=2-\frac{u}{1-u(1-u)}.$$ For every positive integer $x$, one can deduce from the exact formula above that $$\frac{x}{1+u}-\frac{2u^2}{(1+u)^2}\le
\mathbb{E}(\eta_x)\le\frac{x}{1+u}+\frac{2u}{(1+u)^2}.$$ The width of the interval delimited by the upper and the lower bounds of $\mathbb{E}(\eta_x)$ above is $2u/(1+u)\le1$.
Bounds on $\mathbb{E}(\eta_x)$, depending on the parity of $x$, are as follows. For every odd $x$, $$\mathbb{E}(\eta_x)\ge x/(1+u),$$ and for every even $x$, $$\mathbb{E}(\eta_x)\le x/(1+u)+u(1-u)/(1+u)\le(x+1/4)/(1+u).$$ These refined bounds yield intervals around $\mathbb{E}(\eta_x)$, which depend on the parity of $x$, and whose width is always at most $2u/(1+u)^2\le1/2$.
Fix a positive integer $x$, a real number $u$ in $]0,1[$, and let $r=\varrho(u)$. Let $p_x^y=\mathbb{P}(\sigma_y=x)$. Then $\eta_x(\mathbb{P})$ is proportional to the measure $\displaystyle\sum_yp_x^y\,\delta_y$ and, for every positive $y$, $$\sum_{x=y}^{2y}p_x^y\,t^x=\left((1-u)\,t+u\,t^2\right)^y.$$ Hence the distribution of $\eta_x$ is $\mu_\eta(r,x)$, where $\mu_\eta(r,x)=\nu_\eta(r,x)/|\nu_\eta(r,x)|$ and $$\nu_\eta(r,x)=\sum_{y=h(x)}^x\binom{y}{x-y}\,4^yr^y\,\delta_y.$$ When $u=0$, $r=\infty$ and $\mu_\eta(\infty,x)$ is the Dirac distribution at $x$. When $u=1$, $r=0$ and $\mu_\eta(0,x)$ is the Dirac distribution at $h(x)$. For the first values of $x$, the distributions $\mu_\eta(r,x)$ are as follows: $\mu_\eta(r,1)=\delta_1$, $$\mu_\eta(r,2)=\frac{\delta_1+4r\delta_2}{1+4r},\quad
\mu_\eta(r,3)=\frac{2\delta_2+4r\delta_3}{2+4r},\quad
\mu_\eta(r,4)=\frac{\delta_2+12r\delta_3+16r^2\delta_4}{1+12r+16r^2}.$$ This implies that, for every positive $x$, $$\mathbb{E}(\eta_x)=r\,g'_x(r)/g_x(r),\quad
g_x(r)=\sum_{y=h(x)}^x\binom{y}{x-y}\,4^yr^y.$$ To study the generating functions $g_x$, we introduce $g_0(r)=1$ and $$G(r,z)=\sum_{x\ge0}g_x(r)\,z^x.$$ Summing first over $y\le x\le 2y$, then over $y\ge0$, one gets $$G(r,z)=\sum_{y\ge0}(4rz)^y\,(1+z)^y=1/(1-4rz(1+z))=C_1(r,z).$$ From the proof of lemma \[l.cl\], one knows that the poles of $C_1(r,z)$ are $z=\gamma(r)$ and $z=-\gamma_2(r)$ with $\gamma_2(r)=\gamma(r)+1$, hence, $$G(r,z)=\frac1{\gamma(r)+\gamma_2(r)}\left(\frac{\gamma_2(r)}{1-z/\gamma(r)}+\frac{\gamma(r)}{1+z/\gamma_2(r)}\right).$$ This shows that, for every nonnegative $x$, $$g_x(r)=\frac{\gamma(r)\gamma_2(r)}{\gamma(r)+\gamma_2(r)}
\left(\gamma(r)^{-(x+1)}-(-\gamma_2(r))^{-(x+1)}\right).$$ From here, the expression of $\gamma(r)$ as a function of $r$ and tedious computations of derivatives yield the result.
Comparison with a naïve estimator
---------------------------------
For a given value $u$ in $]0,1[$ and for a branching process $X_{0:\infty}$ with offspring distribution $(1-u)\delta_1+u\delta_2$, when $n$ converges to infinity, $$S_n\sim X_n(1+1/(1+u)+1/(1+u)^2+\cdots)=X_n(1+1/u)\quad\mbox{almost surely},$$ hence $B(X_{0:\infty})=u$ almost surely. The naïve pointwise prediction of the mean initial population conditional on $X_1=x$, namely $N_u(x)=x/(1+u)$, should be compared to the Bayesian prediction $\mathbb{E}_u(\xi_x)$ for $r=\varrho(u)$. For $x=2$, one gets $$\frac{\mathbb{E}_u(\xi_2)}{N_u(2)}=\frac{(4u+6(1-u)^2)\,(1+u)}{2(4u+3(1-u)^2)}.$$ This ratio is $1$ when $u=0$ or $u=1$, greater than $1$ for every $u$ in $]0,\frac13[$, and smaller than $1$ for every $u$ in $]\frac13,1[$. Hence the naïve and Bayesian predictions cannot be easily compared, at least on $X_1=x$ for a given finite $x$.
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|
[**TSL/ISV-98-0198\
October 1998**]{}
[**Higher Partial Waves in near Threshold** ]{}
[ H. Calén $^a$, J. Dyring $^a$, G. Fäldt $^a$, K. Fransson $^a$, L. Gustafsson $^a$, S. Häggström $^a$, B. Höistad $^a$, J. Johanson $^a$, A. Johansson $^a$, T. Johansson $^a$, S. Kullander $^a$, A. Mörtsell $^a$, R. Ruber $^a$, J. Złomańczuk $^a$, C. Ekström $^b$, K. Kilian $^c$, W. Oelert $^c$, T. Sefzick $^c$, R. Bilger $^d$, W. Brodowski $^d$, H. Clement $^d$, G.J. Wagner $^d$, A. Bondar $^e$, A. Kuzmin $^e$, B. Shwartz $^e$, V. Sidorov $^e$, A. Sukhanov $^e$, V. Dunin $^f$, B. Morosov $^f$, A. Povtorejko $^f$, A. Sukhanov $^f$, A. Zernov $^f$, A. Kupsc $^g$, P. Marciniewski $^g$, J. Stepaniak $^g$, J. Zabierowski $^h$, A. Turowiecki $^i$, Z. Wilhelmi $^i$, C. Wilkin $^j$\
]{} $^a$ Department of Radiation Science, Uppsala University, S-751 21 Uppsala, Sweden\
$^b$ The Svedberg Laboratory, S-751 21 Uppsala, Sweden\
$^c$ IKP, Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany\
$^d$ Physikalisches Institut, Tübingen University, D-72076 Tübingen, Germany\
$^e$ Institute of Nuclear Physics, Novosibirsk 630 090, Russia\
$^f$ Joint Institute for Nuclear Research Dubna, 101000 Moscow, Russia\
$^g$ Institute for Nuclear Studies, PL-00681 Warsaw, Poland\
$^h$ Institute for Nuclear Studies, PL-90137 Lódz, Poland\
$^i$ Institute of Experimental Physics, Warsaw University, PL-0061 Warsaw, Poland\
$^j$ Physics & Astronomy Dept., University College London, London WC1E 6BT, U.K.\
[**Abstract:**]{} Exclusive measurements of the production of $\eta$-mesons in the $pp\to pp\eta$ reaction have been carried out at excess energies of 16 and 37 MeV above threshold. The deviations from phase space are dominated by the proton-proton final state interaction and this influences particularly the energy distribution of the $\eta$ meson. However, evidence is also presented at the higher energy for the existence of an anisotropy in the angular distributions of the $\eta$-meson and also of the final proton-proton pair, probably to be associated with $D$-waves in this system interfering with the dominant $S$-wave term. The sign of the $\eta$ angular anisotropy suggests that $\rho$-exchange is important for this reaction.\
4ex
The production of $\eta$ mesons in proton-proton collisions near threshold has been measured in recent years by three different groups [@SPESIII; @PINOT; @TSL1] and a fairly consistent picture has emerged. In comparison to the analogous pion case [@IUCF; @TSL2], $S$-wave production is more dominant near threshold, and this is generally ascribed to the presence of the N$^*$(1535) S$_{11}$ isobar, whose width overlaps the $\eta-p$ threshold and which has large branching fractions into both $\eta-p$ and $\pi-p$ [@PDG]. Dalitz plots obtained for the $pp\to pp\eta$ reaction at low energy show strong deviations from phase space due to the presence of the proton-proton final state interaction (FSI) [@TSL1]. In addition there are residual effects but the precision of these measurements was limited, in part, by the uncertainty in the determination of the proton angles. This defect has been overcome in the present experiment through the inclusion of a tracking device covering the forward angles and operating in conjunction with the apparatus previously used. The combination allowed us to investigate the influence of higher partial waves in the angular distributions and to identify dependences on both the angles of the $\eta$ and of the proton-proton relative momentum, which become stronger with energy. This behaviour is the first indication of effects from higher partial waves in the $pp\to pp\eta$ reaction. The shape of the $\eta$ angular variation is sensitive to the basic production mechanism and the data suggest that $\rho$-exchange provides a more important driving term here than $\pi$-exchange.
The experiment was carried out using the PROMICE/WASA facility at the CELSIUS storage ring of the The Svedberg Laboratory at beam energies of 1296 and 1350 MeV, corresponding to centre-of-mass excess energies $Q = 16$ and 37 MeV respectively. Using a cluster gas jet target, integrated luminosities of about 50 nb$^{-1}$ and 200 nb$^{-1}$ were obtained for the two energies in a total of 30 hours of running, and these yielded about 300 and 750 good $pp\eta$ events in the final sample. Details of the detector system are given in Refs. [@PROMICE; @Dyring], and only the main points are discussed here.
The forward-going protons were measured in a detector system covering polar angles between 4$^{\circ}$ and 20$^{\circ}$ with respect to the beam direction. It consists of a tracking detector, followed by a three-layer scintillator hodoscope and a four-layer scintillator calorimeter. A second hodoscope is placed at the end of the detector system to register penetrating particles. The tracker consists of two planes, each with four layers of thin-walled cylindrical drift chambers, so-called “straw chambers”, oriented in the vertical and horizontal directions. This arrangement allows the proton scattering angles to be reconstructed to a precision of better than 1$^{\circ}$ (FWHM). The $\eta$’s are identified from their 2$\gamma$ decay channel, where the $\gamma$’s are detected in two CsI(Na) arrays situated on either side of the scattering chamber. Scintillator hodoscopes are placed in front of the CsI arrays to veto charged particles. The 2$\gamma$ invariant mass resolution obtained at the $\eta$-meson mass is 20 MeV (RMS).
In the off-line analysis, only those events with an identified $\eta$ meson together with two energetic protons in the forward detector were retained and, in these cases, a kinematical fit was applied in order to extract the energy-momentum vectors for the particles. To avoid problems in accounting for losses through nuclear reactions in the detector material, only the proton directions were used, and this results in a fit with three constraints (3C). The number of background events with two uncorrelated $\gamma$’s arising from different pions in 2$\pi^0$ production becomes negligible after applying a lower cut at 5% on the confidence level of the fit. In the subsequent analysis of our data, we exploit the fact that at low energies only a few amplitudes are allowed and that these lead to distributions in which the angular and momentum variables are intimately linked. It is only through the introduction of such a functional form into Monte Carlo simulations that we can draw any firm conclusions on the physics of the process. The detector response and acceptance calculations for the experiment were made with events generated from a full Monte Carlo simulation using GEANT3 [@Geant]. The resulting geometrical acceptance for this type of event is about 0.5% and 2% at $Q = 16$ and 37 MeV respectively. The data were checked for internal consistency by applying different geometrical cuts.
The sole production amplitude which survives at threshold corresponds to the transition $^{3\!}P_0\to\, ^{1\!}S_0\,s$, where we are using the standard $^{2S+1\!}L_J$ notation for the $pp$ system, with the lower case letter denoting the angular momentum of the $\eta$-meson with respect to the $pp$ system. At slightly higher energies, amplitudes corresponding to the production of $P$- and $D$-wave $pp$ pairs introduce dependences on the $pp$ angles with similar momentum threshold factors since the $D$-wave term can interfere with the threshold $S$-wave amplitude. However, given that $\eta$ production in the $^{1\!}S_0\,p$ state is forbidden by selection rules, the first non-trivial $\eta$ angular dependence is expected to come from the interference of the $s$- and $d$-wave amplitudes. These considerations lead us to take the following simple form for the low energy $pp\to pp\eta$ amplitude: $$M=\tilde{A}_{Ss}\:\phi_{f}^{\dagger}\,(\hat{p}\cdot\vec{\varepsilon}_{i})
+A_{Sd}\:\phi_{f}^{\dagger}\,
(\hat{p}\cdot\vec{k})(\vec{k}\cdot\vec{\varepsilon}_{i})
+A_{Ps}\:\phi_i\,(\vec{q}\cdot\vec{\varepsilon}_f^{\,\dagger})
+A_{Ds}\:\phi_f^{\dagger}\,
(\hat{p}\cdot\vec{q})(\vec{q}\cdot\vec{\varepsilon}_i)\:.$$ The momenta of the initial proton and final $\eta$ in the overall c.m. system are denoted by $\vec{p}$ and $\vec{k}$ respectively, $2\vec{q}$ is the relative momentum in the final two-proton system, $\vec{\varepsilon}_i$ ($\vec{\varepsilon}_f$) the spin-one polarisation vector of the initial (final) proton-proton pair, and $\phi_i$ ($\phi_f$) the corresponding spin-zero functions. The indices on the amplitudes denote the dominant final angular momentum state in a term. It should however be noted that the strict $Ss$ partial wave amplitude is given by $$\label{Relation}
A_{Ss} = \tilde{A}_{Ss} +\fmn{1}{3}\,k^2\,A_{Sd} + \fmn{1}{3}\,q^2\,A_{Ds}\:.$$
Keeping only terms up to order $k^2$ or $q^2$, [*i.e.*]{} a total of two units of angular momentum in the final intensity, the spin-averaged matrix element squared becomes $$\label{M-squared}
\overline{|M|^2}= \frac{1}{4}\left[|\tilde{A}_{Ss}|^2 +
2\,k^2\,Re\left\{\tilde{A}_{Ss}^*A_{Sd}\right\}\,\cos^2\theta_{\eta} +
2\,q^2\,Re\left\{\tilde{A}_{Ss}^*A_{Ds}\right\}\,\cos^2\theta_{pp}
+q^2\,|A_{Ps}|^2\right]\:,$$ where $\theta_{\eta}$ and $\theta_{pp}$ are the angles that the $\eta$ and the $pp$ relative momentum make with respect to the beam direction.
In a model where the N$^*(1535)$ isobar is excited through one pion exchange [@GW], it is expected that the $|A_{Ps}|^2$ term should be smaller than the $Re\left\{\tilde{A}_{Ss}^*A_{Ds}\right\}$ of the $Ss$-$Ds$ interference by a factor of the order of $\mu/4m_p$, where $m_{\eta}$ and $m_p$ are the the $\eta$ and proton masses respectively, and $\mu = 2m_pm_{\eta}/(2m_p+m_{\eta})$ is the reduced mass of the final state. Since the $Ps$ term has no characteristic angular dependence, it would be difficult to isolate a small effect as compared to a possible energy dependence of $\tilde{A}_{Ss}$, and so any such contribution is ignored.
The $^{1\!}S_0$ state of the final $pp$ system is subject to a very strong FSI which is central to any analysis of low energy production. All the terms in Eq. (\[M-squared\]) are influenced by the FSI, with the exception of the $P$-wave contribution, and this will reduce even further the relative importance of $|A_{Ps}|^2$. Without knowing the radial dependence of the $\eta$-production operator, the FSI effect is slightly model-dependent. We estimate it by taking the ratio of the Paris $^{1\!}S_0$ wave function [@Paris] at its maximum at $r=1$ fm to the corresponding plane wave function; numerical calculations with the Paris wave functions show that Coulomb effects are negligible under the conditions of the present experiment. There is a common enhancement factor $F_{SS}(q)$ for the $(Ss)^2$ and $Ss$-$Sd$ interference terms and an analogous $F_{SD}(q)$ for the $Ss$-$Ds$ interference. They may be parametrised in a similar manner to that given in ref. [@FW] $$\begin{aligned}
\nonumber
F_{SS}(q)&=&0.440+\frac{151.7}{1+q^2/\alpha^2}\:,\\
F_{SD}(q)&=&0.968+\frac{11.5}{1+q^2/\alpha^2} \:,\end{aligned}$$ with $\alpha=-0.053$ fm$^{-1}$.
We make the assumption that, apart from the FSI, the amplitudes $\tilde{A}_{Ss}$, $A_{Sd}$ and $A_{Ds}$ appearing in Eq. (\[M-squared\]) are constant. The resulting form of the differential cross section is $$\label{sigma}
d\sigma = \frac{N}{p}\:\left(F_{SS}(q)
+ a\,\frac{k^2}{\mu m_p}\:F_{SS}(q) \cos^2\theta_{\eta}
+ b\,\frac{q^2}{\mu m_p}\: F_{SD}(q) \cos^2\theta_{pp}\right)
\mbox{\it dLips}\:,$$ where $N$ is a normalization constant and [*dLips*]{} is the invariant three-body phase space distribution. Typical momentum factors $\mu m_p$ are explicitly shown to leave two dimensionless constants $a$ and $b$ to be determined from the shapes of the angular and energy distributions. It must be stressed that the functional form of Eq. (\[sigma\]) does not depend upon the details of a specific dynamical model but on the assumption of constant amplitudes.
To extract differential distributions from the data, we fitted the Monte Carlo simulations, with events weighted according to Eq. (4), to the experimental results. Following this procedure at $Q = 37$ MeV, a combined fit to the $\theta_{\eta}$ and $\theta_{pp}$ angular distributions yields the parameter values $a= -15 \pm 5$ and $b= 22\pm 7$, which are correlated, and the overall normalisation constant. In addition to the given statistical errors, there are systematic errors, arising mainly from uncertainties in the geometrical alignment of the apparatus, which are of the order 10% for the $a$ parameter and 20% for the $b$ parameter. These values were then used in the Monte Carlo simulation to make the acceptance corrections needed to unfold the data; the resulting corrected experimental data and fitted angular distributions are shown in Figs. 1a and 2a. It should be noted that the $a$ and $b$ parameters were not deduced by fitting to the corrected data shown in the figures.
The distribution in the $\eta$ kinetic energy $T_{\eta}$, shown in Fig. 3a, is shifted towards higher energies with respect to phase space. This is a direct consequence of the strongly attractive $pp$ FSI, which enhances events where the $\eta$ recoils against the two protons which have low excitation energy. The dashed curve, which corresponds to phase space modified by the FSI, produces too big an effect. The solid curve is the prediction with the values of $a$ and $b$ as determined by the angular distributions. The non-vanishing of the averages of $\cos^2\theta$ leaves $k^2$ and $q^2$ terms in Eq. (\[sigma\]) which yield a much better fit.
At $Q =16$ MeV, S-waves are even more dominant and the angular distributions are broadly compatible with phase space modified by the pp FSI. Nevertheless the shapes are in fact marginally better reproduced with the parameter values derived from the higher energy data and the corresponding predictions are shown in Figs. 1b and 2b. The $T_{\eta}$ data shown in Fig. 3b, while again demonstrating the effect of the $pp$ FSI, are well described by the parametrisation. The experimental points do depend, to some extent, on the shape of the assumed differential distribution, but this is well within the statistical uncertainty of the data. The poor quality of the data for $\cos\theta_{pp} > 0.8$ and at the upper end of the $T_{\eta}$ distribution reflects mainly the loss of events due to the beam pipe so that discrepancies with the parametrisation should not be taken seriously in these regions.
The numerical values of the differential cross sections are given in Tables 1–3. The data were normalised to the values of the total $pp\to pp\eta$ cross sections given in Ref. [@TSL1], namely $\sigma(T_p=1293~\mbox{\rm MeV}) = (2.11\pm 0.32)~\mu$b and $\sigma(T_p=1352~\mbox{\rm MeV}) = (4.92\pm 0.74)~\mu$b.
All existing theoretical models describing $\eta$-production in proton-proton scattering are broadly similar [@GW; @Theory], consisting of a meson exchange exciting a nucleon isobar, dominantly the $S_{11}$ N$^*$(1535), which decays into an $s$-wave $\eta$-proton pair. The models differ mainly in their choice of mesons exchanged and, in particular, the relative importance of $\pi$ and $\rho$ exchange [@GW; @Theory]. If, for simplicity, only the pion exchange term is retained then it follows from the expansion of the corresponding propagator and vertex function that $a=1$ and $b=4$ [@FW7]. Our experimental values are significantly larger than these.
Higher partial wave N$^*$ resonances are potentially very important for the $\eta$ angular distribution. Away from threshold there is evidence for significant $d$-wave production of the $\eta$ meson in the $\pi^-p\to \eta n$ reaction [@Deinet] and this is confirmed in the preliminary high statistics data from the Crystal Ball collaboration [@Ben]. Whilst having no appreciable effect on the $pp$ angular distribution, the inclusion of such $d$-wave production in a one pion exchange model would contribute about $a\approx +7$ in the $\eta$ angular distribution which, though of the right order of magnitude, is of opposite sign to what we have deduced from our data. Thus, in contrast to our findings presented in Fig. 1b, such a term would favour production towards $\cos\theta_{\eta} =\pm 1$.
However the sign of the $\cos^2\theta_{\eta}$ term is negative in the photoproduction $\gamma p\to\eta p$ a little above threshold [@Krusche] and, using vector dominance ideas, this is likely to be true for $\rho p\to\eta
p$ as well. In a pure $\rho$-exchange model, the elementary distribution would contribute $a\approx -2.5$ which, though too small, is of the same sign as the one apparent in our data. In the original estimation of this process [@GW] it was claimed that $\rho$-exchange should be stronger than $\pi$-exchange and that the interference between them was mainly destructive. If this were indeed the case, the $a$ coefficient could be enhanced significantly because of the negative sign between the $\rho$ and $\pi$ amplitudes. Our data would support such a conclusion. This would also lead to the prediction that the $pn\to
d\eta$ should show a much flatter distribution since the $\rho$ and $\pi$ amplitudes add in this case [@GW].
In conclusion, we have presented the first experimental evidence for non-isotropy in the $pp \to pp\eta$ angular distributions close to threshold. The signals are generally small in the data as compared for example to the proton-proton final state interaction, which has overwhelming importance. Nevertheless, a clear sign of an $\eta$ angular anisotropy has been found which could be the first direct indication of $\rho$-dominance in $\eta$ production.
An economic description of all our distributions, sufficient for acceptance corrections, has been given in terms of the two free parameters of Eq. (\[sigma\]) by taking the amplitudes $\tilde{A}_{Ss}$, $A_{Sd}$ and $A_{Ds}$ to be constant. If, for example, we assume instead that $A_{Ss}$ of Eq. (\[Relation\]) is constant then the $\cos^2\theta$’s in Eq. (\[sigma\]) are replaced by Legendre polynomials $P_2(\cos\theta)$. The fits to the angular distributions in Figs. 1 and 2 are very similar but, due to the vanishing of the angular average of $P_2(\cos\theta)$, the results in Fig. 3 are identical to the broken curve, which represents just phase space and FSI. To restore the previous good agreement requires an additional free $q^2$ or $k^2$ term in the fitting and, when this is introduced, the results are essentially identical to the solid curve of Fig. 3. The shapes of the angular distributions are unaffected by such a modified procedure.
A significant improvement of the statistics would be welcome to tie down the model parameters in this area. Such an improvement is anticipated through the use of the WASA detector [@WASA], which is designed for the study of rare decays of the $\eta$ meson and which will have an almost $4\pi$ coverage of the photons from the $\eta$ decay. The effects of higher partial waves should increase strongly with beam energy but to exploit this would, in our case, require an energy upgrade of CELSIUS or the study of quasi-free production on the deuteron.
We are grateful to the TSL/ISV personnel for their continued help during the course of this work. Discussions with M. Garçon on the analysis of this experiment were very helpful. Financial support for this experiment and its analysis was provided by the Swedish Natural Science Research Council, the Swedish Royal Academy of Science, the Swedish Institute, the Bundesministerium für Bildung und Forschung (06TU886), Deutsche Forschung Gesellschaft (Mu 705/3 Graduiertenkolleg), the Polish State Committee for Scientific Research, the Russian Academy of Science, and the European Science Exchange Programme. The data presented here are based upon the work of J. Dyring, in partial fulfillment of the Ph.D. requirements [@Dyring].
3ex
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--------------------- -------------------------------------- --------------------- --------------------------------------
$\cos\theta_{\eta}$ $d\sigma/d\Omega_{\eta}$ ($\mu$b/sr) $\cos\theta_{\eta}$ $d\sigma/d\Omega_{\eta}$ ($\mu$b/sr)
$-0.9$ $0.170\pm 0.029$ $-0.9$ $0.234\pm 0.027$
$-0.7$ $0.168\pm 0.033$ $-0.7$ $0.317\pm 0.040$
$-0.5$ $0.123\pm 0.029$ $-0.5$ $0.392\pm 0.049$
$-0.3$ $0.219\pm 0.039$ $-0.3$ $0.552\pm 0.061$
$-0.1$ $0.186\pm 0.036$ $-0.1$ $0.479\pm 0.057$
$\phantom{-}0.1$ $0.262\pm 0.044$ $\phantom{-}0.1$ $0.490\pm 0.057$
$\phantom{-}0.3$ $0.164\pm 0.032$ $\phantom{-}0.3$ $0.434\pm 0.060$
$\phantom{-}0.5$ $0.157\pm 0.031$ $\phantom{-}0.5$ $0.554\pm 0.069$
$\phantom{-}0.7$ $0.178\pm 0.034$ $\phantom{-}0.7$ $0.425\pm 0.061$
$\phantom{-}0.9$ $0.100\pm 0.024$ $\phantom{-}0.9$ $0.188\pm 0.038$
--------------------- -------------------------------------- --------------------- --------------------------------------
: Differential cross section with respect to the $\eta$ c.m.angle for the $pp\to pp\eta$ reaction at $Q=16$ and 37 MeV ($T_p=1296$ and 1350 MeV). In addition to the statistical error, there is an overall systematic uncertainty of about 20%.
------------------- ------------------------------------ ------------------- ------------------------------------
$\cos\theta_{pp}$ $d\sigma/d\Omega_{pp}$ ($\mu$b/sr) $\cos\theta_{pp}$ $d\sigma/d\Omega_{pp}$ ($\mu$b/sr)
$0.05$ $0.196\pm 0.039$ $0.05$ $0.331\pm 0.043$
$0.15$ $0.150\pm 0.029$ $0.15$ $0.308\pm 0.041$
$0.25$ $0.166\pm 0.028$ $0.25$ $0.343\pm 0.043$
$0.35$ $0.182\pm 0.026$ $0.35$ $0.405\pm 0.047$
$0.45$ $0.169\pm 0.024$ $0.45$ $0.344\pm 0.042$
$0.55$ $0.190\pm 0.029$ $0.55$ $0.355\pm 0.042$
$0.65$ $0.125\pm 0.029$ $0.65$ $0.473\pm 0.050$
$0.75$ $0.163\pm 0.043$ $0.75$ $0.391\pm 0.050$
$0.85$ $0.169\pm 0.084$ $0.85$ $0.508\pm 0.070$
— — $0.95$ $0.88\pm 0.19$
------------------- ------------------------------------ ------------------- ------------------------------------
: Differential cross section with respect to the proton-proton angle for the $pp\to pp\eta$ reaction at $Q=16$ and 37 MeV. In addition to the statistical error, there is an overall systematic uncertainty of about 20%.
------------ ---------------------------------- ------------ ----------------------------------
$T_{\eta}$ $d\sigma/dT_{\eta}$ ($\mu$b/MeV) $T_{\eta}$ $d\sigma/dT_{\eta}$ ($\mu$b/MeV)
$ 0.5$ $0.046\pm 0.010$ $ 1.0$ $0.055\pm 0.010$
$ 1.5$ $0.074\pm 0.013$ $ 3.0$ $0.110\pm 0.014$
$ 2.5$ $0.089\pm 0.015$ $ 5.0$ $0.115\pm 0.015$
$ 3.5$ $0.114\pm 0.018$ $ 7.0$ $0.142\pm 0.017$
$ 4.5$ $0.114\pm 0.019$ $ 9.0$ $0.164\pm 0.020$
$5.5$ $0.146\pm 0.025$ $11.0$ $0.175\pm 0.022$
$6.5$ $0.173\pm 0.033$ $13.0$ $0.160\pm 0.021$
$7.5$ $0.191\pm 0.045$ $15.0$ $0.244\pm 0.029$
$8.5$ $0.059\pm 0.029$ $17.0$ $0.183\pm 0.028$
$9.5$ $0.237\pm 0.090$ $19.0$ $0.155\pm 0.030$
$10.5$ $0.196\pm 0.098$ $21.0$ $0.216\pm 0.041$
— — $23.0$ $0.214\pm 0.047$
— — $25.0$ $0.212\pm 0.059$
— — $27.0$ $0.160\pm 0.080$
------------ ---------------------------------- ------------ ----------------------------------
: Differential cross section with respect to the $\eta$ c.m.kinetic energy for the $pp\to pp\eta$ reaction at $Q=16$ and 37 MeV. In addition to the statistical error, there is a typical systematic uncertainty of about 20%, which increases at the upper end of the spectrum.
[**Figure Captions**]{}\
Fig. 1. Differential cross section in the $\eta$-production angle for the $pp\to pp\eta$ reaction at (a) $Q=37$ MeV ($T_p =1350$ MeV) and (b) $Q=16$ MeV ($T_p=1295$ MeV). The dashed curves represent the Monte Carlo predictions of phase space modified by the proton-proton final state interaction, whereas the solid curve includes also the angular dependence of Eq. (5) with $a=-15$ and $b=22$.\
Fig. 2. Differential cross section in the proton-proton production angle for the $pp\to pp\eta$ reaction at (a) $Q=37$ MeV and (b) $16$ MeV with curves as described in Fig. 1.\
Fig. 3. Distribution in $\eta$ kinetic energies from the $pp\to pp\eta$ reaction at (a) $Q=37$ MeV and (b) $16$ MeV. The short-dashed curves represent the Monte Carlo predictions of phase space and the long-dashed shows the influence of the proton-proton final state interaction. The solid curve includes also the modifications induced by the angular and momentum dependence of Eq. (5) with $a=-15$ and $b=22$. Since the experimental acceptance changes somewhat according to the Monte Carlo generator used, the experimental points would be slightly lowered if we had extracted them using $a=b=0$.
|
---
author:
- Alfredo Guevara
bibliography:
- 'references.bib'
title: Holomorphic Classical Limit for Spin Effects in Gravitational and Electromagnetic Scattering
---
Introduction {#sec:intro}
============
Since the early days of QFT, the use of effective methods to describe the low energy regime of more fundamental theories [@HeisenbergEuler; @Weinberg1; @Weinberg2] has proven extremely successful [@SMEFT; @GiesQED-EFT; @SchererChiral]. One of the most powerful applications of Effective Field Theories (EFTs) is the case where the high energy completion of the underlying theory is unknown. In this direction, the problem of General Relativity as an EFT has been studied as a tool for obtaining predictions whenever the relevant scales are much smaller than $M_{Planck}$ [@Porto:16; @Donoghue:17]. For this regime the methods of QFT can be safely applied to compute both classical and quantum long range observables. In this context, the motivation for these problems stems from the always increasing interest in the measurement of gravitational waves as definitive tests of GR, which has led to the acclaimed first detection by LIGO in 2016 [@LIGO; @LIGO2]. Specifically, the binary inspiral stage, defined by the characteristic scale $v^{2}\sim Gm/r$, has been the subject of extensive research since it can be addressed with analytical methods [@Blanchet:02BinaryInsp; @Futamase2007; @Rothstein-Binary].
The key object in the study of the binary inspiral problem is the effective potential associated to a two-body system. This potential admits a non-relativistic expansion in powers of $v^{2}\sim Gm/r$, known as the post-Newtonian (PN) expansion. Pioneered by the seminal work of Einstein-Infeld-Hoffman long ago [@EIH], several attempts have been made to evaluate the potential at higher PN orders. The EFT approach is based on using Feynman diagrammatic techniques and treating the PN expansion as a perturbative loop expansion [@Duff; @Donoghue:1994dn; @Muzinich:1995uj; @Akhundov:1996]. A standard setup is the $2\rightarrow2$ scattering of massive objects $m_{a}$ and $m_{b}$, interacting through the exchange of multiple gravitons (Fig. \[fig:intro\]). In this case the classical potential can be obtained from the long range behavior of the amplitude after implementing the Born approximation [@Feinberg:1988yw; @Holstein:2008sw; @Vaidya:14]. This classical piece is in turn extracted by setting the COM (Center of Mass) frame, in which the momentum transfer reads $|\vec{q}|=\sqrt{-t}$ and corresponds to the Fourier conjugate of the distance $r$. Calculations in this framework have proved extremely long and tedious, even though there have been remarkable simplifications in the context of non-relativistic approaches [@Goldberger:2004jt; @Rothstein-Binary; @Gilmore:2008gq; @Porto:2005ac; @Foffa:2016rgu]. In addition, the electromagnetic analog of the effective potential has been also discussed in [@Feinberg:1988yw; @Holstein:2008sw; @Holstein:2016fxh; @Iwasaki] in the context of classical corrections to Coulomb scattering. As expected the long range behavior of this potential, i.e. the $\frac{1}{r^{n}}$ falloff, is identical to the gravitational case. The computations are simpler in general and thus it also serves as a toy model for the PN problem.
\[fig:intro\]
One of the distinctive characteristics of the PN expansion is the treatment of the binary system as localized sources endowed with a tower of multipole moments. The evaluation of higher multipole moments starting at 1.5PN requires to incorporate spin into the massive particles involved in the scattering process [@Porto:2005ac; @Portospin2; @Vaidya:14], along with radiative corrections. These spin contributions account for the internal angular momentum of the objects in the macroscopic setting [@Barker1979; @Vaidya:14]. The universality of the gravitational coupling implies that it is enough to consider massive particles of spin $S$ to evaluate the spin multipole effects up to order $2S$ in the spin vector $|\vec{S}|$. Such computation was first done up to 1-loop by Holstein and Ross in [@Holstein:08] and then by Vaidya in [@Vaidya:14], leading to $|\vec{S}|^{2}$ and $|\vec{S}|^{4}$ results, respectively. The electromagnetic counterpart has also been discussed up to $|\vec{S}|^{2}$ [@Holstein:2008sw]. Higher spin multipole moments are characterized by containing higher powers of the momentum transfer $|\vec{q}|$ and $|\vec{S}|$. Thus, in order to evaluate classical spin effects an expansion of the amplitude to arbitrarily subleading orders in $|\vec{q}|=\sqrt{-t}$ is required. This, together with the natural increase in difficulty for manipulating higher spin degrees of freedom in loop QFT processes [@Holstein:08], renders the computation virtually doable only within the framework of intrinsic non-relativistic approaches along with the aid of a computer for higher PN orders [@Porto:2008jj; @Kol:2010ze; @Levi:2014sba; @Levi:2017kzq].
In this paper we find that the combination of several new methods can bypass some of the aforementioned difficulties. We provide fully relativistic formulas for the classical part of the amplitude valid for any spin at both tree and 1-loop level. The difficulty in extracting arbitrarily subleading momentum powers is avoided by noting that the $t\rightarrow0$ and $|\vec{q}|\rightarrow0$ expansions can be disentangled outside the COM frame. That is, we evaluate the classical piece in a covariant way by selecting the *leading* order in the limit $t\rightarrow0$, which we approach by using complexified momenta. We find that the multipole terms are fully visible at leading order, and propose Lorentz covariant expressions for them in terms of the momentum transfer $K^{\mu}$. These expressions can then be analytically extended to the COM frame by putting $K=(0,\vec{q})$. This is what we call the *holomorphic classical limit* (HCL).
To bypass the intrinsic complications due to the evaluation of higher spin loop processes we draw upon a battery of modern techniques based on the analytic structure of scattering amplitudes. In fact, techniques such as spinor helicity formalism, on-shell recursion relations (BCFW), and unitarity cuts have proven extremely fruitful for both computations of gravity and gauge theory amplitudes [@Bern:1994cg; @Bern:1994zx; @Witten:2003nn; @Cachazo:2004kj; @Bern:2010ue; @Naculich:2014naa]. In this context, several simplifications in the computation of the 1-loop potential have already been found for scalar particles in [@Neill:2013wsa; @Bjerrum-Bohr:2013; @Holstein:2016fxh]. Pioneered by the work of Bjerrum-Bohr et al. [@Bjerrum-Bohr:2014zsa] these methods were applied to the light-bending case [@Bjerrum-Bohr:2014lea; @Bjerrum-Bohr:2016hpa; @Bai:2016ivl; @Holstein:2016cfx; @AmpforAstro], where one of the external particle carries helicity $|h|\in\{0,\frac{1}{2},1\}$, and universality with respect to $|h|$ was found. Here we extend these approaches by considering two more techniques, both very recently developed as a natural evolution of the previously mentioned. The first one appeared in [@LS], where Cachazo and the author proposed to use a generalized form of unitarity cuts, known as the Leading Singularity (LS), in order to extract the classical part of gravitational amplitudes leading to the effective potential. It was shown that while at tree level this simply corresponds to computing the t channel residue, at 1-loop the LS associated to the triangle diagram leads to a fully relativistic form containing the 1PN correction for scalar particles, through a multidispersive treatment in the t channel. The second technique was proposed by Arkani-Hamed et al. in [@Nima:16] and gives a representation for *massive* states of arbitrary spin completely built from spinor helicity variables. Hence we use such construction to compute the LS associated to both the gravitational and electromagnetic triangle diagram as well as the respective tree level residues, this time including higher spin in the external particles. The combination of these techniques with the HCL leads to a direct evaluation of the 1-loop correction to the classical piece. The result is expressed in a compact and covariant manner in terms of spinor helicity operators, which are then matched to the standard spin operators of the EFT. As a crosscheck we recover the results for both gravity and EM presented in [@Holstein:08; @Holstein:2008sw; @Neill:2013wsa; @Vaidya:14] for $S\leq1$. By suitably defining the massless limit, we are also able to address the light-like scattering situation and check the proposed universality of light bending phenomena.
As an important remark, in this work we restrict our attention to spinning particles minimally coupled to gravity or EM. This is what is needed to reproduce the effective potential and intrinsic multipole corrections associated to point-like sources, corresponding to black hole processes. As a consequence we find various universalities with respect to spin which are manifest in spinor helicity variables, and were previously argued in [@Holstein:08; @Bjerrum-Bohr:2013; @Vaidya:14]. The non-minimal extension, relevant for evaluating finite size effects, is left for future work.
This paper is organized as follows. In section \[sec:pre\] we review the kinematics and spin considerations associated to the $2\rightarrow2$ process, which motivates the holomorphic classical limit. We then proceed to give a short overview of the notation and conventions used along the work, specifically those regarding manipulations of spinor helicity variables. Next, in section \[sec:scalar\] we review scalar scattering and implement the HCL to extract the electromagnetic and gravitational classical part from leading singularities at tree and 1-loop level, including the light bending case. Next, in section \[sec:spin\] we introduce the new spinor helicity representation for massive kinematics, leaving the details to appendix \[sec:spinrep\], and use it to extend the previous computations to spinning particles. In section \[sec:discussion\] we discuss the applications of these results as well as possible future directions. Finally, in appendix \[sec:comframe\] we provide a prescription to match our results to the standard form of EFT operators appearing in the effective potential for the cases $S=\frac{1}{2},1$.
Preliminaries {#sec:pre}
=============
Kinematical Considerations and the HCL {#sub:HNRL}
--------------------------------------
In the EFT framework, the off-shell effective potential can be extracted from the S-matrix element associated to the process depicted in Fig. \[fig:intro\], see e.g. [@Neill:2013wsa]. The standard kinematical setup for this computation is given by the Center of Mass (COM) coordinates, which are defined by $\vec{P_{1}}+\vec{P_{3}}=0$. We can check that 4-particle kinematics for this setup imply $$(P_{1}+P_{3})\cdot(P_{1}-P_{2})=0\,,\label{eq:x-1}$$ which means that the momentum transfer vector $K:=(P_{1}-P_{2})$ has the form $$K=(0,\vec{q})\,,\quad t=K^{2}=-\vec{q}^{2}\,,\label{eq:tcom}$$ in the COM frame. For completeness, we also define here the average momentum $\vec{p}$ as $$\frac{P_{1}+P_{2}}{2}=(E_{a},\vec{p})\,,\quad\frac{P_{3}+P_{4}}{2}=(E_{b},-\vec{p})\,,\label{eq:average}$$ where $E_{a},$$E_{b}$ are the respective energies for the COM frame, while $\vec{p}^{2}\propto v^{2}$ gives the characteristic velocity of the problem. From these definitions we can solve for the explicit form of the momenta $P_{i}$, $i\in\{1,2,3,4\}$, and also easily check the transverse condition $\vec{p}\cdot\vec{q}=0$. In the non-relativistic limit $\frac{\sqrt{-t}}{m}=\frac{\text{|\ensuremath{\vec{q}}|}}{m}\rightarrow0$, the center of mass energy $\sqrt{s}$ can be parametrized as a function of $\vec{p}^{\,2}$. In fact, $$\begin{aligned}
s & = & (P_{1}+P_{3})^{2}\,,\nonumber \\
& = & (E_{a}+E_{b})^{2}\nonumber \\
& = & (m_{a}+m_{b})^{2}\left(1+\frac{\vec{p}^{2}}{m_{a}m_{b}}+O(\vec{p}^{\,4})\right)+O(\vec{q}^{\,2})\label{eq:nrls}\end{aligned}$$ Note that the remaining kinematic invariant may be obtained as $u=2(m_{a}^{2}+m_{b}^{2})-t-s\,.$ In practice, we regard the amplitude for Fig. \[fig:intro\] as a function $M(t,s)$, which may contain poles and branch cuts in both variables. At this point we can also introduce the spin vector $S^{\mu}$, which will be in general constructed from polarization tensors associated to the spinning particles, see e.g. [@Vaidya:14]. Suppose for instance that the particle $m_{b}$ carries spin, then the spin vector satisfies the transversal condition
$$S^{\mu}(P_{3}+P_{4})_{\mu}=0\,,$$
implying that in the non-relativistic regime $\vec{p}\rightarrow0$ the 4-vector becomes purely spatial, i.e. $S^{\mu}\rightarrow(0,\vec{S})$.
The PN expansion and the corresponding EM analog then proceed by extracting the classical (i.e. $\hbar$-independent) part of the scattering amplitude $M(t,s)$ expressed in these coordinates. This is done by selecting the lowest order in $|\vec{q}|$ for fixed powers of $G$, spin $|\vec{S}|$ and $\vec{p}^{2}$ [@Neill:2013wsa; @Vaidya:14]. This claim is argued by dimensional analysis, where it is clear that for a given order in $G$ each power of $|\vec{q}|$ carries a power of $\hbar$ unless a spin factor $|\vec{S}|$ is attached [@Holstein:2004dn; @Porto:2005ac; @Neill:2013wsa]. Here $G$ is equivalent to 1PN order and acts as a loop counting parameter, while the latter quantities can be counted as 1PN corrections each [@Vaidya:14]. For a given number of loops and fixed value of $s$, the expansion around $t=-\vec{q}^{2}=0$ used to select the classical pieces coincides with the non-relativistic limit $\frac{\vec{q}}{m}\rightarrow0$. Additionally, in the COM frame the $2^{2n}$-pole and $2^{2n-1}$-pole interactions due to spin emerge in the form [@Porto:2005ac; @Holstein:2008sw; @Vaidya:14] $$V_{S}=c_{1}(|\vec{p}|)S_{1}^{i_{1}\ldots i_{2n}}q_{i_{1}}\ldots q_{i_{2n}}+c_{2}(|\vec{p}|)S_{2}^{i_{1}\ldots i_{2n}}q_{i_{1}}\ldots q_{i_{2n-1}}p_{i_{2n}}=O(|\vec{q}|^{2n-1})\,,\label{eq:multipole}$$ where $S_{j}^{i_{1}\ldots i_{2n}}$, $j=1,2,$ are constructed from polarization tensors of the scattered particles in such a way that the powers of $|\vec{S}|$ exactly match the powers of $|\vec{q}|$ in $V_{S}$. They are, in consequence, classical contributions and correspond to the so-called mass ($j=1$) and current $(j=2$) multipoles [@Vines:2016qwa]. These terms arise in the scattering amplitude when one of the external particles, for instance the one with mass $m_{a}$, carries spin $S_{a}\geq n$. Note that in order to evaluate spin effects a non-relativistic expansion to arbitrary high orders in $|\vec{q}|$ is required. To deal with this difficulty we note that is obtained, through the non-relativistic expansion, from the generic covariant form
$$S^{\mu_{1}\cdots\mu_{m}}K_{\mu_{1}}\cdots K_{\mu_{k}}(P_{a_{k+1}})_{\mu_{k+1}}\cdots(P_{a_{m}})_{\mu_{m}}\,,\quad a_{i}\in\{1,3\}.\label{eq:covmultipole}$$
where $k=2n$ for mass multipoles and $k=2n+1$ for current multipoles. These spin forms are characteristic of the multipole interactions in the sense that they are partly determined by general constraints[^1] and they emerge already in the tree level amplitude, being consistently reproduced at the loop level [@Holstein:08]. We give explicit examples of these for $S=\frac{1}{2},1$ in appendix \[sec:comframe\]. Once the non-relativistic limit is taken by expanding with respect to $\vec{q}$ and $\vec{p}$, these terms lead to the structures present in $V_{S}$, i.e. they capture the complete spin-dependent couplings, together with some higher powers of $|\vec{q}|$ which are quantum in nature. The advantage of writing the multipole terms in the covariant form is that these are completely visible once the limit $t=K^{2}\rightarrow0$ is taken, that is, at leading order in the $t$ expansion. All the neglected pieces, i.e. subleading orders in $t$, which are not captured by these multipole forms simply correspond to quantum corrections. Thus our strategy is to compute the coefficients associated to these EFT operators[^2] in the $t\rightarrow0$ limit. This is done by examining the leading order of an arbitrary linear combination of them and performing the match with the classical piece of the amplitude, obtained by computing the leading singularity [@LS]. The explicit matching procedure is demonstrated in appendix \[sec:comframe\], where we use spinor helicity variables to write the multipole terms. The idea is that at $t=0$ the expression is not well defined but is. This means that we can write our answer for the EFT potential in terms of and then proceed to analytically continue it to the region $t\neq0$, which is easily achieved by putting $K=(0,\vec{q})$ and the corresponding expressions for $P_{i}$. The evaluation of the classical piece near $t=0$ is the holomorphic classical limit (HCL).
A few final remarks regarding the HCL are in order. First, as anticipated the term holomorphic stems from the on-shell condition $P_{i}\cdot K=\pm K^{2}$, $i\in\{1,...,4\}$, which for $t\rightarrow0$ yields $P_{i}\cdot K\rightarrow0$. In turn, this implies that the external momenta $P_{i}$ must be complexified. Hence, in order to reach the $t=0$ configuration we must consider an analytic trajectory in the kinematic space, which we can parametrize in terms of a complex variable $\beta$. We introduce such trajectory explicitly in section \[sub:scalar-Loop-amplitude\], where we also evaluate the amplitude as $\beta\rightarrow1$. Second, we stress that just the HCL is enough to recover the classical potential with arbitrary multipole corrections. The complete non-relativistic limit can be further obtained by expanding around $s\rightarrow(m_{a}+m_{b})^{2}$, i.e. expanding in $\vec{p}^{2}$ for a given power of $|\vec{q}|$. These corrections in $\vec{p}^{\,2}$ account for higher PN corrections when implemented through the Born approximation, which at 1-loop also requires to subtract the iterated tree level potential. We perform the procedure only at the level of the amplitude and refer to [@Feinberg:1988yw; @Holstein:08; @Vaidya:14; @Neill:2013wsa] for details on iterating higher PN corrections. As the expressions we provide for the classical piece correspond to all the orders in $\vec{p}^{\,2}$ encoded in a covariant way, we regard the HCL output as a fully relativistic form of the classical potential. In fact, the construction is covariant since it is based on the null condition for $K$, which will also prove useful when defining the massless limit of external particles for addressing light-like scattering. Finally, the soft behavior of the momentum transfer $K\rightarrow0$, which is the equivalent of $\frac{\vec{q}}{m}\rightarrow0$ for COM coordinates, is not needed and we find that it does not lead to further insights on the behavior of the potential.
Conventions
-----------
Before proceeding to the computation of scattering processes, we set the conventions that will be used extensively throughout the paper. The constructions are based on the acclaimed spinor helicity variables, see e.g. [@Elvang; @AmpforAstro] for a review. Here we just stress some of the notation.
Using a mostly minus $(+,-,-,-)$ signature, a generic 4-momentum $P^{\mu}$, with $P^{2}=m^{2}$, can be written as $$P_{\alpha\dot{\alpha}}=P^{\mu}(\sigma_{\mu})_{\alpha\dot{\alpha}},\quad\bar{P}^{\dot{\alpha}\alpha}=P^{\mu}(\bar{\sigma}_{\mu})^{\dot{\alpha}\alpha}\,,$$ where $\sigma^{\mu}=(\mathbb{I},\sigma^{i})$ and indices are lowered/raised from the left via the $\epsilon$ tensor[^3], for instance $\bar{P}^{\dot{\alpha}\alpha}=\epsilon^{\dot{\alpha}\dot{\beta}}\epsilon^{\alpha\beta}P_{\dot{\beta}\beta}$ or simply $\bar{P}=\epsilon P\epsilon^{T}$. We will also use $P$ to refer both to the 4-vector $P^{\text{\ensuremath{\mu}}}$ and the bispinor $P_{\alpha\dot{\alpha}}$. For instance, $$\begin{aligned}
P_{\alpha\dot{\alpha}}\bar{P}^{\dot{\alpha}\beta}=m^{2}\delta_{\alpha}^{\beta}\,, & \textsc{{\rm or}} & P\bar{P}=m^{2}\mathbb{I}\,,\label{eq:identities}\\
P_{\alpha\dot{\alpha}}\bar{Q}^{\dot{\alpha}\alpha} & ={\rm Tr}(P\bar{Q})= & 2P\cdot Q\,.\nonumber \end{aligned}$$ A massless momentum satisfies $\det(K)=0$ and hence can be written as $$K_{\alpha\dot{\alpha}}=|\lambda]_{\dot{\alpha}}\langle\lambda|_{\alpha}\,,{\rm or}\,\,{\rm simply}\,\,\,K=|\lambda]\langle\lambda|.$$ The conjugates are defined by $[\lambda|=\epsilon|\lambda]$ and $|\lambda\rangle=\langle\lambda|\epsilon^{T}$. With these definitions $\bar{K}=|\lambda\rangle[\lambda|.$ The bilinears $[\lambda\eta]=[\lambda|^{\dot{\alpha}}|\eta]_{\dot{\alpha}}$ and $\langle\lambda\eta\rangle=\langle\lambda|_{\alpha}|\eta\rangle^{\alpha}$ are then naturally defined as the corresponding contractions. From Eq. we have $$[\lambda\eta]\langle\eta\lambda\rangle=2K\cdot R\,,$$ where $R=|\eta]\langle\eta|$. This also motivates the notation
$$[\lambda|P|\lambda\rangle=\langle\lambda|\bar{P}|\lambda]\,,$$
for the contraction $[\lambda|^{\dot{\alpha}}P_{\beta\dot{\alpha}}|\lambda\rangle^{\beta}$. In the following we may omit the spinor indices $(\alpha,\dot{\alpha})$ when possible and deal with $2\times2$ operators. In appendix \[sec:spinrep\] we use these variables to construct the representation for massive states of arbitrary spin, first introduced in [@Nima:16].
Scalar Scattering {#sec:scalar}
=================
In this section we recompute the Leading Singularity for gravitational scattering of both tree and 1-loop level amplitudes for the no spinning case, as first presented in [@LS]. This time we embed the computation into the framework of the HCL, which will lead directly to the classical contribution. We also present, without additional effort, the analogous results for the EM case. Along the way we introduce new variables which will prove helpful for the next sections.
Let us first introduce a dimensionless variable which will be well suited to describe the internal helicity structure of the scattering. Motivated by the $2\rightarrow2$ process described in section \[sub:HNRL\], we start by considering two massive particles interacting with a massless one. If both massive particles have the same mass $m$, the on-shell condition for the process implies $[k|P|k\rangle=0$, where $P$ is one of the (incoming) massive momenta and $K=|k]\langle k|$ corresponds to the momenta of the massless particle. Thus, as proposed in [@Nima:16], it is natural to introduce dimensionless variables $x$ and $\bar{x}$ such that $$\begin{split}[k|P=mx\langle k|\,,\quad\langle k|\bar{P}=m\bar{x}[k|\,.\end{split}
\label{eq:3ptparam}$$ The condition $P\bar{P}=m^{2}$ yields $x\bar{x}=1$. Note that $x$ carries helicity weight $h=+1$ under the little group transformations of $K$. Furthermore, $mx$ precisely corresponds to the stripped 3pt amplitude for the case in which the massive particle is a scalar and the massless particle has $h=1$[^4]. For higher helicity one simply finds ($h>0$) [@Nima:16]
$$A_{scalar}^{(+h)}=\alpha(mx)^{h}\,,\qquad A_{scalar}^{(-h)}=\alpha(m\bar{x})^{h}\,.\label{eq:3ptscalar}$$
The (minimal) coupling constant $\alpha$ has to be chosen according to the theory under consideration, determined once the helicity $|h|$ is given, i.e. $h=\pm1$ for EM and $h=\pm2$ for gravity. Regarding the gravitational interaction, its universal character allows us to fix the coupling by $\alpha=\frac{\kappa}{2}=\sqrt{8\pi G}$ irrespective of the particle type, whereas for EM it will depend on the electric charge carried by such particle.
Tree Amplitude\[sub:scalar-Tree-amplitude\]
-------------------------------------------
Let us start by computing the tree level contributions to the classical potential. As explained in [@LS], these can be directly obtained from the Leading Singularity, which for tree amplitudes is simply the residue at $t=0$. Here, it is transparent that the analytic expansion around such pole will yield subleading terms $t^{n}$, $n\geq0$, which are ultralocal (e.g. quantum) once the Fourier transform is implemented in COM coordinates $t=-\vec{q}^{2}$ [@Holstein:08]. Furthermore, by unitarity this residue precisely corresponds to the product of on-shell 3pt amplitudes (see Fig. \[fig:treeex\]), that is to say, we can use the leading term in the HCL to evaluate the classical potential. Note that, even though there exist different couplings contributing to the s and u channel, these correspond to contact interactions between the different particles and do not lead to a long-range potential [@LS].
With these considerations we proceed to compute the leading contribution to the Coulomb potential by considering the one-photon exchange diagram. Summing over both helicities and using we find
![A one photon/graviton exchange process. In the HCL the internal particle is on-shell and the two polarizations need to be considered.\[fig:treeex\]](tree)
$$M_{(0,0,1)}^{(0)}=\frac{1}{t}\left(A_{3}^{(+1)}(P_{1})A_{3}^{(-1)}(P_{3})+A_{3}^{(-1)}(P_{1})A_{3}^{(+1)}(P_{3})\right)=\alpha^{2}\dfrac{m_{a}m_{b}}{t}\left(x_{1}\bar{x}_{3}+\bar{x}_{1}x_{3}\right)\,.\label{eq:phexchange}$$
Here we have used $M_{(S_{a},S_{b},|h|)}^{(0)}$ to denote the classical piece of the $2\rightarrow2$ amplitude, as opposed to the notation $A_{n}(P_{i})$ which we reserve for the $n$ pt amplitudes used as building blocks. The index $(0)$ indicates leading order (tree level), which will be equivalent to 0PN for the gravitational case. The subindex $(S_{a},S_{b},|h|)=(0,0,1)$ denotes spinless particles exchanging a photon.
The variables $x_{1}$$(\bar{x}_{1})$ and $x_{3}$$(\bar{x}_{3})$ are now associated to $P_{1}$ and $P_{3}$ respectively, through . An explicit form can be obtained in terms of the null momentum transfer $K=P_{4}-P_{3}=|k]\langle k|$, but it is not needed here. At this stage we introduce the kinematic variables
$$u:=m_{a}m_{b}x_{1}\bar{x}_{3}\,,\quad v:=m_{a}m_{b}\bar{x}_{1}x_{3}.\label{eq:uvdef}$$
Note that these variables are defined only in the HCL, i.e. for $t=0$. Each of these carries no helicity, i.e. it is invariant under little group transformations of the internal particle. However, they represent the contribution from the two polarizations in the exchange of Fig. \[fig:treeex\], and as such they are swapped under parity. In appendix \[sec:comframe\] we give explicit expressions for $u$ and $v$ in terms of their parity even and odd parts. Nevertheless, we stress that for this and the remaining sections the only identities which are needed can be stated as
$$uv=m_{a}^{2}m_{b}^{2}\,,\quad u+v=2P_{1}\cdot P_{3}\,,\label{eq:uv-1}$$
and readily follow from their definition and . We then regard the new variables as a (parity sensitive) parametrization of the $s$ channel emerging in the HCL. Further expanding in the non-relativistic limit yields $u,v\rightarrow m_{a}m_{b}$.
With these definitions, we can now proceed to write the result in a parity invariant form as
$$M_{(0,0,1)}^{(0)}=\alpha^{2}\dfrac{u+v}{t}=\alpha^{2}\dfrac{s-m_{a}^{2}-m_{b}^{2}}{t}\,.\label{eq:phexchange-1}$$
After implementing COM coordinates and including the proper relativistic normalization, this leads to the Coulomb potential in Fourier space, which can be expanded in the limit $s\rightarrow(m_{a}+m_{b})^{2}$. In fact, assuming both particles carry the same electric charge $e=\frac{\alpha}{\sqrt{2}}$, we can use , to write $$\frac{M_{(0,0,1)}^{(0)}}{4E_{a}E_{b}}=-\frac{e^{2}}{\vec{q}^{\,2}}\left(1+\frac{\vec{p}^{\,2}}{m_{a}m_{b}}+...\right)\,.$$
We are now in position to easily compute the one-graviton exchange diagram. The answer is again given by the parity invariant expression
$$M_{(0,0,2)}^{(0)}=\alpha^{2}\dfrac{u^{2}+v^{2}}{t}=\frac{\kappa^{2}}{4}\dfrac{(s-m_{a}^{2}-m_{b}^{2})^{2}-2m_{a}^{2}m_{b}^{2}}{t}\,.\label{eq:grexchange}$$
Again, this leads to a relativistic expression for the Newtonian potential, and can be put into the standard form by using the dictionary provided in subsection \[sub:HNRL\] $$\frac{M_{(0,0,2)}^{(0)}}{4E_{a}E_{b}}=4\pi G\frac{m_{a}m_{b}}{\vec{q}^{\,2}}\left(1+\frac{(3m_{a}^{2}+8m_{a}m_{b}+3m_{b}^{2})}{2m_{a}^{2}m_{b}^{2}}\vec{p}^{\,2}+...\right),$$ in agreement with the computations in [@Duff; @BjerrumBohr:2002kt; @Neill:2013wsa; @LS].
Two final remarks are in order. First, it is interesting that the gravitational result can be directly obtained by squaring the $u,v$ variables, i.e. squaring both contributions from the EM case. This will be a general property that we will encounter again for the discussion of the Compton amplitude in the next section, as was already pointed out in [@Bjerrum-Bohr:2013] in relation with the double-copy construction. Second, it is worth noting that up to this point no parametrization of the external momenta was needed. In other words, the tree level computation was done solely in terms of (pseudo)scalar variables. As we will see now, the 1-loop case can be addressed with the help of an external parametrization specifically designed for the HCL. This parametrization will provide an extension of the variables $u$ and $v$ in a sense that will become clear.
1-Loop Amplitude: Triangle Leading Singularity\[sub:scalar-Loop-amplitude\]
---------------------------------------------------------------------------
Here we proceed to compute the triangle LS [@LS] in order to obtain the first classical correction to the potential. This leading singularity is associated to the 1-loop diagram arising from two photons/gravitons exchange, Fig. \[fig:loopex\]. As explained in the previous work, the LS of the triangle diagram captures the second discontinuity of the amplitude as a function of $t$, which is precisely associated to the non-analytic behavior $\frac{1}{\sqrt{-t}}=\frac{1}{|\vec{q}|}$. In the gravitational case this accounts for $G^{2}$ corrections or equivalently 1PN. In order to track exclusively this contribution we proceed to discard higher (analytic and non-analytic) powers of $t$ by appealing to the HCL. This can be implemented to any order in $t$ by means of the following parametrization of the external kinematics
![The triangle diagram used to compute the leading singularity, corresponding to the $b$- topology. The $a$-topology is obtained by reflection, i.e. by appropriately exchanging the external particles. \[fig:loopex\]](triangle)
$$\begin{split}P_{3} & =|\eta]\langle\lambda|+|\lambda]\langle\eta|\,,\\
P_{4} & =\beta|\eta]\langle\lambda|+\frac{1}{\beta}|\lambda]\langle\eta|+|\lambda]\langle\lambda|\,,\\
\frac{t}{m_{b}^{2}} & =\frac{(\beta-1)^{2}}{\beta}\,,\\
\langle\lambda\eta\rangle & =[\lambda\eta]=m_{b}\,.
\end{split}
\label{eq:param}$$
The parametrization is constructed by first defining a complex null vector $K=|\lambda]\langle\lambda|$ orthogonal to $P_{3}$ and $P_{4}$. Then the bispinors $(P_{3})_{\alpha\dot{\alpha}}$ and $(P_{4})_{\alpha\dot{\alpha}}$ are expanded in a suitably constructed basis, which also provides the definition of $|\eta]_{\dot{\alpha}}$ and $\langle\eta|_{\alpha}$ up to a scale which is fixed by the fourth condition. As explained in appendix \[sec:spinrep\] (following the lines of [@Nima:16]) this basis also provides a representation for the little group associated to massive states. The dimensionless parameter $\beta$ was called $x$ in [@LS] and was introduced as a natural description of the t channel. In this sense, this parametrization should be regarded as an extension of the one presented there, which can be recovered for $\beta^{2}\neq1$ by means of the shift $$|\eta]\rightarrow|\eta]+\frac{\beta}{1-\beta^{2}}|\lambda]\,,\quad\langle\eta|\rightarrow\langle\eta|-\frac{\beta}{1-\beta^{2}}\langle\lambda|\,.$$ However, in this case we are precisely interested in the degenerate point $\beta=1$, i.e. $t=0$, in order to define the HCL. For this point we have $P_{4}-P_{3}=K=|\lambda]\langle\lambda|$ as the null momentum transfer. As opposed to the tree level case, such momentum is not associated to any particle in the exchange of Fig. , but distributed between the internal photons/gravitons. In general for $\beta\neq1$, $K$ is just an auxiliary vector and thus we need not to consider little group transformations for $|\lambda],\langle\lambda|$, i.e. these are fixed spinors. Finally, we also provide a parametrization for the $s$ channel by extending the definitions for $t\neq0$
$$\begin{split}u & =[\lambda|P_{1}|\eta\rangle\,,\quad v=[\eta|P_{1}|\lambda\rangle\,,\end{split}
\label{eq:uv}$$
such that $u+v=2P_{1}\cdot P_{3}$ and $uv\rightarrow m_{a}^{2}m_{b}^{2}\quad\text{as}\quad\beta\rightarrow1.$
We are now well equipped to evaluate the triangle Leading Singularity. Here we sketch the computation of the contour integral and refer to [@LS] for further details. It is given by
$$\begin{split}M_{(0,0,|h|)}^{(1,b)} & =\frac{1}{4}\sum_{h_{3},h_{4}=\pm|h|}\int_{\Gamma_{{\rm LS}}}d^{4}L\,\delta(L^{2}-m_{b}^{2})\,\delta(k_{3}^{2})\,\delta(k_{4}^{2})\\
& \times A_{4}(P_{1},-P_{2},k_{3}^{h_{3}},k_{4}^{h_{4}})\times A_{3}(P_{3},-L,-k_{3}^{-h_{3}})\times A_{3}(-P_{4},L,-k_{4}^{-h_{4}})\,,
\end{split}
\label{eq:int}$$
where the superscript $(1,b)$ denotes the (1-loop) triangle $b$-topology depicted in Fig. \[fig:loopex\]. The $a$-topology is simply obtained by exchanging particles $m_{a}$ and $m_{b}$: We leave the explicit procedure for the appendix and in the following we deal only with $M_{(0,0,|h|)}^{(1,b)}$. In we denote by $A_{3}$ and $A_{4}$ to the respective tree level amplitudes entering the diagram (note the minus sign for *outgoing* momenta), and $$\begin{split}k_{3} & =-L+P_{3}\,,\quad\end{split}
k_{4}=L-P_{4}\,.\label{eq:k3k4-1}$$ The sum is performed over propagating internal states and enforces matching polarizations between the 3pt and 4pt amplitudes. $\Gamma_{LS}$ is a complex contour defined to enclose the emerging pole in . This pole will be explicit after a parametrization for the loop momenta $L$ is implemented and the triple-cut corresponding to the three delta functions is performed. This will leave only a 1-dimensional contour integral for a suitably defined $z\in\mathbb{C}$, where $L=L(z)$. We now use the previously defined basis of spinors to parametrize $$\begin{split}L(z) & =z\ell+\omega K\,,\\
\ell & =A|\eta]\langle\lambda|+B|\lambda]\langle\eta|+AB|\lambda]\langle\lambda|+|\eta]\langle\eta|\,,
\end{split}
\label{eq:L}$$ where one scale in $\ell$ has been absorbed into $z$ and we have further imposed the condition $\ell^{2}=0$. Using Eqs. , we find that implementing the triple-cut in fixes $\omega(z)=-\frac{1}{z}$, while $A(z),B(z)$ become simple rational functions of $z$ and $\beta$. The integral then takes the form $$\begin{split}M_{(0,0,|h|)}^{(1,b)}=\sum_{h_{3},h_{4}}\frac{\beta}{16(\beta^{2}-1)m_{b}^{2}} & \int_{\Gamma_{{\rm LS}}}\frac{dy}{y}A_{4}(P_{1},-P_{2},k_{3}^{h_{3}}(y),k_{4}^{h_{4}}(y))\,\\
& \times A_{3}(P_{3},-L(y),-k_{3}^{-h_{3}}(y))\times A_{3}(-P_{4},L(y),-k_{4}^{-h_{4}}(y))\,,
\end{split}
\label{eq:int2}$$ where $y:=-\frac{z}{(\beta-1)^{2}}$ and we now define the contour to enclose the emergent pole at $y=\infty$, i.e. $\Gamma_{{\rm LS}}=S_{\infty}^{1}$[^5] The internal massless momenta are given by
$$\begin{split}k_{3}(y) & =\underbrace{\frac{1}{\beta+1}\left(|\eta](\beta^{2}-1)y+|\lambda](1+\beta y)\right)}_{|k_{3}]}\,\underbrace{\frac{1}{\beta+1}\left(\langle\eta|(\beta^{2}-1)-\frac{1}{y}\langle\lambda|(1+\beta y)\right)}_{\langle k_{3}|}\,,\\
k_{4}(y) & =\underbrace{\frac{1}{\beta+1}\left(-\beta|\eta](\beta^{2}-1)y+|\lambda](1-\beta^{2}y)\right)}_{|k_{4}]}\,\underbrace{\frac{1}{\beta+1}\left(\frac{1}{\beta}\langle\eta|(\beta^{2}-1)+\frac{1}{y}\langle\lambda|(1-y)\right)}_{\langle k_{4}|}\,.
\end{split}
\label{eq:k3k4}$$
As $\frac{\beta}{\beta^{2}-1}\rightarrow\frac{m_{b}}{2\sqrt{-t}}$ for the HCL, we find that the expression already contains the required classical correction when the leading term of the integrand, around $\beta=1$, is extracted. We can straightforwardly evaluate the 3pt amplitudes at $\beta=1$, giving finite contributions. This simplification will indeed prove extremely useful for the $S>0$ cases in section \[sec:spin\]. On the other hand, for the 4pt amplitude the limit $\beta\rightarrow1$ is needed to obtain a finite answer, since it contains a pole in the t channel.
Explicitly, at $\beta=1$ the internal momenta are given by $$\begin{split}k_{3}^{0}(y) & =\underbrace{\frac{1}{2}|\lambda](1+y)}_{|k_{3}]}\,\underbrace{\frac{-1}{2y}\langle\lambda|(1+y)}_{\langle k_{3}|}\,,\\
k_{4}^{0}(y) & =\underbrace{\frac{1}{2}|\lambda](1-y)}_{|k_{4}]}\,\underbrace{\frac{1}{2y}\langle\lambda|(1-y)}_{\langle k_{4}|}\,.
\end{split}
\label{eq:k3k4NRL}$$ We thus note that in the HCL both internal momenta are collinear and aligned with the momentum transfer $K$. For the standard non-relativistic limit defined in the COM frame the condition $\beta\rightarrow1$ certainly implies the soft limit $K\rightarrow0$ and in general leads to vanishing momenta for the gravitons and vanishing 3pt amplitudes at $\beta=1$.
Now, using the expression for the momenta $P_{3}$ and (outgoing) $P_{4}$, we readily find $$\begin{split}x_{3} & =x_{4}=-y\,,\end{split}
\label{eq:x3x4}$$ such that the 3pt amplitudes are given (at $\beta=1$) by
$$\begin{split}A_{3}(P_{3},-L(y),-k_{3}^{+|h|}(y))A_{3}(-P_{4},L(y),-k_{4}^{-|h|}(y))\Big|_{\beta=1} & =\alpha^{2}m_{b}^{2}\,\\
A_{3}(P_{3},-L(y),-k_{3}^{+|h|}(y))A_{3}(-P_{4},L(y),-k_{4}^{+|h|}(y))\Big|_{\beta=1} & =\alpha^{2}m_{b}^{2}(y^{2})^{|h|}\,.
\end{split}
\label{eq:3ptNRL}$$
We note that for $h_{3}=-h_{4}$ the contribution from the 3pt amplitudes is invariant under conjugation. In fact, as can be already checked from the conjugation is induced by $y\rightarrow-y$. Even though the full contribution from the triangle leading singularity requires to sum over internal helicities, in the HCL $\beta\rightarrow1$ the conjugate configuration $h_{3}=-h_{4}=-|h|$ yields the same residue, while the configurations $h_{3}=h_{4}$ yield none as we explain below. This means that the full result can be obtained by evaluating the case $h_{3}=-h_{4}=+|h|$ and inserting a factor of $2$. Returning to the computation, now reads
$$M_{(0,0,|h|)}^{(1,b)}=\frac{\alpha^{2}}{16}\left(\frac{m_{b}}{\sqrt{-t}}\right)\int_{\infty}\frac{dy}{y}A_{(4,|h|)}^{(-+)}(\beta\rightarrow1)\,,\label{eq:int3}$$
where $A_{(4,|h|)}^{(-+)}(\beta\rightarrow1)$ is the leading order of the 4pt. Compton-like amplitude, given by
$$A_{(4,|h|)}^{(-+)}=\alpha^{2}\begin{cases}
\dfrac{\langle k_{3}|P_{1}|k_{4}]^{2}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=1\\
\\
\frac{1}{t}\times\dfrac{\langle k_{3}|P_{1}|k_{4}]^{4}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=2
\end{cases}\label{eq:A4i}$$
We note that the stripped Compton amplitudes exhibit the double-copy factorization $A_{(4,2)}=4\frac{(k_{3}\cdot P_{1})(k_{3}\cdot P_{2})}{t}(A_{(4,1)})^{2}$ as explained in [@Bjerrum-Bohr:2013]. We will come back at this point in section \[sec:spin\]. By considering the definitions , and using together with momentum conservation constraints, we find the HCL expansions $$\begin{split}\langle k_{3}|P_{1}|k_{4}] & =(\beta-1)\left(u\frac{1-y}{2}+v\frac{1+y}{2}+\frac{(v-u)(1-y^{2})}{4y}\right)+O(\beta-1)^{2}\,,\\
\langle k_{3}|P_{1}|k_{3}] & =\langle k_{3}|P_{2}|k_{3}]+O(\beta-1)^{2}=(\beta-1)\frac{(v-u)(1-y^{2})}{4y}+O(\beta-1)^{2}\,.
\end{split}
\label{eq:expansion}$$ where it is understood that $u,v$ are evaluated at $\beta=1$. We note that the conjugation $y\rightarrow-y$ is equivalent to change $u\leftrightarrow v$, as expected. Also, we can now argue why the Compton amplitude gives a finite answer in the limit $\beta\rightarrow1$. Consider for instance the gravitational case. By unitarity, such limit induces a t channel factorization into a 3-graviton amplitude and a scalar-scalar-graviton amplitude $A_{3}$. Because of the collinear configuration at $\beta=1$, the 3-graviton amplitude vanishes at the same rate as the t channel propagator $\sim(\beta-1)^{2}$, yielding a finite result. Note that, for this factorization, regular terms in $t$ will contribute to the result and hence these 3pt factors are not enough to compute the HCL answer.
At this stage we exhibit for completeness the expressions for the Compton amplitude in the case of same helicities. It is given by
$$A_{(++)}^{(4,|h|)}=\alpha^{2}\begin{cases}
\dfrac{[k_{3}k_{4}]^{2}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=1\\
\\
\frac{1}{t}\times\dfrac{[k_{3}k_{4}]^{4}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=2
\end{cases}\label{eq:A4i-1}$$
By expanding $[k_{3}k_{4}]$ in an analogous form to and, together with , inserting it back into we easily find that this gives indeed vanishing residue. In fact, this can also be checked to any order in $(\beta-1),$ i.e. with no expansion at all [@LS]. As anticipated, the configurations $h_{3}=h_{4}$ simply do not lead to a classical potential.
Finally, by inserting into we find that the residue is trivial ($\text{Res}_{\infty}=1$) for $|h|=1$, while for $|h|=2$ we have
$$M_{(0,0,2)}^{(1,b)}=\frac{3\alpha^{4}m_{b}}{2^{7}\sqrt{-t}}(5u^{2}+6uv+5v^{2})\,.\label{eq:int4}$$
The expression is indeed symmetric in $u,v$, as expected by parity invariance. By using we can write in an analogous form to its tree level counterpart
$$M_{(0,0,2)}^{(1,b)}=G^{2}\pi^{2}\frac{3m_{b}}{2\sqrt{-t}}\left(5(s-m_{a}^{2}-m_{b}^{2})^{2}-4m_{a}^{2}m_{b}^{2}\right)\,.\label{eq:int4-2}$$
The contribution $M_{(0,0,2)}^{(1,a)}$ is obtained by exchanging $m_{a}\leftrightarrow m_{b}$. After implementing the Born approximation as explained in [@Feinberg:1988yw; @Holstein:08], this indeed recovers the 1PN form of the effective potential including the corrections in $\vec{p}^{\,2}$ [@BjerrumBohr:2002kt; @Holstein:08; @LS; @Neill:2013wsa; @Vaidya:14].
Massless Probe Particle\[sub:Massless-probe-particle\]
------------------------------------------------------
Here we show that the massless case $m_{a}=0$ can be regarded as a smooth limit defined in the variables $u,v$. In this case such limit is natural to define since both massless and massive scalar fields contain the same number of degrees of freedom. In appendix we show, however, how to extend this construction to representations with nonzero spin. In the following we focus for simplicity in the gravitational case, the electromagnetic analog being straightforward. Moreover, the gravitational case is motivated by the study of light bending phenomena within the framework of EFT, see [@Bjerrum-Bohr:2014zsa; @AmpforAstro].
In order to discuss the massless limit, it is convenient to absorb the mass into the definition of $x,\bar{x}$ given in , i.e. these quantities now carry units of energy. Then, the massless condition $P_{3}\bar{P}_{3}=0$ is equivalent to $x_{3}\bar{x}_{3}=0$, thus one of the helicity configurations in must vanish at $\beta=1$. This choice corresponds to selecting one of the graviton polarizations to give vanishing contribution, that is either $u=0$ or $v=0$. Due to parity invariance the election is not relevant, hence we put $v=0$ and find from
$$u=s-m_{b}^{2}\,,\label{eq:masslessu}$$
which in turn yields
$$\begin{split}M_{m_{a}=0}^{(0)} & =\alpha^{2}\dfrac{u^{2}}{t}\\
& =\alpha^{2}\dfrac{(s-m_{b}^{2})^{2}}{t}\,.
\end{split}
\label{eq:masslessgrex}$$
Analogously, for the 1-loop correction we find
$$\begin{split}M_{m_{a}=0}^{(1,b)} & =\frac{3\alpha^{4}m_{b}(5u^{2})}{2^{7}\sqrt{-t}}\,\\
& =\frac{15\alpha^{4}}{2^{7}}\times\frac{m_{b}(s-m_{b}^{2})^{2}}{\sqrt{-t}}\,.
\end{split}
\label{eq:masslessloop}$$
After including the normalization factor $(4E_{a}E_{b})^{-1}\approx(4E_{a}m_{b})^{-1}$ we find that this recovers the 1PN correction of the effective potential for a massless probe particle [@Holstein:2016cfx; @Bjerrum-Bohr:2014zsa]. It is important to note that in this result only the $b-$topology LS contributes, i.e. no symmetrization is needed. This is because the triangle LS scales with the mass, i.e. for the $a-$topology is proportional to $\frac{m_{a}}{\sqrt{-t}}$ and thus vanishes in this case. In fact, classical effects require at least one massive propagator entering the loop diagram [@Holstein:2004dn], see also discussion. We will again resort to this fact in section \[sub:lightbend\], where we construct the massless limit for spinning particles.
HCL for Spinning Particles {#sec:spin}
==========================
In this section we proceed to consider the case of particles with nonzero spin. That is, we extend the computation of the triangle leading singularity presented in section \[sec:scalar\] but this time for external particles with masses $m_{a}$, $m_{b}$ and spins $S_{a}$, $S_{b}$ respectively. By using the Born approximation, the LS leads to the 1-loop effective potential arising in gravitational or electromagnetic scattering of spinning objects, already computed in [@Holstein:08] for $S_{a},S_{b}\in\{0,\frac{1}{2},1\}$. Here we provide an explicit expression for the tree level LS and a contour integral representation for the 1-loop correction, both valid for any spin. We explicitly expand the contour integral for $S_{a}\leq1$, $S_{b}$ arbitrary. In appendix \[sec:comframe\] we explain how to recover the results of [@Holstein:08] by projecting our corresponding expression in the HCL to the standard EFT operators.
We start by explaining a novel spinor helicity representation for the little group of a massive particle of spin $S$, first introduced by Arkani-Hamed et al. [@Nima:16]. The space is spanned by $2S+1$ polarization states, corresponding to the spin $S$ representation of $SU(2)$. Following the lines of section \[sec:scalar\] we will focus on the 3pt. amplitudes $A_{3}(P_{3},P_{4},K)$ as operators acting on in this space, which will then serve as building blocks for the leading singularities. In our case, it will be natural to take advantage of the parametrization of the previous section, $$\begin{split}P_{3} & =|\eta]\langle\lambda|+|\lambda]\langle\eta|\,,\\
P_{4} & =\beta|\eta]\langle\lambda|+\frac{1}{\beta}|\lambda]\langle\eta|+|\lambda]\langle\lambda|\,,
\end{split}
\label{eq:param-1}$$ to construct the little group representation for momenta $P_{3}$ and $P_{4}$ (carrying the same spin $S$) in a simultaneous fashion. We will denote the respective $2S+1$ dimensional Hilbert spaces by $V_{3}^{S}$ and $\bar{V}_{4}^{S}$. In appendix \[sec:spinrep\] we explicitly construct $V_{3}^{\frac{1}{2}}$ and $\bar{V}_{4}^{\frac{1}{2}}$ starting from the well known Dirac spinor representation. For general spin, a basis for these spaces is given by the $2S$-th rank tensors [^6]
$$\begin{split}|m\rangle & =\frac{1}{[\lambda\eta]^{S}}\underbrace{|\lambda]\odot\ldots\odot|\lambda]}_{m}\odot\underbrace{|\eta]\odot\ldots\odot|\eta]}_{2S-m}\,\in V_{3}^{S}\,,\\
\langle n| & =\frac{1}{[\lambda\eta]^{S}}\underbrace{[\lambda|\odot\ldots\odot[\lambda|}_{n}\odot\underbrace{[\eta|\odot\ldots\odot[\eta|}_{2S-n}\,\in\bar{V}_{4}^{S}\,,
\end{split}
\label{eq:reps}$$
with $m,n=0,\ldots,2S$. Here the symbol $\odot$ denotes the symmetrized tensor product. The normalization is chosen for latter convenience, i.e. $$\begin{aligned}
\eta_{\dot{\alpha}}\odot\lambda_{\dot{\beta}} & = & \frac{\eta_{\dot{\alpha}}\lambda_{\dot{\beta}}+\eta_{\dot{\beta}}\lambda_{\dot{\alpha}}}{\sqrt{2}}\,,\label{eq:normalization}\\
\eta_{\dot{\alpha}}\odot\lambda_{\dot{\beta}}\odot\lambda_{\dot{\gamma}} & = & \frac{\eta_{\dot{\alpha}}\lambda_{\dot{\beta}}\lambda_{\dot{\gamma}}+\eta_{\dot{\beta}}\lambda_{\dot{\alpha}}\lambda_{\dot{\gamma}}+\eta_{\dot{\gamma}}\lambda_{\dot{\alpha}}\lambda_{\dot{\beta}}}{\sqrt{3}}\,,\nonumber \end{aligned}$$ etc. As we explicitly show below, in this framework we regard the 3pt amplitudes as operators $A_{S}:\,\bar{V}_{4}^{S}\otimes V_{3}^{S}\rightarrow\mathbb{\mathbb{C}}$, that is, they are to be contracted with the states given in for both particles. The representation is symmetric and anti-chiral in the sense that it is spanned by symmetrizations of the anti-chiral spinors $|\lambda]$, $|\eta]$. Further details on the choice of basis and the chirality are given in appendix \[sec:spinrep\] (see also [@Nima:16]).
Consider then the 3pt amplitudes for two particles of momenta $P_{\text{3}}$, $P_{4}$ and spin $S$ interacting with a massless particle of momenta $K=P_{4}-P_{3}$ and helicity $\pm h$. From we see that the on-shell condition $K^{2}=0$ sets $\beta=1$, i.e. $K=|\lambda]\langle\lambda|$. For the massless particle, we choose the standard representation in terms of the spinors $\langle k|=\frac{\langle\lambda|}{\sqrt{x}}$ and $[k|=\sqrt{x}[\lambda|$, where $x$ carries helicity weight $+1$ and agrees with the definition for our parametrization. Note that $[\lambda|$ and $\langle\lambda|$ remain fixed under little group transformations. With these conventions the minimally coupled 3pt amplitudes are given by the operators
$$\begin{split}A_{S}^{(+h)} & =\alpha(mx)^{h}\left(1-\frac{|\lambda][\lambda|}{m}\right)^{\otimes2S}=\alpha(mx)^{h}\left(1-\frac{|\lambda][\lambda|}{m}\right)\otimes\ldots\otimes\left(1-\frac{|\lambda][\lambda|}{m}\right)\,,\\
A_{S}^{(-h)} & =\alpha(m\bar{x})^{h}=\text{\ensuremath{\alpha}}\left(\frac{m}{x}\right)^{h}\,.
\end{split}
\label{eq:3pts}$$
These expressions represent extensions of the ones given in . Note that we have omitted trivial tensor structures (i.e. the identity operator) in . For example, in the second line the explicit index structure is $$\left(A_{S}^{(-h)}\right)_{\dot{\beta}_{1}\ldots\dot{\beta}_{2S}}^{\dot{\alpha}_{1}\ldots\dot{\alpha}_{2S}}=\text{\ensuremath{\alpha}}\left(\frac{m}{x}\right)^{h}\left(\mathbb{I}^{\otimes2S}\right)_{\dot{\beta}_{1}\ldots\dot{\beta}_{2S}}^{\dot{\alpha}_{1}\ldots\dot{\alpha}_{2S}}=\text{\ensuremath{\alpha}}\left(\frac{m}{x}\right)^{h}\delta_{\dot{\beta}_{1}}^{\dot{\alpha}_{1}}\ldots\delta_{\dot{\beta}_{2S}}^{\dot{\alpha}_{2S}}\,.$$ The value for the amplitude is now obtained as the matrix element $\langle n|A_{S}^{(\pm h)}|m\rangle$. This contraction is naturally induced by the bilinear product $[\,,\,]$ of spinors. For instance, consider the matrix element associated to the transition of particle of momenta $P_{3}$ and polarization $|m\rangle$ to momenta $P_{4}$ and polarization $|n\rangle$, while absorbing a graviton: $$A^{m+(-h)\rightarrow n}=\langle n|A_{S}^{(-h)}|m\rangle=\alpha\left(\frac{m_{b}}{x}\right)^{h}\langle n|m\rangle\,,$$ where the contraction
$$\langle n|m\rangle=(-1)^{m}\delta_{m+n,2S}\,\label{eq:innerp}$$
is induced by . The relation of this contraction with the inner product, and the corresponding normalizations, are discussed in appendix \[sec:spinrep\]. We note further that for helicity $-h$ the only non trivial amplitudes are of the form $\langle n|A_{S}^{(-h)}|2S-n\rangle$ and correspond to the scalar amplitude. This is a consequence of choosing the anti-chiral basis. For $+h$ helicity this is not the case, but the fact that $A_{S}^{(+h)}$ is to be contracted with totally symmetric states of $V_{3}^{S}$ and $\bar{V}_{4}^{S}$ allows us to commute any two factors in the tensor product of . That is, we can expand without ambiguity
$$\begin{aligned}
A_{S}^{(+h)} & = & \alpha(mx)^{h}\left(1-\frac{|\lambda][\lambda|}{m}\right)^{\otimes2S}\nonumber \\
& = & \alpha(mx)^{h}\left(1-2S\frac{|\lambda][\lambda|}{m}+\binom{2S}{2}\frac{|\lambda][\lambda|\otimes|\lambda][\lambda|}{m^{2}}+\ldots\right)\,,\label{eq:spinexp}\end{aligned}$$
where we again omitted the trivial operators in the tensor product. As we explain in appendix \[sec:spinrep\], $|\lambda][\lambda|$ is proportional to the spin vector, hence we call it *spin operator* hereafter (see also [@Nima:16]). Here we can see that in general the contraction $\langle0|A_{S}|2S\rangle$ projects out the spin operator, again recovering the scalar amplitude.
Tree Amplitudes
---------------
We follow the lines of section \[sec:scalar\] and evaluate the $2\rightarrow2$ t channel residue. This time we assign spins $S_{a}$, $S_{b}$ to the particles of mass $m_{a}$, $m_{b}$, respectively. However, in order to construct the corresponding $SU(2)$ representation for the momenta $P_{1},P_{2}$, we need to repeat the parametrization for $P_{3}$ and $P_{4}$ given in . This time we have
$$\begin{split}P_{1} & =|\hat{\eta}]\langle\hat{\lambda}|+|\hat{\lambda}]\langle\hat{\eta}|\,,\\
P_{2} & =\beta|\hat{\eta}]\langle\hat{\lambda}|+\frac{1}{\beta}|\hat{\lambda}]\langle\hat{\eta}|+|\hat{\lambda}]\langle\hat{\lambda}|\,,
\end{split}
\label{eq:param2}$$
together with the normalization $[\hat{\lambda}\hat{\eta}]=m_{a}$. Both parametrizations can be matched in the HCL, effectively reducing the apparent degrees of freedom. In fact, $\beta\rightarrow1$ yields $|\lambda]\langle\lambda|\rightarrow-|\hat{\lambda}]\langle\hat{\lambda}|$. Recall that at $\beta=1$ the tree level process of Fig. $\ref{fig:treeex}$ consists of a photon/graviton exchange, with corresponding momentum $K=|\lambda]\langle\lambda|$. For this internal particle we choose the spinors
$$|K]=|\hat{\lambda}]=\frac{|\lambda]}{\gamma}\,,\,|K\rangle=|\hat{\lambda}\rangle=-\gamma|\lambda\rangle\,,\label{eq:compat}$$
for some $\gamma\in\mathbb{C}$. By using the definitions $\eqref{eq:3ptparam}$ for both $P_{1}$ and $P_{3}$ we find $x_{1}=1\,,\quad\bar{x}_{3}=-\gamma^{2}\,$, Using $\eqref{eq:uvdef}$ we can then solve for $\gamma$, completely determining $|\hat{\lambda}]$ and $\langle\hat{\lambda}|$: $$\gamma^{2}=-\frac{u}{m_{a}m_{b}}=-\frac{m_{a}m_{b}}{v}.$$
After this detour, we are ready to compute the tree level residue. The $2\rightarrow2$ amplitude is here regarded as the operator $M_{(S_{a},S_{b},|h|)}^{(0)}:V_{1}^{S_{a}}\otimes\bar{V}_{2}^{S_{a}}\otimes V_{3}^{S_{b}}\otimes\bar{V}_{4}^{S_{b}}\rightarrow\mathbb{C}$, where $V_{1}^{S_{a}},\bar{V}_{2}^{S_{a}}$ are constructed in analogous manner to . Using the expansion we find our first main result $$\begin{split}M_{(S_{a},S_{b},|h|)}^{(0)} & =\alpha^{2}\dfrac{(m_{a}m_{b})^{h}}{t}\left((x_{1}\bar{x}_{3})^{h}\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)^{2S_{a}}+(\bar{x}_{1}x_{3})^{h}\left(1-\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\right)\,\\
& =\dfrac{\alpha^{2}}{t}\left(u^{h}\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)^{2S_{a}}+v^{h}\left(1-\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\right)\\
& =\dfrac{\alpha^{2}}{t}\left(u^{h}-2u^{h}S_{a}\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\otimes\mathbb{I}_{b}+S_{a}(2S_{a}-1)\frac{|\tilde{\lambda}]|\hat{\lambda}][\hat{\lambda}|[\tilde{\lambda|}}{m_{a}^{2}}\otimes\mathbb{I}_{b}\right.\\
& \quad\left.+v^{h}-2v^{h}S_{b}\,\mathbb{I}_{a}\otimes\frac{|\lambda][\lambda|}{m_{b}}+\ldots\right)\,,
\end{split}
\label{eq:spinexchange}$$ where $h=1$ for Electromagnetism and $h=2$ for Gravity. In the third and fourth line we exhibited explicitly the identity operators for both representations to emphasize that the spin operators act on different spaces and hence cannot be summed. In appendix \[sec:spinrep\] it is argued, by examining the $S=\frac{1}{2}$ and $S=1$ case, that the binomial expansion is in direct correspondence with the expansion in multipoles moments and hence to the PN expansion for the gravitational case. That is to say we can match the operators $|\hat{\lambda}][\hat{\lambda}|^{\otimes2n}$, $|\hat{\lambda}][\hat{\lambda}|^{\otimes2n-1}$ to the spin operators in the HCL and compute the respective coefficients in the EFT expression, as we demonstrate in appendix \[sec:comframe\] for the cases in the literature, i.e. $S\leq1$. Note further that we can easily identify universal multipole interactions as predicted by [@Holstein:08; @Bjerrum-Bohr:2013] for the minimal coupling, the leading one corresponding to scalar (orbital) interaction. Here we emphasize again that all these multipole interactions can be easily seen at $\beta=1$, in contrast with the COM frame limit.
Finally, note that the parametrization that we introduced did not seem relevant in order to obtain . However, it is indeed implicit in the choice of basis of states needed to project the operator $M_{(S_{a},S_{b},h)}^{(0)}$ into a particular matrix element. Next we compute the 1-loop correction for this process, which requires extensive use of the parametrization.
1-Loop Amplitude {#sub:spin1loop}
----------------
We now compute the triangle LS for the case in which the external particles carry spin. We explicitly expand the contour integral in the HCL for the case $S_{a}\leq1$ and $S_{b}$ arbitrary. The limitation for $S_{a}$ simply comes from the fact that for $S_{a}\leq1$ the four point Compton amplitude has a well known compact form [@Bjerrum-Bohr:2013] both for EM and gravity. Let us remark that the expression for higher spins is also known in terms of the new spinor helicity formulation [@Nima:16], but we will leave the explicit treatment for future work. Additionally, the case $S_{a}\leq1$ is enough to recover all the 1-loop results for the scattering amplitude in the literature [@Holstein:2008sw; @Holstein:08], and suffices here to demonstrate the effectiveness of the method (see appendix \[sec:comframe\]). Note that the final result is obtained by considering the two triangle topologies for the leading singularity, which can be obtained by symmetrization as we explain below.
In the following we regard the 3pt and 4pt amplitudes entering the integrand as $2\times2$ operators equipped with the natural multiplication. Analogous to the scalar case, only the opposite helicities contribute to the residue and both configurations give the same contribution, hence we focus only on $(+-)$. Furthermore, the 3pt amplitudes can also be readily obtained at $\beta=1$, by using into . They give
$$\begin{split}A_{3}(P_{3},-L(y),k_{3}^{+i}(y))A_{3}(-P_{4},L(y),k_{4}^{-i}(y))\Big|_{\beta=1} & =\alpha^{2}m_{b}^{2}\left(1-\frac{|k_{3}][k_{3}|}{ym_{b}}\right)^{2S_{b}}\,,\\
& =\alpha^{2}m_{b}^{2}\left(1-\frac{(1+y)^{2}}{4y}\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\,.
\end{split}
\label{eq:3ptNRLspin}$$
This time note that the $y$ variable carries helicity weight $+1$, as can be seen from plugging $k_{3}$ and $P_{3}$ in . This means that we needed to restore the helicity factor $y$ in the first line in order to account for little group transformations of $k_{3}$. As in the tree level case, eq. corresponds to an expansion in terms of spin structures that survive the limit $\beta=1$.
We now proceed to compute the 4pt Compton amplitude in the limit $\beta\rightarrow1$. For this, consider
$$A_{(4,|h|)}^{(S_{a})}=\alpha^{2}\begin{cases}
\Gamma^{\otimes2S_{a}}\dfrac{\langle k_{3}|P_{1}|k_{4}]^{2-2S_{a}}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=1,\,\\
\\
\Gamma^{\otimes2S_{a}}\frac{1}{t}\times\dfrac{\langle k_{3}|P_{1}|k_{4}]^{4-2S_{a}}}{\langle k_{3}|P_{1}|k_{3}]\langle k_{3}|P_{2}|k_{3}]} & |h|=2,
\end{cases}\label{eq:A4spin}$$
for $S_{a}\in\{0,\frac{1}{2},1\}$. Here we have defined the $2\times2$ matrix [@Nima:16]
$$\Gamma=|k_{4}]\langle k_{3}|P_{1}+P_{2}|k_{3}\rangle[k_{4}|\,.\label{eq:Gamma}$$
As anticipated, the 4pt. amplitude takes a compact form for $S_{a}\leq1$, and exhibits remarkable factorizations relating EM and gravity [@Bjerrum-Bohr:2013]. Furthermore, we have already computed the expansions , hence we only need to compute the leading term in $\Gamma$! Using the parametrizations , , together with , we find
$$\begin{split}\Gamma & =(\beta-1)\left(\hat{u}\frac{(1-y)}{2}+v\frac{(1+y)}{2}+(v-\hat{u})\frac{1-y^{2}}{4y}\right)+O(\beta-1)^{2}\,,\end{split}
\label{eq:expansiongamma}$$
where $$\hat{u}=u\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)\,.$$ We see that the expansion effectively attaches a spin factor $\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)$ to $u$ in the expression . This is expected since the $A_{(4,i)}^{(S_{a})}$ is built from the 3pt amplitudes , which can be obtained from the scalar case by promoting $x_{1}^{h}\rightarrow x_{1}^{h}\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)^{S_{a}}$ while $\bar{x}_{1}$ remains the same. Consequently, the expression precisely reduces to its scalar counterpart once the spin operator is projected out: Comparing both expansions we find $${\rm Tr}(\Gamma)=2\langle k_{3}|P_{1}|k_{4}]\,,$$ as required by . The conjugation $y\rightarrow-y$ in $\Gamma$ effectively swaps $\tilde{u}\leftrightarrow v$. This time this transformation also modifies the contribution from the 3pt amplitudes , but once the residue is computed the leading singularity is still invariant (in the HCL).
Finally, considering the contribution $h_{3}=-h_{4}=-2$ in eq. :
$$\begin{split}M_{(S_{a},S_{b},2)}^{(1,b)}=\frac{\beta}{8(\beta^{2}-1)m_{b}^{2}} & \int_{\Gamma_{{\rm LS}}}\frac{dy}{y}A_{4}(P_{1},-P_{2},k_{3}^{-2}(y),k_{4}^{+2}(y))\,\\
& \otimes A_{3}(P_{3},-L(y),-k_{3}^{+2}(y))A_{3}(-P_{4},L(y),-k_{4}^{-2}(y))\,,
\end{split}
\label{eq:int2-1}$$
and inserting , , together with , we find our second main result for the classical piece associated to spinning particles
$$\begin{split}M_{(S_{a},S_{b},2)}^{(1,b)} & =\frac{\alpha^{4}}{16}\frac{m_{b}}{\sqrt{-t}(v-u)^{2}}\int_{\infty}\frac{dy}{y^{3}(1-y^{2})^{2}}\left(\hat{u}y(1-y)+vy(1+y)+\left(v-\hat{u}\right)\frac{1-y^{2}}{2}\right)^{\otimes2S_{a}}\,\\
& \,\quad\times\left(uy(1-y)+vy(1+y)+\frac{(v-u)(1-y^{2})}{2}\right)^{4-2S_{a}}\otimes\left(1-\frac{(1+y)^{2}}{4y}\frac{|\lambda][\lambda|}{m_{b}}\right)^{\otimes S_{b}}
\end{split}$$
together with the analogous expression for $|h|=1$. The residue can then be computed and expanded as a polynomial in spin operators. Evidently, the factor $\Gamma^{\otimes2S_{a}}$ is responsible for these higher multipole interactions, together with the spin operators coming from the 3pt amplitudes . Finally, symmetrization is needed in order to derive the classical potential. This means that we need to consider the triangle topology obtained by exchanging particles $m_{a}$ and $m_{b}$. This can be easily done since our expressions are general as long as $S_{a}$, $S_{b}$ $\leq1$. In appendix \[sec:comframe\] we explicitly show how to construct the full answer for $S_{a}=S_{b}=\frac{1}{2}$ in terms of the standard EFT operators, and find full agreement with the results in [@Holstein:08]. This time it can be readily checked that the Electromagnetic case also leads to analogous spin corrections, which coincide with those given in [@Holstein:2008sw].
Light Bending for Arbitrary Spin {#sub:lightbend}
--------------------------------
We will now implement the construction of appendix \[sub:Massless-representation\] to obtain the massless limit in a similar fashion as we did for the scalar case in sec. \[sub:Massless-probe-particle\]. We will again focus on the gravitational case since it is of interest for studying light bending phenomena, addressed in detail in [@Bjerrum-Bohr:2016hpa; @Bai:2016ivl] for particles with non trivial helicity.
Let us then proceed to take the massless limit of the parametrization (at $\beta=1$) corresponding to $\tau|\hat{\eta}]\rightarrow0$. This yields $x_{1}\rightarrow0$, which is in turn equivalent to $u\rightarrow0$. We get from , using
$$\begin{split}M_{(h_{a},S_{b},2)}^{(0)} & =\alpha^{2}\frac{v^{2}}{t}\left(1-\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\\
& =\alpha^{2}\dfrac{(s-m_{b}^{2})^{2}}{t}\left(1-\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\,,
\end{split}
\label{eq:spinexchangemassless}$$
where $S_{a}=h_{a}$ now corresponds to the helicity of particle $a$. This operator is to be contracted with the states $|0\rangle$, $|2h_{a}\rangle$ associated to momenta $P_{3}$ and the corresponding ones for $P_{4}$, which carry the information of the polarizations. It is however trivial in the sense that it is proportional to the identity for such states, in particular being independent of $h_{a}$. In the non-relativistic limit we find $s-m_{b}^{2}\rightarrow2m_{b}E$, with $E\ll m_{b}$ corresponding to the energy of the massless particle. This shows how the low energy effective potential obtained from is independent of the type of massless particle, as long as it is minimally coupled to gravity. This is the universality of the light bending phenomena previously proposed in [@Bjerrum-Bohr:2016hpa]. It may seem that this claim depends on the choice $u=0$ or $v=0$ for defining the massless limit, since for $v=0$ the operator $\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{[\hat{\lambda}\hat{\eta}]}\right)^{2h_{a}}$ would certainly show up in the result. However, as argued in the appendix \[sub:Massless-representation\], the choice $v=0$ is supplemented by the choice of a different basis of states for the massless representation, such that this operator is again proportional to the identity and hence independent of $h_{a}$.
To argue for the universality at the 1-loop level, we consider the massless limit of , given by $$\Gamma\rightarrow(\beta-1)\left(v(1+y)+v\frac{1-y^{2}}{2y}\right)\,,$$ which is precisely the massless limit of $\langle k_{3}|P_{1}|k_{4}]$, i.e. the corresponding factor for the scalar case. The conclusion is that the behavior of $A_{(4,2)}^{(S_{a})}$ is again independent of $S_{a}=h_{a}$, hence showing the universality. The LS for gravity now reads
$$M_{(h_{a},S_{b},2)}^{(1,b)}=\left(\frac{\alpha^{4}}{2^{8}}\right)\frac{(s-m_{b})^{2}m_{b}}{\sqrt{-t}}\int_{\infty}\frac{dy\,(1+y)^{6}}{y^{3}(1-y)^{2}}\left(1-\frac{(1+y)^{2}}{4y}\frac{|\lambda][\lambda|}{m_{b}}\right)^{2S_{b}}\,.$$
This leading singularity is all what is needed to compute the classical potential for the massless case, since as explained in subsection \[sub:Massless-probe-particle\] the $a-$topology has vanishing LS. Thus, we note that because there is no need to symmetrize there is no restriction on $S_{a}$ at all. This means that, up to 1-loop, we have access to all orders of spin corrections for a massless particle interacting with a rotating point-like source. The expression can be used in principle to recover the multipole expansion of the Kerr black hole solution up to order $G^{2}$, see discussion.
Discussion {#sec:discussion}
==========
In this work we have proposed the implementation of a new set of techniques in order to extract in a direct manner the classical behavior of a variety of scattering amplitudes, including arbitrarily high order spin effects. This classical piece can then be used to construct an effective field theory for long range gravitational or electromagnetic interactions. It was shown in [@LS] that for the gravitational case the 1-loop correction to such interaction is completely encoded into the triangle leading singularity. In this work we have reproduced this result and extended the argument to the electromagnetic case in a trivial fashion. The reason this is possible is because the triangle LS captures the precise non-analytic dependence of the form $t^{-\frac{1}{2}}$, which carries the subleading contribution to the potential. As explained in [@Holstein:2004dn], this structure arises from the interplay between massive and massless propagators entering the loop diagrams. This is the case whenever massive particles exchange multiple massless particles which mediate long range forces, such as photons or gravitons.
We have also included the tree level residues for both cases, which correspond to the leading Newtonian and Coulombian potentials. In this case, both computations were completely analogous and the gravitational contribution could be derived by “squaring” the electromagnetic one. This is reminiscent of the double copy construction, which has been shown to be realized even for the case where massive particles are involved [@Bjerrum-Bohr:2013; @Bjerrum-Bohr:2014lea]. At 1-loop level, such construction is most explicitly realized in the factorization properties of the Compton amplitude. In the overall picture, this set of relations between gravity and EM amplitudes renders the computations completely equivalent. Even though the latter carries phenomenological interest by itself, it can also be regarded as a model for understanding long range effects arising in higher PN corrections, including higher loop and spin orders.
The HCL was designed as a suitable limit to extract the relevant orders in $t$ from the complete classical leading singularities introduced in [@LS]. When embedded in this framework, the computation of the triangle LS proves not only simpler but also leads directly to $t^{-\frac{1}{2}}$ contribution including all the spin interactions. As explained in section \[sub:HNRL\] and explicitly shown in appendix \[sec:comframe\], the covariant form of these interactions allows us to discriminate them from the purely quantum higher powers of $t$, which appear merged in the COM frame. In order to distinguish them we resorted to the following criteria: For a given power of $G$, a subleading order in $|\vec{q}|$ can be classical if it appears multiplied by the appropriate power of the spin vector $|\vec{S}|$. In the HCL framework this is easily implemented since the combination $|\vec{q}||\vec{S}|$ will always emerge from a covariant form which does not vanish for $t\rightarrow0$. For instance, for $S=\frac{1}{2}$, the spin-orbit interaction only arises from $\epsilon_{\alpha\beta\gamma\delta}P_{1}^{\alpha}P_{3}^{\beta}K^{\gamma}S^{\delta}$ and can be tracked directly at leading order.
In striking contrast with previous approaches, the evaluation of spin effects does not involve increased difficulty with respect to the scalar case and can be put on equal footing. This is a direct consequence of implementing the massive representation with spinor helicity variables, which certainly bypasses all the technical difficulties associated to the manipulation of polarization tensors. As an important outcome we have proved that the forms of the higher multipole interactions are independent of the spin we assign to the scattered particles. This is a consequence of the equivalence principle, which we have implemented by assuming minimally coupled amplitudes. The expressions have been explicitly shown to agree with the previous results in the literature for the lowest spin orders, corresponding to $S=1$ and $S=\frac{1}{2}$, yielding spin-orbit, quadrupole and spin-spin interactions. We emphasize, however, that the proposed expressions correspond to a relativistic completion of these results, in the sense that they contain the full $\vec{p}^{\,2}$ expansion.
At this point one could argue that the former difficulty of the diagrammatic computations has been transferred here to the difficulty in performing the matching to the EFT operators. In fact, in order to obtain the effective potential (in terms of vector fields) it is certainly necessary to translate the spinor helicity operators to their standard forms, as was done in appendix \[sec:spinrep\] for $S=\frac{1}{2}$ and $S=1$. We do not think that this should be regarded as a complication. First, as a consequence of the universality we have found, it is clear that we only need to perform the translation once and for a particle of a given spin $S$, as high as the order of multipole corrections we require. Second and more important, we think that this work along with e.g. [@Neill:2013wsa; @Bjerrum-Bohr:2013; @Bjerrum-Bohr:2014zsa; @Holstein:2016fxh; @AmpforAstro; @LS] will serve as a further motivation towards a complete reformulation of the EFT framework which naturally integrates recent developments in scattering amplitudes. For instance, one could aim for a reformulation of the effective potential, or even better, its replacement by a gauge invariant observable, solely in terms of spinor helicity variables so that no translation is needed to address the dynamics of astrophysical objects.
Next we give some proposals for future work along these lines.
The most pressing future direction is the extension of the leading singularity techniques in the context of classical corrections at higher loops [@LS]. This is supported by the fact that higher orders in the PN expansion are associated to characteristic non-analytic structures in the t channel [@Neill:2013wsa], which are precisely what the LS captures. By consistency with the PN expansion such higher orders would require to include spin multipole corrections, so that both the HCL and the new spin representation emerge as promising additional tools for such construction. One could hope that with these methods the scalar and the spinning case will be again on equal footing. Additionally, the PN expansion also requires to incorporate radiative corrections and finite-size effects. The latter may be included within the spin representation presented here by introducing non-minimal couplings, see e.g. [@Levi:2015msa].
![The matrix element of the stress-energy tensor $\langle T_{\mu\nu}(K)\rangle$ corresponds to the 3 point function associated to a pair of massive particles and an external off-shell graviton. The coupling to internal gravitons emanating from the massive source yields, in the long-range behavior, higher corrections in $G$. \[fig:tuv\]](tuv)
The first consistency check for higher loop classical corrections is to reproduce known solutions to Einstein equations. In the spirit of [@Duff; @Neill:2013wsa] and the more modern implementations [@Luna:2016hge; @Goldberger:2017frp] we could argue that this work indeed represent progress towards the derivation of classical spacetimes from scattering amplitudes. As argued by Donoghue [@Donoghue:1994dn; @Donoghue:2017pgk] a way to obtain the spacetime metric is to compute the long-range behavior of the off-shell expectation value $\langle T_{\mu\nu}(K)\rangle$ illustrated in Fig. \[fig:tuv\], which yields the Schwarzschild/Kerr solutions through Einstein equations. At first glance it would seem that is not possible to compute this matrix element using the on-shell methods here exposed. However, this is simply analogous to the fact that we require an off-shell two-body potential for the PN problem. The solution is, of course, to attach another external particle to turn Fig. \[fig:tuv\] into the scattering process of Fig. \[fig:intro\]. In this way we can get information about off-shell subprocesses by examining the $2\rightarrow2$ amplitude.
A simple way to proceed in that direction is to incorporate probe particles whose backreaction can be neglected. In fact, the massless case explored in subsections \[sub:Massless-probe-particle\] and \[sub:lightbend\] can be regarded as a probe particle choice. The lack of backreaction is realized in the fact that only one triangle topology is needed for obtaining the classical piece of the amplitude, which in turn can be thought of containing the process of Fig. \[fig:tuv\]. Furthermore, this piece has no restriction in the spin $S$ of the massive particle, i.e. we can compute both the tree level and 1-loop potential to arbitrarily high multipole terms. By extracting the matrix element $\langle T_{\mu\nu}(K)\rangle$ we could recover both leading and subleading orders in $G$ to arbitrary order in angular momentum of the Kerr solution, see also [@Vaidya:14]. In fact, it was recently proposed [@Vines:2016qwa] that by examining a probe particle in the Kerr background the generic form of the multipole terms entering the 2-body Hamiltonian can be extracted at leading order in $G$ and arbitrary order in spin.
Of course, it is also tempting to explore the opposite direction, outside the probe particle limit. One could try to obtain an expression for the effective (i.e. long-range) vertex of Fig. \[fig:tuv\], including higher couplings with spin, expressed in terms of spinor variables. Then an effective potential could be constructed in terms of several copies of these vertices, for instance to address the n-body problem in GR [@Chu:2008xm; @Galaviz:2010te; @Hartung:2010jg; @Cannella:2011wv].
I would like to thank Freddy Cachazo for proposing this problem and supervising this work as part of the Perimeter Scholars International program. I also thank Sebastian Mizera for useful discussions and comments on the manuscript. I acknowledge the PSI program for providing me with an academically enriching year, as well as CONICYT for financial support. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.
Spinor Helicity Variables for Massive Kinematics {#sec:spinrep}
================================================
Here we construct the $SU(2)$ states and their respective operators written in terms of anti-chiral spinors, first proposed in [@Nima:16] as a presentation of the massive little group. In we considered two massive particles (with same mass $m_{b}$ and spin $S$) and constructed the spaces $V_{3}^{S}$, $\bar{V}_{4}^{S}$ associated to their respective states. We also introduced the contraction between these states which will naturally occur in the matrix element of the scattering processes:
$$\langle n|m\rangle=(-1)^{m}\delta_{m+n,2S}\,.$$ This follows from the normalization explained in . It is also natural to define an inner product for each space if we identify $\bar{V}_{4}^{S}=\left(V_{3}^{S}\right)^{*}$, i.e. as providing a dual basis for $V_{3}^{S}$ [^7]. With these conventions, we can expand any operator $O\in\left(V_{3}^{S}\right)^{*}\otimes\left(\bar{V}_{4}^{S}\right)^{*}$ as
$$O=\sum_{n,m\leq2S+1}(-1)^{n+m-2S}|\bar{n}\rangle\langle\bar{m}|\,\langle n|O|m\rangle\,,\label{eq:changebasis}$$
where $\bar{m}=2S-m,\bar{n}=2S-n$. Of course, this expansion is general for any choice of basis as long as $|\bar{n}\rangle,\langle\bar{m}|$ are defined as the duals. It is even possible to use different states for $V_{3}^{S}$ and $V_{4}^{S}$, spanned by different spinors $\{|\lambda],|\eta]\}$ and $\{|\bar{\lambda}],|\bar{\eta}]\}$. However, it is natural to use the basis as it coincides for both massive particles entering the 3pt amplitude, and also coincides with the dual basis up to relabelling. Next we explicitly illustrate the natural map between the states and the well known Dirac spinor representation for $S=\frac{1}{2}$. We also show how to construct the chiral presentation in terms of angle spinors, in which the basis for both particles turn out to be different.
First, consider the parametrization . The basis of solutions for the (momentum space) Dirac equation are given in terms of the spinors
$$\begin{split}u_{3}^{+}=\begin{pmatrix}\langle\lambda|\\{}
[\lambda|
\end{pmatrix}\,, & \qquad u_{3}^{-}=\begin{pmatrix}-\langle\eta|\\{}
[\eta|
\end{pmatrix}\,,\\
\\
\bar{u}_{4}^{+}=(-\beta|\lambda\rangle\,\,\ |\lambda])\,, & \qquad\bar{u}_{4}^{-}=(\frac{|\eta\rangle}{\beta}+|\lambda\rangle\,\,\ |\eta])\,.
\end{split}
\label{eq:diracsols}$$
(For $\beta=1$, note that follows from the Dirac equation with $x=1$). Thus it is now natural to use $|\eta]$ and $|\lambda]$ to construct the $S=\frac{1}{2}$ representation for the (outgoing) particle $P_{4}$, and similarly for $P_{3}$. This yields an anti-chiral representation of $SU(2)$. From the definition we find (slightly abusing the notation)
$$|+\rangle=\frac{|\lambda]}{\sqrt{m_{b}}}\,,\quad|-\rangle=\frac{|\eta]}{\sqrt{m_{b}}}\,\,\,\in V_{3}^{\frac{1}{2}}\,.\label{eq:normex}$$
and analogously for $\langle\pm|\in\bar{V}_{4}^{\frac{1}{2}}$ . The expansion leads to the $2\times2$ operator
$$O=\frac{1}{m_{b}}\left(-|\lambda][\lambda|\,O_{(--)}+|\lambda][\eta|\,O_{(-+)}+|\eta][\lambda|\,O_{(+-)}-|\eta][\eta|\,O_{(++)}\right)\,.\label{eq:changebasischiral}$$
Had we used the chiral part, we would have selected a different basis for each of the massive particles. In fact, the chiral part is obtained by acting with $P_{3}$, $P_{4}$ on the anti-chiral states, respectively. This means that the change of basis (for $S=\frac{1}{2}$) is given by
$$\bar{O}=\frac{\bar{P}_{3}OP_{4}}{m^{2}}\,,\label{eq:changebasisantichiral}$$
where we have used matrix multiplication, with the extension to higher values of spin being straightforward.
For completeness we present here some useful expressions obtained at $\beta=1$, even though they can easily be computed in general
$$\begin{split}m^{2}\bar{u}_{4}\gamma_{\mu}u_{3}\rightarrow m^{2}\gamma_{\mu} & =2(P_{4})_{\mu}|\eta][\lambda|-2(P_{3})_{\mu}|\lambda][\eta|-2v_{\mu}|\lambda][\lambda|\\
& =2m(P_{3})_{\mu}+2K_{\mu}|\eta][\lambda|-2v_{\mu}|\lambda][\lambda|\,,\\
\bar{u}_{4}u_{3}\rightarrow\mathbb{I}_{2\times2} & =\dfrac{(P_{3})^{\mu}}{m}\gamma_{\mu}=2-\frac{|\lambda][\lambda|}{m}\,,\\
\frac{m^{2}}{2}\bar{u}_{4}\gamma_{5}\gamma_{\mu}u_{3}\rightarrow m^{2}S_{\mu} & =2K_{\mu}|\eta][\eta|-2(R_{\mu}+\frac{1}{2}v_{\mu})|\lambda][\lambda|\\
& \qquad+2(u_{\mu}-v_{\mu}+K_{\mu})|\eta][\lambda|+2(u_{\mu}-v_{\mu})|\lambda][\eta|\,,
\end{split}
\label{eq:translation}$$
where $$\begin{split}2v_{\mu}=[\eta|\sigma_{\mu}|\lambda\rangle\,, & \quad2u_{\mu}=[\lambda|\sigma_{\mu}|\eta\rangle\,,\\
v_{\mu}+u_{\mu}=(P_{3})_{\mu}\,, & \quad2R_{\mu}=[\eta|\sigma_{\mu}|\eta\rangle\,.
\end{split}
\label{eq:vecuv}$$ Here $\mathbb{I}_{2\times2}$ is the identity operator for Dirac spinors, projected into the two-dimensional subspaces spanned by the wavefunctions $u^{\pm}$. On the other hand, in the second line we used the identity $$1=\dfrac{|\eta][\lambda|-|\lambda][\eta|}{[\lambda\eta]}\,.\label{eq:spinid}$$
From the fourth line of , using $2q\cdot K=-m^{2}$ we find in the HCL
$$S_{\mu}K^{\mu}=|\lambda][\lambda|\,.\label{eq:spinop-1}$$
This is the reason we call $|\lambda][\lambda|$ a spin operator. One may wonder why the spin operator appears in the expansion of $\mathbb{I}_{2\times2}$, which contains the scalar contribution. Even though $\mathbb{I}$ and $\gamma_{5}\gamma_{\mu}$ are orthogonal as Dirac matrices, this does not hold once they are projected into the 2D subspace of physical states. This is consistent with the non-relativistic expansions of [@Holstein:08], where the form $\bar{u}_{4}u_{3}$ indeed contributes to the spin interaction. In fact, this is also true for higher spin generalizations as we now show.
Motivated by the manifest universality found in section \[sec:spin\], i.e. expression , we consider the following extensions for arbitrary spin $S_{b}$ (not to be confused with the spin vector $S_{\mu}$) $$\begin{aligned}
S_{\mu}K^{\mu} & = & 2S_{b}|\lambda][\lambda|\,,\label{eq:claim}\\
\mathbb{I}_{(2S_{b}+1)} & = & 2\left(1-S_{b}\frac{|\lambda][\lambda|}{m}\right)\,,\nonumber \end{aligned}$$ As explained in the discussion after Eq. , we omit the trivial part of the operators on the RHS. This allows to keep the expressions compact and makes the universality manifest. Let us briefly perform a nontrivial check of equations for higher spins. We do so by examining the representation for $S_{b}=1$, which in the standard framework is given by polarization vectors satisfying $\epsilon^{(i)}\cdot P=0$, $i=1,2,3$, for a given momentum $P^{2}=m_{b}^{2}$. In terms of spinor helicity variables the polarization vectors $\epsilon_{3}$ and $\epsilon_{4}$ are represented as operators acting on $V_{3}^{1}$ and $\bar{V}_{4}^{1}$ respectively. Explicitly, we can choose [^8]
$$\begin{aligned}
\frac{m_{b}^{2}(\epsilon_{3})_{\mu}}{2} & \rightarrow & [\lambda|[\lambda|R_{\mu}-[\lambda|[\eta|(u-v+K)_{\mu}-[\eta|[\eta|K_{\mu}\,,\\
\frac{m_{b}^{2}(\epsilon_{4})_{\mu}}{2} & \rightarrow & |\lambda]|\lambda](R+\frac{1}{2}P_{3})_{\mu}-|\lambda]|\eta](u-v-K)_{\mu}-|\eta]|\eta]K_{\mu}\,,\end{aligned}$$
Using this expression it is easy to check the validity of for $S_{b}=1$, with [@Holstein:08] $$\begin{aligned}
\epsilon_{3}\cdot\epsilon_{4} & \rightarrow & \mathbb{\mathbb{I}}_{3}\,,\nonumber \\
\frac{1}{2m_{b}}\epsilon_{\mu\alpha\beta\gamma}\epsilon_{4}^{\alpha}\epsilon_{3}^{\beta}(P_{3}+P_{4})^{\gamma} & \rightarrow & S_{\mu}\,.\label{eq:spin1op}\end{aligned}$$ Also, we can now derive the form of the quadrupole interaction. It is given by $$(\epsilon_{4}\cdot K)(\epsilon_{3}\cdot K)=|\lambda]|\lambda]\otimes[\lambda|[\lambda|\,.\label{eq:quadrupole}$$ We will use this expression in appendix \[sec:comframe\] to translate the leading singularity into standard EFT operators.
For illustration purposes, let us close this section by constructing the representation of the 3pt amplitudes for $S=\frac{1}{2}$ massive fields with a graviton. Let the polarization of the massless particle be described by $|\bar{\lambda}]=\sqrt{x}|\lambda]$, $\langle\bar{\lambda}|=\frac{\langle\lambda|}{\sqrt{x}}$, where $x$ carries helicity $1$ (recall $|\lambda]$ is fixed) and agrees with . The 3pt amplitude is given by [@Vaidya:14]
$$\begin{split}A_{\frac{1}{2}}^{(+2)} & =\frac{\alpha m}{2}\gamma_{\mu}\dfrac{[\bar{\lambda}|\sigma^{\mu}|\eta\rangle[\bar{\lambda}|P_{3}|\eta\rangle}{\langle\eta\bar{\lambda}\rangle^{2}}\,,\\
A_{\frac{1}{2}}^{(-2)} & =\frac{\alpha m}{2}\gamma_{\mu}\dfrac{[\eta|\sigma^{\mu}|\bar{\lambda}\rangle[\eta|P_{3}|\bar{\lambda}\rangle}{[\eta\bar{\lambda}]^{2}}\,.
\end{split}$$
Here we have fixed the reference spinor entering in the 3pt. amplitudes to be $\eta$. Using together with we find
$$\begin{split}A_{\frac{1}{2}}^{(+2)} & =\alpha(mx)^{2}\left(1-\frac{|\lambda][\lambda|}{m}\right)\,,\\
A_{\frac{1}{2}}^{(-2)} & =\alpha\left(\dfrac{m}{x}\right)^{2}\,,
\end{split}
\label{eq:appt}$$
precisely agreeing with for $|h|=2$. Furthermore, in the chiral representation we find, using
$$\begin{split}\bar{A}{}_{\frac{1}{2}}^{(+2)} & =\alpha\left(\frac{m}{\bar{x}}\right)^{2}\,,\\
\bar{A}_{\frac{1}{2}}^{(-2)} & =\alpha(m\bar{x})^{2}\left(1-\frac{|\lambda\rangle\langle\lambda|}{m}\right)\,.
\end{split}$$
where $\bar{x}$ is defined in .\
Massless Representation\[sub:Massless-representation\]
------------------------------------------------------
We can extend the treatment described in section \[sub:Massless-probe-particle\] in order to construct the states for massless particles. The idea is to use the two highest weight states $|0\rangle$, $|2S\rangle$ of the massive representation as the physical polarizations of the massless one, after the limit is taken. The massless case can be formally defined by introducing a variable $\tau$ in the parametrization , i.e. by putting either $|\eta]\mapsto\tau|\eta]$ or $|\eta\rangle\mapsto\tau|\eta\rangle$ and then proceed to take the limit $\tau\rightarrow0$. This parametrizes the mass of both $P_{3}(\tau)$ and $P_{4}(\tau)$ as $m^{2}(\tau)=\tau m^{2}$. Next we proceed to sketch the procedure leading to the massless 3pt. amplitudes[^9] and study both choices $\tau|\eta]\rightarrow0$ and $\tau|\eta\rangle\rightarrow0$. As these amplitudes correspond to the building blocks for both the tree level residue and the triangle LS in section \[sec:spin\], showing that they can be recovered from our expressions proves the equivalence with the standard spinor helicity approach for massless particles. This approach was recently implemented in [@Bjerrum-Bohr:2016hpa].
In the following we will consider $\beta=1$. Indeed, the massless deformation of the momenta is only consistent in the HCL since $t=\tau\frac{(\beta-1)^{2}}{\beta}m_{b}^{2}$$\rightarrow0$ as $\tau\rightarrow0$. This is enough for our purposes in section \[sec:spin\] since we evaluate both the tree level residue and triangle LS by neglecting subleading contributions in $t$. For the choice $|\eta]\mapsto\tau|\eta]$ we thus have $$\begin{split}P_{3} & =\tau|\eta]\langle\lambda|+|\lambda]\langle\eta|\,\longrightarrow|\lambda]\langle\eta|\,,\\
P_{4} & =\tau|\eta]\langle\lambda|+|\lambda](\langle\eta|+\langle\lambda|)\,\longrightarrow|\lambda](\langle\eta|+\langle\lambda|)\,,\\
K & =|\lambda]\langle\lambda|\,.
\end{split}
\label{eq:massless1}$$ In the following we choose $|\lambda]$,$\langle\lambda|$ to represent the polarizations of the particle $K$. As explained in section , it is convenient to reabsorb the mass into the definition of $x$ , thus we have
$$x=\tau[\lambda\eta]=\tau m\,,\quad\bar{x}=\langle\lambda\eta\rangle=m\,.$$
This means $\tau|\eta]\rightarrow0$ is equivalent to the limit $x\rightarrow0$, keeping $\bar{x}$ fixed. As the reference spinor $|\eta]$ is also fixed, we can assume that the neither the basis nor the operators depend on $\tau$ in any other way that is not through $x$. With these considerations, we find for the massless limit $$\begin{split}A_{S}^{(h)}=0\,, & \quad A_{S}^{(-h)}=\alpha\bar{x}^{h}\,,\end{split}
\label{eq:3ptslim}$$ where at this stage $\bar{x}=\langle\lambda\eta\rangle$ is not restricted since the original mass $m$ is not relevant after the limit is taken. We then note that all the positive helicity amplitudes vanish. In fact, these amplitudes can be described in terms of square brackets, thus it is expected that they vanish for the $\tau=0$ limit in . Now, the negative helicity amplitudes in the standard spinor helicity notation read [@Elvang]
$$\begin{aligned}
A(3^{+S},4^{-S},K^{-h}) & = & \alpha\frac{\langle K3\rangle^{h-2S}\langle K4\rangle^{h+2S}}{\langle43\rangle^{h}}\label{eq:3ptmasslesss}\\
& = & \alpha\bar{x}^{h}\,.\nonumber \end{aligned}$$
Note that this derivation is also valid for $A(3^{-S},4^{+S},K^{-h})$ up to a possible sign. Also, the configuration $A(3^{+S},4^{+S},K^{-h})$ together with its conjugate do not correspond to the minimal coupling and thus vanish. In order to interpret these amplitudes as matrix elements of , we need to specify the basis of states for the massless particles. It turns out that just the highest weight states in are enough for this purpose. That is, we find $$\begin{aligned}
A(3^{+S},4^{-S},K^{-h})=\langle2S|A_{S}|0\rangle & , & A(3^{-S},4^{+S},K^{-h})=\langle0|A_{S}|2S\rangle\,,\label{eq:expval}\\
A(3^{+S},4^{+S},K^{-h})=\langle2S|A_{S}|2S\rangle & , & A(3^{+S},4^{+S},K^{-h})=\langle2S|A_{S}|2S\rangle\,,\nonumber \end{aligned}$$ therefore showing the equivalence of both approaches for massless particles. Here we note that a somehow more straightforward approach is to define the massless limit directly in the expectation values , following [@Nima:16]. Instead, we have opted for constructing the corresponding operators , since our integral expressions in section \[sec:spin\] are given in terms of them. These operators are defined for the basis built from the fixed spinors $|\lambda]$ and $|\eta]$, which are reminiscent of the massive representation in .
The choice $|\eta\rangle\mapsto\tau|\eta\rangle$ is completely analogous and yields
$$\begin{split}A_{S}^{(h)}=\alpha x^{h}\left(1-\frac{|\lambda][\lambda|}{[\lambda\eta]}\right)^{S}\,, & \quad A_{S}^{(-h)}=0\end{split}
\,,\label{eq:3ptslim-1}$$
i.e. vanishing negative helicity amplitudes. This is expected since we have
$$\begin{split}P_{3} & =|\eta]\langle\lambda|+\tau|\lambda]\langle\eta|\,\longrightarrow|\eta]\langle\lambda|\,,\\
P_{4} & =|\eta]\langle\lambda|+\tau|\lambda]\langle\eta|+|\lambda]\langle\lambda|\,\longrightarrow(|\lambda]+|\eta])\langle\lambda|\,.
\end{split}
\label{eq:massless1-1}$$
However, this time we note that the natural basis of spinors for $P_{4}$ is given by $|\bar{\eta}]:=|\lambda]+|\eta]$ and $|\lambda]$. When expressed in terms of this basis, the expression takes a form analogous to . Hence we construct the states $\langle0|$,$\langle2S|$ in $\bar{V}_{4}^{S}$ in terms of these spinors.
Matching the Spin Operators {#sec:comframe}
===========================
Here we explain how to recover the standard form of the potential in terms of generic spin operators , starting from the results of section \[sec:spin\]. As usual throughout this work, we focus on the gravitational case since it presents greater difficulty in the standard approaches. We give two examples which illustrate how the procedure works. First, we present the tree level result for the case $S_{a}=0$, $S_{b}=1$, which yields both a spin-orbit and a quadrupole interaction. Second, we discuss the matching at 1-loop level for the case $S_{a}=S_{b}=\text{\ensuremath{\frac{1}{2}}}$. Both computations were done in [@Holstein:08] using standard Feynman diagrammatic techniques, which lead to notably increased difficulty with respect to the scalar case. Here we find that the computations are straightforward using the techniques introduced throughout this work. In fact, all the computations in [@Holstein:08] can be redone in a direct way and we leave them as an exercise for the reader. The same can be done for the EM case in order to recover the results previously presented in [@Holstein:2008sw].
The starting point for both cases are the explicit expressions for the variables $u,v$ that we used to construct the amplitudes. We can easily solve them from Eq. . We find $$\begin{aligned}
2u & = & s-m_{a}^{2}-m_{b}^{2}+\sqrt{\left(s-m_{a}^{2}-m_{b}^{2}\right)^{2}-4m_{a}^{2}m_{b}^{2}}\,,\label{eq:uvexplicit}\\
2v & = & s-m_{a}^{2}-m_{b}^{2}-\sqrt{\left(s-m_{a}^{2}-m_{b}^{2}\right)^{2}-4m_{a}^{2}m_{b}^{2}}\,,\nonumber \end{aligned}$$ where the square root corresponds to the parity odd piece. From the definition it is clear that under the exchange $P_{1}\leftrightarrow P_{3}$ (which we perform below), $u$ and $v$ must also be exchanged. Now, to keep the notation compact, let us write $$P_{1}\cdot P_{3}=rm_{a}m_{b}\,,\quad r>1.$$ Note that in the non-relativistic regime we have $r\rightarrow1$. Now we can write \[eq:uvexplicit\] as $$\begin{aligned}
u & =m_{a}m_{b}\left(r+\sqrt{r^{2}-1}\right)\,,\quad & v=m_{a}m_{b}\left(r-\sqrt{r^{2}-1}\right)\,.\label{eq:uvcompact}\end{aligned}$$ Consider now the case $S_{a}=0$, $S_{b}=1$. Let us construct a linear combination of the EFT operators associated to scalar, spin-orbit, and quadrupole interaction, that is [@Holstein:08; @Vaidya:14]
$$\bar{M}_{(0,1,2)}^{(1)}=\alpha^{2}\dfrac{(m_{a}m_{b})^{2}}{t}\left(c_{1}(r)\epsilon_{3}\cdot\epsilon_{4}+c_{2}(r)\frac{\epsilon_{\mu\alpha\beta\gamma}K^{\mu}P_{1}^{\alpha}P_{3}^{\beta}S^{\gamma}}{m_{a}m_{b}^{2}}+c_{3}(r)\frac{(\epsilon_{4}\cdot K)(\epsilon_{3}\cdot K)}{m_{b}^{2}}\right)\,.\label{eq:ansatz}$$
The reason we call $\epsilon_{3}\cdot\epsilon_{4}$ a scalar interaction is because, as will be evident in a moment, it is the only piece surviving the contraction $\langle0|\bar{M}_{(0,1,2)}^{(1)}|2\rangle$, which we identified as the scalar amplitude (see discussion below Eq. ).
Note that we have not assumed the non-relativistic limit in the $u,v$ variables, only the HCL $t=0$ which selects the classical contribution. The operators can now be expanded using , . For this, it is enough to note that in the HCL the spin-orbit piece takes the form
$$\epsilon_{\mu\alpha\beta\gamma}K^{\mu}P_{1}^{\alpha}P_{3}^{\beta}S^{\gamma}=\frac{K\cdot S}{2}\sqrt{\left(s-m_{a}^{2}-m_{b}^{2}\right)^{2}-4m_{a}^{2}m_{b}^{2}}=m_{a}m_{b}(K\cdot S)\sqrt{r^{2}-1}\,.\label{eq:dethcl}$$
We then find
$$\bar{M}_{(0,1,2)}^{(0)}=\alpha^{2}\dfrac{(m_{a}m_{b})^{2}}{t}\left(2c_{1}-2\frac{|\lambda][\lambda|}{m_{b}}(c_{1}-c_{2}\sqrt{r^{2}-1})+c_{3}\frac{|\lambda]|\lambda]\otimes[\lambda|[\lambda|}{m_{b}^{2}}\right)\,.$$ Comparing now with the expression , which in this case reads
$$\begin{aligned}
M_{(0,1,2)}^{(0)} & = & \dfrac{\alpha^{2}}{t}\left(u^{2}+v^{2}\left(1-\frac{|\lambda][\lambda|}{m_{b}}\right)^{2}\right)\\
& = & \dfrac{\alpha^{2}}{t}\left(u^{2}+v^{2}-2v^{2}\frac{|\lambda][\lambda|}{m_{b}}+v^{2}\frac{|\lambda]|\lambda]\otimes[\lambda|[\lambda|}{m_{b}^{2}}\right)\\
& = & \alpha^{2}\dfrac{(m_{a}m_{b})^{2}}{t}\left((4r^{2}-2)-2\left(2r^{2}-1-2r\sqrt{r^{2}-1}\right)\frac{|\lambda][\lambda|}{m_{b}}\right.\\
& & \left.+\left(2r^{2}-1-2r\sqrt{r^{2}-1}\right)\frac{|\lambda]|\lambda]\otimes[\lambda|[\lambda|}{m_{b}^{2}}\right)\,,\end{aligned}$$
we find $$\begin{aligned}
c_{1} & = & 2r^{2}-1\,,\\
c_{2} & = & 2r\,,\\
c_{3} & = & 2r^{2}-1-2r\sqrt{r^{2}-1}\,.\end{aligned}$$ The result in [@Holstein:08] can then be recovered by imposing the non-relativistic limit $s\rightarrow s_{0}$, which in this case reads $r\rightarrow1$[^10]. Even though we computed the residue in at $t=0$, it is evident that this expression can be analytically extended to the region $t\neq0$ in which the COM frame can be imposed, as described in . This is precisely done in [@Holstein:08] where the effective potential is obtained from this expression after implementing the Born approximation.
The 1-loop result for $S_{a}=0$, $S_{b}=1$ can be computed in the same fashion, by using the expressions provided in section . Expectedly, the EFT operators are exactly the same that appeared at tree level, but the behavior of the coefficients $c_{1}$, $c_{2}$ and $c_{3}$ as functions of $r$ differs in the sense that it can involve poles at $r=1$. We now illustrate all this by considering the more complex case also addressed in [@Holstein:08], namely $S_{a}=S_{b}=\frac{1}{2}$.
For $S=\frac{1}{2}$ the multipole operators are restricted to the scalar and spin-orbit interaction. They read [@Holstein:08] $$\begin{aligned}
\mathcal{U}=\bar{u}_{4}u_{3} & ,\, & \mathcal{E}=\epsilon_{\alpha\beta\gamma\delta}P_{1}^{\alpha}P_{3}^{\beta}K^{\gamma}S^{\delta}\,.\end{aligned}$$ In our case we will consider two copies of these operators, one for each particle. That is to say we propose the following form for the 1-loop leading singularity
$$\begin{aligned}
\bar{M}_{(\frac{1}{2},\frac{1}{2},2)}^{(1)} & = & \left(\frac{\alpha^{4}}{16}\right)\frac{\left(m_{a}m_{b}\right)^{2}}{\sqrt{-t}}\left(c_{11}\mathcal{U}_{a}\mathcal{U}_{b}+c_{12}\frac{\mathcal{U}_{a}\mathcal{E}_{b}}{m_{b}^{2}m_{a}}+c_{21}\frac{\mathcal{E}_{a}\mathcal{U}_{b}}{m_{a}^{2}m_{b}}+c_{22}\frac{\mathcal{E}_{a}\mathcal{E}_{b}}{m_{b}^{3}m_{a}^{3}}\right)\,\label{eq:ansatzspin1/2}\\
& = & \alpha^{4}\frac{\left(m_{a}m_{b}\right)^{2}}{4\sqrt{-t}}\left(c_{11}-\frac{c_{11}-c_{12}\sqrt{r^{2}-1}}{2}\left(\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)-\frac{c_{11}+c_{21}\sqrt{r^{2}-1}}{2}\left(\frac{|\lambda][\lambda|}{m_{b}}\right)\right.\nonumber \\
& & \left.+\frac{\left(c_{11}-(c_{12}-c_{21})\sqrt{r^{2}-1}-c_{22}(r^{2}-1)\right)}{4}\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\otimes\frac{|\lambda][\lambda|}{m_{b}}\right)\,.\nonumber \end{aligned}$$
Here we have used , and . A minus sign was introduced when implementing for particle $m_{a}$, which arises from the mismatch between both parametrizations in the HCL, i.e. $K=|\lambda]\langle\lambda|=-|\hat{\lambda}][\hat{\lambda}|$. We proceed to compare this with the sum of the two triangle leading singularities given by , using the results of section \[sub:spin1loop\]. The result can be written
$$M_{(\frac{1}{2},\frac{1}{2},2)}^{(1,{\rm full})}=M_{(\frac{1}{2},\frac{1}{2},2)}^{(1,b)}+M_{(\frac{1}{2},\frac{1}{2},2)}^{(1,a)}\,,$$ where $M_{(\frac{1}{2},\frac{1}{2},2)}^{(1,a)}$ is obtained by exchanging $m_{a}\leftrightarrow m_{b}$, $|\hat{\lambda}][\hat{\lambda}|\leftrightarrow|\lambda][\lambda|$ and $u\leftrightarrow v$ in
$$\begin{split}M_{(\frac{1}{2},\frac{1}{2},2)}^{(1,b)} & =\left(\frac{\alpha^{4}}{16}\right)\frac{m_{b}}{\sqrt{-t}(v-u)^{2}}\int_{\infty}\frac{dy}{y^{3}(1-y^{2})^{2}}\left(\hat{u}y(1-y)+vy(1+y)+\left(v-\hat{u}\right)\frac{1-y^{2}}{2}\right)\,\\
& \,\quad\quad\times\left(uy(1-y)+vy(1+y)+\frac{(v-u)(1-y^{2})}{2}\right)^{3}\otimes\left(1-\frac{(1+y)^{2}}{4y}\frac{|\lambda][\lambda|}{m_{b}}\right)\,,
\end{split}
\label{eq:3ptNRLspin-1}$$
with $\hat{u}=u\left(1-\frac{|\hat{\lambda}][\hat{\lambda}|}{m_{a}}\right)$. After computing the contour integral, we can easily solve for the coefficients $c_{ij}$, $i,j\in\{1,2\}$. In order to compare with the results in the literature, we first perform the non-relativistic expansion
$$\begin{aligned}
c_{11} & = & 6(m_{a}+m_{b})+\ldots\nonumber \\
c_{12} & = & \frac{4m_{a}+3m_{b}}{2(r-1)}+11\left(m_{a}+\frac{3}{4}m_{b}\right)+\ldots\nonumber \\
c_{21} & = & \frac{3m_{a}+4m_{b}}{2(r-1)}+11\left(\frac{3}{4}m_{a}+m_{b}\right)+\ldots\label{eq:coefs}\\
c_{22} & = & \frac{m_{a}+m_{b}}{4(r-1)^{2}}+\frac{9(m_{a}+m_{b})}{2(r-1)}+\ldots\nonumber \end{aligned}$$
Note that even though the coefficients present poles, they are parity invariant in the sense that they do not contain square roots. To put the result in the same form as [@Holstein:08], we need to further extract the standard spin-spin interaction term from our operator $\mathcal{E}_{a}\mathcal{E}_{b}$. This accounts for extracting the classical piece, which can be obtained by returning to the physical region $t=K^{2}\neq0$. Using we find $$\mathcal{E}_{a}\mathcal{E}_{b}=m_{a}m_{b}(r^{2}-1)\left((S_{a}\cdot K)(S_{b}\cdot K)-K^{2}(S_{a}\cdot S_{b})\right)+rK^{2}(P_{1}\cdot S_{b})(P_{3}\cdot S_{a})+O(K^{3})\,,$$ where $O(K^{3})=O(|\vec{q}|^{3})$ denotes quantum contributions, i.e. higher orders in $|\vec{q}|$ for a fixed power of spin $|\vec{S}|$. However, we note the presence of the couplings $P_{i}\cdot S\sim\vec{v}\cdot\vec{S}$ which certainly do not appear in the effective potential [@Barker1979; @Holstein:08; @Vaidya:14]. In fact, in the standard EFT framework these couplings are dropped by the so-called Frenkel-Pirani conditions or Tulczyjew conditions [@Frob:2016xte][^11]. In our case they have emerged due to our bad choice of ansatz . In fact, the right choice is now clearly obtained by replacing $$\mathcal{E}_{a}\mathcal{E}_{b}\rightarrow m_{a}m_{b}(r^{2}-1)\left((S_{a}\cdot K)(S_{b}\cdot K)-K^{2}(S_{a}\cdot S_{b})\right)\,,$$ corresponding to the correct spin-spin interaction term [@Porto:2005ac], which is already visible at tree level [@Holstein:2008sw; @Holstein:08; @Vaidya:14]. Note, however, that this does not modify the HCL of this operator, which comes solely from the first term. Consequently, our results are still valid and indeed they agree with the ones in the literature [@Holstein:08]. They can be regarded as a fully relativistic completion leading to the effective potential up to order $G^{2}$.
[^1]: For instance, they vanish whenever the momentum transfer $K$ is orthogonal to the polarization tensors $K_{\mu_{1}}\epsilon^{\mu_{1}\ldots\mu_{S}}=0$ as can be checked in [@Vaidya:14], or equivalently, when it is aligned with the spin vector.
[^2]: Hereafter we may refer to the multipole terms , as EFT operators indistinctly. This is in order to contrast them with the spinor operators to be defined in section \[sec:spin\], which will be then matched to EFT operators.
[^3]: Such that $\epsilon^{12}=-\epsilon_{12}=1$.
[^4]: For real momenta we find that $x$ corresponds to a phase. It also induces non-local behavior in the 3pt amplitudes [@Nima:16]. However, we ignore these physical restrictions for now since we are describing generic 3pt amplitudes which will be used to construct the leading singularities.
[^5]: Also the choice $y=0$ is permitted for the contour, i.e. $\Gamma_{{\rm LS}}=S_{0}^{1}$. This choice does not matter in the HCL since the leading piece in is invariant under the inversion of the contour [@LS].
[^6]: The notation $|m\rangle$ for the states may seem unfortunate since it is similar to the one for angle (chiral) spinors. However, as we will be mostly using the anti-chiral representation for spinors, the risk of confusion is low.
[^7]: The contraction $\langle n|m\rangle$, as defined, is antisymmetric for fermions. This is reminiscent of the spin-statistics theorem, as such form is proportional to the minimally coupled 3pt amplitude. On the other hand, in order to interpret this contraction as an inner product it is necessary to introduce the dual map $\zeta:V^{S}\rightarrow\left(V^{S}\right)^{*}$. For instance, defining $\zeta:|n\rangle\mapsto(-1)^{2s}\langle n|$ leads to a symmetric expression.
[^8]: Here we use the notation $[\lambda|[\eta|$ to account for the standard tensor product, i.e. not symmetrized. Of course, we can replace $[\lambda|[\eta|\rightarrow\frac{1}{\sqrt{2}}[\lambda|\odot[\eta|$, where $\odot$ involves the normalization .
[^9]: At this level we keep the discussion general for $S$ and $h$. Of course, (interacting) massless higher spin particles are known to be inconsistent by very fundamental principles, thus effectively restricting our choices to $S,h\leq2$.
[^10]: There are, however, some discrepancies in conventions which may be fixed by replacing $-\epsilon_{f}^{b*}\rightarrow\epsilon_{4}$, $iS_{b}\rightarrow S_{b}$ in [@Holstein:08]. We find our conventions more appropriated since the sign in the scalar interaction is the same for any spin.
[^11]: They can arise, however, when including non-minimal couplings corresponding to higher dimensional operators, see e.g. [@Levi:2015msa]
|
---
abstract: |
\
I discuss some theoretical expectations for the synchrotron emission from a relativistic blast-wave interacting with the ambient medium, as a model for GRB afterglows, and compare them with observations. An afterglow flux evolving as a power-law in time, a bright optical flash during and after the burst, and light-curve breaks owing to a tight ejecta collimation are the major predictions that were confirmed observationally, but it should be recognized that light-curve decay indices are not correlated with the spectral slopes (as would be expected), optical flashes are quite rare, and jet-breaks harder to find in Swift X-ray afterglows.
The slowing of the early optical flux decay rate is accompanied by a spectral evolution, indicating that the emission from ejecta (energized by the reverse shock) is dominant in the optical over that from the forward shock (which energizes the ambient medium) only up to 1 ks. However, a long-lived reverse shock is required to account for the slow radio flux decays observed in many afterglows after $\sim 10$ day.
X-ray light-curve plateaus could be due to variations in the average energy-per-solid-angle of the blast-wave, confirming to two other anticipated features of GRB outflows: energy injection and angular structure. The latter is also the more likely origin of the fast-rises seen in some optical light-curves. To account for the existence of both chromatic and achromatic afterglow light-curve breaks, the overall picture must be even more complex and include a new mechanism that dominates occasionally the emission from the blast-wave: either late internal shocks or scattering (bulk and/or inverse-Compton) of the blast-wave emission by an outflow interior to it.
author:
- 'A. Panaitescu'
title: 'Gamma-Ray Burst afterglows: theory and observations'
---
[ address=[ ISR-1, Los Alamos National Laboratory, Los Alamos, NM 87545, USA]{} ]{}
Introduction
============
A relativistic motion of GRB sources was advocated by [@paczynski86; @goodman86] from that the energies released exceed by many orders of magnitude the Eddington luminosity for a stellar-mass object, especially if GRBs are at cosmological distances (see also [@shemi90; @meszaros92]).
The detection by CGRO/EGRET of photons with energy above 1 MeV during the prompt burst emission (e.g. [@hurley94]) shows that GRB sources are optically thin to such photons. Together with the sub-MeV burst isotropic-equivalent output of $10^{52}-10^{54}$ ergs (e.g. [@bloom01]) and the millisecond burst variability timescale, the condition for optical thickness to high energy photons gives another reason why GRBs must arise from ultra-relativistic sources, moving at Lorentz factor $\Gamma \simg 100$ (e.g. [@fenimore93; @lithwick01]).
The same conclusion is enforced by the measurement of a relativistic expansion of the radio afterglow source. That expansion was either measured directly, as for GRB 030329 ($z=0.17$), whose size increased at an apparent speed of 5c, indicating a source expanding at $\Gamma \siml 6$ at 1–2 months [@taylor04], or was inferred from the rate at which interstellar scintillation [@goodman97] quenches owing to the increasing source size, as for GRB 970508, whose expansion speed is inferred to be close to $c$ at 1 month [@waxman98]. The adiabatic dynamical evolution of a blast-wave, $\Gamma^2 M = const$, where $M$ is the mass of the ambient medium, leads to $\Gamma \propto t^{-3/8}$ for a homogeneous medium ($t$ being the observer-frame photon arrival time). Then, $\Gamma
(30\,{\rm d}) = 2$ extrapolated to the burst time implies $\Gamma(100\,{\rm s}) \sim 100$. Extrapolating to such early times is justified by that most optical afterglow light-curves display a power-law decay starting after the burst, which sets a lower limit on the source Lorentz factor at that time.
Whether the GRB ejecta are a cold baryonic outflow accelerated by the adiabatic losses of fireball’s initial thermal/radiation energy (e.g. [@meszaros93; @piran93]), or relativistic pairs formed through magnetic dissipation in a Poynting outflow, as in the electromagnetic model of [@lyutikov06], their interaction with ambient medium will drive two shocks: a reverse shock crossing the ejecta and a forward-shock sweeping the circumburst medium, as illustrated in Figure \[bw\]. Both shocks energize their respective media, accelerate relativistic particles and generate magnetic fields through some plasma-instability related process, such as the two-stream Weibel instability driven by an anisotropic particle distribution function [@medvedev99]. The original magnetic field of the fireball at $10^7$ cm becomes too weak by the time the fireball reaches the $10^{15}-10^{17}$ cm radius (where the burst and afterglow emissions are produced) for the synchrotron emission to account for the sub-MeV burst emission and for the longer-wavelength, ensuing afterglow emission, even if the fireball was initially magnetically dominated [@meszaros93; @medvedev99].
The evolution of the synchrotron and inverse-Compton fluxes produced by the blast-wave at a fixed frequency (i.e. the light-curve) is determined by how the characteristics of the spectrum (break frequencies and peak flux) change with time. Figure \[spek\] shows the expected afterglow synchrotron spectrum, whose characteristics depend on the blast-wave radius, number of radiating electrons, their distribution with energy, magnetic field strength, and Lorentz factor.
If the typical electron energy and magnetic field energy correspond to some fixed fraction of the post-shock energy, or if they start from such a fixed fraction and then evolve adiabatically (as for adiabatically colling ejecta), then the afterglow light-curve depends on (1) the evolution of the blast-wave Lorentz factor, the blast-wave radius being $R \simeq \Gamma^2 ct$ (with $t$ the photon-arrival time measured since burst trigger) (2) the spectrum of the blast-wave emission (i.e. the distribution of electrons with energy), and, in the case of the reverse-shock, (3) the evolution of the incoming mass. The power-law deceleration of the blast-wave ($\Gamma \propto r^{-(3-s)/2} \propto
t^{-(3-s)/(8-4s)}$ for $s<3$, where $n \propto r^{-s}$ is the radial stratification of the ambient medium density) and the power-law afterglow spectrum ($F_\nu \propto \nu^{-\beta}$) are two factors which lead to a power-law afterglow light-curve ($F_\nu \propto t^{-\alpha}$), with the decay index $\alpha$ being a linear function of the spectral slope $\beta$. These are the only two factors at work for the forward-shock emission and the ejecta emission during the adiabatic cooling phase (which starts when the reverse shock has crossed the ejecta shell), the two models that yield power-law afterglow light-curves in the most simple and natural way.
In contrast, for the reverse-shock emission (i.e. the ejecta emission while the shock exists), the light-curve depends also on the radial distribution of ejecta mass and of their Lorentz factor, thus the observed power-law light-curves require additional properties to be satisfied by the relativistic ejecta. Such properties seem [*ad-hoc*]{} when it comes to explaining single power-law afterglows whose flux displays an unchanged decay over 2–4 decades in time (such as the X-ray afterglows of GRBs 050801, 050820A, 06011B, 060210, 060418, 061007), but they also provide the flexibility required to account for the prevalent X-ray afterglow light-curves that exhibit one or more breaks.
Afterglow light-curves
======================
I consider first the afterglow emission at early times, when the blast-wave is sufficiently relativistic that the observer receives boosted emission from a region of half-angle opening $\Gamma^{-1}$ (as seen from the center of the blast-wave) that is smaller than the half-aperture $\theta_{jet}$ of the collimated outflow. In that case, the observer does not “see” yet the angular boundary of the outflow and the received emission is as bright as for a spherical blast-wave. The evolution of the spectral characteristics of the emission from ejecta and the swept-up ambient medium are presented below, the resulting power-law decay indices of the synchrotron flux being listed in Table 1.
Ejecta emission (energized by reverse shock or cooling adiabatically)
---------------------------------------------------------------------
For a short-duration ejecta release, the reverse shock crosses the ejecta shell over an observer-frame time that depends primarily on the ejecta Lorentz factor $\Gamma_0$: $t_{dec} = 350\, (z+1) (E_{53}/n_0\Gamma_{0,2}^8)^{1/3}$ s, where $E_{53}$ is the isotropic-equivalent ejecta kinetic energy in $10^{53}$ erg, $n_0$ is the ambient medium density in protons per ${\rm cm^3}$, and $\Gamma_{0,2} = \Gamma_0/100$. In this case, the reverse shock is semi-relativistic and, more likely, radiates below the optical. After $t_{dec}$, the input of energy into the shocked structure ceases and the blast-wave begins to decelerate.
If the ejecta release is an extended process, the deceleration timescale depends primarily on the duration $\tau$ over which the ejecta are expelled: $t_{dec} = 0.7\,(z+1) \tau$. In this case, the reverse shock is relativistic and could produce a bright optical emission. The separation between these two cases is set by $t_{dec}(\Gamma) = t_{dec}(\tau)$, “short-duration ejecta release” meaning $\tau < \tilde{\tau} \equiv 500\, (E_{53}/n_0
\Gamma_{0,2}^8)^{1/3}$ s. For a wind medium, the deceleration timescale is $t_{dec} = 3\, (z+1) E_{53}/ (A_*
\Gamma_{0,2}^4)$ s for a short-duration ejecta release, where $A_*$ is the wind density parameter, normalized to that resulting for $10^{-5}\, M_\odot/{\rm yr}$ being ejected at a terminal velocity of 1000 km/s, $t_{dec}$ being the same as for a homogeneous medium in the case of a long-duration ejection (which occurs for $\tau > \tilde{\tau} \equiv 4
\, E_{53}/(A_* \Gamma_{0,2}^4)$ s).
At the deceleration radius, $\sim 1/3$ of the ejecta energy has been transferred to the swept-up ambient medium, which moves at $\Gamma \simeq (2/3) \Gamma_0$ for $\tau < \tilde{\tau}$ and a lower $\Gamma$ for $\tau > \tilde{\tau}$. Taking into account that the energy per particle in the post forward-shock gas is $\Gamma$, it follows that, at $t_{dec}$, the ejecta mass is larger than that of the forward shock by at most a factor $\simeq \Gamma_0$. This implies that, at $t_{dec}$, the peak flux of the reverse-shock emission spectrum is a factor $\simg 100$ larger than the peak flux of the forward-shock spectrum, hence, the optical flash from the reverse shock could be up to 5 magnitudes brighter than the optical emission from the forward shock.
As mentioned above, when there is a reverse shock crossing the ejecta, its emission flux should depend on the density and Lorentz factor of the incoming ejecta. Semi-analytical calculations of the ejecta synchrotron emission when there is a reverse shock have been done by [@uhm07] and [@genet07] for a density and Lorentz factor of the incoming ejecta tailored to produce X-ray light-curve plateaus, hydrodynamical calculations of the reverse-shock dynamics have been presented by [@kobayashi00], and calculations of the ejecta emission after the reverse shock has crossed the ejecta (i.e. during adiabatic cooling) have been published by [@meszaros97; @sari99; @kobayashi00A; @alin04]. However, analytical calculations of the ejecta emission while there is a reverse shock [@kobayashi03] have yet to be done.
Assuming a uniform ejecta density & Lorentz factor and an extended ejecta release, for which the reverse shock is relativistic and the shocked ejecta are slightly decelerated even before $t_{dec}$, owing to the progressive dilution of the incoming ejecta, I find that the [*reverse-shock*]{} peak flux $F_p$ and break frequencies $\nu_i$ (injection) and $\nu_c$ (cooling) evolve as $F_p \propto t^{-1/5}$, $\nu_i \propto
t^0$, $\nu_c \propto t^{-4/5}$ for a homogeneous medium and $F_p \propto t^{-1/3}$, $\nu_i \propto t^{-2/3}$, $\nu_c \propto t^{2/3}$ for a wind. As for the [*ejecta*]{} emission decay during the adiabatic cooling phase, the evolution of the spectral characteristics is approximately $F_p \propto t^{-0.67}$, $\nu_i,\nu_c \propto t^{-1.2}$ for a homogeneous medium, and $F_p \propto t^{-0.80}$, $\nu_i,\nu_c \propto t^{-1.5}$ for a wind. For $\nu_{obs} < \nu_i$, the cooling ejecta flux should decay with an index $\alpha = 0.3$, for either type of medium. Above $\nu_i$ but below $\nu_c$, the decay index is $\alpha = 1.19 \beta - 0.67$ for a homogeneous medium and $\alpha = 1.47 \beta - 0.80$ for a wind. Depending on the treatment of the ejecta dynamics and adiabatic cooling, other researchers reached slightly different results $\alpha = a\beta + k$ with $a \simeq 1.3$ and $k \simeq 0.8$ in [@meszaros97], $a \simeq 1.5$ and $k \simeq 1.0$ in [@sari99], $a \in (1.37,1.66)$ and $k \in (0.75,0.96)$ in [@kobayashi00A].
After $\nu_c$ falls below $\nu_{obs}$ owing to adiabatic cooling, the observer receives no emission from the area of angular opening $\Gamma^{-1}$ moving directly toward the observer (because of the exponential cut-off of the synchrotron emissivity above the synchrotron peak) but receives emission from the fluid moving at increasing angles larger than $\Gamma^{-1}$. That emission (called [*large-angle*]{} emission, lacking a better name) was released at the same time as the emission from angles less than $\Gamma^{-1}$, but arrives later at observer because of the spherical curvature of the emitting surface and finite speed of light, and is less beamed relativistically. As shown by [@kumar00], the large-angle emission is characterized by $F_p \propto t^{-2}$, $\nu_i,\nu_c \propto
t^{-1}$, the power-law decay index being $\alpha = \beta + 2$ (see also [@fenimore97]) These are general results, arising only from relativistic effects, and independent of the emission process. The only assumption made in its derivation is that the surface emissivity properties are angle-independent.
Forward-shock emission (energized ambient medium)
-------------------------------------------------
Before deceleration of the blast-wave begins, the shocked ambient medium moves at a constant $\Gamma \simeq (2/3)\Gamma_0$, if the reverse shock is semi-relativistic, or is slowly decelerating as $\Gamma \propto t^{-(3-s)/(10-2s)}$, if the reverse shock is relativistic.
For a [*semi-relativistic*]{} reverse shock, the spectral characteristics of the [*pre-deceleration*]{} forward-shock synchrotron emission evolve as $F_p \propto t^3$, $\nu_i = const$, $\nu_c \propto t^{-2}$ for a homogeneous medium, and $F_p = const$, $\nu_i \propto t^{-1}$, $\nu_c \propto t$ for a wind. For a [*relativistic*]{} reverse shock, the above scalings become $F_p \propto t^{3/5}$, $\nu_i \propto t^{-6/5}$, $\nu_c \propto t^{-4/5}$ for $s=0$, and $F_p \propto t^{-1/3}$, $\nu_i \propto t^{-4/3}$, $\nu_c \propto t^{2/3}$ for $s=2$. The forward-shock synchrotron emission [*after deceleration*]{} has received the most attention (e.g. [@meszaros97; @sari98; @alin98; @chevalier99; @granot99; @granot02]). Under the usual assumptions of constant blast-wave energy and micro-physical electron and magnetic field parameters, the forward-shock peak flux and spectral break frequencies evolution is $F_p \propto t^0$, $\nu_i \propto t^{-3/2}$, $\nu_c \propto t^{-1/2}$ for a homogeneous medium, and $F_p \propto t^{-1/2}$, $\nu_i \propto t^{-3/2}$, $\nu_c \propto t^{1/2}$ for a wind. From here, it follows that the flux below $\nu_i$ should rise slowly as $F_\nu \propto
t^{1/2}$ for a homogeneous medium or be constant for a wind medium. For $\nu_i < \nu_{obs}$, the forward-shock flux decay is a power-law of index $\alpha =
(3/2)\beta + k$, with $k=1/2$ if ambient medium has a wind-like stratification (as expected for a massive stellar long-GRB progenitor) and if $\nu_{obs} < \nu_c$, $k=0$ for a homogeneous medium (which, surprisingly, is more often found to be compatible with the observed afterglows than a wind) if $\nu_{obs} < \nu_c$, and $k=-1/2$ if $\nu_c < \nu_{obs}$, for any type of medium. All the above results hold for a spherical outflow or a collimated one before the jet boundary becomes visible to the observer. At the jet-break time $t_{jet}$, when deceleration lowers the jet Lorentz factor to $\Gamma = \theta_{jet}^{-1}$, the emission from the jet edge is no longer relativistically beamed away from the direction toward the observer. At $t > t_{jet}$, the lack of emitting fluid at angles larger than $\theta_{jet}$ leads to a steepening of the afterglow decay by $\Delta \alpha = 3/4$ for a homogeneous medium and $\Delta \alpha = 1/2$ for a wind. Simultaneously, the lateral spreading of the jet becomes important and leads to a faster deceleration of the jet, which switches from a power-law in the blast-wave radius to an exponential [@rhoads99], yielding an extra steepening of the afterglow decay of magnitude smaller or comparable to $\Delta \alpha$ above. Together, these two [*jet effects*]{} lead to $F_p \propto t^{-1}$, $\nu_i \propto t^{-2}$, $\nu_c \propto t^0$, rather independent of the ambient medium stratification, and a post jet-break forward-shock flux decay of index $\alpha = 1/3$ below $\nu_i$, while for $\nu_{obs} > \nu_i$, one obtains $\alpha = 2\beta + 1$ below $\nu_c$ and $\alpha = 2\beta$ above $\nu_c$.
[|l|lllll|lllll|]{}\
& & & &\
MODEL & $\nu<\nu_i<\nu_c$&$\nu<\nu_c<\nu_i$&$\nu_i<\nu<\nu_c$&$\nu_c<\nu<\nu_i$&$\nu_{i,c}<\nu_i$ & $\nu<\nu_i<\nu_c$&$\nu<\nu_c<\nu_i$&$\nu_i<\nu<\nu_c$&$\nu_c<\nu<\nu_i$&$\nu_{i,c}<\nu_i$\
RS(1) & 1/5 & 1/15 & 1/5 & 3/5 & 3/5 & 1/9 & 5/9 & $2/3\beta+1/3$ & 0 & $2\beta/3-1/3$\
RS(2) & 0.3 & 0.3 & $1.2\beta+0.7$ &–& $\beta+2$ & 0.3 & 0.3 & $1.5\beta+0.8$ &–& $\beta+2$\
FS(1) & -3 & -11/3 & -3 & -2 & -2 & -1/3 & 1/3 & $\beta$ & -1/2 & $\beta-1$\
FS(2) & -1 & -19/15 & $1.2\beta-0.6$ & -1/5 & $1.2\beta-0.4$ & -1/9 & 5/9 & $4/3\beta+1/3$ & 0 & $4/3\beta-2/3$\
FS(3) & -1/2 & -1/6 & $1.5\beta$ & 1/4 & $1.5\beta-0.5$ & 0 & 2/3 & $1.5\beta+0.5$ & 1/4 & $ 1.5\beta - 0.5$\
Afterglow observations
======================
[@paczynski93] were the first to predict the existence of radio afterglows following the burst phase. [@meszaros97] have analyzed two models for the reverse shock emission and one for the forward shock, predicting long-lived optical afterglows with a flux decaying as a power of time. The first detection of an afterglow and measurement of a power-law flux decay followed soon (GRB 970228 [@wijers97]), with many other optical [@kann07] and X-ray afterglows [@obrien06; @willingale07] having been observed until today.
In general, the broadband (radio, optical, X-ray) emission of GRB afterglows display the expected power-law spectra and light-curves, as well as other features, which, in chronological order of their [*prediction*]{} are: radio scintillation ([@goodman97] & [@waxman98]), optical counterpart flashes ([@sari99] & [@akerlof99]), jet-breaks ([@rhoads99] & [@kulkarni99]), dimmer afterglows for short bursts ([@alin01] & [@kann08]). GRB afterglows display sufficient diversity (e.g. wide luminosity distributions at all observing frequencies, non-universal shock micro-physical parameters) and puzzling features (slowly-decaying radio fluxes, X-ray light-curve plateaus, chromatic X-ray light-curve breaks) to challenge the standard external-shock model and warrant various modifications. Below, I discuss some of these issues.
Light-curves and spectra
------------------------
According to the temporal scalings identified in the previous section, the light-curves of GRB afterglows should display rises in the early phase, if the forward-shock emission is dominant (because the reverse-shock flux is most often expected to decay), followed by a decay, both being power-laws in time. Furthermore, the broadband afterglow spectra are expected to be rising at (radio) frequencies below the spectrum peak and fall-off at higher (X-ray) photon energies. Also expected is that the light-curve decay indices $\alpha$ and spectral slopes $\beta$ satisfy one or more closure relationships and that, there is a positive correlation between $\alpha$ and $\beta$ (from that $d\alpha/d\beta
\in [1,2]$).
Figure \[aglows\] illustrates the flux power-law decays and power-law spectra typically observed for GRB afterglows. The light-curves chosen there display long-lived power-law decays, but many afterglows exhibit more diversity, their light-curves showing two or three power-law decays, sometimes even rising at earlier times, rarely exhibiting brightening episodes (in optical or X-ray) or sudden drops (in X-ray).
The broadband spectrum of GRB afterglow 030329 shown in Figure \[aglows\] displays an optically thick part (to self-absorption) in the radio, at earlier times, a constant peak flux up to 10 days, during which the radio flux rises slowly, followed by a decreasing peak flux and decreasing radio flux. For $\nu_{obs} < \nu_i$, as required by the radio spectrum (right panel), the rise of the radio flux, $F_{GHz} \propto t^{0.54}$ (left panel), is consistent with that expected from a decelerating forward-shock interacting with a homogeneous medium ($F_\nu \propto t^{1/2}$), and marginally consistent with the forward-shock pre-deceleration emission for either a semi-relativistic reverse shock and wind medium ($F_\nu \propto t^{1/3}$) or a relativistic reverse shock and homogeneous medium ($F_\nu
\propto t^1$).
The evolution of spectral breaks is generally hard to determine observationally and use for identifying the correct afterglow model: self-absorption affects only the early radio emission, when the large flux fluctuations are caused by interstellar scintillation, while the cooling break is too shallow and evolves too slowly to be well measured even if it fell in the optical or X-ray bands. The best prospects for this test is offered by the injection break, which should cross the radio domain at tens of days, when the scintillation amplitude is reduced by the larger source size.
Sufficient radio coverage to construct radio afterglow spectra at many epochs and determine the peak frequency $\nu_i$ and flux $F_p$ is rarely achieved. GRB 030329 is one such case (Figure \[radiopk\]), the evolution of $\nu_i$ being slower than expected for a spherical blast-wave or a jet that does not expand (yet or ever) laterally (for either, $\nu_i \propto
t^{-3/2}$ for any medium stratification), while that of $F_p$ is close to that expected for a jet spreading laterally (for which $F_p \propto t^{-1}$). Thus, the evolutions of $\nu_i$ and $F_p$ for GRB afterglow 030329 seem mutually inconsistent. The slower-than-expected evolution of $\nu_i$ requires that shock micro-physical are not constant (as assumed in the standard model), but the evolution of $F_p$ can be accounted for by a spherical blast-wave provided that the ambient medium density decreases as $n \propto r^{-2.5}$.
Rising optical light-curves have been seen for more than a dozen afterglows up to 1 ks after trigger (Figure \[rises\]). If interpreted as the due to the pre-deceleration emission from the forward shock (e.g. [@molinari07]), they require smaller than average initial ejecta Lorentz factors (if the reverse shock is semi-relativistic) or longer-lasting ejections. The existence of energetic ejecta with a significantly smaller ejecta Lorentz factor could also explain the late (1–2 d) sharp rise displayed by the optical afterglow of GRB 970508 (Figure \[aglows\]).
However, late-rising afterglows could also be due to a structured outflow endowed with two “hot spots”, one moving directly toward the observer and giving the prompt GRB emission, and another one moving slightly off the direction toward the observer [@granot02A], at an angle $\theta_{obs} > \Gamma_0^{-1}$, its emission becoming visible when the outflow Lorentz factor decreases to $\Gamma = \theta_{obs}^{-1}$. The same result could be accomplished with an axially symmetric outflows having a bright core that yields the burst emission and a bright ring that produced the rising afterglow when it becomes visible. Thus, a late-rising afterglow may be a relativistic effect rather than the signature of some ejecta with a lower initial Lorentz factor, either shock (reverse or forward) being a possible origin of the rising afterglow. In fact, this model is found to account better for the peak luminosity–peak time anti-correlation exhibited by a dozen optical afterglows with early, fast rises than the pre-deceleration external-shock model [@alin08], although it should be noted that only half of that correlation is real (i.e. optical peaks do not occur later and are not brighter than a certain linear limit in log-log space) while the other half is just an observational bias, as there are many optical afterglows exhibiting decaying fluxes from first measurement that fall below the peak flux–peak time relation found for the fast-rising afterglows (Figure \[rises\]).
Most afterglow observations were made during the decay phase, where $\alpha$ and $\beta$ are expected to be correlated and satisfy one or more closure relationships. The left panel of Figure \[ab\] shows the temporal and spectral indices of optical and X-ray afterglows measured before the jet break and the expectations for the (post-deceleration) forward-shock model (as it seems more likely that the reverse shock dominates the afterglow emission only until at most 1 ks). Surprisingly, no significant correlation can be seen between $\alpha$ and $\beta$, which may be taken either as indication that the standard forward-shock model does not account for the diversity of afterglows (e.g. departures from its assumptions of constant shock parameters would be required to explain the decays below the “S1” model, which are too slow) or that more than one variant of it realized which, combined with a small baseline in $\alpha$ and $\beta$, requires a much larger sample to reveal the expected underlying correlation.
Early optical afterglows
------------------------
A bright optical emission arising from the reverse shock was predicted by [@sari99] and may have been observed for the first time in the optical counterpart (i.e. during the burst) accompanying GRB 990123 [@akerlof99] and in the early afterglow emission following GRB 021211, but without any further candidates until recently. Lacking a continuous, long-lived injection of new ejecta, and because of the adiabatic cooling, the reverse-shock emission should be confined to the early afterglow. Then, the early afterglow emissions of GRB 990123 and 021211 being brighter than the extrapolation of the later flux and decaying faster are two reasons for attributing those two early optical emissions to the reverse shock. The larger brightness (by 2.5 mag for 990123 and by 1 mag for 021211) could be explained by that the number of ejecta electrons is, at deceleration (i.e. around burst end), larger by a factor $\Gamma_0 \sim 100$ than in the forward shock, and by a smaller factor at later times (as for 021211), with some relative dimming of the reverse shock optical flux attributed to the peak of the reverse-shock synchrotron spectrum being lower than that of the forward shock.
However, both the above reasons disappear if the origin of time is not at trigger but sometime later, e.g. at 30–40 s (corresponding to the peak of GRB 990123 optical flash). Then, the early optical emission appears as a small deviation of the extrapolation of the late flux and the entire afterglow light-curve is consistent with a single power-law, indicating a unique dissipation mechanism. Thus, absent spectral information, the evidence for a reverse shock origin of the early optical emissions of GRB 990123 and 021211 is circumstantial.
Such spectral information has been acquired only recently, for the early optical emission of GRB afterglows 061126 and 080319B. For the former, [@perley08] finds that the steeper-decaying early (up to 200 s) optical emission is harder than at later times (after 1 ks). The indices $\alpha \simeq 2.0$ and $\beta \simeq 0.9$ of the early optical emission of GRB 061126 are consistent with the closure relation expected for adiabatic cooling ejecta. The spectral evolution observed simultaneously with the slowing of light-curve decay suggests the emergence of a different component after 1 ks.
The optical afterglow of GRB 080319B also displayed a spectral hardening simultaneous with the reduction in the flux decay rate [@wozniak08], supporting a reverse-shock origin of the early fast-decay phase and forward-shock origin of the later slower-decaying emission. However, the decay at early times is too fast (for the measured spectral slope) to be attributed to the adiabatic cooling of ejecta. In fact, the decay index $\alpha
\simeq 2 +\beta$ is consistent with the expectations for the large-angle emission released during the burst. However, that does not exclude a reverse-shock origin of the early optical flux, as the cooling frequency may have fallen below the optical, revealing the large-angle emission.
The slow softening of the optical spectrum of GRB afterglow 080319B after 1 ks was a surprise. If the rather flat ($\beta = 0$) spectral slope at 1 ks were due to the peak energy of the forward-shock synchrotron spectrum being in the optical, then a much faster softening is expected, given that the injection frequency evolution $\nu_i \propto t^{-3/2}$ is also fast. Energy injection in the blast-wave or an increasing electron/magnetic shock parameters could account for slow decrease of $\nu_i$ required by the slow spectral softening of GRB 080319B afterglow optical emission.
Jet-breaks
----------
A tight collimation of GRB ejecta, into a jet of half-aperture less than 10 degrees, is desirable to reduce the isotropic-equivalent GRB output, reaching $10^{54.5}$ erg, to lower values, below $10^{52.5}$ erg, compatible with what the mechanisms for production of relativistic jets by solar-mass black-holes can yield. Besides the $\alpha-\beta$ closure relationship being satisfied self-consistently (i.e. by a forward-shock model with same features before and after the jet-break), [*achromaticity*]{} of the break (i.e. simultaneous occurrence at all frequencies) is an essential test of this model.
The steepening of the afterglow flux decay due to collimation of ejecta was predicted by [@rhoads99] and was observed for the first time in the optical emission of GRB afterglow 990123 [@kulkarni99]. About 3/4 of well-monitored pre-Swift optical afterglows displayed jet-breaks at 0.5–3 day, as shown in the compilation of [@zeh06]. The X-ray coverage of pre-Swift afterglow extended over at most 1 decade in time and was insufficient to test for the existence of an X-ray light-curve break simultaneous with that seen in the optical.
A smaller fraction, between 1/3 and 2/3, of Swift X-ray afterglows also exhibit jet-breaks [@alin07A], defined as a steepening occurring after 0.1 day from a power-law decay with $\alpha \siml 1$ to one with $\alpha > 1.5$. Comparing the post jet-break temporal and spectral indices with the expectations for the forward-shock model (right panel of Figure \[ab\]) shows that that model accounts for observations of post jet-break decays if jets are both both spreading and conical. However, just as for pre jet-break decays, the expected $\alpha-\beta$ correlation is not seen.
While there are many examples of potential jet-breaks in the X-ray light-curve monitored by Swift, few are sufficiently well-monitored in the optical to test for the achromaticity of the break. Figure \[jets\] shows the only 3 afterglows sufficiently sampled and followed sufficiently late to search for achromatic light-curve breaks. Besides the simultaneity of the optical and light-curve breaks, note the equality of the pre and post-break decay indices.
The smaller fraction of Swift X-ray afterglows that exhibit jet-breaks, relative to that of pre-Swift such optical afterglows, could be due to Swift detecting and localizing afterglows that are fainter than those followed in the optical prior to Swift. The argument here is that, if all jets had the same energy (e.g. [@frail01]), then the afterglow flux and jet-break time should be anti-correlated: $F_\nu \propto dE/d\Omega \propto \theta_{jet}^{-2}$ and $t_{jet} \propto (dE/d\Omega) \theta_{jet}^4 \propto \theta_{jet}^2$ (for a wind-like medium), leading to $F_\nu \propto t_{jet}^{-1}$, where $dE/d\Omega$ is the forward-shock’s kinetic energy per solid angle, thus dimmer afterglows should display later jet-breaks.
As shown in Figure \[ox\], the afterglows with jet-breaks at 0.3–10 days are brighter by a factor $\sim 10$ than those without jet-breaks detected until 10 days, thus the anti-correlation between afterglow flux and jet-break time expected for a universal jet energy is confirmed, even though jet energies inferred from the timing of afterglow light-curve breaks have a broad distribution (e.g. [@ghirlanda04]). That ratio of 10 between the average brightness of afterglows with breaks and of those without breaks until 10 days and the above-derived $F_\nu \propto t_{jet}^{-1}$ imply that the latter type of afterglows should display a break at 3–100 days, which could be missed if monitoring does not extend for sufficiently long times. For this reason, some of the dimmer X-ray afterglows detected by Swift may have breaks that are too late to be observed, leading to an apparent paucity of Swift X-ray afterglows with jet-breaks, as noted by [@burrows06].
Slowly-decaying radio afterglows
--------------------------------
The long-time monitoring of radio afterglows showed that often there is an incompatibility between the radio and optical flux decays. After the peak of the forward-shock synchrotron spectrum falls below the radio domain, which should happen within $\sim 10$ days and is, indeed, observed in the radio spectra of GRB afterglows 970508 [@frail00], 021004, and 030329 (Figure \[radiopk\]), the radio and optical flux decays are expected to be similar, up to a difference $\delta \alpha = \pm 1/4$ that could occur if the cooling break is in between radio and optical and if the jet is not laterally spreading.
In a set of nine pre-Swift afterglows with long temporal coverage at both frequencies, I find that the above expectation is met by only four: GRB afterglows 980703, 970508, 000418, and 021004, but that the radio flux of GRB afterglows 991208, 991216, 000301C, 000926, and 010222 decay much slower than in the optical, with $\alpha_{opt} - \alpha_{rad} = 0.8, 0.8,
1.3, 1.6, 1.5$, respectively [@alin04A] (see also [@frail04]). For all the above three cases for which the optical and radio decays are well-coupled, the decays are slower than $\alpha = 1.5$, indicating a wide jet, while for all the five cases of decoupled radio and optical decays, the optical displays a decay steeper than $\alpha = 1.5$ after a $\sim 1$ day break that could be interpreted as a jet-break. Thus, whenever the optical flux decays fast, there seems to be a mechanism which produces radio emission in excess of that expected for the forward-shock model. An example of each type of radio afterglow is shown in Figure \[or\].
Because the slow radio flux decay is observed [*at the same time*]{} as the faster optical decay, the decoupling of radio and optical light-curves cannot be attributed to energy injection in the blast-wave, to a structured outflow, or to evolving micro-physical parameters (mechanisms which have also been used to explain the slow early decays seen in Swift X-ray afterglow plateaus), nor to the transition to non-relativistic dynamics. Instead, the decoupled radio and optical light-curve decays may indicate that these emissions arise from different parts of the relativistic outflow.
That could happen if the outflow endowed with angular structure (in the sense that its kinetic energy per solid angle is anisotropic), with a core that dominates the optical emission, yielding the jet-break, and an outer, wider envelope that produces the radio emission. The problem with this model [@alin04A] is that, optical and radio afterglows being long-lived, the spectral break frequencies of the core and envelope emissions evolve substantially, making it impossible for their emissions to be dominant over such long timescales at only one frequency, i.e. without “interfering” with the emission of the other part of the outflow. Shortly put, it is quite likely that the emission from the radio envelope would soon dominate the optical emission from the core and change the initially steep optical flux decay into a slower one. (While that is a general issue for explaining decoupled afterglow light-curves with a structured outflow, [@racusin08] shows that it can be avoided for GRB 080319B, whose optical and X-ray light-curve decays are decoupled for until 1 day).
Another possibility is that the optical emission arises in the forward shock while the radio is from the reverse shock. For adiabatic cooling, the ejecta emission should decay slowly, even when observations are at a frequency below that of the spectral peak, thus a reverse shock energizing the ejecta is required to account for the flat or slowly rising part of radio light-curves (up to about 10 days). In this case, the light-curve decay depends on the law governing the injection of fresh ejecta into the reverse shock, the observed radio light-curve indices being close to the closure relations derived in the previous section for a uniform radial distribution of the incoming ejecta mass.
In the reverse-forward shock model for afterglows with different radio and optical light-curve decays, the cross-interference issue may also exist, as the forward-shock synchrotron peak flux, being larger than $\sim 0.1$ mJy at 1 day (to account for the $\sim$ 20 magnitude optical flux), could over-shine the reverse-shock radio emission at some later time, when the peak energy of the forward-shock emission spectrum reaches the radio domain. This issue is best addressed with numerical calculations of the blast-wave dynamics and radiation. In this way, I found [@alin05] that the most likely solution for the decoupled radio and optical light-curves is that the radio afterglow emission is dominated by the reverse shock during the first decade in time, with the forward-shock emission peaking in the radio at about 100 days, overtaking that from the reverse shock sometime during the second decade. By itself, each component would display a decay faster than observed, but their sum resembles a shallow power-law over two decades in time.
X-ray plateaus and chromatic breaks
-----------------------------------
The Swift satellite has opened a new temporal window for observations of X-ray afterglows, which previously were monitored by BSAX only after 8 hours after trigger. The major surprise (i.e. a feature not predicted) in Swift observations was that, although it appeared that the X-ray afterglow emission at hours and days extrapolated back to the burst time would match the GRB flux, implying a smooth transition from counterpart to afterglow emission, the X-ray flux from burst end to several hours is much less than that back-extrapolation, displaying at 0.3–10 ks a phase of slow decay, with $\alpha \in (0,3/4)$. In fact, that should have been a partial surprise because BSAX has observed a sharply decaying GRB tail in at least three cases, indicating that a phase of slow X-ray decay must exist at the burst end. Figure \[0315\] illustrates the “plateau” phase observed for GRB afterglow 050315.
In the simplest form of the blast-wave model, the magnitude of the light-curve decay steepening at the end of the plateau requires the peak of the synchrotron spectrum to fall below the X-ray band at the end of the plateau. However, that explanation is ruled out by that, observationally, the plateau end is most often not accompanied by the a spectral evolution [@nousek06; @willingale07; @liang07], although exceptions exist [@butler07].
Because the plateau phase is followed by a “normal” decay, compatible with the expectations of the standard forward-shock model, it is natural to think that departures from the assumptions of that standard model are the cause of X-ray plateaus: (1) increase of the average energy per solid angle of the blast-wave area visible to the observer by means of (1a) energy injection in the blast-wave owing to some late ejecta catching-up with the forward-shock [@nousek06; @zhang06; @alin06], or by absorbing low-frequency electromagnetic radiation from a millisecond pulsar [@zhang06], (1b) an anisotropic outflow [@alin06; @eichler06], (2) evolving shock micro-physical parameters [@fan06; @ioka06], and (3) blast-wave interacting with an “altered” ambient medium, shaped by a GRB precursor [@ioka06]. The effect on the afterglow flux decay of the above mechanisms for a variable “apparent” kinetic energy of the blast-wave were first investigated by [@meszaros98; @rees98], the X-ray plateaus discovered by Swift several years later providing the first tentative confirmation that those mechanisms may be at work.
However, the discovery of [*chromatic*]{} light-curve breaks at the end of the X-ray plateau [@fan06; @watson06; @alin06A], which are not seen in the optical as well, soon showed that neither of the above mechanisms for X-ray plateaus provide a complete picture of the afterglow phenomenon, as in all those models the break should be [*achromatic*]{}, manifested at all frequencies. Evolution of shock parameters for electron and magnetic field energies could “iron out” the optical light-curve break produced by the other mechanisms listed above, provided that the cooling frequency is between optical and X-ray (to allow a way of decoupling the optical and X-ray light-curves), however there is no reason for their evolution to conspire and hide the optical light-curve break so often (universality of the required micro-physical parameter evolutions with blast-wave Lorentz factor would provide some support to this contrived model).
To date, I find that there are 11 good cases of chromatic X-ray breaks, 6 good cases of achromatic breaks, and 3 afterglows displaying well-coupled optical and X-ray light-curves: a single power-law decay, of same decay index at both frequencies, extending over over three decades in time. Figure \[xbr\] shows two examples of achromatic breaks, one with discrepant post-break optical and X-ray decay indices, and one chromatic X-ray break.
To explain the chromatic X-ray breaks, [@uhm07] and [@genet07] have proposed that the [*entire*]{} afterglow emission is produced by the reverse shock and have shown that, by placing either the injection or the cooling frequency between optical and X-ray, decoupled light-curves can be obtained. Just as energy injection in the blast-wave, this mechanism relies on the existence of a long-lived central engine that expels ejecta until the last afterglow measurement (the existence of a reverse shock at days is also required by the slow decays seen in a couple of radio afterglows). On the other hand, the transition observed in two optical afterglows from a fast-decaying phase to one of slower decay at 1 ks, accompanied by spectral evolution, argue in favour of the reverse-shock emission being dominant only up to 1 ks, after which it seems more natural to attribute the afterglow emission to the forward-shock. Furthermore, the results shown by [@uhm07] show a softening of the reverse-shock optical spectrum at the transition from fast to slow decay, which is in contradiction with the hardening observed in GRB afterglows 061126 and 080319B. Thus, the reverse-shock model may not provide a correct description of the entire afterglow emission.
On average, the X-ray to optical flux ratio is larger for afterglows with chromatic X-ray breaks than for afterglows with coupled optical and X-ray light-curves (i.e. with achromatic breaks or single power-law decays), as shown in Figure \[fxfo\]. This indicates that chromatic X-ray breaks are due to a mechanism whose emission over-shines an underlying one only in the X-rays, but not in the optical. Thus, it seems that the diversity of optical vs. X-ray light-curve behaviours should be attributed to the existence of two mechanisms for afterglow emission and not to a unique origin.
So far, three proposals along that line have been put forth: dust-scattering of the blast-wave emission, bulk and inverse-Compton scattering of the same emission, and a central-engine mechanism that could be the same internal shocks in a variable wind that are believed to produce the prompt burst emission [@rees94].
[@shao07] have proposed that X-ray plateaus result from scattering by dust in the host galaxy, much like the expanding rings produced by dust-scattering in our Galaxy in GRB afterglows 031203 [@vaughan04] and 050713A [@tiengo06]. However, for dust-scattering, harder photons are those scattered at a smaller angle, thus they arrive earlier at observer, leading to a strong spectral softening of the X-ray light-curve plateau, of $\delta \beta \simeq 2$, and to a strong dependence of the plateau duration on the photon energy, $\Delta t \propto \nu^{-2}$, both of which are clearly refuted by afterglow observations [@shen08].
[@ghisellini08] have proposed that, in some afterglows, the “central engine” makes a substantial contribution to the afterglow X-ray flux. This model requires a central engine that operates until the last afterglow detection; the dissipation mechanism may be shocks in a variable outflow, which can also account for the bright and short-lived flares observed in many Swift X-ray afterglows (e.g. [@burrows07; @chincarini07]).
Given that the forward-shock model with cessation of energy injection at the plateau end can explain the achromatic breaks and that the standard forward-shock model accounts for the coupled single power-law light-curves, it would be more desirable to identify a mechanism for producing decoupled X-ray and optical light-curves that is related to the forward shock and which dominates its emission only occasionally.
Bulk and inverse-Compton scattering of the forward-shock photons by an outflow interior to the blast-wave is such a mechanism. In this model [@alin08A], all the afterglow emission originates in the forward shock, which explains so naturally the long-lived, power-law decay of GRB afterglows, coupled optical and X-ray light-curves resulting when the scattered emission is dimmer than the forward-shock’s, while chromatic X-ray light-curve breaks occur when the scattered emission is dominant in the X-ray. For this model to work, the scattering outflow must be almost purely leptonic, to ensure a sufficiently high (sub-unity) optical depth to electron scattering to account for the observed X-ray flux, assuming that the kinetic energy of the scattering outflow is not much larger than that of the forward shock. The same outflow also injects energy into the blast-wave and, if that energy is larger than the forward shock’s, then it mitigates the blast-wave deceleration, producing a light-curve plateau ending with an achromatic break when the injected energy falls below that of the forward shock and stops being dynamically important. Therefore, a delayed outflow is the origin of both chromatic and achromatic light-curve breaks, the former occurring when the scattered emission is dominant, while the latter happening when the forward-shock emission is dominant. Still, the achromatic break of GRB 050730 (Figure \[xbr\]), followed by an X-ray flux decay much steeper than that of the optical cannot be explained with only cessation of energy injection at the time of the break, and requires an extra feature, that the shock micro-physical parameters are not constant.
The above scattering model also explains late X-ray flares, which arise from dense or hot (i.e. with relativistic electrons) sheets within the outflow. When dominant, the scattered X-ray emission received at time $t$ reflects the properties (density, Lorentz factor) of the outflow at $ct/(z+1)$ behind the forward shock, sharp drops of the X-ray flux as that observed for GRB afterglow 070110 at 30 ks being due to a gap in the scattering outflow.
For an instantaneous release of all the ejecta, the kinematics of outflow radial-spreading owing to different initial Lorentz factors, followed by deceleration of the forward-shock, leads to that, when the ejecta of Lorentz factor $\Gamma_{sc}$ catch up with the forward shock, moving at $\Gamma_{fs}$, the Lorentz factor contrast is $\Gamma_{sc}/\Gamma_{fs} = \sqrt{4-s}$ (i.e. 2 for a homogeneous medium and $\sqrt{2}$ for a wind). That ratio is too small for bulk-scattering to boost enough the forward-shock emission to dominate that arriving directly from the forward shock. But, if the scattering outflow was energized by internal shocks, inverse-Compton scatterings by relativistic electrons (of comoving frame energy $\gamma_e
m_e c^2$) can achieve that goal. In fact the properties of the scattered emission depend only on the product $\Gamma_{sc} \gamma_e$. Thus, for a sudden release of ejecta to lead to a sufficiently bright scattered emission, the scattering outflow should be hot. Alternately, if the scattering outflow is cold, then the larger ratio $\Gamma_{sc}/\Gamma_{fs}
\simg 100$ necessary for the scattered flux to over-shine that from the forward shock requires a long-lived engine.
As a general test of all models that explain decoupled optical and X-ray afterglows by attributing them to different mechanisms, there should not be any afterglows whose optical and X-ray light-curves evolve from decoupled (i.e. with a chromatic break) to coupled (i.e. with an achromatic break), or vice-versa, unless one of the light-curves displays a sudden flux or spectral change that would indicate a second mechanism becoming dominant. So far, I find only two cases of afterglows whose optical and X-ray light-curves evolve from decoupled to coupled (GRB afterglows 070110 and 080319B), but their X-ray light-curves display, indeed, a sharp drop (at 20 ks, in both cases).
Conclusions
===========
The temporal and spectral properties of GRB afterglows are, in general, consistent with those predicted for the synchrotron emission from the blast-wave produced when highly relativistic ejecta (initial Lorentz factor above 100) interact with the ambient medium.
As expected from shocks accelerating particles with a power-law distribution with energy, power-laws are observed in the optical and X-ray afterglow continua. A spectral softening (i.e. decrease of peak frequency) is expected owing to the blast-wave deceleration and is observed in radio afterglows spectra and in the behaviour of afterglow radio light-curves, which rise slowly until the synchrotron spectrum peak reaches the radio domain and fall-off afterward.
Rising afterglow light-curves are seen at early times in the optical, and are consistent with the pre-deceleration emission from the forward shock, although a structured outflow with a hot-spot that gradually becomes visible to the observer is also possible, in which case the rising afterglow emission could also be explained with the reverse shock.
Much more often, afterglow light-curves display power-law decays of indices that are not correlated with the spectral slopes, as would be expected for the external-shock model. That inconsistency could be due to a substantial intrinsic scatter in decay indices and spectral slopes, owing to more than one variant of the blast-wave model occurring in GRB afterglows, combined with a small range of those indices being realized.
Bright optical flashes accompanying the burst emission were predicted to arise from the reverse shock, owing to the larger number of ejecta electrons than in the forward shock. Fast-falling optical light-curves have been observed at 100–1000s in two afterglows (991023 and 021211), followed by a slower of the decay, which was taken as evidence for the reverse-shock emission dominating the early afterglow, although spectral information was not available to test that hypothesis. More recently, the early optical spectral slopes were measured for two afterglows (061126 and 080319B), the decay of the former being consistent with that from adiabatically-cooling ejecta, while the later is faster than expected and consistent with it being the large-angle emission released at an earlier time.
For the above two optical afterglows with spectral information at early times, the slowing of the optical flux decay at 1 ks is accompanied by a spectral evolution, which indicates the transition from one mechanism to another. Most naturally, that is the transition from ejecta emission to forward-shock emission, with the reverse-shock emission being relevant for the optical afterglow only during its early phase.
Evidence for a reverse-shock emission is also provided by the slow radio flux decays observed after 10 day in several afterglows. Adiabatically cooling ejecta would yield a decay faster than observed, particularly if the synchrotron cooling frequency were to fall below the radio, thus a reverse-shock accelerating ejecta electrons is required to operate for days and produce a radio emission decaying much slower than the optical at the same time, the latter being attributed to the forward shock. Together with the above conclusion regarding the contribution of the reverse shock to the early optical afterglow, this suggests that the reverse shock is the main afterglow source for a duration that decreases with observing frequency, perhaps never being dominant in the X-rays and having no connection with the chromatic X-ray light-curve breaks seen at $\sim 10$ ks in most afterglows.
That the X-ray-to-optical flux-ratio is larger (by a factor 5) for afterglows with chromatic X-ray light-curve breaks than for those with coupled optical and X-ray light-curves (i.e. with achromatic breaks or similar power-law decays), indicates the existence of a different mechanism producing the X-ray emission of afterglows with chromatic X-ray breaks, coupled light-curves resulting when the emission from that novel mechanism is negligible. Long-lived internal shocks or scattering of the blast-wave emission by an outflow located behind it are two possibilities that could explain chromatic X-ray breaks, as well as the flares seen in many X-ray light-curves. The latter mechanism is also related to energy injection in the blast-wave, whose cessation accounts naturally for achromatic light-curve breaks. Thus, in the scattering model, the diversity of optical and X-ray light-curve relative behaviours is attributed to the interplay between the scattered and direct blast-wave emissions, combined with the changing dynamics of the blast-wave produced when the scattering outflow brings into the shock more energy than already existing.
On energetic grounds, GRB outflows should be collimated into jets narrower than 10 degrees. The steepening of the afterglow flux decay when the jet boundary becomes visible to the observer was another major prediction confirmed by observations. Just as for the pre jet-break phase, more than one jet model is required to account for the measured decay indices (given the observed spectral slopes). Swift X-ray afterglows display light-curve jet-breaks less often than pre-Swift optical afterglows, which could be due to that the former afterglows (being dimmer) arise from wider jets whose jet-breaks occur later and could, thus, be missed more often.
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abstract: 'In this work, we describe a set of rules for the design and initialization of well-conditioned neural networks, guided by the goal of naturally balancing the diagonal blocks of the Hessian at the start of training. Our design principle balances multiple sensible measures of the conditioning of neural networks. We prove that for a ReLU-based deep multilayer perceptron, a simple initialization scheme using the geometric mean of the fan-in and fan-out satisfies our scaling rule. For more sophisticated architectures, we show how our scaling principle can be used to guide design choices to produce well-conditioned neural networks, reducing guess-work.'
author:
- |
Aaron Defazio & Léon Bottou\
Facebook AI Research New York
bibliography:
- 'icml2019cond.bib'
nocite: '[@*]'
title: 'Scaling Laws for the Principled Design, Initialization and Preconditioning of ReLU Networks'
---
Introduction
============
The design of neural networks is often considered a black-art, driven by trial and error rather than foundational principles. This is exemplified by the success of recent architecture random-search techniques [@NAS; @Li2019RandomSA], which take the extreme of applying no human guidance at all. Although as a field we are far from fully understanding the nature of learning and generalization in neural networks, this does not mean that we should proceed blindly.
This work derives various scaling laws by investigating a simple guiding principle:
> All else being equal, the diagonal blocks of the Hessian corresponding to each weight matrix should have similar average singular values.
This condition is important when the stochastic gradient algorithm is used and can help even when adaptive optimization methods are used, as they have no notion of the correct conditioning right at initialization.
In this work we define a scaling quantity $\gamma_{l}$ for each layer $l$ that approximates the average singular value, involving the second moments of the forward-propagated values and the second moments of the backward-propagated gradients. We argue that networks with constant $\gamma_{l}$ are better conditioned than those that don’t, and we analyze how common layer types affect this quantity. We call networks that obey this rule *preconditioned* neural networks, in analogy to preconditioning of linear systems.
As an example of some of the possible applications of our theory, we:
- Propose a *principled* weight initialization scheme that can often provide an improvement over existing schemes;
- Show which common layer types automatically result in well-conditioned networks;
- Show how to improve the conditioning of common structures such as bottlenecked residual blocks by the addition of fixed scaling constants to the network (Detailed in Appendix \[sec:resnet\]).
Notation
========
We will use the multilayer perceptron (i.e. a classical feed-forward deep neural network) as a running example as it is the simplest non-trivial deep neural network structure. We use ReLU activation functions, and use the following notation for layer $l$ (following [@kaiming-rectifiers2015]): $$y_{l}=W_{l}x_{l}+b_{l},$$ $$x_{l+1}=ReLU(y_{l}),$$ where $W_{l}$ is a $n_{l}^{\text{out}}\times n_{l}^{\text{in}}$ matrix of weights, $b_{l}$ is the bias vector, $y_{l}$ the preactivation vector and $x_{l}$ is the input activation vector for the layer. The quantities $n_{l}^{\text{out}}$ and $n_{l}^{\text{in}}$ are called the fan-out and fan-in of the layer respectively. We also denote the gradient of a quantity with respect to the loss (i.e. the back-propagated gradient) with the prefix $\Delta$. We initially focus on the least-squares loss. Additionally, we assume that each bias vector is initialized with zeros unless otherwise stated.
Conditioning by balancing the Hessian
=====================================
Our proposed approach focuses on the spectrum of the diagonal blocks of the Hessian. In the case of a MLP network, each diagonal block corresponds to the weights from a single weight matrix $W_{i}$ or bias vector $b_{i}$. This block structure is used by existing approaches such as K-FAC and variants [@martens-grosse-2015; @kronconv; @martens2017; @george2018fast], which correct the gradient step using estimates of second-order information. In contrast, our approach modifies the network to improve the Hessian without modifying the step.
When using the ReLU activation function, as we consider in this work, a neural network is no longer a smooth function of its inputs, and the Hessian becomes ill-defined at some points in the parameter space. Fortunately, the spectrum is still well-defined at any twice-differentiable point, and this gives a local measure of the curvature. ReLU networks are typically twice-differentiable almost everywhere, which is the case when none of the activations or weights are exactly 0 for instance. We assume this throughout the remainder of this work.
GR scaling: A measure of Hessian average conditioning
-----------------------------------------------------
Our analysis will proceed with batch-size 1 and a network with $k$ outputs. We consider the network at initialization, where weights are centered, symmetric and i.i.d random variables, and biases are set to zero.
ReLU networks have a particularly simple structure for the Hessian with respect to any set of activations, as the network’s output is a piecewise-linear function $g$ fed into a final layer consisting of a loss. This structure results in greatly simplified expressions for diagonal blocks of the Hessian with respect to the weights.
We will consider the output of the network as a composition two functions, the current layer $g$, and the remainder of the network $h$. We write this as a function of the weights, i.e. $f(W_{l})=h(g(W_{l}))$. The dependence on the input to the network is implicit in this notation, and the network below layer $l$ does not need to be considered.
Let $R_{l}=\nabla_{y_{l}}^{2}h(y_{l})$ be the Hessian of $h$, the remainder of the network after application of layer $l$ (recall $y_{l}=W_{l}x_{l}$). Let $J_{l}$ be the Jacobian of $y_{l}$ with respect to $W_{l}$. The Jacobian has shape $J_{l}:n_{l}^{\text{out}}\times\left(n_{l}^{\text{out}}n_{l}^{\text{in}}\right)$. Given these quantities, the diagonal block of the Hessian corresponding to $W_{l}$ is equal to: $$G_{l}=J_{l}^{T}R_{l}J_{l}.$$ The *$l$th diagonal block of the (Generalized) Gauss-Newton* matrix $G$ [@martens-insights]. We will use this fact to simplify our analysis. We discuss this decomposition further in Appendix \[subsec:gauss-newton\]. Note that each row of $J_{l}$ has $n_{l}^{\text{in}}$ non-zero elements, each containing a value from $x_{l}$. This structure can be written as a block matrix, $$J_{l}=\left[\begin{array}{ccc}
x_{l} & 0 & 0\\
0 & x_{l} & 0\\
0 & 0 & \ddots
\end{array}\right],\label{eq:j-matrix}$$ Where each $x_{l}$ is a $1\times n_{l}^{\text{in}}$ row vector. This can also be written as a Kronecker product with an identity matrix as $I_{n_{l}^{\text{out}}}\otimes x_{l}$.
Our quantity of interest is the average squared singular value of $G_{l}$, which is simply equal to the (element-wise) second moment of the product of $G$ with a i.i.d normal random vector $r$: $$E[\left(G_{l}r\right)^{2}]=E[\left(J_{l}^{T}R_{l}J_{l}r\right)^{2}].$$
(The GR scaling) Under the assumptions outlined in Appendix \[subsec:assumptions1\], $E[\left(G_{l}r\right)^{2}]$ is equal to the following quantity, which we call the GR scaling for MLP layers: $$\text{(GR scaling) \, }\gamma_{l}\doteq n_{l}^{\text{in}}E\left[x_{l}^{2}\right]^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.$$ We define a “balanced” or “preconditioned” network as one in which $\gamma_{l}$ is equal for all $l$ (full derivation in Appendix \[sec:GR-scaling-derivation\]).
![\[fig:ratio-lenet\]Distributions of the ratio of theoretical scaling to actual for a strided LeNet network](boxplots_random2){width="100.00000%"}
Balancing this theoretically derived GR scaling quantity in a network will produce an initial optimization problem for which the blocks of the Hessian are expected to be approximately balanced with respect to their average singular value.
Due the the large number of approximations needed for this derivation, including complete independence between forward and backward signals (Appendix \[subsec:assumptions1\]), we don’t claim that this theoretical approximation is accurate, or that the blocks will be closely matched in practice. Rather, we make the lesser claim that a network with very disproportionate values of $\gamma_{l}$ between layers is likely to have more convergence difficulties during the early stages of optimization then one for which the $\gamma_{l}$ are balanced.
To check the quality of our approximation, we computed the ratio of the convolutional version of the GR scaling equation (Equation \[eq:conv-gnr\]) to the actual $E[\left(G_{l}r\right)^{2}]$ product for a strided (rather than max-pooled, see Table \[tab:scalingtable\]) LeNet model, where we use random input data and a random loss (i.e. for outputs $y$ we use $y^{T}Ry$ for an i.i.d normal matrix $R$), with batch-size 1024, and $32\times32$ input images. The results are shown in Figure \[fig:ratio-lenet\] for 100 sampled setups; there is generally good agreement with the theoretical expectation.
Preconditioning balances weight-to-gradient ratios
==================================================
We provide further motivation for the utility of $GR$ preconditioning by comparing it to another simple quantity of interest. Consider at network initialization, the ratio of the (element-wise) second moments of each weight-matrix gradient to the weight matrix itself: $$\nu_{l}\doteq\frac{E[\Delta W_{l}^{2}]}{E[W_{l}^{2}]}.$$ This ratio approximately captures the relative change that a single SGD step with unit step size on $W_{l}$ will produce. We call this quantity the weight-to-gradient ratio. When $E[\Delta W_{l}^{2}]$ is very small compared to $E[W_{l}^{2}]$, the weights will stay close to their initial values for longer than when $E[\Delta W_{l}^{2}]$ is large. In contrast, if $E[\Delta W_{l}^{2}]$ is very large compared to $E[W_{l}^{2}]$, then learning can be expected to be unstable, as the sign of the elements of $W$ may change rapidly between optimization steps.
In either case the global learning rate can be chosen to correct the step’s magnitude, however this affects all weight matrices equally, possibly making the step too small for some weight matrices and too large for others. By matching $\nu_{l}$ across the network, we avoid this problem. Remarkably, this weight-to-gradient ratio turns out to be equivalent to the GR scaling for MLP networks:
(Appendix \[sec:variance-ratio\]) $\nu_{l}$ is equal to the GR scaling $\gamma_{l}$ for i.i.d mean-zero randomly-initialized multilayer perceptron layers under the independence assumptions of Appendix \[subsec:assumptions1\].
Preconditioning of neural networks via initialization {#sec:precond-geom}
=====================================================
For ReLU networks with a classical multilayer-perceptron (i.e. non-convolutional, non-residual) structure, we show in this section that initialization using i.i.d mean-zero random variables with second moment inversely proportional to the geometric mean of the fans: $$E[W_{l}^{2}]=\frac{c}{\sqrt{n_{l}^{\text{in}}n_{l}^{\text{out}}}},\label{eq:geom-init}$$ for some fixed constant $c$, gives a constant GR scaling throughout the network.
\[prop:feedforward\]Let $W_{0}:m\times n$ and $W_{1}:p\times m$ be weight matrices satisfying the geometric initialization criteria of Equation \[eq:geom-init\], and let $b_{0},b_{1}$ be zero-initialized bias parameters. Then consider the following sequence of two layers where $x_{0}$ and $\Delta y_{1}$ are i.i.d, mean 0, uncorrelated and symmetrically distributed: $$y_{0}=W_{0}x_{0}+b_{0},\quad x_{1}=ReLU(y_{0}),\quad y_{1}=W_{1}x_{1}+b_{1}.$$ Then $\nu_{0}=\nu_{1}$ and so $\gamma_{0}=\gamma_{1}$.
Note that the ReLU operation halves both the forward and backward second moments, due to our assumptions on the distributions of $x_{0}$ and $\Delta y_{1}$. So: $$E[x_{1}^{2}]=\frac{1}{2}E[y_{0}^{2}],\quad E[\Delta y_{0}^{2}]=\frac{1}{2}E[\Delta x_{1}^{2}].\label{eq:relu-fowardhalve}$$ Consider the first weight-gradient ratio, using $E[\Delta W_{l}^{2}]=E[x_{l}^{2}]E[\Delta y_{l}^{2}]$: $$\frac{E[\Delta W_{0}^{2}]}{E[W_{0}^{2}]}=\frac{1}{c}E[x_{0}^{2}]E[\Delta y_{0}^{2}]\sqrt{nm}.$$ Under our assumptions, backwards propagation to $\Delta x_{1}$ results in $E[\Delta x_{1}^{2}]=pE[W_{1}^{2}]E[\Delta y_{1}^{2}]$ , so: $$\begin{aligned}
E[\Delta y_{0}^{2}] & =\frac{1}{2}E[\Delta x_{1}^{2}]=\frac{1}{2}pE[W_{1}^{2}]E[\Delta y_{1}^{2}]=\frac{1}{2}p\frac{c}{\sqrt{mp}}E[\Delta y_{1}^{2}],\end{aligned}$$ So: $$\begin{aligned}
\frac{E[\Delta W_{0}^{2}]}{E[W_{0}^{2}]} & =\frac{1}{2c}p\frac{c}{\sqrt{mp}}\sqrt{nm}E[x_{0}^{2}]E[\Delta y_{1}^{2}]=\frac{1}{2}\sqrt{np}E[x_{0}^{2}]E[\Delta y_{1}^{2}].\label{eq:linear-ratio-1}\end{aligned}$$ Now consider the second weight-gradient ratio: $$\frac{E[\Delta W_{1}^{2}]}{E[W_{1}^{2}]}=\frac{1}{c}\sqrt{pm}E[x_{1}^{2}]E[\Delta y_{1}^{2}].$$ Under our assumptions, applying forward propagation gives $E[y_{0}^{2}]=nE[W_{0}^{2}]E[x_{0}^{2}],$ and so from Equation \[eq:relu-fowardhalve\] we have: $$\begin{aligned}
E[x_{1}^{2}] & =\frac{1}{2}nE[W_{0}^{2}]E[x_{0}^{2}]=\frac{1}{2}n\frac{c}{\sqrt{nm}}E[x_{0}^{2}],\end{aligned}$$ $$\begin{aligned}
\therefore\frac{E[\Delta W_{1}^{2}]}{E[W_{1}^{2}]} & =\frac{1}{2c}\sqrt{pm}\cdot n\frac{c}{\sqrt{nm}}E[x_{0}^{2}]E[\Delta y_{1}^{2}]\\
& =\frac{1}{2}\sqrt{np}E[x_{0}^{2}]E[\Delta y_{1}^{2}],\end{aligned}$$ which matches Equation \[eq:linear-ratio-1\], so $\nu_{0}=\nu_{1}$.
\[rem:feedforward\]This relation also holds for sequences of (potentially) strided convolutions, but only if the same kernel size is used everywhere, and a nonzero-padding scheme is used such as reflective padding.
Traditional Initialization schemes
----------------------------------
The most common approaches are the *Kaiming* [@kaiming-rectifiers2015] and *Xavier* [@xavierglorot2010] initializations. The Kaiming technique for ReLU networks is actually one of two approaches:
$$(\text{fan-in)}\,\text{Var}[W_{l}]=\frac{2}{n_{l}^{\text{in}}}\qquad\text{or}\qquad(\text{fan-out)}\,\text{Var}[W_{l}]=\frac{2}{n_{l}^{\text{out}}}.\label{eq:fan-in}$$
For the feed-forward network above, assuming random activations, the forward-activation variance will remain constant in expectation throughout the network if fan-in initialization of weights [@efficientbackprop2012] is used, whereas the fan-out variant maintains a constant variance of the back-propagated signal. The constant factor 2 in the above expressions corrects for the variance-reducing effect of the ReLU activation. Although popularized by @kaiming-rectifiers2015, similar scaling was in use in early neural network models that used tanh activation functions [@bottou-88b].
These two principles are clearly in conflict; unless $n_{l}^{\text{in}}=n_{l}^{\text{out}}$, either the forward variance or backward variance will become non-constant, or as more commonly expressed, either *explode* or *vanish*. No *prima facie* reason for preferring one initialization over the other is provided. Unfortunately, there is some confusion in the literature as many works reference using Kaiming initialization without specifying if the fan-in or fan-out variant is used.
The Xavier initialization [@xavierglorot2010] is the closest to our proposed approach. They balance these conflicting objectives using the arithmetic mean: $$\text{Var}[W_{l}]=\frac{4}{n_{l}^{\text{in}}+n_{l}^{\text{out}}},\label{eq:xavier}$$ to “... approximately satisfy our objectives of maintaining activation variances and back-propagated gradients variance as one moves up or down the network”. This approach to balancing is essentially heuristic, in contrast to the geometric mean approach that our theory directly guides us to.
Geometric initialization balances biases
----------------------------------------
We can use the same proof technique to compute the GR scaling for the bias parameters in a network. Our update equations change to include the bias term: $y_{l}=W_{l}x_{l}+b_{l}$, with $b_{l}$ assumed to be initialized at zero. We show in Appendix \[sec:bias-scaling-appendix\] that: $$\gamma_{l}^{b}=\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.$$ It is easy to show using the techniques of Section \[sec:precond-geom\] that the biases of consecutive layers have equal GR scaling as long as geometric initialization is used. However, unlike in the case of weights, we have less flexibility in the choice of the numerator. Instead of allowing all weights to be scaled by $c$ for any positive $c$, we require that $c=2$, so that: $$E[W_{l}^{2}]=\frac{2}{\sqrt{n_{l}^{\text{in}}n_{l}^{\text{out}}}}.\label{eq:fixedc_init}$$
(Appendix \[sec:bias-scaling-appendix\]) Consider the setup of Proposition \[prop:feedforward\]. As long as the weights are initialized following Equation \[eq:fixedc\_init\] and the biases are initialized to 0, we have that $\gamma_{0}^{b}=\gamma_{1}^{b}.$
Network input scaling balances weights against biases
-----------------------------------------------------
It is traditional to normalize a dataset before applying a neural network so that the input vector has mean 0 and variance 1 in expectation. This principle is rarely quested in modern neural networks, despite the fact that there is no longer a good justification for its use in modern ReLU based networks. In contrast, our theory provides direct guidance for the choice of input scaling. We show that the second moment of the input effects the GR scaling of bias and weight parameters differently, and that they can be balanced by careful choice of the initialization.
Consider the GR scaling values for the bias and weight parameters in the first layer of a ReLU-based multilayer perceptron network, as considered in previous sections. We assume the data is already centered. Then the scaling factors for the weight and bias layers are: $$\gamma_{0}=n_{0}^{\text{in}}k_{0}^{2}\rho_{0}^{2}E\left[x_{0}^{2}\right]^{2}\frac{E[\Delta y_{0}^{2}]}{E[y_{0}^{2}]},\qquad\gamma_{0b}=\rho_{0}^{2}\frac{E[\Delta y_{0}^{2}]}{E[y_{0}^{2}]}.$$ We can cancel terms to find the value of $E\left[x_{l}^{2}\right]$ that makes these two quantities equal: $$E\left[x_{0}^{2}\right]=\frac{1}{\sqrt{n_{0}^{\text{in}}k_{0}^{2}}}.$$ In common computer vision architectures such as VGG (detailed below), the input planes are the 3 color channels, and the kernel size is $k=3$, giving $E\left[x_{0}^{2}\right]\approx0.2$. Using the traditional variance-one normalization will result in the effective learning rate for the bias terms being lower than that of the weight terms. This will result in potentially slower learning of the bias terms than for the input scaling we propose.
Output second moments
---------------------
A neural network’s behavior is also very sensitive to the second moment of the outputs. For a convolutional network without pooling layers (but potentially with strided dimensionality reduction), if geometric-mean initialization is used the activation second moments are given by: $$\begin{aligned}
E[x_{l+1}^{2}] & =\frac{1}{2}k^{2}n_{l}^{in}E[W_{l}^{2}]E[x_{l}^{2}]=\sqrt{\frac{n_{l}^{in}}{n_{l}^{out}}}E[x_{l}^{2}].\end{aligned}$$ The application of a sequence of these layers gives a telescoping product: $$\begin{aligned}
E[x_{L+1}^{2}] & =\left(\prod_{l=0}^{L}\sqrt{\frac{n_{l}^{in}}{n_{l}^{out}}}\right)E[x_{0}^{2}]=\sqrt{\frac{n_{0}^{in}}{n_{L}^{out}}}E[x_{0}^{2}].\end{aligned}$$ We potentially have independent control over this second moment at initialization, as we can insert a fixed scalar multiplication factor at the end of the network that modifies it. This may be necessary when adapting a network architecture that was designed and tested under a different initialization scheme, as the success of the architecture may be partially due to the output scaling that happens to be produced by that original initialization. We are not aware of any existing theory guiding the choice of output variance at initialization for the case of log-softmax losses, where it has a non-trivial effect on the back-propagated signals, although output variances of 0.01 to 0.1 appear to work well. The output variance should **always** be checked and potentially corrected when switching initialization schemes.
Designing well-conditioned neural networks
==========================================
[|c|c|>p[0.55]{}|]{} & [Maintains Scaling]{} & [Notes]{}[\
]{} & & [\
]{} & & [As above, but only if all kernel sizes are the same]{}[\
]{} & & [Operations in residual blocks will be scaled correctly against each other, but not against non-residual operations]{}[\
]{} & & [\
]{} & & [\
]{} & & [\
]{} & & [Any positively-homogenous function with degree 1 ]{}[\
]{} & & [\
]{} & & [Maintains scaling if entirely within the linear regime]{}[\
]{}
Our scaling principle can be used for the design of more complex network structures as well. In this section, we detail the general principles that can be used to design well-conditioned networks with more complicated structures.
Convolutional networks {#subsec:full-case}
----------------------
The concept of GR scaling may be extended to convolutional layers with kernel width $k_{l}$, batch-size $b$, and output resolution $\rho_{l}\times\rho_{l}$ . A straight-forward derivation gives expressions for the convolution weight and biases of: $$\gamma_{l}=bn_{l}^{\text{in}}k_{l}^{2}\rho_{l}^{2}E\left[x_{l}^{2}\right]^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]},\qquad\gamma_{l}^{b}=\rho_{l}^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.\label{eq:conv-gnr}$$ This requires an assumption of independence of the values of activations within a channel that is not true in practice, so $\gamma_{l}$ tends to be further away from empirical estimates for convolutional layers than for non-convolutional layers, although it is still a useful guide. The effect of padding is also ignored here. The standard technique of padding with zeros will only cause a modest decrease in output variance, and so it is typically safe to ignore this additional complication except in extremely deep networks. Sequences of convolutions are well scaled against each other along as the kernel size remains the same. The scaling of layers involving differing kernel sizes can be corrected using the addition of constants into the network (Appendix \[sec:resnet\]).
Maintaining conditioning {#subsec:activation-variance}
------------------------
Consider a network where $\gamma_{l}$ is constant throughout. We may add an additional layer between any two existing layers without affecting this conditioning, as long as the new layer maintains the activation-gradient second-moment product: $$E[\Delta x_{l+1}^{2}]E[x_{l+1}^{2}]=E[\Delta x_{l}^{2}]E[x_{l}^{2}],$$ and dimensionality; this follows from Equation \[eq:conv-gnr\]. For instance, adding a simple scaling layer of the form $x_{l+1}=2x_{l}$ doubles the second moment during the forward pass and doubles the backward second moment during back-propagation, which maintains this product: $$E[\Delta x_{l+1}^{2}]E[x_{l+1}^{2}]=\frac{1}{2}E[\Delta x_{l}^{2}]\cdot2E[x_{l}^{2}].$$ When spatial dimensionality changes between layers we can see that the GR scaling is no longer maintained just by balancing this product, as $\gamma$ depends directly on the square of the spatial dimension. Instead, a pooling operation that changes the forward and backwards signals in a way that counter-acts the change in spatial dimension is needed. The use of stride-2 convolutions, as well as average pooling results in the correct scaling, but other types of pooling generally do not. Table \[tab:scalingtable\] lists operations that preserve scaling when inserted into an existing preconditioned network. Operations such as linear layers preserve the scaling of existing layers but are only themselves well-scaled if they are initialized correctly. For an architecture such as ResNet-50 that uses operations that break scaling, some constants should be introduced into the network to correct scaling. In a ResNet-50, residual connections, max-pooling and varying kernel sizes need to be corrected for (we describe this procedure in Appendix \[sec:resnet\]).
Conditioning multipliers {#sec:multipliers}
------------------------
We can change the value of $\gamma_{l}$ for a single layer without modifying the forward or backward propagated signals in the network via reparametrization [@diagonal-rescaling]. If we introduce an additional scalar $\alpha_{l}$: $$y_{l}=\alpha_{l}W_{l}x_{l}+b_{l},$$ and modify the initialization of $W_{l}$ such that $y_{l}$ is unchanged, then the backward signal $\Delta x_{l}$ is unchanged, however the value of $\Delta W$ changes and the GR scaling is then multiplied by $\alpha_{l}^{4}$: $$\gamma_{l}=\alpha_{l}^{4}bn_{l}^{\text{in}}k_{l}^{2}\rho_{l}^{2}E\left[x_{l}^{2}\right]^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.\label{eq:full-gnr}$$ By introducing these untrained conditioning constants we may modify the GR scaling of each block independently, and potentially improve the initial conditioning of any given network.
Experimental Results
====================
[|>p[0.2]{}|>p[0.35]{}|>p[0.15]{}|>p[0.15]{}|]{} Method & Average Normalized loss $(\pm0.01)$ & Worst in \# & Best in \# [\
]{} Arithmetic mean & 0.90 & 14 & 3 [\
]{} Fan in & 0.84 & 3 & 5 [\
]{} Fan out & 0.88 & 9 & **12** [\
]{} Geometric mean & **0.81** & **0** & 6 [\
]{}
We considered a selection of dense and moderate-sparsity multi-class classification datasets from the LibSVM repository, 26 in total. The same model was used for all datasets, a non-convolutional ReLU network with 3 weight layers total. The inner two layers were fixed at 384 and 64 nodes respectively. These numbers were chosen to result in a larger gap between the optimization methods, very little difference could be expected if a more typical $2\times$ gap was used.
For every dataset, learning rate and initialization combination we ran 10 seeds and picked the median loss after 5 epochs as the focus of our study (The largest differences can be expected early in training). Learning rates in the range $2^{1}$ to $2^{-12}$ (in powers of 2) were checked for each dataset and initialization combination, with the best learning rate chosen in each case based off of the median of the 10 seeds. Training loss was used as the basis of our comparison as we care primarily about convergence rate, and are comparing identical network architectures. Some additional details concerning the experimental setup and which datasets were used is available in the Appendix.
Table \[tab:scalingtable\] shows that geometric initialization is the most consistent of the initialization approaches considered. It has the lowest loss, after normalizing each dataset, and it is never the worst of the 4 methods on any dataset. Interestingly, the fan out method is most often the best method, but consideration of the per-dataset plots (Appendix \[sec:full-results\]) shows that it often completely fails to learn for some problems, which pulls down its average loss and results in it being the worst for 9 datasets.
Conclusion
==========
Although not a panacea, by using the scaling principle we have introduced, neural networks can be designed with a reasonable expectation that they will be optimizable by stochastic gradient methods, minimizing the amount of guess-and-check neural network design. As a consequence of our scaling principle, we have derived an initialization scheme that automatically preconditions common network architectures. Most developments in neural network theory attempt to explain the success of existing techniques post-hoc. Instead, we show the power of the scaling law approach by deriving a new initialization technique from theory directly.
Assumptions {#subsec:assumptions1}
===========
The following assumptions are used in the derivation of the GR scaling:
(A1) The input and target values are drawn element-wise i.i.d from a centered symmetric distribution with known variance.
(A2) The Hessian of the remainder of the network above each block, with respect to the output, has Frobenius norm much larger than $1$. We make this assumption so that we can neglect all but the highest order terms that are polynomial in this norm.
(A3) All activations, pre-activations and gradients are independently distributed element-wise. In practice due to the mixing effect of multiplication by random weight matrices, only the magnitudes of these quantities are correlated, and the effect is small for wide networks due to the law of large numbers. Independence assumptions of this kind are common when approximating second-order methods; the block-diagonal variant of K-FAC [@martens-grosse-2015] makes similar assumptions for instance.
Assumption A2 is the most problematic of these assumptions, and we make no claim that it holds in practice. However, we are primarily interested in the properties of blocks and their scaling with respect to each other, not their absolute scaling. Assumption A2 results in very simple expressions for the scaling of the blocks without requiring a more complicated analysis of the top of the network. Similar theory can be derived for other assumptions on the output structure, such as the assumption that the target values are much smaller than the outputs of the network.
GR scaling derivation\[sec:GR-scaling-derivation\]
==================================================
To compute the second moment of the elements of $G_{l}r$, we can calculate the second moment of matrix-random-vector products against $J_{l}$, $R_{l}$ and $J_{l}^{T}$ separately since $R$ is uncorrelated with $J_{l}$, and the back-propagated gradient $\Delta y_{l}$ is uncorrelated with $y_{l}$ (Assumption A3).
#### Jacobian products $J_{l}$ and $J_{l}^{T}$ {#jacobian-products-j_l-and-j_lt .unnumbered}
Recall that each row of $J_{l}$ has $n_{l}^{\text{in}}$ non-zero elements (Equation \[eq:j-matrix\]), each containing a value from $x_{l}$. The value $x_{l}$ is i.i.d random at the bottom layer of the network (Assumption A1). For layers further up, the multiplication by a random weight matrix from the previous layer ensures that the entries of $x_{l}$ are identically distributed (Assumption A3). So we have: $$E\left[\left(J_{l}r\right)^{2}\right]=n_{l}^{\text{in}}E[r^{2}]E[x_{l}^{2}]=n_{l}^{\text{in}}E[x_{l}^{2}].$$ Note that we didn’t assume that the input $x_{l}$ is mean zero, so $Var[x_{l}]\neq E[x_{l}^{2}].$ This is needed as often the input to a layer is the output from a ReLU operation, which will not be mean zero.
For the transposed case, we have a single entry per column, so: $$E\left[\left(J_{l}^{T}(R_{l}J_{l}r)\right)^{2}\right]=E[\left(R_{l}J_{l}r\right)^{2}]E[x_{l}^{2}].$$
#### Upper Hessian $R_{l}$ product {#upper-hessian-r_l-product .unnumbered}
Instead of using $R_{l}u$, for any random $u$, we will instead compute it for $u=y_{l}/E[y_{l}^{2}]$, it will have the same expectation since both $J_{l}r$ and $y_{l}$ are uncorrelated with $R_{l}$ (Assumption A3). The piecewise linear structure of the network above $y_{l}$ with respect to the $y_{l}$ makes the structure of $R_{l}$ particularly simple. It is a least-squares problem $g(y_{l})=\frac{1}{2}\left\Vert \Phi y_{l}-t\right\Vert ^{2}$ for some $\Phi$ that is the linearization of the remainder of the network. The gradient is $\Delta y=\Phi^{T}\left(\Phi y-t\right)$ and the Hessian is simply $R=\Phi^{T}\Phi$. So we have that $$\begin{aligned}
E\left[\Delta y_{l}^{2}\right] & =E\left[\frac{1}{n_{l}^{\text{out}}}\left\Vert \Phi^{T}\left(\Phi y-t\right)\right\Vert ^{2}\right]\\
& =E\left[\frac{1}{n_{l}^{\text{out}}}\left\Vert \Phi^{T}\Phi y\right\Vert ^{2}\right]+E\left[\frac{1}{n_{l}^{\text{out}}}\left\Vert \Phi^{T}t\right\Vert ^{2}\right]\:(\text{Uncorr. A1)}\\
& \approx E\left[\frac{1}{n_{l}^{\text{out}}}\left\Vert \Phi^{T}\Phi y\right\Vert ^{2}\right]=E\left[\left(R_{l}y_{l}\right)^{2}\right].\qquad(\text{Assumption A2)}\end{aligned}$$ Applying this gives: $$\begin{aligned}
E\left(R_{l}u\right)^{2} & =E[u^{2}]E[\left(R_{l}y_{l}\right)^{2}]/E[y_{l}^{2}]=E[u^{2}]E[\Delta y_{l}^{2}]/E[y_{l}^{2}].\end{aligned}$$
The Gauss-Newton matrix {#subsec:gauss-newton}
-----------------------
Standard ReLU classification and regression networks have a particularly simple structure for the Hessian with respect to the input, as the network’s output is a piecewise-linear function $g$ feed into a final layer consisting of a convex log-softmax operation, or a least-squares loss. This structure results in the Hessian with respect to the input being equivalent to its *Gauss-Newton* approximation. The Gauss-Newton matrix can be written in a factored form, which is used in the analysis we perform in this work. We emphasize that this is just used as a convenience when working with diagonal blocks, the GN representation is not an approximation in this case.
The (Generalized) Gauss-Newton matrix $G$ is a positive semi-definite approximation of the Hessian of a non-convex function $f$, given by factoring $f$ into the composition of two functions $f(x)=h(g(x))$ where $h$ is convex, and $g$ is approximated by its Jacobian matrix $J$ at $x$, for the purpose of computing $G$: $$G=J^{T}\left(\nabla^{2}h(g(x))\right)J.$$ The GN matrix also has close ties to the Fisher information matrix [@martens-insights], providing another justification for its use.
Surprisingly, the Gauss-Newton decomposition can be used to compute diagonal blocks of the Hessian with respect to the weights $W_{l}$ as well as the inputs [@martens-insights]. To see this, note that for any activation $y_{l}$, the layers above may be treated in a combined fashion as the $h$ in a $f(W_{l})=h(g(W_{l}))$ decomposition of the network structure, as they are the composition of a (locally) linear function and a convex function and thus convex. In this decomposition $g(W_{l})=$$W_{l}x_{l}+b_{l}$ is a function of $W_{l}$ with $x_{l}$ fixed, and as this is linear in $W_{l}$, the Gauss-Newton approximation to the block is thus not an approximation.
The Weight gradient ratio is equal to GR scaling for MLP models
===============================================================
\[sec:variance-ratio\] The weight-gradient ratio $\nu_{l}$ is equal to the GR scaling $\gamma_{l}$ for i.i.d mean-zero randomly-initialized multilayer perceptron layers under the independence assumptions of Appendix \[subsec:assumptions1\].
To see the equivalence, note that under the zero-bias initialization, we have from $y_{l}=W_{l}x_{l}$ that: $$E[y_{l}^{2}]=n_{l}^{\text{in}}E[W_{l}^{2}]E[x_{l}^{2}],\label{eq:linear-forward-1}$$ and so: $$E[W_{l}^{2}]=\frac{E[y_{l}^{2}]}{n_{l}^{\text{in}}E[x_{l}^{2}]}.$$ The gradient of the weights is given by $\Delta W_{ij}=\Delta y_{li}x_{lj}$ and so its second moment is: $$E[\Delta W_{l}^{2}]=E[x_{l}^{2}]E[\Delta y_{l}^{2}].\label{eq:linear-back-1}$$ Combining these quantities gives: $$\nu_{l}=\frac{E[\Delta W_{l}^{2}]}{E[W_{l}^{2}]}=n_{l}^{\text{in}}E\left[x_{l}^{2}\right]^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.$$
Bias scaling {#sec:bias-scaling-appendix}
============
We consider the case of a convolutional neural network with spatial resolution $\rho\times\rho$ for greater generality. Consider the Jacobian of $y_{l}$ with respect to the bias. It has shape $J_{l}^{b}:(n_{l}^{out}\rho_{l}^{2})\times(n_{l}^{out})$. Each row corresponds to a $y_{l}$ output, and each column a bias weight. As before, we will approximate the product of $G$ with a random i.i.d unit variance vector $r$: $$G_{l}^{b}r=J_{l}^{bT}R_{l}J_{l}^{b}r,$$
The structure of $J_{l}^{b}$ is that each block of $\rho^{2}$ rows has the same set of 1s in the same column. Only a single 1 per row. It follows that: $$E\left[\left(J_{l}^{b}r\right)^{2}\right]=1.$$ The calculation of the product of $R_{l}$ with $J_{l}^{b}r$ is approximated in the same way as in the weight scaling calculation. For the $J^{bT}$ product, note that there is an additional $\rho^{2}$ as each column has $\rho^{2}$ non-zero entries, each equal to 1. Combining these three quantities gives: $$\gamma_{l}^{b}=\rho^{2}\frac{E[\Delta y_{l}^{2}]}{E[y_{l}^{2}]}.$$
Consider the setup of Proposition \[prop:feedforward\], with the addition of biases: $$y_{0}=W_{0}x_{0}+b_{0},$$ $$x_{1}=ReLU(y_{0}),$$ $$y_{1}=W_{1}+b_{1}.$$
As long as the weights are initialized following Equation \[eq:fixedc\_init\] and the biases are initialized to 0, we have that $$\gamma_{0}^{b}=\gamma_{1}^{b}.$$
We will include $c=2$ as a variable as it clarifies it’s relation to other quantities. We reuse some calculations from Proposition \[prop:feedforward\]. Namely that: $$E[y_{0}^{2}]=c\sqrt{\frac{n}{m}}E[x_{0}^{2}],$$ $$E[\Delta y_{0}^{2}]=\frac{1}{2}c\sqrt{\frac{p}{m}}E[\Delta y_{1}^{2}].$$ Plugging these into the definition of $\gamma_{0}^{b}$: $$\gamma_{0}^{b}=\frac{E[\Delta y_{0}^{2}]}{E[y_{0}^{2}]}=\frac{\frac{1}{2}c\sqrt{\frac{p}{m}}E[\Delta y_{1}^{2}]}{c\sqrt{\frac{n}{m}}E[x_{0}^{2}]}=\frac{\sqrt{p}E[\Delta y_{1}^{2}]}{2\sqrt{n}E[x_{0}^{2}]}.$$ For $\gamma_{1}^{b}$, we require the additional quantity: $$\begin{aligned}
E[y_{1}^{2}] & =mE[x_{1}^{2}]E\left[W_{1}^{2}\right]\\
& =m\left(\frac{1}{2}c\sqrt{\frac{n}{m}}E[x_{0}^{2}]\right)\left(\frac{c}{\sqrt{mp}}\right)\\
& =\frac{c^{2}}{2}\sqrt{\frac{n}{p}}E[x_{0}^{2}].\end{aligned}$$ Again plugging this in: $$\begin{aligned}
\gamma_{1}^{b} & =\frac{E[\Delta y_{1}^{2}]}{E[y_{1}^{2}]}\\
& =\frac{E[\Delta y_{1}^{2}]}{\frac{c^{2}}{2}\sqrt{\frac{n}{p}}E[x_{0}^{2}]}\\
& =\frac{\sqrt{p}E[\Delta y_{1}^{2}]}{\frac{c^{2}}{2}\sqrt{n}E[x_{0}^{2}]}.\end{aligned}$$ So comparing these expressions for $\gamma_{0}^{b}$ and $\gamma_{1}^{b}$, we see that $\gamma_{0}^{b}=\gamma_{1}^{b}$ if and only if $c=2.$
Conditioning of ResNets Without Normalization Layers {#sec:resnet}
====================================================
There has been significant recent interest in training residual networks without the use of batch-normalization or other normalization layers [@fixup]. In this section, we explore the modifications that are necessary to a network for this to be possible and show how to apply our preconditioning principle to these networks.
The building block of a ResNet model is the residual block: $$z_{l+1}=\text{ReLU}\left(F(z_{l})+z_{l}\right),$$ where in this notation $F$ is a composition of layers. Unlike classical feedforward architectures, the pass-through connection results in an exponential increase in the variance of the activations in the network as the depth increases. A side effect of this is the output of the network becomes exponentially more sensitive to the input of the network as depth increases, a property characterized by the Lipschitz constant of the network [@boris2018].
This exponential dependence can be reduced by the introduction of scaling constants $s_{l}$ to each block: $$z_{l+1}=\text{ReLU}\left(s_{l}F(z_{l})+z_{l}\right).$$
The introduction of these constants requires a modification of the block structure to ensure constant conditioning between blocks. A standard bottleneck block, as used in the ResNet-50 architecture, has the following form: $$\begin{aligned}
y_{0} & =C_{0}(x_{0}),\\
x_{1} & =\text{ReLU}(y_{0}),\\
y_{1} & =C_{1}(x_{1}),\\
x_{2} & =\text{ReLU}(y_{1}),\\
y_{2} & =C_{2}(x_{2}),\\
x_{3} & =\text{ReLU}(y_{2}+x_{0}).\end{aligned}$$
In this notation, $C_{0}$ is a $1\times1$ convolution that reduces the number of channels 4 fold, $C_{1}$ is a $3\times3$ convolution with equal input and output channels, and $C_{2}$ is a $1\times1$ convolution at increases the number of channels back up 4 fold to the original input count.
If we introduce a scaling factor $s_{l}$ to each block $l$, then we must also add conditioning multipliers $\beta_{l}$ to each convolution to change their GR scaling, as we described in Section \[sec:multipliers\]. The correct scaling constant depends on the scaling constant of the previous block. A simple calculation gives the equation: $$\beta_{l}^{2}=\beta_{l-1}^{2}\frac{1+s_{l}^{2}}{1+s_{l-1}^{2}}.$$ Since scaling is relative, the first block may be scaled with $\beta_{0}=1$ and $s_{0}=1$. We recommend using a flat $s_{l}=s$ for all $l$ to avoid having to introduce the $\beta_{l}$ factors. The block structure including the $\beta_{l}$ factors is: $$\begin{aligned}
y_{0} & =\frac{1}{\beta}C_{0}(x_{0}),\\
x_{1} & =\text{ReLU}(y_{0}),\\
y_{1} & =\frac{1}{\sqrt{3}\beta}C_{1}(x_{1}),\\
x_{2} & =\text{ReLU}(y_{1}),\\
y_{2} & =\frac{1}{\beta}C_{2}(x_{2}),\\
x_{3} & =\text{ReLU}\left(sy_{2}+x_{0}\right)\end{aligned}$$ The weights of each convolution must then be initialized with the standard deviation modified such that the combined convolution-scaling operation gives the same output variance as would be given if the geometric-mean initialization scheme is used without extra scaling constants. For instance, the initialization of the $C_{0}$ convolution must have standard deviation scaled down by dividing by $\frac{1}{\beta}$ so that the multiplication by $\frac{1}{\beta}$ during the forward pass results in the correct forward variance. The $1/\sqrt{3}$ factor corrects for the change in kernel shape for the middle convolution.
Correction for mixed residual and non-residual blocks
-----------------------------------------------------
Since the initial convolution in a ResNet-50 model is also not within a residual block, it’s GR scaling is different from the convolutions within residual blocks. Consider the composition of a non-residual followed by a residual block, without max-pooling or ReLUs for simplicity of exposition: $$y_{0}=\alpha C_{0}(x_{0}),\quad x_{1}=y_{0},$$ $$y_{1}=s_{1}C_{1}(x_{1}),\quad z_{1}=y_{1}+x_{1}.$$
Without loss of generality, we assume that $E\left[x_{0}^{2}\right]=1$, and assume a single channel input and output.
Our goal is to find a constant $\alpha$, so that $\gamma_{0}=\gamma_{1}$. Note that the initialization of $C_{0}$ is tied to $\alpha$ inversely, so that the variance of $y_{0}$ is independent of the choice of $\alpha$. Our scaling factor will also depend on the kernel sizes used in the two convolutions, so we must include those in the calculations.
From Equation \[eq:full-gnr\], the GR scaling for $C_{0}$ is
$$\begin{aligned}
\gamma_{0} & =\alpha^{4}n_{l}^{\text{in}}k_{0}^{2}E\left[x_{0}^{2}\right]^{2}\frac{E[\Delta y_{0}^{2}]}{E[y_{0}^{2}]}\\
& =\alpha^{4}k_{0}^{2}E[\Delta y_{0}^{2}].\end{aligned}$$
Note that $E[\Delta y_{0}^{2}]=\left(1+s_{1}^{2}\right)E[\Delta z_{1}^{2}]$ so:
$$\gamma_{0}=\left(1+s_{1}^{2}\right)\alpha^{4}k_{0}^{2}E[\Delta z_{1}^{2}],$$
For the residual convolution, we need to use a modification of the standard GR equation due to the residual branch. The derivation of $\gamma$ for non-residual convolutions assumes that the remainder of the network above the convolution responds linearly (locally) with the scaling of the convolution, but here due to the residual connection, this is no longer the case. For instance, if the weight were scaled to zero, the output of the network would not also become zero (recall our assumption of zero-initialization for bias terms). This can be avoided by noting that the ratio $E[\Delta y_{1}^{2}]/E[y_{1}^{2}]$ in the GR scaling may be computed further up the network, as long as any scaling in between is corrected for. In particular, we may compute this ratio at the point after the residual addition, as long as we include the factor $s_{1}^{4}$ to account for this. So we in fact have: $$\begin{aligned}
\gamma_{1} & =s_{1}^{4}n_{l}^{\text{in}}k_{1}^{2}E\left[x_{1}^{2}\right]^{2}\frac{E[\Delta z_{1}^{2}]}{E[z_{1}^{2}]}\\
& =s_{1}^{4}k_{1}^{2}\frac{E[\Delta z_{1}^{2}]}{1+s_{1}^{2}}.\end{aligned}$$ We now equate $\gamma_{0}=\gamma_{1}$: $$s_{1}^{4}k_{1}^{2}\frac{E[\Delta z_{1}^{2}]}{1+s_{1}^{2}}=\left(1+s_{1}^{2}\right)\alpha^{4}k_{0}^{2}E[\Delta z_{1}^{2}],$$ $$\frac{k_{1}^{2}}{k_{0}^{2}}\cdot\frac{s_{1}^{4}}{\left(1+s_{1}^{2}\right)^{2}}=\alpha^{4}.$$ Therefore to ensure that $\gamma_{0}=\gamma_{1}$ we need: $$\alpha^{2}=\frac{k_{1}}{k_{0}}\cdot\frac{s_{1}^{2}}{\left(1+s_{1}^{2}\right)}.$$
Final layer {#final-layer .unnumbered}
-----------
A similar calculation applies when the residual block is before the non-residual convolution, as in the last layer linear in the ResNet network, giving a scaling factor for the linear layer (effective kernel size 1) of: $$\alpha^{2}=\frac{s_{L-1}^{2}}{\left(1+s_{L-1}^{2}\right)}k_{L-1}.$$
Full experimental results
=========================
Details of input/output scaling {#details-of-inputoutput-scaling .unnumbered}
-------------------------------
To prevent the results from being skewed by the number of classes and the number of inputs affecting the output variance, the logit output of the network was scaled to have standard deviation 0.05 after the first minibatch evaluation for every method, with the scaling constant fixed thereafter. LayerNorm was used on the input to whiten the data. Weight decay of 0.00001 was used for every dataset. To aggregate the losses across datasets we divided by the worst loss across the initializations before averaging.
Plots {#sec:full-results .unnumbered}
-----
Plots show the interquartile range (25%, 50% and 75% quantiles) of the best learning rate for each case.
{width="100.00000%"}
|
---
abstract: |
Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.
[**Keywords**]{}: pattern avoidance, generating trees, kernel method
address:
- 'Department of Statistics, University of Wisconsin, Madison, WI 53706'
- 'Department of Mathematics, University of Haifa, 3498838 Haifa, Israel'
author:
- David Callan
- Toufik Mansour
title: 'Enumeration of small Wilf classes avoiding 1324 and two other $4$-letter patterns'
---
Introduction
============
In recent decades pattern avoidance has received a lot of attention. It has a prehistory in the work of MacMahon [@macmahon1915] and Knuth [@K], but the current interest was sparked by a paper of Simion and Schmidt [@SiS]. They thoroughly analyzed 3-letter patterns in permutations, including a bijection between 123- and 132-avoiding permutations, thereby explaining the first (nontrivial) instance of what is, in modern terminology, a Wilf class. Since then the problem has been addressed on several other discrete structures, such as compositions, $k$-ary words, and set partitions; see, e.g., the texts [@SHM; @TM] and references contained therein.
Permutations avoiding a single 4-letter pattern have been well studied (see, e.g., [@St0; @St; @W; @wikipermpatt]), and the latter form 7 symmetry classes and 3 Wilf classes. As for pairs of 4-letter patterns, there are 56 symmetry classes, for all but 5 of which the avoiders have been enumerated [@wikipermpatt]. Le [@L] established that these $56$ symmetry classes form $38$ distinct Wilf classes.
The $\binom{24}{3}= 2024$ triples of 4-letter patterns split into 317 symmetry classes. It is known [@CMS3patI; @CMS3patII] that the 317 symmetry classes split into $242$ Wilf classes, 32 of which are large (a Wilf class is called large if it contains at least two symmetry classes, and small if it consists of a singleton symmetry class) and the large Wilf classes are all explicitly enumerated in [@CMS3patI; @CMS3patII], where it is shown that each has an algebraic generating function.
Our goal here is to enumerate (with one exception, see [@gp2015] and [@SlA257562]) all the small Wilf classes that contain the pattern 1324. Running the INSENC algorithm (regular insertion encoding, see [@ALR; @V]) over all the 210 small Wilf classes determines the generating function for 126 of them, as presented in the Appendix. The remaining small classes that contain 1324 are listed in Table \[tabgf1324\] along with their generating functions, where the numbering follows that of Table 2 in the appendix to[@HYL], based on lex order of counting sequences.
Section \[prelim\] contains some preliminary remarks, and Section \[proofs\] gives the proofs for all the results in Table \[tabgf1324\] not proved elsewhere.
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Preliminaries {#prelim}
=============
We say a permutation is *standard* if its support set is an initial segment of the positive integers, and for a permutation $\pi$ whose support is any set of positive integers, St($\pi$) denotes the standard permutation obtained by replacing the smallest entry of $\pi$ by 1, the next smallest by 2, and so on. Typically, for a a given triple $T$, we consider cases and analyze the structure of a $T$-avoider in each case to the point where we say that $T$-avoiders have such and such a form in that case. It is always to be understood that we are also asserting, without explicit mention, that a permutation of the specified form is a $T$-avoider, and this enables us to determine the various “contributions” to the $F_T(x)$ for $T$-avoiders, yielding an equation for $F_T(x)$. The equation may be an explicit expression for $F_T(x)$ or an algebraic or functional equation. For all but one symmetry class, the turns out to be algebraic of degree $\le 4$. For the exceptional class (Case 237, where $\{1432, 1324, 1243\}$ and $\{4123, 4231, 4312\}$ are representative triples), the is conjectured not to satisfy any ADE (algebraic differential equation), see [@gp2015] and [@Sl Seq. A257562].
A permutation $\pi$ expressed as $\pi=i_1\pi^{(1)}i_2\pi^{(2)}
\cdots i_m\pi^{(m)}$ where $i_1<i_2<\cdots<i_m$ and $i_j>\max(\pi^{(j)})$ for $1 \leq j \leq m$ is said to have $m$ *left-right maxima* (at $i_1,i_2,\ldots,i_m$). Given nonempty sets of numbers $S$ and $T$, we will write $S<T$ to mean $\max(S)<\min(T)$ (with the inequality vacuously holding if $S$ or $T$ is empty). In this context, we will often denote singleton sets simply by the element in question. Also, for a number $k$, $S-k$ means the set $\{s-k:s\in S\}$.
Throughout, $C(x)=\frac{1-\sqrt{1-4x}}{2x}$ denotes the generating function for the Catalan numbers $C_n:=\frac{1}{n+1}\binom{2n}{n}=\binom{2n}{n}-\binom{2n}{n-1}$. As is well known [@K; @wikipermpatt], $C(x)$ is the generating function for $(|S_n(\pi)|)_{n\ge 0}$ where $\pi$ is any one of the six 3-letter patterns. The identity $C(x)=\frac{1}{1-xC(x)}$ or, equivalently, $xC(x)^2=C(x)-1$ is used to simplify some of our results. Also throughout, $L(x)=\frac{1-x}{1-2x}$ denotes the generating function for $\{213,231\}$-avoiders (resp. $\{213,123\}$-avoiders, resp. $\{132,123\}$-avoiders), see [@SiS], and $K(x),K'(x)$ etc. are variously used for other known generating functions.
Proofs
======
Case 49: $\{1324,2341,4123\}$
-----------------------------
For this case, we need the following lemmas.
\[lem49a1\] Let $T=\{1324,2341,4123\}$. The generating function for the number of permutation $(n-1)\pi'n\pi''\in S_n(T)$ is given by $$H(x)=\frac{x^3C(x)^3}{1-x}+\frac{x^2}{1-x}+\frac{x^4}{(1-x)(1-2x)}+\frac{x^5}{(1-x)^4}.$$
Let us write an equation for $H(x)$. Let $\pi=(n-1)\pi'n\pi''\in S_n(T)$. If $n=2$ then we have a contribution of $x^2$. So let us assume that $n>2$, so there are two cases, either $n-2$ belongs to $\pi'$ or to $\pi''$.
- $n-2$ belongs to $\pi'$: If $\pi''=\emptyset$ then we have a contribution of $x^3(F_{\{123,132\}}-1)=\frac{x^3}{1-2x}$, see [@SiS]. So, we can assume that $\pi''\neq\emptyset$. If $\pi'$ has a letter between $n-1$ and $n-2$, then $\pi$ can be written as $$\pi=(n-1)(i-1)(i-2)\cdots
j(n-2)(j-1)(j-2)\cdots1n(n-3)(n-4)\cdots i,$$ which counted by $\frac{x^5}{(1-x)^3}$. Otherwise, $\pi'$ has no letter between $n-1$ and $n-2$, which gives a contribution of $xH(x)-\frac{x^3(1-x)}{1-2x}$.
- $n-2$ belongs to $\pi''$: In this case, we have a contribution of $x(C(x)-1-xC(x))=x^3C(x)^3$, where $C(x)$ counts the $\{123\}$-avoiders.
Hence, by adding all the contributions, we have $$H(x)=x^2+\frac{x^3}{1-2x}+\frac{x^5}{(1-x)^3}+xH(x)-\frac{x^3(1-x)}{1-2x}+x^3C(x)^3,$$ which completes the proof.
\[lem49a2\] Let $T=\{1324,2341,4123\}$. The generating function for the number of $T$-avoiders with exactly $2$ left-right maxima is given by $$G_2(x)=\frac{1}{1-x}\left(x^4C(x)^5+\frac{x^5}{(1-x)^5}+\frac{x^5}{(1-x)^4}+\frac{x^4}{1-2x}+H(x)\right),$$ where $H(x)$ is given in Lemma $\ref{lem49a1}$.
Let us write an equation for $G_2(x)$. Let $\pi=i\pi'n\pi''\in
S_n(T)$ be a permutation with exactly $2$ left-right maxima. We consider the following cases:
- $i=n-1$: We have a contribution of $H(x)$ as defined in Lemma \[lem49a1\]. So from now, we assume that $\pi''$ contains the letter $n-1$.
- $\pi''=(n-1)\pi'''$: We have a contribution of $xG_2(x)$. So we can assume that there is at least one letter between $n$ and $n-1$. Since $\pi''$ avoids $1324$ and $4123$, we see that there are exactly at most one letter between $n$ and $n-1$ that greater than $i$.
- if there is a letter in $n\pi''$ between $n$ and $n-1$ that it is greater than $i$, then $\pi$ can be written as $\pi=i\pi'nk\beta(n-1)\cdots(k+1)\alpha$ such that $\beta<i$ and $\alpha>i$. By considering either $\beta$ is empty or not, we obtain the contributions $\frac{x^4}{1-x}L(x)^2$ and $\frac{x^5}{(1-x)^4}$, respectively. Recall $L(x)=\frac{1-x}{1-2x}$ is the generating function for $\{213,231\}$-avoiders (also for $\{213,123\}$-avoiders).
- Thus, we can assume that $\pi=\pi'n\beta(n-1)\alpha$ such that $\alpha$ contains the subsequence $(n-1)(n-2)\cdots(i+1)$, $\beta<i$ and $\beta$ is decreasing (since $\pi$ avoids $4123$, and $\beta$ is not empty). Suppose that $\beta=ee'\beta'$ then $e>e'>\beta'$ and $\pi'>e'$. If $\pi'$ has a letter between $e$ and $e'$, then easy to see that the contribution is given by $\frac{x^{4+d}}{(1-x)^4}$, where $d$ is the number of the letters in $\pi''$ that are greater than $i$. Otherwise, the contribution is given by $x^{3+d}C(x)^{3+d}$, where $d$ is the number of the letters in $\pi''$ that are greater than $i$. Therefore, we have a contribution of $$\sum_{d\geq1}\frac{x^{4+d}}{(1-x)^4}+\sum_{d\geq1}x^{3+d}C(x)^{3+d},$$ which equals $$\frac{x^{5}}{(1-x)^5}+\frac{x^{4}C(x)^{4}}{1-xC(x)}.$$
Hence, the various contributions give $$G_2(x)=xG_2(x)+x^4C(x)^5+\frac{x^5}{(1-x)^5}+\frac{x^5}{(1-x)^4}+\frac{x^4}{1-2x}+H(x),$$ which completes the proof.
\[th49a\] Let $T=\{1324,2341,4123\}$. Then $$F_T(x)=C(x)+\frac{x^3 - 3 x^4 +3 x^5 -5 x^6 + 9 x^7 -4 x^8}{(1-x)^6 (1-2x)^2}\, .$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_{\{123\}}(x)=xC(x)$, see [@K]. By Lemmas \[lem49a1\] and \[lem49a2\], we have that $$G_2(x)=\frac{1}{1-x}\left(x^4C(x)^5+\frac{x^5}{(1-x)^5}+\frac{x^5}{(1-x)^4}+\frac{x^4}{1-2x}+H(x)\right),$$ where $$H(x)=\frac{x^3C(x)^3}{1-x}+\frac{x^2}{1-x}+\frac{x^4}{(1-x)(1-2x)}+\frac{x^5}{(1-x)^4}.$$ Now, let us write an equation for $G_3(x)$. Let $\pi=i_1\pi'i_2\pi''n\pi'''\in S_n(T)$ with exactly $3$ left-right maxima ($i_1,i_2,n$). Since $\pi$ avoids $1324$ and $2341$, then $\pi'>\pi''$ and $\pi'''=\beta\alpha$ such that $\beta>i_2>\alpha>i_1$. By considering the cases $\alpha,\beta$ are empty or not, we obtain the contributions $x^3L(x)^2$, $x^3(L(x)-1)L(x)/(1-x)$, $x^3(L(x)-1)L(x)/(1-x)$, and $x^5/(1-x)^4$. Hence, $$G_3(x)=x^3L(x)^2+\frac{2x^3}{1-x}(L(x)-1)L(x)+\frac{x^5}{(1-x)^4}$$.
By very similar techniques as in case $G_3(x)$, we obtain that $$G_4(x)=x^4(L(x)+(L(x)-1)/(1-x))^2.$$
Now, let us write an equation for $G_m(x)$ and $m\geq5$. Let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $T$, we see that $\pi^{(s)}=\emptyset$ for all $s=3,4,\ldots,m-1$, $\pi^{(1)}>\pi^{(2)}$ and $\pi^{(m)}=\beta\alpha$ such that $\beta>i_{m-1}>\alpha>i_{m-2}$. Thus, $G_m(x)=xG_{m-1}(x)$, for all $m\geq5$. Therefore, $$F_T(x)-1-xC(x)-G_2(x)-G_3(x)=\frac{G_4(x)}{1-x}.$$ By substituting the expressions for $G_2(x),G_3(x),G_4(x)$ and simplifying, we obtain the stated generating function.
Case 69: $\{1234, 1324, 3412\}$
-------------------------------
A permutation $\pi=\pi_1\pi_2\cdots\pi_n\in S_n$ determines $n+1$ positions, called *sites*, between its entries. The sites are denoted $1,2,\dots,n+1$ left to right. In particular, site $i$ is the space between $\pi_{i-1}$ and $\pi_{i}$, $2\le i \le n$. Site $i$ in $\pi$ is said to be [*active*]{} (with respect to $T$) if, by inserting $n+1$ into $\pi$ in site $i$, we get a permutation in $S_{n+1}(T)$, otherwise *inactive*.
Say $j$ is an *ascent index* for a permutation $\pi=\pi_1\pi_2 \cdots \pi_n$ of $[n]$ if $\pi_j<\pi_{j+1}$, and then $\pi_j$ is an *ascent bottom*.
To construct the generating forest for $T$-avoiders, we first specify the labels. For $n\ge 2$, define the *label* of $\pi \in S_n(T)$ to be $(k,s)$, where $k$ is the number of active sites in $\pi$ and $s$ is the number of active sites greater than the largest ascent index (LAI for short) with LAI taken to be 0 if there are no ascents, that is, if $\pi$ is decreasing.
For instance, the active sites for $\pi=12$ are $\{1,2,3\}$ and LAI $=1$, so the label for 12 is $(3,2)$. Also, 12 has three children $312$, $132$ and $123$ with active sites $\{1,3,4\},\,\{1,2,3\}$ and $\{1,2,3\}$, respectively, and LAI $ = 2,1,2$, respectively; hence labels $(3,2),(3,2)$, and $(3,1)$. Similarly, all 3 sites for $21$ are active and and LAI $=0$, so its label is $(3,3)$, and it has three children $321$, $231$ and $213$ with active sites $\{1,2,3,4\}$ in all three cases, and LAI $ = 0,1,2$, respectively; hence labels $(4,4),(4,3)$, and $(4,2)$. An avoider $\pi\in S_n(T)$ has a label $(k,s)$ with $k=s$ only if $\pi$ is decreasing, in which case $k=s=n+1$. Otherwise, $0\le s < k$.
\[prop69\] The roots for the generating forest $\mathcal{T}$ of $S_n(T)$ are $12$ and $21$ with labels $(3,2)$ and $(3,3)$ respectively, and the succession rules for the labels of children, in order of increasing insertion site, are given by $$\begin{array}{llll}
(k,s) & \rightsquigarrow & (1,0)\ (2,0)\ \dots \ (k,0) & \textrm{\quad for $s=0$ and $k\ge 1$,}\\[1mm]
& \rightsquigarrow & (2,1)\ (2,0)\ (3,0)\ \dots\ (k-1,0)\ (k,1) & \textrm{\quad for $s=1$ and $k\ge 2$,}\\[1mm]
& \rightsquigarrow & (s+1,s)\ (3,1)\ (4,1)\ \dots \ (k-s+1,1)\ & \\
& & (k-s+2,2)\ (k-s+2,1)\ (k-s+3,1)\ \dots \ (k,1) & \raisebox{1.5ex}[0pt]{\textrm{\quad for $2\le s \le k-1$,}} \\[1mm]
& \rightsquigarrow & (k+1,k+1)\ (k+1,k) \ \dots \ (k+1,2) & \textrm{\quad for $s=k$,}
\end{array}$$
As an example (bullets denote active sites, an underscore denotes last ascent bottom), the label of $\pi=\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 3 \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,
\underline{2}\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 6 \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,
5\, 4\, 1 \in S_6(T)$ is $(k,s)=(4,2)$; its children are $\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,7\, 3 \,
\underline{2}\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 6 \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,
5\, 4\, 1 $, $\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,3\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 7 \,
\underline{2} \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 6 \,
5\, 4\, 1 $, $\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,3\, \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,
\underline{2} \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 7 \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 6 \,
5\, 4\, 1 $, $\,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,3\, \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\,
2 \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, \underline{6} \,\textrm{\raisebox{.2ex}{\tiny{$\bullet$}}}\, 7 \,
5\, 4\, 1 $, in that order, with labels $(3,2), (3,1),\,(4,2),\,(4,1)$ respectively.
The proof of Proposition \[prop69\] is based on induction by a routine checking of cases, and is left to the reader. We note a few properties of the active sites for $\pi \in S_n(T)$. Site 1 is always active. If site $n+1$ is active, then so is site $n$. The active sites always form either an interval of integers (necessarily an initial segment of the positive integers) or a pair of intervals of integers. In the latter case, at least one of the intervals is of length 1. For example, $5 3 2 4 1 6 \in S_6(T)$ has active sites $\{1,4,5,6\}$ and $5 3 4 2 7 6 1 \in S_7(T)$ has active sites $\{1,2,5\}$.
[**Enumeration**]{}: Let $A_{k,s}=A_{k,s}(t)$ be the generating function for the number of vertices labeled $(k,s)$ in level $n$ in the generating forest $\mathcal{T}$, where the roots 12 and 21 are at level $2$. Define $$A(u,v)=A(t;u,v)=\sum_{k\geq1,s\geq0}A_{k,s}u^kv^s,\qquad L(u,v)=\frac{u^3v^3t^2}{1-uvt}.$$ Proposition \[prop69\] leads to $$\begin{aligned}
A(u,v)&=t^2u^3v^2(1+v)+\frac{ut}{1-u}\left(A(1,0)-A(u,0)\right)+vt\frac{d}{dv}A(u,v)\mid_{v=0}\notag\\
&+\frac{t}{1-u}\left(u^2\frac{d}{dv}A(1,v)\mid_{v=0}-\frac{d}{dv}A(u,v)\mid_{v=0}\right)+utA(1,uv)-utL(1,uv)-utA(1,0)\notag\\
&+\frac{vu^3t}{1-u}\left(A(1,1)-A(1,0)-\frac{d}{dv}A(1,v)\mid_{v=0}-L(1,1)\right)\label{eq69m1}\\
&-\frac{uvt}{1-u}\left(A(u,1)-A(u,0)-\frac{d}{dv}A(u,v)\mid_{v=0}-L(u,1)\right)\notag\\
&+u^2v^2t\left(A(u,1/u)-A(u,0)-\frac{1}{u}\frac{d}{dv}A(u,v)\mid_{v=0}-L(1,1)\right)\notag\\
&+\frac{u^4v^2t^3}{(1-v)(1-ut)}-\frac{u^4v^5t^3}{(1-v)(1-uvt)}.\notag\end{aligned}$$ By substituting $v=1/u$ into , we obtain $$\begin{aligned}
A(u,1/u)=\frac{t(t-u^2t+uA(1,1)-A(u,1))}{(1-u)(1-x)},\end{aligned}$$ which implies $$\begin{aligned}
A(u,v)&=t^2u^3v^2(1+v)+\frac{ut}{1-u}\left(A(1,0)-A(u,0)\right)+vt\frac{d}{dv}A(u,v)\mid_{v=0}\notag\\
&+\frac{t}{1-u}\left(u^2\frac{d}{dv}A(1,v)\mid_{v=0}-\frac{d}{dv}A(u,v)\mid_{v=0}\right)+utA(1,uv)-utL(1,uv)-utA(1,0)\notag\\
&+\frac{vu^3t}{1-u}\left(A(1,1)-A(1,0)-\frac{d}{dv}A(1,v)\mid_{v=0}-L(1,1)\right)\label{eq69m2}\\
&-\frac{uvt}{1-u}\left(A(u,1)-A(u,0)-\frac{d}{dv}A(u,v)\mid_{v=0}-L(u,1)\right)\notag\\
&+u^2v^2t\left(\frac{t(t-u^2t+uA(1,1)-A(u,1))}{(1-u)(1-x)}-A(u,0)-\frac{1}{u}\frac{d}{dv}A(u,v)\mid_{v=0}-L(1,1)\right)\notag\\
&+\frac{u^4v^2t^3}{(1-v)(1-ut)}-\frac{u^4v^5t^3}{(1-v)(1-uvt)}.\notag\end{aligned}$$ Substitute $v=0$ into and into the derivative of respect to $v$. Solve the resulting system for the variables $\frac{d}{dv}A(u,v)\mid_{v=0}$ and $\frac{d}{dv}A(1,v)\mid_{v=0}$ to get $$\begin{aligned}
\frac{d}{dv}A(u,v)\mid_{v=0}&=\frac{2ut-2u+1}{1-2t}A(u,0)-\frac{ut}{(1-u)(1-2t)}A(u,1)-\frac{(1-u)ut}{1-2t}A(1,0)\label{eq69m3}\\
&+\frac{u^3t}{(1-u)(1-2t)}-\frac{u^3t^3}{(1-t)(1-2t)(1-ut)}.\notag\end{aligned}$$ Thus, by using twice, can be written as [$$\begin{aligned}
A(u,v)&=\frac{(uvt^2-(t-1)^2)uvt}{(1-2t)(1-t)(1-u)}A(u,1)+\frac{(uv-1)(uvt-vt+2t-1)}{1-2t}A(u,0)+tuA(1,uv)\\
&+\frac{(uvt(t-1)+vt(1-2t)+(t-1)^2)tvu^3}{(1-t)(1-u)(1-2t)}A(1,1)+\frac{ut(1-uv)(uvt-vt+2t-1)}{1-2t}A(1,0)\\
&-\frac{(uv(3uv+2v-1)t^3-((u^2+5u+2)v^2+uv+1)t^2+(2(1+u)v^2+(v+1))t-v(1+v))u^3vt^2}{(1-ut)(1-t)(1-uvt)(1-2t)}\end{aligned}$$]{} By a routine computer check, the solution of this equation is given by $$\begin{aligned}
A(u,v)&=\frac{t^2uK(u,v)}{(1-uvt)(1-t)^6(1-ut)^2(1-2t)^2},\end{aligned}$$ where [$$\begin{aligned}
K(u,v)&=u^2v^2(1+v)-u^2v(2(u+5)v^2+(u+8)v-1)t\\
&+u(u(u^2+19u+43)v^3+u(6u+29)v^2+(1-5u)v+1)t^2\\
&+(1-2u-u(3u^2-6u+1)v+u^2(u+3)(u-21)v^2-u^2(9u^2+80u+104)v^3)t^3\\
&+((1+u)(1-4u)+u(16u^2-3u-4)v-u^2(3u^2-39u-86)v^2+u^2(39u^2+192u+155)v^3)t^4\\
&+\bigl(14u^2+u-5+u(2u^3-26u^2+11u+2)v-3u^2(18u+25)v^2\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-2u^2(u^3+49u^2+142u+73)v^3\bigr)t^5\\
&+\bigl(2u^3-12u^2+8u-1-u(8u^3-12u^2+7u-5)v-u^2(2u^3-4u^2-43u-46)v^2\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+u^2(10u^3+147u^2+265u+85)v^3\bigr)t^6\\
&-u\bigl(6u^2-2u-3+(1-u^2)(8u-1)v-u(u+1)(8u^2-7u-20)v^2\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+u(18u^3+133u^2+154u+28)v^3\bigr)t^7\\
&+u^2(4u-4+(2u^2+2u-3)v-(10u^3+3u^2-16u-4)v^2+(14u^3+73u^2+52u+4)v^3)t^8\\
&-4u^3v((u^2+6u+2)v^2+(1-u^2)v+u-1)t^9+4u^4v^3t^{10}.\end{aligned}$$]{} Since $A(1,1)=\sum_{n\geq2}|S_n(T)|t^n$ and so $F_T(t)=1+t+A(1,1)$, we get the following result.
\[th69a\] Let $T=\{1234, 1324, 3412\}$. Then $$F_T(x)=\frac{1-9x+35x^2-75x^3+98x^4-78x^5+34x^6-10x^7}{(1-2x)^2(1-x)^6}.$$
Case 72: $\{1243,1324,3412\}$
-----------------------------
First, we look at $G_2(x)$. For $\pi=i\pi' n \pi''$ with 2 left-right maxima and $d\ge 0$ letters in $\pi''$, let $H_d(x)$ and $J_d(x)$ denote the generating functions in the repective cases $i=n-1$ and $i<n-1$. Note $d\ge 1$ in case $i<n-1$. Thus, with $H(x):=\sum_{d\ge 0}H_d(x)$ and $J(x):=\sum_{d\ge 1}J_d(x)$, we have $G_2(x)=H(x)+J(x)$.
\[lem72a1\] $$H(x)=\frac{x^2(1-9x+34x^2-69x^3+80x^4-54x^5+21x^6-3x^7)}{(1-x)^4(1-2x)^2(1-3x+x^2)}\,.$$
Clearly, $H_0(x)=x^2K(x)$ where $K(x)=F_{\{132,3412\}}(x)$, so by [@Sl Seq. A001519] we have that $H_0(x)=\frac{x^2(1-2x)}{1-3x+x^2}$. For $d\geq1$, by considering the position of the letter $n-2$ in $\pi''$ (either leftmost, rightmost, or in the middle), we obtain $$H_d(x)=xH_d(x)+x(H_d(x)-x^{d+2})+\frac{x^{d+2}}{(1-x)^d}+\frac{(K(x)-1)x^{d+4}}{1-x}+\frac{dx^{d+5}}{(1-x)^2}\,.$$ Note that the last equation also holds for $d=0$. Now sum over $d\geq0$.
By similar arguments, one can obtain the the following result for $J(x)$.
\[lem72a2\] We have $$J(x)=\frac{x^3(1-4x+9x^2-11x^3+4x^4-2x^5)}{(1-x)^5(1-2x)^2}.$$
\[th72a\] Let $T=\{1243,1324,3412\}$. Then $$F_T(x)=1+\frac{x(1-11x+54x^2-152x^3+268x^4-311x^5+237x^6-109x^7+30x^8-4x^9)}{(1-x)^6(1-2x)^2(1-3x+x^2)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$. By Lemma \[lem72a1\] and Lemma \[lem72a2\], we have that $$\begin{aligned}
G_2(x)=H(x)+J(x)=\frac{x^2(1-9x+36x^2-81x^3+107x^4-88x^5+50x^6-14x^7+x^8)}{(1-x)^5(1-2x)^2(1-3x+x^2)}.\label{eq72a1}\end{aligned}$$ Now, let us write an equation for $G_m(x)$ with $m\geq3$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$, $\pi^{(s)}<i_1$ for all $s=1,2,\ldots,m-1$ and, since $\pi$ avoids $1243$, $\pi^{(m)}<i_2$. If $\pi^{(m-1)}=\emptyset$ then we have a contribution of $xG_{m-1}(x)$. Otherwise, since $m\geq3$ and $\pi$ avoids $3412$, we see that $\pi^{(m)}<i_1$. Moreover, since $\pi$ avoids $3412$ and $1324$, we see that $\pi^{(1)}>\pi^{(2)}>\cdots>\pi^{(m)}$ and $\pi^{(2)}\cdots\pi^{(m)}$ is decreasing, while $\pi^{(m-1)}$ is not empty, and $\pi^{(1)}$ avoids $132$ and $3412$. Therefore, we have a contribution of $x^{m+1}K(x)/(1-x)^{m-1}$, where $K(x)$ is given in the proof of Lemma \[lem72a1\]. Hence, $$G_m(x)=xG_{m-1}(x)+\frac{x^{m+1}}{(1-x)^{m-1}}\frac{1-2x}{1-3x+x^2}.$$ Summing over all $m\geq3$ and using the expressions for $G_0(x)$ and $G_1(x)$, we obtain $$F_T(x)-1-xF_T(x)-G_2(x)=x(F_T(x)-1-xF_T(x))+\frac{x^4}{(1-x)(1-3x+x^2)}\, .$$ Solve for $F_T(x)$ using to complete the proof.
Case 75: $\{1243,1324,4231\}$
-----------------------------
We make use of the following generating functions: $$\begin{aligned}
F_{\{132,231\}}(x)&=\frac{1-x}{1-2x},\\
F_{\{132,213,4231\}}(x)&=1+\frac{x}{(1-x)^2},\\
F_{\{231,1243,1324\}}(x)&=1+\frac{(x^4-6x^3+7x^2-4x+1)x}{(1-2x)(1-x)^4},\end{aligned}$$ denoted, respectively, $L$, $A$ and $B$ for short (all three follow from the main result in [@MV]).
Let $G_m(x)$ be the generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xB$. To get an explicit formula for $G_m(x)$, we look at the cases $m=2$ and $m\geq3$, and in the latter case we consider whether $\pi^{(m)}$ has a letter between $i_1$ and $i_2$ or not. We split the rather lengthy treatment into subsections.
### $H_m(x)$ and $H'_m(x)$
Define $H_m(x)$ to be the generating function for permutations $$\pi=(n-m)\pi^{(0)}(n-m+1)\pi^{(1)}\cdots n\pi^{(m)}\in S_n(T),\quad 1\le m \le n-1,$$ and define $H'_m(x)$ to be the generating function for permutations $$\pi=(n-m)\pi^{(0)}(n-m+1)\pi^{(1)}\cdots n\pi^{(m)}(n-m-1)\in S_n(T),\quad 1\le m \le n-2.$$
\[lem75a0\] For $m\ge 1$, $H'_m(x)=x^{m+2}L^{m+1}$ and $H_m(x)$ satisfies $$\begin{aligned}
H_m(x)&=x^{m+1}+\sum_{j=0}^{m-1}(x^{j+1}H_{m-j}(x))+\frac{mx^{m+3}A}{1-x}+mx^{m+2}(L-1/(1-x))\\
&+H'_m(x)+x^{m+2}(1/(1-x)^{m+1}-1)(A-1)+x^{m+2}(B-1).\end{aligned}$$
First, we treat $H'_m(x)$. Let $\pi=(n-m)\pi^{(0)}(n-m+1)\pi^{(1)}\cdots n\pi^{(m)}(n-m-1)\in S_n(T)$. In the case $n=m+2$, we have a contribution of $x^{m+2}$. Otherwise, the letter $n-m-2$ belongs to $\pi^{(j)}$, where $j=0,1,\ldots,m$. For each $j=0,1,\ldots,m-1$, we have a contribution $x^{j+1}H'_{m-j}$. For $j=m$, $\pi^{(m)}$ can be written as $\alpha(n-m-2)\beta$ where $\pi^{(0)}\cdots\pi^{(m-1)}\alpha<\beta<n-m-2$. The contribution for the case $\beta=\emptyset$ is $xH'_m(x)$ and for the case $\beta\neq\emptyset$ is $x^{m+3}(L-1)$. Hence, $$H'_m(x)=x^{m+2}+xH'_m(x)+\cdots+x^mH'_1(x)+xH'_m(x)+x^{m+3}(L-1).$$ Setting $m=1$, we can solve for $H'_1(x)$ and then the result follows by induction on $m$.
Next, $H_m(x)$. Let $\pi=(n-m)\pi^{(0)}(n-m+1)\pi^{(1)}\cdots n\pi^{(m)}\in S_n(T)$. In the case $m=n-1$, we have a contribution of $x^{m+1}$. Otherwise, the letter $n-m-1$ belong to $\pi^{(j)}$ for some $j\in \{0,1,\ldots,m\}$. If this $j$ is $<m$, then $\pi{(0)}=\cdots=\pi^{(j-1)}=\emptyset$, and $\pi^{(j)}$ can be written as $\alpha(n-m-1)\beta$ where $\alpha<\beta\pi^{(j+1)}\cdots\pi^{(m)}$. Then we have the contributions $x^{j+1}H_{m-j}$, $\frac{x^{m+3}}{1-x}A$ and $x^{m+2}(L-1/(1-x))$ for the cases $\alpha=\emptyset$, $\alpha$ is nonempty decreasing, and $\alpha$ contains a rise, respectively. If the letter $n-m-1$ belongs to $\pi^{(m)}$ then $\pi^{(m)}$ has the form $\alpha(n-m-1)\beta$ and $\pi^{(0)}\cdots\pi^{(m-1)}\alpha$ is both decreasing and $<\beta$. So we have the contributions $H'_m(x)$, $x^{m+2}(1/(1-x)^{m+1}-1)(A-1)$, and $x^{m+2}(B-1)$, for the cases $\beta=\emptyset$, $\beta\neq\emptyset$ and $\pi^{(0)}\cdots\pi^{(m-1)}\alpha=\emptyset$, $\beta\neq\emptyset$ and $\pi^{(0)}\cdots\pi^{(m-1)}\alpha$ is not empty, respectively. Hence, $H_m(x)$ satisfies the claimed relation,
### $D_k^m(x)$ and ${D'}_k^m(x)$
Define
- $D_k^m(x)$ to be the generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ such that $i_2=i_1+k+1$ and $\pi^{(m)}$ contains the subsequence $(i_1+k)(i_1+k-1)\cdots(i_1+1)$, for all $k\geq1$.
- ${D'}_k^m(x)$ to be the generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ such that $i_2=i_1+k+3$ and $\pi^{(m)}$ contains the subsequence $(i_1+k+2)(i_1+k+1)\cdots(i_1+3)(i_1+1)(i_2+2)$, for all $k\geq0$.
\[lem75a1\] For all $k\geq1$, $$D_k^2(x)=x^{k+2}+xD_k^2(x)+\frac{(k-1)x^{k+3}}{1-x}+E_k(x)+F_k(x),$$ where $$\begin{aligned}
E_k(x)&=E'_k(x)+x^{k+3}(L-1),\\
E'_k(x)&=x^{k+3}+xE'_k(x)+\frac{(k-1)x^{k+4}}{1-x}+xE_k(x),\\
F_k(x)&=F'_k(x)+x^{k+3}(B-1)+\frac{x^{k+4}(A-1)(1+(k+1)\frac{x}{1-x})}{1-x}+\frac{(k+1)x^{k+4}(A-1)}{1-x},\\
F'_k(x)&=x^{k+3}+xF'_k(x)+\frac{kx^{k+4}}{1-x}+xF'_k(x)+x^{k+4}(L-1).\end{aligned}$$
The proof is based on considering all possible positions of the next smallest element whose position we have not yet fixed. Let us write an equation for $D_k^2(x)$. Let $\pi=i\pi'n\pi''\in S_n(T)$ such that $\pi''$ contains the subsequence $(i+k)(i+k-1)\cdots(i+1)$ and $n=i+k+1$. If $i=1$, then we have a contribution of $x^{k+2}$. Otherwise, let $i>1$. If the letter $i-1$ belongs to $\pi'$ then it is the leftmost letter of $\pi'$, so we have a contribution of $xD_k^2(x)$. If the letter $i-1$ belongs to $\pi''$ and is on the left side of $i+2$, then there exists $j\in [3,k-1]$ such that $\pi=in(i+k)\cdots(i+j)(i-1)\cdots1(i+j-1)\cdots(i+1)$, which gives a contribution of $\frac{x^{k+3}}{1-x}$, for all $j=3,4,\ldots,k-1$. Let us denote the contribution of the case when the letter $i-1$ is between the letters $i+2$ and $i+1$ (resp. on right side of $i+1$) by $E_k(x)$ (resp. $F_k(x)$). Then $$D_k^2(x)=x^{k+2}+xD_k^2(x)+\frac{(k-1)x^{k+3}}{1-x}+E_k(x)+F_k(x).$$
To write an equation for $E_k(x)$, let $\pi=i\pi'n\pi^{(k)}(i+k)\cdots\pi^{(2)}(i+2)\pi^{(1)}(i+1)\pi^{(0)}\in S_n(T)$ with $n=i+k+1$ and $\pi^{(1)}=\alpha(i-1)\beta$. Define $E'_k(x)$ to be the contribution in case $\beta=\emptyset$. If $\beta\neq\emptyset$, then $\pi=in(i+k)\cdots(i+2)(i-1)\beta(i+1)$ which implies a contribution of $x^{k+3}(L-1)$. Thus $$E_k(x)=E'_k(x)+x^{k+3}(L-1)\, .$$ By considering the possible positions of the letter $i-2$ in $\pi=i\pi'n\pi^{(k)}(i+k)\cdots\pi^{(1)}(i+1)\in S_n(T)$ with $n=i+k+1$ and $\pi^{(1)}=\alpha(i-1)$, we obtain the equation $$E'_k(x)=x^{k+3}+xE'_k(x)+\frac{(k-1)x^{k+4}}{1-x}+xE_k(x)\,.$$
To write an equation for $F_k(x)$, let $\pi=i\pi'n\pi^{(k)}(i+k)\cdots\pi^{(1)}(i+1)\pi^{(0)}\in S_n(T)$ with $n=i+k+1$ and $\pi^{(0)}=\alpha(i-1)\beta$. Define $F'_k(x)$ to be the contribution of the case $\beta=\emptyset$. When $\beta\neq\emptyset$, by considering the possible positions of the letter $i-2$ in $\pi$, we obtain the contributions $x^{k+3}(B-1)$ when $\pi'\pi^{(k)}\cdots\pi^{(1)}\alpha=\emptyset$, $\frac{x^{k+4}}{1-x}\big(1+(k+1)\frac{x}{1-x}\big)(A-1)$ when $\pi'$ is nonempty (and necessarily decreasing), and $\frac{(k+1)x^{k+4}}{1-x}(A-1)$ when $\pi'=\empty$ and $\pi^{(k)}\cdots\pi^{(1)}\alpha\neq\emptyset$ (here either $\alpha\neq\emptyset$ and $\pi^{(k)}\cdots\pi^{(1)}=\emptyset$, or $\alpha=\emptyset$ and there exists $j\in [1,k]$ such that $\pi^{(j)}\neq\emptyset$ and $\pi^{(i)}=\emptyset$ for all $i\neq j$). Thus, $$F_k(x)=F'_k(x)+x^{k+3}(B-1)+\frac{x^{k+4}(A-1)\big(1+(k+1)\frac{x}{1-x}\big)}{1-x}+\frac{(k+1)x^{k+4}(A-1)}{1-x}.$$
By considering the possible positions of the letter $i-2$ in $\pi=i\pi'n\pi^{(k)}(i+k)\cdots\pi^{(1)}(i+1)\in S_n(T)$ with $n=i+k+1$ and $\pi^{(0)}=\alpha(i-1)$, we find that $$F'_k(x)=x^{k+3}+xF'_k(x)+\frac{kx^{k+4}}{1-x}+xF'_k(x)+x^{k+4}(L-1)\, ,$$ which completes the proof.
The next lemma, giving ${D'}_k^2(x)$, can be established by similar arguments.
\[lem75a2\] Let $P_k(x)$ be the generating function for the number of permutations $\pi=i\pi'n\pi^{(k)}(i+k+2)\cdots\pi^{(1)}(i+3)\pi^{(0)}\in S_n(T)$ such that $n=i+k+3$, $\pi^{(0)}$ contains the subsequence $(i+1)(i+2)$. For all $k\geq0$, ${D'}_k^2(x)=\frac{1}{1-x}P_k(x)$, where $$P_k(x)=x^{k+4}+xP_k(x)+\frac{kx^{k+5}}{1-x}+E''_k(x)+x^{k+4}(L-1)$$ and the generating function $E''_k(x)$ for the number of permutations $\pi=i\pi'n\pi^{(k)}(i+k+2)\cdots\pi^{(1)}(i+3)\pi^{(0)}\in S_n(T)$ such that $n=i+k+3$ and $\pi^{(0)}=\alpha(i-1)\beta(i+1)(i+2)$ satisfies $E''_k(x)=E'''_k(x)+x^{k+5}(L-1)$, where $E'''_k(x)$ is the generating function for the number of permutations $\pi=i\pi'n\pi^{(k)}(i+k+2)\cdots\pi^{(1)}(i+3)\pi^{(0)}\in S_n(T)$ such that $n=i+k+3$ and $\pi^{(0)}=\alpha(i-1)(i+1)(i+2)$ which satisfies satisfies $E'''_k(x)=x^{k+5}+xE'''_k(x)+kx^{k+6}/(1-x)+xE''_k(x)$.
Now, we are ready to write a formula for the generating function $G_2(x)$ for $T$-avoiders with $2$ left-right maxima.
\[pro75a1\] The generating function $G_2(x)$ is given by $$G_2(x)=\frac{x^2(1-8x+31x^2-71x^3+100x^4-93x^5+64x^6-32x^7+11x^8-2x^9)}{(1-x)^7(1-2x)^4}.$$
Let $\pi=i\pi'n\pi''\in S_n(T)$ with $2$ left-right maxima. If $n=i+1$, then the contribution is $H_1(x)$ by Lemma \[lem75a0\]. If $n>i$, then the letters of $\pi''$ that are greater than $i$ form either a decreasing sequence $(i+k)(i+k-1)\cdots(i+1)$ or a decreasing sequence followed by an increasing sequence of at least two terms $(i+k)(i+k-1)\cdots(i+1+s)(i+1)(i+2)\cdots(i+s)$ with $s\ge 2$. By Lemmas \[lem75a1\] and \[lem75a2\], we obtain the contributions $D_k^2(x)$ with $k\geq1$ and ${D'}_k^2(x)$ with $k\geq0$, respectively. Thus, $$G_2(x)=H_1(x)+\sum_{k\geq1}D_k^2(x)+\sum_{k\geq0}{D'}_k^2(x).$$ After working out explicit expressions for $H_1(x)$, $\sum_{k\geq1}D_k^2(x)$ and $\sum_{k\geq0}{D'}_k^2(x)$ from Lemmas \[lem75a0\], \[lem75a1\] and \[lem75a2\], respectively, we complete the proof.
Lemmas \[lem75a1\] and \[lem75a2\] suggest a method to study the generating functions $D_k^m(x),\ k\geq1$ and ${D'}_k^m(x),\ k\geq0$.
### Formula for $D_1^m(x)$
\[lem75b1\] Let $m\geq3$. Then $D_1^m(x)=x^{m+1}L^m+(S_m+S'_m)/(1-x)$, where $$S_m=\frac{1}{1-2x}(x^{m+2}+x^{m+3}\sum_{j=1}^{m-1}(1-x)^{-j}+x^{m+3}(L-1))$$ and $$S'_m=x^{m+2}(B-1)+\frac{x^{m+3}(A-1)}{1-x}+x^{m+2}(1/(1-x)^m-1)(A-1)+\frac{x^{m+4}}{(1-x)^2}(A-1).$$
Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima such that $\pi^{(m)}$ has exactly one letter between $i_1$ and $i_2$. So, $i_j=i_1+j$ for all $j=2,3,\ldots,m$ and $\pi^{(m)}=\alpha(i_1+1)\beta$. We denote the generating function for permutations $\pi$ with $\beta=\emptyset$ by $E_1^m$, and we denote the generating function for permutations $\pi$ with $\beta=\emptyset$ and $i_1-1 \in \pi^{(j)}$ by $E_1^{m,j}$, for $j=2,3,\ldots,m$. Then, by considering the position of letter $i-1$ in $\pi$ with $\beta=\emptyset$, we obtain $$E_1^m=x^{m+1}+xE_1^m+\sum_{j=2}^{m-1}E_1^{m,j}+E_1^{m,m},$$ where $$E_1^{m,s}=x^{m+2}+xE_1^{m,s}+\cdots+xE_1^{m,m-1}+x^{m+3}(L-1)$$ with $E_1^{m,m}=xE_1^m+x^{m+2}(L-1)$. Hence, by induction on $s$, we have $$E_1^{m,s}=\sum_{i=0}^{m-s-1}\binom{m-s-1}x^{i+1}(1-2x)^{-i-1}(x^{m+1}+x^{m+2}(L-1)).$$ Thus, $E_1^m=x^{m+1}L^m$.
Denote the generating function for permutations $\pi$ with $\beta=\beta'(i-1)$ by $S_m$, and the generating function for permutations $\pi$ with $\beta=\beta'(i-1)\beta''$ with $\beta''\neq\emptyset$ by $S'_m$. Clearly, $D_1^m=E_1^m+(S_m+S'_m)/(1-x)$. By considering the position of the letter $i-2$ in the permutations counted by $S_m$, we obtain $$S_m=x^{m+2}+xS_m+x^{m+3}\sum{j=1}^{m-1}(1-x)^{-j}+xS_m+x^{m+3}(L-1).$$ By considering the four possibilities where $\pi^{(1)}\cdots\pi^{(m-1)}\alpha$ and $\beta'$ are empty or not, we complete the proof.
### A formula for ${D'}_0^m$
\[lem75c0\] Let $m\geq3$. Then ${D'}_0^m(x)=\frac{x^{m+2}}{1-x}(L+x(L+L^2+\cdots+L^m-1))$.
As in the proof of Lemma \[lem75b1\], by considering the position of the letter $i_1-1$ in permutations $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima and $\pi^{(m)}=\alpha(i_1+1)\beta(i_1+2)$, we obtain $${D'}_0^m(x)=x^{m+2}+x{D'}_0^m+\sum_{j=2}^m{D'}_0^{m,j},$$ where ${D'}_0^{m,j}$ is the generating function for the $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima and $\pi^{(m)}=\alpha(i_1+1)\beta(i_1+2)$ such that the letter $i_1-1$ belongs to $\pi^{(j)}$. Again, by considering the position of the letter $i_1-2$, we have $${D'}_0^{m,j}=x^{m+3}+x{D'}_0^{m,j}+x{D'}_0^{m,j+1}+...+x{D'}_0^{m,m-1}+x^{m+4}(L-1)$$ with (see Lemma \[lem75a0\]) $${D'}_0^{m,m}=x^{m+3}(L-1)+x^{m+2}(L-1)+x^2H'_{m-1}(x)=x^{m+3}(L-1)+x^{m+2}(L-1)+x^{m+3}L^m.$$ By induction on $s$, we see that $${D'}_0^{m,s}=x^{m+3}L^{m-s+1}.$$ Thus, by substituting the expression for ${D'}_0^{m,j}$ into the equation for ${D'}_0^m$, we complete the proof.
### A formula for $D_k^m$ and ${D'}_k^m$
In the next two lemmas we study the generating functions $D_k^m(x)$ with $k\geq2$ and ${D'}_k^m$ with $k\geq1$.
As before, by considering all possible positions of the next smallest element whose position we have not yet fixed, we obtain the following results.
\[lem75b2\] Let $m\geq3$ and $k\geq2$. Define $S_k^{m,m-1}$ (resp. $S_k^{m,m}$, ${E}_k^m$, ${E'}_k^m$) to be the generating function for the $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima such that the letters between $i_1$ and $i_2$ in $\pi^{(m)}$ form a decreasing sequence $j_1j_2\cdots j_k$ and the letter $i_1-1$ lies between $j_{k-1}$ and $j_k$ (resp. $i_1-1$ lies on the right side of $j_k$, $\pi^{(m)}$ has no letter smaller than $i_1$ on the right side of the letter $j_1$, $\pi^{(m)}$ has at least one letter smaller than $i_1$ on the right side of the letter $j_1$). Then $D_k^m(x)=E_k^m+{E'}_k^m$, where $E_k^m=\frac{x^{m+k}}{(1-x)^m}$, $$\begin{aligned}
{E'}_k^m&=x{E'}_k^m+\frac{(k-2)x^{m+k+1}}{1-x}+S_k^{m,m-1}+{S'}_k^{m,m},\\
S_k^{m,m-1}&=x^{m+k+1}(L-1)+T_k^m,\\
T_k^m&=x^{m+k+1}+xT_k^m+x^{m+k+2}\sum_{j=2}^{m-1}(1-x)^{-j}+\frac{(k-1)x^{m+k+2}}{1-x}+xT_k^m+x^{m+k+2}(L-1),\\
S_k^{m,m}&=x^{m+k+1}(B-1)+x^{m+k+1}\frac{(k+1)x}{1-x}(A-1)+\frac{x^{m+k+2}}{1-x}(A-1)\left(1+\frac{(k+1)x}{1-x}\right)\\
&+x^{m+k+1}\left(\frac{1}{(1-x)^{m-2}}-1\right)\frac{A-1}{1-x}+\frac{x^{m+k+2}(A-1)}{(1-x)^2}\left(\frac{1}{(1-x)^{m-2}}-1\right)+{T'}_k^m,\\
{T'}_k^m&=x^{m+k+1}+x{T'}_k^m+x^{m+k+2}\sum_{j=2}^{m-1}(1-x)^{-j}+\frac{kx^{m+k+2}}{1-x}+x{T'}_k^m+x^{m+k+2}(L-1).\end{aligned}$$
\[lem75c1\] Let $m\geq3$ and $k\geq1$. Define $U_k^m$ to be the generating function for the $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima such that the letters between $i_1$ and $i_2$ in $\pi^{(m)}$ form a subsequence $(i_2-1)\cdots(i_1+3)(i_1-1)(i_1+1)(i_1+2)$ of $k+1$ terms. Then $$\begin{aligned}
{D'}_k^m&=x^{m+k+2}+x{D'}_k^m+x^{m+k+3}\sum_{j=2}^{m-1}(1-x)^{-j}+\frac{kx^{m+k+3}}{1-x}\\
&+x^{m+k+3}(L-1)+U_k^m+x^{m+k+2}(L-1),\\
U_k^m&=x^{m+k+3}+xU_k^m+x^{m+k+4}\sum_{j=2}^{m-1}(1-x)^{-j}+\frac{kx^{m+k+4}}{1-x}\\
&+x^{m+k+4}(L-1)+xU_k^m.\end{aligned}$$
### A formula for $F_T(x)$
Now, we are ready to find an explicit formula for the generating function $F_T(x)$ by getting a formula for $G_m(x)$.
\[th75a\] Let $T=\{1243,1324,4231\}$. Then $$F_T(x)=\frac{x}{1-3x+x^2}-\frac{2-4x-4x^2-x^3}{(1-2x)^2}
+\frac{3-20x+55x^2-83x^3+74x^4-38x^5+12x^6-2x^7}{(1-x)^8}.$$
Fix $m\geq3$. Let $G'_m(x)$ be the generating function for the $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima and $i_2>i_1+1$. Let $\gamma$ be the subsequence of letters of $\pi^{(m)}$ that are larger than $i_1$ and smaller than $i_2$. Since $\pi$ avoids $T$, $\gamma$ can be written as either $(i_2-1)(i_2-2)\cdots(i_1+1)$ or $(i_2-1)(i_2-2)\cdots(i_1+s+1)(i_1+1)(i_1+2)\cdots(i_1+s)$ for $s\geq2$. It follows that $$G_m(x)=H_{m-1}(x)+\sum_{k\geq0}D_k^m+\frac{1}{1-x}\sum_{k\geq1}{D'}_k^m.$$ Thus, $$\sum_{m\geq3}G_m(x)=\sum_{m\geq2}H_m(x)+\sum_{m\geq3}\sum_{k\geq0}D_k^m+\frac{1}{1-x}\sum_{m\geq3}\sum_{k\geq1}{D'}_k^m.$$ By Lemmas \[lem75a0\] and \[lem75b1\]-\[lem75c1\], we obtain after simplifying $$\sum_{m\geq3}G_m(x)=\frac{x^3p(x)}{(1-x)^8(1-2x)^2(1-3x+x^2)},$$ where $p(x)=1-10x+49x^2-149x^3+296x^4-403x^5+408x^6-322x^7+187x^8-74x^9+18x^{10}-2x^{11}$. Since $$F_T(x)=1+xB+G_2(x)+\sum_{m\geq3}G_m(x),$$ and we have determined each of the summands, the result follows.
Case 76: $\{3412,1324,2341\}$
-----------------------------
Here we use the fact that $$F_{\{132,2341,3412\}}(x)=F_{\{213,2341,3412\}}(x)=\frac{1-3x+x^2}{(1-x)^2(1-2x)},$$ denoted $A$ for short (followed from the main result of see [@MV]).
\[lem76a1\] The generating function $G_2(x)$ for $T$-avoiders with $2$ left-right maxima is given by $$G_2(x)=\frac{x^2(1-7x+23x^2-40x^3+39x^4-22x^5+9x^6-4x^7)}{(1-x)^6(1-2x)^2}.$$
Suppose $\pi=i\al n\be \in S_n(T)$ with $2$ left-right maxima. If $i=1$, then $\pi=1n\be$ and where $\be$ avoids $\{213,2341,3412\}$, giving a contribution to $G_2(x)$ of $x^2A$. Now suppose $i>1$. If $i-1$ is the leftmost letter of $\al$, we obtain a contribution of $xG_2(x)$ by deleting $i-1$. If $i-1 \in \al$ and is not the leftmost letter, then $\pi$ decomposes as $\pi=i\alpha'(i-1)\alpha''n\beta'\beta''$, where $\beta'>i>i-1>\beta''>\alpha'>\alpha''$ (to avoid 1324) and $\beta''$ is decreasing (to avoid 3412), as in Figure \[figAK2\].
(4,3.6) (4,2)(4,6.5) (2,4)(6,4) (0,2)(0,4)(2,4)(2,0)(4,0)(4,2)(0,2) (4,6.5)(6,6.5)(6,8.5)(4,8.5)(4,6.5) (6,4)(8,4)(8,6)(6,6)(6,4) (0,6.5)(2,6)(4,8.5) (1,3) (3,1) (5,7.5) (7,5) (-.4,6.8) (1.2,6.0) (3.6,8.8)
Consider four cases according as $\alpha'',\beta''$ are empty or not:
- $\alpha'',\,\beta''$ both empty. Note that $\al'$ is nonempty and avoids $\{132,2341,3412\}$ and $\be'$ avoids $\{213,2341,3412\}$. So the contribution is $x^3(A-1)A$.
- $\alpha''=\emptyset,\ \beta''\ne \emptyset$. Here, $\be'$ is decreasing (to avoid 2341), while $\al'$ is nonempty and avoids $\{132,2341,3412\}$. So the contribution is $x^4(A-1)/(1-x)^2$.
- $\alpha''\ne\emptyset,\ \beta'' = \emptyset$. Here, $\al'$ is decreasing (to avoid 2341) and nonempty, $\al''$ is also decreasing (to avoid 3412) and nonempty, and $\be'$ avoids $\{213,2341,3412\}$. So the contribution is $x^5A/(1-x)^2$.
- $\alpha''\ne\emptyset,\ \beta'' \ne \emptyset$. Here, $\al'$ and $\be'$ are decreasing (both to avoid 2341) and $\al'$ is nonempty, $\al''$ is also decreasing (to avoid 3412) and nonempty, and $\be''$ is nonempty (and decreasing, as always). So the contribution is $x^6/(1-x)^4$.
Next, suppose $i>1$ and $i-1 \in \be$ so that $\pi=i \al n \be' (i-1) \be''$. Then $\be'>i$ since $j \in \be'$ with $j<i$ makes $inj(i-1)$ a $3412$. For $d\ge 0$, let $B_d$ be the generating function for such permutations where $\be'$ has $d$ letters.
\[lem76aB0\] We have $$B_0=\frac{x^3(1-5x+12x^2-13x^3+4x^4+4x^5-4x^6)}{(1-x)^4(1-2x)^2}\,.$$
First, let $K$ be the generating function for $T$-avoiders $i(i-2)(i-4)\pi' (i+1)(i-1)(i-3)\pi''$ with $2$ left-right maxima such that $i\geq5$ where $n=i+1$. Let us write an equation for $K$. When $i=5$ the contribution is $x^6$. Otherwise, we consider the position of $i-5$. If $i-5$ is either the leftmost letter of $\pi'$ or of $\pi''$ then we have contribution of $xK$. Otherwise $\pi'=\alpha'(i-5)\alpha''$ with $\alpha'\neq\emptyset$ such that $i-5>\pi''>\alpha'>\alpha''$ with $\alpha''$ and $\pi''$ are decreasing. By consider whether $\alpha''$ is empty or not, we obtain the contribution $x^7(A-1+x^2/(1-x)^2)/(1-x)$. Thus $$K=2xK+x^6+x^7(A-1+x^2/(1-x)^2)/(1-x)\,,$$ which leads to $$K=\frac{(1-4x+6x^2-2x^3)x^6}{(1-2x)^2(1-x)^2}\,.$$
Next, let $K'$ be the generating function for $T$-avoiders $i(i-2)\pi'n(i-1)(i-3)\pi''$ with $2$ left-right maxima such that $n>i\geq4$. Let us write an equation for $K'$. When $i=4$ we have a contribution of $x^5A$. If $i-4$ is the leftmost letter of $\pi'$ then we have a contribution $K$, and if $i-4$ is leftmost letter of $\pi''$ then we have a contribution $xK'$. Otherwise $\pi'=\alpha'(i-4)\alpha''$ with $\alpha'\neq\emptyset$ such that $i-4>\pi''>\alpha'>\alpha''$ with $\alpha''$ and $\pi''$ decreasing. By considering whether $\alpha''$ is empty or not, we obtain the contribution $x^6(A-1+x^2/(1-x)^2)/(1-x)$. Thus $K'=x^5A+xK'+K+x^6(A-1+x^2/(1-x)^2)/(1-x)$, which leads to $$K'=\frac{(1-4x+6x^2-2x^3-2x^4)x^5}{(1-2x)^2(1-x)^3}\,.$$
Next, let $K''$ be the generating function for $T$-avoiders $i(i-2)\pi'n(i-1)\pi''$ with $2$ left-right maxima such that $n>i\geq3$. By using similar techniques as for finding formulas $K$ and $K'$, we have $$\begin{aligned}
K''=x^4A+xK''+K'+x^5A(A-1)+\frac{x^7A}{(1-x)^2}+\frac{x^6(A-1)}{1-x}+\frac{x^8}{(1-x)^3}\,,\end{aligned}$$ which implies $$K''=\frac{(1-3x+4x^2)x^4}{(1-2x)^2(1-x)^2}\,.$$
Lastly, let $K'''$ be the generating function for $T$-avoiders $i\pi'n(i-1)j(i-2)\pi''$ with $2$ left-right maxima such that $n>j>i\geq3$. By using similar techniques as for finding formulas $K$ and $K'$, we obtain $$K'''=\frac{(1-x+2x^3)x^5}{(1-2x)(1-x)^2}\,.$$
Now, we are ready to write an equation for the generating function $B_0$ for $T$-avoiders $i\pi'n(i-1)\pi''$ with $2$ left-right maxima such that $n>i\geq2$. Again, we decompose the structure by looking at position of $i-2$. If $i=2$ then we have a contribution of $x^3A$, if $i-2$ is leftmost letter of $\pi'$ then we have a contribution of $K''$, if $i-2$ belongs to $\pi''$ and is not the leftmost letter of it, we have a contribution of $x^4A(A-1)+\frac{x^6A}{(1-x)^2}+\frac{x^5(A-1)}{1-x}+\frac{x^7}{(1-x)^3}$, and if $i-2$ belongs to $\pi''$ we have a contribution of $K'''$. Thus, $$B_0=x^3A+K''+x^4A(A-1)+\frac{x^6A}{(1-x)^2}+\frac{x^5(A-1)}{1-x}+\frac{x^7}{(1-x)^3}+K'''\,,$$ which completes the proof.
One can similarly show (details omitted) that $$B_1=\frac{x^4(1-2x+2x^2)(1-2x+2x^2-2x^4)}{(1-x)^4(1-2x)^2}\, ,$$ and $B_d=xB_{d-1}$ for all $d\geq2$. Hence, the total contribution for the case $i>1$ and $i-1 \in \be$ is given by $B$, where $B-B_1-B_0=x(B-B_0)$, which leads to $$B=\frac{x^3(1-5x+13x^2-17x^3+9x^4+2x^5-4x^6)}{(1-x)^5(1-2x)^2}.$$
Adding all the contributions, $$G_2(x)=x^2A+xG_2(x)+x^3(A-1)A+ \frac{x^4(A-1)}{(1-x)^2} +\frac{x^5 A}{(1-x)^2}+\frac{x^6}{(1-x)^4}+B,$$ with solution for $G_2(x)$ as claimed.
\[th76a\] Let $T=\{3412,1324,2341\}$. Then $$G_T(x)=\frac{1-10x+44x^2-110x^3+173x^4-176x^5+114x^6-45x^7+12x^8-4x^9}{(1-x)^7(1-2x)^2}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$, $G_1(x)=xF_T(x)$, and $G_2(x)$ is given above.
Now, let us write an equation for $G_m(x)$ with $m\geq3$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $2341$, we can write $\pi$ as $$\pi=i_1\pi^{(1)}i_2\pi^{(2)}i_3i_4\cdots i_m\gamma\beta,$$ where $\pi^{(1)}>\pi^{(2)}$ and $\gamma>i_{m-1}>\beta>i_{m-2}$. By considering the four possibilities for whether $\pi^{(2)},\beta$ are empty of not, we obtain the contributions $x^mA^2$ (both empty), $\frac{x^{m+1}}{(1-x)^2}A$ ($\pi^{(2)}$ empty, $\be$ nonempty), $\frac{x^{m+1}}{(1-x)^2}A$ ($\pi^{(2)}$ nonempty, $\be$ empty) and $\frac{x^{m+2}}{(1-x)^4}$ (both nonempty). Thus, $$G_m(x)=x^m\left(A+\frac{x}{(1-x)^2}\right)^2.$$ Therefore, $$F_T(x)-1-xF_T(x)-G_2(x)=\sum_{m\geq3}G_m(x)=\frac{x^3}{1-x}\left(A+\frac{x}{(1-x)^2}\right)^2.$$ Substituting for $G_2(x)$ and solving for $F_T(x)$ completes the proof.
Case 80: $\{1324,2341,3421\}$
-----------------------------
In order to study this case, we need the following lemmas.
\[lem80a1\] Let $T=\{1324,2341,3421\}$. Let $G_2(x)$ be the generating function for the number of permutations $\pi=i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima. Then $$G_2(x)=\frac{x^2(1-8x+28x^2-52x^3+50x^4-22x^5+5x^6)}{(1-x)^4(1-2x)^2(1-3x+x^2)}.$$
In order to find a formula for the generating function $G_2(x)$, we refine it as follows. Let $G_2(x;d)$ be the generating function for the number of permutations $\pi=i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima and where $\pi''$ has exactly $d$ points smaller than $i$. Clearly, $$G_2(x;0)=x^2F_{\{132,2341,3421\}}(x)F_{\{213,2341,3421\}}(x)=x^2K(x)^2,$$ where $K(x)=\frac{1-3x+3x^2}{(1-x)^2(1-2x)}$ (where we leave the proof to the reader). For $d=1$, our permutations can be written as $i(i-1)\cdots (i'+1)\alpha n(n-1)\cdots i''i'\beta$ with $\alpha<i'$ and $i<\beta<i''$. By considering whether $\alpha$ and $\beta$ are empty or not, one can show that $$G_2(x;1)=\frac{x^3}{(1-x)^2}+\frac{x^3(K(x)-1)^2}{(1-x)^2}+\frac{2x^4(1-x+x^2)}{(1-x)^4(1-2x)}.$$
For $d\geq2$, our permutations can be written as $$\pi=i\alpha^{(1)}\cdots\alpha^{(d)}n\beta^{(1)}j_1\cdots\beta^{(d)}j_d\beta^{(d+1)}$$ such that all the letters that are greater than $j_1$ in $i\alpha^{(1)}\cdots\alpha^{(d)}$ are decreasing and $n\beta^{(1)}\cdots\beta^{(d)}$ is decreasing. We denote all the letters between $j_1$ and $j_2$ in $\pi'$ by $\delta$. Now, let us write an equation for $G_2(x;d)$. If $\delta=\beta^{(2)}=\emptyset$, then we have a contribution of $xG_2(x;d-1)$. If $\delta\neq\emptyset$, then $\pi$ can be written as $$\pi=i\alpha^{(1)}\cdots\alpha^{(d-1)}\gamma'\gamma''n\beta^{(1)}j_1\cdots\beta^{(d)}j_d\beta^{(d+1)}$$ such that $\alpha^{(1)}\alpha^{(2)}\cdots\alpha^{(d-1)}\gamma'$ is decreasing, $j_{d+2-s}>\alpha^{(s)}>j_{d+1-s}$ for all $s=1,2,\ldots,d-1$ with $j_{d+1}=i$, $j_2>\gamma'=\delta>j_1$ and $j_1>\gamma''$ where $\gamma''$ avoids $132,2341,3421$ and $\beta^{(d+1)}$ avoids $213,2341,3421$. Thus, we have a contribution of $\frac{x^{d+3}}{(1-x)^{2d}}K(x)^2$. Otherwise, $\delta=\emptyset$ and $\beta^{(2)}\neq\emptyset$, so similarly, we have a contribution of $\frac{x^{d+3}}{(1-x)^{2d-1}}K(x)^2$. Therefore, for all $d\geq2$, $$G_2(x;d)=xG_2(x;d-1)+\frac{x^{d+3}}{(1-x)^{2d}}K(x)^2+\frac{x^{d+3}}{(1-x)^{2d-1}}K(x)^2.$$ By summing over all $d\geq2$, we obtain $$G_2(x)-G_2(x;0)-G_2(x;1)=x(G_2(x)-G_2(x;0))+\frac{x^5K(x)^2}{(1-x)^2(1-3x+x^2)}+\frac{x^5K(x)^2}{(1-x)(1-3x+x^2)}.$$ By using the values of $G_2(x;0)$ and $G_2(x;1)$, and then solving for $G_2(x)$, we complete the proof.
\[th80a\] Let $T=\{1324,2341,3421\}$. Then $$F_T(x)=\frac{1-7x+20x^2-29x^3+25x^4-10x^5+2x^6}{(1-x)^5(1-3x+x^2)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$. Note that the generating function $G_2(x)$ is given in Lemma \[lem80a1\].
Let $m=3$. Each permutation of $\pi\in S_n(T)$ with exactly $3$ left-right maxima can be represented as either
- $\pi=i_1\pi^{(1)}(i_1+1)n\pi^{(3)}$ with $\pi^{(1)}<i_1$ and $\pi^{(3)}>i_1+1$, where $\pi^{(1)}$ avoids $132,2341,3421$ and $\pi^{(3)}$ avoids $213,2341,3421$.
- $\pi=i_1\pi^{(1)}i_2n(n-1)\cdots(i_2+1)(i_1+1)(i_1+2)\cdots(i_2-1)$, where $i_1+1<i_2$ and $\pi^{(1)}$ avoids $132,2341,3421$.
- $\pi=i_1(i_1-1)\cdots i'(i_1+1)12\cdots(i'-1)n\pi^{(3)}$, where $i'>1$ and $\pi^{(3)}$ avoids $213,2341,3421$.
- $\pi=i_1(i_1-1)\cdots i'i_212\cdots(i'-1)n(n-1)\cdots(i_2+1)(i_1+1)(i_1+2)\cdots(i_2-1)$, where $i_1+1<i_2$ and $i'>1$.
By finding the corresponding generating functions, we obtain $$G_3(x)=x^3K(x)^2+\frac{2x^4K(x)}{(1-x)^2}+\frac{x^5}{(1-x)^4}=\frac{x^3}{(1-2x)^2}.$$
Now, let us write an equation for $G_m(x)$ with $m\geq4$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $2341$, we see that $\pi^{(s)}<i_1$ for all $s=2,3,\ldots,m-1$ and $\pi^{(m)}>i_{m-2}$. Hence, we have a contribution of $xG_{m-1}(x)$ (by removing the letter $i_{m-2}$). Thus, $G_m(x)=x^{m-3}G_3(x)$ for all $m\geq3$. Therefore, $$F_T(x)-G_0(x)-G_1(x)-G_2(x)=\sum_{m\geq3}G_m(x)=\frac{1}{1-x}G_3(x),$$ which, by substituting the values of $G_j(x)$, $j=0,1,2,3$, we complete the proof.
Case 84: $\{4231,1324,2341\}$
-----------------------------
\[lem84a1\] Let $\pi=i\alpha n\beta\in S_n(T)$ with exactly $2$ left-right maxima, $i\geq2$ and $i-1$ in $\alpha$. Then the generating function $H(x)$ for such permutations is given by $$H(x)=xG_2(x)+x^3(L(x)-1)\left(\frac{L(x)}{1-x}+L(x)-\frac{1}{1-x}\right),$$ where $G_2(x)$ is the generating function for permutations in $S_n(T)$ with exactly $2$ left-right maxima and $L(x)=\frac{1-x}{1-2x}$.
Let us write an equation for $H(x)$. If $i-1$ is the first letter of $\alpha$, then the contribution is $xG_2(x)$. Otherwise, $i-1$ is the last letter of $\al$ (to avoid 1324 and 4231), and $\pi$ can be written as $\pi=i\alpha'(i-1)n\beta'\beta''$ such that $\emptyset\neq\alpha'<\beta''<i<\beta'$, where $\alpha'$ avoids both $132$ and $231$ and $\be''$ avoids both $213$ and $231$. Thus, by considering whether $\beta'=n(n-1)\cdots(i+1)$ or not, we get contributions of $x^3\big(L(x)-1\big)L(x)/(1-x)$ and $x^3\big(L(x)-1\big)\big(L(x)-1/(1-x)\big)$ (recall that $F_{\{132,231\}}=F_{\{213,231\}}=L(x)$). Hence, $$H(x)=xG_2(x)+x^3\big(L(x)-1\big)\left(\frac{L(x)}{1-x}+L(x)-\frac{1}{1-x}\right),$$ as required.
\[lem84a2\] Let $\pi=i\alpha n\beta\in S_n(T)$ with exactly $2$ left-right maxima, $i\geq2$, and $i-1$ in $\beta$. Then the generating function $H'(x)$ for such permutations is given by $$H'(x)=\frac{x^3(1-4x+9x^2-12x^3+6x^4-x^5)}{(1-3x+x^2)(1-2x)^2(1-x)^2}.$$
In order to write an equation for $H'(x)$, we define $A_d(x)$ to be the generating function for permutations $\pi=i\alpha n\beta'(i-1)\beta''\in S_n(T)$ with exactly 2 left-right maxima, $i\geq2$, and such that $\beta'$ has $d$ letters that are greater than $i$ (these $d$ letters form a decreasing subsequence because $\pi$ avoids $4231$). We leave to the reader to show that $A_0(x)=\frac{x^3(1-3x+5x^2-4x^3)}{(1-x)(1-2x)^3}$, $A_1(x)=\frac{x^4(1-3x+4x^2-5x^3+4x^4)}{(1-x)(1-2x)^4}$, and for $d\geq2$, $$A_d(x)-xA_{d-1}(x)=\frac{x^{d+4}(1-x)^d}{(1-2x)^{d+3}}.$$ Summing over all $d\geq0$, we obtain $$H'(x)=\frac{x^3(1-4x+9x^2-12x^3+6x^4-x^5)}{(1-3x+x^2)(1-2x)^2(1-x)^2},$$ as required.
\[lem84a3\] The generating function $G_2(x)$ for permutations in $S_n(T)$ with exactly $2$ left-right maxima is given by $$G_2(x)=\frac{x^2(1-7x+22x^2-35x^3+29x^4-16x^5+6x^6-x^7)}{(1-3x+x^2)(1-2x)^2(1-x)^3}.$$
Permutations in $S_n(T)$ with exactly 2 left-right maxima whose first letter is 1 (and hence second letter is $n$) have the generating function $x^2L(x)$, where $L(x)=F_{\{231,213\}}(x)=\frac{1-x}{1-2x}$. Thus, by Lemmas \[lem84a1\] and \[lem84a2\], we obtain $$\begin{aligned}
G_2(x)&=x^2L(x)+xG_2(x)+x^3(L(x)-1)\left(\frac{L(x)}{1-x}+L(x)-\frac{1}{1-x}\right)\\
&+\frac{x^3(1-4x+9x^2-12x^3+6x^4-x^5)}{(1-3x+x^2)(1-2x)^2(1-x)^2},\end{aligned}$$ and solving for $G_2(x)$ completes the proof.
\[th84a\] Let $T=\{4231,1324,2341\}$. Then $$F_T(x)=\frac{1-9x+33x^2-62x^3+64x^4-36x^5+7x^6}{(1-3x+x^2)(1-2x)^2(1-x)^3}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_{\{231,1324\}}(x)=
x \frac{1 - 4 x + 5 x^2 - x^3)}{(1 - 2 x)^2 (1 - x)}$ [@MV], and $G_2(x)$ is given by Lemma \[lem84a3\]. Now suppose $m\geq3$ and let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $2341$, we see that $\pi^{(3)}\cdots\pi^{(m-1)}=\emptyset$, $\pi^{(1)}>\pi^{(2)}$, and $\pi^{(m)}=\alpha\beta$ with $\alpha>i_{m-1}>\beta>i_{m-2}$. By considering the four possibilities, $\pi^{(2)},\,\beta$ empty or not, we obtain the contributions $x^mL(x)^2$, $\frac{x^m}{1-x} L(x)\big(L(x)-1\big)$, $\frac{x^m}{1-x} L(x)\big(L(x)-1\big)$ and $\frac{x^m}{(1-x)^2}\big(L(x)-1\big)^2$. Thus, $$G_m(x)=x^m\left(L(x)+\frac{L(x)-1}{1-x}\right)^2.$$ Summing over $m\geq3$, we obtain $$F_T(x)-1-G_1(x)-G_2(x)=\frac{x^3}{1-x}\left(L(x)+\frac{L(x)-1}{1-x}\right)^2\,.$$ Using the expressions for $G_1(x),G_2(x)$, we complete the proof.
Case 86: $\{3412,2431,1324\}$
-----------------------------
In this subsection, let $A=\frac{1-2x}{1-3x+x^2}$ denote the generating function for $F_{\{132,3412\}}(x)$ and let $B=\frac{1-3x+3x^2}{(1-x)^2(1-2x)}$ denote the generating function for $F_{\{213,2431,3412\}}(x)$ (they can be derived from results in [@MV]).
\[lem86a1\] Let $H(x)$ be the generating function permutations in $S_n(T)$ whose first letter is $n-1$. Then $$H(x)=\frac{x^2(x^4-x^3+5x^2-4x+1)}{(1-x)(1-2x)(1-3x+x^2)}.$$
Refine $H(x)$ to $H_d(x)$, the generating function for $T$-avoiders $\pi=(n-1)\pi'n\pi''$ where $\pi''$ has $d$ letters. Since $\pi$ avoids $3412$, $\pi''$ is a decreasing subsequence say $j_1j_2\cdots j_d$. Clearly, $H_0(x)=x^2A$. If $d\geq2$, then there is no letter between $j_1$ and $j_d$ in $\pi'$, otherwise $\pi$ contains $2431$. Thus $H_d(x)=x^{d-1}H_1(x)$, for all $d\geq1$. Now, let us compute $H_1(x)$. If $j_1=1$, then we have a contribution of $x^3A$. So, we can assume that $j_1>1$. Define $H_1(x;e,d)$ to be the generating function for $T$-avoiders $\pi=(n-1)\alpha1\beta nj_1$, where $\beta$ has $e$ letters smaller than $j_1$ and $d$ letters greater than $j_1$. Clearly, $\beta$ is an increasing subsequence. Now let us examine the following cases:
- $e=d=0$. We have a contribution of $xH_1(x)$.
- $d=0$ and $e\geq1$. Here we see that $\alpha=\alpha'\alpha''$ with $\alpha'$ a decreasing subsequence and $\alpha''<e_1$, where $e_1$ is the smallest letter in $\beta$. So we have a contribution of $\frac{x^{e+4}}{(1-x)^{e+1}}A$.
- $e=0$ and $d\geq1$. Here we see that either $\alpha=\alpha'\alpha''$ with $\alpha'$ decreasing and $\alpha''<j_1$, or $\alpha=\alpha'\alpha''(j_1+1)\alpha'''$ with $\alpha'$ decreasing, $j_1>\alpha''>\alpha'''$. So we have a contribution of $x^4\left(\frac{A}{1-x}+\frac{x(A-1)}{(1-x)^2}\right)\frac{x^d}{(1-x)^d}$.
- $e,d\geq1$. Here we see that $\alpha=\alpha'\alpha''$ with $\alpha'$ decreasing and $\alpha''<e_1$, where $e_1$ is the smallest letter in $\beta$. So we have a contribution of $\frac{x^{d+e+6}A}{(1-x)^{d+e+1}}$.
Hence, $$(1-x)H_1(x)=x^3A+\sum_{e\geq1}\frac{x^{e+4}}{(1-x)^{e+1}}A+\sum_{d\geq1}x^4\left(\frac{A}{1-x}+\frac{x(A-1)}{(1-x)^2}\right)\frac{x^d}{(1-x)^d}+
\sum_{d,e\geq1}\frac{x^{d+e+6}A}{(1-x)^{d+e+1}},$$ which leads to $$H_1(x)=\frac{x^3(1-3x+3x^2+x^3)}{(1-2x)(1-3x+x^2)}.$$ Since $H(x)=x^2A(x)+\sum_{d\geq1}H_d(x)=x^2A(x)+\frac{1}{1-x}H_1(x)$, the result follows.
\[lem86a2\] The generating function for $T$-avoiders with $2$ left-right maxima is given by $$G_2(x)=\frac{x^2(1-5x+10x^2-8x^3+x^4+x^5-x^6)}{(1-x)^3(1-2x)(1-3x+x^2)}.$$ The generating function for $T$-avoiders with $3$ left-right maxima is given by $$G_3(x)=\frac{x^3(1-4x+4x^2+3x^3-5x^4-3x^5+4x^6-x^7)}{(1-2x)(1-x)^4(1-3x+x^2)}.$$
We treat $G_2(x)$ and leave the similar derivation of $G_3(x)$ to the reader. Let $\pi=i\pi'n\pi''\in S_n(T)$ with 2 left-right maxima. By Lemma \[lem86a1\], the contribution for the case $i=n-1$ is $H(x)$. So let us assume that $i<n-1$, that is, $\pi''$ contains the letter $i+1$. Since $\pi$ avoids $2431$, we can write $\pi$ as $\pi=i\alpha'n\alpha''\beta'$ where $\alpha'\alpha''<i<\beta'$, $\beta'\neq\emptyset$ and $\alpha''$ is decreasing. If $\alpha''=\emptyset$, then we have contribution of $x^2A(B-1)$. If $\alpha''$ has exactly one letter then we have a contribution of $\frac{x^3}{1-x}A(B-1)+x^3\big(A-1/(1-x)\big)(B-1)$ corresponding to the cases $\alpha'$ decreasing or not. If $\alpha''$ has at least two letters, then $\beta'>i>\alpha'>\alpha''$, so the contribution is $x^4A(B-1)/(1-x)$. By adding all the contributions, we complete the proof.
\[th86a\] Let $T=\{3412,2431,1324\}$. Then $$F_T(x)=\frac{1-7x+19x^2-24x^3+16x^4-4x^5-x^6+2x^7}{(1-x)^3(1-2x)(1-3x+x^2)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1,\ G_1(x)=xF_T(x)$, and $G_2(x),\,G_3(x)$ are given in Lemma \[lem86a2\]. Now let us write an equation for $G_m(x)$ with $m\geq4$. Suppose $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ we see that $\pi^{(j)}<i_1$ for all $j=1,2,\ldots,m-1$. With $\alpha=\pi^{(m-1)}$ and $\beta$ the subsequence of all letters between $i_{m-2}$ and $i_{m-1}$ in $\pi^{(m)}$, we have contributions as follows:
- If $\alpha=\beta=\emptyset$, then we have $xG_{m-1}(x)$;
- If $\alpha=\emptyset$ and $\beta\neq\emptyset$, then we have $x^{m+1}A/(1-x)^{m-2}$;
- If $\alpha\neq\emptyset$ and $\beta=\emptyset$, then we have $x^{m+1}AB/(1-x)^{m-1}$;
- If $\alpha\neq\emptyset$ and $\beta\neq\emptyset$, then we have $x^{m+2}A/(1-x)^{m-1}$.
Thus, $G_m(x)=xG_{m-1}(x)+x^{m+1}A/(1-x)^{m-2}+x^{m+1}AB/(1-x)^{m-1}+x^{m+2}A/(1-x)^{m-1}$, for all $m\geq4$. Summing over $m\geq4$, we obtain $$\begin{aligned}
&F_T(x)-1-xF_T(x)-G_2(x)-G_3(x)\\
&=x\big(F_T(x)-1-xF_T(x)-G_2(x)\big)+\frac{x^5A}{(1-x)(1-2x)}+\frac{x^5AB}{(1-x)^2(1-2x)}+\frac{x^6A}{(1-x)^2(1-2x)}.\end{aligned}$$ The result follows by substituting the expressions for $G_2(x)$, $G_3(x)$, $A$ and $B$, and solving for $F_T(x)$.
Case 88: $\{3412,3421,1324\}$
-----------------------------
For this case we outline a proof based on a labelled generating forest and also give a proof based on left-right maxima.
### Labelled generating forest
The table in the following lemma both recursively defines labels and gives valid succession rules. The $j$-th entry on the right hand side in the rules gives the label when $n+1$ is inserted into the $j$-th active site left to right. A label $k^i,\ 1\le i \le 5$, always indicates $k$ active sites. The proof is omitted.
\[lem88a1\] The generating forest $\mathcal{F}$ is given by $$\begin{array}{ll}
\mbox{\bf Roots: }&3^1,3^3\\
\mbox{\bf Rules: }&k^1\rightsquigarrow 3^12^53^5\cdots(k-2)^5(k+1)^1(k+1)^2,\quad k\geq3,\\ &k^2\rightsquigarrow3^32^53^5\cdots(k-3)^5(k-1)^4k^4(k+1)^3,\quad k\geq4,\\
&k^3\rightsquigarrow3^32^53^5\cdots(k-2)^5k^4(k+1)^3,\quad k\geq3,\\
&2^3\rightsquigarrow2^33^4,\\
&k^4\rightsquigarrow2^32^53^5\cdots(k-1)^5(k+1)^4,\quad k\geq3,\\
&k^5\rightsquigarrow1^52^53^5\cdots k^5,\quad k\geq1\,.\\
\end{array}$$
\[th88a\] Let $T=\{3412,3421,1324\}$. Then $$F_T(x)=\frac{(1-x)^2(1-5x+7x^2+x^3)}{(1-2x)^4}.$$
Let $a_k(x)$, $b_k(x)$, $c_k(x)$, $d_k(x)$ and $e_k(x)$ be the generating functions for the number of permutations in the $n$th level of the generating forest $\mathcal{F}$ with label $k^1$, $k^2$, $k^3$, $k^4$ and $k^5$, respectively. By Lemma \[lem88a1\], we have $$\begin{aligned}
a_k(x)&=xa_{k-1}(x),\quad k\geq4,\\
b_k(x)&=xa_{k-1}(x),\quad k\geq4,\\
c_k(x)&=x(b_{k-1}+c_{k-1}(x)),\quad k\geq4,\\
d_k(x)&=x(d_{k-1}(x)+c_k(x)+b_k(x)+b_{k+1}(x)),\quad k\geq4,\\
e_k(x)&=xe_k(x)+x\sum_{j\geq k+1}(d_j(x)+e_j(x))+x\sum_{j\geq k+2}(a_j(x)+c_j(x))+x\sum_{j\geq k+3}b_j(x),\quad k\geq2\end{aligned}$$ with $a_3(x)=x^2+x\sum_{j\geq3}a_j(x)$, $c_2(x)=xc_2(x)+x\sum_{j\geq3}d_j(x)$, $c_3(x)=x^2+xc_3(x)+x\sum_{j\geq4}(b_j(x)+c_j(x))$, $d_3(x)=xc_2(x)+xc_3(x)+xb_4(x)$ and $e_1(x)=x\sum_{j\geq1}e_j(x)$.
Now let $A(x,v)=\sum_{k\geq3}a_k(x)v^k$, $B(x,v)=\sum_{k\geq3}b_k(x)v^k$, $C(x,v)=\sum_{k\geq3}c_k(x)v^k$, $D(x,v)=\sum_{k\geq2}d_k(x)v^k$ and $E(x,v)=\sum_{k\geq2}e_k(x)v^k$.
By the above recurrences, we see that $A(x,v)-(x^2+xA(x,1))v^3=xvA(x,v)$. Thus, by substituting $v=1$, we obtain $A(x,1)=\frac{x^2}{1-2x}$, which implies $A(x,v)=\frac{x^2v^3(1-x)}{(1-xv)(1-2x)}$. Thus, by the equation for $b_k(x)$, we have $B(x,v)=\frac{x^3v^4(1-x)}{(1-xv)(1-2x)}$.
By the equations for $c_k(x),d_k(x),e_k(x)$, we have $$\begin{aligned}
&C(x,v)-c_3(x)v^3-c_2(x)v^2=xv(B(x,v)+C(x,v)-c_2(x)v^2),\\
&D(x,v)-d_3(x)=x(vD(x,v)+C(x,v)-c_3(x)v^3-c_2(x)v^2+B(x,v)+(B(x,v)-b_4(x)v^4)/v),\\
&E(x,v)-e_1(x)v=\frac{x}{1-v}(v^2E(x,1)-vE(x,v)+v^2D(x,1)-D(x,v)+v^3C(x,1)-C(x,v))\\
&+xv^2c_2(x)+\frac{x}{1-v}(v^4B(x,1)-B(x,v)+v^3A(x,1)-A(x,v))\end{aligned}$$ with $c_2(x)=\frac{x}{1-x}D(x,1)$, $c_3(x)=x^2+xB(x,1)+xC(x,1)-\frac{x^2}{1-x}D(x,1)$, $d_3(x)=\frac{x^2}{(1-x)^2}D(x,1)$ and $e_1(x)=xE(x,1)v$. Note that $b_4(x)=\frac{x^3(1-x)}{1-2x}$ (from $B(x,v)$).
Substituting $v=1$ in the first two equations, and solving for $C(x,1)$ and $D(x,1)$, we obtain $$C(x,1)=\frac{x^2(1-4x+7x^2-4x^3-2x^4)}{(1-2x)^3},\quad
D(x,1)=\frac{x^3(1-x)(1-2x^2)}{(1-2x)^3},$$ which implies $$\begin{aligned}
C(x,v)&=\frac{x^2v^2(x^2(1-2x^2)+(1-5x+4x^5+9x^2-8x^3)v)}{(1-xv)^2(1-2x)^3},\\
&+\frac{x^3(1-x)v^4((x^2+2x-1)(2x^2-2x+1)+x(1-2x)^2v)}{(1-xv)^2(1-2x)^3},\\
D(x,v)&=\frac{x^3v^3(1-4x+5x^2+2x^3-6x^4+x^2(8x^3-6x^2+2x-1)v+x^4(1-2x^2)v^2)}{(1-xv)^3(1-2x)^3}.\end{aligned}$$
We solve the equation for $E(x,v)$ by using the kernel method (see, e.g., [@HM] for an exposition), taking $v=\frac{1}{1-x}$. Using the expressions for $A(x,v)$, $B(x,v)$, $C(x,v)$ and $D(x,v)$, this gives $$E(x,1)=\frac{x^4(x+1)(3-4x)}{(1-2x)^4}.$$ Since $F_T(x)=1+x+A(x,1)+B(x,1)+C(x,1)+D(x,1)+E(x,1)$, the result follows.
Case 93: $\{1324,2413,3421\}$
-----------------------------
In order to study this case, we need the following lemmas.
\[lem93a1\] Let $T=\{1324,2413,3421\}$. Let $H_m(x)$ be the generating function for the number of $T$-avoiders of the form $\pi=(n+1-m)\pi^{(1)}(n+2-m)\pi^{(2)}\cdots n\pi^{(m)}$ such that $\pi(n+1)\in S_{n+1}(T)$. Then $$H_m(x)=x^mK(x)+\frac{(m-1)x^{m+1}L(x)}{1-x},$$ where $K(x)=F_{\{132,3421\}}(x)=1+\frac{x(1-3x+3x^2)}{(1-x)(1-2x)^2}$ (see [@Sl Seq. A005183]) and $L(x)=\frac{1-x}{1-2x}$.
Let us write a formula for $H_m(x)$. Let $\pi=(n+1-m)\pi^{(1)}(n+2-m)\pi^{(2)}\cdots n\pi^{(m)}$ such that $\pi(n+1)\in S_{n+1}(T)$. Since $\pi$ avoids $1324$, we see that $\pi^{(1)}>\cdots>\pi^{(m)}$. If $\pi^{(2)}=\cdots=\pi^{(m)}=\emptyset$, then $\pi^{(1)}$ avoids $132$ and $3412$, which gives a contribution of $x^mK(x)$. Otherwise, since $\pi$ avoids $3421$, there exists a unique $j$ such that $\pi^{(j)}\neq\emptyset$ with $2\leq j\leq m$. Moreover, $\pi^{(1)}$ avoids $132,231$. Thus, we have a contribution of $\frac{x^{m+1}L(x)}{1-x}$. Hence, $H_m(x)=x^mK(x)+\frac{(m-1)x^{m+1}L(x)}{1-x}$, as required.
\[lem93a2\] Let $T=\{1324,2413,3421\}$. Let $H'_m(x)$ be the generating function for the number of permutations $\pi=(n+1-m)\pi^{(1)}(n+2-m)\pi^{(2)}\cdots n\pi^{(m)}\in S_{n}(T)$. Then $$H'_m(x)=xH'_{m-1}(x)+\frac{x^{m+1}}{(1-x)(1-2x)},$$ with $H'_2(x)=\frac{x^2(1-4x+5x^2)}{(1-2x)^3}$.
We leave the formula for $H'_2(x)$ to the reader. Now we write a formula for $H'_m(x)$ with $m\geq3$. Let $\pi=(n+1-m)\pi^{(1)}(n+2-m)\pi^{(2)}\cdots n\pi^{(m)}\in S_n(T)$. If $\pi^{(m-1)}=\emptyset$, then we have a contribution of $xH'_{m-1}(x)$. Thus, we can assume that $\pi^{(m-1)}\neq\emptyset$. Since $\pi$ avoids $1324$ and $2413$, we have that $\pi^{(2)}=\cdots = \pi^{(m-2)}=\emptyset$, $\pi^{(1)}>\pi^{(m)}>\pi^{(m-1)}$ and $\pi^{(m-1)}\pi^{(m)}$ is increasing. Note $\pi^{(1)}$ avoids $132$ and $231$. Thus, we have a contribution of $\frac{x^{m+1}}{(1-x)^2}L(x)$. Hence, $H'_m(x)=xH'_{m-1}(x)+\frac{x^{m+1}}{(1-x)(1-2x)}$, as required.
Similar consideration yield the following result.
\[lem93a3\] Let $T=\{1324,2413,3421\}$. Let $H''_m(x)$ be the generating function for the number of permutations $\pi=(n+1-m)\pi^{(1)}(n+2-m)\pi^{(2)}\cdots n\pi^{(m)}\in S_{n}(T)$ such that $\pi^{(m)}$ has a letter smaller than $n+1-m$. Then $$H''_m(x)=xH''_{m-1}(x)+\frac{x^{m+2}}{(1-x)(1-2x)},$$ with $H''_2(x)=\frac{x^3(1-4x+6x^2-2x^3)}{(1-x)(1-2x)^3}$.
Now we are ready to state the formula for $F_T(x)$.
\[th93a\] Let $T=\{1324,2413,3421\}$. Then $$F_T(x)=\frac{1-10x+42x^2-94x^3+120x^4-86x^5+31x^6-3x^7}{(1-x)^3(1-2x)^4}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Now let us write an equation for $G_m(x)$ with $m\geq2$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $2413$, we see that $\pi^{(s)}<i_1$ for all $s=1,2,\ldots,m-1$ and $\pi^{(m)}=\beta^{(m)}\cdots\beta^{(1)}$ such that $i_{s-1}<\beta^{(s)}<i_{s}$ for all $s=1,2,\ldots,m$ (with $i_0=0$). We consider cases:
- $\beta^{(2)}=\cdots\beta^{(j-1)}=\emptyset$ and $\beta^{(j)}\neq\emptyset$ with $j=2,3,\ldots,m-1$. Since $\pi$ avoids $T$, we see that $\pi^{(s)}=\emptyset$ for all $s=j,j+1,\ldots,m-1$ and $\beta^{(s)}=\emptyset$ for all $s\neq j,m$, where $\beta^{(m)}$ avoids $132$ and $231$, and $\beta^{(j)}$ is nonempty increasing. Thus, by Lemma \[lem93a1\], we have a contribution of $\frac{x^{m+2-j}H_{j-1}(x)L(x)}{1-x}$.
- $\beta^{(2)}=\cdots\beta^{(m-1)}=\emptyset$ and $\beta^{(m)}\neq\emptyset$. If $\beta^{(1)}=\emptyset$, then we have a contribution of $xH_{m-1}(x)(K(x)-1)$, see Lemma \[lem93a1\]. Otherwise, we have a contribution of $H''_m(x)(L(x)-1)$, see Lemma \[lem93a3\].
- $\beta^{(2)}=\cdots\beta^{(m)}=\emptyset$. For this case, we have a contribution of $H'_m(x)$, see Lemma \[lem93a2\].
By summing over all contributions, we obtain $$G_m(x)=H'_m(x)+(xH_{m-1}(x)(K(x)-1)+H''_m(x)(L(x)-1)+\sum_{j=1}^{m-2}\frac{x^{m+1-j}H_j(x)L(x)}{1-x},$$ for all $m\geq2$. Hence, by summing over all $m\geq2$ and using the initial conditions $G_0(x)$ and $G_1(x)$, we have $$F_T(x)-1-xF_T(x)=H'(x)+(L(x)-1)H''(x)+\left(\frac{x^3}{(1-x)^2}L(x)+x(K(x)-1)\right)H(x),$$ where, by Lemmas \[lem93a1\], \[lem93a2\] and \[lem93a3\], $$\begin{aligned}
H(x)&=\sum_{m\geq1}H_m(x)=\frac{xK(x)}{1-x}+\frac{x^3}{(1-x)^2(1-2x)}=\frac{x(1-3x+3x^2)}{(1-x)(1-2x)^2},\\
H'(x)&=\sum_{m\geq2}H'_m(x)=\frac{H'_2(x)}{1-x}+\frac{x^4}{(1-x)^3(1-2x)}=\frac{x^2(1-3x+3x^2)^2}{(1-x)^3(1-2x)^3},\\
H''(x)&=\sum_{m\geq2}H''_m(x)=\frac{H''_2(x)}{1-x}+\frac{x^5}{(1-x)^3(1-2x)}=\frac{x^3(1-3x+3x^2)(1-2x+2x^2)}{(1-x)^3(1-2x)^3}.\end{aligned}$$ Hence, by solving for $F_T(x)$, we complete the proof.
Case 99: $\{1324,3142,4231\}$
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Throughout this case, we abbreviate $F_{\{132,4231\}}(x)$ by $K(x)$. Recall $K(x)=1+\frac{x(1-3x+3x^2)}{(1-x)(1-2x)^2}$ [@Sl Seq. A005183]. To find $F_T(x)$, we use the following lemmas.
\[lem99a1\] Let $T=\{1324,3142,4231\}$. Define $J_m(x)$ to be the generating function for the number of permutations $\pi=i_1i_2\cdots i_m\pi'\in S_n(T)$ with exactly $m$ left-right maxima, all occurring at the start, such that $i_1>1$. Set $J(x)=\sum_{m\geq2}J_m(x)$. Then $J_2(x)=\frac{x^3(1-2x+3x^2-x^3)}{(1-x)^2(1-2x)^2}$ and $$J(x)=\frac{x^3(1-2x+4x^2)}{(1-x)(1-2x)^3}\, .$$
Let us write an equation for $J_m(x)$ with $m\geq3$. Let $\pi=i_1i_2\cdots i_m\pi'\in S_n(T)$ with exactly $m$ left-right maxima. If $i_{m-1}=n-1$, then we have a contribution of $xJ_{m-1}(x)$. Thus, we can assume that $i_{m-1}<n-1$. If $\pi'=(n-1)\pi''$, then we have a contribution of $xJ_m(x)$, otherwise, since $\pi$ avoids $T$ (note that $1$ belongs to $\pi'$), we can write $\pi$ as $\pi'=i'(i'-1)\cdots(i_{m-1}+1)\alpha\beta$, such that $\alpha<i_1$, $i'<\beta<n$ and $n-1$ belongs to $\beta$. Note that $\beta$ avoids $213$ and $231$, and $\alpha$ avoids $132$ and $231$. Thus, we have a contribution of $x^m(1+x)(L(x)-1)^2$. Hence, $$J_m(x)=xJ_{m-1}(x)+xJ_m(x)+x^m(1+x)(L(x)-1)^2.$$
By very similar techniques, we find that $J_2(x)=x(K(x)-1)+xJ_2(x)+x^2(1+x)(L(x)-1)^2$. Thus, $J_2(x)=\frac{x^3(1-2x+3x^2-x^3)}{(1-x)^2(1-2x)^2}$ and the displayed recurrence for $J_m$ then readily yields the stated expression for $J(x)$.
\[lem99a2\] Let $T=\{1324,3142,4231\}$. Define $J'_m(x)$ to be the generating function for the number of permutations $\pi=i_1i_2\cdots i_m\pi'\in S_n(T)$ with exactly $m$ left-right maxima such that $i_1=1$ (and $i_m=n$). Then $$J'(x):=\sum_{m\geq3}J'_m(x)=\frac{x^3(1-2x+2x^2)}{(1-x)(1-2x)^2}.$$
Let us write an equation for $J'_m(x)$ with $m\geq3$. Let $\pi=i_1i_2\cdots i_m\pi'\in S_n(T)$ with exactly $m$ left-right maxima such that $i_1=1$. By considering whether $i_2=2$ or not (we leave the details to the reader), we obtain the contributions $xJ'_{m-1}(x)$ and $\frac{x^{m+1}L(x)}{(1-x)^{m-1}}$. Hence, $$J'_m(x)=xJ'_{m-1}(x)+\frac{x^{m+1}L(x)}{(1-x)^{m-1}}$$ with $J'_2(x)=x^2L(x)$. The result follows by solving this recurrence.
\[th99a\] Let $T=\{1324,3142,4231\}$. Then $$F_T(x)=\frac{1-8x+25x^2-36x^3+23x^4-4x^5+x^6}{(1-x)(1-2x)^4}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_{\{132,4231\}}(x)=xK(x)$.
Now, let us write an equation for $G_m(x)$ with $m\geq2$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$, we see that $\pi^{(s)}<i_1$ for all $s=2,3,\ldots,m-1$. We consider the following cases:
- $\pi^{(m)}$ has a letter smaller than $i_1$. Since $\pi$ avoids $4231$, we see that $\pi^{(1)}\cdots\pi^{(m-1)}$ is decreasing. Assume that $\pi^{(j)}\neq\emptyset$ and $\pi^{(j+1)}=\cdots\pi^{(m-1)}=\emptyset$. Since $\pi$ avoids $3142$, we have that $\pi^{(m)}$ has no letters between $i_1$ and $i_j$. Thus, we have a contribution of $\frac{1}{1-x}J_m(x)$ when $j=1$, and $\frac{x^j}{(1-x)^j}J_{m+1-j}(x)$ when $j=2,3,\ldots,m-1$, where $J_d(x)$ is defined in Lemma \[lem99a1\]. Hence, the total contribution for this case is given by $$\frac{1}{1-x}J_m(x)+\sum_{j=2}^{m-1}\frac{x^j}{(1-x)^j}J_{m+1-j}(x).$$
- $\pi^{(m)}>i_1$ and $\pi^{(2)}=\cdots\pi^{(m-1)}=\emptyset$. Then we have a contribution of $L(x)J'_m(x)$, where $J'_m(x)$ is given in Lemma \[lem99a2\].
- $\pi^{(m)}>i_1$ and there exists $j$ such that $\pi^{(j)}\neq\emptyset$ and $\pi^{(j+1)}=\cdots\pi^{(m-1)}=\emptyset$, where $j=2,3,\ldots,m-1$. As before, we obtain the contributions $\frac{x^{j-1}(L(x)-1)}{(1-x)^{j-1}}J'_{m+1-j}(x)$ when $j\leq m-2$, and $\frac{x^mL(x)(L(x)-1)}{(1-x)^{m-2}}$ when $j=m-1$. Hence, the total contribution from this case is $$\frac{x^mL(x)(L(x)-1)}{(1-x)^{m-2}}+\sum_{j=1}^{m-3}\frac{x^{j}(L(x)-1)}{(1-x)^{j}}J'_{m-j}(x).$$
Hence, $$\begin{aligned}
G_m(x)&=L(x)J'_m(x)+\frac{x^mL(x)(L(x)-1)}{(1-x)^{m-2}}+\sum_{j=1}^{m-3}\frac{x^j(L(x)-1)}{(1-x)^j}J'_{m-j}(x)\\
&+\frac{1}{1-x}J_m(x)+\sum_{j=2}^{m-1}\frac{x^j}{(1-x)^j}J_{m+1-j}(x).\end{aligned}$$ Summing over $m\geq2$ and using the expressions for $G_0(x)$ and $G_1(x)$, we obtain $$\begin{aligned}
F_T(x)-1-xK(x)&=\sum_{m\geq2}G_m(x)\\
&=x^2L(x)^2+\frac{1}{1-x}J_2(x)+L(x)J'(x)+\frac{x^3L(x)(L(x)-1)}{1-2x}\\
&+\frac{x(L(x)-1)J'(x)}{1-2x}+\frac{1}{1-x}(J(x)-J_2(x))+\frac{x^2}{(1-x)(1-2x)}J(x),\end{aligned}$$ where $J(x)=\sum_{d\geq2}J_d(x)$, $J_2(x)$ and $J'(x)=\sum_{d\geq3}J'_d(x)$ are given in Lemmas \[lem99a1\] and \[lem99a2\]. After several algebraic operations, we complete the proof.
Case 132: $\{1324,2341,2413\}$
------------------------------
\[th132a\] Let $T=\{1324,2341,2413\}$. Then $$F_T(x)=\frac{1-8x+23x^2-27x^3+12x^4-4x^5+x^6}{(1-3x+x^2)^3}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Let us write an equation for $G_2(x)$. Let $\pi=i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima. The contribution of the case $\pi''>i$ is $x^2K(x)^2$, where $K(x)=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{132,2341\}$- (or $\{213,2341\}$-) avoiders (for example, see [@MV]). If there is a letter in $\pi''$ smaller than $i$, we can write $\pi$ as $\pi=i\pi'n(n-1) \cdots(i+1)\pi''$ such that $\pi''$ is not empty. Thus, we have a contribution of $\frac{x}{1-x}\big(F_T(x)-1-xK(x)\big)$, where $\frac{x^2}{1-x}K(x)$ counts the permutations of the form $i\pi'n(n-1)\cdots(i+1)$. Hence, $G_2(x)=x^2K(x)^2+\frac{x}{1-x}\big(F_T(x)-1-xK(x)\big)$.
Next, let us write an equation for $G_3(x)$. Let $\pi=i_1\pi'i_2\pi''i_3\pi'''\in S_n(T)$ with exactly $3$ left-right maxima. Since $\pi$ avoids $1324$ and $2413$, we can write $\pi=i_1\pi'i_2\pi''i_3\alpha\beta$ such that $\pi''<\pi'<i_1<\beta<i_2<\alpha<i_3$. By considering the four cases where $\pi'',\beta$ are empty or not (if both are nonempty we get an occurrence of $2413$, so we actually have three cases), we obtain $$G_3(x)=x^3K(x)^2+\frac{x^3}{1-x}K(x)\big(K(x)-1\big)+
\frac{x^3}{1-x}K(x)\big(K(x)-1\big).$$
Finally, let us write an equation for $G_m(x)$ with $m\geq4$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $T$, we see that $i_1>\pi^{(1)}>\pi^{(2)}$, $\pi^{(j)}=\emptyset$ for $j=3,4,\ldots,m-1$, and $\pi^{(m)}=\alpha\beta$ such that $i_m>\alpha>i_{m-1}>\beta$. Again, by considering the four cases where $\pi^{(2)},\beta$ are empty or not, we obtain that $$G_m(x)=x^mK(x)^2+\frac{x^m}{1-x}K(x)\big(K(x)-1\big)+\frac{x^m}{1-x}K(x)\big(K(x)-1\big)+\frac{x^m}{(1-x)^2}(K(x)-1)^2.$$
By summing for $m\geq4$ and using the expressions for $G_0(x),G_1(x),G_2(x)$ and $G_3(x)$, we obtain $$\begin{aligned}
F_T(x)&=1+xF_T(x)+\frac{x}{1-x}\big(F_T(x)-1-xK(x)\big)+x^2K(x)^2 \\
&+x^3K(x)^2+\frac{2x^3}{1-x}K(x)\big(K(x)-1\big) +\frac{x^4}{1-x}\left(K(x)+\frac{K(x)-1}{1-x}\right)^2,\end{aligned}$$ which, by solving for $F_T(x)$, completes the proof.
Case 150: $\{1324,3421,3241\}$
------------------------------
In this section, define $L=F_{\{213,231\}}(x)=\frac{1-x}{1-2x}$ (see [@SiS]). We also have, by the simple decomposition $\pi'n\pi''$ of $132$-avoiders, $$A:=F_{\{132,3421,3241\}}(x)=\frac{1-3x+3x^2}{(1-x)^2(1-2x)}$$ and $$B:=F_{\{213,3421\}}(x)=F_{\{132,4312\}}(x)=\frac{1-4x+5x^2-x^3}{(1-x)(1-2x)^2}$$.
\[lem150a1\] The generating function for $T$-avoiders of the form
- $(d+1)n\pi'(d+2)$ with $n-3\geq d\geq1$ is given by $$E_d(x)=\frac{x^{d+3}+\sum_{j=2}^{d+1}\frac{x^{d+4}L}{(1-x)^j}}{1-x/(1-x)}.$$
- $(d+1)n\pi'$ with $n-2\geq d\geq1$ is given by $$D_d(x)=\frac{x^{d+2}+\sum_{j=2}^d\frac{x^{d+3}L}{(1-x)^j}+\frac{E_d(x)}{1-x}+\frac{x^{d+3}}{(1-x)^{d+1}}(B-1/(1-x))}{1-x/(1-x)}.$$
Moreover, $$E(x):=\sum_{d\geq1}E_d(x)=\frac{(1-3x+4x^2)x^4}{(1-2x)^3}\mbox{ and }D(x):=\sum_{d\geq1}D_d(x)=\frac{(1-5x+10x^2-5x^3)x^3}{(1-2x)^4}.$$
Let us write an equation for $E_d(x)$. Let $\pi=(d+1)n\pi'(d+2)\in S_n(T)$. If $n=d+3$, we have a contribution of $x^{d+3}$, so let us assume that $n\geq d+4$ and consider 3 cases:
- If $d+3$ on left side of letter $1$, $\pi$ can be written as $(d+1)n(n-1)\cdots n'(d+3)\pi'(d+2)$ with $\pi'<n'$, so we have $\frac{x}{1-x}E_d(x)$.
- If $d+3$ between the letters $j$ and $j+1$, $j=1,2,\ldots,d-1$, then $\pi$ can be written as $$(d+1)n\alpha^{(1)}1\alpha^{(2)}2\cdots\alpha^{(j)}j\alpha^{(j+1)}(d+3)\pi'(j+1)(j+2)\cdots d(d+2)$$ such that the subsequence $n\alpha^{(1)}\cdots\alpha^{(j+1)}$ is decreasing and greater than $\pi'$ and $\pi'>d+3$, where $\pi'$ avoids $\{231,213\}$. So, we have $\frac{x^{d+4}}{(1-x)^j}L(x)$.
- If $d+3$ between the letters $d$ and $d+2$, then $\pi$ can be written as $$(d+1)n\alpha^{(1)}1\alpha^{(2)}2\cdots\alpha^{(d)}d\alpha^{(d+1)}(d+3)\pi'(d+2)$$ such that the subsequence $n\alpha^{(1)}\cdots\alpha^{(d+1)}$ is decreasing and greater than $\pi'$ and $\pi'>d+3$, where $\pi'$ avoids $\{231,213\}$. So, we have $\frac{x^{d+4}}{(1-x)^{d+1}}L(x)$.
Thus, $E_d(x)=x^{d+3}+\frac{x}{1-x}E_d(x)+\sum_{j=2}^{d+1}\frac{x^{d+4}L}{(1-x)^j}$, as claimed.
Similarly, we can write an equation for $D_d(x)$ and obtain $$D_d(x)=x^{d+2}+\frac{x}{1-x}D_d(x)+\sum_{j=2}^d\frac{x^{d+3}L}{(1-x)^j}+\frac{E_d(x)}{1-x}+\frac{x^{d+3}}{(1-x)^{d+1}}(B-1/(1-x)),$$ where $x^{d+2}$, $\frac{x}{1-x}D_d(x)$, $\frac{x^{d+3}L}{(1-x)^{j+1}}$ and $\frac{E_d(x)}{1-x}+\frac{x^{d+3}}{(1-x)^{d+1}}(B-1/(1-x))$ are the respective contributions for the cases $n=d+2$, the letter $d+2$ on the left side of the letter $1$, the letter $d+2$ between the letters $j$ and $j+1$ with $j=1,2,\ldots,d-1$, and the letter $d+2$ on the right side of the letter $d$.
By summing over $d\geq1$, we complete the proof.
\[lem150a2\] For $m\ge 3$, the generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m$ left-right maxima and $\pi^{(1)}\neq\emptyset$ is given by $$G_{m,1}(x)=\frac{x^{m+1}}{(1-x)^2}(A-1)L+x^m(A-1)B+(m-2)\frac{x^{m+1}}{1-x}(A-1)L.$$
Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima and $\pi^{(1)}\neq\emptyset$. By the definitions we see that $\pi^{(2)}\cdots\pi^{(m-1)}=\emptyset$. We have the following contributions to $G_{m,1}(x)$:
- If $\pi^{(m)}>i_{m-1}$, then we have $x^m(A-1)B$.
- If $\pi^{(m)}$ has a letter smaller than $i_1$, then $\pi^{(1)}<\pi^{(m)}=\pi'(k+1)(k+2)\cdots(i_1-1)(i_1+1)\cdots(i_2-1)$, $\pi'>i_{m-1}$, $\pi'$ avoids $\{213,231\}$ and $\pi^{(1)}$ avoids $\{132,3421,3241\}$. This leads to a contribution of $\frac{x^{m+1}}{(1-x)^2}(A-1)L$.
- If $\pi^{(m)}>i_1$ and $\pi^{(m)}$ has a letter between $i_j$ and $i_{j+1}$, where $j=1,2,\ldots,m-2$, then $\pi^{(1)}<i_1<\pi^{(m)}$, where $\pi^{(m)}=\pi'(i_j+1)\cdots(i_{j+1}-1)$ with $\pi'$ a $\{213,231\}$-avoider, and $\pi^{(1)}$ avoids $\{132,3421,3241\}$. This leads to a contribution of $\frac{x^{m+1}}{1-x}(A-1)L$.
Adding these contributions yields the result.
Using similar arguments, we obtain the following result.
\[lem150a3\] The generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m$ left-right maxima and $\pi^{(1)}=\cdots=\pi^{(s-1)}=\emptyset$ and $\pi^{(s)}\neq\emptyset$, $2\leq s\leq m-2$, is given by $$G_{m,s}(x)=\frac{x^{m+2}L}{(1-x)^2}(1+sx/(1-x))+\frac{x^{m+1}}{1-x}B+(m-2)\frac{x^{m+2}L}{(1-x)^2}.$$
\[lem150a4\] Let $T=\{1324,3421,3241\}$. Then the generating function for $T$-avoiders $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m$ left-right maxima and $\pi^{(1)}=\cdots=\pi^{(m-2)}=\emptyset$ and $\pi^{(m-1)}\neq\emptyset$ is given by $$G_{m,m-1}(x)=N_m+N'_m+M_m+M'_m,$$ where $$\begin{aligned}
N_m&=x^{m+1}B/(1-x),\\
N'_m&=(m-2)x^{m+2}L/(1-x)^2,\\
M_m&=\frac{xN_m/(1-x)+x^{m+2}(L-1/(1-x))/(1-x)^3}{1-x/(1-x)} ,\\ M'_m&=\frac{xN_m'+(m-2)x^{m+3}(L-1)/((1-x)^2(1-2x))}{1-x}.\end{aligned}$$
To write an equation for $G_{m,m-1}(x)$, suppose $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m$ left-right maxima and $\pi^{(1)}=\cdots=\pi^{(m-2)}=\emptyset$ and $\pi^{(m-1)}\neq\emptyset$. Let $\alpha$ be the subsequence of $\pi^{(m)}$ consisting of letters smaller than $\pi^{(1)}$, and let $\beta$ be the subsequence of $\pi^{(m)}$ consisting of letters between $i_1$ and $i_{m-1}$. Note that $\pi^{(m)}<\alpha\beta$. Now let us consider the following four cases:
- $\alpha=\beta=\emptyset$: Here $\pi^{(m-1)}<i_1$ and $\pi^{(m)}>i_{m-1}$. So we have a contribution $N_m$.
- $\alpha=\emptyset$ and $\beta\neq\emptyset$: Here $\pi^{(m-1)}<i_1$, and there exits $1\leq j\leq m-2$ such that $\pi^{(m)}=\pi'(i_j+1)\cdots(i_{j+1}-1)$ and $\pi'$ avoids $\{213,231\}$. So we have a contribution $N'_m$.
- $\alpha\neq\emptyset$ and $\beta=\emptyset$. In this case $\pi^{(m)}$ can be written as $$\beta^{(s)}(i_1-s)\beta^{(s-1)}(i_1-s+1)\cdots \beta^{(1)}(i_1-1)\beta^{(0)},$$ where $\beta^{(s)}>\beta^{(s-1)}>\cdots>\beta^{(0)}>i_{m-1}$ and $\pi^{(m-1)}<i_1-s$. Let $M_m$ be the contribution of this case. By considering whether $\beta^{(s)}$ is a decreasing sequence (including the empty case) or contains an ascent, we obtain the contributions $\frac{x}{1-x}(N_m+M_m)$ and $x^{m+2}(L-1/(1-x))/(1-x)^3$. Thus, $$M_m=\frac{x}{1-x}(N_m+M_m)+x^{m+2}(L-1/(1-x))/(1-x)^3,$$ with solution for $M_m$ as above.
- $\alpha\neq\emptyset$ and $\beta\neq\emptyset$. In this case, there exists $j$, $1\leq j\leq m-2$ such that $\pi^{(m)}$ can be written as $$\beta^{(s)}(i_1-s)\beta^{(s-1)}(i_1-s+1)\cdots\beta^{(1)}(i_1-1)\beta^{(0)}(i_j+1)(i_j+2)\cdots(i_{j+1}-1),$$ where $\beta^{(s)}>\beta^{(s-1)}>\cdots>\beta^{(0)}>i_{m-1}$ and $\pi^{(m-1)}<i_1-s$. Let $M'_m$ be the contribution of this case. By considering whether $\beta^{(0)}$ is empty or not, we obtain the contributions $x(N'_m+M'_m)$ and $(m-2)x^{m+3}(L-1)/((1-x)^2(1-2x))$. Thus, $$M'_m=x(N'_m+M'_m)+(m-2)x^{m+3}(L-1)/((1-x)^2(1-2x)),$$ with solution for $M'_m$ as above.
This completes the proof.
\[th150a\] Let $T=\{1324,3421,3241\}$. Then $$F_T(x)=\frac{1-11x+52x^2-136x^3+214x^4-204x^5+111x^6-28x^7}{(1-x)^3(1-2x)^3(1-3x+2x^2)}.$$
First, we study the generating function $G'_m(x)$ for $T$-avoiders $\pi$ with $m$ left-right maxima such that $\pi_1=1$. Clearly, $G'_m(x)/x$ is the generating function for the $\{213,3421,3241\}$-avoiders with $m-1$ left-right maxima. It is not hard to see that $$G'_m(x)=(m-2)\,\frac{x^{m+1}}{1-x}\,L+x^mB.$$
Now let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Let us find the generating function $G_2(x)$. Suppose $\pi=i\pi'n\pi''\in S_n(T)$ with 2 left-right maxima. If $\pi''>i$, then we have a contribution of $x^2(A-1)B$. Otherwise, we can write $\pi$ as $$\pi=i\pi'n\beta^{(s)}(i-s)\beta^{(s-1)}(i-s+1)\cdots \beta^{(1)}(i-1)\beta^{(0)},$$ where $\beta^{(s)}>\beta^{(s-1)}>\cdots>\beta^{(0)}>i$ and $\pi'<i-s$. Let us denote the contribution of this case by $K$. By considering whether $\beta^{(s)}$ is decreasing or contains a rise, we obtain the contributions of $\frac{x}{1-x}(K+x^2(A-1)B)$ and $\frac{x^3}{(1-x)^2}(A-1)(L-\frac{1}{1-x})$. Thus, $$K=\frac{x}{1-x}(K+x^2(A-1)B)+\frac{x^3}{(1-x)^2}(A-1)\left(L-\frac{1}{1-x}\right),$$ and $G_2(x)-G'_2(x)=x^2(A-1)B+K$.
Now let us find an explicit formula for $G_m(x)$. By Lemmas \[lem150a1\]–\[lem150a4\], we see that $$G_m(x)-G'_m(x)=G_{m,1}(x)+\sum_{j=2}^{m-2}G_{m,j}(x)+G_{m,m-1}(x)+x^{m-2}(D_m(x)+(m-2)E_m(x)/(1-x)),$$ where $G_{m,m}(x)$ counted by $x^{m-2}(D_m(x)+(m-2)E_m(x)/(1-x))$. Thus, $$F_T(x)=1+xF_T(x)+\sum_{m\geq2}(G_m(x)-G'_m(x))+\sum_{m\geq2}G'_m(x).$$ By substituting the expressions for $G_m(x)-G'_m(x)$ and solving for $F_T(x)$, we complete the proof.
Case 151: $\{1324,1342,3421\}$
------------------------------
Here, we focus on the number of left-right maxima and begin with the case where $n$ is the second letter. Let $J_d(x)$ be the generating for permutations of the form $(d+1)n\alpha\in S_n(T)$.
\[lem151a1\] $J_0(x)=x^2L(x)=\frac{x^2(1-x)}{1-2x}$ and for all $d\geq1$, $$J_d(x)=\frac{x^{d+2}+x^2\left(\frac{x}{1-x}\right)^{d+1}L(x)+\sum_{j=0}^{d-2}x^{3+j}\left(\frac{x}{1-x}\right)^{d-j}}{1-\frac{x}{1-x}}\,.$$
Clearly, $J_0(x)=x^2F_{\{213,231\}}(x)=x^2L(x)$. Thus, we assume that $d\geq1$ and let us write an equation for $J_d(x)$. If $d=n-2$ then we have a contribution of $x^{d+2}$. Otherwise, $d\leq n-3$, and consider the position of the letter $d+2$. Note that the letters $1,2,\dots,d$ occur in that order since $\pi$ starts $(d+1)n$ and avoids 3421.
- The letter $d+2$ appears on the left side of the letter $1$. In this case, $(d+1)n\alpha=(d+1)n(n-1)\cdots(n'+1)(d+2)\alpha'$, which leads to a contribution of $\frac{x}{1-x}J_d(x)$.
- The letter $d+2$ appears between the letters $p$ and $p+1$, where $1\leq p\leq d-1$. Then, $(d+1)n\alpha=(d+1)n\alpha'(d+2)(p+1)(p+2)\cdots d$, where all the letters in $\alpha'$ which are $>d+2$ (resp. $<p+1$) are decreasing (resp. increasing). Thus, we have a contribution of $$x^{2+d-p}\left(\frac{x}{1-x}\right)^{p+1}\,.$$
- The letter $d+2$ appear on the right side of $d$. In this case, $\pi$ has the form $(d+1)n\alpha'(d+2)\beta$ where all the letters in $\alpha'$ greater than $d+2$ form a decreasing subsequence $\gamma$ with $\gamma>\beta>d+2$, and $\beta$ avoids $\{213,231\}$. Thus, we have a contribution of $$x^2\left(\frac{x}{1-x}\right)^{d+1}L(x)\,.$$
Hence, by adding all the contributions, we have $$J_d(x)=x^{d+2}+\frac{x}{1-x}J_d(x)+x^2\left(\frac{x}{1-x}\right)^{d+1}L(x)+\sum_{p=1}^{d-1}x^{2+d-p}\left(\frac{x}{1-x}\right)^{p+1},$$ and the stated expression for $J_d(x)$ follows.
Now set $J(x)=\sum_{d\geq0}J_d(x)$, the for $T$-avoiders whose largest letter occurs in second position.
\[cor151a1\] $$J(x)=\frac{(1-5x+9x^2-5x^3-x^4)x^2}{(1-2x)^3(1-x)}\,.$$
\[lem151a3\] Let $G_2(x)$ be the generating function for permutations in $S_n(T)$ with exactly two left-right maxima. Then $$G_2(x)=\frac{1-8x+26x^2-40x^3+25x^4-2x^5-x^6)x^2}{(1-3x+x^2)(1-2x)^4}.$$
Define $G_2(x;d)$ to be the generating function for permutations in $i\pi'n\pi''\in S_n(T)$ with two left-right maxima such that $\pi''$ contains $d$ letters smaller than $i$. If $d=0$ so that $\pi$ has the form $i\al n\be $ with $\al<i<\be$, then $\al$ avoids $\{132,3421\}$ and $\be$ avoids $\{213,231\}$. Hence $G_2(x;0)=x^2A(x)L(x)$, where $A(x)=1+\frac{x(1-3x+3x^2)}{(1-x)^2(1-2x)^2}$ is the generating function for $S_n(\{132,3421\})$ [@MV]. From now on, we assume that $d\geq1$. The letters $j_1,\dots,j_d$ in $\pi''$ that are less than $i$ are increasing, and so $\pi$ can be decomposed as $$\pi=i\alpha^{(1)}\cdots\alpha^{(d+1)}n\pi'',$$ where $i>\alpha^{(1)}>j_d>\alpha^{(2)}>\cdots>\alpha^{(d)}>j_1>\alpha^{(d+1)}$. Now, we treat the following cases:
- $\alpha^{(j)}=\emptyset$ for all $j=2,3,\ldots,d+1$. In this case, we have a contribution of $L(x)J_d(x)$, see Lemma \[lem151a1\].
- $\alpha^{(s)}\neq\emptyset$ and $\alpha^{(j)}=\emptyset$ for all $j=s+1,s+2,\ldots,d+1$, where $s=2,3,\ldots,d+1$. In this case, $\alpha^{(j)}$ is a decreasing sequence for $j=1,2,\ldots,s-1$, $\alpha^{(s)}$ avoids $\{132,231\}$, and $\pi''=\beta^{(1)}j_1\cdots\beta^{(d)}j_d\gamma$, where $\beta=\beta^{(1)}\cdots\beta^{(d)}$ is a decreasing subsequence such that $\beta>\gamma$ and $\gamma$ avoids $\{213,231\}$. So, we have a contribution of $$x^2\left(\frac{x}{1-x}\right)^dL(x)\frac{1}{(1-x)^{s-1}}\big(L(x)-1\big)$$ for $s=2,3,\ldots,d$, and $$x^2\left(\frac{x}{1-x}\right)^dL(x)\frac{1}{(1-x)^d}\big(A(x)-1\big)$$ for $s=d+1$.
- $\alpha^{(d+1)}\neq\emptyset$ and $\alpha^{(s)}$ contains a rise (a rise of $\pi$ is an index $i$ such that $\pi_i<\pi_{i+1}$) with $s=1,2,\ldots,d$. In this case, $\alpha^{(1)}\cdots\alpha^{(s-1)}$ is a decreasing sequence, $\alpha^{(s+1)}\cdots\alpha^{(d)}$ is empty, $\alpha^{(d)}$ is increasing and $\pi''$ decomposes as in the previous bullet. So the contribution is $$x^2\left(\frac{x}{1-x}\right)^dL(x)\left(L(x)-\frac{1}{1-x}\right)\frac{1}{(1-x)^{s-1}}\frac{x}{1-x}.$$
Hence, for $d\geq1$, $$\begin{aligned}
G_2(x;d)&=L(x)J_d(x)+\sum_{j=1}^{d-1}\frac{x^{d+2}L(x)\big(L(x)-1\big)}{(1-x)^{j+d}}\\
&+\frac{x^{d+3}L(x)\big(A(x)-1\big)}{(1-x)^{2d}}
+\sum_{j=0}^{d-1}\frac{x^{d+3}\big(L(x)-1/(1-x)\big)L(x)}{(1-x)^{d+1+j}}.\end{aligned}$$ Summing over $d\geq0$ and using Lemma \[lem151a1\], we obtain the stated formula for $G_2(x)$.
\[lem151a2\] Let $G_3(x)$ be the generating for the number of permutations in $S_n(T)$ with exactly three left-right maxima. Then $$G_3(x)=\frac{(1-4x+5x^2+x^3-5x^4)x^3}{(1-2x)^4}.$$
We consider permutations $\pi=i_1\pi^{(1)}i_2\pi^{(2)}i_3\pi^{(3)}\in S_n(T)$ with exactly three left-right maxima. Since $\pi$ avoids $1324$ and $1342$, we see that $\pi^{(1)}>\pi^{(2)}\beta$ where $\beta$ is all the letters in $\pi^{(3)}$ smaller than $i_1$. If $\pi^{(2)}=\emptyset$ and $\beta=\emptyset$ then we have a contribution of $x^3A(x)L(x)$. If $\pi^{(2)}=\emptyset$ and $\beta\neq\emptyset$, then we have a contribution of $x^3L(x)\big(J(x)-x^2L(x)\big)$. If $\pi^{(2)}\neq\emptyset$, we see that $\pi^{(1)}>\pi^{(2)}\beta$, $\pi^{(1)}$ avoids $\{132,231\}$, $\pi^{(2)}\beta$ is an increasing sequence. Suppose $\beta=j_1j_2\cdots j_d$, then $\pi^{(3)}=\beta^{(1)}j_1\cdots\beta^{(d)}j_d\beta^{(d+1)}$, where $\beta^{(1)}\cdots\beta^{(d)}$ is a decreasing sequence which is greater than $\beta^{(d+1)}$, and $\beta^{(d+1)}$ avoids $\{213,231\}$. Thus, we have a contribution of $x^3L(x)\frac{x}{1-x}L(x)\frac{1}{1-x/(1-x)}$. Hence, $$G_3(x)=x^3A(x)L(x)+x^3L(x)(J(x)-x^2L(x))+x^3L(x)\frac{x}{1-x}L(x)\frac{1}{1-x/(1-x)},$$ and, using Corollary \[cor151a1\], the result follows.
\[th151a\] Let $T=\{1324,1342,3421\}$. Then $$F_T(x)=\frac{1-12x+61x^2-169x^3+275x^4-263x^5+136x^6-29x^7+x^8}{(1-3x+x^2)(1-2x)^4(1-x)^2}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$, and $G_2(x)$ and $G_3(x)$ are given by Lemmas \[lem151a3\] and \[lem151a2\], respectively. Now let $m\geq4$ and let us write an equation for $G_m(x)$. Suppose $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. If $\pi^{(2)}$ is empty then we have a contribution of $xG_{m-1}(x)$. Otherwise, $\pi^{(j)}=\emptyset$ for all $j=3,4,\ldots,m-1$, $\pi^{(m)}>i_{m-1}$ and $\pi^{(1)}>\pi^{(2)}$, where $\pi^{(m)}$ avoids $\{213,231\}$, $\pi^{(1)}$ avoids $\{132,231\}$, and $\pi^{(2)}$ is an increasing sequence. Recall that $F_{\{213,231\}}(x)=F_{\{132,231\}}(x)=\frac{1-x}{1-2x}$ [@SiS]. Thus, we have $$G_m(x)=xG_{m-1}(x)+\frac{x^{m+1}}{1-x}\frac{1-x}{1-2x},$$ for all $m\geq4$. Summing over $m\geq4$, we obtain $$F_T(x)-1-G_1(x)-G_2(x)-G_3(x)=x(F_T(x)-1-G_1(x)-G_2(x))+\frac{x^5}{(1-x)(1-2x)}\,.$$ Solving for $F_T(x)$ using the expressions above for $G_1(x),G_2(x),G_3(x)$, we complete the proof.
Case 153: $\{4231,1324,1342\}$
------------------------------
Here, we use the same techniques as in Case 84.
\[lem153a1\] Let $m\geq3$. The generating function for permutations $\pi=\pi_1\pi_2\cdots\pi_n\in S_n(T)$ with exactly $m$ left-right maxima and $\pi_1=1$ is given by $\frac{x^m(1-x)}{1-2x}$.
Such permutations can be written as $12\cdots(m-1)n\pi'$ where $\pi'$ avoids $213$ and $231$, and the result follows.
\[lem153a2\] Let $m\geq3$. The generating function $G_m(x)$ for permutations $\pi=\pi_1\pi_2\cdots\pi_n\in S_n(T)$ with exactly $m$ left-right maxima satisfies the recurrence $$G_m(x)=x^mL^2(x)+\frac{x}{1-x}\big(G_{m-1}(x)-x^{m-1}L(x)\big)\, .$$
To write an equation for $G_m(x)$, let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima. Since $\pi$ avoids $1324$ and $1342$ we see that $\pi^{(2)}\cdots\pi^{(m)}$ has no letter between $i_1$ and $i_{m-1}$. If $\pi^{(2)}\cdots\pi^{(m)}$ has no letter smaller than $\pi^{(1)}$, the contribution is $x^mF_{\{132,231\}}(x)F_{\{213,231\}}(x)=x^mL^2(x)$. Otherwise, $\pi^{(1)}$ is a decreasing sequence, and then we have a contribution of $$\frac{x}{1-x}\big(G_{m-1}(x)-x^{m-1}L(x)\big)$$ (see Lemma \[lem153a1\]). Now add the two contributions.
The next two lemmas give the generating function for permutations in $S_n(T)$ with exactly $2$ left-right maxima. We leave the proof to the diligent reader (very similar to Case 84).
\[lem153a3\] Let $\pi=i\alpha n\beta\in S_n(T)$ with exactly $2$ left-right maxima and $i\geq2$.
- If $i-1$ is the left most letter of $\alpha$, then the generating function for such permutations $\pi$ is given by $xG_2(x)$, where $G_2(x)$ is the generating function for permutations in $S_n(T)$ with exactly $2$ left-right maxima.
- If $i-1$ is in $\alpha$ but not the first letter in $\alpha$, then the generating function for such permutations $\pi$ is given by $x^3L(x)\big((L(x)-1\big)$.
- If $i-1$ is a letter in $\beta$ such that there are exactly $d$ letters between $n$ and $i-1$ in $\pi$ that are greater than $i$. We denote the generating function for the number of such permutations by $A_d(x)$. Then $A_0(x)=\frac{x^3(1-3x+5x^2-4x^2)}{(1-x)(1-2x)^3}$, $A_1(x)-xA_0(x)=\frac{x^5(2-5x+4x^2)}{(1-2x)^5}$, and for all $d\geq2$, $$A_d(x)-xA_{d-1}(x)=\frac{x^{d+4}(1-x)^d}{(1-2x)^{d+3}}.$$
\[lem153a4\] Let $G_2(x)$ be the generating function for permutations in $S_n(T)$ with exactly $2$ left-right maxima. Then $$G_2(x)=\frac{x^2(1 -7x+22x^2-36x^3 +33x^4 -20x^5 + 7x^6 -x^7)}{(1-3x+x^2)(1-x)^3(1-2x)^2}.$$
With the definitions introduced in Lemma \[lem153a3\], we have $$M(x):=\sum_{d\geq0}A_d(x)=\frac{x^3(1 -4x+9x^2-12x^3+6x^4-x^5)}{(1-3x+x^2)(1-x)^2(1-2x)^2}.$$ Thus, by Lemma \[lem153a3\], $$G_2(x)=\frac{M(x)+x^3L(x)(L(x)-1)+x^2L(x)}{1-x},$$ where $x^2L(x)$ counts the permutations $1n\pi'\in S_n(T)$, and the result follows.
\[th153a\] Let $T=\{4231,1324,1342\}$. Then $$F_T(x)=\frac{1-10x+41x^2-87x^3+101x^4-61x^5+15x^6-x^7}{(1-x)^2(1-2x)^3(1-3x+x^2)}.$$
Note that the number of permutations in $S_n(T)$ with leftmost letter is $n$ is given by $G_1(x)=xF_{\{231,1324\}}(x)=xF_{\{132,4231\}}(x)=\frac{x(1+x(L(x)-1)L(x))}{1-x}$, where $L(x)=\frac{1-x}{1-2x}$, see [@MV]. Thus, by Lemma \[lem153a2\] and Lemma \[lem153a4\], we obtain $$F_T(x)-1-G_1(x)=\frac{x^3}{1-x}L^2(x)+\frac{x}{1-x}(F_T(x)-G_1(x)-1)-\frac{x^3}{(1-x)^2}L(x),$$ which, by solving for $F_T(x)$, completes the proof.
Case 156: $T=\{1324,2341,2431\}$
--------------------------------
\[th156a\] Let $T=\{1324,2341,2431\}$. Then $$F_T(x)=\frac{1-8x+23x^2-25x^3+3x^4+7x^5}{(1-2x)^2(1-3x+x^2)(1-2x-x^2)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Let us write an equation for $G_2(x)$. Let $\pi=i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima. The contributions for the cases $n-2\geq i=1$ and $i=n-1\geq 1$ are $x^2(M(x)-1)$ and $x\big(F_T(x)-1\big)$ respectively, where $M(x)=\frac{1-x-x^2}{1-2x-x^2}$ is the generating function for $\{213,2341,2431\}$-avoiders (for example, see [@MV]). Denote the contribution for the case $2\leq i\leq n-2$ by $H(x)$. Then $G_2(x)=x^2\big(M(x)-1\big)+x\big(F_T(x)-1\big)+H(x)$. To find a formula for $H(x)$ we consider the position of $i-1$ in $\pi$, which leads to either $\pi=i\alpha(i-1)\beta'n\beta''\beta'''$ with $\beta'''>i\alpha>\beta'\beta''$ or $\pi=i\alpha'n\alpha''(i-1)\beta'\beta''$ with $\beta''>i>\alpha'\alpha''>\beta'$. By examining the four possibilities, $\alpha,\beta'\beta''$ either empty or not, in the first case and examining the four possibilities for $\alpha'\alpha'',\beta'$ in the second case, we obtain that $$\begin{aligned}
H(x)&=x^3\big(M(x)-1\big)+x^3\big(K(x)-1\big)\big(M(x)-1\big)+\frac{x^4}{1-x}\big(K(x)-1\big)\big(M(x)-1\big)\\
&+x^3\big(M(x)-1\big)+x^3\big(K(x)-1\big)\big(M(x)-1\big)+\frac{x^4}{1-x}\big(K(x)-1\big)\big(M(x)-1\big)+2xH(x),\end{aligned}$$ where $K(x)=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{132,2341\}$-avoiders. Thus, $$H(x)=\frac{2x^4(1-x)^2}{(1-2x)(1-3x+x^2)(1-2x-x^2)}.$$
Next, let us write an equation for $G_3(x)$. Let $\pi=i_1\pi'i_2\pi''i_3\pi'''\in S_n(T)$ with exactly $3$ left-right maxima. Since $\pi$ avoids $T$, we can write $\pi=i_1\pi'i_2\pi''i_3\beta\alpha$ where $\pi''<\pi'<i_1<\beta<i_2<\alpha<i_3$. By considering the four cases where $\pi'',\beta$ are empty or not, we obtain $$G_3(x)=x^3\left(K(x)M(x)+K(x)\big(M(x)-1\big)+\frac{\big(K(x)-1\big)M(x)}{1-x}
+\frac{\big(K(x)-1\big)\big(M(x)-1\big)}{1-x}\right).$$
Finally, let us write an equation for $G_m(x)$ with $m\geq4$. Let $\pi=i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $T$, we see that $i_1>\pi^{(1)}>\pi^{(2)}$, $\pi^{(j)}=\emptyset$ for $j=3,4,\ldots,m-1$, and $\pi^{(m)}$ has the form $\beta\alpha$ where $i_m>\alpha>i_{m-1}>\beta$. Thus, $\pi$ avoids $T$ if and only if the permutation that is obtained from $\pi$ by removing $i_{m-2}$ avoids $T$. Hence $G_m(x)=xG_{m-1}(x)=\dots =x^{m-3}G_3(x)$, for $m\geq3$.
By summing over $m\geq4$ and using the expressions for $G_0(x),G_1(x),G_2(x)$ and $G_3(x)$, we obtain $$\begin{aligned}
&F_T(x)=1+xF_T(x)+x\big(F_T(x)-1\big)+x^2\big(M(x)-1\big)+H(x)\\ &\,+\frac{x^3}{1-x}\left(K(x)M(x)+K(x)\big(M(x)-1\big)+\frac{\big(K(x)-1\big)M(x)}{1-x}+\frac{\big(K(x)-1\big)\big(M(x)-1\big)}{1-x}\right),\end{aligned}$$ which, by solving for $F_T(x)$, completes the proof.
Case 158: $\{1324,1342,3412\}$
------------------------------
Here, it is convenient to consider $J(x)$, the generating function for $T$-avoiders whose maximal letter occurs in second position, and its refinement to $J_d(x)$, the generating function for $T$-avoiders whose maximal letter occurs in second position and whose first letter is $d$, that is, permutations of the form $dn\pi''\in S_n(T)$.
\[lem158a1\] With the preceding notation, $$J_1(x)=\frac{x^2(1-x)}{1-2x},\quad J_2(x)=\frac{x^3(1-2x+2x^2)}{(1-2x)^2}, \quad J(x)=\frac{x^2(1-2x)}{(1-x)(1-3x)}\, .$$
Suppose $\pi \in S_n(T)$ has the form $dn\pi''$. Since $\pi$ avoids $3412$, we see that $\pi''$ contains the subsequence $(d-1)(d-2)\cdots1$. Therefore, if $d=n-1$, the contribution is $x^{d+1}$ and otherwise, by considering whether $n-1$ occurs before or after the string $(d-1)(d-2)\cdots1$, we find $$J_d(x)=x^{d+1}+\sum_{j=1}^dx^jJ_{d+1-j}+\sum_{j=1}^dx^j(J_{d+1-j}-x^{d+2-j}),$$ for all $d\geq1$. Solving this recurrence successively for $d=1$ and $d=2$ gives the first two stated expressions. Also, summing over all $d\geq1$ yields $$J(x)=\frac{x^2}{1-x}+\frac{2x}{1-x}J(x)-\frac{x^3}{(1-x)^2},$$ from which the expression for $J(x)$ follows.
\[lem158a2\] Let $G_2(x)$ be the generating function for the number of $T$-avoiders with exactly $2$ left-right maxima. Then $G_2(x)=\frac{x^2(1-7x+19x^2-23x^3+9x^4)}{(1-x)^2(1-2x)(1-3x)(1-3x+x^2)}$.
Again, we refine $G_2(x)$ by defining $G_2(x;d)$ to be the generating function for the number of permutations $i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima where $\pi''$ has $d$ letters smaller than $i$. Since $\pi$ avoids $3412$, we see that $\pi''$ is decreasing. Now, let us write an equation for $G_2(x;d)$. Clearly, $G_2(x;0)=x^2F_{\{132,3412\}}(x)F_{\{213,231\}}(x)=\frac{x^2(1-x)}{1-2x}K(x)$, where $F_{\{132,3412\}}(x)=K(x)=\frac{1-2x}{1-3x+x^2}$ (see [@Sl Seq A001519]) and $F_{\{213,231\}}(x)=\frac{1-x}{1-2x}$ (see [@SiS]).
For $d=1$, let $j$ denote the letter in $\pi''$ smaller than $i$. Thus $i>2$ and $\pi$ can be written, according as $j>1$ or $j=1$, as either (1) $\pi=i(i-1)\cdots(j+1)\alpha^{(1)}n(n-1)\cdots (i'+1)i'j\beta^{(1)}$ where $1 \in \alpha^{(1)}$ and $j>\alpha^{(1)}$, $i'>\beta^{(1)}$ and $\beta^{(1)}$ avoids $213$ and $231$, and $\alpha^{(1)}$ avoids $132$ and $3412$, or (2) $\pi=i\alpha n\beta$ where $\alpha$ consists of the letters $2,3,\dots,i-1$ in some order and $\alpha$ avoids $132$ and $3412$. Hence, $$G_2(x;1)=\frac{x^3}{(1-x)(1-2x)}(K(x)-1)+K(x)J_2(x),$$ where $J_2(x)$ is defined above.
Now, let $d\geq2$. Since $\pi$ avoids $1324$ and $1342$, we can express $\pi$ as $$\pi=i\alpha^{(1)}\alpha^{(2)}\cdots\alpha^{(d+1)}n\beta^{(1)}j_1\cdots\beta^{(d)}j_d\beta^{(d+1)}$$ where $i>\alpha^{(d+1)}>j_1>\alpha^{(d)}>j_2>\cdots>\alpha^{(2)}>j_d>\alpha^{(1)}$. We consider three cases:
- $\alpha^{(s)}\neq\emptyset$ and $\alpha^{(s+1)}=\cdots=\alpha^{(d+1)}=\emptyset$ with $s=3,4,\ldots,d+1$. In this case $\alpha^{(1)},\ldots,\alpha^{(s-1)}$ are decreasing and $\alpha^{(s)}$ avoids $132$ and $3412$, while $\beta^{(2)}=\cdots=\beta^{(d+1)}=\emptyset$ and $\beta^{(1)}$ is decreasing. Hence, we have a contribution of $\frac{x^{d+2}}{(1-x)^{s}}\big(K(x)-1\big)$.
- $\alpha^{(3)}=\cdots=\alpha^{(d+1)}=\emptyset$ and $\alpha^{(2)}\neq\emptyset$. In this case, $\alpha^{(1)}$ and $\beta^{(1)}$ are decreasing, and $\beta^{(1)}>\beta^{(2)}> \cdots >\beta^{(d+1)}$. Thus, we have a contribution of $\frac{x}{(1-x)^2}\big(K(x)-1\big)J_d(x)$.
- $\alpha^{(2)}=\cdots=\alpha^{(d+1)}=\emptyset$. In this case, we have a contribution of $K(x)J_{d+1}(x)$.
Hence, for all $d\geq2$, $$G_2(x;d)=\sum_{s=3}^{d+1}\frac{x^{d+2}}{(1-x)^s}\big(K(x)-1\big)+\frac{x}{(1-x)^2}\big(K(x)-1\big)J_d(x)+K(x)J_{d+1}(x).$$ Summing over $d\geq 0$ and using Lemma \[lem158a1\], we obtain the stated expression for $G_2(x)$.
\[th158a\] Let $T=\{1324,1342,3412\}$. Then $$F_T(x)=\frac{1-10x+40x^2-81x^3+88x^4-50x^5+11x^6}{(1-x)^3(1-2x)(1-3x)(1-3x+x^2)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$, and $G_2(x)$ is given by Lemma \[lem158a2\].
Now, let us write an equation for $G_m(x)$ with $m\geq3$. A $T$-avoider $i_1\pi^{(1)}\cdots i_m\pi^{(m)}$ with $m\ge 3$ left-right maxima $i_1,i_2,\dots,i_m$ has the restricted form shown in Figure \[figAK3\],
(4,4.2) (0,1)(5,1)(5,8.5)(0,8.5)(0,1) (1,2)(4,2)(4,5)(1,5)(1,2) (0,5)(1,5)(1,6)(0,6)(0,5) (4,1)(5,1)(5,2)(4,2)(4,1) (4,7.5)(5,7.5)(5,8.5)(4,8.5)(4,7.5) (0,6)(1,6.5)(3,7.5)(4,8.5) (.5,5.5) (2.5,3.5) (4.5,1.5) (4.5,8) (-.3,6.2) (.7,6.7) (2.5,7.8) (3.7,8.9) (2.,7)
where $i_1,i_2,\dots,i_{m-1}$ are increasing consecutive integers, $A$ is a list of zero or more decreasing consecutive integers, $\pi^{(m)}$ is composed of $B_1$ and $B_2$, and regions not marked are empty. Furthermore, $\pi^{(1)}$ avoids $\{132,3412\}$ and, of course, $i_1 i_m \pi^{(m)}$ avoids $T$. Conversely, every permutation of this form with $m\ge 3$ satisfying the latter two conditions is a $T$-avoider. Hence we get contributions as follows: $K(x)$ from $\pi^{(1)}$; $J(x)$ from $i_1 i_m \pi^{(m)}$; $x^{m-2}$ from $i_2,\dots,i_{m-1}$; and $1/(1-x)^{m-2}$ from $A$. So $G_m(x)=x^{m-2}K(x)J(x)/(1-x)^{m-2}$. By summing over $m\geq3$, we obtain $$F_T(x)-1-xF_T(x)-G_2(x)=\frac{x^3(1-2x)}{(1-x)(1-3x)(1-3x+x^2)}\, .$$ Now solve for $F_T(x)$ using Lemma \[lem158a2\] to complete the proof.
Case 180: $\{1342,2314,4231\}$
------------------------------
\[th180a\] Let $T=\{1342,2314,4231\}$. Then $$F_T(x)=\frac{1-7 x +18 x^2 -22 x^3 +16 x^4 -6 x^5 +x^6 - \left(x- 5 x^2+8 x^3 - 2 x^4 -2 x^5 +x^6\right)C(x) }{(1-2 x) (1-x)^2 \left(1-5 x+4 x^2-x^3\right)}\,.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_{\{231\}}(x)=xC(x)$ (see [@K]). Now let $m\geq3$ and let us write equation for $G_m(x)$. Let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $2314$ and 1342, we see that $\pi^{(j)}>i_{j-1}$ for all $j=2,3,\ldots,m-1$. If $\pi^{(m)}$ has a letter smaller than any letter in $\pi^{(1)}$, then $i_j\pi^{(j)}=i_j(i_j-1)\cdots(i_{j-1}+1)$ for all $j=1,2,\ldots,m-1$, and $\pi^{(m)}=i_m(i_m-1)\cdots(i_{m-1}+1)\beta$ such that $\beta$ is non empty permutation in $S_{i_0}(231)$. Hence, we get a contribution of $\frac{x^m}{(1-x)^m}(C(x)-1)$. Otherwise, $\pi^{(j)}>i_{j-1}$ for all $j=2,3,\ldots,m$, which gives a contribution of $x^mC^m(x)$. Hence, $$G_m(x)=\frac{x^m}{(1-x)^m}(C(x)-1)+x^mC^m(x),$$ for all $m\geq3$.
Now, let us focus on the case $m=2$. First, let $H$ be the generating function for permutations $i\pi'n\pi''\in S_n(T)$ with exactly $2$ left-right maxima and containing the subsequence $n(n-1)\cdots(i+1)$. Let us write an equation for $H$. If $i=1$, then we have a contribution of $x^2/(1-x)$. Otherwise, $n-1\geq i\geq2$ and consider the position of $i-1$. If $(i-1)$ is the first letter in $\pi'$ then we have a contribution of $xH$. If $(i-1)\in \pi'$ but is not the first letter in $\pi'$, then $\pi$ must have the form $i\alpha(i-1)\beta n(n-1)\cdots(i+1)$ with $\alpha<\beta<i-1$, $\alpha \ne \emptyset$, and $\alpha,\beta$ both $231$-avoiders, which implies a contribution of $x^3\big(C(x)-1\big)C(x)/(1-x)$. Otherwise, $i-1 \in \pi''$ and then $\pi$ can be decomposed as in Figure \[figAK4\].
(5,4.3) (0,0)(2,0)(2,2)(0,2)(0,0) (2,6)(6,6)(6,8)(2,8)(2,6) (2,0)(6,0)(6,3.5)(10,3.5)(10,2)(2,2)(2,0) (10,4)(12,4)(12,6)(10,6)(10,4) (0,4)(2,8)(6,3.5) (1,1) (4,7) (4,1) (8,3.1) (8,2.4) (11,5) (-.5,4.2) (5.1,3.6) (1.5,8.2)
where $\downarrow$ indicates a region of decreasing entries. Since St($iAnB$) ($B$ is spread over two regions) is of the type counted by $H$ and $D$ contributes $C(x)$, we get a contribution of $x/(1-x)C(x)H$. Hence, $$H=x^2/(1-x)+xH+x^3(C(x)-1)C(x)/(1-x)+xC(x)H/(1-x),$$ which leads to $$H=\frac{x^2/(1-x)+x^3(C(x)-1)C(x)/(1-x)}{1-x-xC(x)/(1-x)}.$$
Now let us write an equation for $G_2(x)$. Let $\pi=i\pi'n\pi''$ with exactly $2$ left-right maxima. If $i=1$, then we have a contribution of $x^2C(x)$. Otherwise, $n-1\geq i\geq2$ and again consider the position of $i-1$. If $\pi'=(i-1)\pi'''$, then we have a contribution of $xG_2(x)$. If $\pi'=\alpha(i-1)\beta$ such that $\alpha$ is not empty then $\pi=i\alpha(i-1)\beta n\pi''$ with $\alpha<\beta<i-1<\pi''<n$, which gives a contribution of $x^3\big(C(x)-1\big)C(x)^2$. Thus, we can assume that $i-1$ belongs to $\pi''$. In this case, $\pi$ can be written as $\pi=i\pi'n\alpha'(i-1)\alpha''\alpha'''$ such that $i\pi'n\alpha'$ has $2$ left-right maxima and contains the subsequence $n(n-1)\cdots (j+1)$, each letter in $\alpha''$ is greater than each letter smaller than $i$ in $\pi'\alpha'$, and $i<\alpha'''<j+1$, where $\alpha'',\alpha'''$ avoids $231$ and $i\pi'n\alpha'$ avoids $T$. Thus, we have a contribution of $xHC(x)^2$. Hence, $$G_2(x)=x^2C(x)+xG_2(x)+ x^3\big(C(x)-1\big)C(x)^2+xHC(x)^2,$$ which implies $$G_2(x)=\frac{x^2C(x)+x^3\big(C(x)-1\big)C(x)^2+xHC(x)^2}{1-x}.$$
Summing over $m\geq0$, we obtain $$F_T(x)=1+xC(x)+\frac{x^2C(x)+x^3\big(C(x)-1\big)C^2(x)+xHC(x)^2}{1-x}+\frac{x^3\big(C(x)-1\big)}{(1-x)^2(1-2x)}+\frac{x^3C^3(x)}{1-xC(x)}\,,$$ and this expression simplifies to the stated form.
Case 184: $\{1324,2431,3241\}$
------------------------------
\[th184a\] Let $T=\{1324,2431,3241\}$. Then $$F_T(x)=\frac{1-8x+24x^2-32x^3+19x^4-3x^5}{(1-x)(1-2x)(1-3x+x^2)^2}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Let us write an equation for $G_m(x)$ with $m\geq2$. Suppose $\pi\in S_n(T)$ has exactly $m$ left-right maxima. Since $\pi$ avoids $3241$ and $2431$, we can express $\pi$ as $$\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_{m-1}\pi^{(m-1)}
i_m\pi^{(m)}\rho^{(1)}\rho^{(2)}\cdots\rho^{(m-1)}$$ where $\pi^{(1)}<\cdots<\pi^{(m)}<i_1<\rho^{(1)}<i_2<\rho^{(2)}<\cdots<i_{m-1}<\rho^{(m-1)}<i_m$. Since $\pi$ avoids $1324$ we see that at most one element of $L:=\{\pi^{(1)},\ldots,\pi^{(m-1)}\}$ is nonempty and at most one element of $R:=\{\rho^{(1)},\ldots,\rho^{(m-1)}\}$ is nonempty. Thus, we have the following cases:
- if $L$ and $R$ are both lists of empty words, then $\pi$ avoids $T$ if and only if $\pi^{(m)}$ avoids $T$, so we have a contribution $x^mF_T(x)$.
- if, say, $\pi^{(s)}\in L$ is nonempty and $R$ is a list of empty words, then $\pi$ avoids $T$ if and only if $\pi^{(s)}$ avoids $132$ and $3241$, and $\pi^{(m)}$ avoids $213$ and $2431$. Thus, we have a contribution of $x^mK(x)\big(K(x)-1\big)$, where $K(x)=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{132,3241\}$-avoiders [@Sl Seq. A001519]. By symmetry, $K(x)$ is also the generating function for $\{213,2431\}$-avoiders since $R\circ C \circ I(\{213,2431\} )= \{132,3241\})$, where $R=$ reverse, $C=$ complement, and $I=$ inverse on permutations.
- if $\rho^{(t)}\in R$ is nonempty and $L$ is a list of empty words, then, similarly, we have the same contribution of $x^mK(x)\big(K(x)-1\big)$.
- lastly, suppose $\pi^{(s)}\in L$ and $\rho^{(t)}\in R$ are nonempty. If $t\geq s+1$, then $\pi$ avoids $T$ if and only if $\pi^{(m)}=\emptyset$, $\pi^{(s)}$ avoids $132$ and $3241$, and $\rho^{(t)}$ avoids $213$ and $2431$, which gives a contribution of $x^m\big(K(x)-1\big)^2$. Otherwise, $1\leq t\leq s$, and we have the same conditions except $\pi^{(m)}$ must be increasing rather than empty, which leads to a contribution of $\frac{x^m}{1-x}\big(K(x)-1\big)^2$.
By adding all the contributions, we find that for all $m\geq2$, $$G_m(x)=x^mF_T(x)+2(m-1)x^m\big(K(x)-1\big)K(x)+x^m\left(\binom{m-1}{2}+\binom{m}{2}\frac{1}{1-x}\right)\big(K(x)-1\big)^2.$$ Summing over $m\geq2$, we obtain $$F_T(x)-1-xF_T(x)=\frac{x^2}{1-x}F_T(x)+\frac{2x^3(1-3x+2x^2)}{(1-x)^2(1-3x+x^2)^2}+\frac{x^4(1+x-x^2)}{(1-x)^2(1-3x+x^2)^2},$$ and solving for $F_T(x)$ completes the proof.
Case 187: $\{1324,2314,2431\}$
------------------------------
\[th187a\] Let $T=\{1324,2314,2431\}$. Then $$F_T(x)=\frac{1-9x+31x^2-49x^3+34x^4-7x^5}{(1-3x+x^2)^2(1-2x)^2}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Let us write an equation for $G_2(x)$. Suppose $\pi \in S_n(T)$ has 2 left-right maxima so that $\pi=i\pi'n\pi''$. If $i=n-1$, then the contribution is given by $x(F_T(x)-1)$. Otherwise, we denote the contribution by $H$. So $G_2(x)=x(F_T(x)-1)+H$. For $H$, $\pi$ can be written as $\pi=i\pi'n\alpha\beta$ with $1\le i \le n-2$ and $\pi'\alpha<i<\beta$ and hence $\beta\neq\emptyset$. Note that $\beta$ avoids both $213$ and $2431$ and $\pi'\alpha$ avoids both $231$ and $132$. By considering whether $\pi'\alpha$ is empty or not and whether $i-1$ belongs to $\pi'$ or to $\alpha$, we find that $$H=x^2(L-1)+\big(xH+x^3K(L-1)\big)+\big(xH+x^3K(L-1)\big),$$ where $L=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{213,2431\}$-avoiders [@Sl Sequence A001519] and $K=\frac{1-x}{1-2x}$ is the generating function for $\{132,231\}$-avoiders [@SiS]. Hence, $H=\frac{x^3(1-x)(1-2x+2x^2)}{(1-3x+x^2)(1-2x)^2}$, which implies $$G_2(x)=x\big(F_T(x)-1\big)+\frac{x^3(1-x)(1-2x+2x^2)}{(1-3x+x^2)(1-2x)^2}.$$
Now, let us write an equation for $G_m(x)$ with $m\geq3$. Suppose $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $2314$, we see that $\pi^{(2)}=\cdots=\pi^{(m-1)}=\emptyset$. We consider the following two cases:
- $\pi^{(m)}<i_{m-1}$. In this case, by removing the letter $n$, we obtain a bijection between such permutations of length $n$ and $T$-avoiders with $m-1$ left-right maxima of length $n-1$. Therefore, we have a contribution of $xG_{m-1}(x)$.
- $\pi^{(m)}$ contains a letter between $i_{m-1}$ and $i_m=n$. In this case, since $\pi$ avoids $1324$ and $2314$, we see that $\pi^{(m)}>i_{m-1}$. So $\pi$ avoids $T$ if and only if $\pi^{(m)}$ avoids $213$ and $2431$ and $\pi^{(1)}$ avoids $231$ and $132$. So the contribution for this case is $x^mK(L-1)$.
Thus, $G_m(x)=xG_{m-1}(x)+x^mK(L-1)$. Summing over $m\geq3$, we obtain $$F_T(x)-G_2(x)-G_1(x)-G_0(x)=x(F_T(x)-G_1(x)-G_0(x))+\frac{x^3}{1-x}K(L-1).$$ Hence, using the expressions for $G_2(x)$, $G_1(x)$ and $G_0(x)$, we obtain $$(1-2x)F_T(x)-\frac{(2x^3+6x^2-5x+1)(1-x)^3}{(1-3x+x^2)(1-2x)^2}
=x(1-x)F_T(x)-\frac{x(1-5x+7x^2-3x^3+x^4)}{(1-3x+x^2)(1-2x)},$$ and solving for $F_T(x)$ completes the proof.
Case 193: $\{1324,2431,3142\}$
------------------------------
Observe that each pattern here contains 132. So if a permutation avoids 132, then it certainly avoids $T$.
\[th193a\] Let $T=\{1324,2431,3142\}$. Then $$F_T(x)=\frac{x-1+ \left(x^2-5 x+2\right)C(x)}{1-3 x+x^2}\, .$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
Now let us write an equation for $G_m(x)$ with $m\geq2$. Let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. We see that $i_1>\pi^{(1)}>\pi^{(2)}>\cdots>\pi^{(m-1)}$ (to avoid 1324) and $\pi^{(m)}$ can be written as $\alpha^{(1)}\alpha^{(2)}\cdots\alpha^{(m)}$ such that $\pi^{(m-1)}>\alpha^{(1)}$ and $i_1<\alpha^{(2)}$ (to avoid 3142) and $\alpha^{(2)}<i_2<\alpha^{(3)}<\cdots<i_{m-1}<\alpha^{(m)}<i_m$ (to avoid 2431). Furthermore, $\pi^{(j)}$ avoids $132$ for all $j=1,2,\ldots,m-1$ (or $i_m$ is the 4 of a 1324), and at most one of $\alpha^{(2)}, \ldots,\alpha^{(m)}$ is nonempty (to avoid 1324). We consider two cases according as $\alpha^{(2)}=\cdots=\alpha^{(m)}=\emptyset$ or not:
- $\alpha^{(2)}=\cdots=\alpha^{(m)}=\emptyset$. Here, $\alpha^{(1)}$ only needs to avoid $T$ and we have a contribution of $x^mC(x)^{m-1}F_T(x)$
- There is a unique $j$ in the interval $[2,m]$ such that $\alpha^{(j)}\ne \emptyset$. Here, $\alpha^{(1)}$ must avoid $132$ (or $\alpha^{(j)}$ contains the 4 of 1324), and $\alpha^{(j)}$ must avoid both 213 $\approx 324$ (or $i_1$ is the 1 of a 1324) and $2431$. Also, $\pi^{(j)}= \cdots = \pi^{(m)}=\emptyset$ (to avoid 3142). So $\pi^{(1)},\ldots,\pi^{(j-1)},\alpha^{(1)}$ each contribute $C(x)$ and we get a contribution of $x^m C(x)^j \big(K(x)-1\big)$ where $K(x)=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{213,2431\}$ avoiders [@Sl Seq. A001519].
Thus, $$G_m(x)=x^mC(x)^{m-1}F_T(x)+\sum_{j=2}^m x^m C(x)^j\big(K(x)-1\big)\,.$$
Summing over $m\geq2$ and using the expressions for $G_1(x)$ and $G_0(x)$, we obtain $$F_T(x)=1+xF_T(x)+\sum_{m\geq2}\left(x^mC(x)^{m-1}F_T(x)+\sum_{j=2}^mx^mC(x)^j(K(x)-1)\right)\, .$$ Solving for $F_T(x)$ gives the stated after simplification.
Case 195: $\{1324,2341,1243\}$
------------------------------
In this subsection, let $A=\frac{1-2x}{1-3x+x^2}$ and $B=\frac{1-x+x^3}{(1-x)(1-x-x^2)}$ denote the generating functions for $F_{\{213,2341\}}(x)$ and $F_{\{132,213,2341\}}(x)$ (they can be derived from results in [@MV]). The first few lemmas refer to permutations with exactly 2 left-right maxima.
\[lem195a1\] The generating function $J_m(x)$ for permutations of the form $$(n-m-1)\alpha^{(m)}n\alpha^{(m-1)}(n-1)\cdots\alpha^{(0)}(n-m)\in S_n(T)$$ with $2$ left-right maxima satisfies $$J_m(x)=x^{m+2}+xJ_m(x)+\frac{x^{m+3}}{(1-x)^{m+1}}(A-1)+\sum_{j=1}^mx^jJ_{m+1-j}(x)\,.$$ Moreover, if $J(x,w)=\sum_{m\geq1}J_m(x)w^{m-1}$, then $$J(x,w)=\frac{x^3((1+2w)x^3-(5+3w)x^2+(4+w)x-1)}{(wx+x-1)(x^2-3x+1)(wx^2-wx-2x+1)}\,.$$
Let us write an equation for $J_m(x)$. Let $\pi=(n-m-1)\alpha^{(m+1)}n\alpha^{(m-1)}(n-1)\cdots\alpha^{(0)}(n-m)\in S_n(T)$. If $n=m+2$ then we have a contribution of $x^{m+2}$. Otherwise, we can consider the position of the letter $n-m-2$. If the letter $n-m-2$ belongs to $\alpha^{(s)}$ with $s=1,2,\ldots,m$, then there is no letter smaller than $n-m-2$ on its left side (otherwise, $\pi$ contains $1243$), so we have a contribution of $x^{m+1-s}J_{s}$. So, we can assume that the letter $n-m-2$ belongs to $\alpha^{(0)}$, that is, $\alpha^{(0)}=\gamma(n-m-2)\gamma'$. Since $\pi$ avoids $1324$, we have $\alpha^{(m)}\cdots\alpha^{(1)}\gamma>\gamma'$. In the case $\gamma'=\emptyset$ then we have a contribution of $xJ_m(x)$. Otherwise, since $\pi$ avoids $2341$, we have that $\alpha^{(m)}\cdots\alpha^{(1)}\gamma$ is decreasing and $\gamma'$ avoids $\{132,2341\}$, which gives a contribution of $\frac{x^{m+3}}{(1-x)^{m+1}}(A-1)$. Hence, by adding all contributions, we obtain $$J_m(x)=x^{m+2}+xJ_m(x)+\frac{x^{m+3}}{(1-x)^{m+1}}(A-1)+\sum_{j=1}^mx^jJ_{m+1-j}(x)\,.$$ Multiplying by $w^{m-1}$ and summing over $m\geq1$, we complete the proof.
\[lem195a2\] The generating function $K_m(x)$ for $T$-avoiders of the form $$\pi=i\pi'n\alpha^{(1)}(i+m)\cdots\alpha^{(m)}(i+1)$$ with $2$ left-right maxima satisfies $K_1(x)=J_1(x)+K_2(x)$ and for all $m\geq2$, $$K_m(x)=J_m(x)+K_m(x)+\frac{x^{m+3}}{(1-x)(1-2x)(1-x-x^2)},$$ where $J_m(x)$ is defined in Lemma \[lem195a1\]. Moreover, $$K_1(x)=\frac{x^3(3x^7-10x^6+x^5+30x^4-42x^3+26x^2-8x+1)}{(2x-1)(x^2-3x+1)^2(x-1)^2(x^2+x-1)}\,.$$
Let us write an equation for $K_m(x)$. Clearly, $K_1(x)=J_1(x)+K_2(x)$. Let $m\geq2$, and let $\pi=i\pi'n\alpha^{(1)}(i+m)\cdots\alpha^{(m)}(i+1)\in S_n(T)$ with 2 left-right maxima. If $i=n-m-1$ then we have a contribution of $J_m(x)$ (see Lemma \[lem195a1\]). Otherwise, since $\pi$ avoids $1243$ and $1324$, the letter $i+m+1$ belongs to either $\alpha^{(1)}$ or $\alpha^{(2)}$. The former case gives contribution of $K_{m+1}(x)$. The latter case gives a contribution of $\frac{x^{m+2}}{(1-x)(1-2x)(1-x-x^2)}$, where the proof details are left to the reader. Thus, for all $m\geq2$, $$K_m(x)=J_m(x)+K_m(x)+\frac{x^{m+3}}{(1-x)(1-2x)(1-x-x^2)}\,.$$ Summing over $m\geq2$, we obtain $$K_2(x)=\sum_{m\geq2}J_m(x)+\frac{x^5}{(1-x)^2(1-2x)(1-x-x^2)}\,.$$ Hence, since $K_1(x)=J_1(x)+K_2(x)$, we have $$K_1(x)=\sum_{m\geq1}J_m(x)+\frac{x^5}{(1-x)^2(1-2x)(1-x-x^2)}\,,$$ where $\sum_{m\geq1}J_m(x)=J(x,1)$ is given in Lemma \[lem195a1\].
\[lem195a3\] The generating function $M(x)$ for permutations of the form $i\pi'n\pi''(i+1)\pi'''(i+2)\in S_n(T)$ with $2$ left-right maxima is given by $$\begin{aligned}
M(x)&=xK_1(x)+\frac{x^4}{(1-x)(1-2x)}(A-1)\\
&=\frac{x^4(2x^7-6x^6-x^5+22x^4-30x^3+20x^2-7x+1)}{(2x-1)(x^2-3x+1)^2(x-1)^2(x^2+x-1)}.\end{aligned}$$
Let us write an equation for $M(x)$. By Lemma \[lem195a2\], the case $\pi'''=\emptyset$ gives a contribution of $xK_1(x)$. Otherwise, $\pi'\pi''>\pi'''$ and the subsequence of $\pi'\pi''$ consisting of letters smaller than $i$ is decreasing, and the subsequence of $\pi''$ consisting of letters greater than $i$ is also decreasing, and $\pi'''$ is a nonempty permutation that avoids $\{132,2341\}$. Hence, we have a contribution of $\frac{x^4}{(1-x)(1-2x)}(A-1)$. By adding all the contributions, we complete the proof.
\[lem195a4\] The generating function $N(x)$ for permutations of the form $i\pi'n\pi''(i+2)\pi'''(i+3)(i+1)\in S_n(T)$ with $2$ left-right maxima is given by $$\begin{aligned}
N(x)&=\frac{x^5}{(2x-1)(x^2+x-1)}\,.\end{aligned}$$
Let us write an equation for $N(x)$. If $\pi'''\neq\emptyset$, then, as in the previous lemma, the subsequence of $\pi'\pi''$ consisting of letters smaller than $i$ is decreasing, and the subsequence of $\pi''$ consisting of letters greater than $i$ is also decreasing. Thus, we have a contribution of $\frac{x^6}{(1-x)(1-2x)}$. If $\pi'''=\emptyset$, we get a contribution of $\frac{x^5(x^3+(1-x)^2)}{(1-x)(1-2x)(1-x-x^2)}$ (proof omitted). Adding the two contributions, we complete the proof.
\[lem195a5\] Define the following generating functions for $T$-avoiders with $2$ left-right maxima:
- $B_m(x)$ for permutations of the form $i\pi'n\pi''\in S_n(T)$ such that $\pi''$ contains the subsequence $(i+m)(i+m-1)\cdots(i+1)$;
- $H_m(x)$ for permutations of the form $$(n-m-1)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m)\alpha^{(0)}\in S_n(T);$$
- $E_m(x)$ for permutations of the form $$(n-m-1)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m)\alpha^{(0)}\in S_n(T)$$ such that the letter $(n-m-2)$ belongs to $\alpha^{(1)}$;
- $D_m(x)$ for permutations of the form $$(n-m-1)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m)\alpha^{(0)}\in S_n(T)$$ such that the letter $(n-m-2)$ belongs to $\alpha^{(0)}$.
Then for all $m\geq1$, $$\begin{aligned}
(a)\,\,& B_m(x)=H_m(x)+B_{m+1}(x)+\frac{x^{m-2}}{1-x}N(x)\mbox{ with }B_1(x)=H_1(x)+B_2(x)+\frac{1}{1-x}M(x);\\
(b)\,\,& H_m(x)=x^{m+2}+E_m(x)+D_m(x)+\sum_{j=1}^mx^jH_{m+1-j}(x);\\
(c)\,\,& D_m(x)=H_{m+1}(x);\\
(d)\,\,& E_m(x)=x^{m+2}(F_T(x)-1)+\frac{x^{m+4}B}{1-x}(t^{m+1}-1)+x^{m+3}(t^{m+1}-1)(A-1/(1-x))\\
&\qquad\qquad+x(J_m(x)-x^{m+2})+x^{m+3}(B-1)(t^{m+1}-1),\end{aligned}$$ with $t=1/(1-x)$, where $M(x),N(x),J_m(x)$ are defined in the three preceding lemmas.
\(a) To write an equation for $B_1(x)$, suppose $\pi=i\pi'n\pi''\in S_n(T)$ with 2 left-right maxima. If $n=i+1$, then we have $H_1(x)$. Otherwise, the letter $i+2$ appears either on the left side of $i+1$, which leads to a contribution of $B_2(x)$, or on right side of $i+1$. In the latter case $\pi$ can be written as $\pi=i\pi'n\gamma(i+1)\gamma'(i+2)(i+3)\cdots(i+s)$ with $s\geq2$, which gives a contribution of $\frac{1}{1-x}M(x)$, see Lemma \[lem195a3\]. Thus, $B_1(x)=H_1(x)+B_2(x)+\frac{1}{1-x}M(x)$.
Let us write an equation for $B_m(x)$ with $m\geq2$. As in the case $B_1(x)$, we obtain $$B_m(x)=H_m(x)+B_{m+1}(x)+\frac{x^{m-2}}{1-x}N(x)\,,$$ where $N(x)$ is defined in Lemma \[lem195a4\].
\(b) The recurrence for the generating function $H_m(x)$ can be obtained by using very similar techniques as in Lemma \[lem195a1\] using the definitions of $E_m(x)$ and $D_m(x)$.
\(c) By mapping each permutation $(n-m-1)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m)\alpha^{(0)}$ to $(n-m-2)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m-1)\alpha^{(0)}$, we obtain the required relation.
\(d) Let us write an equation for $E_m(x)$. Let $$(n-m-1)\alpha^{(m+1)}n\alpha^{(m)}(n-1)\cdots\alpha^{(1)}(n-m)\alpha^{(0)}\in S_n(T)$$ with 2 left-right maxima and $\alpha^{(1)}=\gamma(n-m-2)\gamma'$ where the letter $n-m-2$ belongs to $\alpha^{(1)}$. The contribution of the case $\alpha'=\alpha^{(m+1)}\cdots\alpha^{(2)}\gamma=\emptyset$ is $x^{m+2}(F_T(x)-1)$. So we can assume that $\alpha'\neq\emptyset$. If $\gamma'\neq\emptyset$ then by considering whether $\gamma'$ is decreasing or not, we get contributions of $\frac{x^{m+4}}{1-x}(t^{m+1}-1)B$ and $x^{m+1}(t^{m+1}-1)(A-1/(1-x))$, respectively. Thus, we can assume that $\alpha'\neq\emptyset$ and $\beta=\emptyset$, and by considering either $\alpha^{(0)}$ is empty or not, we obtain the contributions $x(J_m(x)-x^{m+2})$ and $x^{m+3}(t^{m+1}-1)(B-1)$, respectively. Sum all contributions to complete the proof.
By Lemma \[lem195a5\] (b) and (c), we have $$H_m(x)=x^{m+2}+E_m(x)+H_{m+1}(x)+\sum_{j=1}^mx^jH_{m+1-j}(x)\,.$$ Define $H(x,w)=\sum_{m\geq1}H_m(x)w^{m-1}$ and $E(x,w)=\sum_{m\geq1}E_m(x)w^{m-1}$. Thus, this recurrence can be written as $$H(x,w)=\frac{x^3}{1-xw}+\frac{x}{1-xw}H(x,w)+E(x,w)+\frac{1}{w}\big(H(x,w)-H(x,0)\big)\, ,$$ which implies $$\left(w-1-\frac{xw}{1-xw}\right)H(x,w)=\frac{x^3w}{1-xw}+E(x,w)-H(x,0)\, .$$ This type of functional equation can be solved systematically using the [*kernel method*]{} (see, e.g., [@HM] for an exposition) by taking $w=C(x)$. We find $$\begin{aligned}
H(x,0)=\frac{x^3C(x)}{1-xC(x)}+E(x,C(x))=x^3C^2(x)+E(x,C(x))\, . \label{eq195H1}\end{aligned}$$ Lemma \[lem195a5\](d) and some mathematical programming yields $$\begin{aligned}
E(x,w)&=\frac{x^3}{1-wx}(F_T(x)-1)\\
&-\frac{x^5(-3x^7+11x^6-11x^5-5x^4+20x^3-22x^2+11x-2)}{(x-1)^3(wx+x-1)(x^2-3x+1)(wx^2-wx-2x+1)(wx-1)(x^2+x-1)}\\
&-\frac{x^6w((x^2-5x+3)(x^3-x^2-2x+1)+x(x-1)(x^3-x^2-2x+1)w)}{(x-1)^3(wx+x-1)(x^2-3x+1)(wx^2-wx-2x+1)(wx-1)}\, ,\end{aligned}$$ which leads to $$\begin{aligned}
E(x,C(x))&=x^3C(x)(F_T(x)-1)\notag\\
&-\frac{x^5C^6(x)(-3x^7+11x^6-11x^5-5x^4+20x^3-22x^2+11x-2)}{(x-1)^3(x^2-3x+1)(x^2+x-1)}\label{eq195E1}\\
&-\frac{x^6C^7(x)((x^2-5x+3)(x^3-x^2-2x+1)+x(x-1)(x^3-x^2-2x+1)C(x))}{(x-1)^3(x^2-3x+1)}.\notag\end{aligned}$$
Now we are ready to give the main result of this subsection.
\[th195a\] Let $T=\{1324,2341,1243\}$. Then $F_T(x)=$ $$\frac{ (1 - 7 x + 19 x^2 - 25 x^3 + 13 x^4 + 4 x^5 - 5 x^6 + x^7)C(x) -1 + 7 x - 19 x^2 + 23 x^3 - 7 x^4 - 7 x^5 + 4 x^6}{x(1 - x)^2 (1 - 3 x + x^2) (1 - x - x^2)}\, .$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
First, we treat $G_2(x)$. Each $\pi=i\pi'n\pi''\in S_n(T)$ with 2 left-right maxima can be decomposed as either $(n-1)\pi'n\pi''$ or $i\pi'n\pi''$ with $1\leq i<n-2$ with respective contributions to $G_2(x)$ of $x(F_T(x)-1)$ and $B_1(x)$. From Lemma \[lem195a5\](a) we have $$B_1(x)=\sum_{m\geq1}H_m(x)+\frac{M(x)}{1-x}+\frac{xN(x)}{(1-x)^2},$$ which implies $$B_1(x)=H(x,1)+\frac{M(x)}{1-x}+\frac{xN(x)}{(1-x)^2},$$ where $M(x)$ and $N(x)$ are given in Lemmas \[lem195a3\] and \[lem195a4\], respectively, and $$H(x,1)=\frac{1-x}{x}\left(H(x,0)-\frac{x^3}{1-x}-E(x,1)\right),$$ where $$E(x,1)=\frac{x^3\big(F_T(x)-1\big)}{1-x}-\frac{x^5(2x^7-11x^6+15x^5+2x^4-21x^3+25x^2-12x+2)}
{(x-1)^3(2x-1)(x^2-3x+1)^2(x^2+x-1)}$$ and $H(x,0)$ and $E(x,C(x))$ are given in and .
Now, let us write a formula for $G_3(x)$. Let $\pi=i_1\pi'i_2\pi''i_3\pi'''\in S_n(T)$ with 3 left-right maxima. then $i_1<\pi'''<i_2$ and $\pi'>\pi''$. By considering whether $\pi',\pi''$ are empty or not, we obtain the contributions $x^3A$ ($\pi'=\pi''=\emptyset$), $x^3(A-1)/(1-x)$ ($\pi'\neq\emptyset,\pi''=\emptyset$), $x^3B/(1-x)^2$ ($\pi'=\emptyset,\pi''\neq\emptyset$) and $0$. Thus, $$G_3(x)=x^3A+\frac{x^3(A-1)}{1-x}+\frac{x^3B}{(1-x)^2}\,.$$
Now, let us write a formula for $G_m(x)$ with $m\geq4$. Let $i_1\pi^{(1)}\cdots i_m\pi^{(m)}\in S_n(T)$ with $m$ left-right maxima. Then $\pi^{(3)}=\cdots=\pi^{(m)}=\emptyset$ and $\pi^{(1)}>\pi^{(2)}$. By considering whether $\pi^{(2)}$ is empty or not, we have $$G_m(x)=x^mA+\frac{x^m(A-1)}{1-x}\,.$$
Summing over $m\geq0$, we obtain $$F_T(x)=1+xF_T(x)+G_2(x)+\frac{x^3}{1-x}A+\frac{x^3}{(1-x)^2}(A+B-1).$$ Substituting the formula for $G_2(x)$, we get $$\begin{aligned}
F_T(x)&=1+xF_T(x)+x(F_T(x)-1)+\frac{1-x}{x}H(x,0)-x^2F_T(x)\\
&+\frac{x^4(2x^7-11x^6+15x^5+2x^4-21x^3+25x^2-12x+2)}{(x-1)^2(2x-1)(x^2-3x+1)^2(x^2+x-1)}\\
&+\frac{M(x)}{1-x}+\frac{xN(x)}{(1-x)^2}+\frac{x^3}{1-x}A+\frac{x^3}{(1-x)^2}(A+B-1).\end{aligned}$$ where $M(x)$ and $N(x)$ are given in Lemmas \[lem195a3\] and \[lem195a4\], respectively, and $H(x,0)$ and $E(x,C(x))$ are given in and , respectively. Solving for $F_T(x)$ and simplifying the result, we complete the proof.
Case 210: $\{1243,1324,2431\}$
------------------------------
\[th210a\] Let $T=\{1243,1324,2431\}$. Then $$F_T(x)=\frac{\left(1-6 x+13 x^2-11 x^3+4 x^4\right)}{x^2(1-x)^2 }C(x)-
\frac{1-6 x+12 x^2-8 x^3+2 x^4}{x^2(1-x) (1-2 x)}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
For $G_m(x)$ with $m\geq2$, suppose $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$, $\pi^{(s)}<i_1$ for all $s=1,2,\ldots,m-1$. Since $\pi$ avoids $1243$, $\pi^{(m)}<i_2$.
(7,3.1) (6,6)(6,9) (2,8)(12,8) (6,0)(6,6)(0,6)(0,3) (3,4.5) (10.5,4.5) (13.5,7) (7.5,2) (7.5,1) (0,6)(2,8)(6,9) (-.5,6.3) (1.5,8.3) (3.6,9) (5.4,9.3) (6,0)(9,0)(9,6)(15,6)(15,8)(12,8)(12,3)(0,3)
First, suppose $\pi^{(m)}$ has a letter greater than $i_1$ and consider two cases:
- all the letters between $i_1$ and $i_2$ in $\pi^{(m)}$ are increasing. Here, $\pi^{(1)}\cdots\pi^{(m-1)}$ is decreasing ($\pi$ avoids $1243$), and $\pi^{(m)}$ can be decomposed as ($\pi$ avoids $2431$) $\alpha(i_1+1)\cdots(i_2-1)$ and, writing $\pi^{(1)}\cdots\pi^{(m-1)}$ as $j_sj_{s-1}\cdots j_1$, $\alpha$ can be further decomposed as $\alpha^{(0)}\alpha^{(1)}\cdots\alpha^{(s)}$ with $\alpha^{(0)}<j_1<\alpha^{(1)}<\cdots<j_s<\alpha^{(s)}<i_1$, where $\alpha^{(0)}$ avoids $132$ and $\alpha^{(i)}$ is an increasing subword for all $i=1,2,\ldots,s$. In short, $\pi$ has the schematic form shown in the Figure \[figTK1\] where $\downarrow$ denotes decreasing, $\uparrow$ denotes increasing, and blank regions are empty. If $A$ is empty, then so is $C$ and, in any case, the position of $i_m$ is determined by $AC$. From Figure \[figTK1\], $\pi$ can be written as $i_1 A_1i_m BCD$, where $A_1$ consists of $A$ and the left-right maxima $i_2,\dots,i_{m-1}$, and $\pi$ is determined by St($A_1C$), St($B$), and St($D$). The latter two contribute $C(x)$ and $\frac{x}{1-x}$ respectively. Now $AC$ is a $\{132,231\}$-avoider and St($A_1C$) can be viewed as a $\{132,231\}$-avoider decorated with $m-2$ dots (for $i_2,\dots,i_{m-1}$) placed in the spaces between its entries but not after the first ascent. For example, the 4 boxes shown accept dots in $\Box 5 \Box 2 \Box1\Box34$. The for such constructs with $M$ dots, each dot counting as a letter, can be shown to be $x^M L(x)^{M+1}$ where $L(x)=\frac{1-x}{1-2x}$ is the for $\{132,231\}$-avoiders. Thus, with $M=m-2$, the contribution of this case is $\frac{x^{m+1}}{1-x}L(x)^{m-1}C(x)$.
- the letters between $i_1$ and $i_2$ in $\pi^{(m)}$ do not form an increasing sequence, where the sequence $\pi^{(1)}\cdots\pi^{(m-1)}\alpha$ is decreasing, and $\pi^{(m)}=\alpha\beta$ with $\alpha<i_1<\beta<i_2$. Thus, we have a contribution of $\frac{x^m}{(1-x)^m}\big(L(x)-\frac{1}{1-x}\big)$.
Hence, for all $m\geq2$, $$G_m(x)=H_m(x)+\frac{x^{m+1}}{1-x}L(x)^{m-1}C(x)+\frac{x^m}{(1-x)^m}\left(L(x)-\frac{1}{1-x}\right),$$ where $H_m(x)$ is the generating function for those permutations $\pi\in S_n(T)$ with $\pi^{(m)}<i_1$ (in other words $i_1=n+1-m$).
Now let us write an equation for $H_m(x)$. Let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima such that $i_j=n+j-m$ for all $j$. If $\pi^{(1)}=\emptyset$ then we have a contribution of $xH_{m-1}(x)$. Otherwise, if we assume that $\pi^{(1)}$ has exactly $s$ left-right maxima, it is not hard to see that those permutations are bijection with the permutations with exactly $m+s-1$ left-right maxima. Thus, $$H_m(x)=xH_{m-1}(x)+x(G_m(x)+G_{m+1}(x)+G_{m+2}(x)+\cdots)$$ with $H_1(x)=G_1(x)$ (by definitions).
Define $G(x;t)=\sum_{m\geq0}G_m(x)t^m$ and $H(x;t)=\sum_{m\geq1}H_m(x)t^m$. Note that $G(x;1)=F_T(x)$. Hence, the above recurrences can be written as $$\begin{aligned}
H(x;t)&=xtF_T(x)+xtH(x;t)+\frac{xt^2}{1-t}(F_T(x)-1)-\frac{xt}{1-t}(G(x;t)-1),\\
G(x;t)&=1+H(x;t)+\frac{x^3t^2L(x)C(x)}{(1-x)(1-xt)}+\frac{x^2t^2}{(1-x)(1-x-xt)}(L(x)-1/(1-x)).\end{aligned}$$ Hence, $$\begin{aligned}
\left(1+\frac{xt}{(1-t)(1-xt)}\right)G(x;t)&=\frac{1}{1-xt}+\frac{xt}{(1-t)(1-xt)}F_T(x)\\
&+\frac{x^3t^2L(x)C(x)}{(1-x)(1-xt)}+\frac{x^2t^2}{(1-x)(1-x-xt)}(L(x)-1/(1-x)).\end{aligned}$$ This equation can be solved by the kernel method (see, e.g., [@HM] for an exposition) taking $t=C(x)$ and leads, after simplification, to our theorem.
Case 211: $\{1234,1324,2341\}$
------------------------------
All three patterns contain 123 and so $T$-avoiders consist of 123-avoiders together with $T$-avoiders that contain a 123. The former are counted by $C(x)$. To count the latter, let $abc$ at positions $i,j,k$ be the leftmost 123 pattern (smallest $i$, then smallest $j$, then smallest $k$). This 123 pattern divides the permutation diagram into rectangular regions as shown in Figure \[figAK6\] where the bullets represent the leftmost 123.
(6,7) (6,0)(8,0)(8,2)(6,2)(6,0) (4,6)(6,6)(6,8)(4,8)(4,6) (6,6)(8,6)(8,8)(6,8)(6,6) (2,2)(6,2)(6,6)(4,6)(4,4)(2,4)(2,2) (2,0)(2,4)(0,4)(0,0)(2,0) (0,0)(8,0)(8,8)(0,8)(0,0) (0,2)(8,2) (0,4)(8,4) (4,6)(8,6) (2,0)(2,8) (4,0)(4,8) (6,0)(6,8) (2,2)[0.1]{}(4,4)[0.1]{}(6,6)[0.1]{} (.9,5.9)[$\al$]{} (2.8,5.9)[$\al'$]{} (6.7,5.1)[$\be\,\downarrow$]{} (6.4,4.4)[$(1234)$]{} (6.8,2.9)[ $\be'$]{} (2.7,1.1)[$\ga\,\downarrow$]{} (2.4,0.4)[$(1234)$]{} (4.9,0.9)[$\ga'$]{} (.5,0.8)[(min)]{} (.5,2.8)[(min)]{} (2.5,2.8)[(min)]{} (4.4,2.8)[$(1324)$]{} (4.5,4.8)[(min)]{} (4.5,6.8)[(min)]{} (6.4,6.8)[$(1234)$]{} (6.4,0.8)[$(2341)$]{} (2,-.5)[$i$]{} (4,-.5)[$j$]{} (6,-.5)[$k$]{} (8.3,2)[$a$]{} (8.3,4)[$b$]{} (8.3,6)[$c$]{}
Shaded regions are empty for the indicated reason, where (min) refers to the minimal, that is, leftmost property of the 123 pattern, and unshaded regions are labeled $\al,\al'$ etc. A down arrow ($\downarrow$) indicates necessarily decreasing entries, again for the indicated reason. Also, $\al\al'$ avoids 123 (or the middle bullet ends a 2341), while $\be$ lies to the left of $\be'$ and $\ga>\ga'$ (both to avoid 1324). We consider 4 cases according as $\be',\ga'$ are empty or not:
- $\be',\ga'$ both empty. Here, St($\al\, a\, \al'\, c$) is a a 123-avoider $\tau$ of length $\ge 2$ with last entry $>1$. Say $\tau$ has length $n$, with 1 in position $i<n$ and last entry $\ell\ge 2$. Now $\be$ serves to “decorate” $\tau$ with 0 or more dots inserted arbitrarily in the $\ell-1$ spaces between 1 and $\ell$. Similarly, $\ga$ amounts to 0 or more dots inserted in the $n-i$ spaces between $i$ and $n$. Thus, the contribution of this case is $$x \sum_{n\ge 2}\sum_{i=1}^{n-1}\sum_{\ell=2}^{n}C(n,i,\ell)x^n \sum_{u,v\ge 0}\binom{u+\ell-2}{u}\binom{v+n-i-1}{v}\,,$$ where $C(n,i,\ell)$ is the number of $123$-avoiders of length $n$ with 1 in position $i$ and last entry $\ell$. It is known that $C(n,i,\ell)=\frac{n-2-i+\ell}{n+i-\ell}\binom{n+i-\ell}{n-1}$, a generalized Catalan number, and the displayed sum evaluates to the compact expression $x^3 C(x)^5$.
- $\be'$ empty, $\ga'$ nonempty. Here, $\ga'$ avoids 123 and 132 and so the contribution is $x^3 C(x)^5 (L-1)=x^3 C(x)^5\frac{x}{1-2x}$ where $L=\frac{1-x}{1-2x}$ is the for $\{123,132\}$-avoiders.
- $\be'$ nonempty, $\ga'$ empty. Here, $\be'$ avoids 123 and 213 and so the contribution is $x^3 C(x)^5 (L-1)= x^3 C(x)^5\frac{x}{1-2x}$ since $L=\frac{1-x}{1-2x}$ is also the for $\{123,213\}$-avoiders.
- $\be',\ga'$ both nonempty. In case $\ga'$ is decreasing, the only restriction on $\be'$ is to avoid 123 and 213 (since $\ga' \ne \emptyset$). So $\ga'$ contributes $\frac{x}{1-x}$ and $\be'$ contributes $L-1$. In case $\ga'$ is not decreasing, $\be'$ *is* decreasing (to avoid 1234) and $\ga'$ contributes $L-\frac{1}{1-x}$ while $\be'$ contributes $\frac{x}{1-x}$. So, $\be',\ga'$ together contribute $\frac{x}{1-x}(L-1) + \big(L-\frac{1}{1-x}\big)\frac{x}{1-x}= \frac{x^2}{(1-x)^2 (1-2x)}$ for an overall contribution of $x^3 C(x)^5 \frac{x^2}{(1-x)^2 (1-2x)}$
Hence, summing over 123-avoiders and the 4 cases, $$F_T(x)=C(x)+x^3 C(x)^5\left(1 + \frac{x}{1-2x} + \frac{x}{1-2x} + \frac{x^2}{(1-x)^2 (1-2x)}\right)\, .$$ Simplifying this expression, we have established
\[th211a\] Let $T=\{1234,1324,2341\}$. Then $$F_T(x)=\frac{(1 - 4 x + 5 x^2 - 3 x^3)\, C(x) - (1 - 4 x + 6 x^2 - 4 x^3)}{x (1 - x)^2 (1 - 2 x)}\, .$$
Case 212: $\{1324,2413,2431\}$
------------------------------
Note that each pattern contains 132.
\[th212a\] Let $T=\{1324,2413,2431\}$. Then $$F_T(x)=1+\frac{x(1-4x+4x^2-x^3-x(1-4x+2x^2)C(x))}{(1-3x+x^2)(1-3x+x^2-x(1-2x)C(x))}.$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$. For $m=2,$ $\pi$ has the form $i \al n \be$ with $\al<i$, and $\be$ cannot contain both letters $>i$ and $<i$ (2413 and 2431). If $\be$ has no letter $>i$, then $i=n-1$ and deleting it gives a contribution of $x\big(F_T(x)-1\big)$. Otherwise, $\be>i$ and $\al$ avoids 132 (due to $n$) and $\be$ avoids both 213 (due to $i$) and 2431 and is nonempty, contributing $x^2 C(x)\big(K(x)-1\big)$, where $K(x)=\frac{1-2x}{1-3x+x^2}$ is the generating function for $\{213,2431\}$-avoiders [@Sl Seq. A001519]. So $G_2(x)=x\big(F_T(x)-1\big)+x^2C(x)\big(K(x)-1\big)$.
Now let us write equation for $G_m(x)$ for $m\geq3$. Suppose $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$, we have $\pi^{(s)}<i_1$ for all $s=1,2,\ldots,m-1$. Since $\pi$ avoids $2413$ and $2431$, either $\pi^{(2)}$ is empty or $\pi^{(m)}$ has no letter between $i_1$ and $i_2$ (or both). Thus, we have three cases
- $\pi^{(2)}=\emptyset$ and $\pi^{(m)}$ has no letter between $i_1$ and $i_2$. Here, $i_2$ and its position is determined by the rest of the avoider, and we have a contribution of $xG_{m-1}(x)$.
- $\pi^{(2)}=\emptyset$ and $\pi^{(m)}$ has a letter between $i_1$ and $i_2$. Here, $\pi^{(3)}=\cdots=\pi^{(m-1)}=\emptyset$ (or they would contain the 1 of a 2413), and since $\pi$ avoids both $2413$ and $2431$, $i_1<\pi^{(m)}$. Additionally, $\pi^{(m)}$ is not empty and avoids both $213$ (or $i_1$ is the 1 of a 1324) and $2431$, and $\pi^{(1)}$ avoids $132$ (or $i_m$ is the 4 of a 1324). Hence, the contribution is $x^mC(x)\big(K(x)-1\big)$.
- $\pi^{(2)}\neq\emptyset$ and $\pi^{(m)}$ has no letter between $i_1$ and $i_2$. Since $\pi$ avoids $2413$, we see that $i_1>\pi^{(1)}>\pi^{(s)}$ for all $s=2,3,\ldots,m-1$. Also, $\pi^{(1)}$ avoids 132 and $i_2=i_1+1$. So $\pi$ can be recovered from St($i_1 \pi^{(1)}$) and St($i_2\pi^{(2)}\cdots i_m\pi^{(m)}$), giving respective contributions of $xC(x)$ and $G_{m-1}(x)-G'_{m-1}(x)$ where $G'_{m-1}(x)$ counts the $\pi^{(2)}$ empty case, that is, $G'_m(x)$ is the generating function for $T$-avoiders with $m$ left-right maxima in which the first letter is smaller than the second. Thus, the contribution is $xC(x)\big(G_{m-1}(x)-G'_{m-1}(x)\big)$.
Hence, for all $m\geq3$, $$\begin{aligned}
G_m(x)=xG_{m-1}(x)+x^mC(x)\big(K(x)-1\big)+xC(x)\big(G_{m-1}(x)-G'_{m-1}(x)\big)\, .\label{eq212a1}\end{aligned}$$
For $G'_2(x)$, a $T$-avoider has the form $\pi= i n \pi'$. If $\pi'$ is empty, the contribution is $x^2$. So suppose $\pi'\neq\emptyset$. If $i=n-1$ then we have $x^2(F_T(x)-1)$, but if $i<n-1$, then, since $\pi$ avoids 2413 and 2431 we see that $i=1$ and $\pi=1n\pi'$ with $\pi'\neq\emptyset$ avoiding 213 and 2431, giving a contribution of $x^2(K(x)-1)$. So $$\label{212g2p}
G'_2(x)=x^2\big(K(x)+F_T(x)-1\big)\,.$$
For $G'_m(x)$ with $m\ge 3$, using the same three cases as above, we find by similar arguments the recurrence $$\begin{aligned}
G'_m(x)& = & xG'_{m-1}(x)+x^m(K(x)-1)+x\big(G_{m-1}(x)-G'_{m-1}(x)\big) \notag \\
&=& x^m \big(K(x)-1\big)+xG_{m-1}(x)\, .\label{eq212a2}\end{aligned}$$
Substituting and into , we have $$G_m(x)=x\big(C(x)+1\big)G_{m-1}(x)-x^2C(x)G_{m-2}(x)$$ for $m\ge 3$, with $G_2,G_1,G_0$ as above. Summing over $m\geq3$, we obtain $$F_T(x)-1-xF_T(x)-G_2(x)=x\big(C(x)+1\big)\big(F_T(x)-1-xF_T(x)\big)-
x^2C(x)\big(F_T(x)-1\big)\, .$$ Solving for $F_T(x)$ completes the proof.
Case 213: $\{2431,1324,1342\}$
------------------------------
\[th213a\] Let $T=\{2431,1324,1342\}$. Then $$F_T(x)=\frac{ (1 - 5 x + 8 x^2 - 5 x^3)\,C(x)-1 + 4 x - 4 x^2 + x^3}{x^2(1 - 2 x) }\, .$$
Let $G_m(x)$ be the generating function for $T$-avoiders with $m$ left-right maxima. Clearly, $G_0(x)=1$ and $G_1(x)=xF_T(x)$.
First, we find a formula for $G_2(x)$. Since $\pi$ avoids $2431$, we can write a $T$-avoider $\pi$ with 2 left-right maxima as $\pi=i\alpha n\beta\gamma$ where $\alpha\beta<i<\gamma$, and $\gamma$ avoids both 132 and 231 (or $i$ is the 1 of a 1324 or 1342). If $\gamma=\emptyset$, then $\pi=n-1\, \al n\be$ and, by deleting $n-1$, the contribution is $x\big(F_T(x)-1\big)$. Now suppose $\gamma\neq\emptyset$ and let $H$ denote the contribution of such permutations to $G_2(x)$. If $\alpha\beta=\emptyset$, then $\pi=1n\gamma$ with $\gamma=\emptyset$ and the contribution is $x^2(J-1)$ where $J=\frac{1-x}{1-2x}$ is the generating function for $\{213,231\}$-avoiders. Henceforth, $\alpha\beta\ne\emptyset$, and so $i>1$ and consider 3 cases:
- $i-1 \in \al$. Here, $\pi=i\al'\, i-1\,\al''n\be \gamma$ where $\al'>\al''\beta\ (1324)$ and $\al'$ avoids 132. So the contribution is $xC(x)H$.
- $i-1$ is the last letter of $\be$. Deleting $i-1$ gives a one-size-smaller avoider and the contribution is $xH$.
- $i-1\in \be$ but is not the last letter of $\be$. Here $\al=\emptyset$ (2431) and $\pi=in\be'\,i-1\,\be'' \gamma$ where $\be''\ne \emptyset$, $\be'>\be''$ (1324) and $\be',\be''$ each avoid 132. So, with $C(x),C(x)-1,J-1$ respectively from $\be',\be'',\gamma$, the contribution is $x^3C(x)\big(C(x)-1\big)(J-1)$.
Thus, $H = x^2(J-1)+ xC(x)H +xH+ x^3 C(x)\big(C(x)-1\big)(J-1)$ and so $$H=\frac{x^2(J-1)+x^3\big(C(x)-1\big)C(x)(J-1)}{1-x-xC(x)},$$ which leads to $$G_2(x)=x\big(F_T(x)-1\big)+H=x\big(F_T(x)-1\big)+\frac{x^2(J-1)+x^3\big(C(x)-1\big)C(x)(J-1)}{1-x-xC(x)}.$$
Now let $m\geq3$ and let us write an equation for $G_m(x)$. Let $\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_m\pi^{(m)}\in S_n(T)$ with exactly $m$ left-right maxima. Since $\pi$ avoids $1324$ and $1342$, we can express $\pi$ as $$\pi=i_1\pi^{(1)}i_2\pi^{(2)}\cdots i_{m-2}\pi^{(m-2)}i_{m-1}\alpha i_m\beta\gamma$$ such that $i_1>\pi^{(1)}>\cdots>\pi^{(m-2)}>\alpha\beta$ and $i_{m-1}<\gamma$. Note that $\pi$ avoids $T$ if and only if $\pi^{(j)}$ avoids $132$ for $j=1,2,\ldots,m-2$ and the standard form of $i_1\alpha n\beta\gamma$ is a $T$-avoider with two left-right maxima. Hence, $$G_m(x)= x^{m-2} C(x)^{m-2}G_2(x).$$
By summing for $m\geq2$ and using the expressions for $G_1(x)$ and $G_0(x)$, we obtain $$F_T(x)=1+xF_T(x)+\frac{G_2(x)}{1-xC(x)}\, ,$$ with solution as stated.
For the next two cases we set $a_T(n)=|S_n(T)|$ and let $a_T(n;j_1,j_2,...,j_s)$ denote the number of permutations in $S_n(T)$ whose first $s$ letters are $j_1j_2...j_s$.
Case 231: $\{1324,1342,2341\}$
------------------------------
Set $b(n;j)=a_T(n;j,j+1)$.
\[lem231a1\] For $1\leq j\leq n-2$, $$\begin{aligned}
a_T(n;j)&=a_T(n-1;1)+\cdots+a_T(n-1;j)+b(n;j)\end{aligned}$$ and $a_T(n;n)=a_T(n;n-1)=a_T(n;n-2)=a_T(n-1)$.
The initial conditions $a_T(n;n)=a_T(n;n-1)=a_T(n-1)$ easily follow from the definitions. For $1\leq j\leq n-2$, we have $$\begin{aligned}
a_T(n;j)&=\sum_{i=1}^{j-1}a_T(n;j,i)+\sum_{i=j+1}^na_T(n;j,i).\end{aligned}$$ So assume $1 \leq j\leq n-2$ and let $\pi=ji\pi'$ be a member of $S_n(T)$. We consider several cases on $i$. Since $\pi$ avoids $1324$ and $1342$, we have that either $1\leq i\leq j+1$ or $i=n$. If $i=n$, then $\pi$ avoids $T$ if and only if $j\pi'$ avoids $T$, so $a_T(n;j,n)=a_T(n-1,j)$. So, $$\begin{aligned}
a_T(n;j)&=\sum_{i=1}^{j-1}a_T(n;j,i)+a_T(n-1;j)+b(n;j).\end{aligned}$$ Let $1\leq i\leq j-1$. If $\pi$ avoids $T$ then $i\pi'$ avoids $T$. On other hand, if $\pi$ contains $1324$ (resp. $1342$), then $i\pi'$ contains $1324$ (resp. $1324$). Also, if $\pi$ contains $2341$ where $j$ does not occur in the corresponding occurrence, then $i\pi'$ contains $2341$. Thus, we assume that $\pi$ contains $jabc$ with $c<j<a<b$. If $c<i$, then $i\pi'$ contains $2341$, otherwise $i\pi'$ contains $1342$. Therefore, $i\pi'$ avoids $T$ if and only if $\pi$ avoids $T$, which implies $a_T(n-1;j,i)=a_T(n;i)$. Hence, $$\begin{aligned}
a_T(n;j)&=\sum_{i=1}^{j-1}a_T(n-1;i)+a_T(n-1;j)+b(n;j),\end{aligned}$$ as required.
Define $A_T(n;v)=\sum_{j=1}^na_T(n;j)v^{j-1}$ and $B(n;v)=\sum_{j=1}^{n-2}b(n;j)v^{j-1}$. Then Lemma \[lem231a1\] can be written as $$A_T(n;v)=\frac{1}{1-v}(A_T(n-1;v)-v^nA_{T}(n-1;1))+B(n;v)$$ with $A_T(0;v)=A_T(1;v)=1$.
Define $A_T(x,v)=\sum_{n\geq0}A_T(n;v)x^n$ and $B(x,v)=\sum_{n\geq3}B(n;v)x^n$. Then, the above recurrence can be written as $$\begin{aligned}
A(x,v)=1+\frac{x}{1-v}(A(x,v)-vA(xv,1))+B(x,v).\label{eq231a1}\end{aligned}$$
\[lem231a2\] $$B(x,v)=\frac{x^3(1-2xv)}{(1-3xv+x^2v^2)(1-2x)}.$$
Let $\pi=j(j+1)\pi'\in S_n(T)$ with $1\leq j\leq n-2$. Since $\pi$ avoids $2341$, we can express $\pi$ as $\pi=j(j+1)\alpha\beta$ with $\alpha<j<j+1<\beta$. Note that $\pi$ avoids $T$ if and only if $\alpha$ avoids both $132,2341$ and $\beta$ avoids both $213,231$. By [@Sl Seq. A001519] we have that $F_{\{132,2341\}}(x)=\frac{1-2x}{1-3x+x^2}$, and by [@SiS] we have that $F_{\{213,231\}}(x)=\frac{1-x}{1-2x}$, yielding $$\begin{aligned}
B(x,v)&=\sum_{n\geq3}\sum_{j=1}^{n-2}|S_{j-1}(132,2341)||S_{n-1-j}(213,231)|v^{j-1}x^n\\
&=x^2F_{\{132,2341\}}(xv)(F_{\{213,231\}}(x)-1),\end{aligned}$$ which leads to $B(x,v)=\frac{x^3(1-2xv)}{(1-3xv+x^2v^2)(1-2x)}$, as required.
By and Lemma \[lem231a2\], we obtain $$A(x/v,v)=1+\frac{x}{v(1-v)}(A(x/v,v)-vA(x,1))+\frac{x^3(1-2x)}{v^2(1-3x+x^2)(v-2x)}.$$ To solve the preceding functional equation, we apply the kernel method (see, e.g., [@HM] for an exposition) and take $v=\frac{1+\sqrt{1-4x}}{2}=1/C(x)$. Then $$F_T(x)=A(x,1)=C(x)+\frac{x^3(1-2x)C^4(x)}{(1-3x+x^2)(1-2xC(x))}\,.$$ After simplification, this gives the following result.
\[th231a\] Let $T=\{1324,1342,2341\}$. Then $$F_T(x)=\frac{(1 - 3 x) \big(1 - 2x-xC(x)\big)}{(1 - 4 x) (1 - 3 x + x^2)}$$
Case 241: $\{1324,1243,1234\}$
------------------------------
Set $b(n;j)=a_T(n;j,n-1)$.
\[lem241a1\] For $1\leq j\leq n-2$, $$\begin{aligned}
a_T(n;j)&=a_T(n-1;1)+\cdots+a_T(n-1;j)+b(n;j),\\
b(n;j)&=b(n-1;1)+\cdots+b(n-1;j-1)+a_T(n-2;j)\end{aligned}$$ and $a_T(n;n)=a_T(n;n-1)=a_T(n;n-2)=a_T(n-1)$, $b(n;n)=b(n;n-2)=a_T(n-2)$ and $b(n;n-1)=0$.
The initial conditions $a_T(n;n)=a_T(n;n-1)=a_T(n;n-2)=a_T(n-1)$, $b(n;n)=b(n;n-2)=a_T(n-2)$ and $b(n;n-1)=0$ easily follow from the definitions. For $1\leq j\leq n-2$, we have $$\begin{aligned}
a_T(n;j)&=\sum_{i=1}^{j-1}a_T(n;j,i)+\sum_{i=j+1}^{n}a_T(n;j,i).\end{aligned}$$ So assume $1 \leq j\leq n-3$ and let $\pi=ji\pi'$ be a member of $S_n(T)$. We consider several cases for $i$. If $i=n$, then $a_T(n;j,n)=a_T(n-1;j)$. If $1\leq i<j$ then $ji\pi'$ (respectively, $jn\pi'$) avoids $T$ if and only if $i\pi'$ (respectively, $j\pi'$) avoids $T$, so $a_T(n;j,i)=a_T(n-1;i)$ for all $i=1,2,\ldots,j-1$ and $a_T(n;j,n)=a_T(n-1;j)$. Since $\pi$ avoids $1234$ and $1243$, we see that either $i<j$ or $j>n-2$, and $a_T(n;j,n)=a_T(n-1;j)$. Thus, $$\begin{aligned}
a_T(n;j)&=\sum_{i=1}^{j-1}a_T(n-1;i)+a_T(n-1;j)+b(n;j),\end{aligned}$$ which completes the proof of the first recurrence relation.
For the second relation, by similar reasons, we have $$\begin{aligned}
b(n;j)&=\sum_{i=1}^{j-1}a_T(n;j,n-1,i)+\sum_{i=j+1}^{n-2}a_T(n;j,n-1,i)+a_T(n;j,n-1,n).\end{aligned}$$ Clearly, $a_T(n;j,n-1,n)=a_T(n-2,j)$. The permutation $j(n-1)(n-2)\pi''$ contains $1324$, so $a_T(n;j,n-1,n-2)=0$. The permutation $j(n-1)i\pi''$ with $j+1\leq i\leq n-3$ contains either $1234$ or $1243$, so $a_T(n;j,n-1,i)=0$. Thus, $$\begin{aligned}
b(n;j)&=\sum_{i=1}^{j-1}a_T(n;j,n-1,i)+a_T(n-2;j).\end{aligned}$$ Let $j(n-1)i\pi''\in S_n$. If $i(n-1)\pi''$ contains a pattern in $T$ then $j(n-1)i\pi''$ contains the same pattern in $T$. Now, suppose $i(n-1)\pi''$ avoids $T$, so if $\pi=j(n-1)i\pi''$ contains $1234$ or $1243$, then $i(n-1)\pi''$ contains $1234$ or $1243$, which implies that $\pi$ avoids $1234$ and $1243$. If $\pi$ contains $1324$, then we can assume that $1324$ occurs in $\pi$ as a subsequence $jabc$ with $j<b<a<c$, otherwise $i(n-1)\pi'$ contains $1324$. Also, we can assume that $b=n-1$ which gives $c=n$, otherwise $iabc$ occurs in $i(n-1)\pi''$. Since $j\leq n-3$, $\pi$ has an element $d$ such that either $j<d<b$ or $b<d<n-1$. Thus, $i(n-1)\pi''$ contains either $idba$ or $ibda$ or $ibad$, that is $i(n-1)\pi''$ does not avoid $T$. Therefore, $\pi$ avoids $T$. Hence, $$a_T(n;j,n-1,i)=a_T(n-1;i,n-2)=b(n-1;i),$$ for all $i=1,2,\ldots,j-1$, which implies $$\begin{aligned}
b(n;j)&=\sum_{i=1}^{j-1}b(n-1;i)+a_T(n-2;j)\, .\end{aligned}$$ This completes the proof.
Define $A_T(n;v)=\sum_{j=1}^na_T(n;j)v^{j-1}$ and $B(n;v)=\sum_{j=1}^nb(n;j)v^{j-1}$. Then Lemma \[lem241a1\] can be written as $$\begin{aligned}
A_T(n;v)&=A_T(n-1;1)(v^{n-1}+v^{n-2}+v^{n-3})+\frac{1}{1-v}(A_T(n-1;v)-v^{n-3}A_T(n-1;1))\\
&+B(n;v)-A_T(n-2;1)v^{n-1},\\
B(n;v)&=A_T(n-2;1)(v^{n-1}+v^{n-3})+\frac{1}{1-v}(vB(n-1;v)-B(n-1;1)v^{n-3})\\
&+A_T(n-2;v)+A_T(n-3;1)v^{n-2}\end{aligned}$$ with $A_T(0;v)=A_T(1;v)=1$, $A_T(2;v)=1+v$, $B(0;v)=B(1;v)=0$ and $B(2;v)=v$.
Define $A_T(x,v)=\sum_{n\geq0}A_T(n;v)x^n$ and $B(x,v)=\sum_{n\geq0}B(n;v)x^n$. Then, the above recurrence can be written as $$\begin{aligned}
A_T(x,v)&=1+\left(x+\frac{x}{v}+\frac{x}{v^2}\right)A_T(xv,1)\nonumber\\
&+\frac{x}{1-v}\left(A_T(x,v)-\frac{1}{v^2}A_T(xv,1)\right)+B(x,v)-x^2vA_T(xv,1)\label{eq241a1}\\
B(x,v)&=\frac{x}{1-v}\left(vB(x,v)-\frac{1}{v^2}B(xv,1)\right)+x^3vA_T(xv,1)\nonumber\\
&+\frac{x^2(1+v)}{v}(A_T(xv,1)-1)+x^2vA_T(xv,1).\label{eq241a2}\end{aligned}$$ By multiplying by $v^2(1-v)^2\left(1-\frac{xv}{1-v}\right)$ and by $v^2(1-v)^2$, then adding the results, we obtain $$\begin{aligned}
K(x,v)A_T(x/v;v)&=\frac{x(v^3(1-x-v)-x(1-v)(1-v+xv))}{v}A_T(x;1)+\frac{x(1-v)}{v}B(x;1)\\
&+\frac{(1-v)(x^2(1-v^2)-v^3(1-x-v))}{v},\end{aligned}$$ where $K(x,v)=xv(1-v^2)+x^2(1-3v+v^2)-v^2(1-v)^2$.
To solve the preceding functional equation, we apply the kernel method (see, e.g., [@HM] for an exposition) and take $$\begin{aligned}
v&=v_-=\frac{2+(\sqrt{5}-1)x+\sqrt{4-12x+6x^2-4\sqrt{5}x-2\sqrt{5}x^2}}{4},\,\mbox{and}\\
v&=v_+=\frac{2-(\sqrt{5}+1)x+\sqrt{4-12x+6x^2+4\sqrt{5}x+2\sqrt{5}x^2}}{4},\\\end{aligned}$$ which satisfies $K(x;v_+)=K(x,v_-)=0$. Hence, since $F_T(x)=A_T(x,1)$, we obtain the following result.
\[th241a\] Let $T=\{1324,1243,1234\}$. Then $F_T(x)$ is given by $$\begin{aligned}
\frac{(v_--1)(v_+-1)\,\big((v_-+v_+)(v_-^2+v_+^2-x^2)+(x-1)(v_-^2+v_+^2+v_-v_+)\big)}{x\,\big(x-(v_--1)(v_+-1)(v_-^2+v_+^2+v_-v_++x(v_-+v_++2-x))\big)}\,.\end{aligned}$$
**[Acknowledgement]{}**
We thank Richard Mather for pointing out some inaccuracies in the previous version.
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[|l|c|c|]{}
\
\
No. &$T$&$F_T(x)$\
\
No. & $T$&$F_T(x)$\
\
1&$\{4321,3412,1234\}$&$73x^9+ 199x^8+ 240x^7+ 162x^6+ 69x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
2&$\{4321,3142,1234\}$&$85x^9+ 221x^8+ 252x^7+ 164x^6+ 69x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
3&$\{2143,4312,1234\}$&$\frac{18x^7+31x^6+22x^5+8x^4+2x^3+2x^2-2x+1}{(1-x)^{3}}$\
4&$\{4231,2143,1234\}$&$\frac{2x^{10}-6x^9+6x^8+4x^7+4x^6+8x^5+6x^4-4x^3+7x^2-4x+1}{(1-x)^{5}}$\
5&$\{2143,3412,1234\}$&$\frac{2x^5+10x^4-11x^3+11x^2-5x+1}{(1-x)^6}$\
7&$\{3421,4312,1234\}$&$\frac{-9x^7+24x^6+23x^5+8x^4+2x^3+2x^2-2x+1}{(1-x)^3}$\
8&$\{2431,4213,1234\}$&$\frac{26+21x+15x^2}{2(1-x-x^2-x^3)}+\frac{4x^{10}(1+3x)-62x^9-28x^8+66x^7+27x^6-15x^5-53x^4+41x^3+51x^2-75x+24}{2(x-1)^3(1-x-x^2)^2}$\
9&$\{2134,4312,1243\}$&$\frac{-3x^7-5x^6+3x^5+10x^4-11x^3+11x^2-5x+1}{(1-x)^6}$\
10&$\{4213,1432,1234\}$&$\frac{2x^{11}+4x^{10}+10x^9+12x^8+6x^7-19x^6-19x^5-7x^4-x^3+2x-1}{(x-1)(x^5+3x^4+2x^3+x^2+x-1)(x^3+x^2+x-1)}$\
11&$\{4231,1432,1234\}$&$\frac{5x^9-2x^8-x^7+9x^6+9x^5+6x^4-4x^3+7x^2-4x+1}{(1-x)^5}$\
12&$\{2341,4312,1324\}$&$\frac {x^{10}-4x^9+3x^8+ 5x^7-7x^5+ 21x^4- 22x^3+ 16x^2- 6x+ 1}{(1-x)^{7}}$\
13&$\{3214,1432,1234\}$&$-\frac{x^5+x^3+x^2+x-1}{x^{12}+16x^{11}+10x^{10}+17x^9+25x^8+25x^7-7x^6-14x^5-5x^4-2x^3-x^2-2x+1}$\
14&$\{4231,2134,1243\}$&$\frac{4x^9-11x^8+10x^7+2x^6-7x^5+21x^4-22x^3+16x^2-6x+1}{(1-x)^7}$\
16&$\{2314,1432,4123\}$&$\frac{2x^9-x^8+x^7+3x^6+6x^5-6x^4+11x^3-13x^2+6x-1}{(x^2-3x+1)(x^3+x^2+x-1)(1-x)^3}$\
17&$\{2341,2143,4123\}$&$\frac{x^7- 13x^5+ 25x^4 - 29x^3+ 20x^2- 7x + 1}{(x^2- 3x + 1)(1-x)^5}$\
18&$\{2341,1432,4123\}$&$\frac{x^{10}-7x^9+19x^8-25x^7+12x^6+10x^5-20x^4+25x^3-19x^2+7x-1}{(x^2+1)(x^2-3x+1)(x^3-x^{2}-2x+1)(x-1)^3}$\
19&$\{2431,4312,1234\}$&$\frac{6x^9- 7x^8- 7x^7 + 4x^6+ 10x^5+ 6x^4- 4x^3+ 7x^2- 4x+ 1}{(1-x)^5}$\
20&$\{4312,1432,1234\}$&$\frac{(x + 1)(2x^9-18x^8+33x^7-20x^6+12x^5-22x^4+16x^3-12x^2+5x-1)}{(x-1)^5}$\
21&$\{4312,3142,1234\}$&$\frac{2x^9- 3x^8- 2x^{6} - 6x^5+ 21x^4- 22x^3+ 16x^2- 6x + 1}{(1-x)^7}$\
22&$\{2134,4312,1432\}$&$\frac {x^6+ 6x^5- 21x^4+22x^3- 16x^2+ 6x - 1}{(x - 1)^7}$\
23&$\{2431,4132,1234\}$&$\frac{1-2x}{x^2-3x+1}+\frac{(24x^9-116x^8+213x^7-158x^6+9x^5+37x^4-9x^3+x^2-3x+1)x^3}{(x-1)^5(2x-1)^3}$\
24&$\{4231,3412,1234\}$&$\frac{2x^6- 6x^5+ 21x^4- 22x^3+ 16x^2- 6x + 1}{(1-x)^7}$\
25&$\{3412,4132,1234\}$&$\frac{3x^6- 6x^5+ 21x^4- 22x^3+ 16x^2- 6x + 1}{(1-x)^7}$\
26&$\{2134,4312,1342\}$&$-\frac {x^8- 9x^6+ 27x^5- 43x^4+ 38x^3- 22x^2+ 7x - 1}{(1-x)^8}$\
27&$\{2314,4312,1432\}$&$-\frac{3x^9-x^8-18x^7+17x^6+15x^5-44x^4+47x^3-27x^2+8x-1}{(2x-1)(x^2+x-1)(x-1)^6}$\
28&$\{4231,3142,1234\}$&$\frac{2x^8- 10x^7+ 40x^6- 70x^5+ 81x^4- 60x^3+ 29x^2- 8x + 1}{(1-x)^9}$\
31&$\{2314,4312,1342\}$&$\frac{5x^{10}-22x^9+12x^8+89x^7-249x^6+354x^5-316x^4+179x^3-62x^2+12x-1}{(x^2-3x+1)(2x-1)^3(x-1)^4}$\
32&$\{2134,1432,4123\}$&$\frac{x^{10}-4x^9+4x^8-x^6-5x^5+6x^4-11x^3+13x^2-6x+1}{(x^2-3x+1)(x^3+x^2+x-1)(x-1)^3}$\
33&$\{2134,3412,4132\}$&$\frac {2x^7- 16x^5+ 36x^4 - 42x^3+ 26x^2- 8x + 1}{(2x - 1)^{3}(x - 1)^3}$\
34&$\{2143,4132,1234\}$&$-\frac{x^9-2x^8-x^7+4x^6-x^5-2x^4+3x^3-8x^2+5x-1}{(x^2-3x+1)(x-1)(2x-1)}$\
36&$\{3412,3124,1432\}$&$-\frac{x^8+2x^7-26x^6+62x^5-83x^4+69x^3-34x^2+9x-1}{(x-1)^{5}(x^2-3x+1)(2x-1)}$\
37&$\{3142,1432,1234\}$&$\frac {(x^3- 2x^2+ 3x - 1)^2}{x^8- x^7+ 4x^6- 7x^5+ 19x^4- 24x^3 + 18x^2- 7x+1}$\
38&$\{4321,1423,1234\}$&$147x^9+ 359x^8+ 367x^7+ 198x^6+72x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
39&$\{4321,4123,1234\}$&$185x^9+ 400x^8+ 396x^7+ 205x^6+72x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
40&$\{2341,4312,1234\}$&$\frac{x^9- 5x^8+ 6x^7+ x^6+ 5x^5- 21x^4+ 22x^3- 16x^2+ 6x - 1}{(x - 1)^7}$\
41&$\{4312,1342,1234\}$&$-\frac {2x^7- 8x^6+ 26x^5- 43x^4+ 38x^3- 22x^2+ 7x - 1}{(x - 1)^8}$\
42&$\{2341,4132,1234\}$&$\frac{4x^8- 5x^7- 7x^{6} - 7x^5+ 22x^4- 28x^3+ 20x^2- 7x + 1}{(2x-1)^2(x-1)^4}$\
43&$\{2314,4213,1432\}$&$-\frac{9x^6-35x^5+54x^4-49x^3+27x^2-8x+1}{(3x^3-5x^2+4x-1)(2x-1)(x-1)^3}$\
44&$\{4213,1342,1234\}$&$\frac{x^{10}-6x^9+9x^8+9x^7-54x^6+94x^5-104x^4+76x^3-35x^2+9x-1}{(x^3-2x^2+3x-1)(2x-1)(x-1)^5}$\
45&$\{4213,2134,1432\}$&$\frac{x^{10}-2x^9-x^8-13x^7+54x^6-99x^5+108x^4-77x^3+35x^2-9x+1}{(x-1)^{2}(3x^3-5x^2+4x-1)^2}$\
46&$\{2341,4132,1324\}$&$\frac{2x^7+5x^6-3x^{5}+3x^4+6x^3-12x^2+6x-1}{(x-1)(2x-1)(x^2-3x+1)(x^2+x-1)}$\
47&$\{2413,4132,1234\}$&$-\frac{3x^6-21x^5+40x^4-43x^3+26x^2-8x+1}{(2x-1)(x-1)^{4}(x^2-3x+1)}$\
48&$\{4312,3124,1342\}$&$-\frac{x^9-15x^8+73x^7-175x^6+247x^5-228x^4+138x^3-52x^2+11x-1}{(x^2-3x+1)^{2}(x-1)^6}$\
51&$\{4213,3124,1432\}$&$\frac{x^6- 7x^4+ 12x^3-13x^2+ 6x - 1}{(x^2- 3x + 1)(3x^3- 5x^2+4x - 1)}$\
52&$\{1432,4123,1234\}$&$-\frac{x^8- 4x^7+ 3x^6+ 4x^5- 11x^4+ 20x^3- 18x^2+ 7x - 1}{(x-1)^2(x^2-3x+1)^2}$\
53&$\{2134,4132,1243\}$&$\frac{x^{10}-4x^9-6x^8+68x^7-186x^6+291x^5-283x^4+170x^3-61x^2+12x-1}{(2x-1)^2(x^2-3x+1)^2(x-1)^3}$\
54&$\{3124,1432,1234\}$&$\frac{(1-x)^3(2x^3-2x^2+3x-1)}{2x^9-7x^8+7x^7-10x^6+16x^5-27x^4+29x^3-19x^2+7x-1}$\
57&$\{2143,1432,1234\}$&$\frac {x^7+ x^6- x^5+ 3x^3+ 2x^2+ 2x - 1}{x^7+ x^6- x^5- x^4+ 2x^3+x^2+3x-1}$\
58&$\{4321,1243,1234\}$&$144x^9+ 396x^8+ 382x^7+ 202x^6+73x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
59&$\{4321,1324,1234\}$&$334x^9+ 669x^8+ 484x^7+ 215x^6+73x^5+ 21x^4+ 6x^3+ 2x^2+ x + 1$\
60&$\{4312,4132,1234\}$&$\frac{x^7+16x^6+12x^5+6x^4-4x^3+7x^2-4x+1}{(1-x)^5}$\
61&$\{4312,1243,1234\}$&$\frac{x^{10} -4x^9+ 3x^8+2x^7+ x^6+ 4x^5- 21x^4+ 22x^3- 16x^{2}+ 6x - 1}{(x-1)^7}$\
62&$\{4231,4312,1234\}$&$\frac{3x^8- 8x^7+ 4x^6 - 4x^5+ 21x^4- 22x^3+ 16x^2- 6x + 1}{(1-x)^7}$\
63&$\{4312,1324,1234\}$&$\frac{x^{10}-5x^9+ 6x^8+2x^7- 5x^6+ 4x^5- 21x^4+ 22x^3- 16x^2 + 6x -1}{(x-1)^7}$\
64&$\{4312,3412,1234\}$&$\frac {3x^7+ 5x^6- 4x^5 + 21x^4- 22x^3+ 16x^2- 6x + 1}{(1-x)^7}$\
65&$\{4213,4132,1234\}$&$-\frac{3x^8+5x^7+13x^6+7x^5+2x^4+x^3+5x^2-4x+1}{(x^2+x-1)(x^3+x^2+x-1)(x-1)^3}$\
66&$\{4231,4132,1234\}$&$\frac{2x^7+8x^6-4x^5+21x^4-22x^3+16x^2-6x+1}{(1-x)^7}$\
67&$\{4312,1324,1243\}$&$\frac{2x^{10}-7x^8+65x^7-187x^6+274x^5-248x^4+145x^3-53x^2+11x-1}{(x-1)^6(2x-1)^3}$\
68&$\{4312,1342,1243\}$&$\frac{3x^7-4x^6-14x^5+36x^4-42x^3+26x^2-8x+1}{(x-1)^3(2x-1)^3}$\
70&$\{4312,3124,1243\}$&$-\frac{11x^7-62x^6+128x^5-146x^4+102x^3-43x^2+10x-1}{(2x-1)^3(x-1)^5}$\
71&$\{4231,1243,1234\}$&$-\frac{4x^8-2x^7-17x^6+25x^5-43x^4+38x^3-22x^2+7x-1}{(x-1)^8}$\
73&$\{4231,1324,1234\}$&$-\frac{x^{10}-15x^8+55x^7-111x^6+149x^5-141x^4+89x^3-37x^2+9x-1}{(x-1)^{10}}$\
79&$\{2134,4132,1234\}$&$\frac{2x^{11}+x^{10}-10x^9-9x^8+12x^7+17x^6-30x^5+2x^4+28x^3-24x^2+8x-1}{(x^2+2x-1)(2x-1)(x-1)^3(x^2+x-1)^2}$\
81&$\{2431,4312,1324\}$&$\frac{145x^3+11x-1-248x^4-193x^6+274x^5-53x^2-x^9-13x^8+80x^7)}{(2x-1)^3(x-1)^6}$\
82&$\{4312,3142,1243\}$&$\frac{x^7+2x^6-27x^5+59x^4-61x^3+33x^2-9x+1}{(x-1)^{2}(2x-1)^4}$\
83&$\{4312,3412,1243\}$&$\frac{x^7+2x^6+4x^5-23x^4+36x^3-25x^2+8x-1}{(x-1)(2x-1)^4}$\
85&$\{2314,4132,1432\}$&$-\frac{x^5+5x^4-11x^3+13x^2-6x+1}{(2x-1)(x^2-3x+1)(x-1)^2}$\
87&$\{4312,3124,1432\}$&$-\frac{2x^9-46x^7+143x^6-226x^5+221x^4-137x^3+52x^2-11x+1}{(x-1)^5(x^2-3x+1)(2x-1)^2}$\
89&$\{3142,4132,1234\}$&$-\frac{4x^5-16x^4+24x^3-19x^2+7x-1}{(2x-1)(x^2-3x+1)(x-1)^3}$\
91&$\{4213,1342,1243\}$&$-\frac{4x^5-14x^4+17x^3-14x^2+6x-1}{(3x-1)(x^2-x+1)(x-1)^3}$\
92&$\{2314,3124,1432\}$&$\frac{(x^3-2x^2+3x-1)(x^2+x-1)(1-x)^3}{x^9-2x^8+6x^7-4x^6-7x^5+32x^4-40x^3+25x^2-8x+1}$\
95&$\{2314,4132,1342\}$&$-\frac{4x^6-25x^5+51x^4-56x^3+32x^2-9x+1}{(2x-1)(x-1)^2(x^2-3x+1)^2}$\
96&$\{2134,4132,1342\}$&$-\frac{4x^8+6x^7-45x^6+100x^5-126x^4+95x^3-42x^2+10x-1}{(2x-1)^2(x^2-3x+1)(x-1)^4}$\
97&$\{2341,4312,4123\}$&$-\frac{(x-1)^4(x^3-2x^2+3x-1)}{x^8-4x^7+18x^6-35x^5+51x^4-47x^3+26x^2-8x+1}$\
98&$\{2134,3124,1432\}$&$\frac{(x-1)^{3}(x^3+2x-1)}{4x^6-7x^5+9x^4-15x^3+13x^2-6x+1}$\
100&$\{4312,1342,4123\}$&$-\frac{4x^6-16x^5+30x^4-31x^3+20x^2-7x+1}{(x-1)^3(2x^4-7x^3+8x^2-5x+1)}$\
101&$\{3124,4132,1342\}$&$-\frac{4x^6-16x^5+30x^4-31x^3+20x^2-7x+1}{(x-1)^3(2x^4-7x^3+8x^2-5x+1)}$\
102&$\{2413,3142,1234\}$&$-\frac{(x-1)^{3}(x^3-2x^2+3x-1)}{x^7-4x^6+12x^5-23x^4+28x^3-19x^2+7x-1}$\
104&$\{2134,4132,1423\}$&$-\frac{3x^7-4x^6+17x^5-46x^4+55x^3-32x^2+9x-1}{(x-1)(x^2-3x+1)(2x-1)^3}$\
105&$\{4213,2134,1342\}$&$-\frac{7x^6-25x^5+51x^4-56x^3+32x^2-9x+1}{(2x-1)(x-1)^2(x^2-3x+1)^2}$\
107&$\{4213,3412,1342\}$&$-\frac{(x^2-x+1)(2x-1)^3}{(4x^3-7x^2+5x-1)(x-1)^3}$\
110&$\{2134,3142,1432\}$&$-\frac{(x-1)(3x^3-5x^2+4x-1)^2}{x^9+2x^8-27x^7+86x^6-144x^5+150x^4-100x^3+42x^2-10x+1}$\
111&$\{2143,3142,1234\}$&$\frac{(x^3-2x^2+3x-1)^2}{(2x^3-3x^2+4x-1)(x-1)^3}$\
113&$\{2134,1432,1234\}$&$\frac{2x^5-x^4-3x^3-2x^2-2x+1}{2x^5-2x^3-x^2-3x+1}$\
114&$\{4312,1423,1234\}$&$-\frac{x^{10}-3x^9+2x^8+4x^7-9x^6+24x^5-43x^4+38x^3-22x^2+7x-1}{(x-1)^8}$\
115&$\{4231,1423,1234\}$&$-\frac{2x^{10}-17x^9+66x^8-158x^7+256x^6-289x^5+230x^4-126x^3+46x^2-10x+1}{(x-1)^{11}}$\
116&$\{4312,4123,1243\}$&$\frac{x^9-23x^8+133x^7-315x^6+419x^5-350x^4+188x^3-63x^2+12x-1}{(2x-1)^4(x-1)^5}$\
117&$\{3124,4132,1234\}$&$\frac{x^9-6x^8+22x^7-53x^6+92x^5-104x^4+76x^3-35x^2+9x-1}{(x^3-2x^2+3x-1)(2x-1)(x-1)^5}$\
119&$\{4312,1432,1324\}$&$\frac{-350x^4-63x^2+419x^5-26x^8+138x^7-317x^6+188x^3+x^9-1+12x}{(2x-1)^4(x-1)^5}$\
120&$\{4132,1423,1234\}$&$-\frac{x^8+4x^7-41x^6+99x^5-126x^4+95x^3-42x^2+10x-1}{(x^2-3x+1)(2x-1)^2(x-1)^4}$\
122&$\{4213,1432,1324\}$&$-\frac{5{x}^{5}-19{x}^{4}+25{x}^{3}-19{x}^{2}+7x-1}{(x-1)({x}^{2}-3x+1)(3{x}^{3}-5{x}^{2}+4x-1)}$\
123&$\{4132,1342,1234\}$&$-\frac{3x^8-46x^7+141x^6-225x^5+221x^4-137x^3+52x^2-11x+1}{(x^2-3x+1)(2x-1)^{2}(x-1)^5}$\
124&$\{2341,4132,4123\}$&$-\frac{(2x-1)(x-1)^4}{2x^6-8x^5+19x^4-27x^3+19x^2-7x+1}$\
128&$\{2341,3142,4123\}$&$\frac{\left({x}^{2}-3x+1\right)\left(x-1\right)^{5}}{4{x}^{7}-23{x}^{6}+55{x}^{5}-78{x}^{4}+66{x}^{3}-33{x}^{2}+9x-1}$\
135&$\{1432,4123,1243\}$&$-\frac{5x^5-14x^4+22x^3-18x^2+7x-1}{(x-1)^4(2x^2-4x+1)}$\
136&$\{4213,1342,4123\}$&$\frac{x^5-3x^3+4x^2-4x+1}{x^5+x^4-6x^3+7x^2-5x+1}$\
137&$\{3124,1432,1342\}$&$-\frac{(x^2-3x+1)(x^2+2x-1)}{(x-1)(x^4-2x^3-5x^2+5x-1)}$\
138&$\{2134,3142,1243\}$&$-\frac{(x^2-3x+1)(x^2+2x-1)}{(x-1)(x^4-2x^3-5x^2+5x-1)}$\
139&$\{2143,3124,1342\}$&$-\frac{(x^2-3x+1)(x^2+2x-1)}{(x-1)(x^4-2x^3-5x^2+5x-1)}$\
140&$\{3124,1432,1243\}$&$\frac{(3x-1)(x-1)^3}{9x^4-19x^3+17x^2-7x+1}$\
141&$\{2143,1423,1234\}$&$\frac{2x^4-4x^3+7x^2-5x+1}{4x^4-9x^3+11x^2-6x+1}$\
142&$\{1432,1342,4123\}$&$\frac{x^3+3x-1}{x^3-2x^2+4x-1}$\
143&$\{4312,4123,1234\}$&$-\frac{x^8-3x^7-12x^6+23x^5-43x^4+38x^3-22x^2+7x-1}{(x-1)^8}$\
144&$\{4231,4123,1234\}$&$-\frac{3x^8-15x^7+40x^6-66x^5+81x^4-60x^3+29x^2-8x+1}{(x-1)^9}$\
145&$\{4312,1423,1243\}$&$\frac{2x^7-2x^6-25x^5+59x^4-61x^3+33x^2-9x+1}{(x-1)^2(2x-1)^4}$\
146&$\{4132,1243,1234\}$&$\frac{x^9-4x^8+20x^6-58x^5+83x^4-69x^3+34x^2-9x+1}{(2x-1)(x^2-3x+1)(x-1)^5}$\
147&$\{4132,1324,1234\}$&$-\frac{13x^{10}-45x^9+83x^8-38x^7-141x^6+308x^5-306x^4+178x^3-62x^2+12x-1}{(x^2-3x+1)(x^2+x-1)(2x-1)^2(x-1)^5}$\
148&$\{2134,4132,1324\}$&$-\frac{5x^8-51x^7+172x^6-288x^5+283x^4-170x^3+61x^2-12x+1}{(2x-1)^2(x^2-3x+1)^2(x-1)^3}$\
152&$\{4231,2341,4123\}$&$\frac{(x-1)^{6}(x^2-3x+1)}{5x^8-31x^7+83x^6-134x^5+144x^4-99x^3+42x^2-10x+1}$\
154&$\{4312,1342,1423\}$&$-\frac{3x^5-14x^4+21x^3-18x^2+7x-1}{(x-1)(2x^3-4x^2+4x-1)(x^2-3x+1)}$\
155&$\{3124,4132,1243\}$&$-\frac{3x^5-14x^4+21x^3-18x^2+7x-1}{(x-1)(2x^3-4x^2+4x-1)(x^2-3x+1)}$\
160&$\{4312,1432,1342\}$&$\frac{2x^5-4x^4-10x^3+16x^2-7x+1}{(x-1)(3x-1)(2x-1)(x^2+2x-1)}$\
161&$\{4312,4132,1342\}$&$-\frac{7x^5-22x^4+33x^3-24x^2+8x-1}{(x^3-3x^2+4x-1)(x-1)(2x-1)^2}$\
167&$\{3142,3124,1432\}$&$-\frac{(2x-1)(x-1)(x^2-3x+1)}{x^5-7x^4+18x^3-17x^2+7x-1}$\
168&$\{3124,1432,1423\}$&$-\frac{(x^2-3x+1)(2x-1)^{2}}{(x-1)(x^4-13x^3+16x^2-7x+1)}$\
169&$\{3142,1423,1234\}$&$-\frac{(x^2-3x+1)(2x-1)^2}{(x-1)(x^4-13x^3+16x^2-7x+1)}$\
179&$\{2134,1432,1423\}$&$-\frac{2x^5-8x^4+12x^3-12x^2+6x-1}{(x^4-5x^3+10x^2-6x+1)(x^2-x+1)}$\
181&$\{2143,1324,1234\}$&$-\frac{2x^3+3x-1}{x^4-2x^3+2x^2-4x+1}$\
183&$\{4132,4123,1234\}$&$\frac{x^8-8x^7+31x^{6}-75x^5+98x^4-75x^3+35x^2-9x+1}{(2x-1)^2(x-1)^6}$\
186&$\{4132,4123,1243\}$&$-\frac{-27{x}^{5}+55{x}^{4}-57{x}^{3}+32{x}^{2}-9x+1+4{x}^{6}}{\left(3x-1\right)\left({x}^{2}-3x+1\right)\left(x-1\right)^{4}}$\
189&$\{2143,2134,1432\}$&$-\frac{x^4-7x^3+8x^2-5x+1}{x^5-5x^4+13x^3-12x^2+6x-1}$\
200&$\{2143,3124,1243\}$&$\frac{{x}^{3}-6{x}^{2}+5x-1}{\left(x-1\right)\left(5{x}^{2}-5x+1\right)}$\
202&$\{1432,1423,1234\}$&$\frac{x^4-4x^3+10x^{2}-6x+1}{3x^4-11x^3+15x^2-7x+1}$\
205&$\{1432,1324,1234\}$&$-\frac{x^7-2x^6+4x^5-17x^4+24x^3-18x^2+7x-1}{2x^6-14x^5+34x^4-38x^3+24x^2-8x+1}$\
206&$\{1432,1243,1234\}$&$-\frac{x^6+5x^4-12x^3+12x^2-6x+1}{2x^5-13x^4+21x^3-17x^2+7x-1}$\
|
---
author:
- Johannes Broedel
- Claude Duhr
- Falko Dulat
- Brenda Penante
- Lorenzo Tancredi
bibliography:
- 'bib.bib'
title: Elliptic polylogarithms and Feynman parameter integrals
---
‘=11 =manfnt @tchout\[\#1\][[tempcnta\#1 whilenumtempcnta>@]{}]{} @tchout dubious\[\#1\]
tempboxa tempdimatempboxa
W@tchout\#1[@tchout\[\#1\]]{} ‘=12
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank the Mainz Institute for Theoretical Physics (MITP) in the context of the workshop “High Time for Higher Orders: From Amplitudes to Phenomenology” and the Galileo Galilei Institute in Florence in the context of the workshop “Amplitudes in the LHC era” for their hospitality and partial support during different phases of this work. This work was completed at the ETH Institute for Theoretical Studies in the framework of the program “Periods, modular forms and scattering amplitudes”. We thank the institute for its hospitality. This research was supported by the ERC grant 637019 “MathAm”, and the U.S. Department of Energy (DOE) under contract DE-AC02-76SF00515.
|
---
abstract: 'In the first part of this note, we review and compare various instances of the notion of *twisted coefficient system*, a.k.a. *polynomial functor*, appearing in the literature. This notion hinges on how one defines the *degree* of a functor from ${\mathcal{C}}$ to an abelian category, for various different structures on ${\mathcal{C}}$. In the second part, we focus on twisted coefficient systems defined on *partial braid categories*, and explain a functorial framework for this setting.'
author:
- 'Martin Palmer$/\!\!/$23 February 2019'
bibliography:
- 'acotcs.bib'
title: '**A comparison of twisted coefficient systems**'
---
Introduction {#sec:introduction}
============
This note is concerned with *twisted coefficient systems*, by which we mean simply functors from a category ${\mathcal{C}}$ equipped with a certain structure to an abelian category ${\mathcal{A}}$. The structure on ${\mathcal{C}}$ depends on the precise situation that one wishes to study, and is used to define a notion of *degree* for any functor ${\mathcal{C}}\to {\mathcal{A}}$. The main goal of this note is to compare different structures on the source category ${\mathcal{C}}$, and the resulting notions of degree.
Twisted coefficient systems of *finite degree*, also known as *polynomial functors*, are often used to study the homology of interesting spaces or groups (or other abelian invariants, such as the filtration quotients in the lower central series of a group), for example automorphism groups of free groups and congruence groups ([*cf*. ]{}[@DjamentVespa2019FoncteursFaiblementPolynomiaux §5]). Indeed, polynomial functors first appeared in the paper [@EilenbergMac1954groupsHnII] of Eilenberg and MacLane (see §9), where they were used to compute the homology, in a certain range of degrees, of the Eilenberg-MacLane spaces $K(A,n)$ for $n{\geqslant}2$. This involves assembling the objects of interest (e.g. the homology of congruence subgroups) into a polynomial functor, and leveraging the fact that it has finite degree to study them.
On the other hand, polynomial functors may also appear as the *coefficients* in homology groups: one may also be interested in the homology of a family of spaces or groups, with respect to a corresponding family of local coefficient systems *that assemble into a polynomial functor* – it is in this guise that polynomial functors are more commonly referred to as *twisted coefficient systems* (of finite degree). Many families of groups or spaces are known to be *homologially stable* with respect to (appropriately-defined) finite-degree twisted coefficient systems, including the symmetric groups, braid groups, configuration spaces, general linear groups, automorphism groups of right-angled Artin groups and mapping class groups of surfaces and of $3$-manifolds.[^1]
#### Cross-effects vs. endofunctors. {#cross-effects-vs.endofunctors. .unnumbered}
There are two common approaches to defining the *degree* of a functor ${\mathcal{C}}\to {\mathcal{A}}$. One approach uses certain structure on ${\mathcal{C}}$ to define the *cross-effects* of a given functor, which consists of an ${\mathbb{N}}$-graded set of objects of ${\mathcal{A}}$, and the degree is then determined by the vanishing of these objects. The idea is that the cross-effects encode information about the functor; if they vanish above a certain level, then this information is concentrated in an essentially finite amount of data, which makes it possible to prove certain things about the given functor (such as homological stability with coefficients in this functor). For example, this is the approach taken in the paper [@Palmer2018Twistedhomologicalstability], from which this note arose, to prove homological stability for configuration spaces with finite-degree twisted coefficients.
The second approach is a recursive definition, depending on the choice of an endofunctor $s$ of ${\mathcal{C}}$ and a natural transformation $\mathrm{id} \to s$. This allows one to prove things about functors of finite degree (in this sense) by induction on the degree. For example, the main theorem of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], providing a “machine” for proving twisted homological stability for families of groups, is an inductive proof of this kind, and the degree of their twisted coefficient systems is defined recursively.
#### Degree and height. {#degree-and-height. .unnumbered}
In this note, we will use the word *degree* to refer to a definition of the second kind, i.e., a recursive definition, and we will use the word *height* to refer to a definition of the first kind, given by the vanishing of certain *cross-effects* above a certain “height”.
In sections \[sec:inductive-degree\] and \[sec:cross-effects\] (respectively) we compare various notions of *degree* and *height* appearing in the literature. We will focus on comparisons between [@DjamentVespa2019FoncteursFaiblementPolynomiaux], [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], [@Ivanov1993homologystabilityTeichmuller], [@CohenMadsen2009Surfacesinbackground], [@Boldsen2012Improvedhomologicalstability] and [@Palmer2018Twistedhomologicalstability] for the degree, and between [@DjamentVespa2019FoncteursFaiblementPolynomiaux], [@HartlVespa2011Quadraticfunctorspointed], [@HartlPirashviliVespa2015Polynomialfunctorsalgebras], [@CollinetDjamentGriffin2013Stabilitehomologiquepour] and [@Palmer2018Twistedhomologicalstability] for the height. We do not pursue here the relationship between the height and the degree of a functor ${\mathcal{C}}\to {\mathcal{A}}$ (when both are defined) in the greatest possible generality (although this is discussed in certain special cases, see Remark \[rmk:summary\] for a summary). Rather, we focus on comparing (and unifying) *with each other* the various different notions of *degree* appearing in the literature, and similarly for the various different notions of *height* in the literature.
#### Historical remarks. {#historical-remarks. .unnumbered}
The notion of what we term the *height* of a functor was first introduced by Eilenberg and MacLane (who used the name *degree*) in [@EilenbergMac1954groupsHnII §9], where it was used to compute the integral homology of Eilenberg-MacLane spaces in a range of degrees. Somewhat later, it was used by Dwyer [@Dwyer1980Twistedhomologicalstability] to formulate and prove a twisted homological stability theorem for general linear groups. The *height* of a functor also appears in [@Pirashvili2000DoldKantype] (see §2.3) and was used in [@Betley2002Twistedhomologyof] to prove a twisted homological stability theorem for the symmetric groups. More recently, it also appears in [@HartlVespa2011Quadraticfunctorspointed], [@HartlPirashviliVespa2015Polynomialfunctorsalgebras], [@CollinetDjamentGriffin2013Stabilitehomologiquepour] (in which it is used to prove a homological stability theorem for automorphism groups of free products of groups) and [@DjamentVespa2019FoncteursFaiblementPolynomiaux] (which also introduces the notion of *weak* polynomial functors, in contrast to *strong* polynomial functors – see also Definition \[def:degree-general\] below, which is inspired by their definition).
The notion of what we term the *degree* of a functor appeared first (as far as the author is aware) implicitly in the work of Dwyer [@Dwyer1980Twistedhomologicalstability] and (slightly later) explicitly in the work of van der Kallen [@Kallen1980Homologystabilitylinear] (see §5.5).[^2] This notion was also used by Ivanov [@Ivanov1993homologystabilityTeichmuller] to formulate and prove a twisted homological stability theorem for mapping class groups, and analogous (not quite identical) definitions were used also by [@CohenMadsen2009Surfacesinbackground] and [@Boldsen2012Improvedhomologicalstability] in similar contexts ([*cf*. ]{}§\[para:degree-mcg\]). The notion also appears in the work of Djament and Vespa [@DjamentVespa2019FoncteursFaiblementPolynomiaux], who use both *degree*-like ([*cf*. ]{}D[é]{}finitions 1.5 and 1.22) and *height*-like ([*cf*. ]{}Proposition 2.3) descriptions of their polynomial functors. It is also used by Randal-Williams and Wahl [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] in their general framework for proving twisted homological stability theorems for sequences of groups.
#### Outline. {#outline. .unnumbered}
In section \[sec:inductive-degree\] we describe a general framework for defining the *degree* of a functor ${\mathcal{C}}\to {\mathcal{A}}$ using the structure of an endofunctor $s$ of ${\mathcal{C}}$ together with a natural transformation $\mathrm{id} \to s$ (more generally, a collection of such data), and specialise this to several settings in the literature, including symmetric monoidal categories (§\[para:degree-DV\]), labelled braid categories (§\[para:partial-braid-categories\]) and categories of decorated surfaces (§\[para:degree-mcg\]). In section \[sec:cross-effects\] we set up a general framework for defining the degree of a functor using cross-effects (which we call the *height* of the functor), and specialise this to various settings in the literature, including symmetric monoidal categories (§§\[para:specialise-DV\]–\[para:finite-coproducts\]), wreath products of categories (§\[para:specialise-CDG\]) and labelled braid categories (§\[para:specialise-this-paper\]). In sections \[sec:functorial-configuration-spaces\] and \[sec:representations-of-categories\] we consider the special case of labelled braid categories ${\mathcal{C}}= {\mathcal{B}}(M,X)$ in more detail, and describe a functorial (in $M$ and $X$) setting for the notions of degree and height of functors ${\mathcal{B}}(M,X) \to {\mathcal{A}}$.
Recursive degree {#sec:inductive-degree}
================
To relate different notions of *degree* in the literature, we use a notion of *category with stabilisers*, which is roughly a category ${\mathcal{C}}$ equipped with endofunctors $s_i \colon {\mathcal{C}}\to {\mathcal{C}}$ and natural transformations $\mathrm{id} \to s_i$. These are the objects of a category ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ (see Definition \[def:degree-general\]). There is then a natural notion of *degree* for any functor ${\mathcal{C}}\to {\mathcal{A}}$ with ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ and ${\mathcal{A}}$ an abelian category. There is a functor ${\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}}\to {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ which is compatible with the definition of [@DjamentVespa2019FoncteursFaiblementPolynomiaux] of the degree of a functor with source a monoidal category with initial unit object (§\[para:degree-DV\]). This construction also generalises to left modules over such a monoidal category (Remark \[rmk:modules-over-monoidal-categories\]). There is another functor ${\mathcal{B}}\colon {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\to {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, which we will define later in §\[sss:some-functors\],[^3] where ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ is a category whose objects are smooth manifolds-with-boundary equipped with a collar neighbourhood and a basepoint on the boundary. The operation of boundary connected sum of manifolds gives ${\mathcal{B}}({\mathbb{D}}^n)$ the structure of a monoidal category (with initial unit object) and ${\mathcal{B}}(M)$ the structure of a left module over it, so ${\mathcal{B}}(M)$ may also be viewed as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ using the previous construction. These two objects of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ are not equal, but nevertheless result in the same definition of the *degree* of a functor ${\mathcal{B}}(M) \to {\mathcal{A}}$ (this is Proposition \[p:two-degrees-agree\]). See §§\[para:gen-def-degree\]–\[para:partial-braid-categories\] for the details of this brief summary. In §§\[para:degree-WRW\] and \[para:degree-mcg\] we also discuss how the general Definition \[def:degree-general\] relates to the notion of degree used in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] and the notions of degree used in relation to mapping class groups. Throughout this section, ${\mathcal{A}}$ will denote a fixed abelian category.
A general definition. {#para:gen-def-degree}
---------------------
\[def:degree-general\] Let ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ be the category whose objects are small $1$-categories ${\mathcal{C}}$ equipped with a collection $\{s_i \colon {\mathcal{C}}\to {\mathcal{C}}\}_{i \in I}$ of endofunctors and natural transformations $\{\imath_i \colon \mathrm{id} \to s_i\}_{i \in I}$. We call such an object a *category with stabilisers*. A morphism in ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ from $({\mathcal{C}},I,s,\imath)$ to $({\mathcal{D}},J,t,\jmath)$ is a functor $f \colon {\mathcal{C}}\to {\mathcal{D}}$ together with a function $\sigma \colon I \to J$ and a collection of natural isomorphisms $\{ \psi_i \colon f \circ s_i \to t_{\sigma(i)} \circ f \}_{i \in I}$ such that $\jmath_{\sigma(i)} * \mathrm{id}_f = \psi_i \circ (\mathrm{id}_f * \imath_i)$ for all $i \in I$. We denote by ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ the full subcategory on objects where $\lvert I \rvert = 1$ (i.e., categories with just one stabiliser – we will restrict to this subcategory later, in §\[sec:functorial-configuration-spaces\]). It also contains $\mathsf{Cat}$ as the full subcategory on objects where $I = \varnothing$, but this will not be relevant for us.
We define the *degree* of functors from ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ to the abelian category ${\mathcal{A}}$ as follows. The function $\mathrm{deg} \colon \mathsf{Fun}({\mathcal{C}},{\mathcal{A}}) \to \{-1,0,1,\ldots,\infty\}$ is the largest function such that $\mathrm{deg}(0) = -1$ and such that for non-zero $T$ we have $\mathrm{deg}(T) {\leqslant}d$ if and only if $$\label{eq:degree-condition}
\mathrm{deg}(\mathrm{coker}(T\imath_i \colon T \to Ts_i)) {\leqslant}d-1,$$ for all $i$.
We may also vary the definition slightly and define the *split degree* to be the largest function $\mathrm{sdeg}\colon \mathsf{Fun}({\mathcal{C}},{\mathcal{A}}) \to \{-1,0,1,\ldots,\infty\}$ such that $\mathrm{sdeg}(0) = -1$ and such that for non-zero $T$ we have $\mathrm{sdeg}(T) {\leqslant}d$ if and only if $$\label{eq:split-degree-condition}
\begin{gathered}
T\imath_i \colon T \to Ts_i \text{ is a split monomorphism in } \mathsf{Fun}({\mathcal{C}},{\mathcal{A}}) \text{ and } \\ \mathrm{sdeg}(\mathrm{coker}(T\imath_i \colon T \to Ts_i)) {\leqslant}d-1,
\end{gathered}$$ for all $i$. In between these two definitions, there is the *injective degree* $\mathrm{ideg}(T)$, where the condition that $T\imath_i$ is a split monomorphism in $\mathsf{Fun}({\mathcal{C}},{\mathcal{A}})$ is weakened to the condition that $\mathrm{ker}(T\imath_i) = 0$.
Another variation of the definition is inspired by the notion of weak degree (*degr[é]{} faible*) introduced by Djament and Vespa [@DjamentVespa2019FoncteursFaiblementPolynomiaux]. Note that $\mathrm{ker}(T\imath_i \colon T \to Ts_i)$ is a subobject of $T$ in the abelian category $\mathsf{Fun}({\mathcal{C}},{\mathcal{A}})$ for all $i$, and therefore so is the sum $\sum_i \mathrm{ker}(T\imath_i \colon T \to Ts_i)$, which we denote by $\kappa(T)$, following the notation of [@DjamentVespa2019FoncteursFaiblementPolynomiaux]. We then define the *weak degree* to be the largest function $\mathrm{wdeg}\colon \mathsf{Fun}({\mathcal{C}},{\mathcal{A}}) \to \{-1,0,1,\ldots,\infty\}$ such that $$\mathrm{wdeg}(T) = -1 \qquad\text{if and only if}\qquad \kappa(T) = T,$$ and otherwise we have $\mathrm{wdeg}(T) {\leqslant}d$ if and only if $\mathrm{wdeg}(\mathrm{coker}(T\imath_i \colon T \to Ts_i)) {\leqslant}d-1$ for all $i$.
\[rmk:4-definitions-of-deg\] A simple inductive argument shows that $$\mathrm{wdeg}(T) {\leqslant}\mathrm{deg}(T) {\leqslant}\mathrm{ideg}(T) {\leqslant}\mathrm{sdeg}(T)$$ for all functors $T\colon {\mathcal{C}}\to {\mathcal{A}}$. Moreover, if ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ has the property that each $\imath_i$ has a left-inverse, i.e., a natural transformation $\pi_i \colon s_i \to \mathrm{id}$ such that $\pi_i \circ \imath_i = \mathrm{id}_{\mathrm{id}}$, then all four types of degree are equal for all functors $T \colon {\mathcal{C}}\to {\mathcal{A}}$.
\[rmk:gen-of-degree-under-composition\] In §\[sss:degree\] below we discuss the question of when the degree of a functor ${\mathcal{C}}\to {\mathcal{A}}$ is preserved under precomposition, in the setting where ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\subset {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$.[^4] That discussion extends easily to the setting of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, and also to the other variations of *degree* defined above, so, for completeness, we mention the general statement here. Let $f = (f,\sigma,\psi) \colon {\mathcal{C}}\to {\mathcal{D}}$ be a morphism in ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. Lemma \[l:degree-under-composition\] generalises to say that for any functor $T \colon {\mathcal{D}}\to {\mathcal{A}}$ we have ${\ensuremath{\mathsf{x}}}\mathrm{deg}(Tf) {\leqslant}{\ensuremath{\mathsf{x}}}\mathrm{deg}(T)$, with equality if $f$ is essentially surjective on objects and $\sigma$ is surjective, for ${\ensuremath{\mathsf{x}}}\in \{ \varnothing, \text{i}, \text{s} \}$. For the weak degree we have $\text{wdeg}(Tf) {\leqslant}\text{wdeg}(T)$ if $\sigma$ is surjective, and we have equality $\text{wdeg}(Tf) = \text{wdeg}(T)$ if $\sigma$ is bijective and $f$ is essentially surjective on objects. We may then generalise Definition \[def:braidable\] by saying that an object $({\mathcal{C}},I,s,\imath)$ of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ is *braidable* if there are certain natural isomorphisms $\Psi_i \colon s_i \circ s_i \to s_i \circ s_i$ for each $i \in I$. Corollary \[coro:braidable\] generalises exactly as stated to objects of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$.
\[rmk:degree-under-composition-DV\] The above remark, specialised to the setting of Djament and Vespa (see below) and with ${\ensuremath{\mathsf{x}}}= \varnothing$, recovers Proposition 1.7 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux]. With ${\ensuremath{\mathsf{x}}}= \text{w}$, it implies the analogous statement for the weak degree of functors from a monoidal category with initial unit object. In the notation of [@DjamentVespa2019FoncteursFaiblementPolynomiaux], this says that if $\alpha \colon {\mathcal{M}}\to {\mathcal{M}}^\prime$ is a strict monoidal functor between strict monoidal categories whose unit objects are initial, and if $\alpha$ is moreover surjective on objects, then it induces a functor $\mathcal{P}\mathit{ol}_n({\mathcal{M}}^\prime,{\mathcal{A}}) \to \mathcal{P}\mathit{ol}_n({\mathcal{M}},{\mathcal{A}})$.
Specialising to the setting of Djament and Vespa. {#para:degree-DV}
-------------------------------------------------
In the article [@DjamentVespa2019FoncteursFaiblementPolynomiaux], Djament and Vespa work with the category ${\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}}$ whose objects are small strict symmetric monoidal categories whose unit object is initial, and whose morphisms are strict monoidal functors. Now, one may define a functor $$\Psi \colon {\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}}\longrightarrow {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$$ as follows: the underlying category of $\Psi({\mathcal{M}})$ is just ${\mathcal{M}}$ and the indexing set for the collection of endofunctors is the set $\mathrm{ob}({\mathcal{M}})$ of objects of ${\mathcal{M}}$. For each $x \in \mathrm{ob}({\mathcal{M}})$, the endofunctor $s_x \colon {\mathcal{M}}\to {\mathcal{M}}$ is $x \oplus -$ and the natural transformation $\imath_x \colon \mathrm{id} \to s_x$ consists of the morphisms $i_x \oplus \mathrm{id}_y \colon y = 0 \oplus y \to x \oplus y$, where $i_x \colon 0 \to x$ is the unique morphism from the initial object $0$ to $x$. If $F \colon {\mathcal{M}}\to {\mathcal{N}}$ is a strict monoidal functor, then $\Psi(F) \colon \Psi({\mathcal{M}}) \to \Psi({\mathcal{N}})$ is simply the functor $F$, together with the function $\mathrm{ob}(F)$ from the indexing set $\mathrm{ob}({\mathcal{M}})$ of $\Psi({\mathcal{M}})$ to the indexing set $\mathrm{ob}({\mathcal{N}})$ of $\Psi({\mathcal{N}})$, and the natural isomorphisms are identities.
Given ${\mathcal{M}}\in \mathrm{ob}({\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}})$, an abelian category ${\mathcal{A}}$ and a functor $T \colon {\mathcal{M}}\to {\mathcal{A}}$, we may view ${\mathcal{M}}$ as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ via the functor $\Psi$, and therefore obtain notions of *degree* $\mathrm{deg}(T)$ and *weak degree* $\mathrm{wdeg}(T)$. These coincide with the definitions of *strong degree* and *weak degree*, introduced in [@DjamentVespa2019FoncteursFaiblementPolynomiaux], respectively ([*cf*. ]{}D[é]{}finition 1.5 for the strong degree, and for the weak degree see D[é]{}finitions 1.10, 1.16 and 1.22, as well as Proposition 1.19, which provides the key property – using the notation of [@DjamentVespa2019FoncteursFaiblementPolynomiaux] – that $\delta_x$ and $\pi_{\mathcal{M}}$ commute).
The article [@DjamentVespa2019FoncteursFaiblementPolynomiaux] in fact sets up a detailed theory of *weak polynomial functors* (those with finite weak degree) by considering the quotient category $\mathsf{Fun}({\mathcal{M}},{\mathcal{A}})/{\ensuremath{\mathcal{S}\mathrm{n}}}({\mathcal{M}},{\mathcal{A}})$, where ${\ensuremath{\mathcal{S}\mathrm{n}}}({\mathcal{M}},{\mathcal{A}})$ is the full subcategory of functors $T$ with $\mathrm{wdeg}(T)<0$. Since the notion of weak degree may be described very generally, whenever the source category is an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, it may be interesting to try to export this theory from ${\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}}$ to other settings to which the general definition for ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ specialises, such as twisted coefficient systems for mapping class groups ([*cf*. ]{}§\[para:degree-mcg\] below).
We note that the construction $\Psi$ above does not use the symmetry of ${\mathcal{M}}\in {\ensuremath{\mathcal{M}\mathrm{on}_{\mathit{ini}}}}$, and in fact works equally well for any strict monoidal category whose unit object is initial. Another remark is that, if the unit object of ${\mathcal{M}}$ is null, i.e., initial and terminal, then the natural transformations $\imath_x \colon \mathrm{id} \to s_x$ have left-inverses $\pi_x \colon s_x \to \mathrm{id}$ given by the morphisms $t_x \oplus \mathrm{id}_y \colon x \oplus y \to 0 \oplus y = y$, where $t_x \colon x \to 0$ is the unique morphism from $x$ to the terminal object $0$. So for functors ${\mathcal{M}}\to {\mathcal{A}}$ from a monoidal category with null unit object, the three types of degree coincide, by Remark \[rmk:4-definitions-of-deg\].
\[rmk:modules-over-monoidal-categories\] Recall that a strict left-module over a strict monoidal category ${\mathcal{M}}$ is a category ${\mathcal{C}}$ and a functor ${\oplus} \colon {\mathcal{M}}\times {\mathcal{C}}\to {\mathcal{C}}$ such that ${\oplus} \circ (\mathbf{1}_{{\mathcal{M}}} \times \mathrm{id}_{{\mathcal{C}}}) = \mathrm{id}_{{\mathcal{C}}}$ and ${\oplus} \circ (\mathrm{id}_{{\mathcal{M}}} \times {\oplus}) = {\oplus} \circ ({\oplus} \times \mathrm{id}_{{\mathcal{C}}})$, where $\mathbf{1}_{{\mathcal{M}}} \colon * \to {\mathcal{M}}$ takes the unique object to the unit object $I_{{\mathcal{M}}}$ of ${\mathcal{M}}$. If $I_{{\mathcal{M}}}$ is initial in ${\mathcal{M}}$, then any strict left-module ${\mathcal{C}}$ over ${\mathcal{M}}$ naturally has the structure of an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, generalising exactly the construction above, which is the case of ${\mathcal{M}}$ as a module over itself: the indexing set is $\mathrm{ob}({\mathcal{M}})$, the endomorphisms are defined by $x \oplus -$ and the natural transformations are formed using the fact that $I_{{\mathcal{M}}}$ is initial.[^5]
Partial braid categories. {#para:partial-braid-categories}
-------------------------
In §\[sec:functorial-configuration-spaces\] below we define another functor $${\mathcal{B}}\colon {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}\longrightarrow {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\subset {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$$ sending a manifold $M$ (equipped with a collar neighbourhood and a basepoint on its boundary) and a space $X$ to the (labelled) *partial braid category* ${\mathcal{B}}(M,X)$, whose objects are the non-negative integers. See §§\[sss:some-categories\] and \[sss:some-functors\] for the full details of this construction (alternatively §\[para:degree-WRW\] for a description of the underlying category ${\mathcal{B}}(M,X)$, without the functoriality or the structure as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$). For the next few paragraphs we will denote this object instead by ${\mathcal{B}}(M,X)^{{\ensuremath{\dagger}}}$ in order to distinguish it from a different structure (which we will define next) on the same underlying category, also making it into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$.
For $n{\geqslant}2$ let ${\mathbb{D}}^n$ denote the closed unit disc in Euclidean $n$-space, equipped with a collar neighbourhood and basepoint on its boundary. This is an object of ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$, and for any $X \in {\ensuremath{\mathsf{Top}_{\circ}}}$ the category ${\mathcal{B}}({\mathbb{D}}^n,X)$ can be made into a strict monoidal category with the number zero as its (null) unit object. For any object $M$ of ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ of dimension $n$, the category ${\mathcal{B}}(M,X)$ then has the structure of a strict left-module over ${\mathcal{B}}({\mathbb{D}}^n,X)$. Both the monoidal and the module structure are induced by the operation of boundary connected sum of two manifolds in ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$. Thus, by Remark \[rmk:modules-over-monoidal-categories\] above, there is another structure on ${\mathcal{B}}(M,X)$ making it into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, coming from this module structure. Denote this object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ by ${\mathcal{B}}(M,X)^{{\ensuremath{\ddagger}}}$.
The objects ${\mathcal{B}}(M,X)^{{\ensuremath{\dagger}}}$ and ${\mathcal{B}}(M,X)^{{\ensuremath{\ddagger}}}$ of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ have the same underlying category ${\mathcal{B}}(M,X)$, so any functor $T \colon {\mathcal{B}}(M,X) \to {\mathcal{A}}$ has a degree with respect to each of these structures; denote these by $\mathrm{deg}^{{\ensuremath{\dagger}}}(T)$ and $\mathrm{deg}^{{\ensuremath{\ddagger}}}(T)$ respectively.
\[p:two-degrees-agree\] In this setting, for any functor $T \colon {\mathcal{B}}(M,X) \to {\mathcal{A}}$, we have $\mathrm{deg}^{{\ensuremath{\dagger}}}(T) = \mathrm{deg}^{{\ensuremath{\ddagger}}}(T)$.
We will prove this as a corollary of a more general statement about modules over monoidal categories. For any object $({\mathcal{C}},I,s,\imath) \in {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ and functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$, we have a degree $\mathrm{deg}(T)$. But for any element $x \in I$ we may also forget part of the structure, considering just the object $({\mathcal{C}},s_x,\imath_x) \in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$, and compute the degree of $T$ with respect to this structure – denote this by $\mathrm{deg}^x(T)$. An easy inductive argument shows that $\mathrm{deg}^x(T) {\leqslant}\mathrm{deg}(T)$.
\[p:two-degrees-agree-2\] Let ${\mathcal{C}}$ be a strict left-module over a strict braided monoidal category ${\mathcal{M}}$, whose unit object $I_{{\mathcal{M}}}$ is null, and which is generated by $x \in \mathrm{ob}({\mathcal{M}})$, in the sense that every object of ${\mathcal{M}}$ is isomorphic to $x^{\oplus n}$ for some non-negative integer $n$. Consider ${\mathcal{C}}$ as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ as in Remark \[rmk:modules-over-monoidal-categories\] and let $T \colon {\mathcal{C}}\to {\mathcal{A}}$ be a functor. Then $\mathrm{deg}^x(T) = \mathrm{deg}(T)$.
We prove an analogous comparison result for heights in Proposition \[prop:compare-two-heights-special-case\]. See Remark \[rmk:summary\] for a summary of how these facts are related. Also see Remark \[rmk:pre-braided\] for generalisations of Proposition \[p:two-degrees-agree-2\] and references to related results.
First note that the monoidal category ${\mathcal{B}}({\mathbb{D}}^n,X)$ is braided (since $n{\geqslant}2$) and is generated by the object $1$. Thus the category ${\mathcal{C}}= {\mathcal{B}}(M,X)^{{\ensuremath{\ddagger}}}$ satisfies the hypotheses of Proposition \[p:two-degrees-agree-2\], which implies that $\mathrm{deg}^{{\ensuremath{\ddagger}}}(T) = \mathrm{deg}(T) = \mathrm{deg}^{1}(T) = \mathrm{deg}^{{\ensuremath{\dagger}}}(T)$.[^6]
To prove Proposition \[p:two-degrees-agree-2\], we first establish a lemma, which will allow us to apply Corollary \[coro:braidable\] from §\[sec:functorial-configuration-spaces\] below in the present setting. Let ${\mathcal{C}}$ be as in the statement of the proposition, considered as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, i.e., equipped with an endofunctor and natural transformation $\iota_y \colon \mathrm{id}_{{\mathcal{C}}} \Rightarrow y \oplus -$ for each object $y$ of ${\mathcal{M}}$. Write ${\mathcal{C}}^x$ for the object $({\mathcal{C}},x \oplus -,\iota_x)$ of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$, where we have forgotten all but one of the endofunctors. (For example if ${\mathcal{C}}= {\mathcal{B}}(M,X)^{{\ensuremath{\ddagger}}}$ and $x = 1$ then ${\mathcal{C}}^x = {\mathcal{B}}(M,X)^{{\ensuremath{\dagger}}}$.)
\[lem:braidable\] The object ${\mathcal{C}}^x \in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ is braidable in the sense of Definition \[def:braidable\].
We need to find a certain natural automorphism $\Psi$ of the functor $s \circ s = x \oplus x \oplus -$. Note that $\imath * \mathrm{id}_s$ and $\mathrm{id}_s * \imath$ are the natural transformations $s \Rightarrow s \circ s$ consisting of the morphisms $x \oplus c \to x \oplus x \oplus c$, for each $c \in \mathrm{ob}({\mathcal{C}})$, given by the matrices $$\label{eq:two-matrices}
\left(\,
\begin{matrix}
0 & 0 \\
\mathrm{id}_x & 0 \\
0 & \mathrm{id}_c
\end{matrix}
\,\right)
\qquad\text{and}\qquad
\left(\,
\begin{matrix}
\mathrm{id}_x & 0 \\
0 & 0 \\
0 & \mathrm{id}_c
\end{matrix}
\,\right)$$ respectively. We need to show that these differ by a natural automorphism $\Psi$. This may be constructed from the braiding of ${\mathcal{M}}$, as follows. Write $i$ for the inclusion ${\mathcal{C}}\to {\mathcal{M}}\times {\mathcal{M}}\times {\mathcal{C}}$ given by $c \mapsto (x,x,c)$ and write $f$ for the flip functor ${\mathcal{M}}\times {\mathcal{M}}\to {\mathcal{M}}\times {\mathcal{M}}$ given by $(y,z) \mapsto (z,y)$. Then the braiding of ${\mathcal{M}}$ is a natural isomorphism $b \colon {\oplus} \Rightarrow {\oplus} \circ f \colon {\mathcal{M}}\times {\mathcal{M}}\to {\mathcal{M}}$. Taking products with ${\mathcal{C}}$ and identities, this induces a natural isomorphism $b \times \mathrm{id} \colon {\oplus} \times \mathrm{id}_{{\mathcal{C}}} \Rightarrow ({\oplus} \circ f) \times \mathrm{id}_{{\mathcal{C}}} \colon {\mathcal{M}}\times {\mathcal{M}}\times {\mathcal{C}}\to {\mathcal{M}}\times {\mathcal{C}}$. Then we may take $\Psi$ to be the automorphism $\oplus * (b \times \mathrm{id}) * i$ of $x \oplus x \oplus -$. Diagrammatically: $$\label{eq:natural-aut}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (l) at (-5,0) {${\mathcal{C}}$};
\node (ml) at (20,0) {${\mathcal{M}}\times {\mathcal{M}}\times {\mathcal{C}}$};
\node (m) at (60,-10) {${\mathcal{M}}\times {\mathcal{M}}\times {\mathcal{C}}$};
\node (mr) at (100,0) {${\mathcal{M}}\times {\mathcal{C}}$};
\node (r) at (120,0) {${\mathcal{C}}.$};
\draw[->] (l) to node[above,font=\small]{$i$} (ml);
\draw[->] (ml.east) to node[above,font=\small]{$\oplus \times \mathrm{id}_{\mathcal{C}}$} (mr.west);
\draw[->] (ml.south east) to[out=-45,in=180] (m.west);
\draw[->] (m.east) to[out=0,in=225] (mr.south west);
\node at (88,-8) [anchor=west] {$\oplus \times \mathrm{id}_{\mathcal{C}}$};
\node at (36,-8) [anchor=east] {$f \times \mathrm{id}_{\mathcal{C}}$};
\draw[->] (mr) to node[above,font=\small]{$\oplus$} (r);
\draw[double,double equal sign distance,-implies] (60,-2) to node[right,font=\small]{$b \times \mathrm{id}$} (m);
\end{tikzpicture}
\end{split}$$ In components, we may write this as the collection of morphisms $b_{x,x} \oplus \mathrm{id}_c$ for $c \in \mathrm{ob}({\mathcal{C}})$, where $b_{x,x}$ denotes the braiding of ${\mathcal{M}}$ on the object $x$. The fact that $\imath * \mathrm{id}_s$ and $\mathrm{id}_s * \imath$ differ by $\Psi$ then follows from the equation: $$\label{eq:natural-aut-equation}
\left(\,
\begin{array}{ccc}
\multicolumn{2}{c}{\multirow{2}{*}{$b_{x,x}$}} & 0 \\
&& 0 \\
0 & 0 & \mathrm{id}_c
\end{array}
\,\right)
\cdot
\left(\,
\begin{matrix}
\mathrm{id}_x & 0 \\
0 & 0 \\
0 & \mathrm{id}_c
\end{matrix}
\,\right)
\quad = \quad
\left(\,
\begin{matrix}
0 & 0 \\
\mathrm{id}_x & 0 \\
0 & \mathrm{id}_c
\end{matrix}
\,\right) ,$$ where we are using the matrix notation of .
It is always true that $\mathrm{deg}^x(T) {\leqslant}\mathrm{deg}(T)$, as observed just before the statement of Proposition \[p:two-degrees-agree-2\]. So we just need to prove, for all $d{\geqslant}-1$, that, if $\mathrm{deg}^x(T) {\leqslant}d$, then $\mathrm{deg}(T) {\leqslant}d$. The proof will be by induction on $d$. The base case, when $d=-1$, is clear, since both statements are equivalent to $T$ being equal to the zero functor.
Now let $d{\geqslant}0$ and assume by induction that the implication is true for smaller values of $d$. We assume that $\mathrm{deg}^x(T) {\leqslant}d$ and we need to show that $\mathrm{deg}(T) {\leqslant}d$. We showed in Lemma \[lem:braidable\] that ${\mathcal{C}}^x = ({\mathcal{C}},s_x,\imath_x) \in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ is braidable, so Corollary \[coro:braidable\] implies that $\mathrm{deg}^x(T \circ s_x) {\leqslant}\mathrm{deg}^x(T)$. Here we are writing $s_x$ as shorthand for $x \oplus -$. Iterating this argument, we see that $$\mathrm{deg}^x(T \circ (s_x)^i) {\leqslant}\mathrm{deg}^x(T) {\leqslant}d$$ for all $i{\geqslant}0$. By the recursive definition of $\mathrm{deg}^x(-)$, this means that $$\mathrm{deg}^x \bigl( \mathrm{coker} \bigl( T \circ (s_x)^i * \imath_x \colon T \circ (s_x)^i \longrightarrow T \circ (s_x)^{i+1} \bigr) \bigr) {\leqslant}d-1$$ for all $i{\geqslant}0$. Since the unit object $I_{{\mathcal{M}}}$ of ${\mathcal{M}}$ is null, not just initial, we know that the natural transformations $\imath_y$, for objects $y \in \mathrm{ob}({\mathcal{M}})$, are all split-injective. Now, for any $n{\geqslant}0$, the natural transformation $T * \imath_{x^{\oplus n}}$ is equal to the composition $$\bigl( T \circ (s_x)^{n-1} * \imath_x \bigr) \circ \quad\cdots\cdots\quad \circ \bigl( T \circ (s_x)^2 * \imath_x \bigr) \circ \bigl( T \circ s_x * \imath_x \bigr) \circ \bigl( T * \imath_x \bigr) .$$ This is a composition of split-injective morphisms in the abelian category $\mathsf{Fun}({\mathcal{C}},{\mathcal{A}})$, so we have $$\label{eq:decomposition-of-cokernels}
\mathrm{coker}(T * \imath_{x^{\oplus n}}) \;\cong\; \bigoplus_{i=0}^{n-1} \; \mathrm{coker}(T \circ (s_x)^i * \imath_x)$$ and hence $$\mathrm{deg}^x(\mathrm{coker}(T * \imath_{x^{\oplus n}})) \;=\; \max_{i=0,\ldots,n-1} \bigl( \mathrm{deg}^x (\mathrm{coker}(T \circ (s_x)^i * \imath_x)) \bigr) {\leqslant}d-1.$$ Now let $y$ be any object of ${\mathcal{M}}$. By assumption, $y$ is isomorphic to $x^{\oplus n}$ for some $n{\geqslant}0$. Thus there is a natural isomorphism $\Phi \colon T \circ s_{x^{\oplus n}} \to T \circ s_y$ such that $\Phi \circ (T * \imath_{x^{\oplus n}}) = T * \imath_y$, and so $$\mathrm{coker}(T * \imath_y) \;\cong\; \mathrm{coker}(T * \imath_{x^{\oplus n}}).$$ Thus, for any object $y$ of ${\mathcal{M}}$, we have $\mathrm{deg}^x(\mathrm{coker}(T * \imath_y)) {\leqslant}d-1$. By the inductive hypothesis we therefore also have, for any $y \in \mathrm{ob}({\mathcal{M}})$, $$\mathrm{deg}(\mathrm{coker}(T * \imath_y)) {\leqslant}d-1.$$ By the recursive definition of $\mathrm{deg}(-)$, this implies that $\mathrm{deg}(T) {\leqslant}d$.
\[rmk:pre-braided\][^7] For Lemma \[lem:braidable\], and thus for Proposition \[p:two-degrees-agree-2\], it is possible to weaken the assumption that ${\mathcal{M}}$ is braided to the assumption that it is *pre-braided* (a notion that was introduced in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism Definition 1.5]). By definition, this means that its underlying groupoid ${\mathcal{M}}^{\sim}$ is braided and the braiding $b_{x,y} \colon x \oplus y \to y \oplus x$ of ${\mathcal{M}}^{\sim}$ satisfies the equation $$\label{eq:pre-braided}
\left(\,
\begin{array}{cc}
\multicolumn{2}{c}{\multirow{2}{*}{$b_{x,y}$}} \\
&
\end{array}
\,\right)
\cdot
\left(\,
\begin{matrix}
\mathrm{id}_x \\
0
\end{matrix}
\,\right)
\quad = \quad
\left(\,
\begin{matrix}
0 \\
\mathrm{id}_x
\end{matrix}
\,\right) \; \colon \; x \longrightarrow y \oplus x,$$ for any two objects $x,y$ of ${\mathcal{M}}$. The existence of the braiding on ${\mathcal{M}}^{\sim}$ allows one to construct the automorphism $\Psi$ (replace each appearance of ${\mathcal{M}}$ with ${\mathcal{M}}^{\sim}$ in the diagram ) and the relation implies the relation . By the same reasoning, we could dually weaken the assumption that ${\mathcal{M}}$ is braided to the assumption that it is *pre^op^-braided*, meaning that ${\mathcal{M}}^{\sim}$ is braided and its braiding satisfies the equation $$\label{eq:preop-braided}
\left(\,
\begin{array}{cc}
\multicolumn{2}{c}{\multirow{2}{*}{$b_{x,y}$}} \\
&
\end{array}
\,\right)
\cdot
\left(\,
\begin{matrix}
0 \\
\mathrm{id}_y
\end{matrix}
\,\right)
\quad = \quad
\left(\,
\begin{matrix}
\mathrm{id}_y \\
0
\end{matrix}
\,\right) \; \colon \; y \longrightarrow y \oplus x,$$ for any two objects $x,y$ of ${\mathcal{M}}$.
The assumption that $I_{{\mathcal{M}}}$ is null in Proposition \[p:two-degrees-agree-2\] was convenient to make the homological algebra simpler, by giving us the decomposition , but we expect the proposition to hold more generally whenever $I_{{\mathcal{M}}}$ is initial ([*cf*. ]{}Proposition 1.8 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux]; see also Proposition 3.9 of [@Soulie2017LongMoodyconstruction]). One can of course also generalise this proposition to the setting in which ${\mathcal{M}}$ has a given *set* of objects that generate it, instead of a single object ([*cf*. ]{}the two references just cited).
When $I_{{\mathcal{M}}}$ is not null, the four versions of degree defined in §\[para:gen-def-degree\] do not necessarily coincide, so one may ask whether Proposition \[p:two-degrees-agree-2\] is also true if deg is replaced by ${\ensuremath{\mathsf{x}}}\text{deg}$ for ${\ensuremath{\mathsf{x}}}\in \{ \text{i} , \text{s} , \text{w} \}$. For the weak degree (${\ensuremath{\mathsf{x}}}= \text{w}$) this is true, by Proposition 1.24 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux] (their statement is for a symmetric monoidal category, rather than a left-module over a pre-braided monoidal category, but their methods should extend to this more general setting too), and for ${\ensuremath{\mathsf{x}}}= \text{i or s}$ the above proof goes through with minor modifications, making use of Remark \[rmk:gen-of-degree-under-composition\].
Relation to the degree of Randal-Williams and Wahl. {#para:degree-WRW}
---------------------------------------------------
In their paper [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], Randal-Williams and Wahl use a notion of degree of twisted coefficient systems which is slightly different to that of [@DjamentVespa2019FoncteursFaiblementPolynomiaux], and which they remark is inspired by the work of Dwyer [@Dwyer1980Twistedhomologicalstability], van der Kallen [@Kallen1980Homologystabilitylinear] and Ivanov [@Ivanov1993homologystabilityTeichmuller].
#### Setting. {#setting. .unnumbered}
The starting point in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] is a *homogeneous category* ${\mathcal{C}}$ – meaning a monoidal category whose unit object is initial, satisfying two axioms H1 and H2 described in Definition 1.3 of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] – which is also *pre-braided* – see Remark \[rmk:pre-braided\] above – together with two objects $a$ and $x$ of ${\mathcal{C}}$. Let ${\mathcal{C}}_{a,x}$ denote the full subcategory on the objects $x^{\oplus m} \oplus a \oplus x^{\oplus n}$. There is an endofunctor of this category given by $x \oplus -$ and a natural transformation $\mathrm{id} \to (x \oplus -)$ since the unit of ${\mathcal{C}}$ is initial, so ${\mathcal{C}}_{a,x}$ is in this way an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\subset {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. A twisted coefficient system in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] is a functor $T \colon {\mathcal{C}}_{a,x} \to {\mathcal{A}}$. For each $N {\geqslant}0$ they define a notion of *degree at $N$* and *split degree at $N$* for $T$ (see Definition 4.10); when $N=0$ these correspond to the injective degree and the split degree of $T$ as defined in §\[para:gen-def-degree\].
If we denote by ${\mathcal{C}}_x$ the full (monoidal) subcategory of ${\mathcal{C}}$ on the objects $x^{\oplus n}$ for $n{\geqslant}0$, then ${\mathcal{C}}_{a,x}$ is a left-module over ${\mathcal{C}}_x$, so there is a notion of (injective, split, etc.) degree of functors ${\mathcal{C}}_{a,x} \to {\mathcal{A}}$ coming from Remark \[rmk:modules-over-monoidal-categories\]. If the unit object of ${\mathcal{C}}$ is null,[^8] this exactly coincides with the degree (at $N=0$) of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism]. To see this, note first that the degree of $T$ (at $N=0$) according to [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] is precisely $\mathrm{deg}^x(T)$ in the notation of Proposition \[p:two-degrees-agree-2\].[^9] Then Proposition \[p:two-degrees-agree-2\] plus Remark \[rmk:pre-braided\] imply that this is equal to the degree of $T$ according to the structure of ${\mathcal{C}}_{a,x}$ as a module over ${\mathcal{C}}_x$.
#### Injective braid categories. {#injective-braid-categories. .unnumbered}
As mentioned above (§\[para:partial-braid-categories\]), we define in §\[sec:functorial-configuration-spaces\] below the *partial braid category* ${\mathcal{B}}(M,X)$ associated to a manifold $M$ and space $X$, which is naturally a category with stabiliser, in other words, an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$. It may be described as follows: its objects are finite subsets $c$ of $M$ equipped with a function $\ell \colon c \to X$ (“labelled by $X$”). A morphism from $\ell \colon c \to X$ to $m \colon d \to X$ is a *braid between subconfigurations of $c$ and $d$ labelled by paths in $X$*. More precisely, it is a path $\gamma$ in the configuration space $C_k(M,X)$ for some integer $k$, up to endpoint-preserving homotopy, such that $\gamma(0)$ is the restriction of $\ell$ to some subset of $c$ and $\gamma(1)$ is the restriction of $m$ to some subset of $d$.
In fact, this defines a slightly larger category $\hat{{\mathcal{B}}}(M,X)$, of which ${\mathcal{B}}(M,X)$ is a skeleton. Both $M$ and $X$ are assumed to be path-connected, so the isomorphism classes of the objects $\ell \colon c \to X$ of $\hat{{\mathcal{B}}}(M,X)$ are determined by the cardinality $\lvert c \rvert$. Then ${\mathcal{B}}(M,X)$ is the full subcategory on the objects $c_n \to \{x_0\} \subseteq X$, where $x_0$ is the basepoint of $X$ and $c_n$ is a certain nested sequence of subsets of $M$ of cardinality $n$. We may therefore think of the objects of ${\mathcal{B}}(M,X)$ as the non-negative integers. See §§\[sss:some-categories\] and \[sss:some-functors\] for more details, including the functoriality of this definition with respect to $M$ and $X$ and the structure making ${\mathcal{B}}(M,X)$ into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$.
There is a subcategory of ${\mathcal{B}}(M,X)$, denoted ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ and called the *injective braid category*, also with the non-negative integers as objects, but with only those morphisms (using the description of the previous paragraph) where $\gamma(0)=\ell$. Morphisms in ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ may be thought of as “fully-defined injective braids on $M$”, whereas those in ${\mathcal{B}}(M,X)$ are “partially-defined injective braids on $M$”. The stabiliser (endofunctor plus natural transformation) of ${\mathcal{B}}(M,X)$ restricts to ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$, making it into a subobject in the category ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$.
The simplest example corresponds to taking $X$ a point and $M={\mathbb{R}}^n$ for $n{\geqslant}3$, in which case ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ is equivalent to the category FI of finite sets and injections, and ${\mathcal{B}}(M,X)$ is equivalent to the category $\text{FI}\sharp$ of finite sets and partially-defined injections.
#### Which braid categories are homogeneous? {#which-braid-categories-are-homogeneous .unnumbered}
One may wonder whether the categories ${\mathcal{B}}(M,X)$ and ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ are pre-braided homogeneous. First of all, if $M$ splits as $M = {\mathbb{R}}\times M^\prime$, they are both monoidal with initial unit object, and if moreover $M^\prime$ also splits as $M^\prime = {\mathbb{R}}\times M^{\prime\prime}$ they are braided (and hence pre-braided). The category ${\mathcal{B}}(M,X)$ is, however, never homogeneous: it fails axiom H1 for homegeneity. On the other hand, the category ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ always satisfies axiom H1, and it satisfies axiom H2 if and only if $M = {\mathbb{R}}^2 \times M^\prime$ has dimension at least $3$, i.e., $\mathrm{dim}(M^\prime) {\geqslant}1$.
In particular, the category ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}({\mathbb{R}}^2)$ is not homogeneous. The “natural” pre-braided homogeneous category whose automorphisms groups are the braid groups is denoted $U\beta$ in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], and comes with a natural functor $U\beta \to {\ensuremath{\mathcal{B}_{\mathsf{f}}}}({\mathbb{R}}^2)$. Using the graphical calculus for $U\beta$ described in §1.2 of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], this functor may be described as taking a braid diagram representing a morphism of $U\beta$ and forgetting all strands with “free” ends.
#### Comparison of twisted homological stability results. {#comparison-of-twisted-homological-stability-results. .unnumbered}
As an aside, we discuss briefly the overlap between the twisted homological stability results of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] and those of [@Palmer2018Twistedhomologicalstability] (where this note originated). For the purposes of this paragraph, a sequence of (based, path-connected) spaces $X_n$ indexed by non-negative integers is *homologically stable* if for each $i$ the group $H_i(X_n)$ is independent of $n$ (up to isomorphism) once $n$ is sufficiently large. (Given a sequence of groups $G_n$ we consider their classifying spaces $X_n = BG_n$.) If ${\mathcal{C}}$ is a category whose objects are non-negative integers and $\mathrm{Aut}_{\mathcal{C}}(n) = \pi_1(X_n)$, then a functor $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$ determines a local coefficient system on each $X_n$, and the sequence is homologically stable *with coefficients in $T$* if the corresponding local homology groups stabilise.
Theorem A of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] says that the groups $\mathrm{Aut}_{{\mathcal{C}}}(a \oplus x^{\oplus n})$ are homologically stable with coefficients in any finite-degree twisted coefficient system on ${\mathcal{C}}_{a,x}$, as long as ${\mathcal{C}}$ is pre-braided homogeneous and a certain simplicial complex built out of ${\mathcal{C}}_{a,x}$ is highly-connected.
Taking ${\mathcal{C}}= {\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ and $M = {\mathbb{R}}^2 \times M^\prime$ for a manifold $M^\prime$ of dimension at least one, we saw above that ${\mathcal{C}}$ is pre-braided homogeneous. Taking $a = 0$ and $x = 1$, we have ${\mathcal{C}}_{a,x} = {\mathcal{C}}$, which is equivalent to the category $\mathrm{FI}_G$ of [@SamSnowden2014RepresentationscategoriesG] with $G=\pi_1(M\times X)$. As noted in [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] (at the bottom of page 596), the associated simplicial complex is known to be highly-connected by a result of [@HatcherWahl2010Stabilizationmappingclass], and so Theorem A of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] applies in this setting. In fact, it yields a particular case of their Theorem D, saying that the sequence of fundamental groups $G_n = \pi_1(C_n(M,X)) \cong \pi_1(M\times X) \wr \Sigma_n$ satisfies twisted homological stability for finite-degree coefficient systems on the category ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X) = U(\sqcup_n G_n)$. On the other hand, in this setting, Theorem A of [@Palmer2018Twistedhomologicalstability] says that the sequence of (*non-aspherical*) spaces $C_n(M,X)$ satisfies twisted homological stability for finite-degree coefficient systems on the larger category ${\mathcal{B}}(M,X)$.
If $M=S$ is a surface and $X=BG$ is an aspherical space, then the configuration spaces $C_n(S,BG)$ are also aspherical with fundamental groups $G_n = G\wr \beta_n^S$, where $\beta_n^S$ denotes the $n$th surface braid group. In this case Theorem A of [@Palmer2018Twistedhomologicalstability] says that this sequence of groups satisfies twisted homological stability for finite-degree coefficient systems on ${\mathcal{B}}(S,BG)$. In this setting, Theorem D of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] also says that this sequence of groups satisfies twisted homological stability, but for finite-degree coefficient systems on the category $U(\sqcup_n G_n)$. This is more general, since there is a natural functor $U(\sqcup_n G_n) \to {\ensuremath{\mathcal{B}_{\mathsf{f}}}}(S,BG) \subset {\mathcal{B}}(S,BG)$, and precomposition by this functor preserves the degree of twisted coefficient systems ([*cf*. ]{}Lemma \[l:degree-under-composition\]).
When $M$ has dimension greater than $2$ or when $X$ has non-trivial higher homotopy groups, the spaces $C_n(M,X)$ are not aspherical, so in this setting the twisted homological stability result of [@Palmer2018Twistedhomologicalstability] is not comparable to the results of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism], since the latter paper is concerned only with sequences of *groups*. On the other hand, the framework of [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism] has been generalised by Krannich [@Krannich2017Homologicalstabilitytopological] to a topological setting, which includes the setting of configuration spaces, even when they are not aspherical. See Remark 1.5 of [@Palmer2018Twistedhomologicalstability] for a comparison.
Degree of twisted coefficient systems on mapping class groups. {#para:degree-mcg}
--------------------------------------------------------------
There are several different settings that have been considered for twisted coefficient systems on mapping class groups and their degrees, all using the notion of “split degree” (or slight variations thereof) described in §\[para:gen-def-degree\]. We will describe and compare these different settings, using the language of §\[para:gen-def-degree\], without defining in all details the categories involved.
There is a certain category ${\mathcal{C}}$, introduced by Ivanov [@Ivanov1993homologystabilityTeichmuller], whose objects are compact, connected, oriented surfaces $F$ equipped with an embedded arc in $\partial F$. Morphisms are, roughly, embeddings together with a path between the midpoints of the two arcs, all considered up to ambient isotopy. There is an endofunctor $t\colon {\mathcal{C}}\to {\mathcal{C}}$ and a natural transformation $\mathrm{id} \to t$ defined by Ivanov, which on objects takes the boundary connected sum with $F_{1,1}$, the torus with one boundary component. There is another such endofunctor $a\colon {\mathcal{C}}\to {\mathcal{C}}$, introduced by Cohen and Madsen [@CohenMadsen2009Surfacesinbackground], which instead takes the boundary connected sum with an annulus.
The coefficient systems of Ivanov are indexed on ${\mathcal{C}}$ and his degree is the *split degree* (as defined in §\[para:gen-def-degree\]), considering ${\mathcal{C}}$ as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ using just the endomorphism $t$. Cohen and Madsen use a slight variation of ${\mathcal{C}}$ to index their coefficient systems, and their degree is again the split degree, but this time using both $t$ and $a$ to turn ${\mathcal{C}}$ into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. As a side note, their definition very slightly deviates from this, in fact. They do not require that the splittings of $T \to Tt$ and of $T \to Ta$ are functorial, i.e., they do not have to be natural transformations. They only require that $T(F) \to Tt(F)$ and $T(F) \to Ta(F)$ split for each $F$, and that these splittings are equivariant for the action of the automorphism group of the object $F$ in ${\mathcal{C}}$ (which is the mapping class group of $F$). In other words, $T \to Tt$ and $T \to Ta$ are only required to be split mono natural transformations after restricting ${\mathcal{C}}$ to the subcategory ${\mathcal{C}}_{\mathrm{aut}} \subset {\mathcal{C}}$ of all automorphisms in ${\mathcal{C}}$.
Boldsen [@Boldsen2012Improvedhomologicalstability] uses the same ${\mathcal{C}}$ as Cohen and Madsen and the same two endofunctors, and he also introduces another functor $p \colon {\mathcal{C}}(2) \to {\mathcal{C}}$, defined on a certain subcategory ${\mathcal{C}}(2)$ of ${\mathcal{C}}$ in which objects all have at least two boundary components, which glues a pair of pants onto two boundary components of a given surface. His coefficient systems are indexed on ${\mathcal{C}}$, as for Cohen and Madsen. The endofunctors $t$ and $a$ turn ${\mathcal{C}}$ into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, and therefore give a notion of split degree. However, Boldsen’s definition of degree is slightly stricter: the recursive condition is modified to say that $T \to Tt$ and $T \to Ta$ must be split mono and $\mathrm{sdeg}(\mathrm{coker}(T\to Tt)) {\leqslant}d-1$ and $\mathrm{sdeg}(\mathrm{coker}(T\to Ta)) {\leqslant}d-1$, and also $T|_{{\mathcal{C}}(2)} \to Tp$ must also be split mono, in $\mathsf{Fun}({\mathcal{C}}(2),{\mathcal{A}})$.
Randal-Williams and Wahl also consider mapping class groups as an example of their general twisted homological stability machine, and their setup is again slightly different to the previous settings. They consider two subcategories of ${\mathcal{C}}$ separately. One is the full subcategory on surfaces with any genus but a fixed number of boundary components, to which the endofunctor $t$ restricts. They then consider coefficient systems indexed on this subcategory, and define the split degree of such coefficient systems by using the restriction of $t$ to view the subcategory as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. (For simplicity we are taking $N=0$ in their definition of split degree.) Separately, they consider the subcategory on surfaces with a fixed genus and any number of boundary components, to which the endofunctor $a$ restricts. They then consider coefficient systems indexed on this subcategory, and define the split degree by using the restriction of $a$ to view it as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. Finally, they also consider a non-orientable analogue of Ivanov’s category ${\mathcal{C}}$, with objects all non-orientable surfaces with a given fixed number of boundary components and any (non-orientable) genus. This admits an endofunctor $m$ defined on objects by taking the boundary connected sum with a M[ö]{}bius band, and they then consider coefficient systems indexed on this category, with the split degree defined by using $m$ to view it as an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$.
Vanishing of cross-effects {#sec:cross-effects}
==========================
In this section, we give a general definition of the *height* of a functor ${\mathcal{C}}\to {\mathcal{A}}$, for an abelian category ${\mathcal{A}}$ and a category ${\mathcal{C}}$ equipped with certain structure,[^10] and relate it to various notions of *height* appearing in the literature, including that of [@DjamentVespa2019FoncteursFaiblementPolynomiaux] (much of this section has been directly inspired by the definitions given in that paper). In particular, this encompasses the setting where ${\mathcal{C}}$ is monoidal and its unit object is either initial or terminal (see §\[para:specialise-DV\] and §\[para:specialise-HPV\]), and also the setting where ${\mathcal{C}}$ is any category equipped with a functor ${\mathcal{I}}\to {\mathcal{C}}$, where ${\mathcal{I}}$ is the category defined just below at the beginning of §\[para:first-def\] (see §\[para:specialise-this-paper\]). In §\[para:compare-two-heights\] we study the intersection between these two settings. This is analogous to §\[para:partial-braid-categories\] above (which is concerned with the intersection between two different ways of defining the *degree* of a functor with source ${\mathcal{C}}$); see in particular Remark \[rmk:summary\].
Throughout this section ${\mathcal{A}}$ will denote a fixed abelian category. In proofs we will often assume that ${\mathcal{A}}$ is a category of modules over a ring, so that its objects have elements, which is justified by the Freyd-Mitchell embedding theorem.
First definition. {#para:first-def}
-----------------
Let ${\mathcal{I}}$ be the category whose objects are the non-negative integers, and whose morphisms $m \to n$ are subsets of $\{ 1,\ldots,\mathrm{min}(m,n) \}$, with composition given by intersection. The endomorphism monoid $\mathrm{End}_{{\mathcal{I}}}(n)$ is denoted ${\ensuremath{\mathcal{I}_{n}}}$, and is the monoid of subsets of ${\ensuremath{\underline{n}}}= \{1,\ldots,n\}$ under the operation $\cap$ with neutral element ${\ensuremath{\underline{n}}}$ itself. We will also think of ${\ensuremath{\mathcal{I}_{n}}}$ as a category on the single object $\bullet$. There is an operation $\mathrm{cr}(-)$ that takes a functor $f\colon {\ensuremath{\mathcal{I}_{n}}} \to {\mathcal{A}}$ as input and produces the following object of ${\mathcal{A}}$: $$\mathrm{cr}(f) \;=\; \mathrm{im} \biggl(\, \sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} f({\ensuremath{\underline{n}}}\smallsetminus S) \colon f(\bullet) \longrightarrow f(\bullet) \biggr)$$ as output. Now suppose that we are given a category ${\mathcal{C}}$ equipped, for each $n{\geqslant}0$, with a collection of functors $\{ f_j\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}_{j\in J_n}$. Then the *height* $\mathrm{ht}(T)\in\{-1,0,1,2,\ldots,\infty\}$ of a functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ is defined by the criterion that $\mathrm{ht}(T){\leqslant}h$ if and only if for all $n>h$ and all $j\in J_n$, $\mathrm{cr}(T\circ f_j)=0$.
Second definition. {#para:second-def}
------------------
Let ${\ensuremath{\bar{\mathcal{I}}_{n}}}$ denote the set of all subsets of ${\ensuremath{\underline{n}}}$, considered as a partially-ordered set – and thus as a category – under the relation of inclusion of subsets. There is an operation ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(-)$ taking a functor $f\colon {\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{A}}$ as input and producing the following object of ${\mathcal{A}}$: $${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f) \;=\; \mathrm{coker} \biggl(\, \bigoplus_{S \subsetneq {\ensuremath{\underline{n}}}} f(S\hookrightarrow{\ensuremath{\underline{n}}}) \colon \bigoplus_{S \subsetneq {\ensuremath{\underline{n}}}} f(S) \longrightarrow f({\ensuremath{\underline{n}}}) \biggr)$$ as output. Now suppose that we are given a category ${\mathcal{C}}$ equipped, for each $n{\geqslant}0$, with a collection of functors $\{ f_j\colon {\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}_{j\in J_n}$. Then the *height* ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T)\in\{-1,0,1,2,\ldots,\infty\}$ of a functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ is defined by the criterion that ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T){\leqslant}h$ if and only if for all $n>h$ and all $j\in J_n$, ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(T\circ f_j)=0$.
Relationship between the definitions. {#para:two-definitions}
-------------------------------------
There is a functor $z\colon {\ensuremath{\bar{\mathcal{I}}_{n}}}\to {\ensuremath{\mathcal{I}_{n}}}$ given by sending each morphism $S\subseteq T$ in ${\ensuremath{\bar{\mathcal{I}}_{n}}}$ to the morphism ${\ensuremath{\underline{n}}}\smallsetminus (T\smallsetminus S)$ in ${\ensuremath{\mathcal{I}_{n}}}$. (More generally, any lattice $L$ may be viewed as a monoid $L^{\wedge}$ under the meet operation, and there is an analogous functor $L\to L^{\wedge}$ if $L$ is a Boolean algebra.) This relates the two constructions above as follows:
\[lem:two-definitions\] For any functor $f\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{A}}$ the objects $\mathrm{cr}(f)$ and ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z)$ are isomorphic.
A category ${\mathcal{C}}$ equipped with collections of functors $\{{\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}$ as in the first definition may be viewed via $z$ as a category equipped with collections of functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}$ as in the second definition. Hence – *a priori* – functors $T\colon{\mathcal{C}}\to{\mathcal{A}}$ have two possibly different heights, $\mathrm{ht}(T)$ and ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T)$. But the above lemma implies that these coincide, i.e. $\mathrm{ht}(T)={\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T)$. The second definition is therefore more general, reducing to the first definition in the special case where the given functors ${\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}$ all factor through $z\colon {\ensuremath{\bar{\mathcal{I}}_{n}}}\to {\ensuremath{\mathcal{I}_{n}}}$.
This is proved exactly as Proposition 2.9 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux]. We will give the details here, in order to identify (for later; see §\[para:semi-functors\]) where we use the fact that $f$ preserves the identity. First of all we will show that: $$\label{eq:ker-im-identity}
\mathrm{ker}(g) \;=\; \sum_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{im}(f(S)) \qquad\text{where}\qquad g = \displaystyle\sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} f({\ensuremath{\underline{n}}}\smallsetminus S).$$
$(\supseteq):$ Let $x=f(T)(y)$ for $y\in f(\bullet)$ and $T\subsetneq{\ensuremath{\underline{n}}}$. Choose $i\in{\ensuremath{\underline{n}}}\smallsetminus T$ and write $$g(x) \;\;= \sum_{S\subseteq {\ensuremath{\underline{n}}}\smallsetminus\{i\}} \Bigl( (-1)^{\lvert S\rvert} f({\ensuremath{\underline{n}}}\smallsetminus S)f(T)(y) + (-1)^{\lvert S\rvert +1} f(({\ensuremath{\underline{n}}}\smallsetminus S)\smallsetminus \{i\})f(T)(y) \Bigr) .$$ Since $(T\smallsetminus S)\smallsetminus\{i\} = T\smallsetminus S$ the terms cancel pairwise and $x\in\mathrm{ker}(g)$.
$(\subseteq):$ Suppose $x\in f(\bullet)$ and $g(x)=0$. Since $f$ preserves the identity, i.e. $f({\ensuremath{\underline{n}}})=\mathrm{id}$, we may write $$\begin{aligned}
x \;\;&= \sum_{\varnothing \neq S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert +1} f({\ensuremath{\underline{n}}}\smallsetminus S)(x) \\
&= \sum_{S\subsetneq{\ensuremath{\underline{n}}}} (-1)^{n-\lvert S\rvert +1} f(S)(x) \quad \in \quad \sum_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{im}(f(S)).\end{aligned}$$
Now note that the right-hand side of (the left-hand equation of) is equal to the image of $$h = \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(S) \colon \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(\bullet) \longrightarrow f(\bullet).$$ Hence we have $\mathrm{cr}(f) = \mathrm{im}(g) \cong f(\bullet)/\mathrm{ker}(g) = f(\bullet)/\mathrm{im}(h) = \mathrm{coker}(h) = {\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z)$.
Specialising to the setting of Djament and Vespa. {#para:specialise-DV}
-------------------------------------------------
Let ${\mathcal{C}}$ be a symmetric monoidal category whose monoidal unit is null (simultaneously initial and terminal). In [@DjamentVespa2019FoncteursFaiblementPolynomiaux], Djament and Vespa define the notion of a *strong polynomial* functor ${\mathcal{C}}\to{\mathcal{A}}$ of *degree* $d$. Their definition is recovered by the first definition of a *functor of height $d$* above by equipping ${\mathcal{C}}$ with the following collections of functors $\{{\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}$. Take $J_n$ to be the set of $n$-tuples $(x_1,\ldots,x_n)$ of objects of ${\mathcal{C}}$. The associated functor ${\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}$ sends the unique object $\bullet$ to $\bigoplus_{i=1}^n x_i$ and a subset $S\subseteq{\ensuremath{\underline{n}}}$ to the endomorphism $\bigoplus_{i=1}^n \phi_i$ where $\phi_i=\mathrm{id}$ for $i\in S$ and $\phi_i=0$ otherwise.
More generally, let ${\mathcal{C}}$ be a symmetric monoidal category whose monoidal unit is initial. The general definition of Djament and Vespa is for this setting, and corresponds to the second definition of a *functor of height $d$* above by equipping ${\mathcal{C}}$ with the following collections of functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}$. Take $J_n$ to be the set of $n$-tuples $(x_1,\ldots,x_n)$ of objects of ${\mathcal{C}}$ as before. The associated functor ${\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}$ sends the object $S\subseteq{\ensuremath{\underline{n}}}$ to $\bigoplus_{i\in S}x_i$ and the inclusion $S\subseteq T$ to the canonical morphism $\bigoplus_{i\in S} x_i \cong \bigoplus_{i\in T} y_i \to \bigoplus_{i\in T} x_i$ where $y_i=x_i$ if $i\in S$ and $y_i=0$ otherwise.
Of course, our general definition of a *functor of height $d$* introduced above specialises very naturally to this setting as it was directly inspired by the work of Djament and Vespa.[^11] Soon we will generalise it slightly (§\[para:semi-functors\]) so that it also recovers the notion of *height* used in [@Palmer2018Twistedhomologicalstability]. First we describe the dual of our second definition of height and specialise it to the setting of [@HartlPirashviliVespa2015Polynomialfunctorsalgebras].
Third definition. {#para:third-def}
-----------------
There is an operation ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(-)$ that takes a functor $f\colon {\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}\to{\mathcal{A}}$ as input and produces the following object of ${\mathcal{A}}$: $${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(f) \;=\; \mathrm{ker} \biggl(\, \bigoplus_{S \subsetneq {\ensuremath{\underline{n}}}} f(S\hookrightarrow{\ensuremath{\underline{n}}}) \colon f({\ensuremath{\underline{n}}}) \longrightarrow \bigoplus_{S \subsetneq {\ensuremath{\underline{n}}}} f(S) \biggr)$$ as output. Suppose that we are given a category ${\mathcal{C}}$ equipped, for each $n{\geqslant}0$, with a collection of functors $\{ f_j\colon {\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}\to{\mathcal{C}}\}_{j\in J_n}$. The *height* ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}^\prime(T)\in\{-1,0,1,2,\ldots,\infty\}$ of a functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ is defined by the criterion that ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}^\prime(T){\leqslant}h$ if and only if for all $n>h$ and all $j\in J_n$, ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(T\circ f_j)=0$.
Relation between all three definitions. {#para:three-definitions}
---------------------------------------
This may be related to the first and second definitions as follows. There is a functor $z^\prime\colon {{\ensuremath{\bar{\mathcal{I}}_{n}}}}^{\mathrm{op}}\to {\ensuremath{\mathcal{I}_{n}}}$ given by sending each morphism $S\subseteq T$ in ${\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}$ to the morphism ${\ensuremath{\underline{n}}}\smallsetminus (T\smallsetminus S)$ in ${\ensuremath{\mathcal{I}_{n}}}$. Using this and the functor $z\colon {\ensuremath{\bar{\mathcal{I}}_{n}}}\to {\ensuremath{\mathcal{I}_{n}}}$ from above, any functor ${\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{A}}$ induces functors ${\ensuremath{\bar{\mathcal{I}}_{n}}} \to {\mathcal{A}}$ and ${\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}} \to {\mathcal{A}}$.
\[lem:three-definitions\] For any functor $f\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{A}}$ we have isomorphisms ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z) \cong \mathrm{cr}(f) \cong {\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(f\circ z^\prime)$.
A category ${\mathcal{C}}$ equipped with collections of functors $\{{\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}$ as in the first definition may be viewed as a category equipped either with collections of functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}$ as in the second definition or collections of functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}\to{\mathcal{C}}\}$ as in the third definition. The above lemma implies that in this situation the three possible notions of height for functors ${\mathcal{C}}\to{\mathcal{A}}$ all coincide.
By Lemma \[lem:two-definitions\] it suffices to prove that ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z) \cong {\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(f\circ z^\prime)$, in other words: $$\label{eq:coker-and-ker}
\mathrm{coker} \biggl( \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(S) \colon f(\bullet)^{n!-1} \longrightarrow f(\bullet) \biggr)
\;\cong\;
\mathrm{ker} \biggl( \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(S) \colon f(\bullet) \longrightarrow f(\bullet)^{n!-1} \biggr)$$ where we have written $f(\bullet)^{n!-1}$ to denote $\bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}}f(\bullet)$. Since the morphisms $f(S)$ are idempotent and pairwise commute, there is a decomposition $$f(\bullet) \;\;\cong \bigoplus_{T\subseteq {\mathcal{P}}^\prime(n)} \biggl( \bigcap_{S\in T} \mathrm{ker} (f(S)) \cap \bigcap_{S\in{\mathcal{P}}^\prime(n)\smallsetminus T} \mathrm{im} (f(S)) \biggr) ,$$ where ${\mathcal{P}}^\prime(n)$ denotes the set of proper subsets of ${\ensuremath{\underline{n}}}$.[^12] The direct sum of all components except the one corresponding to $T={\mathcal{P}}^\prime(n)$ is equal to $\sum_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{im}(f(S))$, so we have: $$\begin{aligned}
f(\bullet) \;&\cong \,\bigcap_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{ker}(f(S)) \;\oplus\; \sum_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{im}(f(S)) \\
&\cong \;\mathrm{ker} \biggl( \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(S) \colon f(\bullet) \longrightarrow f(\bullet)^{n!-1} \biggr) \;\oplus\; \mathrm{im} \biggl( \bigoplus_{S\subsetneq{\ensuremath{\underline{n}}}} f(S) \colon f(\bullet)^{n!-1} \longrightarrow f(\bullet) \biggr) ,\end{aligned}$$ which implies the isomorphism , as desired.
Specialising to the setting of Hartl-Pirashvili-Vespa. {#para:specialise-HPV}
------------------------------------------------------
Let ${\mathcal{C}}$ be a monoidal category whose monoidal unit is null, and which is not necessarily symmetric. In [@HartlPirashviliVespa2015Polynomialfunctorsalgebras], Hartl, Pirashvili and Vespa define the notion of a *polynomial* functor ${\mathcal{C}}\to{\mathcal{A}}$ of *degree* $d$. (When ${\mathcal{C}}$ is symmetric it agrees with the definition of [@DjamentVespa2019FoncteursFaiblementPolynomiaux].) Their definition is recovered by our third definition of a *functor of height $d$* by equipping ${\mathcal{C}}$ with the following collections of functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}\to{\mathcal{C}}\}$.[^13] As before, take $J_n$ to be the set of $n$-tuples $(x_1,\ldots,x_n)$ of objects of ${\mathcal{C}}$. The associated functor ${\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}\to{\mathcal{C}}$ sends the object $S\subseteq{\ensuremath{\underline{n}}}$ to the object $\bigoplus_{i\in S}x_i$ and the inclusion $S\subseteq T$ to the canonical morphism $\bigoplus_{i\in T} x_i \to \bigoplus_{i\in T} y_i \cong \bigoplus_{i\in S} x_i$ where $y_i=x_i$ if $i\in S$ and $y_i=0$ otherwise. Note: since ${\mathcal{C}}$ is not symmetric, to correctly define $\bigoplus_{i\in S}x_i$ we must consider ${\ensuremath{\underline{n}}}$ as a totally-ordered set and use the inherited ordering of each subset $S\subseteq{\ensuremath{\underline{n}}}$.
We note that the definition of [@HartlPirashviliVespa2015Polynomialfunctorsalgebras] only requires the monoidal unit to be terminal. Also, the definition given earlier (§\[para:specialise-DV\]) for a symmetric monoidal category with initial unit works equally well when the monoidal structure is not symmetric, as long as one is careful, as in the previous paragraph, to use the natural total ordering on ${\ensuremath{\underline{n}}}$. Thus there is a general notion of *height* for functors ${\mathcal{C}}\to{\mathcal{A}}$ whenever ${\mathcal{C}}$ is monoidal and its unit is either initial or terminal, and these notions coincide when the unit is null.
Categories with finite coproducts; relation to the Taylor tower. {#para:finite-coproducts}
----------------------------------------------------------------
In [@HartlVespa2011Quadraticfunctorspointed] there is a definition of *polynomial* functor ${\mathcal{C}}\to{\mathcal{A}}$ of *degree* $d$ in the setting where ${\mathcal{C}}$ has a null object and finite coproducts, and where ${\mathcal{A}}$ is either $\mathsf{Ab}$ or $\mathsf{Grp}$, the category of groups. When ${\mathcal{A}}=\mathsf{Ab}$ this is a special case of the definition of [@HartlPirashviliVespa2015Polynomialfunctorsalgebras], since ${\mathcal{C}}$ has a monoidal structure given by the coproduct. When ${\mathcal{A}}=\mathsf{Grp}$ it falls outside the scope of the discussion in this section, since $\mathsf{Grp}$ is not an abelian category. It is, however, a *semi-abelian category* (see [@JanelidzeMarkiTholen2002Semiabeliancategories; @Borceux2004surveysemiabelian]), which suggests that it would be interesting to try to extend the general notion of the *height* of a functor ${\mathcal{C}}\to{\mathcal{A}}$ in this section to the case where ${\mathcal{A}}$ is only semi-abelian (for example the category of groups or the category of non-unital rings).
As an aside, we recall that when the monoidal structure on ${\mathcal{C}}$ is given by the coproduct, one can do more than just define the height of a functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$: one can also approximate it by functors of smaller height, and these approximations form its so-called *Taylor tower*. The key property of the coproduct that allows this is that its universal property equips us with “fold maps” $c + \cdots + c \to c$. In the next paragraph, we recall briefly the construction from [@HartlVespa2011Quadraticfunctorspointed], using the terminology of the present section.
Recall that the structure on ${\mathcal{C}}$ used to define the height of a functor defined on it is a collection of functors $f_{(c_1,\ldots,c_n)} \colon {\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}} \to {\mathcal{C}}$, one for each $n$-tuple of of objects in ${\mathcal{C}}$ (and for each $n{\geqslant}0$), and the cross-effect ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(Tf_{(c_1,\ldots,c_n)})$ is a subobject of $T(c_1 + \cdots + c_n)$, where $+$ denotes the coproduct in ${\mathcal{C}}$. Now take $c_1 = \cdots = c_n = c$. The universal property of the coproduct gives us a morphism $c + \cdots + c \to c$, to which we may apply $T$ and then compose with the inclusion of the cross-effect to obtain a morphism ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(Tf_{(c,\ldots,c)}) \to T(c)$. Define $p_{n-1} T(c)$ to be the cokernel of this morphism. This construction is functorial in $c$ and defines a functor $p_{n-1} T$ of height ${\leqslant}n-1$, which is to be thought of as the best approximation of $T$ by such a functor. There are also natural transformations $p_0 T \leftarrow p_1 T \leftarrow \cdots \leftarrow p_{n-1} T \leftarrow p_n T \leftarrow \cdots$ and $T \to \mathrm{lim}(p_\bullet T)$, which between them constitute the “Taylor tower” of $T$.
Specialising to the setting of Collinet-Djament-Griffin. {#para:specialise-CDG}
--------------------------------------------------------
Let ${\ensuremath{\mathsf{Se}^{\mathsf{fin}}}}$ denote the category of finite sets and partially-defined functions and let $\Sigma$ denote its subcategory of finite sets and partially-defined injections. For any intermediate category $\Sigma {\leqslant}\Lambda {\leqslant}{\ensuremath{\mathsf{Se}^{\mathsf{fin}}}}$ and any category ${\mathcal{C}}$ we may define the *wreath product* ${\mathcal{C}}\wr \Lambda$ to have finite tuples of objects of ${\mathcal{C}}$ as objects, and a morphism from $(c_1,\ldots,c_m)$ to $(d_1,\ldots,d_n)$ to consist of a morphism $\phi\colon m \to n$ of $\Lambda$ together with morphisms $\alpha_i \colon c_i \to d_{\phi(i)}$ of ${\mathcal{C}}$ for each $i$ on which $\phi$ is defined. We write this morphism as $(\phi \mathbin{;} \{\alpha_i\}_{i\in \mathrm{dom}(\phi)})$.
We may then equip ${\mathcal{C}}\wr \Lambda$ with collections of functors $\{ {\ensuremath{\mathcal{I}_{n}}} \to {\mathcal{C}}\wr \Lambda \}$, as follows. As before, take the indexing set $J_n$ to be the set of $n$-tuples of objects of ${\mathcal{C}}$. The functor ${\ensuremath{\mathcal{I}_{n}}} \to {\mathcal{C}}\wr \Lambda$ associated to the $n$-tuple $(c_1,\ldots,c_m)$ takes the unique object $\bullet$ of ${\ensuremath{\mathcal{I}_{n}}}$ to $(c_1,\ldots,c_m)$ and a subset $S \subseteq {\ensuremath{\underline{n}}}$ to the endomorphism $(r_S \mathbin{;} \{\mathrm{id}_{c_i}\}_{i\in S})$ where $r_S(i)=i$ for $i\in S$ and $r_S(i)$ is undefined otherwise.
This defines a notion of *height* for any functor $T \colon {\mathcal{C}}\wr \Lambda \to {\mathcal{A}}$ into an abelian category ${\mathcal{A}}$, using the first definition (§\[para:first-def\]) above. This exactly recovers the definition of *height* given by Collinet, Djament and Griffin [@CollinetDjamentGriffin2013Stabilitehomologiquepour] in this setting. To see this, we may by Lemma \[lem:three-definitions\] use the third definition (§\[para:third-def\]) above instead. Unravelling this definition, we see that it is precisely the definition of [@CollinetDjamentGriffin2013Stabilitehomologiquepour], given in D[é]{}finitions 2.5 together with the sentence before Proposition 2.11.[^14]
Semi-functors. {#para:semi-functors}
--------------
The construction in §\[para:first-def\] taking a functor $f\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{A}}$ as input and returning an object $\mathrm{cr}(f)$ of ${\mathcal{A}}$ works also if $f$ is just a *semi-functor*, in other words preserving composition but not necessarily identities.[^15] So if ${\mathcal{C}}$ is a category equipped, for each $n{\geqslant}0$, with a collection of semi-functors $\{ f_j\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}_{j\in J_n}$, then we may define the *height* of a semi-functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ exactly as before: $\mathrm{ht}(T){\leqslant}h$ if and only if for all $n>h$ and all $j\in J_n$, $\mathrm{cr}(T\circ f_j)=0$. The second (§\[para:second-def\]) and third (§\[para:third-def\]) definitions of height generalise in the same way: if ${\mathcal{C}}$ is a category equipped with collections of semi-functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}_{j\in J_n}$ or $\{{\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}} \to{\mathcal{C}}\}_{j\in J_n}$ then we have a notion of the *height* of any semi-functor ${\mathcal{C}}\to{\mathcal{A}}$, defined exactly as in the case of functors.
Lemma \[lem:two-definitions\] is no longer true for semi-functors: the fact that $f$ preserves the identity was used to prove one of the two inclusions for the equality . However, the rest of the proof goes through and shows that there is an exact sequence ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z) \to \mathrm{cr}(f) \to 0$ in this setting. The proof of Lemma \[lem:three-definitions\] does not use that $f$ preserves the identity, so we still have that ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(f\circ z) \cong {\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}^\prime(f\circ z^\prime)$ when $f$ is a semi-functor. As a result, if ${\mathcal{C}}$ is a category equipped with collections of semi-functors $\{ {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}$ and $T\colon {\mathcal{C}}\to{\mathcal{A}}$ is a semi-functor, then: $$\mathrm{ht}(T) {\leqslant}{\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T) = {\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}^\prime(T).$$ In fact, ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}(T)$ is often infinite when the semi-functors $\{{\ensuremath{\bar{\mathcal{I}}_{n}}}\to{\mathcal{C}}\}$ are not functors ([*cf*. ]{}Proposition \[prop:htbar-is-infinite\]) so the right notion in this case is $\mathrm{ht}(T)$, which we will use in the next subsection.
Specialising to partial braid categories. {#para:specialise-this-paper}
-----------------------------------------
As before, we denote by ${\mathcal{I}}$ the category with objects $0,1,2,\ldots$ and morphisms $m \to n$ corresponding to subsets of $\{ 1,\ldots,\mathrm{min}(m,n) \}$, with composition given by intersection. We will sometimes think of these morphisms $m \to n$ as partially-defined functions $\{1,\ldots,m\} \to \{1,\ldots,n\}$ that are the identity wherever they are defined. We will usually abbreviate $\{1,\ldots,n\}$ as ${\ensuremath{\underline{n}}}$.
Let ${\mathcal{C}}$ be a category equipped with a functor $s\colon {\mathcal{I}}\to {\mathcal{C}}$. For example, ${\mathcal{C}}$ could be an object of ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$, in the notation of §\[sss:height\] below, which in particular includes the case where ${\mathcal{C}}$ is the partial braid category ${\mathcal{B}}(M,X)$ defined in §\[sss:some-functors\] (see also §\[para:degree-WRW\]).
Now equip ${\mathcal{C}}$ with the following collections of semi-functors $\{ f_m\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}\}_{m\in J_n}$. Take the indexing set $J_n$ to be ${\mathbb{N}}\cap [n,\infty)$. Then for $m{\geqslant}n$ let $f_m$ be the composite semi-functor $${\ensuremath{\mathcal{I}_{n}}} \to {\ensuremath{\mathcal{I}_{m}}} = \mathrm{End}_{{\mathcal{I}}}(m) \hookrightarrow {\mathcal{I}}\xrightarrow{\;s\;} {\mathcal{C}},$$ where ${\ensuremath{\mathcal{I}_{n}}}\to {\ensuremath{\mathcal{I}_{m}}}$ takes a subset $S$ of ${\ensuremath{\underline{n}}}$ to the subset $S+m-n = \{s+m-n \mid s\in S\}$ of ${\ensuremath{\underline{m}}}$. This defines a notion of *height* for each semi-functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$. Unwinding the definition, it says that $\mathrm{ht}(T){\leqslant}h$ if and only if for all $m{\geqslant}n>h$ the following subobject of $Ts(m)$ vanishes: $$\label{eq:subobject-of-Tsm}
\mathrm{im} \biggl(\, \sum_{S\subseteq\{m-n+1,\ldots,m\}} (-1)^{\lvert S\rvert} Ts(f_{S\cup\{1,\ldots,m-n\}}) \biggr) .$$ Here $f_T\colon {\ensuremath{\underline{m}}}\to{\ensuremath{\underline{m}}}$ is the partially-defined function that “forgets” $T\subseteq{\ensuremath{\underline{m}}}$, in other words $f_T(i)=i$ if $i\in{\ensuremath{\underline{m}}}\smallsetminus T$ and is undefined if $i\in T$.
\[lem:isomorphism-of-subobjects\] The subobject of $Ts(m)$ is equal to the subobject $$\label{eq:subobject-of-Tsm-2}
\mathrm{im}(Ts(f_{\{1,\ldots,m-n\}})) \cap \bigcap_{i=m-n+1}^m \mathrm{ker}(Ts(f_{\{i\}})).$$
As a corollary, we deduce that the definition of *height* used in the paper [@Palmer2018Twistedhomologicalstability] (see Definition 3.15 of that paper) is recovered when we specialise in this way, taking ${\mathcal{C}}= {\mathcal{B}}(M,X)$ equipped with the canonical functor ${\mathcal{I}}\to {\mathcal{B}}(M,X)$ (see the paragraph below ).
The height of a functor ${\mathcal{B}}(M,X)\to\mathsf{Ab}$ given by of agrees with the definition above, specialised to the case ${\mathcal{C}}={\mathcal{B}}(M,X)$ and ${\mathcal{A}}=\mathsf{Ab}$.
By Definition 3.15 of [@Palmer2018Twistedhomologicalstability], the height of a functor $T\colon{\mathcal{B}}(M,X)\to\mathsf{Ab}$ is bounded above by $h$ if and only if for all $m{\geqslant}n>h$ we have $T_m^n=0$. Looking at the definition of $T_m^n$ (see Definition 3.10 of [@Palmer2018Twistedhomologicalstability]) we see that it is precisely , and therefore , by Lemma \[lem:isomorphism-of-subobjects\].
We think of ${\mathcal{A}}$ as a concrete category of modules over a ring, by the Freyd-Mitchell embedding theorem, so that we may talk about the elements of its objects.
$\bullet\; \eqref{eq:subobject-of-Tsm-2} \subseteq \eqref{eq:subobject-of-Tsm}:$ Suppose that $x$ is an element of , say $x=Ts(f_{\{1,\ldots,m-n\}})(y)$. If $S$ is a non-empty subset of $\{m-n+1,\ldots,m\}$ then we may pick some $i\in S$ and compute that $$\begin{aligned}
Ts(f_{S\cup\{1,\ldots,m-n\}})(y) &= Ts(f_S) \circ Ts(f_{\{i\}}) \circ Ts(f_{\{1,\ldots,m-n\}})(y) \\
&= Ts(f_S) \circ Ts(f_{\{i\}}) (x) = 0.\end{aligned}$$ Hence we deduce that $$\sum_{S\subseteq\{m-n+1,\ldots,m\}} (-1)^{\lvert S\rvert} Ts(f_{S\cup\{1,\ldots,m-n\}}) (y) = x,$$ so in particular $x\in\eqref{eq:subobject-of-Tsm}$.
$\bullet\; \eqref{eq:subobject-of-Tsm} \subseteq \eqref{eq:subobject-of-Tsm-2}:$ Now suppose that we begin with an element $x$ of the form $$x = \sum_{S\subseteq\{m-n+1,\ldots,m\}} (-1)^{\lvert S\rvert} Ts(f_{S\cup\{1,\ldots,m-n\}}) (y).$$ Then for $m-n+1{\leqslant}i{\leqslant}m$ we have $$\begin{aligned}
Ts(f_{\{i\}})(x) = \sum_{S\subseteq\{m-n+1,\ldots,m\}\smallsetminus\{i\}} &(-1)^{\lvert S\rvert} Ts(f_{\{i\}}) \circ Ts(f_{S\cup\{1,\ldots,m-n\}})(y) \\
&+ (-1)^{\lvert S\rvert +1} Ts(f_{\{i\}}) \circ Ts(f_{S\cup\{i\}\cup\{1,\ldots,m-n\}}) (y)\end{aligned}$$ which vanishes since the terms pairwise cancel, so $x\in\mathrm{ker}(Ts(f_{\{i\}}))$. One can similarly show that $Ts(f_{\{1,\ldots,m-n\}})(x)=x$, so $x\in\mathrm{im}(Ts(f_{\{1,\ldots,m-n\}}))$.
Two notions of height on a cyclic monoidal category. {#para:compare-two-heights}
----------------------------------------------------
There is an overlap between the definition in §\[para:specialise-DV\] of the height of a functor ${\mathcal{C}}\to{\mathcal{A}}$ when ${\mathcal{C}}$ is equipped with a monoidal structure[^16] with null unit and the definition in §\[para:specialise-this-paper\] of the height of a functor ${\mathcal{C}}\to{\mathcal{A}}$ when ${\mathcal{C}}$ is equipped with a functor ${\mathcal{I}}\to{\mathcal{C}}$.
Recall that ${\mathcal{I}}$ and $\Sigma$ have objects $0,1,2,\ldots$, morphisms $m \to n$ of $\Sigma$ are partially-defined injections ${\ensuremath{\underline{m}}}\to {\ensuremath{\underline{n}}}$ and morphisms of ${\mathcal{I}}$ are those partially-defined injections that are the identity wherever they are defined. Let ${\mathcal{B}}$ have the same objects and take the morphisms $m \to n$ of ${\mathcal{B}}$ to be partially-defined braided injections from ${\ensuremath{\underline{m}}}$ to ${\ensuremath{\underline{n}}}$. In other words, it is the category ${\mathcal{B}}({\mathbb{R}}^2)$ from §\[sss:some-functors\]. There is an embedding ${\mathcal{I}}\subset {\mathcal{B}}$ and a functor ${\mathcal{B}}\to \Sigma$ which compose to an embedding ${\mathcal{I}}\subset \Sigma$.
Now let ${\mathcal{C}}$ be a strict monoidal category and pick an object $x$ of ${\mathcal{C}}$. There is then a natural functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$ that takes $n$ to $x^{\oplus n}$. If ${\mathcal{C}}$ is braided then $s$ extends to a monoidal functor ${\mathcal{B}}\to {\mathcal{C}}$ and if it is symmetric then $s$ extends further to a monoidal functor $\Sigma \to {\mathcal{C}}$.[^17]
Now assume that the unit object of ${\mathcal{C}}$ is null and that ${\mathcal{C}}$ is *generated* by $x$ in the sense that every object of ${\mathcal{C}}$ is isomorphic to $x^{\oplus n}$ for some (not necessarily unique) non-negative integer $n$. In this sense we may say that ${\mathcal{C}}$ is a *cyclic monoidal category*. For example, if the manifold $M$ splits as ${\mathbb{R}}\times N$ for some manifold $N$, then the category ${\mathcal{B}}(M,X)$ defined in §\[sss:some-functors\] is a cyclic monoidal category generated by the object $1$. The natural functor $s \colon {\mathcal{I}}\to {\mathcal{B}}(M,X)$ taking $1$ to $1$ is exactly the one constructed in the paragraph below . If $N$ splits further as ${\mathbb{R}}\times N^\prime$ then ${\mathcal{B}}(M,X)$ is braided, and if $N = {\mathbb{R}}^2 \times N^{\prime\prime}$ then it is symmetric. One may then define the *height* of a functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ either using the monoidal structure of ${\mathcal{C}}$ as in §\[para:specialise-DV\] – this will be denoted $\mathrm{ht}_\oplus(T)$ – or using the functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$ as in §\[para:specialise-this-paper\] – this will be denoted $\mathrm{ht}_{\mathcal{I}}(T)$.
\[prop:compare-two-heights-special-case\] For any functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$ we have $\mathrm{ht}_{\mathcal{I}}(T) {\leqslant}\mathrm{ht}_\oplus(T)$. If we assume that ${\mathcal{C}}$ is braided we have an equality $\mathrm{ht}_{\mathcal{I}}(T) = \mathrm{ht}_\oplus(T)$.
We will prove this as a corollary of a slightly more general setup.
\[def:o-s-partition\] Fix $m,n{\geqslant}0$. An *ordered shifted partition* $\lambda$ of $m$ of *length* $n$ – written $\lambda\vdash m$ and $\lvert\lambda\rvert = n$ – is an ordered $(n+1)$-tuple $(\lambda_0,\lambda_1,\ldots,\lambda_n)$ of non-negative integers whose sum is $m$. Associated to this there is a semigroup homomorphism $\psi_\lambda \colon {\ensuremath{\mathcal{I}_{n}}} \to {\ensuremath{\mathcal{I}_{m}}}$ taking a subset $S$ of ${\ensuremath{\underline{n}}}$ to the subset $S_\lambda$ of ${\ensuremath{\underline{m}}}$, where $S_\lambda$ is defined as follows: $$S_\lambda = \bigcup_{i\in S}\{i\}_\lambda \qquad\qquad \{i\}_\lambda = \{ \lambda_0 + \cdots + \lambda_{i-1} + 1, \ldots, \lambda_0 + \cdots + \lambda_i \}.$$ We are viewing ${\mathcal{I}}_n$ as a monoid under intersection, with identity element ${\ensuremath{\underline{n}}}$, so $\psi_\lambda$ is a monoid homomorphism if and only if $\lambda_0 = 0$.
\[def:two-types-of-degree\] Now let ${\mathcal{C}}$ be any category and $s \colon {\mathcal{I}}\to {\mathcal{C}}$ a functor. We obtain (semi-)functors $f_\lambda\colon {\ensuremath{\mathcal{I}_{n}}}\to{\mathcal{C}}$ defined by $${\ensuremath{\mathcal{I}_{n}}} \xrightarrow{\, \psi_\lambda \,} {\ensuremath{\mathcal{I}_{m}}} = \mathrm{End}_{\mathcal{I}}(m) \hookrightarrow {\mathcal{I}}\xrightarrow{\; s\;} {\mathcal{C}}.$$ For any functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$, we define $\mathrm{ht}_{\mathcal{I}}(T)$ and $\mathrm{ht}_\oplus(T)$ by the condition that (for $\square = {\mathcal{I}}\text{ or } {\oplus}$) $\mathrm{ht}_\square(T){\leqslant}h$ if and only if for each ordered shifted partition $\lambda\vdash m$ of length $\lvert\lambda\rvert >h$,
with $\lambda_0=0$,
with $\lambda_1 = \cdots = \lambda_n = 1$,
the cross-effect $\mathrm{cr}(Tf_\lambda)$ vanishes. We similarly define ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_\square(T)$ using ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(Tf_\lambda z)$ in place of $\mathrm{cr}(Tf_\lambda)$. In other words, when $\square = {\mathcal{I}}$ we define the height using (for each $n{\geqslant}0$) the collection of semi-functors $\{f_\lambda \colon {\ensuremath{\mathcal{I}_{n}}} \to {\mathcal{C}}\mid \lambda \vdash m, \lvert\lambda\rvert = n, \lambda_1 = \cdots = \lambda_n = 1 \}$ and when $\square = \oplus$ we define the height using the collection of functors $\{f_\lambda \colon {\ensuremath{\mathcal{I}_{n}}} \to {\mathcal{C}}\mid \lambda \vdash m, \lvert\lambda\rvert = n, \lambda_0 = 0 \}$.
In the previous setup, with a cyclic monoidal category ${\mathcal{C}}$ generated by the object $x$, we had a natural functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$ taking $1$ to $x$. For a functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$ we defined $\mathrm{ht}_{\mathcal{I}}(T)$ to be the height of $T$ as defined in §\[para:specialise-this-paper\], using the structure given by the functor $s$. This is exactly the same as the definition of $\mathrm{ht}_{\mathcal{I}}(T)$ given in Definition \[def:two-types-of-degree\]. Moreover, we defined $\mathrm{ht}_\oplus(T)$ to be the height of $T$ as defined in §\[para:specialise-DV\], using the monoidal structure of ${\mathcal{C}}$. Unravelling the definitions, one can see that this is exactly the same as the definition of $\mathrm{ht}_\oplus(T)$ given in Definition \[def:two-types-of-degree\], using just the functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$. (For this fact, it is critical that ${\mathcal{C}}$ is generated by the object $x$.)
Thus Definition \[def:two-types-of-degree\] for a category ${\mathcal{C}}$ equipped with a functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$ generalises the setting described at the beginning of this subsection, where the functor $s$ arose from the structure of ${\mathcal{C}}$ as a cyclic monoidal category.
For the rest of this subsection, unless otherwise stated, we assume that we are in the general setting of a category ${\mathcal{C}}$ equipped with a functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$, and we use the definitions of height from Definition \[def:two-types-of-degree\].
\[rmk:add-assumption-to-defn\] It is not hard to see that if $\lambda_i=0$ for some $i{\geqslant}1$ then $\mathrm{cr}(Tf_\lambda)=0$. Thus, if $\lambda_0=0$ too (so that $f_\lambda$ is a functor and Lemma \[lem:two-definitions\] applies), we have ${\ensuremath{\,\overline{\! \mathrm{cr}\!}\,}}(Tf_\lambda z)=0$. So in Definition \[def:two-types-of-degree\], when $\square = \oplus$, we may assume that $\lambda_1,\ldots,\lambda_n {\geqslant}1$.
When $\square=\oplus$ the $f_\lambda$ involved in the definition are all *functors*, so $\mathrm{ht}_\oplus(T) = {\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_\oplus(T)$, by the discussion following Lemma \[lem:two-definitions\]. When $\square={\mathcal{I}}$ we only know that $\mathrm{ht}_{\mathcal{I}}(T) {\leqslant}{\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T)$, as discussed in §\[para:semi-functors\]. We first show that ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T)$ is in fact almost always infinite.
\[prop:htbar-is-infinite\] Let $T\colon{\mathcal{C}}\to{\mathcal{A}}$ be any functor. Then ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T)>0$ implies that ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T)=\infty$.
So it remains to compare $\mathrm{ht}_{\mathcal{I}}(T)$ and $\mathrm{ht}_\oplus(T)$, which we do after the next definition.
\[def:admits-conjugations\] We say that a functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ *admits conjugations* if the composite functor $Ts\colon{\mathcal{I}}\to{\mathcal{A}}$ extends to some category ${\mathcal{S}}\supset{\mathcal{I}}$ and for each $n{\geqslant}0$ and $R,S\subseteq{\ensuremath{\underline{n}}}$ with $\lvert R\rvert = \lvert S\rvert$ there exists an automorphism $\phi \in \mathrm{Aut}_{\mathcal{S}}(n)$ such that $\phi r_R \phi^{-1} = r_S$, where $r_R\in\mathrm{End}_{\mathcal{I}}(n)$ denotes the endomorphism that restricts to $R$, i.e., is the identity on $R$ and undefined on ${\ensuremath{\underline{n}}}\smallsetminus R$.
\[eg:admitting-conjugations-braiding\] If ${\mathcal{C}}$ is a strict *braided* monoidal category with null unit object, generated by the object $x$, then the natural functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$ extends to ${\mathcal{B}}\supset {\mathcal{I}}$, as explained at the beginning of this subsection. In this case *every* functor ${\mathcal{C}}\to {\mathcal{A}}$ admits conjugations: we may take ${\mathcal{S}}={\mathcal{B}}$ and for $\phi$ choose any braid connecting the points $R$ with the points $S$ and the points ${\ensuremath{\underline{n}}}\smallsetminus R$ with the points ${\ensuremath{\underline{n}}}\smallsetminus S$. In particular, this applies to ${\mathcal{C}}= {\mathcal{B}}(M,X)$ as defined in §\[sss:some-functors\] when $M$ is of the form ${\mathbb{R}}^2 \times N$.
\[eg:admitting-conjugations-BMX\] In fact, for any $M$ (of dimension at least two), if we take ${\mathcal{C}}= {\mathcal{B}}(M,X)$ with the natural functor $s \colon {\mathcal{I}}\to {\mathcal{B}}(M,X)$ ([*cf*. ]{}), then every functor ${\mathcal{C}}\to {\mathcal{A}}$ admits conjugations: we may take ${\mathcal{S}}$ to be ${\mathcal{B}}(M,X)$ itself and for $\phi$ choose any braid on $M$ that connects the points $\{a_i \mid i\in R\}$ with the points $\{a_i \mid i\in S\}$ and the points $\{a_i \mid i\in{\ensuremath{\underline{n}}}\smallsetminus R\}$ with the points $\{a_i \mid i\in{\ensuremath{\underline{n}}}\smallsetminus S\}$.
\[prop:compare-two-heights\] For any functor $T\colon{\mathcal{C}}\to{\mathcal{A}}$ we have $\mathrm{ht}_{\mathcal{I}}(T) {\leqslant}\mathrm{ht}_\oplus(T)$. If $T$ admits conjugations then $\mathrm{ht}_{\mathcal{I}}(T) = \mathrm{ht}_\oplus(T)$. However, in general the inequality may be strict: for any $h\in\{2,\ldots,\infty\}$ there exists a functor $T_h\colon {\mathcal{I}}\to\mathsf{Ab}$ such that $\mathrm{ht}_{\mathcal{I}}(T_h)=1$ but $\mathrm{ht}_\oplus(T_h)=h$.
This now follows from Proposition \[prop:compare-two-heights\] and Example \[eg:admitting-conjugations-braiding\].
\[rmk:compare-heights-for-BMX\] Proposition \[prop:compare-two-heights-special-case\] applied to the cyclic monoidal category ${\mathcal{C}}= {\mathcal{B}}({\mathbb{R}}\times N,X)$ tells us that $\mathrm{ht}_{\mathcal{I}}(-) {\leqslant}\mathrm{ht}_\oplus(-)$ with equality if $N = {\mathbb{R}}\times N^\prime$. But by Proposition \[prop:compare-two-heights\] and Example \[eg:admitting-conjugations-BMX\] we know that in fact $\mathrm{ht}_{\mathcal{I}}(-) = \mathrm{ht}_\oplus(-)$ for ${\mathcal{C}}= {\mathcal{B}}(M,X)$ for *any* $M$ (of dimension at least two). This suggests that it should be possible to generalise Proposition \[prop:compare-two-heights-special-case\] to a setting where ${\mathcal{C}}$ is a left module over a cyclic monoidal category, analogously to Proposition \[p:two-degrees-agree-2\] for degree.
\[rmk:summary\] For a functor $T \colon {\mathcal{C}}\to {\mathcal{A}}$ we have the following square of equalities: $$\label{eq:square-of-equalities}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node at (0,12) {$\mathrm{deg}^{x}(T)$};
\node at (24,12) {$\mathrm{deg}(T)$};
\node at (0,0) {$\mathrm{ht}_{{\mathcal{I}}}(T)$};
\node at (24,0) {$\mathrm{ht}_{\oplus}(T)$};
\node at (12,12) {$=$};
\node at (12,14.5) [font=\footnotesize] {(a)};
\node at (12,0) {$=$};
\node at (12,-2.5) [font=\footnotesize] {(b)};
\node at (0,6) {\rotatebox{90}{$=$}};
\node at (-3,6) [font=\footnotesize] {(c)};
\node at (24,6) {\rotatebox{90}{$=$}};
\node at (27,6) [font=\footnotesize] {(d)};
\end{tikzpicture}
\end{split}$$ (using notation of §\[para:partial-braid-categories\] in the top row), where
holds when ${\mathcal{C}}$ is a left-module over a braided monoidal category with null unit and generating object $x$ (Proposition \[p:two-degrees-agree-2\]);
holds when ${\mathcal{C}}$ is a braided monoidal category with null unit and generating object $x$ (Proposition \[prop:compare-two-heights-special-case\]) or ${\mathcal{C}}= {\mathcal{B}}(M,X)$ (Remark \[rmk:compare-heights-for-BMX\]);
holds when ${\mathcal{C}}= {\mathcal{B}}(M,X)$, by Lemma 3.16 of [@Palmer2018Twistedhomologicalstability]
holds when ${\mathcal{C}}= {\mathcal{B}}({\mathbb{R}}^3 \times N,X)$, by Proposition 2.3 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux] — more generally, they prove this for ${\mathcal{C}}$ a symmetric monoidal category with initial unit.
Putting these together, we see that in fact (d) holds whenever ${\mathcal{C}}= {\mathcal{B}}(M,X)$ for *any* $M$, via (a)–(c). Also, (c) holds whenever ${\mathcal{C}}$ is a symmetric monoidal category with null unit and generating object $x$, via (a), (b) and (d).
This suggests that (c) should generalise to ${\mathcal{C}}$ any left-module over a braided monoidal category with null unit and generating object $x$ and (d) should generalise to ${\mathcal{C}}$ any left-module over a braided monoidal category with initial unit. This would imply that (b) also generalises to ${\mathcal{C}}$ any left-module over a braided monoidal category with null unit and generating object $x$ ([*cf*. ]{}Remark \[rmk:compare-heights-for-BMX\]).
In the remainder of this subsection we prove Propositions \[prop:htbar-is-infinite\] and \[prop:compare-two-heights\].
Let us abbreviate $Ts(n)$ to $T_n$ and for a subset $S\subseteq\{1,\ldots,\mathrm{min}(k,l)\}$ let us write simply $r_S\colon T_k \to T_l$ instead of $Ts(r_S)$. (Recall that $r_S\colon \underline{k} \to \underline{l}$ is the identity on $S$ and undefined elsewhere.)
Now suppose that ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T)<\infty$. In particular this implies that, for some $h{\geqslant}0$ and any $k{\geqslant}0$, $$T_{k+h} = \sum_{S\subsetneq\underline{h}} \mathrm{im}(r_{S+k}).$$ But each $\mathrm{im}(r_{S+k})$ is contained in $\mathrm{im}(r_{\{k+1,\ldots,k+h\}})$, so $r_{\{k+1,\ldots,k+h\}} \colon T_{k+h} \to T_{k+h}$ is surjective. Also note that $r_{\underline{k}} \colon T_{k+h} \to T_k$ is surjective, since it has a right-inverse. The commutative square $$\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tm) at (0,15) {$T_{k+h}$};
\node (tr) at (40,15) {$T_k$};
\node (bm) at (0,0) {$T_{k+h}$};
\node (br) at (40,0) {$T_k$};
\draw[->>] (tm) to node[above,font=\small]{$r_{\underline{k}}$} (tr);
\draw[->>] (bm) to node[left,font=\small]{$r_{\{k+1,\ldots,k+h\}}$} (tm);
\draw[->] (bm) to node[above,font=\small]{$r_\varnothing$} (br);
\draw[->] (br) to node[right,font=\small]{$r_\varnothing$} (tr);
\draw[draw=none,use as bounding box](-20,0) rectangle (60,18);
\end{tikzpicture}
\end{split}$$ therefore tells us that $r_\varnothing \colon T_k \to T_k$ is also surjective. So for any $m{\geqslant}n>0$ we have $$T_m = \mathrm{im}(r_\varnothing) = \sum_{S\subsetneq{\ensuremath{\underline{n}}}} \mathrm{im}(r_{S+m-n}),$$ and so ${\ensuremath{\,\overline{\raisebox{0pt}[1.3ex][0pt]{\ensuremath{\! \mathrm{ht}\!}}}\,}}_{\mathcal{I}}(T){\leqslant}0$.
We first prove the inequality $\mathrm{ht}_{\mathcal{I}}(T){\leqslant}\mathrm{ht}_\oplus(T)$, then give the promised example of $T$ where it is strict, and then finally show that the additional assumption that $T$ admits conjugations rules out this possibility, i.e., that $\mathrm{ht}_{\mathcal{I}}(T)=\mathrm{ht}_\oplus(T)$ for such $T$.
\(a) *Proof of the inequality.* We use the notation of the previous proof, abbreviating $Ts(r_S)$ to $r_S$. Let $m{\geqslant}n>\mathrm{ht}_\oplus(T)$. We need to show that $$\label{eq:proof-of-inequality}
\sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} r_{({\ensuremath{\underline{n}}}\smallsetminus S)+m-n}$$ is the zero morphism. Let $\lambda$ be the ordered shifted partition of length $n$ with $\lambda_0=0$, $\lambda_1=m-n+1$ and $\lambda_i=1$ otherwise. Then is equal to $$\sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} r_{\{m-n+1,\ldots,m\}} \circ r_{S_\lambda} \;=\; r_{\{m-n+1,\ldots,m\}} \circ \biggl( \sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} \circ r_{S_\lambda} \biggr) .$$ Since $\lvert \lambda \rvert = n > \mathrm{ht}_\oplus(T)$, the morphism in brackets on the right-hand side is zero, and so is zero, as required.$\diamond$
\(b) *Example of strict inequality.* For this example we will take ${\mathcal{C}}$ to be ${\mathcal{I}}$ itself, with $s=\mathrm{id}\colon{\mathcal{I}}\to{\mathcal{I}}$. Fix $h\in\{2,3,4,\ldots,\infty\}$ and define a functor $T_h\colon {\mathcal{I}}\to\mathsf{Ab}$ as follows. The object $n$ is taken to the free abelian group $${\mathbb{Z}}\{ S\subseteq{\ensuremath{\underline{n}}}\mid \lvert S\rvert {\leqslant}h \text{ and } S \text{ has no consecutive elements} \}$$ and for $R\subseteq\{1,\ldots,\mathrm{min}(m,n)\}$ the morphism $r_R\colon {\ensuremath{\underline{m}}}\to {\ensuremath{\underline{n}}}$ is taken to the homomorphism $T_h(m)\to T_h(n)$ that sends the basis element $S\subseteq{\ensuremath{\underline{m}}}$ to the basis element $r_R(S) = S\cap R \subseteq {\ensuremath{\underline{n}}}$.
This will turn out to have $\mathrm{ht}_{\mathcal{I}}(T_h)=1<h=\mathrm{ht}_\oplus(T_h)$. The idea is that both $\mathrm{ht}_{\mathcal{I}}(-)$ and $\mathrm{ht}_\oplus(-)$ examine a functor $T$ using certain partitions – but the former only uses partitions in which each piece has size $1$ and is therefore sensitive to “interference” from the condition above that subsets have *no consecutive elements* and therefore measures the “wrong” height, whereas the latter uses partitions with pieces of arbitrary size, and so is insensitive to such interference.
To show that $\mathrm{ht}_\oplus(T_h)=h$ we take $\lambda\vdash m$ with $\lambda_0=0$ and $\lvert\lambda\rvert =n$ and a basis element $R\subseteq{\ensuremath{\underline{m}}}$ for $T_h(m)$, and consider the element $$\label{eq:element-of-Thm}
\sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} (R\smallsetminus S_\lambda)$$ of $T_h(m)$. We need to show that it is always zero when $n>h$, whereas when $n=h$ there exist $\lambda$ and $R$ such that it is non-zero. If $n>h$ there must be some $i\in\{1,\ldots,n\}$ such that $R\cap\{i\}_\lambda = \varnothing$. Then we may write as the sum over $S\subseteq{\ensuremath{\underline{n}}}\smallsetminus\{i\}$ of $(-1)^{\lvert S\rvert} (R\smallsetminus S_\lambda) + (-1)^{\lvert S\rvert +1} ((R\smallsetminus \{i\}_\lambda)\smallsetminus S_\lambda)$, which cancels to zero since $R\smallsetminus\{i\}_\lambda = R$. When $n=h$ we may take $\lambda$ with $\lambda_0=0$ and $\lambda_i=2$ for $i{\geqslant}1$ (so $m=2n$) and $R=\{2,4,\ldots,2n\}$. This completes the proof that $\mathrm{ht}_\oplus(T_h)=h$.
Now we show that $\mathrm{ht}_{\mathcal{I}}(T_h)=1$. To begin with, note that to have $\mathrm{ht}_{\mathcal{I}}(T_h){\leqslant}0$ would require that $T_h(r_\varnothing)=\mathrm{id}$, which is not the case, so instead we have $\mathrm{ht}_{\mathcal{I}}(T_h) {\geqslant}1$. To see that it is exactly equal to $1$ we need to show that, for all $m{\geqslant}n{\geqslant}2$ and any basis element $R$ of $T_h(m)$, the element $$\sum_{S\subseteq\{m-n+1,\ldots,m\}} (-1)^{\lvert S\rvert} (R\smallsetminus S)$$ is zero. The trick is to rewrite this element as the sum over subsets $S\subseteq\{m-n+1,\ldots,m-2\}$ of $$(-1)^{\lvert S\rvert} \Bigl( Q + (Q \smallsetminus \{m-1,m\}) - (Q \smallsetminus \{m-1\}) - (Q \smallsetminus \{m\}) \Bigr)$$ where we have written $Q = R\smallsetminus S$. Since $R$ (and therefore also $Q$) cannot contain both $m-1$ and $m$ (these would be consecutive elements), the four terms above cancel to zero. This completes the proof that $\mathrm{ht}_{\mathcal{I}}(T_h)=1$.$\diamond$
\(c) *Equality when $T$ admits conjugations.* To show this we will use the following fact, which is an immediate generalisation of Lemma \[lem:isomorphism-of-subobjects\].
\[fact:equality-of-two-subobjects\] If $\lambda\vdash m$ is an ordered shifted partition of length $n$ and $T\colon{\mathcal{C}}\to{\mathcal{A}}$ is a functor, then $$\mathrm{im} \biggl( \sum_{S\subseteq{\ensuremath{\underline{n}}}} (-1)^{\lvert S\rvert} Ts(f_{\{1,\ldots,\lambda_0\} \cup S_\lambda}) \biggr) \;=\; \mathrm{im}(Ts(f_{\{1,\ldots,\lambda_0\}})) \cap \bigcap_{i=1}^n \mathrm{ker}(Ts(f_{\{i\}_\lambda})).$$
Let $T\colon{\mathcal{C}}\to{\mathcal{A}}$ be a functor and assume that $T$ admits conjugations. Suppose that $\mathrm{ht}_{\mathcal{I}}(T){\leqslant}h$. Our aim is to show that $\mathrm{ht}_\oplus(T){\leqslant}h$. In detail, this means the following. Fix $\lambda\vdash m$ with $\lambda_0=0$ and $\lambda_i{\geqslant}1$ for $i{\geqslant}1$ ([*cf*. ]{}Remark \[rmk:add-assumption-to-defn\]) and $\lvert\lambda\rvert =n>h$. In the light of Fact \[fact:equality-of-two-subobjects\], our aim is to show that $$\label{eq:compare-two-heights}
\bigcap_{i=1}^n \mathrm{ker}(Ts(f_{\{i\}_\lambda}))$$ is zero. Since $\mathrm{ht}_{\mathcal{I}}(T){\leqslant}h$, we know (using Lemma \[lem:isomorphism-of-subobjects\] and the fact that $T$ admits conjugations) that for any $S\subseteq{\ensuremath{\underline{m}}}$ of size $\lvert S\rvert >h$, $$\mathrm{im}(Ts(f_{{\ensuremath{\underline{m}}}\smallsetminus S})) \cap \bigcap_{i\in S} \mathrm{ker}(Ts(f_{\{i\}})) = 0.$$ We claim that the following equality always holds: $$\label{eq:claim-equality}
\bigcap_{i=1}^n \mathrm{ker}(Ts(f_{\{i\}_\lambda})) \;=\; \bigoplus_{S} \mathrm{im}(Ts(f_{{\ensuremath{\underline{m}}}\smallsetminus S})) \cap \bigcap_{i\in S} \mathrm{ker}(Ts(f_{\{i\}})),$$ where the direct sum on the right-hand side is taken over all subsets $S\subseteq{\ensuremath{\underline{m}}}$ such that for each $i\in\{1,\ldots,n\}$ we have $S\cap\{i\}_\lambda \neq \varnothing$. Any such subset must have size $\lvert S\rvert {\geqslant}\lvert \lambda \rvert =n>h$, so – in our situation – its contribution to the sum vanishes, and therefore is zero, as required. So it just remains to prove the equality .
$\bullet\; (\supseteq) :$ Let $S\subseteq{\ensuremath{\underline{m}}}$ satisfy the condition above and suppose that $Ts(f_{\{i\}})(x)=0$ for all $i\in S$. For each $j\in\{1,\ldots,n\}$ we may choose $i\in S\cap \{j\}_\lambda$ and compute that $$Ts(f_{\{j\}_\lambda})(x) = Ts(f_{\{j\}_\lambda}) \circ Ts(f_{\{i\}})(x) = 0.$$
$\bullet\; (\subseteq) :$ Since the idempotents $Ts(f_{\{i\}})$ on $Ts(m)$ pairwise commute there is a decomposition $$\begin{aligned}
Ts(m) \;&=\; \bigoplus_{S\subseteq{\ensuremath{\underline{m}}}} \bigcap_{i\in S} \mathrm{ker}(Ts(f_{\{i\}})) \cap \bigcap_{i\in{\ensuremath{\underline{m}}}\smallsetminus S} \mathrm{im}(Ts(f_{\{i\}})) \nonumber \\
&=\; \bigoplus_{S\subseteq{\ensuremath{\underline{m}}}} \bigcap_{i\in S} \mathrm{ker}(Ts(f_{\{i\}})) \cap \mathrm{im}(Ts(f_{{\ensuremath{\underline{m}}}\smallsetminus S})). \label{eq:decomposition}\end{aligned}$$ Now suppose that $x\in Ts(m)$ and $Ts(f_{\{i\}_\lambda})(x)=0$ for each $i\in\{1,\ldots,n\}$. We may write $$x=\sum_{S\subseteq{\ensuremath{\underline{m}}}} x_S$$ corresponding to the decomposition . Note that the endomorphism $Ts(f_{\{i\}_\lambda})$ preserves the decomposition . Since it is a decomposition as a *direct* sum, we must have $Ts(f_{\{i\}_\lambda})(x_S)=0$ for each $S\subseteq{\ensuremath{\underline{m}}}$.
Now, to see that $x$ is contained in the right-hand side of we just need to show that if there exists $i\in\{1,\ldots,n\}$ such that $S\cap\{i\}_\lambda = \varnothing$ then $x_S=0$. But we have $x_S\in\mathrm{im}(Ts(f_{{\ensuremath{\underline{m}}}\smallsetminus S}))$ and $\{i\}_\lambda \subseteq {\ensuremath{\underline{m}}}\smallsetminus S$, so $x_S\in\mathrm{im}(Ts(f_{\{i\}_\lambda}))$. Since $Ts(f_{\{i\}_\lambda})$ is idempotent, this means that
$$x_S = Ts(f_{\{i\}_\lambda})(x_S) = 0. \qedhere$$
The injective braid category. {#para:full-braids}
-----------------------------
Recall from §\[para:degree-WRW\] that the *injective braid category* ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$ is the subcategory of ${\mathcal{B}}(M,X)$ having the same objects (the non-negative integers) and where a morphism of ${\mathcal{B}}(M,X)$ – i.e. a (labelled) braid in $M \times [0,1]$ from some subset of $\{a_1,\ldots,a_m\} \times \{0\}$ to some subset of $\{a_1,\ldots,a_n\} \times \{1\}$ – lies in this subcategory if and only if it has precisely $m$ strands.
The equivalence between the different notions of height discussed in this section suggests how one may extend the notion of height for functors $T \colon {\mathcal{B}}(M,X) \to {\mathcal{A}}$ to a notion of height for functors $T \colon {\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X) \to {\mathcal{A}}$. If we take our definition of the height of a functor defined on ${\mathcal{B}}(M,X)$, which uses the first definition (§\[para:first-def\]) of height (see the discussion in §\[para:specialise-this-paper\] above), and reinterpret it using instead the second definition (§\[para:second-def\]) of height, it may be rewritten in such a way that it involves only morphisms from the subcategory ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$. Thus, the height of $T$ depends only on its restriction to ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$, and indeed one may use this observation to directly define the height of a functor $T \colon {\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X) \to {\mathcal{A}}$. Explicitly, the definition unravels to the following: $\mathrm{height}(T){\leqslant}h$ if and only if for all $m{\geqslant}n>h$ we have $$\sum_S \mathrm{coker} \bigl( T(b(\phi_{m,S})) \colon T(\underline{s}) \longrightarrow T({\ensuremath{\underline{m}}}) \bigr) =0,$$ where the sum is taken over all proper subsets $S$ of $\{ m-n+1, \ldots, m \}$ and $\underline{s}$ denotes $\{ 1,\ldots,\lvert S\rvert \}$. The notation $\phi_{m,S}$ means the unique order-preserving injection $\underline{s} \to {\ensuremath{\underline{m}}}$ whose image is equal to $S \subseteq {\ensuremath{\underline{m}}}$. In general, given any order-preserving injection $\phi \colon \underline{s} \to {\ensuremath{\underline{m}}}$, there is a canonical braid $b(\phi)$ in $M \times [0,1]$ from $\{a_1,\ldots,a_s\} \times \{0\}$ to $\{a_{\phi(1)},\ldots,a_{\phi(s)}\} \times \{1\}$ that realises $\phi$, specified as follows. Recall from §\[sss:some-categories\] that the manifold $M$ comes equipped with a collar neighbourhood $c \colon \partial M \times [0,\infty] \hookrightarrow M$ and a basepoint $p \in \partial M$. Let $L$ be the embedded arc $c(\{p\} \times [1,\infty])$ in the interior of $M$. Then $b(\phi)$ is uniquely determined by specifying its endpoints, as above, and that it must be contained in the embedded square $L \times [0,1]$ in $M \times [0,1]$. Labelling each strand of $b(\phi)$ by the constant path at the basepoint $x_0$ of $X$ makes it into a morphism $\underline{s} \to {\ensuremath{\underline{m}}}$ of ${\ensuremath{\mathcal{B}_{\mathsf{f}}}}(M,X)$.
Possible extensions. {#para:generalisations}
--------------------
We finish this section by suggesting potential extensions of the general definitions of *height* given in §§\[para:first-def\]—\[para:third-def\]. One generalisation, which we have already discussed in detail, is to consider categories ${\mathcal{C}}$ equipped with collections of *semi-*functors ${\mathcal{J}}_n \to {\mathcal{C}}$, i.e., “functors” that are not required to preserve identities (the notation ${\mathcal{J}}_n$ denotes any one of ${\ensuremath{\mathcal{I}_{n}}}$, ${\ensuremath{\bar{\mathcal{I}}_{n}}}$ or ${\ensuremath{\bar{\mathcal{I}}_{n}^{\mathsf{op}}}}$). Another potential generalisation, which was mentioned in §\[para:finite-coproducts\], is to consider twisted coefficient systems (i.e., functors or semi-functors) $T\colon {\mathcal{C}}\to {\mathcal{A}}$ whose target is a *semi-*abelian category, such as the category $\mathsf{Grp}$ of groups. This is motivated by the work of Hartl, Pirashvili and Vespa [@HartlPirashviliVespa2015Polynomialfunctorsalgebras], who study functors of the form ${\mathcal{C}}\to \mathsf{Grp}$ when ${\mathcal{C}}$ admits finite coproducts and a null object.
More fundamentally, one could weaken the structure on ${\mathcal{C}}$ by replacing Boolean algebras with posets possessing less structure. If we work in the setting of §\[para:second-def\], then the structure on ${\mathcal{C}}$ is given by collections of functors ${\ensuremath{\bar{\mathcal{I}}_{n}}} \to {\mathcal{C}}$, where ${\ensuremath{\bar{\mathcal{I}}_{n}}}$ is the poset of subsets of $\{1,\ldots,n\}$ under inclusion, which is a Boolean algebra. It would be interesting to set up a theory of polynomial functors ${\mathcal{C}}\to {\mathcal{A}}$ when ${\mathcal{C}}$ is instead equipped with collections of functors $L(n) \to {\mathcal{C}}$, where the $L(n)$ are lattices with less structure than a Boolean algebra, for example orthocomplemented lattices (in which $\vee$ and $\wedge$ do not necessarily distribute over each other). The lattice of closed subspaces of a Hilbert space is an orthocomplemented lattice, for example, so a natural example to consider would be to take $L(n)$ as the lattice of subspaces of the Hilbert space ${\mathbb{C}}^n$.
Partial braid categories {#sec:functorial-configuration-spaces}
========================
The paper [@Palmer2018Twistedhomologicalstability] is concerned with proving twisted homological stability for the labelled configuration spaces $C_n(M,X)$, for $M$ a connected, open manifold and $X$ a path-connected space. Its twisted coefficient systems are indexed by certain *partial braid categories* ${\mathcal{B}}(M,X)$ associated to these data; in that paper they are defined in a slightly ad hoc way, and the *height* and *degree* of a twisted coefficient system on ${\mathcal{B}}(M,X)$ is defined in this specific context. In this section, we explain a natural functorial framework into which these constructions fit.
The notions of degree and height used in this section agree with those discussed in the previous two sections (whenever both are defined), but the domains of definition are slightly different. The degree in this section is simply defined as a special case of the degree of §\[sec:inductive-degree\], assuming that the source category is an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ rather than of the larger category ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$. The height in this section is defined when the source category is an object of ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$. Such an object is in particular a category ${\mathcal{C}}$ equipped with a functor ${\mathcal{I}}\to {\mathcal{C}}$, where ${\mathcal{I}}$ is a certain category ([*cf*. ]{}§\[para:first-def\]). The general definition of height given in §\[sec:cross-effects\] specialises to this case, as described in §\[para:specialise-this-paper\], and it agrees with the definition given in this section (see Remark \[rmk:two-defs-of-height-agree\]).
For this section, we will take the abelian category ${\mathcal{A}}$ to be the category ${\ensuremath{\mathsf{Ab}}}$ of abelian groups. However, this is just in order to preserve notational similarity with [@Palmer2018Twistedhomologicalstability], and in fact everything generalises directly to the setting of an arbitrary abelian category ${\mathcal{A}}$.
Some categories. {#sss:some-categories}
----------------
We first introduce some $(2,1)$-categories that we will consider. Only the first one has non-identity $2$-morphisms; the other two are really just $1$-categories.
$\bullet\; {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$: Objects are smooth manifolds $M$ (of dimension at least two) equipped with a collar neighbourhood and a basepoint on the boundary. The $1$-morphisms are embeddings preserving collar neighbourhoods and basepoints and $2$-morphisms are isotopies of such embeddings.
More precisely, a collar neighbourhood means a proper embedding $$c \colon \partial M \times [0,\infty] \longrightarrow M$$ such that $c(x,0)=x$ for all $x\in\partial M$. A $1$-morphism from $(M,c_M,p)$ to $(N,c_N,q)$ is then an embedding $f \colon M \hookrightarrow N$ taking $p\in\partial M$ to $q\in\partial N$ and commuting with the collar neighbourhoods, meaning that $f(c_M(x,t)) = c_N(f(x),t)$ for all $x\in\partial M$ and $t\in [0,\infty]$. A $2$-morphism from $f_0$ to $f_1$ is an isotopy $f_s$ such that $f_s(p)=q$ and $f_s(c_M(x,t)) = c_N(f_s(x),t)$ for all $x$, $t$ and $s$.
$\bullet\; {\ensuremath{\mathsf{Top}_{\circ}}}$: The category of based, path-connected topological spaces and based continuous maps.
$\bullet\; {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$: Objects are small $1$-categories ${\mathcal{C}}$ equipped with an endofunctor $s\colon {\mathcal{C}}\to {\mathcal{C}}$ and a natural transformation $\imath \colon \mathrm{id} \to s$. A $1$-morphism from $({\mathcal{C}},s,\imath)$ to $({\mathcal{D}},t,\jmath)$ is a functor $f\colon {\mathcal{C}}\to {\mathcal{D}}$ together with a natural isomorphism $\psi \colon f\circ s \to t\circ f$ of functors ${\mathcal{C}}\to {\mathcal{D}}$ such that $\jmath * \mathrm{id}_f = \psi \circ (\mathrm{id}_f * \imath)$, where $*$ denotes horizontal composition of natural transformations. (Note that this is a subcategory of the category ${\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$ defined in §\[para:gen-def-degree\].)
The partial braid functor. {#sss:some-functors}
--------------------------
This is a $2$-functor $$\label{eq:partial-braid-functor}
{\mathcal{B}}\colon {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}\longrightarrow {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$$ such that, for any manifold $M\in{\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ and any space $X\in{\ensuremath{\mathsf{Top}_{\circ}}}$, the object ${\mathcal{B}}(M,X)\in{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ agrees with the category ${\mathcal{B}}(M,X)$ defined in §2.3 of [@Palmer2018Twistedhomologicalstability] together with the extra data defined in §3.1 of [@Palmer2018Twistedhomologicalstability].
The definition is as follows. Given objects $(M,c,p) \in {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ and $(X,x_0) \in {\ensuremath{\mathsf{Top}_{\circ}}}$, first set $a_t = c(p,t) \in M$ for $t\in [0,\infty]$ and define an embedding $e \colon M \hookrightarrow M$ by $e(c(m,t)) = c(m,t+1)$ for $(m,t) \in \partial M \times [0,\infty]$ and by the identity outside of the collar neighbourhood. The objects of ${\mathcal{B}}(M,X)$ are the non-negative integers. A morphism $m \to n$ is a choice of $k{\leqslant}\mathrm{min}(m,n)$ and a path in $C_k(M,X)$, up to endpoint-preserving homotopy, from a subset of $\{(a_1,x_0),\ldots,(a_m,x_0)\}$ to a subset of $\{(a_1,y_0),\ldots,(a_n,y_0)\}$. These may be thought of as braids in $M \times [0,1]$ whose strands have been labelled by loops in $X$ based at $x_0$. Composition is defined by concatenating paths, and then deleting configuration points for which the concatenated path is defined only half-way. For example, omitting the labels, we have the heuristic picture:
$$\label{eComposition}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
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\begin{scope}[xshift=16mm]
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\node at (32,2.5) {$=$};
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\end{tikzpicture}
\end{split}$$
The endofunctor $s \colon {\mathcal{B}}(M,X) \to {\mathcal{B}}(M,X)$ sends the object $n$ to $n+1$ and sends a morphism $\gamma$, which is a path in $C_k(M,X)$, to the morphism $s_k \circ \gamma$, where $s_k \colon C_k(M,X) \to C_{k+1}(M,X)$ sends a configuration $\{(m_1,x_1),\ldots,(m_k,x_k)\}$ to $\{(a_1,x_0),(e(m_1),x_1),\ldots,(e(m_k),x_k)\}$.
The natural transformation $\imath \colon \mathrm{id} \to s$ consists of the morphisms $n \to n+1$ given by the paths $t \mapsto \{(a_{1+t},x_0),\ldots,(a_{n+t},x_0)\}$.
Given $1$-morphisms $\phi \colon (M,c_M,p) \to (N,c_N,q)$ and $f\colon (X,x_0) \to (Y,y_0)$, we need to specify a functor $F \colon {\mathcal{B}}(M,X) \to {\mathcal{B}}(N,Y)$ and a natural isomorphism $\psi \colon F \circ s \to s \circ F$. In fact, we will define $F$ such that $F \circ s = s \circ F$ and take $\psi$ to be the identity. On objects, we define $F$ to be the identity. A morphism $\gamma$ in ${\mathcal{B}}(M,X)$ – represented by a path in $C_k(M,X)$ for some $k$ – is sent by $F$ to the morphism in ${\mathcal{B}}(N,Y)$ represented by the path $C_k(\phi,f) \circ \gamma$, where $C_k(\phi,f) \colon C_k(M,X) \to C_k(N,Y)$ sends a configuration $\{(m_1,x_1),\ldots,(m_k,x_k)\}$ to $\{(\phi(m_1),f(x_1)),\ldots,(\phi(m_k),f(x_k))\}$.
If $\phi^\prime$ is another $1$-morphism (i.e., embedding) that is connected to $\phi$ by a $2$-morphism (i.e., is isotopic to $\phi$ respecting basepoints and collar neighbourhoods), then applying the above construction to $\phi^\prime$ and $f$, instead of $\phi$ and $f$, results in exactly the same functor $F \colon {\mathcal{B}}(M,X) \to {\mathcal{B}}(N,Y)$. Thus ${\mathcal{B}}$ extends to a $2$-functor by sending all $2$-morphisms to identities.
Degree. {#sss:degree}
-------
Definition \[def:degree-general\] from §\[sec:inductive-degree\] specialises to associate a *degree* $${\ensuremath{\mathrm{deg}}}(T)\in\{-1,0,1,2,3,\ldots,\infty\}$$ to any functor $T\colon {\mathcal{C}}\to\mathsf{Ab}$ for any object ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$. In particular, via the functor above, it associates a *degree* to any functor ${\mathcal{B}}(M,X) \to {\ensuremath{\mathsf{Ab}}}$, and recovers the notion of *degree* used in [@Palmer2018Twistedhomologicalstability].
\[l:degree-under-composition\] If $f\colon {\mathcal{C}}\to {\mathcal{D}}$ is a morphism in ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and $T\colon {\mathcal{D}}\to \mathsf{Ab}$ is any functor, then we have the inequality $\mathrm{deg}(Tf) {\leqslant}\mathrm{deg}(T)$. If $f$ is essentially surjective on objects then it is an equality.
We need to show that $\mathrm{deg}(T){\leqslant}n \Rightarrow \mathrm{deg}(Tf){\leqslant}n$ for each $n{\geqslant}-1$, and that the reverse implication also holds if $f$ is essentially surjective on objects. The base case $n=-1$ is clear, since $\mathrm{deg}(T)=-1$ simply means that $T=0$. It is then an exercise in elementary $2$-category theory to show that $(\Delta T)f \cong \Delta (Tf)$, from which the inductive step follows.
\[def:braidable\] Say that a category $({\mathcal{C}},s,\imath) \in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ is *braidable* if there exists a natural isomorphism $\Psi \colon s \circ s \to s \circ s$ such that $\imath * \mathrm{id}_s = \Psi \circ (\mathrm{id}_s * \imath)$. Note that this is equivalent to saying that the endofunctor $s \colon {\mathcal{C}}\to {\mathcal{C}}$ itself may be extended to a morphism of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$.
\[coro:braidable\] If $({\mathcal{C}},s,\imath) \in {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ is braidable, and $T \colon {\mathcal{C}}\to \mathsf{Ab}$ is any functor, then we have the inequality $\mathrm{deg}(Ts) {\leqslant}\mathrm{deg}(T)$, which is an equality if $s$ is essentially surjective on objects.
Uniformly-defined twisted coefficient systems. {#sss:uniform-coeff-systems}
----------------------------------------------
Given ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$, a twisted coefficient system is simply a functor ${\mathcal{C}}\to{\ensuremath{\mathsf{Ab}}}$. More generally, we may start with a diagram $F\colon{\mathcal{D}}\to{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and define a twisted coefficient system for each object of ${\mathcal{D}}$ in a compatible way, as follows. By abuse of notation, write $F$ also for the composition ${\mathcal{D}}\to{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\to\mathsf{Cat}$ of $F$ with the forgetful functor down to the category of small categories. A *uniformly-defined twisted coefficient system for $F$* is then a functor $$T\colon {\mathcal{D}}{\textstyle\int} F \longrightarrow {\ensuremath{\mathsf{Ab}}}$$ with domain the Grothendieck construction of $F$. For each object $d\in{\mathcal{D}}$ there is a natural functor $j_d\colon F(d) \to {\mathcal{D}}{\textstyle\int} F$, so this determines a twisted coefficient system $T_d \colon F(d) \to {\ensuremath{\mathsf{Ab}}}$ for each $d\in{\mathcal{D}}$.
\[l:grothendieck-construction\] The category ${\mathcal{D}}{\textstyle\int} F$ is naturally an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and $j_d$ is a morphism of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$.
Thus we have a well-defined degree $\mathrm{deg}(T)$ of a uniformly-defined twisted coefficient system $T$, and by Lemma \[l:degree-under-composition\] we know that $\mathrm{deg}(T) {\geqslant}\mathrm{deg}(T_d)$, in other words it is an upper bound on the degrees of the individual twisted coefficient systems $T_d \colon F(d)\to{\mathcal{D}}{\textstyle\int}F\to{\ensuremath{\mathsf{Ab}}}$.
For each object $d\in{\mathcal{D}}$ the category $F(d)$ is equipped with an endofunctor, which we will denote $s_d\colon F(d)\to F(d)$, and a natural transformation $\iota_d \colon 1_{F(d)} \Rightarrow s_d$. Recall that ${\mathcal{D}}{\textstyle\int}F$ has objects $(d,x)$ for $d\in{\mathcal{D}}$ and $x\in F(d)$ and morphisms $(f,g) \colon (d,x)\to (d^\prime,x^\prime)$ where $f\colon d\to d^\prime$ in ${\mathcal{D}}$ and $g\colon F(f)(x)\to x^\prime$ in $F(d^\prime)$. One may then define an endofunctor $\bar{s}$ on ${\mathcal{D}}{\textstyle\int}F$ by setting $\bar{s}(d,x) = (d,s_d(x))$ and $\bar{s}(f,g) = (f,s_{d^\prime}(g))$, and a natural transformation $\bar{\iota} \colon 1_{{\mathcal{D}}{\scriptstyle\int}F} \Rightarrow \bar{s}$ by setting $\bar{\iota}_{(d,x)} = (1_d,(\iota_d)_x)$.
This makes ${\mathcal{D}}{\textstyle\int}F$ into an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and the functor $j_d \colon F(d) \to {\mathcal{D}}{\textstyle\int} F$, together with $\psi = \mathrm{id} \colon j_d \circ s_d \to \bar{s} \circ j_d$, into a morphism of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$.
For example, we could take ${\mathcal{D}}$ to be ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}$ and $F$ to be the functor , in which case a “uniformly-defined twisted coefficient system” determines twisted coefficient systems for all *partial braid categories* ${\mathcal{B}}(M,X)$ simultaneously.
\[r:cocone\] Fix $F\colon {\mathcal{D}}\to{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and suppose we are given a cocone on $F$ (i.e. an object ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ and a natural transformation $\alpha\colon F\Rightarrow \mathrm{const}_{\mathcal{C}}$) together with a functor $T\colon {\mathcal{C}}\to{\ensuremath{\mathsf{Ab}}}$. This determines a functor $\mathbb{T}\colon{\mathcal{D}}{\textstyle\int}F\to{\ensuremath{\mathsf{Ab}}}$ given on objects by $\mathbb{T}(d,x)=T(\alpha_d(x))$. One can show inductively that in this setting we have an inequality $\mathrm{deg}(\mathbb{T}){\leqslant}\mathrm{deg}(T)$.
Note that the category $\Sigma$ of finite cardinals and partially-defined injections is naturally an object of ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ if one equips it with the endofunctor taking $n$ to $n+1$ and a morphism $f$ to the morphism defined by $1 \mapsto 1$ and $i \mapsto f(i-1)+1$ for $i{\geqslant}2$, together with the natural transformation given by the collection of morphisms $\iota_n\colon n \to n+1$ defined by $\iota_n(i)=i+1$. We may therefore consider the slice category $({\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\downarrow \Sigma)$. A lift of a functor $F \colon {\mathcal{D}}\to {\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ to the slice category is the same thing as a cocone on $F$ with $\Sigma\in{\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ at its “tip”. So if we fix a functor $F \colon {\mathcal{D}}\to ({\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\downarrow \Sigma)$, any twisted coefficient system on $\Sigma$ (i.e. functor $\Sigma\to{\ensuremath{\mathsf{Ab}}}$) automatically induces a uniformly-defined twisted coefficient system (i.e. functor ${\mathcal{D}}{\textstyle\int}F\to{\ensuremath{\mathsf{Ab}}}$) of the same or smaller degree.
In particular, the functor ${\mathcal{B}}$ naturally lifts to the slice category[^18] ([*cf*. ]{}the construction of below), so a twisted coefficient system for $\Sigma$ induces a uniformly-defined twisted coefficient system for ${\mathcal{B}}$, and thence twisted coefficient systems for each ${\mathcal{B}}(M,X)$.
Height. {#sss:height}
-------
The definition of the *height* of a twisted coefficient system $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$ requires a different structure on the source category ${\mathcal{C}}$.
Recall from §\[para:compare-two-heights\] that ${\mathcal{I}}$ and $\Sigma$ have objects $0,1,2,\ldots$, morphisms $m \to n$ of $\Sigma$ are partially-defined injections ${\ensuremath{\underline{m}}}\to {\ensuremath{\underline{n}}}$ and morphisms of ${\mathcal{I}}$ are those partially-defined injections that are the identity wherever they are defined. Their automorphism groups are the symmetric groups $\Sigma_n$ and trivial respectively. Denote their endomorphism monoids by ${\mathcal{P}}_n = \mathrm{End}_\Sigma(n)$ and ${\mathcal{I}}_n = \mathrm{End}_{\mathcal{I}}(n)$. Note that ${\mathcal{I}}_n$ is the submonoid of ${\mathcal{P}}_n$ of all idempotent elements. It may also be described as the monoid of subsets of $\{1,\ldots,n\}$ under the operation $\cap$ with neutral element $\{1,\ldots,n\}$, or under the operation $\cup$ with neutral element $\varnothing$. The latter identification is given by associating to a subset $S\subseteq\{1,\ldots,n\}$ the morphism $f_{n,S} \in {\mathcal{I}}_n$ that “forgets” $S$, in other words the partial injection from $\{1,\ldots,n\}$ to itself that is undefined on $S$ and the identity on $\{1,\ldots,n\} \smallsetminus S$.
\[def:cati\] Let ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ be the category whose objects are small categories ${\mathcal{C}}$ equipped with functors $s\colon {\mathcal{I}}\to {\mathcal{C}}$ and $\pi \colon {\mathcal{C}}\to \Sigma$ such that $\pi \circ s$ is the inclusion, and such that the following two conditions are satisfied:
The homomorphisms $\pi \colon \mathrm{End}_{\mathcal{C}}(s(n)) \to {\mathcal{P}}_n$ and $\pi \colon \mathrm{Aut}_{\mathcal{C}}(s(n)) \to \Sigma_n$ are surjective.
(“Locality”) Fix $n{\geqslant}0$ and $\phi\in\mathrm{End}_{\mathcal{C}}(s(n))$. For each $i$ there exists $j$ and for each $j$ there exists $i$ such that $$\label{eq:locality}
\phi \circ s(f_{n,\{i\}}) \;=\; s(f_{n,\{j\}}) \circ \phi.$$
Morphisms from $({\mathcal{C}},s,\pi)$ to $({\mathcal{C}}^\prime,s^\prime,\pi^\prime)$ are functors $f \colon {\mathcal{C}}\to {\mathcal{C}}^\prime$ such that $f \circ s = s^\prime$ and $\pi = \pi^\prime \circ f$.
There is an analogue of the functor for this setting, which we denote by the same letter, $$\label{eq:partial-braid-functor-v2}
{\mathcal{B}}\colon {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}\longrightarrow {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}},$$ and which is defined as follows. Given objects $M \in {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ and $X \in {\ensuremath{\mathsf{Top}_{\circ}}}$, the category ${\mathcal{B}}(M,X)$ itself is defined as in §\[sss:some-functors\]. Now we define functors $s \colon {\mathcal{I}}\to {\mathcal{B}}(M,X)$ and $\pi \colon {\mathcal{B}}(M,X) \to \Sigma$. On objects, $s$ is the identity. If $f \colon m \to n$ is the morphism in ${\mathcal{I}}$ that is the identity on $S \subseteq \{1,\ldots,\mathrm{min}(m,n)\}$ and undefined elsewhere, define $s(f)$ to be the (homotopy class of the) constant path in $C_{\lvert S \rvert}(M,X)$ at the point $\{ (a_s,x_0) \mid s \in S \}$. The functor $\pi$ is also the identity on objects. A morphism $m \to n$ in ${\mathcal{B}}(M,X)$ is determined by a path of configurations from some subconfiguration of $\{a_1,\ldots,a_m\}$ to some subconfiguration of $\{a_1,\ldots,a_n\}$ (together with some labels, which we are ignoring). This induces a partial injection from ${\ensuremath{\underline{m}}}$ to ${\ensuremath{\underline{n}}}$, and the functor $\pi$ records precisely this information.
The locality property is satisfied since deleting the $i$th strand of a braid from one end corresponds to deleting the $j$th strand from the other end for some $j$. If there is no $i$th strand, according to the ordering at one end, then we may take $j$ to be a number such that there is no $j$th strand at the other end, so that both sides of are equal to $\phi$. The surjectivity property holds since any (partial) injection may be realised by a (partial) braid on $M$, since manifolds $M \in {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$ are required to have dimension at least two.
#### An alternative viewpoint. {#an-alternative-viewpoint. .unnumbered}
Since a category ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ in particular comes equipped with a functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$, we have from §\[sec:cross-effects\] a definition of the *height* $$\mathrm{height}(T)\in\{-1,0,1,2,3,\ldots,\infty\}$$ of any functor $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$, as described in §\[para:specialise-this-paper\]. In particular, via the functor , it associates a *height* to any functor ${\mathcal{B}}(M,X) \to {\ensuremath{\mathsf{Ab}}}$, and recovers the notion of *height* used in [@Palmer2018Twistedhomologicalstability].
In the next section we describe the definition of the *height* from a different viewpoint, which depends on the full structure of ${\mathcal{C}}$ as an object of ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$, not just on the functor $s \colon {\mathcal{I}}\to {\mathcal{C}}$. This may be summarised as follows. An object ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ has associated categories and faithful functors ${\mathcal{A}}\to {\mathcal{B}}\subseteq {\mathcal{C}}$, together with an ${\mathbb{N}}$-grading of the objects of ${\mathcal{A}}$. We may therefore filter the category ${\mathcal{A}}$ by defining ${\mathcal{A}}^{>n} \subseteq {\mathcal{A}}$ to be the full subcategory on objects with grading more than $n$, for $n \in \{-1,0,1,2,\ldots,\infty\}$. Now given any functor $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$, there is an associated functor $T^\prime \colon {\mathcal{A}}\to {\ensuremath{\mathsf{Ab}}}$ related to $T$ by the fact that the induced functor $\mathrm{Ind}_{{\mathcal{A}}\to {\mathcal{B}}}(T^\prime)$ is isomorphic to $T$ on the subgroupoid ${\mathcal{B}}^{\sim}$ (the underlying groupoid of ${\mathcal{B}}$). The *height* of $T$ is then $$\mathrm{height}(T) = \mathrm{min}\bigl\lbrace n \bigm| T^\prime \equiv 0 \, \text{ on } {\mathcal{A}}^{>n} \bigr\rbrace .$$
The idea is that the functor $T^\prime \colon {\mathcal{A}}\to {\ensuremath{\mathsf{Ab}}}$ records all of the *cross-effects* of $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$ simultaneously, with the grading indicating which cross-effects correspond to which objects of ${\mathcal{C}}$. This viewpoint could perhaps be useful in generalising the notion of *height* to more sophisticated situations, by allowing the structure of the category ${\mathcal{A}}$ indexing the cross-effects to be more complicated (here it is just a disjoint union of monoids).
The details of this alternative viewpoint on the *height* of a functor are given in §\[sss:height-general\], using some facts about induction for representations of categories, which may be of interest in their own right, which we discuss in §§\[sss:induction\]–\[sss:special-cases\]. We explain in Remark \[rmk:two-defs-of-height-agree\] why this alternative viewpoint agrees exactly with the definition from §\[para:specialise-this-paper\] above.
Induction for representations of categories {#sec:representations-of-categories}
===========================================
In this section we give details of the alternative viewpoint on the *height* of a twisted coefficient system $T \colon {\mathcal{C}}\to {\mathcal{A}}$, when ${\mathcal{C}}$ is an object of the category ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ defined in §\[sss:height\] immediately above. We begin with a detour through the notion of induction for representations of categories in §§\[sss:induction\]–\[sss:special-cases\], and then return to the alternative definition of the *height* of a twisted coefficient system in §\[sss:height-general\].
Induction for representations of categories. {#sss:induction}
--------------------------------------------
We will take ${\mathbb{Z}}$ as our ground ring in this section, but everything works equally well over an arbitrary commutative unital ring. Suppose that we have a functor $f\colon {\mathcal{A}}\to {\mathcal{B}}$ and we wish to extend representations of ${\mathcal{A}}$, i.e., functors $g\colon {\mathcal{A}}\to{\ensuremath{\mathsf{Ab}}}= {\mathbb{Z}}\text{-mod}$, along $f$ to ${\mathcal{B}}$. We will define a functor (“induction along $f$”) $$\label{eq:induction-functor}
\mathrm{Ind}_f \colon \mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}}) \longrightarrow \mathsf{Fun}({\mathcal{B}},{\ensuremath{\mathsf{Ab}}})$$ that does this, and prove a few of its properties. We note that what we will be defining is simply the *left Kan extension* operation along the functor $f$, but we would like to have explicit formulas for this, so we will give an elementary definition instead of using this universal characterisation.
First we explain the notion of a category ring. Given any category ${\mathcal{A}}$, its *category ring* ${\mathbb{Z}}{\mathcal{A}}$ is defined as follows: as an abelian group it is freely generated by the morphisms of ${\mathcal{A}}$, and the product of two morphisms is their composition if they are composable and zero otherwise.[^19] Note that ${\mathbb{Z}}{\mathcal{A}}$ is unital if and only if ${\mathcal{A}}$ has finitely many objects, in which case the unit is given by the formal sum of the identities $1_a$ over all objects $a$ of $A$. This definition was given by B. Mitchell in §7 of [@Mitchell1972Ringswithseveralobjects], see also §2 of [@Webb2007introductionrepresentationscohomology]. (We note that the definition of Mitchell is more general: it associates a ring $[{\mathcal{C}}]$ to each *preadditive* (${\ensuremath{\mathsf{Ab}}}$-enriched) category ${\mathcal{C}}$; the above definition of ${\mathbb{Z}}{\mathcal{A}}$ is recovered as $[{\mathcal{A}}_{{\ensuremath{\mathsf{Ab}}}}]$, where ${\mathcal{A}}_{{\ensuremath{\mathsf{Ab}}}}$ denotes the free preadditive category generated by ${\mathcal{A}}$.) Now, to a functor $f\colon {\mathcal{A}}\to {\mathcal{B}}$ and an object $b$ of ${\mathcal{B}}$ we may associate the following right ${\mathbb{Z}}{\mathcal{A}}$-module: $${\mathbb{Z}}(f,b) = {\mathbb{Z}}\Bigl\langle (\beta,a) \bigm| a\in\mathrm{ob}({\mathcal{A}}),\; \beta\colon f(a)\to b \text{ in } {\mathcal{B}}\Bigr\rangle$$ with the ${\mathbb{Z}}{\mathcal{A}}$ action defined as follows: a morphism $\alpha\colon a_1 \to a_2$ sends $(\beta,a)$ to zero if $a\neq a_2$ and to $(\beta\circ f(\alpha),a_1)$ otherwise. (This could be written more compactly in terms of “heteromorphisms” [@Ellerman2007Adjointfunctorsheteromorphisms] as $\bigoplus_a {\mathbb{Z}}\mathrm{Het}_f(a,b)$, but this will not be relevant for us here.)
Given a representation $g\colon {\mathcal{A}}\to{\ensuremath{\mathsf{Ab}}}$ we may define a left ${\mathbb{Z}}{\mathcal{A}}$-module: $$g(\mathrm{ob} {\mathcal{A}}) = \bigoplus_{a\in\mathrm{ob}({\mathcal{A}})}g(a),$$ with $\alpha\colon a_1\to a_2$ sending $x\in g(a)$ to zero if $a\neq a_1$ and to $g(\alpha)(x)\in g(a_2)$ otherwise. We now define $\mathrm{Ind}_f(g)\colon {\mathcal{B}}\to{\ensuremath{\mathsf{Ab}}}$ as follows: $$\label{eq:induction-fg}
\begin{aligned}
&\text{on objects:}\qquad& &\phantom{=}\mathrm{Ind}_f(g)(b) \;=\; {\mathbb{Z}}(f,b) \,\otimes_{{\mathbb{Z}}{\mathcal{A}}}\, g(\mathrm{ob} {\mathcal{A}}) \\
&\text{on morphisms:}\qquad& &\phantom{=}\mathrm{Ind}_f(g)(\gamma\colon b\to b^\prime) \colon (\beta,a)\otimes x \;\mapsto\; (\gamma\circ\beta,a) \otimes x.
\end{aligned}$$ We note that $\mathrm{Ind}_f(g)(b)$ is generated by elements of the form $(\beta,a)\otimes x$ with $x\in g(a)$.[^20] This defines the functor $\mathrm{Ind}_f$ on objects, i.e. representations of ${\mathcal{A}}$. Given a natural transformation $\tau\colon g\Rightarrow g^\prime$ we define the natural transformation $\mathrm{Ind}_f(\tau) \colon \mathrm{Ind}_f(g) \Rightarrow \mathrm{Ind}_f(g^\prime)$ by
&&& \_f()\_b (,a)x (,a)\_[a]{}(x). &&( xg(a))
This completes the definition of the induction functor .
As mentioned above, one can check that this explicit construction is left adjoint to the restriction functor $(-) \circ f$; in other words, it is the left Kan extension operation: $\mathrm{Ind}_f = \mathrm{Lan}_f$.
Comparison to induction for modules over category rings. {#ss:induction}
--------------------------------------------------------
The construction mentioned above, taking a representation $g\colon {\mathcal{A}}\to{\ensuremath{\mathsf{Ab}}}$ to the ${\mathbb{Z}}{\mathcal{A}}$-module $g(\mathrm{ob}{\mathcal{A}})$, in fact defines an embedding $$\mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}}) \longrightarrow {\mathbb{Z}}{\mathcal{A}}\text{-mod}$$ of the representation category of ${\mathcal{A}}$ as a full subcategory of the category of left ${\mathbb{Z}}{\mathcal{A}}$-modules. The image is the full subcategory on those ${\mathbb{Z}}{\mathcal{A}}$-modules $M$ such that for each element $m\in M$ the set $\{a\in\mathrm{ob}({\mathcal{A}}) \mid 1_a \cdot m \neq 0 \}$ is finite. Hence if ${\mathcal{A}}$ has only finitely many objects, this is an equivalence of categories. This is Theorem 7.1 of [@Mitchell1972Ringswithseveralobjects]; see also Proposition 2.1 of [@Webb2007introductionrepresentationscohomology].
A functor $f\colon {\mathcal{A}}\to {\mathcal{B}}$ induces a homomorphism of abelian groups ${\mathbb{Z}}f\colon {\mathbb{Z}}{\mathcal{A}}\to {\mathbb{Z}}{\mathcal{B}}$ that is a homomorphism of (non-unital) *rings* if and only if $f$ is *injective on objects* (see Proposition 2.2.3 of [@Xu2007Representationscategoriesapplications] and Proposition 3.1 of [@Webb2007introductionrepresentationscohomology]). In this case ${\mathbb{Z}}{\mathcal{B}}$ may be considered as a right module over ${\mathbb{Z}}{\mathcal{A}}$ via the ring homomorphism ${\mathbb{Z}}f$ and hence there is an induction functor $$\label{eq:induction-functor-2}
{\mathbb{Z}}{\mathcal{B}}\otimes_{{\mathbb{Z}}{\mathcal{A}}} - \colon {\mathbb{Z}}{\mathcal{A}}\text{-mod} \longrightarrow {\mathbb{Z}}{\mathcal{B}}\text{-mod}.$$ This agrees with our definition above:
When $f$ is injective on objects so that the right-hand vertical arrow below exists, the following square commutes up to natural isomorphism $$\label{eq:comparing-induction-functors}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,15) {$\mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}})$};
\node (tr) at (40,15) {${\mathbb{Z}}{\mathcal{A}}\text{\textup{-mod}}$};
\node (bl) at (0,0) {$\mathsf{Fun}({\mathcal{B}},{\ensuremath{\mathsf{Ab}}})$};
\node (br) at (40,0) {${\mathbb{Z}}{\mathcal{B}}\text{\textup{-mod}}$};
\draw[->] (tl) to node[left,font=\small]{$\mathrm{Ind}_f$} (bl);
\draw[->] (tr) to node[right,font=\small]{${\mathbb{Z}}{\mathcal{B}}\otimes_{{\mathbb{Z}}{\mathcal{A}}}-$} (br);
{\draw[<-,>=right hook] (tl) to ($ (tl)!0.5!(tr) $);
\draw[->,>=stealth'] ($ (tl)!0.5!(tr) $) to (tr);}
{\draw[<-,>=right hook] (bl) to ($ (bl)!0.5!(br) $);
\draw[->,>=stealth'] ($ (bl)!0.5!(br) $) to (br);}
\end{tikzpicture}
\end{split}$$
As a right ${\mathbb{Z}}{\mathcal{B}}$-module, ${\mathbb{Z}}{\mathcal{B}}$ itself is isomorphic to the direct sum $\bigoplus_b {\mathbb{Z}}\mathrm{Hom}_{\mathcal{B}}({\mathcal{B}},b)$ where the sum is over all objects $b$ of ${\mathcal{B}}$ and the notation $\mathrm{Hom}_{\mathcal{B}}({\mathcal{B}},b)$ denotes the disjoint union of the sets $\mathrm{Hom}_{\mathcal{B}}(b^\prime,b)$ over all objects $b^\prime$ of ${\mathcal{B}}$. This may be viewed as an isomorphism of right ${\mathbb{Z}}{\mathcal{A}}$-modules via ${\mathbb{Z}}f$. Moreover, under the hypothesis that $f$ is injective on objects, the right ${\mathbb{Z}}{\mathcal{A}}$-module ${\mathbb{Z}}(f,b)$ is isomorphic to ${\mathbb{Z}}\mathrm{Hom}_{\mathcal{B}}({\mathcal{B}},b)$. Hence we have isomorphisms of right ${\mathbb{Z}}{\mathcal{A}}$-modules $${\mathbb{Z}}{\mathcal{B}}\;\cong\; \textstyle{\bigoplus_b}\, {\mathbb{Z}}(f,b).$$ Now the result of sending a functor $g\colon {\mathcal{A}}\to{\ensuremath{\mathsf{Ab}}}$ clockwise around the diagram is ${\mathbb{Z}}{\mathcal{B}}\otimes_{{\mathbb{Z}}{\mathcal{A}}} g(\mathrm{ob}{\mathcal{A}})$ whereas the result of sending it anticlockwise around the diagram is $\bigoplus_b {\mathbb{Z}}(f,b) \otimes_{{\mathbb{Z}}{\mathcal{A}}} g(\mathrm{ob}{\mathcal{A}})$.
It does not seem clear how to extend $\mathrm{Ind}_f$ to an induction functor ${\mathbb{Z}}{\mathcal{A}}\text{-mod} \to {\mathbb{Z}}{\mathcal{B}}\text{-mod}$ in the case when $f$ is not injective on objects.
Under certain conditions (although certainly not in general) induction followed by restriction is isomorphic to the identity. More precisely, write $\mathrm{Res}_f(-) = (-)\circ f \colon \mathsf{Fun}({\mathcal{B}},{\ensuremath{\mathsf{Ab}}}) \to \mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}})$, so that $\mathrm{Res}_f \circ \mathrm{Ind}_f$ is an endofunctor of $\mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}})$. Then there is a natural transformation $\mathrm{id} \Rightarrow \mathrm{Res}_f \circ \mathrm{Ind}_f$ (the unit of the adjunction $\mathrm{Ind}_f \dashv \mathrm{Res}_f$) with the property that for each $g\in\mathsf{Fun}({\mathcal{A}},{\ensuremath{\mathsf{Ab}}})$ and $a\in {\mathcal{A}}$ its component $g(a) \to \mathrm{Ind}_f(g)(f(a))$ is surjective if $f$ is full and bijective if $f$ is also faithful. So when $f$ is fully faithful the composition $\mathrm{Res}_f \circ \mathrm{Ind}_f$ is isomorphic to the identity. We leave this assertion without proof since we will not use it (but see §3 of [@Webb2007introductionrepresentationscohomology], especially Prop. 3.2(1), for further discussion).
Special cases. {#sss:special-cases}
--------------
We note that the formula for the induced functor simplifies in some special cases. Suppose first that ${\mathcal{A}}$ is a disjoint union of monoids, i.e., has no morphisms between distinct objects. Then ${\mathbb{Z}}{\mathcal{A}}$ splits as a direct sum of rings $\bigoplus_a {\mathbb{Z}}\mathrm{End}_{\mathcal{A}}(a)$. Also, the right ${\mathbb{Z}}{\mathcal{A}}$-module ${\mathbb{Z}}(f,b)$ splits as a direct sum of modules $\bigoplus_a {\mathbb{Z}}\mathrm{Hom}_{\mathcal{B}}(f(a),b)$ and the left ${\mathbb{Z}}{\mathcal{A}}$-module splits as a direct sum of modules $\bigoplus_a g(a)$. The tensor product therefore splits in the same way, and we have: $$\mathrm{Ind}_f(g)(b) \;\cong\; \textstyle{\bigoplus}_a \bigl( {\mathbb{Z}}\mathrm{Hom}_{\mathcal{B}}(f(a),b) \otimes_{{\mathbb{Z}}\mathrm{End}_{\mathcal{A}}(a)} g(a) \bigr).$$ If the category ${\mathcal{B}}$ is also a disjoint union of monoids, then this simplifies further to $$\mathrm{Ind}_f(g)(b) \;\cong\; \textstyle{\bigoplus}_{a\in f^{-1}(b)} \bigl( {\mathbb{Z}}\mathrm{End}_{\mathcal{B}}(b) \otimes_{{\mathbb{Z}}\mathrm{End}_{\mathcal{A}}(a)} g(a) \bigr).$$
Under certain conditions, this may be written purely in terms of automorphism groups, rather than endomorphism monoids, using the following elementary lemma.
\[l:induction-and-restriction-commute\] Suppose that the square of submonoids
\[x=1mm,y=1mm\] (tl) at (0,10) [$C$]{}; (tr) at (20,10) [$D$]{}; (bl) at (0,0) [$A$]{}; (br) at (20,0) [$B$]{}; [(tl) to ($ (tl)!0.5!(tr) $); ($ (tl)!0.5!(tr) $) to (tr);]{} [(bl) to ($ (bl)!0.5!(br) $); ($ (bl)!0.5!(br) $) to (br);]{} [(bl) to ($ (bl)!0.5!(tl) $); ($ (bl)!0.5!(tl) $) to (tl);]{} [(br) to ($ (br)!0.5!(tr) $); ($ (br)!0.5!(tr) $) to (tr);]{}
satisfies the following condition $(*)$ there is a subset $X\subseteq B\times C$ such that the multiplication map $X\to D$ is surjective and whenever $b_1c_1=b_2c_2$ for $(b_i,c_i)\in X$ there exists $a\in A$ such that $b_1=b_2a$ and $ac_1=c_2$. Then for any ${\mathbb{Z}}C$-module $M$ there is an isomorphism of ${\mathbb{Z}}B$-modules $${\mathbb{Z}}D\otimes_{{\mathbb{Z}}C} M \;\cong\; {\mathbb{Z}}B\otimes_{{\mathbb{Z}}A} M.$$
One may also write this as $\mathrm{Res}_B^D (\mathrm{Ind}_C^D (M)) \cong \mathrm{Ind}_A^B (\mathrm{Res}_A^C (M))$.
There is an obvious ${\mathbb{Z}}B$-module homomorphism $i\colon {\mathbb{Z}}B\otimes_{{\mathbb{Z}}A} M \to {\mathbb{Z}}D\otimes_{{\mathbb{Z}}C} M$ given by $b\otimes m \mapsto b\otimes m$. To define an inverse, note that by property $(*)$ there is a well-defined function $D\times M \to {\mathbb{Z}}B\otimes_{{\mathbb{Z}}A} M$ given by sending $(d,m)$ to $b\otimes c\cdot m$, where $(b,c)\in X$ such that $bc=d$. This is linear in the second entry, and it sends $(dc,m)$ and $(d,c\cdot m)$ to the same element for any $c\in C$, so it induces a homomorphism ${\mathbb{Z}}D\otimes_{{\mathbb{Z}}C} M \to {\mathbb{Z}}B\otimes_{{\mathbb{Z}}A} M$. This is an inverse for $i$.
The condition $(*)$ in Lemma \[l:induction-and-restriction-commute\] will be valid in our setting by the following lemma. Let ${\mathcal{P}}_n$ be the monoid of partial bijections of $\{1,\ldots,n\}$ and write ${\mathcal{P}}_k \times {\mathcal{P}}_{n-k}$ for its submonoid of those partial bijections that preserve the partition into $\{1,\ldots,n-k\}$ and $\{n-k+1,\ldots,n\}$. Write $D^\sim$ for the underlying group of a monoid $D$, so for example $({\mathcal{P}}_n)^\sim = \Sigma_n$ is the $n$th symmetric group.
\[l:checking-property-star\] Suppose that $\pi\colon D\to {\mathcal{P}}_n$ is a surjective monoid homomorphism such that the homomorphism of underlying groups $D^\sim \to \Sigma_n$ is also surjective. Define $C=\pi^{-1}({\mathcal{P}}_k \times {\mathcal{P}}_{n-k})$. Then the square of submonoids $$\label{eq:square-of-submonoids}
\centering
\begin{split}
\begin{tikzpicture}
[x=1mm,y=1mm]
\node (tl) at (0,10) {$C$};
\node (tr) at (20,10) {$D$};
\node (bl) at (0,0) {$C^\sim$};
\node (br) at (20,0) {$D^\sim$};
{\draw[<-,>=right hook] (tl) to ($ (tl)!0.5!(tr) $);
\draw[->,>=stealth'] ($ (tl)!0.5!(tr) $) to (tr);}
{\draw[<-,>=right hook] (bl) to ($ (bl)!0.5!(br) $);
\draw[->,>=stealth'] ($ (bl)!0.5!(br) $) to (br);}
{\draw[<-,>=right hook] (bl) to ($ (bl)!0.5!(tl) $);
\draw[->,>=stealth'] ($ (bl)!0.5!(tl) $) to (tl);}
{\draw[<-,>=right hook] (br) to ($ (br)!0.5!(tr) $);
\draw[->,>=stealth'] ($ (br)!0.5!(tr) $) to (tr);}
\end{tikzpicture}
\end{split}$$ satisfies condition $(*)$ of Lemma \[l:induction-and-restriction-commute\].
Write $n=k+l$ and $A=C^\sim$, $B=D^\sim$. First note that the square above is a pullback diagram, i.e., $A=C\cap B$, which follows from the fact that $\Sigma_k \times \Sigma_l = \Sigma_n \cap ({\mathcal{P}}_k \times {\mathcal{P}}_l)$.
Define $X\subseteq B\times C$ as follows: $(b,c)\in X$ if and only if the partial bijection $\pi(b)$ is order-preserving on $\mathrm{im}(\pi(c))^\perp \coloneqq \{1,\ldots,n\} \smallsetminus \mathrm{im}(\pi(c))$. We need to show that (a) every $d\in D$ is of the form $bc$ for $(b,c)\in X$ and that (b) if $b_1 c_1 = b_2 c_2$ for $(b_i,c_i)\in X$ then $b_2 a=b_1$ and $ac_1=c_2$ for some $a\in A$.
(a). Given any $d\in D$, the partial bijection $\pi(d)$ will not in general preserve the partition $\{1,\ldots,l\} \sqcup \{l+1,\ldots,n\}$, but we may find some permutation $\sigma\in\Sigma_n$ such that $\sigma^{-1}\pi(d)$ does preserve it, i.e., lies in the submonoid ${\mathcal{P}}_k \times {\mathcal{P}}_l$. Moreover, it does not matter how $\sigma^{-1}$ acts away from the image of $\pi(d)$ so we may assume that it is order-preserving on $\mathrm{im}(\pi(d))^\perp$. We assumed that the restriction of $\pi$ to underlying groups is surjective, so we may pick $b\in B=D^\sim$ such that $\pi(b)=\sigma$. Now define $c=b^{-1}d$, so of course $bc=d$. Since $\pi(c)=\sigma^{-1}\pi(d) \in {\mathcal{P}}_k \times {\mathcal{P}}_l$ we know that $c\in C$. It remains to show that $(b,c)$ is in $X$, i.e., that $\pi(b)$ is order-preserving on $\mathrm{im}(\pi(c))^\perp$. But we ensured that $\sigma^{-1}$ is order-preserving on $\mathrm{im}(\pi(d))^\perp$, which is equivalent to saying that $\sigma$ is order-preserving on $\mathrm{im}(\sigma^{-1}\pi(d))^\perp$, which is precisely the required condition.
(b). Define $a=b_2^{-1}b_1 \in B$. It then immediately follows that $b_2 a=b_1$ and $ac_1 = c_2$, so we just have to show that $a\in A$. Since $A=C\cap B$ this means we just need to show that $a\in C$, in other words that $\pi(a)\in {\mathcal{P}}_k \times {\mathcal{P}}_l$ — i.e. that $\pi(a)$ preserves the partition $\{1,\ldots,l\} \sqcup \{l+1,\ldots,n\}$. First, since $\pi(a)\pi(c_1)=\pi(c_2)$ with $\pi(c_i)$ both preserving the partition, it follows that $\pi(a)$ restricted to $\mathrm{im}(\pi(c_1))$ preserves the partition. We will now show that $\pi(a)$ restricted to $\mathrm{im}(\pi(c_1))^\perp$ is order-preserving — which will imply that $\pi(a)$ preserves the partition on all of $\{1,\ldots,n\}$. By the definition of $X$, we know that $\pi(b_2)$ is order-preserving on $\mathrm{im}(\pi(c_2))^\perp$. Hence $\pi(b_2)^{-1}$ is order-preserving on $$\pi(b_2)\bigl( \mathrm{im}(\pi(c_2))^\perp \bigr) = \mathrm{im}(\pi(b_2 c_2))^\perp = \mathrm{im}(\pi(b_1 c_1))^\perp = \pi(b_1)\bigl( \mathrm{im}(\pi(c_1))^\perp \bigr).$$ Combined with the fact that $\pi(b_1)$ is order-preserving on $\mathrm{im}(\pi(c_1))^\perp$ this tells us that $\pi(a) = \pi(b_2)^{-1} \pi(b_1)$ is order-preserving on $\mathrm{im}(\pi(c_1))^\perp$, as required. This completes the proof of property (b) of $X\subseteq B\times C$, so the square of submonoids satisfies condition $(*)$ of Lemma \[l:induction-and-restriction-commute\].
Returning to the definition of height. {#sss:height-general}
--------------------------------------
Following on from §\[sss:height\], we give details of the alternative definition of the *height* of a twisted coefficient system with indexing category ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$. For the first step we define a subcategory ${\mathcal{B}}\subseteq {\mathcal{C}}$, a faithful functor ${\mathcal{A}}\to {\mathcal{B}}$ and an ${\mathbb{N}}$-grading of the objects of ${\mathcal{A}}$. For the second step, given a functor $T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}}$, we define the *cross-effect functor* $T^\prime \colon {\mathcal{A}}\to {\ensuremath{\mathsf{Ab}}}$ associated to $T$, and show that $\mathrm{Ind}_{{\mathcal{A}}\to {\mathcal{B}}}(T^\prime)|_{{\mathcal{B}}^{\sim}} \cong\, T|_{{\mathcal{B}}^{\sim}}$, where ${\mathcal{B}}^\sim$ denotes the underlying groupoid of ${\mathcal{B}}$. As stated in §\[sss:height\], the *height* of $T$ is then the smallest $n$ such that $T^\prime$ is supported on the subcategory ${\mathcal{A}}^{{\leqslant}n} \subseteq {\mathcal{A}}$, in other words vanishes on the subcategory ${\mathcal{A}}^{>n} \subseteq {\mathcal{A}}$. In Remark \[rmk:two-defs-of-height-agree\], we explain why this agrees with the *height* of $T$ as defined in §\[para:specialise-this-paper\].
#### The first step. {#the-first-step. .unnumbered}
Recall that ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ comes equipped with functors $s\colon{\mathcal{I}}\to {\mathcal{C}}$ and $\pi\colon {\mathcal{C}}\to \Sigma$. The objects of ${\mathcal{B}}$ are non-negative integers and those of ${\mathcal{A}}$ are pairs of non-negative integers. Both are simply disjoint unions of monoids, i.e. they consist only of endomorphisms, so we just need to specify $\mathrm{End}_{{\mathcal{A}}}(k,l)$ and $\mathrm{End}_{{\mathcal{B}}}(n)$. As in §\[sss:height\] and in Lemma \[l:checking-property-star\] above, let ${\mathcal{P}}_n$ denote the monoid $\mathrm{End}_\Sigma(n)$ of partial bijections of $\{1,\ldots,n\}$ and write $l=n-k$ for convenience. There is a submonoid isomorphic to ${\mathcal{P}}_k \times {\mathcal{P}}_l$ consisting of those partial bijections that respect the partition $\{1,\ldots,l\}\sqcup\{l+1,\ldots,n\}$ wherever they are defined. We now define $$\begin{aligned}
\mathrm{End}_{{\mathcal{B}}}(n) &= \mathrm{End}_{\mathcal{C}}(s(n)) \\
\mathrm{End}_{{\mathcal{A}}}(k,l) &= \text{preimage of } {\mathcal{P}}_k \times {\mathcal{P}}_l \text{ under the map } \pi \colon \mathrm{End}_{{\mathcal{B}}}(n) \longrightarrow \mathrm{End}_\Sigma(n) = {\mathcal{P}}_n.\end{aligned}$$ This completes the definitions of ${\mathcal{B}}$ and ${\mathcal{A}}$. The grading of the objects of ${\mathcal{A}}$ is given by setting $\mathrm{deg}((k,l)) = k$. There is an obvious faithful functor ${\mathcal{A}}\to {\mathcal{B}}$, given on objects by $(k,l)\mapsto k+l$, and an embedding of categories ${\mathcal{B}}\hookrightarrow {\mathcal{C}}$.
#### The second step. {#the-second-step. .unnumbered}
Recall that the monoid ${\mathcal{I}}_n = \mathrm{End}_{{\mathcal{I}}}(n)$, which is the submonoid of ${\mathcal{P}}_n$ consisting of all idempotent elements, is isomorphic to the power set $\mathsf{P}(\{1,\ldots,n\})$, which is a commutative monoid via the operation $\cup$. The correspondence sends a subset $S\subseteq\{1,\ldots,n\}$ to the idempotent element $f_S\in{\mathcal{P}}_n$ that is undefined on $S$ and the identity elsewhere.
Thus, given an object ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ and a functor $T\colon {\mathcal{C}}\to{\ensuremath{\mathsf{Ab}}}$, we have a collection of idempotents $$Ts(f_S) \colon T(s(n)) \longrightarrow T(s(n)) \qquad\text{for } S\subseteq \{1,\ldots,n\}.$$ Write $l=n-k$ for convenience. In order to define the functor $T^\prime \colon {\mathcal{A}}\to {\ensuremath{\mathsf{Ab}}}$ we need to specify an $\mathrm{End}_{{\mathcal{A}}}(k,l)$-module for each pair $(k,l)$ of non-negative integers. As an abelian group, we define it to be $$\label{eq:functorial-cross-effect}
T^\prime(k,l) \;=\; \mathrm{im}\bigl( Ts(f_{\{1,\ldots,l\}}) \bigr) \cap \bigcap_{i=l+1}^n \mathrm{ker} \bigl( Ts(f_{\{i\}}) \bigr) \quad{\leqslant}\quad T(s(n)).$$ The monoid $\mathrm{End}_{\mathcal{C}}(s(n))$ acts on $T(s(n))$ via the functor $T$, and it turns out (see $3$ lines below) that each element $\phi$ of its submonoid $\mathrm{End}_{{\mathcal{A}}}(k,l)$ sends the subgroup $T^\prime(k,l)$ to itself. Hence $T^\prime(k,l)$ is an $\mathrm{End}_{{\mathcal{A}}}(k,l)$-module — and so we have defined the functor $T^\prime \colon {\mathcal{A}}\to{\ensuremath{\mathsf{Ab}}}$.
The claim in the previous paragraph follows from the fact that $\phi$ commutes with the element $s(f_{\{1,\ldots,l\}})$ and with the set of elements $\bigl\lbrace s(f_{\{l+1\}}),\ldots,s(f_{\{n\}})\bigr\rbrace$. This in turn follows from the fact that $\pi(\phi)\in {\mathcal{P}}_k \times {\mathcal{P}}_l$ together with the “locality” property of ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$.
It remains to show the following:
The functors $\mathrm{Ind}_{{\mathcal{A}}\to {\mathcal{B}}}(T^\prime)$ and $T$ are isomorphic on the subgroupoid ${\mathcal{B}}^{\sim}$.
Since ${\mathcal{B}}$ is a disjoint union of monoids, this is just a more elaborate way of saying that for each $n{\geqslant}0$ there is an isomorphism of modules over $\mathrm{Aut}_{\mathcal{C}}(s(n)) = \mathrm{Aut}_{{\mathcal{B}}}(n)$: $$\label{eq:cross-effect-decomp-generalised}
T(s(n)) \;\cong\; \mathrm{Ind}_{{\mathcal{A}}\to {\mathcal{B}}} (T^\prime)(n).$$ The proof of Proposition 3.5 of [@Palmer2018Twistedhomologicalstability] generalises verbatim to this setting to show that the left-hand side of is isomorphic to $$\bigoplus_{k+l=n} \bigl( {\mathbb{Z}}\mathrm{Aut}_{{\mathcal{B}}}(n) \otimes_{{\mathbb{Z}}\mathrm{Aut}_{{\mathcal{A}}}(k,l)} T^\prime(k,l) \bigr).$$ The categories ${\mathcal{A}}$ and ${\mathcal{B}}$ are both disjoint unions of monoids, so, as remarked in §\[sss:special-cases\], the right-hand side of may be written as follows: $$\bigoplus_{k+l=n} \bigl( {\mathbb{Z}}\mathrm{End}_{{\mathcal{B}}}(n) \otimes_{{\mathbb{Z}}\mathrm{End}_{{\mathcal{A}}}(k,l)} T^\prime(k,l) \bigr).$$ To finish the proof we will apply Lemma \[l:induction-and-restriction-commute\], so we need to know that for each $k+l=n{\geqslant}0$ the square of submonoids
\[x=1mm,y=1mm\] (tl) at (0,10) [$\mathrm{End}_{{\mathcal{A}}}(k,l)$]{}; (tr) at (40,10) [$\mathrm{End}_{{\mathcal{B}}}(n)$]{}; (bl) at (0,0) [$\mathrm{Aut}_{{\mathcal{A}}}(k,l)$]{}; (br) at (40,0) [$\mathrm{Aut}_{{\mathcal{B}}}(n)$]{}; [(tl) to ($ (tl)!0.5!(tr) $); ($ (tl)!0.5!(tr) $) to (tr);]{} [(bl) to ($ (bl)!0.5!(br) $); ($ (bl)!0.5!(br) $) to (br);]{} [(bl) to ($ (bl)!0.5!(tl) $); ($ (bl)!0.5!(tl) $) to (tl);]{} [(br) to ($ (br)!0.5!(tr) $); ($ (br)!0.5!(tr) $) to (tr);]{}
satisfies condition $(*)$ of that lemma. This will be given by Lemma \[l:checking-property-star\] as long as the homomorphism $$\pi\colon \mathrm{End}_{{\mathcal{B}}}(n) = \mathrm{End}_{\mathcal{C}}(s(n)) \longrightarrow \mathrm{End}_\Sigma(n) = {\mathcal{P}}_n$$ (as well as its restriction to maximal subgroups) is surjective. But this is true by definition for any ${\mathcal{C}}\in{\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ (see Definition \[def:cati\]).
\[rmk:two-defs-of-height-agree\] Finally, we note that this description of $\text{height}(T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}})$ for an object ${\mathcal{C}}\in {\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$ agrees with the definition of $\text{height}(T \colon {\mathcal{C}}\to {\ensuremath{\mathsf{Ab}}})$, given in §\[para:specialise-this-paper\], for any category ${\mathcal{C}}$ equipped with a functor ${\mathcal{I}}\to {\mathcal{C}}$ (such as any object of ${\ensuremath{\mathsf{Cat}_{\mathcal{I}}}}$). In other words, it is just a different way of packaging the same definition. To see this: the height of $T$, as defined in this section, is the largest $k$ such that is non-zero for some value of $l$, whereas the height of $T$, as defined in §\[para:specialise-this-paper\], is the largest $n$ such that is non-zero for some value of $m-n$. But the objects and are the same, with $k \leftrightarrow n$ and $l \leftrightarrow m-n$.
Whitney’s embedding theorem for manifolds with collared boundary
================================================================
The Whitney Embedding Theorem implies that any (paracompact) smooth manifold without boundary admits an embedding into some Euclidean space. In footnote \[f:Whitney\] on page , the analogous statement for manifolds with collared boundary was used. We could not find an explicit reference for this in the literature, so we explain here briefly how to deduce the statement for manifolds with collared boundary from the statement for manifolds without boundary.
\[lem:Whitney-with-boundary\] Any (paracompact) smooth manifold $M$ equipped with a collar neighbourhood admits a neat embedding into some Euclidean half-space ${\mathbb{R}}^k_+ = \{ (s_1,\ldots,s_k) \in {\mathbb{R}}^k \mid s_k {\geqslant}0 \}$.
A *collar neighbourhood* means a proper embedding $c \colon \partial M \times [0,1] \hookrightarrow M$ such that $c(p,0)=p$. An embedding $f \colon M \hookrightarrow {\mathbb{R}}^k_+$ is *neat* if it takes $\partial M$ into ${\mathbb{R}}^{k-1} = \partial ({\mathbb{R}}^k_+)$ and the interior of $M$ into the interior of ${\mathbb{R}}^k_+$ and, moreover, there is $\varepsilon > 0$ such that for all $(p,t) \in \partial M \times [0,\varepsilon)$ we have $f(c(p,t)) = (f(p),t)$.
First, we may embed $M$ into a manifold without boundary, either by gluing two copies of $M$ together along their common boundary or simply by attaching an open collar to the boundary of a single copy of $M$. By Whitney’s Embedding Theorem we therefore obtain an embedding $g \colon M \hookrightarrow {\mathbb{R}}^{k-1}$ for some $k$. Now choose a smooth embedding $(x,y) \colon [0,1] \hookrightarrow [0,1]^2$ such that
for $0{\leqslant}t{\leqslant}\frac{1}{4}$ we have $x(t)=0$ and $y(t)=t$,
for $\frac{3}{4} {\leqslant}t{\leqslant}1$ we have $x(t)=t$ and $y(t)=1$.
We may then define the required neat embedding $f \colon M \hookrightarrow {\mathbb{R}}^k_+$ as follows. If $p \in M\smallsetminus\mathrm{image}(c)$ then $f(p) = (g(p),1)$. If $p \in \partial M$ and $t\in [0,1]$ then we define $f(c(p,t)) \;=\; (g(c(p,x(t))),y(t))$.
The idea is that most of $M$ – the part far away from its boundary – is embedded into the affine hyperplane ${\mathbb{R}}^{k-1} \times \{1\}$, and its collar neighbourhood is bent smoothly downwards towards the linear hyperplane ${\mathbb{R}}^{k-1} \times \{0\}$, using the functions $x$ and $y$, such that the boundary of $M$ is on this hyperplane and the part of the collar neighbourhood closest to the boundary of $M$ is embedded so that it rises vertically upwards from the hyperplane.
[^1]: Symmetric groups: [@Betley2002Twistedhomologyof]; braid groups: [@ChurchFarb2013Representationtheoryand; @Randal-WilliamsWahl2017Homologicalstabilityautomorphism; @Palmer2018Twistedhomologicalstability]; configuration spaces: [@Palmer2018Twistedhomologicalstability]; general linear groups: [@Dwyer1980Twistedhomologicalstability; @Kallen1980Homologystabilitylinear]; automorphism groups of free groups: [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism]; automorphism groups of right-angled Artin groups: [@GandiniWahl2016Homologicalstabilityautomorphism]; mapping class groups of surfaces: [@Ivanov1993homologystabilityTeichmuller; @CohenMadsen2009Surfacesinbackground; @Boldsen2012Improvedhomologicalstability; @Randal-WilliamsWahl2017Homologicalstabilityautomorphism]; mapping class groups of $3$-manifolds: [@Randal-WilliamsWahl2017Homologicalstabilityautomorphism]. Note that these are references for the proofs of *twisted* homological stability; in many cases, homological stability with untwisted coefficients was known much earlier.
[^2]: Dwyer [@Dwyer1980Twistedhomologicalstability] explicitly defines a *height*-like notion (at the beginning of §3), but there is a *degree*-like notion implicit in his work ([*cf*. ]{}Theorem 2.2 and the proof of Lemma 3.1). Van der Kallen [@Kallen1980Homologystabilitylinear], on the other hand, uses techniques similar to those of [@Dwyer1980Twistedhomologicalstability], but explicitly uses a *degree*-like notion (see §5.5), and remarks that functors of finite degree (in his sense) can be obtained from functors of finite degree in the sense of [@Dwyer1980Twistedhomologicalstability].
[^3]: The functor that we define later in fact has source ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}$ and target ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$, so we are implicitly composing with the inclusions $M \mapsto (M,*) : {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\to {\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}$ and ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\subset {\ensuremath{\mathsf{Cat}_{\mathsf{st}}}}$, where $* \in {\ensuremath{\mathsf{Top}_{\circ}}}$ is the one-point space.
[^4]: In §\[sec:functorial-configuration-spaces\] we also set ${\mathcal{A}}= {\ensuremath{\mathsf{Ab}}}$, but the only reason for this is to preserve notational similarity with [@Palmer2018Twistedhomologicalstability], and everything in that section generalises verbatim to the setting of an arbitrary abelian category ${\mathcal{A}}$.
[^5]: For the author, the idea of generalising from monoidal categories to modules over monoidal categories came from a conversation in 2015 with Aur[é]{}lien Djament.
[^6]: For the final equality $\mathrm{deg}^{1}(T) = \mathrm{deg}^{{\ensuremath{\dagger}}}(T)$ to be valid, one has to be slightly more precise with the definition of the structure of ${\mathcal{B}}(M,X)$ as a module over ${\mathcal{B}}({\mathbb{D}}^n,X)$: it must be induced by the boundary connected sum between ${\mathbb{D}}^n$ and $M$, *using the component of $\partial M$ containing the basepoint*.
[^7]: The author would like to thank Aur[é]{}lien Djament for pointing out an error in an earlier version of this remark.
[^8]: We expect that this assumption is not necessary, since we expect that Proposition \[p:two-degrees-agree-2\] should hold assuming only that $I_{{\mathcal{M}}}$ is initial, and also with deg replaced by either ideg or sdeg (the case of wdeg seems more subtle).
[^9]: The four types of degree coincide since the unit object of ${\mathcal{C}}$ is null ([*cf*. ]{}Remark \[rmk:4-definitions-of-deg\]), which is why we can write $\mathrm{deg}^x(T)$ instead of, say, $\mathrm{ideg}^x(T)$.
[^10]: In fact, we give three definitions, each depending on a slightly different structure on ${\mathcal{C}}$, and show that they agree whenever two are defined (Lemma \[lem:three-definitions\]).
[^11]: To see that it specialises as claimed to the setting of Djament and Vespa, combine D[é]{}finition 2.1, Proposition 2.3, D[é]{}finition 2.6 and Proposition 2.9 of [@DjamentVespa2019FoncteursFaiblementPolynomiaux].
[^12]: [*Cf*. ]{}the first part of the proof of Lemme 2.7 in [@CollinetDjamentGriffin2013Stabilitehomologiquepour].
[^13]: See Definition 3.6 and Proposition 3.3 of [@HartlPirashviliVespa2015Polynomialfunctorsalgebras].
[^14]: A small difference is that they additionally assume that $T(\varnothing)$ is the zero object of ${\mathcal{A}}$. So, for example, a functor ${\mathcal{C}}\wr \Lambda \to {\mathcal{A}}$ taking every object to a fixed object $a \neq 0$ of ${\mathcal{A}}$ and every morphism to $\mathrm{id}_a$ has height zero according to our definition, whereas it does not have any finite height according to the definition of [@CollinetDjamentGriffin2013Stabilitehomologiquepour]. The difference is analogous to the difference between linear and affine functions.
[^15]: In fact, for this construction, there is no need even for it to preserve composition – but we will want this later.
[^16]: In §\[para:specialise-DV\] it was assumed that the monoidal structure is symmetric, but, as remarked in §\[para:specialise-HPV\], the symmetry is not really necessary for the definition.
[^17]: In the notation of §\[sss:some-functors\], $\Sigma$ is ${\mathcal{B}}({\mathbb{R}}^\infty)$ and ${\mathcal{B}}$ is ${\mathcal{B}}({\mathbb{R}}^2)$, whereas ${\mathcal{I}}$ is a (non-monidal) subcategory of ${\mathcal{B}}({\mathbb{R}})$. For any monoidal category ${\mathcal{C}}$ and object $x$ of ${\mathcal{C}}$, there is a unique monoidal functor ${\mathcal{B}}({\mathbb{R}}) \to {\mathcal{C}}$ sending $1$ to $x$; its restriction to ${\mathcal{I}}\subset {\mathcal{B}}({\mathbb{R}})$ is the “natural” functor $s$ to which we are referring.
[^18]: \[f:Whitney\]One may see this claim as follows. The category ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}\times {\ensuremath{\mathsf{Top}_{\circ}}}$ has a *cofinal* subcategory consisting of (collared, basepointed) Euclidean halfspaces of dimension ${\geqslant}3$ in ${\ensuremath{\mathsf{Mfd}_{\mathsf{c}}}}$, together with the one-point space $* \in {\ensuremath{\mathsf{Top}_{\circ}}}$. Cofinality of this subcategory follows from the Whitney Embedding Theorem, or, more precisely, its analogue for manifolds with collared boundary (see Lemma \[lem:Whitney-with-boundary\]). The functor ${\mathcal{B}}$ sends this whole subcategory to the object $\Sigma$ (and its identity morphism) in ${\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}$ ([*cf*. ]{}§2.4 of [@Palmer2018Twistedhomologicalstability]), thus automatically providing a lift of ${\mathcal{B}}$ to $({\ensuremath{\mathsf{Cat}_{\mathsf{s}}}}\downarrow \Sigma)$.
[^19]: More generally, there is a ring associated to any *semigroup with absorbing element*, i.e. semigroup $S$ containing an element $\infty$ such that $s\infty = \infty s = \infty$ for all $s\in S$. This ring is ${\mathbb{Z}}S/{\mathbb{Z}}\{\infty\}$: the free ring without unit ${\mathbb{Z}}S$ generated by $S$ quotiented by the two-sided ideal ${\mathbb{Z}}\{\infty\}$ generated by $\infty$. A category ${\mathcal{C}}$ may be regarded as a *partial semigroup* and then turned into a semigroup with absorbing element ${\mathcal{C}}^\circ$ by adjoining a new element $\infty$: any composition $fg$ that is undefined in ${\mathcal{C}}$ is defined to be $\infty$ in ${\mathcal{C}}^\circ$. This recovers the definition of *category ring* given above. The construction is similar to that of [@Boettger2016Monoidswithabsorbing], which associates a ring to any *partial monoid*, going via a *monoid with absorbing element*, called a *binoid* in the cited paper.
[^20]: This is because it is clearly generated by elements of this form with $x\in g(a^\prime)$ for $a^\prime$ possibly different to $a$, but if $a\neq a^\prime$ this element is in fact zero, since then $(\beta,a)\otimes x = (\beta,a)\otimes g(\mathrm{id}_{a^\prime})(x) = (\beta,a)\cdot \mathrm{id}_{a^\prime} \otimes x = 0\otimes x = 0$.
|
---
abstract: 'Cascading failure of a power transmission system are initiated by an exogenous event that disable a set of elements (e.g., lines) followed by a sequence of interrelated failures (or more precisely, trips) of overloaded elements caused by the combination of physics of power flows in the changed system topology, and controls. Should this sequence accelerate it can lead to a large system failure with significant loss of load. In previous work we have analyzed deterministic algorithms that in an online fashion (i.e., responding to observed data) selectively shed load so as to minimize the amount of lost load at termination of the cascade. In this work we present a rigorous methodology for incorporating noise and model errors, based on the Sample Average Approximation methodology for stochastic optimization.'
author:
- 'Daniel Bienstock$^{1}$ and Guy Grebla$^{2}$[^1] [^2][^3]'
bibliography:
- 'cdc15.bib'
title: '**Robust Control of Cascading Power Grid Failures using Stochastic Approximation** '
---
Introduction
============
We present a rigorous methodology for computing robust algorithms for control of cascading failures of power transmission systems. We focus on the linearized, or DC, approximation for power flows, and on line tripping outages. We note, however, that the same underlying methodology will apply for other models of power flows and other types of equipment outages. This work extends the approach in [@bienstock2011optimal]; related work is consided in [@chertkovpes11], [@carreras2], [@pooyaeppsteinhines], and references therein.
In the DC approximation to power flows (see [@bergenvittal] for background) a transmission system is modeled by given a directed graph $G$ with $n$ buses and $m$ lines. In addition, for each line $e$ we are given its [*reactance*]{} $x_e$ and its [*limit*]{} $u_e$ (we may also refer to a line $e$ in the form $e = pq$ so as to indicate its “from” bus $p$ and ‘its ‘to’’ buse $q$. Additionally, we are given a [*supply-demand*]{} vector $\beta \in {{\cal R}}^n$ with the following interpretation. For a bus $i$, if $\beta_i > 0$ then $i$ is a [*generator*]{} (a source node) while if $\beta_i < 0$ then $i$ is a [*load*]{} (a demand node) and in that case $-\beta_i$ is the [*demand*]{} $d_i$ at $i$. The condition $\sum_i \beta_i = 0$ is assumed to hold. We denote by ${{\cal G}}$ the set of generators and by ${{\cal D}}$ the set of demand nodes. The linearized power flow problem specifies a variable $f_{pq}$ associated with each line $pq$ (active power flow) and a variable $\phi_p$ (phase angle) associated with each bus $p$. The DC approximation is given by the system system of equations: $$\begin{aligned}
&& N f \ = \ \beta, \ \ \ N^T \phi - X f \ = \ 0, \label{linearpowerflow}\\[-15pt]\nonumber\end{aligned}$$ where $N$ denotes the node-arc incidence matrix of $G$ [@AMO93] and $X = {\mathop{\bf diag}}\{x_{ij}\}$.
\[uniqueness\] It can easily be shown that system (\[linearpowerflow\]) is feasible if and only if $\sum_{i \in K} \beta_i = 0$ for each component (“island”) $K$ of $G$, and in that case the solution is unique in the $f$ variables.
In this statement the line limits $u_{pq}$ play no role. It may be the case that in the (unique) solution $f$ to (\[linearpowerflow\]) we have $|f_{pq}| > u_{pq}$. In that case line $pq$ is *at risk*, and, unless some control action is taken, will eventually trip. The “eventually” is an imprecise statement. Certainly, in the context of cascade modeling, the $u_{pq}$ should be the “emergency” line limits. Nevertheless, care should be taken to model the non-precise nature of line tripping. We will comment on this point, again, later.
We can now outline the model for discrete-time cascading failures from [@bienstock2011optimal]. Similar models have been proposed in the literature, see
In this template, each “time-step” models a time increment of length $\Delta t > 0$. In Step 3, when a load imbalance results in some island, we assume that the larger of generation or load are adjusted, downwards, so as to attain balance. The model does not attempt to explain the cascade within resolution finer than $\Delta t$. It should be noted that cascades can be extremely rapid – perhaps lasting seconds. However, several cascading failures of notoriety have been significantly slower, such as the 2004 U.S. Northeast event [@usc]. For the purposes of this paper, the reader should assume that $\Delta t$ is of the order of minutes. We remark that cascades are extremely noisy events, characterized at a fine time scale by a myriad of events, some of which are extremely challenging to model, such as physical contacts of sagging lines with vegetation, and human action (and errors – which have been found to take place in many cascades). Any model of cascading failures which attempts to model time at a very fine scale (perhaps, continuous time) would need incorporate correspondingly fine models for such events. Additionally, there would be a need to resolve “race-conditions,” which take place when many lines can trip within a very short amount of time with potentially very different cascade outcomes (observed by other authors). In this paper, for simplicity, we will therefore assume a relatively large $\Delta t$. We note that one could furthermore argue that a control algorithm that is subject to noise and incomplete information would benefit from relying on a relatively large $\Delta t$.
In [@bienstock2011optimal] we considered a general form of control template, which assumes that the initial exogenous event that sets-off the cascade has been observed, and that $G$ is the post-event network. The control entails load shedding; when load is shed in some connected subnetwork, generation must be correspondingly adjusted in that subnetwork.
[**Input**]{}: a power grid with graph $G$. Set $G^{(1)} = G$.
\[ln:measurements\] \[ln:apply-control\] \[ln:resulting-flows\] \[ln:adjust\]
We stress that $G$ is the [**post-initiation**]{} state of the grid, i.e. the elements disabled in the initial exogenous event are not present. The termination condition ensures that indeed the cascade is stopped at the end of the planning horizon [@bienstock2011optimal]. In general, Step 3 consists of any step where a network controller obtains current data. In Step 4 this data is used to apply the control computed in Step 1 using the measurements as inputs. A completely deterministic version of Framework \[contcasc\] is considered in [@bienstock2011optimal]. This algorithm is characterized by two features:
- At time $t$, if $K$ is an island in $G^{(t)}$, then in Step 4 all loads in $K$ are scaled by a factor $0 \le \lambda^t_K \le 1$ (as are all generator outputs in $K$).
- In Step 6, a line $e$ is tripped if $u_e \le |g^{(t)}_e|$.
The objective is to compute the $\lambda^t$ so that at termination the total load still being delivered is maximized. The main contribution in [@bienstock2011optimal] is an algorithm that solves this problem in polynomial time, for each fixed value of $T$. This fact may appear surprising given the nature of rule (s.i). There are an exponential number of islands $K$ that could be realized at time-step $t$ – how could the algorithm run in polynomial time?
The answer is that the $\lambda^t$ parameters need only be computed for an [**optimal**]{} realization of the controlled cascade. Having chosen the vectors $\lambda^1, \lambda^2, \ldots, \lambda^{t-1}$ then a unique state will be observed in time-step $t$. This is a consequence of the fact that rule (s.ii) is purely deterministic. Hence if we somehow know that $\lambda^1, \lambda^2, \ldots, \lambda^{t-1}$ have been optimally chosen, then all that is needed is for $\lambda^{t}$ to be optimally chosen, as well. Continuing inductively we will obtain an optimal control across all time-steps. To put it in a pedestrian manner, the controlled cascade plays out like a script, with the events that transpire at each time-step $t$ known, precisely, in advance.
Thus, the key in the analysis of the algorithm in [@bienstock2011optimal] is that, indeed we can choose $\lambda^{t}$ optimally (provided that $\lambda^1, \lambda^2, \ldots, \lambda^{t-1}$ have previously been computed optimally). To prove this point [@bienstock2011optimal] relies on a variant of dynamic programming. Below we will consider an updated form of this algorithm.
Modeling stochastics
====================
Even though the algorithm in [@bienstock2011optimal] is provably optimal, it is readily apparent how that algorithm falls short, and in particular may not prove robust. This concerns rule (s.ii) – should this assumption prove inaccurate it is quite likely that the set of lines that trip in Step 6 will be different than anticipated in the “script” mentioned above. Not only that, but the set of islands actually observed at time-step $t$ may be different from those expected by the “script”, and thus the computed control does not even make sense (i.e. we have the wrong parameters $\lambda^t_K$). In this section we consider algorithms that not only bypass these shortcomings, but also attain a form of algorithmic robustness that can be precisely stated.
- We will incorporate stochastics into the line-tripping rule. More precisely, at time $t$ we compute, for each line $i$, the quantity $\tilde{f}^t_i=(1+\epsilon^t_i) |f^{(t)}_i|$, where where $\epsilon^t_i$ is a random variable from a known distribution. Additionally, we assume that $|\epsilon^t_i|$ is bounded by $1 > b\geq 0$. Here, it is assumed that $b << 1$. The *stochastic line-tripping rule* is that line $i$ is tripped if $\tilde f^t_i \ge u_i$. In what follows, we will $\tilde{f}^t_i$ is the *noisy flow* in line $i$ at time-step $t$.
- Under the stochastic line tripping rule, control rule (s.i) does not make sense because we do not know, in advance, the set of islands that will be realized at time-step $t$. In fact, potentially, any island could be realized. Instead, the control will compute a *single* value $\lambda^t$ which will be used to scale all loads at time-step $t$.
We can now state the optimization problem of interest.
[**Optimal Robust Control Problem**]{}. Compute scaling values $\lambda^1, \ldots, \lambda^{T-1}$ such that subject to rules (L.a), (L.b) we maximize the *expected yield*, where by “yield” we mean the load that is served at termination.
[**Remarks.**]{}
- In (L.a) we do not make any assumptions as to the source or nature of the stochastics. In fact, we generically model “noise” in this fashion, so as to be able to capture *any* form of uncertainty that could hamper a control algorithm, so long as the magnitude of the errors is not overly large. As we will see in our experiments, simply allowing stochastic tripping gives rise to a large variety of cascading outcomes. A control that maximizes expected load will thus be robust with respect to many alternative histories that the system could follow. However, there is a specific setting in which rule makes sense – we can use (L.a) to measure *errors* in line measurements.
- The termination rule in Framework \[contcasc\] needs to be revised when rule (L.a) is applied so as to make certain that the final load shedding does terminate the cascade. We do so by redefining $\psi^{(T)} \, \doteq \, \min\left\{ 1 \, , \,
\max_{j} \{ \frac {\tilde f^{(T)}_{j}} { (1-b)u_{j}} \} \right\}$
Since $b$ is small, this amounts to a small correction, at termination.
Solving the optimal robust control problem under a given noise vector $\epsilon$
================================================================================
Some of the algorithmic steps presented here echo some steps in [@bienstock2011optimal]. However, because of (L.a) and (L.b), the underlying mathematical nature of the problem is fundamentally different, as are the actual proofs. First we introduce some notation.
- We define the function $\eta^{(T)}_G(z | \tilde{f})$ as the maximum expected yield given that in the first time-step the noisy flows are $\tilde{f}^{(1)} = z \tilde{f}$.
- We denote $G^t$ the power grid at the beginning of time-step $t$, a random variable under (L.a).
- Denote by $\epsilon$ the $m\times T$ matrix of $\epsilon_i^{(t)}$ values for all $i$ and $1\leq t\leq m$.
- For $z \ge 0$ real, let $\Theta^{(T)}_G(z | f, \epsilon)$ be the [*deterministic*]{} maximum yield obtained if at the start of the first time-step the flows in the grid are equal $z f$, and all the $\epsilon$ is a given, fixed, vector of values $\epsilon_i^{(t)}$ (rather than random). Note that $\Theta^{(T)}_G$ is deterministic since all realizations of the random variables $\epsilon_i^{(t)}$ are provided.
Clearly the following holds, $$\begin{aligned}
\eta^{(T)}_G(z |
\tilde{f}) = E_{\epsilon} \left [ \Theta^{(T)}_G(z | f',
\epsilon) \right ],\end{aligned}$$ where $f'_i = \frac {\tilde{f}_i} {1+\epsilon_i^{(T)}}$.
\[lem:breakpoints2\] $\Theta^{(1)}_G(z | f', \epsilon)$ is a nondecreasing piecewise-linear function of $z$ with two pieces, the second one of which has zero slope.
Note that since $T = 1$, only termination step in Framework \[contcasc\] will be executed. Denoting by $\tilde D$ the sum of demands implied by $f'$ we have as per our cascade termination criterion that the final total demand at the end of $T = 1$ time-step will equal $$\begin{aligned}
z \tilde D, && \mbox{if} \ \ z \, \le \, 1/\psi^{(1)}, \ \ \ \mbox{and} \\
\frac{z}{z \psi^{(1)}} ~ \tilde D \ = \frac{1}{\psi^{(1)}}~ \tilde D, && \mbox{otherwise}.\ \ \hspace*{.2in}
\end{aligned}$$
Using similar ideas to those shown in [@bienstock2011optimal], we now turn to the general case with $T>1$ and we will show the following theorem.
\[th:breakpoints\] $\Theta^{(T)}_G(z | f, \epsilon)$ has at most $\frac {m!} {(m-T+1)!}$ breakpoints.
To prove Theorem \[th:breakpoints\], we will need the following definitions.
\[def:crit\] A [*critical point*]{} is a real $\gamma > 0$, such that for some line $j$, $\gamma \tilde f_{j} = u_{j}$, i.e. $\gamma |f_{j}| = u_{j}/(1 + \epsilon^t_j)$.
Remark: for completeness we should use, in this definition, the superindex $t$, which we have skipped for brevity. Recall that we assume $u_j > 0$ for all $j$; thus let $0 < \gamma_1 < \gamma_2 < \ldots < \gamma_p$ be the set of all distinct critical points. Here $0 \le p \le m$. Write $\gamma_0 = 0$ and $\gamma_{p+1} = +\infty$.
For $1 \le i \le p$ let $F^{(i)} = \{ j: \, \gamma_h \tilde f_{j} = u_{j} \}.$
Assume that the initial flow is $z f$ with $ 1 \geq z > 0$ and let $0 < \lambda^{(1)} \le 1$ be the optimal multiplier used to scale demands in time-step 1. Write $$\begin{aligned}
&& q(z) = {\mathop{\rm argmax}}\{ h \, : \, \gamma_h < z \}. \label{defq}\end{aligned}$$ Thus, $z \le \gamma_{q+1}$, and so $\lambda^{(1)} z \le \gamma_{q+1}$. We stress that these relationships remain valid in the boundary cases $q = 0$ and $q = p$.
Let the index $i$ be such that $\lambda^{(1)} z \in (\gamma_{i-1}, \gamma_{i}]$.
In time-step 1 of Framework \[contcasc\], at line \[ln:apply-control\] we will scale all demands by $\lambda^{(1)}$. We assume the framework is given a connected graph, namely, $G^{(1)}$ is connected. Therefore, in line \[ln:resulting-flows\] we will also scale all supplies by $\lambda^{(1)}$. Thus, for any $h \le i-1$, and any line $j \in F^{(h)}$, we have that after Step \[ln:resulting-flows\] the absolute value of $f_j$ is $\lambda^{(1)} z \tilde f_{j} > \gamma_h \tilde f_{j} =
u_{j}$, and consequently line $j$ becomes outaged in time-step 1. On the other hand, for any line $j \notin \cup_{h \le i-1} F^{(h)}$, the absolute value of the flow on $j$ immediately after Step \[ln:resulting-flows\] is $\lambda^{(1)} z \tilde f_{j} \le \gamma_{i} \tilde f_{j} \le u_{j}$, and so line $j$ does not become outaged in step 1. In summary, the set of outaged lines is $\cup_{h \le i-1}
F^{(h)}$; in other words, we obtain the same network $G^{(2)} = G^{(1)} \setminus \cup_{h
= 1}^{i-1} F^{(h)}$ for every $z$ with $\lambda^{(1)} z \in (\gamma_{i-1},
\gamma_{i}]$.
For an index $j$, write ${{\cal K}}(j)$ = set of components of $G^{(1)}
\setminus \cup_{h = 1}^{j} F^{(h)}$.
Let $H \in {{\cal K}}(i-1)$ and denote the initial supply-demand vector by $\beta$. Prior to line \[ln:adjust\] in step 1, the supply-demand vector for $H$ is precisely the restriction of $\lambda^{(1)} z \beta$ to the buses of $H$, and when we adjust supplies and demands in line \[ln:adjust\], we will proceed as follows
- if $ \sum_{s \in {{\cal D}}\cap H} (-\lambda^{(1)} z \beta_s) \, \ge \, \sum_{s \in {{\cal G}}\cap H} (\lambda^{(1)} z \beta_s)$ then for each demand bus $s \in {{\cal D}}\cap H$ we will reset its demand to $ -r \lambda^{(1)} z \beta_s$, where $$r = \frac{\sum_{s \in {{\cal G}}\cap H} (\lambda^{(1)} z \beta_s)}{\sum_{s \in {{\cal D}}\cap H} (-\lambda^{(1)} z \beta_s)} = -\frac{\sum_{s \in {{\cal G}}\cap H} (\beta_s)}{\sum_{s \in {{\cal D}}\cap H} (\beta_s)},$$ and we will leave all supplies in $H$ unchanged.
- likewise, if $ \sum_{s \in {{\cal D}}\cap H} (-z \lambda^{(1)} \beta_s) \, < \, \sum_{s \in {{\cal G}}\cap H} (\lambda^{(1)} z \beta_s)$ then the supply at each bus $s \in {{\cal G}}\cap H$ will be reset to $ r \lambda^{(1)} z \beta_s$, where $$r = -\frac{\sum_{s \in {{\cal D}}\cap H} (\beta_s)}{\sum_{s \in {{\cal G}}\cap H}} (\beta_s),$$ but we will leave all demands in $H$ unchanged.
Note that in either case, in time-step 2 component $H$ will have a supply-demand vector of the form $\lambda^{(1)} z \beta^H$, where $\beta^H$ is a supply-demand vector which is [*independent*]{} of $z$. The supply-demand vector $\beta^H$ corresponds with flows $f^H$ on the lines in $H$, which are therefore also [*independent*]{} of $z$. The flows $f^{(2)}$ on $G^{(2)}$ are $\cup_{H\in
G^{(2)}} f^H$, and the final total demand will be $$\begin{aligned}
\Theta_{G^{(2)}}^{(T-1)}(\lambda^{(1)} z | f^{(2)}) \label{Hfinal}\end{aligned}$$
For a given value of $i$, since $z\leq 1$, also $i \leq q$ holds. As noted above, by definition (\[defq\]) of $q$ we have that $\gamma_{i} \le
\gamma_{q} < z$. Thus, the expression in (\[Hfinal\]) is maximized when $\lambda^{(1)} = \frac{\gamma_{i}}{z}$, and we obtain final ($T$-step) demand equal to $$\begin{aligned}
D_i & \doteq & \Theta_{G^{(2)}}^{(T-1)}(\gamma_{i} | \cup_{H\in
{{\cal K}}(i-1)} f^H), \label{fixedcase1}\end{aligned}$$
In summary, $$\begin{aligned}
\Theta_{G^{(1)}}^{(T)}(z | f) & = & \max _{1 \leq i\leq q(z)} D_i \label{eq:final}\end{aligned}$$ We are now ready to prove Theorem \[th:breakpoints\]
In Lemma \[lem:breakpoints2\], we showed that for $T=1$ the function has two breakpoints. As can be seen from , the number of breakpoints is at most $q(z)$. It is important to note that (a) $q(z)$ is at most the number of lines in the graph and (b) the number of non-tripped lines decreases by at least $1$ in every time-step (otherwise, the cascade stopped). Therefore, the number of breakpoints for $T$ steps is $m (m-1) \ldots (m-T+1) = \frac
{m!} {(m-T)!}$.
Solving the optimal robust control problem using the Sample Average Approximation Method
========================================================================================
The key methodology that we will reply in order to solve the problem of interest is the Sample Average Approximation (SAA) method [@KS99]. The method begins by taking $m T N$ i.i.d samples (each under the assumed distribution) of $\epsilon$ values. We view these values as arranged into $N$ ensembles, or *realizations*. Here, for $1 \le k \le N$ we let $\epsilon^k$ denote the $k^{th}$ realization, consisting of values $\epsilon^{(t),k}_i$, for $1 \le t \le T$ and all lines $i$.
Given any candidate vector $\Lambda = (\lambda^{(1)}, \lambda^{(2)}, \ldots, \lambda^{(T-1)})$ for solving the robust optimal control problem, denote by $\Theta_G^T(z | f, \Lambda, k)$ to be the yield obtained at termination if at the start of the first time-step the flows in the grid are equal $z f$, if we use $\Lambda$ as the control vector at each time-step, and $\epsilon^{(t),k}_i$ is used for each $t$ and $i$ in the line-tripping rule (L.a).
Thus, $\Theta_G^T(1 | f^{(1)}, \Lambda, k)$ indicates the behavior of the control vector $\Lambda$ under the fixed noise vector $\epsilon^{k}$. This motivates the following definition.
[**(Sample average robust control problem.)**]{} Given the $N$ realizations $\epsilon^{1}, \ldots, \epsilon^{N}$, compute a control vector $\Lambda$ so as to maximize $$\begin{aligned}
&& \frac{1}{N} \sum_{k = 1}^N \Theta_G^T(1 | f^{(1)}, \Lambda, k). \label{saasum}\end{aligned}$$
This definition is appealing in that, clearly, if $N$ is large a control that maximimized (\[saasum\]) should clearly perform well in our stochastic setting – since it maximizes the average outcome over many possible noise scenarios. This observation can in fact be rigorously established by using the Central Limit Theorem (and observing that all random variables under consideration are bounded). See e.g. equation (2.23) from [@KS99]. If we want to solve the Robust Optimal Control problem within additive error $\delta > 0$ with probability at least $1 - \alpha$ one can establish an upper bound of the form $N = O( \frac {3\sigma^2} {\delta^2/4} \log(1/\alpha)$, where $\sigma$ is the standard deviation of the distribution of the $\epsilon$ values [^4] We can establish an analogue of Lemma \[lem:breakpoints2\]. Let $\hat \eta^T_G (z| f) \ \doteq \ \frac{1}{N} \sum_{k = 1}^N \Theta_G^T(z | f, \Lambda, k)$. This is the maximum average yield at termination under all the realizations if we start time-step 1 with flows $z f$.
\[lem:breakpoints\] $\hat \eta^T_G (z| f)$ is nondecreasing, piecewise-linear, with at most $$N \frac {m!} {(m-T)!}$$ breakpoints.
By theorem \[th:breakpoints\], $\Theta^{(T)}_G(z | f, \epsilon)$ is piecewise-linear with at most $\frac {m!} {(m-T+1)!}$ breakpoints. Since $\hat \eta^T_G (z| f)$ is an average taken over $N$ instances of $\Theta^{(T)}_G(z | f, \epsilon)$, the lemma follows.
Using Lemma \[lem:breakpoints\], it follows that we can efficiently determine the control in order to maximize the expected total demand in the face of measurement errors.
Simulation Study
================
In this section we present simulation results using a modified form of the IEEE 118-bus system, with line limits set approximately at $20 \%$ above flow values[^5]. We first describe our implementation and specific noise model (see (L.a) above). We then present and discuss the computed $\hat \eta^T_G (z| f)$, obtained via the SAA, and compare the performance of our robust control algorithm to its non-robust version from [@bienstock2011optimal].
Implementation and Noise Model
------------------------------
As discussed above an approximately optimal load shedding control can be found by computing the function $\hat \eta^T_G (z| f)$, and for this purpose we will rely on Lemma \[lem:breakpoints2\]. Thus, at each every time-step $t$, we solve the DC equations to obtain the flows and compute the critical points $\gamma_1,\ldots,\gamma_p$. An important point is that these are the critical points arising from all realizations (recall Definition \[def:crit\]). Next, for each $i$ with $1 \le i \le p$ we proceed as follows.
Note that any two choices for $\lambda^1$ in the open interval $(\gamma_{i-1}, \gamma_{i})$ will result in exactly the same set of line trips under all realizations. In fact, any two choices for $\lambda^1$ in $(\gamma_{i-1}, \gamma_{i})$ will produce the same graph $G(i)$ (which we can easily compute) but with supply-demand vectors that differ by a constant factor – and thus, in a common flow vector (up to scale) which we denote by $f(i)$. It follows that if we (recursively) compute the function $\hat \eta^{T-1}_{G(i)} (z| f(i))$ (which is a function of the real $z$) we will obtain $\hat \eta^{T}_{G} (z| f)$ in the interval $(\gamma_{i-1}, \gamma_{i})$.
Next we discuss the specific noise model used to implement rule (L.a). We will first describe a specific noise model. Then we will prove it is of the form (L.a). A line $j$ will trip
\[practtrip\] $$\begin{aligned}
\hspace{-.5in}&& \mbox{with probability $1$, if $|f_j| \geq u_j$} \\
\hspace{-.5in}&& \mbox{with probability $1/2$, if $u_j > |f_j| \geq 0.95 u_j$}\\
\hspace{-.5in} && \mbox{with probability $0$, otherwise.}\end{aligned}$$
Rule (\[practtrip\]) is an example of (L.a).
[*Proof.*]{} Define the random variable $\epsilon_j^{(t)}$ for every $j$ and $t$ as follows: $$\begin{aligned}
\epsilon_j^{(t)} = \frac{0.05}{0.95}, && \mbox{with probability } 0.5, \ \ \ \mbox{and} \\
\epsilon_j^{(t)} = 0, && \mbox{with probability 0.5}.\ \ \hspace*{.2in} \end{aligned}$$
Recall that $\tilde{f}_j^{(t)} = (1+\epsilon_j^{(t)}) f_j^t$ and consider the case where $f_j^t \geq 0.95 u_j$. With probability $0.5$, $\epsilon_j^{(t)}=0$ and therefore $\tilde{f}j^{(t)}<u_j$ and line $j$ will not be tripped. But, with probability $0.5$ we get $\epsilon_j^{(t)}=\frac {0.05}{0.95}$ and $\tilde{f}j^{(t)} = (1+\frac{0.05} {0.95})f_j^t \geq (1+\frac{0.05}
{0.95})0.95 u_j = u_j$ and line $j$ will trip.
Results
-------
We set $T=3$ and remove a single line to start a cascading failure. Fig. \[fig:line2\_5\] shows the $\hat \eta (z|\tilde f)$ when the removal of line $2$ triggers the cascade, the SAA is computed over $5$ realizations. From Fig. \[fig:line2\_5\](a) and \[fig:line2\_5\](b) we can infer that $\lambda_1
\approx 0.72$ and $\lambda_2 \approx 0.85$, respectively. Under such load shedding, the expected yield is $\approx 0.62$.
Next, we increase the number of realization for the SAA from $5$ to $500$, the results are depicted in Fig. \[fig:line2\_500\]. We can see that taking into account more realizations affected the average yield. However, in this specific case the optimal value for $\lambda^1$ is the same for both Fig. \[fig:line2\_500\] and Fig. \[fig:line2\_5\].
A natural question is how the non-robust control algorithm from [@bienstock2011optimal] would behave under the noisy model. To answer this question, we compute the control using the algorithm from [@bienstock2011optimal] which yields $\lambda_1 \approx 0.76$ and $\lambda_2 \approx 0.85$. Indeed, these values result with a yield of $0.65$ in the non-robust model, which is higher than $0.62$ (the expected yield in the robust model). However, evaluating the non-robust contrul under the robust model for $5$ realizations results with an expected yield of $0.37$, far worse than what is attained using our robust control.
We tested values of $T=2,3,4,5$. The initial cascade is caused by tripping line $(4,5)$, since tripping this line causes a relatively major cascade in the 118-bus system.
Fig. \[fig:line2\_realizations\] shows the yield vs the scaling in the first time step for a value of $T=3$. In Fig. \[fig:line2\_realizations\](a) the number of realizations used for the Sample Average Approximation (SAA) is $5$ while in Fig. \[fig:line2\_realizations\](b) this number is increased to $120$. As expected, increasing the number of realizations affects the expected yield in the different linear segments. In Fig. \[fig:line2\_realizations\], we see that despite the fact that the graph differ, in this specific case the maximum yield remains similar.
While Fig. \[fig:line2\_realizations\] shows the load shedding required in the first timestep (shedding of $\approx 0.72$ maximize the yield), the figure does not reveal what will be the load shedding at the second time step. Fig. \[fig:line2\_2\_vs\_3\](a) shows the yield as a function of load shedding in the second times tep [*assuming that in the first time step an optimal choice for $\lambda^1$ is used*]{}. As can be observed from Fig. \[fig:line2\_2\_vs\_3\](a), the optimal choice for $\lambda^1$ is $\approx
0.85$. Fig. \[fig:line2\_2\_vs\_3\](b) considers the case where $T=2$, it can be seen that that load shedding is done at only one time-step. While the maximum yield is slightly lower, overall it is close to the maximum yield ($\approx 62\%$) for the case where $T=3$. We will now show that the difference in the maximum resulting from increasing $T$ can be significant.
In Fig. \[fig:line2\_5\_steps\] we set $T=5$. Fig. \[fig:line2\_5\_steps\](a) shows the yield as a function of load shedding in the first time steps and can be used to find the optimal load shedding scaling for the first time step. Fig. \[fig:line2\_5\_steps\](b) shows the yield as a function of load shedding in the second time step given that the optimal load shedding scaling was done in the first time step. We can see that in this case making a small amount of load shedding in the first two time steps results with a very good final yield ($\approx 85\%$) compared to the case of $T=3$. We remark that in the rest of the time steps the optimal control is to carry out very minimal load shedding.
A natural question is what would the non-robust control algorithm from [@bienstock2011optimal] would obtain in the considered model. To answer this question, we compute the control using the algorithm from [@bienstock2011optimal]. As expected, in the non-robust model, the algorithm from [@bienstock2011optimal] is the best, since there is perfect knowledge of the flows and the algorithm from [@bienstock2011optimal] is optimal. However, when measurement errors are present, the algorithm from [@bienstock2011optimal] performs much worse than the robust algorithm presented here. Table \[tab:yields\] shows the yield obtained by both algorithm. Additionally, Table \[tab:yields\] further demonstrates that increasing $T$ improves the yield. We remark that the non-robust model refers to the model from [@bienstock2011optimal] and the robust model refers to the model considered in this paper.
2 3 4 5
---------------------------- --- --- --- ---
Non-robust solution
and [*non-robust model*]{}
Non-robust solution
and [*robust model*]{}
robust solution
and [*robust model*]{}
: Yields obtained from performing the solution obtained by the robust and non-robust algorithms under the robust and non-robust model.[]{data-label="tab:yields"}
We also compute a $95\%$ confidence interval on the yield obtained when performing the optimal load shedding. The confidence interval is computed for different number of realizations and $T=5$ and it is shown in Table \[tab:confidence\]. The obtained confidence interval are overall very good. As expected, we can see improvement in the confidence interval as we take more realizations.
2 5
------------------ ----------------------- ----------------------- -- --
5 realizations \[$62.39\%,63.19\%$\] \[$83.02\%,84.87\%$\]
60 realizations \[$62.7\%,63.08\%$\] \[$83.84\%,84.62\%$\]
120 realizations \[$62.79\%,63.04\%$\] \[$84.05\%,84.64\%$\]
: A $95\%$ confidence interval for optimal expected yield for different number of realizations.[]{data-label="tab:confidence"}
Suggested Changes and Additional Issues
=======================================
1. Most important problem: for a given trajectory, we can guarantee that the cascade will stop. but how can we guarantee it will stop for all trajectories? Maybe scale based on last measurement, but then for the trajectory we might end with different yield.
2. Find a way to make the error [**proportional to the noisy flow**]{}. Although, it is possible that it is sufficient to make it proportional to the real flow because this should be enough for a deterministic path.
3. Model errors on flows instead on the supply-demand, due to the above point. Will need to define $\Theta^{(R)}_G(t | f)$
4. We need to remember that our expected yield in the current model is when noisy over all possible $W$, a solution for a specific failure might be bad. This relates to what we were talking about - taking the minimum value of the yield to show what we can guarantee.
Future Research
===============
1. Acceptable values for $|T|,\ \epsilon,\ N$.
2. Extending to find optimal solution with a certain probability - as in [@KS99].
3. Note that with the current results we do not utilize the measurements collected in the next time-steps. How would doing so improve our results?
[^1]: \*This research was partially funded by LANL award “Grid Science” and DTRA award HDTRA1-13-1-0021.
[^2]: $^{1}$D. Bienstock is with the Departments of Industrial Engineering and Operations Research, and Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027 USA. [email: dano@columbia.edu]{}
[^3]: $^{2}$G. Grebla is with the Department of Electrical Engineering, Columbia University, New York, NY, 10027 USA. [email: guy@ee.columbia.edu]{}
[^4]: Some technical points skipped for brevity.
[^5]: We will present experiments with larger examples at the conference
|
---
abstract: 'We describe a method to identify inclined water maser disks orbiting massive black holes and to potentially use them to measure black hole masses. Due to the geometry of maser amplification pathways, the minority of water maser disks are observable: only those viewed nearly edge-on have been identified, suggesting that an order of magnitude additional maser disks exist. We suggest that inward-propagating masers will be gravitationally deflected by the central black hole, thereby scattering water maser emission out of the disk plane and enabling detection. The signature of an inclined water maser disk would be narrow masers near the systemic velocity that appear to emit from the black hole position, as identified by the radio continuum core. To explore this possibility, we present high resolution (0.07–0.17) Very Large Array line and continuum observations of 13 galaxies with narrow water maser emission and show that three are good inclined disk candidates (five remain ambiguous). In the best case, for CGCG 120$-$039, we show that the maser and continuum emission are coincident to within $3.5\pm1.4$ pc ($6.7\pm2.7$ milliarcsec). Subsequent very long baseline interferometric maps can confirm candidate inclined disks and have the potential to show maser rings or arcs that provide a direct measurement of black hole mass, although the mass precision will rely on knowledge of the size of the maser disk.'
author:
- Jeremy Darling
title: 'How to Detect Inclined Water Maser Disks (and Possibly Measure Black Hole Masses)'
---
Introduction {#sec:intro}
============
Water masers arising from thin disks around massive black holes provide high brightness temperature non-thermal dynamical tracers of gas in Keplerian orbits. As such, water maser disks viewed edge-on provide tracers of the Keplerian potential and enable measurement of the black hole mass, provided a distance is known in order to translate the apparent angular disk size into a physical size [e.g., @miyoshi1995]. Maser accelerations and proper motions can also be observed, and because the circular velocity is known from Doppler shifts, the geometric distance to the black holes (and host galaxies) can be determined [e.g., @herrnstein1998]. Geometric distances obtained from water masers provide a crucial independent measurement of the Hubble constant and can be used to calibrate other distance indicators such as the period-luminosity relation of Cepheids [@riess2016].
These measurements require maser disks that are viewed within a few degrees of edge-on: otherwise, maser beaming directs emission away from the observer because masers propagate along velocity-coherent paths through the disk. For thin disks, this propagation occurs along the radial path along the line of sight toward the black hole and along the disk tangent points. For warped disks, such as that found in NGC 4258, the picture is more nuanced because a warped disk provides numerous sightlines and inclinations that intersect velocity-coherent parts of the disk [@humphreys2013]. Nonetheless, inclined maser disks are generally not seen in water maser surveys. Or are they?
It seems likely that inclined water maser disks have already been detected by single dish surveys, but they have been discarded because they show no high velocity lines that have canonically been used to identify maser disks. In this paper, we propose a mechanism to produce detectable maser emission from inclined disks that may also be used to obtain black hole masses (Section \[sec:thought\]), we present a method to detect inclined maser disks based on extant surveys (Section \[sec:candidates\]), we present Karl G. Jansky Very Large Array (VLA)[^1] observations of candidate inclined disks (Sections \[sec:obs\] and \[sec:results\]), and we provide a list of inclined disk candidates for further study (Sections \[sec:analysis\] and \[sec:discussion\]). In what follows, we assume a flat cosmology with parameters $H_0 = 70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M = 0.27$, and $\Omega_\Lambda = 0.73$, and we calculate distances from redshifts in the CMB rest frame.
A Thought Experiment {#sec:thought}
====================
Consider a typical 22 GHz water maser thin disk in Keplerian orbit around a massive black hole. This thin disk shows systemic velocity masers at the center of the observed disk, along a radial amplification path, and it shows high-velocity red- and blue-shifted masers at the tangent points of the disk, along azimuthal paths, amplifying spontaneous emission and background host galaxy continuum (Figure \[fig:schematic\], left). If a thin disk is inclined by more than a few degrees, the masers are no longer beamed toward the observer and the disk would not be detected. The known water maser disks show an inclination typically within $\sim$5$^\circ$ of edge-on [e.g., @kuo2011], suggesting that an order of magnitude additional maser disks exist, and they are simply beamed in directions we cannot observe. Note that the famous water maser disk in NGC 4258 [e.g., @miyoshi1995; @herrnstein1998] is atypical, with a disk inclination of 72$^\circ$, but the disk warp provides sightlines that are nearly edge-on, and this is where the masers are seen [@humphreys2013].
For the following discussion we will use fiducial parameters similar to those of many of the known maser disks [@kuo2011; @greene2016]. We assume a distance of 10 Mpc, a black hole mass of $M_{BH} = 10^7\ M_\odot$ (and therefore a Schwarzschild radius of $R_s\sim1\ \mu$pc), and a circular rotation speed of 1000 km s$^{-1}$ in the maser-active part of the disk located 0.5 pc from the black hole and spanning 0.2 pc. We assume a disk thickness equal to that of NGC 4258: $5.5\times10^{14}$ cm or 0.18 mpc [1$\sigma$; @argon2007].
Consider how this system appears to an observer at the location of the black hole: the disk describes a plane in the sky. Water masers reaching this observer must be purely radial (propagating inward) but will have a coherent amplification path in all radial directions (Figure \[fig:schematic\], center). All Keplerian Doppler shifts will be across the line of sight, so all maser emission reaching this location will have zero velocity in this reference frame (ignoring the transverse Doppler shift, which is negligible at $\sim$2 km s$^{-1}$). The black hole-frame observer will thus see a ring of maser light or maser spots with low velocity spread. The maser-active part of the disk would subtend $\sim$2.5 arcminutes (1$\sigma$). The expected maser opening angle would be similar, $\sim$3 arcminutes, assuming a maser path-to-size ratio of $\sim$0.2 pc/0.2 mpc $\simeq 1000$, implying maser light spanning $\sim$500 $R_s$ at the black hole.
The black hole Kerr metric would thus be bathed in maser light with a continuous distribution of impact parameters, implying gravitational deflection of maser light into nearly all angles (the deflection at 100 $R_s$ is 1.1$^\circ$ and grows inversely with impact parameter $R$ as $\theta = 2 R_s/R$; @einstein1915). One would therefore expect incoming maser light to be scattered into many angles, representing a gravitational lensing de-focusing of a given incoming beam. But portions of multiple maser beams from around the disk may be deflected into a given direction, making the net flux as a function of angular direction uncertain and in need of numerical modeling.
NGC 4258 has a warped disk [e.g., @humphreys2013], which means that the in-going radial masers can be misaligned and thus be beamed above or below the dynamical center of the disk, and that the continuous distribution of disk inclination angles will fully populate the black hole metric volume with beams from multiple angles, further enhancing the sampling of impact parameters and deflection angles. If warped disks are common, then maser disks with high velocity lines will be detected more often than one would assume based on random inclination alone (in other words, parts of many disks will have inclinations appropriate for detectable maser amplification), and the black hole will be illuminated by in-going masers from above and below the nominal disk plane, producing a range of incoming maser impact parameters and angles with respect to the disk, thereby enhancing the gravitational deflection probability toward any given observer. Regardless of whether disks are warped or not, maser light will be scattered away from the maser disk plane, making inclined disks potentially detectable, but likely faint compared to edge-on disk masers.
Masers amplify seed photons, and these seed photons can be the portion of a radio continuum that matches the maser line frequency, appropriately Doppler-shifted, or they can be spontaneously emitted maser line photons. In known maser disks, the systemic radial masers typically amplify the AGN radio continuum, and the high-velocity tangential masers amplify spontaneous emission or possibly radio continuum from the host galaxy itself. In-going radial masers do not have access to AGN seed photons and must amplify either continuum from the host galaxy or inward-directed spontaneous emission from the outer parts of the disk. In either case, it is reasonable to expect in-going masers to be weaker than the out-going systemic masers from edge-on disks. It is unclear how the amplification of in-going masers would compare to the amplification of the high-velocity masers in edge-on disks because the tangential amplification pathway may or may not have a physically longer or higher column density velocity-coherent path for amplification.
If one were to observe a water maser disk from a direction other than edge-on, would water maser emission be seen? The thought experiment described here suggests that it would, and the inclination of the disk with respect to the observer would select the deflection angle (or equivalently the impact parameter) of the observed maser light. For example, deflection by 10$^\circ$ would allow observation of masers from the back side of a disk with an inclination of 80$^\circ$, requiring an impact parameter of 12 $R_s$. Other parts of the disk could be viewed if light reaches smaller impact parameters. If, as expected, there is a continuous range of impact parameters, then a continuous distribution of maser light from an extended portion of the disk will contribute to a spectrally narrow maser line complex seen by the observer at nearly any inclination (but perhaps with higher likelihood and intensity at higher inclinations).
Observable Signatures
---------------------
The observational signatures of an inclined water maser disk would be:\
1. A narrow line or line complex\
2. at the systemic velocity\
3. at the apparent black hole location.\
The black hole location would be indicated by the radio continuum core, ideally observed at the same frequency as the water maser. The observational signatures of an inclined water maser disk, however, may also arise from other mechanisms. Water masers can be produced in radio jet-molecular cloud interactions [e.g., @gallimore1996; @claussen1998; @peck2003; @henkel2005], in star-forming regions [e.g., @tarchi2002a; @tarchi2002b; @henkel2005; @hofner2006; @darling2008; @brogan2010; @darling2011; @tarchi2011; @amiri2016], and in outflows [e.g., @greenhill2003b; @kondratko2005; @tarchi2011a but note that some objects in the latter survey may be candidates for inclined disk masers]. These are likely to be the main contaminant among an inclined disk survey sample. VLBI identification of the maser with an AGN via spatial coincidence with the core radio continuum — identified by the spectral index — can resolve the ambiguity (Section \[sec:results\]).
In contrast to edge-on disk systemic masers, it is unclear whether inclined disk maser lines will show proper motion or acceleration. This may depend on the clumpiness of maser-emitting regions, on the maser beam sizes, on the physical extent of the disk that is sampled by the observed line, and on the deflection angle to the observer. One might expect lensed masers to show little or no time variability, but the substructure seen in known water maser disks and the natural variability of water masers suggests that this may be a bad assumption.
Many extragalactic water masers detected in previous surveys meet some or all of the above observational criteria: inclined water maser disks may have already been detected! In most cases, when a single systemic velocity line is detected in water maser surveys, there is no interferometric follow-up because distance or black hole mass measurements (traditionally) require edge-on disks, the signature of which are the high velocity lines emitted from the tangent points of the disk. High resolution observations may also be frustrated by weak or variable water masers.
In Section \[sec:obs\] we present interferometric mapping of a sample of narrow-line systemic velocity water masers that appear to be inclined disk maser candidates. If observations show that the masers remain unresolved and are centered at the location of the central massive black hole (as identified by simultaneous radio continuum observations at 20 GHz), then they remain candidates and should be mapped with VLBI.
Black Hole Masses
-----------------
The Einstein radius of a strong gravitational lens is $$\theta_E = \sqrt{2 R_s {D_{LS}\over D_L D_S}}
= 9.2 \sqrt{{R_s \over {\rm mpc}} \, {D_{LS} \over {\rm pc}}} \left( D_L \over {\rm Mpc} \right)^{-1} {\rm mas},$$ where $D_L$, $D_S$, and $D_{LS}$ are the angular diameter distances to the lens, to the source, and between the lens and the source, respectively, and $D_L \simeq D_S$ for the black hole-maser disk configuration [after @einstein1936]. For the fiducial parameters listed above ($R_s = 1$ $\mu$pc, $D_{LS} = 0.5$ pc, and $D_L = 10$ Mpc), $2 \theta_E = 0.041$ mas. Since the angular resolution (HPBW) of the Very Long Baseline Array (VLBA) is 0.3 milliarcseconds at 22.2 GHz (which should be compared to twice the Einstein radius), the Einstein radius for the fiducial maser disk and black hole would require space-based VLBI to resolve. On the other hand, terrestrial VLBI could resolve a water maser Einstein ring for a more massive black hole with a physically larger maser disk: for $R_s = 100$ $\mu$pc ($M_{BH} = 10^9\ M_\odot$), $D_{LS} = 2$ pc, and $D_L = 10$ Mpc, $2 \theta_E = 0.82$ mas. Unfortunately, water maser disks have yet to be identified orbiting $10^9\ M_\odot$ black holes, but it is unclear whether is this a selection effect or a consequence of physics [@vandenbosch2016]. The most massive black hole measured using a water maser disk to date is in NGC 1194 with $M_{BH} = 10^{7.85\pm0.05}\ M_\odot$ [@kuo2011; @greene2016].
Were an Einstein ring observable from a back side in-going maser in an edge-on disk or from an in-going maser from an edge-on portion of an inclined warped disk, then one can infer a black hole mass from the angular size of the ring. One does need independent measurements of the luminosity distance to the black hole ($D_L$) and the size of the maser-emitting disk ($D_{LS}$). $D_L$ can be obtained from an assumed cosmology and the cosmological redshift, but $D_{LS}$ may be more difficult to measure. The precision of black hole masses obtained from this method may be limited by our ability to measure, model, or estimate the radius of the maser-emitting part of the disk.
Einstein rings require linear alignment between the source, the lens, and the observer, which is not germane to the inclined maser disk geometry (for unwarped disks). Instead, one would naïvely expect to see multiple images of the same maser, which can also be related to the black hole mass. This expectation, which is correct for isotropic emitters, is probably incorrect for masers.
An important difference between maser emission and the standard treatment of gravitational lensing is maser beaming: while there may be sightlines in a gravitational lens geometry that land on the emitter, emission may not be seen if light is not beamed along that sightline. For a general pointlike isotropic light source offset from the lens by angle $\theta_S$, there are two solutions to the lens equation: $$\theta_\pm = {1\over2} \left( \theta_S \pm \sqrt{\theta_S^2 + 4 \theta_E^2}\right).$$ $\theta_+$ represents the angle between the lens and the source image appearing outside $\theta_E$, and $\theta_-$ represents the angle of the image appearing inside $\theta_E$ [@narayan1995 Equation 24, Figures 5 and 7]. For an inclined maser disk configuration, no emission is directed significantly out of the disk plane, so no maser emission would be seen in the $\theta_+$ direction. On the other hand, $\theta_-$ may be small enough that this sightline is included in the maser beam passing very close to the central black hole. In this case, the source-observer deflection angle $\alpha_-$ nearly matches the complement of the disk inclination: $\alpha_- \simeq \pi/2 - i$ (Figure \[fig:schematic\], right).
In contrast to canonical lensing, we expect that the maser beaming will produce only one maser spot image, where light is deflected in the manner shown schematically in Figure \[fig:schematic\] (right). This maser image will lie inside the Einstein radius. For isotropic extended emitters, this image would also be demagnified, but masers are not isotropic emitters and can be exceptionally compact (equivalently, they demonstrate high brightness temperatures). The degree of demagnification is therefore unclear and requires numerical ray-tracing to assess.
For the fiducial parameters above, and assuming an inclination of $80^\circ$, $\theta_S = (R_{\rm maser}/D_S) \sin\alpha_- = 1.8$ mas, $\theta_E = 0.021$ mas, and therefore $|\theta_-| = 0.2$ $\mu$as, which is equivalent to 11 $R_s$. The maser beam spans this distance from the black hole, so the maser emission can be lensed toward the observer in this case. For the $10^9\ M_\odot$ black hole with a larger maser disk described above, $\theta_S = (R_{\rm maser}/D_S) \sin\alpha_- = 7.2$ mas, $\theta_E = 0.41$ mas, and therefore $|\theta_-| = 24$ $\mu$as, which is again equivalent to 11 $R_s$ (the impact parameter determines the deflection angle, which is determined by the inclination).
One would therefore expect lensed masers from inclined disks to be faint and appear to arise from the black hole location ($\theta_- \ll 1$ mas). This treatment assumes a single pointlike maser rather than an extended continuous disk of masers or a set of distributed maser spots. In this more realistic scenario, numerical ray-tracing is required to connect the observable lensed maser image to the black hole mass and maser disk configuration.
Lensed masers from inclined disks may appear to be pointlike or they may describe arcs. In-going masers from the far side of an edge-on disk or in-going masers from an inclined but warped disk may produce Einstein rings. Single-epoch VLBI maps can therefore provide black hole masses, but space-based VLBI may be required for the typical $\sim10^7\ M_\odot$ black hole associated with water maser disks. The mass measurement precision will be limited by uncertainty about the maser disk size rather than by the distance to the object (the maser will provide the systemic redshift, which can be converted into a distance, and the distance uncertainty will be dominated by peculiar velocity departures from the Hubble flow).
Back-Side Masers in Edge-on Disks
---------------------------------
If there are in-going radial water masers in maser disks (and there is no compelling reason to think otherwise), then there should be systemic water masers seen in edge-on disks from the back side of the disk, both lensed and unlensed by the black hole. Are such masers in extant data?
If there are also front-side systemic masers in an edge-on disk, then they will be orders of magnitude brighter than back-side masers, even in the presence of strong lensing, simply due to amplification considerations. The front-side masers can amplify AGN radio continuum, whereas the back-side masers will either be driven by stimulated emission (but with an amplification pathway equal to the front-side masers) or by host galaxy continuum, which is substantially weaker than AGN continuum at 22 GHz in many cases. The back-side maser contribution may therefore be confused by the front-side emission, particularly since both types of maser emission will occur at the systemic velocity.
In rare cases, a back-side maser might be distinguishable from the front-side emission either by a position or a velocity offset (there is some spread to systemic velocity masers, e.g. @gao2016). The observational signature of a back-side maser would be an acceleration in the opposite sense of the front-side systemic maser acceleration (i.e., negative acceleration under the convention that positive velocities are redshifted). This is not seen in published systemic maser acceleration measurements [e.g. @greenhill1995; @nakai1995; @braatz2010; @kuo2015; @gao2016], but such a signal could be lost amid the brighter and numerous front-side masers.
If a back-side maser is lensed into an arc or Einstein ring by the central black hole, then it might be extended in VLBI maps. Resolved systemic maser emission would therefore be an additional observable signature of back-side masers.
Candidate Selection {#sec:candidates}
===================
If inclined water maser disks can be detected via gravitational lensing or deflection of in-going masers by massive black holes, then they have likely already been detected in surveys for maser disks. But they were rejected as disk candidates because they lacked high-velocity emission. Inclined maser disks will appear to have maser emission only at the systemic velocity of the galaxy or AGN. We therefore use extant water maser surveys to select inclined disk candidates.
Using extant water maser surveys, most of which favor Seyfert 2 AGN, we examined single-dish spectra compiled by the Megamaser Cosmology Project[^2] to select objects showing a narrow systemic velocity maser or maser complex. We also imposed a 30 mJy line flux limit and excluded objects south of $-20^\circ$ declination. This process identified 16 inclined maser disk candidates (Table \[tab:obs\]), and most candidates (14) have only been observed with a single dish. Those that do have interferometric maps are NGC 3556, which was mapped using the VLA in CnB and DnA configurations [@tarchi2011] with no 22 GHz continuum detected (1$\sigma$ rms noise of $\sim$0.5 mJy beam$^{-1}$), and NGC 3735,which was mapped using A-array [@greenhill1997], but no 22 GHz continuum was detected.
Since the inclinations of known maser disks are nearly edge-on, the number of inclined maser disks must be large, roughly an order of magnitude larger than the number of detected maser disks. However, the gravitational lensing or deflection of detectable in-going radial masers adds substantial uncertainty to the detection expectations. We do not know how common observable lensed inclined maser disks are in the universe because we do not know the opening angle of the masers, the size of the maser spots with respect to the black hole’s Schwarzschild radius, the brightness of in-going masers, which will depend on the 22 GHz seed photons from the host galaxy, or the degree of gravitational lensing demagnification.
Observations and Data Reduction {#sec:obs}
===============================
We observed the 22.23508 GHz $6_{16}-5_{23}$ ortho water maser line and 20 GHz radio continuum toward 16 candidate inclined maser disks (see Section \[sec:candidates\] and Table \[tab:obs\]) using the VLA in A configuration (the highest angular resolution configuration). Observations of program 15A-297 spanned June 19 2015 through September 26 2015 in five sessions. The fifth session occurred during the A-array to D-array reconfiguration. Each session included visits to flux and bandpass calibrators, and observations of each target object were interleaved with nearby complex gain calibrators with a $\sim$4 minute switching cadence.
Spectral line and continuum observations were simultaneous. The spectral line observations were centered on the redshift of the host galaxy, had 1.1–1.9 km s$^{-1}$ spectral resolution, used 1536 channels to span 128 MHz (1700–2900 km s$^{-1}$), and used dual circular polarization and 8-bit sampling. Continuum observations spanned 4 GHz, 18–22 GHz, using 32 spectral windows spanning 128 MHz each using 128 channels in dual circular polarization and 3-bit sampling.
Table \[tab:obs\] lists the details of the observations and the rms noise in the line cubes and continuum maps. Typical beam sizes were 80–100 milliarcseconds. Noise was about 3 mJy beam$^{-1}$ in 1.2 km s$^{-1}$ channels in the spectral line cubes and about 15–20 $\mu$Jy beam$^{-1}$ in the continuum. The exception was the $z\simeq0.66$ water maser J0804+3607 [@barvainis2005] that was redshifted to 13.4 GHz, in Ku band, which necessarily had lower angular and spectral resolution and lower rms noise in the line but higher rms continuum noise. In this case, the continuum was centered on 14 GHz and spanned 12–16 GHz.
The observing session on September 15 2015 had poor 22 GHz weather for A configuration and could not be calibrated or imaged. NGC 3359, NGC 3556, and NGC 3735 are therefore only listed in Table \[tab:obs\] and are not discussed further or included in any analysis of the remaining 13 objects.
All data reduction and analysis was performed using the Common Astronomy Software Applications package [CASA; @mcmullin2007]. Calibration and flagging used a modified CASA pipeline plus additional manual flagging. Imaging used Briggs weighting with robustness 0.5. Spectra were extracted from spectral line cubes using a maser-centered beam, and integrated line maps were restricted to line-emitting channels. All spectra use the optical velocity definition in the Barycentric reference frame.
[cccCrCccc]{} NGC 291 & 2015-06-21 & 23.3 & 12884 & 20.4 & 5033 & 2.6 & 1.2 & 17\
NGC 520b & 2015-06-21 & 23.3 & 10289 & 27.8 & 1412 & 3.1 & 1.1 & 20\
J0350$-$0127 & 2015-06-21 & 23.3 & 10087 & 20.5 & 8170 & 3.1 & 1.2 & 15\
IC 485 & 2015-09-26 & 23.4 & 8482 & 19.6 & 4746 & 2.2 & 1.2 & 18\
J0804+3607 & 2015-09-26 & 23.3 & 151130 & $-$81.3 & 1065917 & 0.9 & 1.9 & 47\
CGCG 120$-$039 & 2015-09-26 & 23.3 & 8581 & 33.7 & 4442 & 2.1 & 1.2 & 20\
J0912+2304 & 2015-09-06 & 23.3 & 9281 & 65.6 & 6759 & 2.9 & 1.2 & 16\
J1011$-$1926 & 2015-09-06 & 23.4 & 17386 & $-$20.8 & 9849 & 5.3 & 1.2 & 19\
NGC 3359 & 2015-09-15 & 23.3 & 11272 & 48.6 & 96 & & 1.1 &\
NGC 3556 & 2015-09-15 & 23.3 & 10472 & 45.0 & 64 & & 1.1 &\
NGC 3735 & 2015-09-15 & 23.3 & 11268 & 36.8 & 2113 & & 1.2 &\
UGC 7016 & 2015-09-06 & 23.3 & 11491 & $-$74.8 & 5443 & 3.8 & 1.2 & 16\
NGC 5256 & 2015-06-19 & 23.3 & 9684 & $-$73.8 & 5548 & 3.0 & 1.2 & 17\
NGC 5691 & 2015-06-19 & 23.4 & 13185 & 39.3 & 159 & 3.7 & 1.1 & 17\
CGCG 168$-$018 & 2015-06-19 & 23.3 & 8682 & 50.0 & 6360 & 2.7 & 1.2 & 15\
J1939$-$0124 & 2015-06-19 & 25.0 & 9687 & 37.7 & 4138 & 3.0 & 1.2 & 16\
Results {#sec:results}
=======
Water masers were detected in 9 out of 13 objects, and 20 GHz continuum emission was detected in 7 out of 13 objects. Only five objects show both maser and continuum emission, and two objects were detected in neither line nor continuum. Figures \[fig:NGC291\]–\[fig:2MASX1939\] show a 1 square field of view of the first moment maser maps and continuum contours and they show spectra of the detected objects (in line, continuum, or both). Table \[tab:positions\] lists the maser and continuum centroids based on two-dimensional Gaussian fits. For the spectral lines, these fits were made to the integrated line maps. The maser emission was universally unresolved, but the continuum emission was formally resolved when deconvolved from the beam in all but two objects, IC 485 and CGCG 168$-$018.
Table \[tab:positions\] also lists the maser-continuum offsets in angular and physical units. Offsets for four of the five objects detected in both maser and continuum are non-significant with $1\sigma$ uncertainties ranging from about 1 to 40 pc. The only object showing a significant offset between the maser and continuum centroids is CGCG 168$-$018, with a $21.6\pm2.7$ pc offset (Figure \[fig:CGCG168-018\]).
Table \[tab:masers\] lists the measured and derived water maser properties: peak and integrated flux densities, luminosity distance, isotropic line luminosity, the range of velocities spanned by the line emission 3$\sigma$ above the noise, the velocity of peak emission, and the adopted systemic velocity. For J0804+3607, we list redshifts rather than velocities. The detected maser velocities are consistent with previous observations, although the masers can be substantially offset from the systemic velocities, which are obtained from optical and HI 21 cm lines. It is unclear whether these velocity offsets are physical (i.e., due to different line-emitting regions genuinely having different velocities, as is seen in shock-induced maser emission), due to obscuration (optical vs. radio lines), or due to measurement error, particularly in optical redshifts. We therefore do not rely on the velocity offset between the maser emission and the adopted systemic velocity as a criterion for assessing the likelihood of a maser arising from an inclined disk. Isotropic maser luminosities range from kilomaser values ($3.05 \pm 0.26\ L_\odot$ in NGC 520b) to the exceptionally luminous, $L_{iso} = (1.8 \pm 0.1) \times 10^4\ L_\odot$ in J0804+3607 (Section \[sec:discussion\]).
Table \[tab:continuum\] shows the 20 GHz radio continuum properties of the seven detected objects. We include the peak flux density, the integrated flux density, the spectral index derived solely from the 18–22 GHz bandpass, and the deconvolved angular size. Among the six continuum sources with enough signal-to-noise to derive a significant spectral index, four are steep spectrum ($\alpha = -1$ to $-2$) and two are flat ($\alpha \simeq -0.2 \pm 0.2$). One of the latter, CGCG 120$-$039, is only marginally resolved.IC 485, which does not have a spectral index measurement, but which shows both maser and continuum emission, does not have a resolved continuum.
[cllllcc]{} NGC 291 & & & 00:53:29.9101(11) & $-$08.46.03.740(14) & &\
NGC 520b & 01:24:34.91412(14) & +03.47.29.7864(22) & 01:24:34.9099(28) & +03.47.29.7783(92) & 64(42) & 8.6(5.6)\
J0350$-$0127 & 03:50:00.352168(29) & $-$01.27.57.39574(53) & & & &\
IC 485 & 08:00:19.752486(74) & +26.42.05.0526(10) & 08:00:19.7515(14) & +26.42.05.050(14) & 13(19) & 7.6(10.5)\
J0804+3607 & 08:04:31.01144(40) & +36.07.18.1937(52) & 08:04:31.01138(12) & +36.07.18.19927(91) & 5.6(5.3) & 39.6(37.2)\
CGCG 120$-$039 & 08:49:14.07078(16) & +23.22.48.9408(20) & 08:49:14.07097(13) & +23.22.48.9346(17) & 6.7(2.7) & 3.5(1.4)\
J0912+2304 & 09:12:46.36659(33) & +23.04.27.2421(28) & & & &\
J1011$-$1926 & 10:11:50.56731(17) & $-$19.26.43.9645(59) & & & &\
UGC 7016 & & & & & &\
NGC 5256 & & & 13:38:17.79219(15) & +48.16.41.1389(19) & &\
& & & 13:38:17.24843(76) & +48.16.32.2095(92) & &\
NGC 5691 & & & & & &\
CGCG 168$-$018 & 16:30:40.90329(19) & +30.29.19.7066(29) & 16:30:40.90134(21) & +30.29.19.7216(20) & 29.3(3.6) & 21.6(2.7)\
J1939$-$0124 & 19:39:38.91545(38) & $-$01.24.33.2553(39) & & & &\
[crrrrcrrc]{} NGC 520b & 35(3) & 166(14) & 28 & 3.05(26) & 2270–2273 & 2272(1) & 2288(8) & 1\
J0350$-$0127 & 347(4) & 7200(150) & 180 & 5181(108) & 12336–12393 & 12369(1) & 12322(18) & 2\
IC 485 & 78(2) & 2470(130) & 125 & 868(46) & 8307–8387 & 8356(1) & 8338(10) & 3\
J0804+3607 & 9(1) & 80(6) & 3997 & $1.8(1)\times10^4$ & 0.66038–0.66051 & 0.66045(1) & 0.65654(37) & 4\
CGCG 120$-$039 & 82(2) & 438(46) & 116 & 133(14) & 7559–7565 & 7565(1) & 7684(26) & 5\
J0912+2304 & 16(3) & 252(44) & 164 & 151(26) & 10855–10878 & 10855(1) & 10861(26) & 6\
J1011$-$1926 & 44(4) & 624(82) & 123 & 213(28) & 8043–8065 & 8048(1) & 8065(31) & 7\
CGCG 168$-$018 & 25(2) & 279(36) & 162 & 163(21) & 11134–11164 & 11140(1) & 11015(29) & 5\
J1939$-$0124 & 30(2) & 519(80) & 93 & 102(16) & 6170–6206 & 6198(1) & 6226(20) & 8\
[ccccCcC]{} NGC 291 & 263(24) & 1.93(20) & $-$2.0(3) & 451(52)146(19) & 51.8(3.4) & 177(20)57(7)\
NGC 520b & 272(31) & 4.31(52) & $-$0.23(19) & 775(100)164(24) & 95.6(2.3) & 105(14)22(3)\
IC 485 & 77(15) &0.180(48)& & < 9388 & $-$69.5 & < 5249\
J0804+3607 & 4247(83) & 4.71(16) & $-$1.5(3) & 62(12)23(13) & 71(17) & 437(85)162(92)\
CGCG 120$-$039 & 482(20) & 0.555(39) & $-$0.27(16) & 44(8)23(14) & 133(33) & 23(4)12(7)\
NGC 5256 & 633(28) & 1.543(68) & $-$2.0(2) & 127(6)96(8) & 148.7(9.6) & 72(3)55(5)\
& 78(17) & 0.48(11) & & 433(23)68(58) & 163.1(3.1) & 247(13)39(33)\
CGCG 168$-$018 & 267(16) & 0.308(30) & $-$0.95(17) & < 8785 & 77.8 & < 6463\
{width="53.50000%"} {width="46.00000%"}
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Analysis {#sec:analysis}
========
Figure \[fig:offsets\] shows the projected physical offset between the water maser line and continuum centroids for the five objects detected in both line and continuum (Table \[tab:positions\]). Only CGCG 168$-$018 shows a significant offset of $29.3\pm3.6$ milliarcseconds or $21.6\pm2.7$ pc, but based on its 18–22 GHz continuum spectral index of $-0.95\pm0.17$ (Table \[tab:continuum\]), the continuum could be jet emission and not the core of the AGN. All other maser-continuum offsets are not significant and are therefore consistent with inclined maser disk expectations.
![Maser-continuum offsets for those objects detected in both. Only CGCG 168$-$018 shows a significant offset. Error bars are plotted symmetrically. \[fig:offsets\]](offsets.eps){width="45.00000%"}
Discussion {#sec:discussion}
==========
Among the 13 water maser hosts, we did not detect water masers in four objects: NGC 291, UGC 7016, NGC 5256, and NGC 5691. The water maser in NGC 520b is likely associated with star formation and is rejected as an inclined maser disk candidate (see \[subsubsec:NGC520b\]). Objects with unresolved masers but not detected in the 20 GHz continuum remain ambiguous (these are J0350$-$0127, J0912+2304, J1011$-$1926, and J1939$-$0124; note that none of these have other sub-arcsecond radio continuum observations), as does CGCG 168$-$018, the only object to show a significant maser-continuum offset (see \[subsubsec:CGCG168-018\]). IC 485, J0804+3607, and CGCG 120$-$039 remain inclined maser disk candidates: they show no significant maser-continuum offset and no high velocity lines (by selection). While the maser emission from IC 485 is broad and multi-component, this type of structure spanning $\sim$100 km s$^{-1}$ is seen in the systemic masers in many disk systems [e.g., @kuo2011]. J0808+3607 and CGCG 120$-$039 have narrow maser lines and are particularly good inclined disk candidates.
Individual Objects
------------------
We discuss the previous water maser observations, general characteristics, and our results for each individual object below.
### NGC 291
NGC 291 is a barred spiral galaxy with a Seyfert 2 nucleus [@kewley2001; @nair2010]. The maser was detected as a narrow $\sim$60 mJy line in 2006 by GBT program GBT06A-009,^\[footnote:maser\_url\]^ but we did not detect it (2.6 mJy beam$^{-1}$ rms per 1.2 km s$^{-1}$ channel; Table \[tab:obs\] and Figure \[fig:NGC291\]) nor did @kondratko2006 in 2002 (15 mJy rms per 1.3 km s$^{-1}$ channel). We detect extended 20 GHz continuum with 18–22 GHz spectral index $-2.0\pm0.3$ (Table \[tab:continuum\]).
### NGC 520b {#subsubsec:NGC520b}
NGC 520b is one galaxy in the colliding pair in NGC 520 [e.g., @stanford1990]. @castangia2008 detected a maser in 2005, measuring a $\sim40$ mJy peak with the VLA and $L_{iso} \sim 1\ L_\odot$. Our VLA detection shows a $35\pm3$ mJy peak and $L_{iso} = 3.05\pm0.26\ L_\odot$ (Figure \[fig:NGC520b\] and Table \[tab:masers\]). Our VLA map shows extended 20 GHz radio continuum emission (Figure \[fig:NGC520b\], bottom) with a flat 18–22 GHz spectral index of $-0.23\pm0.19$ (Table \[tab:continuum\]) and a similar east-west morphology to the @castangia2008 14.9 GHz map. In our map the maser is slightly north of the @castangia2008 maser position. There is a significant maser-continuum peak offset, although the single-component centroid fits listed in Table \[tab:positions\] are consistent because the radio emission is extended. @castangia2008 favor a star formation origin for the water maser in NGC 520b, but cannot rule out a low luminosity AGN. The preponderance of evidence suggests that NGC 520b is a poor candidate for an inclined maser disk.
### J0350$-$0127
J0350$-$0127 is an almost otherwise unknown spiral or possibly irregular galaxy included in the 2MASS Redshift Survey [@huchra2012] and detected in water maser emission by the GBT programs GBT09-051^\[footnote:maser\_url\]^ in 2010 and GBT10C-019^\[footnote:maser\_url\]^ in 2011. The peak flux density was $\sim$350 mJy in 2011, in good agreement with our VLA measurement of $347\pm4$ mJy (Table \[tab:masers\] and Figure \[fig:2MASX0350\]). This is a broad and luminous maser ($L_{\rm iso} = 5181\pm108\ L_\odot$) with no associated 20 GHz continuum, down to an rms noise of 15 $\mu$Jy beam$^{-1}$ (Table \[tab:obs\]). Without a continuum detection, the nature of this maser remains ambiguous, although it is almost certainly associated with an AGN.
### IC 485
IC 485 is a spiral galaxy detected in the water maser line in 2006 in GBT program GBT06C-035.^\[footnote:maser\_url\]^ The maser has a $\sim$80 mJy peak and shows a broad profile. @zhu2011 lists this object as a maser nondetection. Our VLA observations show a broad, multi-component maser with a similar peak ($78\pm2$ mJy) and a high luminosity, $L_{iso} = 868\pm46\ L_\odot$ (Figure \[fig:IC485\] and Table \[tab:masers\]). The 20 GHz continuum is detected but unresolved and faint ($77\pm15$ $\mu$Jy beam$^{-1}$ peak flux density; Table \[tab:continuum\]). This galaxy was also detected at 1.4 GHz (4.4 mJy), and its dominant radio energy source was classified as star formation by @condon2002. This classification does not exclude the presence of an AGN: @liu2011 classify the optical nucleus of IC 485 as a LINER. The maser-continuum offset is not significant — less than 10.5 pc (1$\sigma$; Table \[tab:positions\]) — so this maser remains an inclined disk candidate.
### J0804+3607
This is a type 2 quasar showing luminous water maser emission at $z\simeq0.66$ [@zakamska2003; @barvainis2005]. The isotropic maser luminosity was $2.31\pm0.46\times10^4\ L_\odot$ in 2005 [@barvainis2005 error from the quoted 20% calibration uncertainty] and $1.8\pm0.1\times10^4\ L_\odot$ in 2015 (this work), consistent with no variation. The maser and 20 GHz continuum emission are coincident to within 40 pc (1$\sigma$; Table \[tab:positions\] and Figure \[fig:SDSSJ0804\]), which is less precise than the other objects due to the much larger distance and lower observing frequency. The 12–16 GHz spectral index is $-1.5\pm0.3$ (Table \[tab:continuum\]). This object remains an inclined maser disk candidate.
### CGCG 120$-$039
The water maser in this little-studied galaxy was detected in 2013 in GBT program GBT13A-236.^\[footnote:maser\_url\]^ It had a peak flux density of $\sim$210 mJy, was blueshifted from the systemic velocity, and showed several components. We detect the maser — now at $82\pm2$ mJy peak, but still showing multiple components (Figure \[fig:CGCG120-039\]) — and the 20 GHz continuum at $0.56\pm0.04$ mJy, which are spatially coincident to within a remarkably small $3.5\pm1.4$ pc (Table \[tab:positions\]). The 18–22 GHz spectral index is flat: $\alpha = -0.27\pm0.16$ (Table \[tab:continuum\]). This object is a good inclined maser disk candidate.
### J0912+2304
The water maser in the galaxy J0912+2304 was detected in 2008 by GBT program GBT07A-034,^\[footnote:maser\_url\]^ showing a $\sim$30 mJy peak. @zhu2011 list this as a water maser nondetection. The VLA observations show a $16\pm3$ mJy peak, but no 20 GHz continuum down to an rms noise of 16 $\mu$Jy beam$^{-1}$ (Figure \[fig:2MASX0912\] and Tables \[tab:obs\] and \[tab:masers\]). The provenance of the maser remains ambiguous.
### J1011$-$1926
The maser in this almost unknown galaxy was detected in GBT program GBT07A-066^\[footnote:maser\_url\]^ in 2008. [@zhu2011] list this object as a water maser nondetection. The GBT detection shows a broad line with a $\sim$80 mJy peak. The VLA detection shows a broad multi-component maser with a $44\pm4$ mJy peak in good agreement with the systemic velocity (Figure \[fig:2MASX1011\] and Table \[tab:masers\]). We do not detect the 20 GHz continuum. The nature of this maser therefore remains ambiguous.
### UGC 7016
The water maser in this barred spiral [@RC3] was detected in 2007 by GBT program GBT07A-034,^\[footnote:maser\_url\]^ but @zhu2011 list this as a nondetection. The GBT detection spectrum had a peak flux density of $\sim$55 mJy, but we did not detect the maser or the 20 GHz continuum emission. VLA rms noise values were 3.8 mJy beam$^{-1}$ per 1.2 km s$^{-1}$ channel in the spectral line cube and 16 $\mu$Jy beam$^{-1}$ in the continuum map (Table \[tab:obs\]).
### NGC 5256
@braatz2004 discovered the water maser in this merging luminous infrared galaxy in 2003. It had a peak flux density of 99 mJy ($L_{iso} = 30\ L_\odot$) and was redshifted from the systemic velocity by $\sim$300 km s$^{-1}$. @braatz2004 claim that all maser emission originates from the southern nucleus [a Sey 2 nucleus according to, e.g., @mazzarella1993]. We detect two 20 GHz continuum sources that are consistent with the positions of the two nuclei [Tables \[tab:positions\] and \[tab:continuum\]; @brown2014], but neither location (nor the larger region that includes the overlap region between the two nuclei) shows maser emission down to an rms noise of 3.0 mJy beam$^{-1}$ in 1.2 km s$^{-1}$ channels (Table \[tab:obs\] and Figure \[fig:NGC5256\]). The spectral index of the northern nuclear continuum is $-2.0\pm0.2$ in the 18–22 GHz band (Table \[tab:continuum\]).
### NGC 5691
The water maser in NGC 5691, a barred non-Seyfert spiral galaxy [@mulchaey1997], was detected in 2009 by GBT program AGBT08C-035^\[footnote:maser\_url\]^ with a $\sim$45 mJy peak flux density. It was listed by @zhu2011 as a maser nondetection. We did not detect the maser or any 20 GHz continuum in this galaxy, with rms noise levels of 3.7 mJy beam$^{-1}$ per 1.1 km s$^{-1}$ channel and 17 $\mu$Jy beam$^{-1}$, respectively (Table \[tab:obs\]).
### CGCG 168$-$018 {#subsubsec:CGCG168-018}
CGCG 168$-$018 is a little-studied galaxy classified as an AGN by @schawinski2010 and listed as a water maser nondetection by @zhu2011. Water maser emission was detected by GBT program GBT07A-066^\[footnote:maser\_url\]^ in 2008. The GBT water maser spectrum shows a $\sim$50 mJy peak, detected by this work at $25\pm2$ mJy (Table \[tab:masers\]). The $0.31\pm0.03$ mJy continuum shows a 18-22 GHz spectral index of $-0.95\pm0.17$ (Table \[tab:continuum\]). The maser emission and 20 GHz continuum are unresolved but show a significant relative offset of $21.6\pm2.7$ pc (Table \[tab:positions\] and Figure \[fig:CGCG168-018\]). This is the only object in the sample that shows a significant offset between the maser and continuum emission (Figure \[fig:offsets\]). While the offset suggests that the maser emission is not deflected from an inclined maser disk, the continuum spectral index suggests that the continuum may arise from a jet, and the radio core is not detected. The provenance of the maser therefore remains ambiguous.
### J1939$-$0124
@greenhill2003 detected the water maser in this spiral galaxy hosting a Sey 2 nucleus in 2002 using the Tidbinbilla antenna. The maser showed a peak flux density of $\sim$28 mJy, and @henkel2005 report $L_{iso} \simeq 160\ L_\odot$. No 22 GHz continuum was detected by @greenhill2003 using the VLA ($< 2.8$ mJy). We likewise detect no continuum, with rms noise of 16 $\mu$Jy beam$^{-1}$ (Table \[tab:obs\]), but we do detect the maser emission with peak $30\pm2$ mJy ($L_{iso} = 102\pm16 \ L_\odot$) although with a substantially different maser profile (Figure \[fig:2MASX1939\] and Table \[tab:masers\]). The 6 and 20 cm continua were detected at $5.4\pm0.4$ mJy and $15.5\pm1.0$ mJy, respectively, by @vader1993, and the 6 cm continuum position agrees with the maser position to within $\sim$1($\sim$430 pc). The nature of this maser remains ambiguous.
Conclusions
===========
This paper presents a physical mechanism that may enable detection of inclined water maser disks orbiting massive black holes via the lensing/deflection of in-going systemic masers. The observational signature of an inclined disk is a maser line or line complex with limited Doppler extent that appears to arise at the location of the black hole, as identified by its radio continuum core. With enough angular resolution, it may be possible to measure the black hole mass if the maser emission forms a lensing arc or Einstein ring, but the mass precision will be limited by one’s ability to measure or estimate the size of the maser-emitting portion of the disk.
We suggest that if inclined maser disks can be detected at all, then they have probably already been detected in single-dish surveys but discarded for interferometric follow-up because they did not show high-velocity lines. We present original 0.07–0.17 (4–100 pc) resolution VLA observations of inclined maser disk candidates with the goal of identifying systems where the maser emission is unresolved and is coincident with the 20 GHz continuum emission.
Of the 16 masers observed with the VLA, we obtained useful data for 13, and among these, five were detected in both 22 GHz maser line emission and in 20 GHz continuum. Of these five, one maser is most likely associated with star formation (NGC 520b), and one shows a significant spatial offset between the maser emission and the continuum (CGCG 168$-$018, but it could still host an inclined maser disk — this case is ambiguous). Three objects are good inclined maser disk candidates that merit further study with VLBI: IC 485, J0804+3607, and CGCG 120$-$039. Five maser hosts remain ambiguous, based either on non-detected or offset continua: J0350$-$0127, J0912+2304, J1011$-$1926, CGCG 168$-$018, and J1939$-$0124.
More straightforward methods for measuring black hole masses from molecular lines may be in the offing. For example, @davis2013 and @barth2016a [@barth2016b] have used carbon monoxide kinematics in thin disks that approach or are within the black hole gravitational sphere of influence to obtain black hole mass measurements. @barth2016a, in particular, demonstrate the ability of ALMA to measure black hole masses with $\sim$10% uncertainty. Although they lack the intrinsic brightness of masers that enables VLBI mapping, thermal molecular lines have the advantage of being observable at any disk inclination.
We thank A. Hamilton and M. Eracleous for helpful discussion and B. Butler for assistance with metadata repair of bespoke observing configurations. We also thank the anonymous referee for important feedback. This research has made use of NASA’s Astrophysics Data System Bibliographic Services and the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[^1]: The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
[^2]: \[footnote:maser\_url\]<https://safe.nrao.edu/wiki/bin/view/Main/PrivateWaterMaserList>
|
---
abstract: |
The different level of interest in deploying the new Internet address space across network operators has kept IPv6 tardy in its deployment. However, since the last block of IPv4 addresses has been assigned, Internet communities took the concern of the address space scarcity seriously and started to move forward actively. After the successful IPv6 test on 8 June, 2011 ( [@URL-V6DAY]), network operators and service/content providers were brought together for preparing the next step of the IPv6 global deployment (on 6 June, 2012 [@URL-V6LAUNCH]). The main purpose of the event was to permanently enable their IPv6 connectivity.
In this paper, based on the Internet traffic collected from a large European Internet Exchange Point (IXP), we present the status of IPv6 traffic mainly focusing on the periods of the two global IPv6 events. Our results show that IPv6 traffic is responsible for a small fraction such as 0.5of the total traffic in the peak period. Nevertheless, we are positively impressed by the facts that the increase of IPv6 traffic/prefixes shows a steep increase and that the application mix of IPv6 traffic starts to imitate the one of IPv4-dominated Internet.
author:
-
-
-
bibliography:
- 'IEEEtran.bib'
title: 'Watching the IPv6 Takeoff from an IXP’s Viewpoint'
---
|
---
address:
- |
Institut für Algebra\
TU Dresden\
01062 Dresden\
Germany
- |
Équipe de Logique Mathématique\
Université Diderot - Paris 7\
UFR de Mathématiques\
75205 Paris Cedex 13, France
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Laboratoire d’Informatique (LIX), CNRS UMR 7161\
École Polytechnique\
91128 Palaiseau\
France
bibliography:
- 'homos21.bib'
title: Projective clone homomorphisms
---
0d\_abstract.tex
Introduction {#sect:intro}
============
1\_intro.tex
Results {#sect:results}
=======
2\_results.tex
The Binary Fragment {#sect:binary}
===================
3\_binary.tex
Automatic Continuity to $\bf 1$ {#sect:auto-cont}
===============================
4\_auto-cont.tex
Locally Finite Idempotent Algebras {#sect:locfin}
==================================
5\_locfin.tex
Oligomorphic Clones {#sect:oligo}
===================
6\_oligo.tex
Open Problems {#sect:open}
=============
7\_open.tex
|
addtoreset[equation]{}[section]{}
\
We study the local structure of zero mode wave functions of chiral matter fields in F-theory unification. We solve the differential equations for the zero modes derived from local Higgsing in the 8-dimensional parent action of F-theory 7-branes. The solutions are found as expansions both in powers and derivatives of the magnetic fluxes. Yukawa couplings are given by an overlap integral of the three wave functions involved in the interaction and can be calculated analytically. We provide explicit expressions for these Yukawas to second order both in the flux and derivative expansions and discuss the effect of higher order terms. We explicitly describe the dependence of the couplings on the $U(1)$ charges of the relevant fields, appropriately taking into account their normalization. A hierarchical Yukawa structure is naturally obtained. The application of our results to the understanding of the observed hierarchies of quarks and leptons is discussed.
Introduction
============
The hierarchical structure of fermion masses and mixings is one of the most remarkable properties of the Standard Model (SM). An outstanding challenge of string theory compactifications is to obtain models with the massless spectrum of the SM and reproducing naturally such hierarchical structure. In type IIB orientifold, as well as heterotic, compactifications the Yukawa couplings which govern fermion masses and mixings are in principle calculable, in the large compact volume limit, in terms of overlap integrals [@GSW2; @SW] Y\_[ij]{} = \_[X\_3]{} \_i \_j \_H Here $\psi_{i}$ and $\phi_H$ are internal wave functions associated to the fermions and Higgs fields respectively, taking values in the compact complex threefold $X_3$. These wave functions are zero modes of higher dimensional fields in the compact internal space. The technical problem here is that in general we do not know how to compute the relevant wave functions for arbitrary curved spaces $X_3$. Such a computation has been completely worked out for the relatively simple case of type IIB toroidal orientifolds [@cim1] with constant $U(1)$ fluxes (see also [@cmq; @vechia; @antoniadis]). In this case with a flat geometry the equations of motion can be fully solved to obtain the wave functions which turn out to have a neat expression in terms of Jacobi $\vartheta$-functions. It was found that in the simplest models only one generation of quarks and leptons acquires a non-trivial Yukawa coupling, which is a good starting point to reproduce the observed hierarchies [@cim1]. Having explicit solutions for the wave functions is also useful to study other physical properties of the compactifications such as the effect of closed string fluxes and warping [@marchesano].
![Intersecting matter curves[]{data-label="cuscus"}](yft4n.eps)
Clearly, it would be interesting to obtain wave functions and Yukawa couplings in more complicated curved geometries and for non-constant fluxes. An obvious obstruction is that determining the wave functions seems to require a knowledge of the global geometry of the compact $X_3$ manifold. In fact, the problem may be more tractable within the context of a bottom-up approach as advocated in [@aiqu] (see also [@bjl; @cgqu; @vw; @cmq1]). The idea is that in order to extract the relevant physics of a SM compactification it is enough to have a local description of the geometry of the branes in which the SM fields reside. This is the case for example of models derived from D3-branes at singularities [@aiqu; @bjl; @vw] in which the SM physics only depends on the local geometry around the singularity. This type of structure is also characteristic of local configurations of D7-branes wrapping intersecting 4-cycles inside $X_3$. F-theory [@ftheory] is the natural non-perturbative extension of these local 7-brane configurations. In the last year local F-theory GUT constructions have been proposed [@dw; @bhv1; @bhv2] as a particularly attractive class of bottom-up configurations with a number of phenomenological virtues (see [@w; @aci; @hv3; @fi; @hv4; @mssn; @hv5; @ttw; @blumen; @rs; @bhsv; @bckmq; @hksv; @li; @jlnx], as well as [@httwy; @tatar2; @bourjaily; @dw1; @htv; @dw2; @ac; @hmssnv; @mssn2; @bgjw; @coll; @mssn3] for other recent developments). In this scheme the Yukawa couplings arise again as overlap integrals now of the form Y\_[ij]{} = \_[S]{} \_i \_j \_H \[yukayuka\] in which $S$ is the compact complex twofold wrapped by the GUT F-theory 7-brane. The quark and lepton multiplets of the SM reside at matter Riemann curves $\Sigma_I$ inside $S$, which correspond geometrically to the intersection of $S$ with the world-volume of other $U(1)$ 7-branes. Yukawa couplings come from the triple overlap of these matter curves involving quarks, leptons and Higgs fields (see figure \[cuscus\]). In order to compute the Yukawa coupling (\[yukayuka\]) we need again the internal wave functions. However, in this case given the local geometry of the coupling it would be enough to have a knowledge of the wave functions close to the intersection point. It was pointed out in [@bhv1] that one can determine the profile of these wave functions close to the intersection point in terms of a certain quasi-topological theory in 8. The equations of motion of that theory have solutions corresponding to hypermultiplet zero modes localized along the matter curves with a Gaussian profile. One finds [@hv3] that to leading order, for a compactification having three generations, only one of them gets a non-trivial Yukawa, in analogy with the results in [@cim1]. However the distortion of the wave functions, due to the presence of $U(1)$ gauge fluxes, could be the natural source of the observed hierarchy of masses and mixings of quarks and leptons [@hv3].
In this article we make a systematic study of the solutions of the differential equations of motion of the quasi-topological 8 theory of [@bhv1]. The zero mode solutions give the local wave functions corresponding to the massless particles residing at the matter curves of F-theory unification models. We make an expansion both in powers and derivatives of the $U(1)$ fluxes and explicitly solve the differential equations. Equipped with these wave functions we compute the Yukawa couplings from the overlap integral of the three wave functions involved in the couplings. These integrals may be calculated analytically and we provide explicit expressions for these Yukawa couplings up to fourth order in the flux and derivative expansions. As suggested in [@hv3], a hierarchy of masses for fermions naturally appears. We also study the application of our results to the understanding of the observed hierarchy of masses and mixings in the SM. We find good qualitative agreement with experiment for reasonable ranges of the flux parameters.
The organization of the rest of this article is as follows. In the next section we provide a brief review of the aspects of F-theory models that concern our discussion. In chapter 3 we study the wave functions of the zero modes which are solutions of the quasi-topological 8 field theory equations. We consider both constant and varying fluxes in a general setting of three intersecting matter curves. The details of the solutions are given in appendix A. In chapter 4 we address the explicit computation of the Yukawa couplings by evaluating the overlap integral of the three relevant wave functions. Based on the leading terms in the Yukawa couplings provided in appendix B, we describe the general structure of the flux-induced corrections and their contribution to the Yukawa matrices. In chapter 5 we apply the previous results to the analysis of the fermion mass spectra of $SU(5)$ GUT’s with non-vanishing hypercharge flux breaking the theory down to the SM. We show that reasonable agreement with observed mass hierarchies and mixing may be obtained for appropriate flux parameters. Chapter 6 is devoted to some final comments and discussion.
The computation of Yukawa couplings in analogous settings has been more recently analyzed in [@cchv; @cp], see the note added at the end of this paper.
Review of F-theory unification
==============================
The purpose of this section is to give a short overview of the F-theory formalism developed in [@bhv1] (see also [@dw; @w]).
In the F-theory setup, the 4 supersymmetric gauge theory descends from 7-branes wrapping a compact surface $S$ of complex codimension one in the threefold base of an elliptically-fibered Calabi-Yau fourfold. The gauge group $G_S$ on the 7-branes depends on the singularity type of the elliptic fiber. In turn $G_S$ can be broken by a vev in a subgroup $H_S \subset G_S$. We consider $H_S=U(1)$, typical examples being the hypercharge in $SU(5)$ or $U(1)_{B-L}$ in an $SO(10)$ GUT. The $U(1)$ background breaks the gauge group and gives rise to matter charged under the commutant of $H_S$ in $G_S$. We assume that $S$ is a del Pezzo surface so that gravity decouples from the gauge theory [@bhv1].
The singularity type of the elliptic fiber can be enhanced to group $G_\Sigma$ along a curve $\Sigma \subset S$ of complex codimension two on the threefold base. This curve appears at the intersection of $S$ and another surface $S^\prime$. On the 7-branes wrapping $S^\prime$ there is a gauge theory with group $G_{S^\prime}$ which decouples when $S^\prime$ is non-compact. Based on the knowledge of intersecting D-branes, one expects additional degrees of freedom due to open strings stretching between the 7-branes wound on $S$ and $S^\prime$. The extra fields localized on the matter curve $\Sigma$ must be charged under $G_S \times G_{S^\prime}$. Indeed this is the picture that arises in F-theory [@kv; @bhv1].
We will now review the basic facts about the charged fields originating in the surface $S$ and in the matter curve $\Sigma$. Our discussion is brief and follows mostly [@bhv1].
Bulk fields {#ss:bulk}
-----------
The effective physics of the 7-branes wrapping $S$ is described by 8 twisted super Yang-Mills on ${\mathbbm{R}}^{3,1} \times S$ [@bhv1]. The supersymmetric multiplets include the gauge field, plus a complex scalar $\varphi$ and fermions $(\eta, \psi, \chi)$ in the adjoint. After twisting the scalar and fermions become forms on $S$. Using local coordinates $(z_1, z_2)$ for $S$ the results are summarized by A = A\_dx\^+ A\_m dz\^m + A\_[|m]{} d\^m & ; & =\_[12]{} dz\^1 dz\^2\
\_= \_[|1]{} d|z\^1 + \_[|2]{} d|z\^2 & ; & \_=\_[12]{} dz\^1 dz\^2 \[d8fdefs\] Notice that $\psi$ is a (0,1) form whereas $\varphi$ and $\chi$ are (2,0) forms. The remaining fermion $\eta_\a$ is a (0,0) form. The subscript $\a$, which corresponds to left handed fermions in ${\mathbbm{R}}^{3,1}$, will be dropped hereafter. The 4, 1 theory has gauge multiplet $(A_\mu, \eta)$, together with chiral multiplets $(A_{\bar m}, \psi_{\bar m})$ and $(\varphi_{12}, \chi_{12})$, plus their complex conjugates.
The 8 effective action found in [@bhv1] can be integrated over the compact surface $S$ to obtain the dynamics of the 4 multiplets. In computing couplings of the charged fields the most interesting term will be the superpotential W= M\_\*\^4\_[S]{} ([F\_S\^[(0,2)]{}]{}) = M\_\*\^4\_[S]{} ([|]{}[A]{}) + M\_\*\^4 \_S ([A]{} ) \[wdef\] where $M_*$ is the mass scale characteristic of the supergravity limit of F-theory. Here $\pmb A$ and $\pmb \Phi$ are chiral superfields with components \_[|m]{} & = & A\_[|m]{} + 2 \_[|m]{} +\
\_[12]{} & = & \_[12]{} + 2 \_[12]{} + \[superfi\] where $\cdots$ involves auxiliary fields. Only the (0,2) component of the superstrength appears in (\[superfi\]).
The equations of motion derived from the 8 effective action are the starting point to discuss the zero modes. The part of the action bilinear in fermions, without kinetic terms, is found to be [@bhv1] I\_F = \_[\^[3,1]{}S]{} d\^4x (\_A + 2i2 \_A + + 2 + [h.c.]{}) \[eq451\] where $\omega$ is the fundamental form of $S$. Taking variations with respect to $\eta $, $\psi$ and $\chi$ respectively gives the equations of motion && \_A + 2 \[ |, \]=0 \[zma\]\
&& |\_A - 2i2 - \[, \] = 0 \[zmb\]\
&& |\_A - 2 \[|, \] = 0 \[zmc\] For the bosonic fields it is found that the field strength $F_S$ must have vanishing (2,0) and (0,2) components and verify the BPS condition F\_S + 2 \[, |\] = 0 \[bpscond\] Finally, the complex scalar must satisfy the holomorphicity condition $\bar\partial_A \varphi=0$.
To determine the charged massless multiplets in 4 it is necessary to specify the background for the adjoint scalar $\varphi$ and the gauge field. When $\langle \varphi \rangle=0$, the equations of motion imply that the number of zero modes of $\psi$ and $\chi$ are counted by topological invariants that depend both on $S$ and the gauge bundle of the background [@bhv1].
Fields at intersections {#ss:sigma}
-----------------------
We now want to discuss the degrees of freedom localized on a matter curve $\Sigma$ occurring at the intersection of surfaces $S$ and $S^\prime$. As explained in [@bhv1], to preserve 1 supersymmetry in 4, the theory on ${\mathbbm{R}}^{3,1} \times \Sigma$ must be 6 twisted super Yang-Mills. The 6 twisted supermultiplet, which includes two complex scalars and a Weyl spinor, decomposes into 4 chiral multiplets $(\sigma, \lambda)$ and $(\sigma^c, \lambda^c)$, plus CPT conjugates. The number of zero modes is given by topological invariants depending on $\Sigma$ and the gauge bundle on a background in $G_\Sigma$.
There is a very nice intuitive way of understanding the matter localized at $\Sigma$. The idea, originally given in [@kv] and expanded in [@bhv1], is to start from the 8 theory on $S$ with gauge group $G_\Sigma$ and then turn on a background for the adjoint scalar given by = m\^2 z\_1 Q\_1 \[vev1\] where $z_1$ is a complex coordinate on $S$, and $Q_1$ is a $U(1)_1$ generator in the Cartan subalgebra of $G_\Sigma$. To streamline notation, $\varphi = \varphi_{12}$. We have explicitly introduced a mass parameter $m$ so that $\varphi$ has the standard dimensions. The basic idea is that in presence of $\langle \varphi \rangle$ the 8 fields have zero modes localized at $z_1=0$ that are naturally associated to the fields at the intersection.
When $z_1=0$ the gauge group is unbroken, but when $z_1 \not=0$ the group is broken to $G_S \times U(1)_1$, with $G_S$ being the group whose generators commute with $Q_1$. The locus $z_1=0$ defines the curve $\Sigma$. Thus, on $\Sigma$ the singularity enhances from $G_S$ to $G_\Sigma \supset G_S \times U(1)_1$. The breaking of the gauge group is explained by the deformation of the singularity type from $G_\Sigma$ to $G_S$ due to the background in the Cartan subalgebra [@kv].
For ordinary D7-branes the adjoint scalar corresponds to degrees of freedom in the transverse direction and a non-zero vev means that some branes are separated and the gauge group is broken. For instance, if there are $(K+1)$ D7-branes to begin and one is moved away, the original $SU(K+1)$ is broken to $SU(K) \times U(1)$. Furthermore, the open strings stretching between the two stacks of D7-branes give rise to massless bifundamentals $({\pmb K}, -1) + (\overline{\pmb K}, 1)$ localized at the intersection. For F-theory seven-branes wrapping a surface $S$ one then expects $\langle \varphi\rangle$ to break the original gauge group to some $G_S$. Moreover, there will be massless ‘bifundamentals’ descending from the adjoint of $G_\Sigma$ which decomposes as a direct sum of irreducible representations $(\RR, q_1)$ under $G_S \times U(1)_1$.
Several examples of singularity resolution were worked out in [@kv] and more recently in [@bhv1; @bhv2; @bourjaily]. For illustration let us consider the case $G_\Sigma=E_6$ and $G_S=SO(10)$ that will be of interest later on. Under $SO(10) \times U(1)$ the $E_6$ adjoint decomposes as = (,0) + (1,0) + (,1) + (,-1) \[e6br\] Therefore, there will be chiral multiplets transforming as $\r{16}$ and $\br{16}$ of $SO(10)$. To see how $SO(10)$ is enhanced to $E_6$ on $\Sigma$ it is convenient to represent the Cartan generators as vectors $|Q_i\rangle$ so that $|\varphi \rangle \propto z_1 | Q_1\rangle$ corresponds to the adjoint vev [@kv]. The simple roots are elements $\langle v_j|$ of the dual space. Those roots with $\langle v_j | \varphi\rangle=0$ remain as $SO(10)$ roots while those with $\langle v_j | \varphi\rangle \propto z_1$ become the weights of the $\r{16}$ and $\br{16}$.
The resolution of the singularity by the adjoint vev can be figured out as explained in [@kv]. The generic $E_6$ singularity can be cast as [@km; @bhv1] y\^2 = x\^3 + 14 z\^4 + \_2 x z\^2 + \_5 xz + \_6 z\^2 + \_8 x + \_9 z + \_[12]{} \[e6sing\] where the $\e_i$ are functions that depend on the adjoint vev. More precisely, in [@km] they are given in terms of an arbitrary vector $(t_1, \cdots, t_6)$ in the $E_6$ Cartan subalgebra. In our notation, in the $E_6 \supset SO(10) \times U(1)_1$ example, $t_1=z_1$ while other $t_i$’s vanish. By choosing $t_1=-3t$, and computing the $\e_i$ according to the formulas of [@km], we obtain the deformation y\^2 =x\^3 + 14 z\^4 - 3 t\^2 x z\^2 - 12 t\^5 xz - 6 t\^6 z\^2 - 12 t\^8 x - 16 t\^9 z - 12 t\^[12]{} \[so10sing\] This is the same result found in [@kv], for a different though equivalent choice of vev vector. It can be shown that for $t\not=0$ there is an $SO(10)$ singularity.
So far we have just reviewed how the gauge group on the curve $\Sigma$ is enhanced. We now want to discuss how the matter localized on $\Sigma$ arises from zero modes of the 8 bulk fields. It is enough to look at fermions because the scalars follow by supersymmetry. We then want to solve the 8 equations of motion for the twisted fermions when $\varphi$ has the vev linear in $z_1$, and there is no gauge background. The solutions that are localized at $z_1=0$ can be interpreted as the fermions $\lambda$ and $\lambda^c$ that come from the twisted super Yang-Mills on ${\mathbbm{R}}^{3,1} \times \Sigma$.
We start from the 8 fermionic equations of motion (\[zma\]-\[zmc\]). To show that there are localized solutions it suffices to work locally and assume that the fundamental form of $S$ has the canonical Euclidean form = 2 (dz\^1 d \^1 + dz\^2 d \^2) \[oms\] Notice that the coordinate along $\Sigma$ is $z_2$ whereas the transverse coordinate is $z_1$. To look for localized solutions one can neglect derivatives in $z_2$. The equations of motion reduce then to 2 \_1 - m\^2 z\_1 q\_1 \_[|2]{} = 0 & ; & |\_[|1]{} \_[|2]{} - 2 m\^2 \_1 q\_1= 0 \[etapsi2\]\
\_1 \_[|1]{} - m\^2 \_1 q\_1 = 0 & ; & |\_[|1]{} - m\^2 z\_1 q\_1 \_[|1]{} \[chipsi1\] = 0 where $\chi=\chi_{12}$. Here $q_1$ is the $U(1)_1$ charge of the fermions that belong to a representation $(\RR, q_1)$ of $G_S \times U(1)_1$. From the above equations we see that there are no localized solutions for $\eta$ and $\psi_{\bar 2}$, and indeed it is consistent to set $\eta=0$ and $\psi_{\bar2 }=0$. On the other hand, the coupled system for $\chi$ and $\psi_{\bar 1}$ has solution =f(z\_2) e\^[-e m\^2 |z\_1|\^2]{} ; \_[|1]{} = - f(z\_2) e\^[-e m\^2 |z\_1|\^2]{} \[locsol\] where $f(z_2)$ is an arbitrary holomorphic function of the coordinate along $\Sigma$. We have set $q_1=e$ to take into account normalization of the charges. Clearly the zero modes are peaked around $z_1=0$, with width in $|z_1|^2$ equal to $1/e m^2$. The constant $em^2$ is expected to be of the order of the F-theory mass scale $M_*^2$.
The solutions localized at $z_1=0$ naturally correspond to the fermions $\lambda$ and $\lambda^c$ that appear in the 6 twisted theory. As argued in [@bhv1], the transformations along $\Sigma$ of $(\psi_{\bar 1}, \chi)$ and $(\lambda, \lambda^c)$ do agree.
Zero modes at intersecting matter curves {#s:local}
========================================
As we have reviewed, there are charged fields localized on a matter curve $\Sigma$ where the singularity type is enhanced. In this section we want to study the situation in which there are three matter curves $\Sigma_I$, $I=a,b,c$, occurring at the intersection of surfaces $S$ and $S^\prime_I$. In turn the three matter curves intersect at a point. On each curve there is a gauge group $G_{\Sigma_I}$ that enhances to $G_p$ at the common point of intersection [@bhv1].
To obtain the wave functions of the fermionic zero modes living at intersections we follow again the approach of [@bhv1]. The strategy is to consider the fermionic equations of motion of the 8 theory on $S$ with a non-trivial background for the adjoint scalar $\varphi$ that determines the curves. One then looks for solutions that are localized on a particular matter curve.
In the previous section we have seen that the equations that can give rise to localized fermionic zero modes are given by \_A + 2 \[|, \] = 0 ; |\_A -\[, \] = 0 \[zm1\] We have set $\eta=0$ because only fermions that appear in 4 chiral multiplets are expected to have localized modes on $\Sigma$. Notice then that equation (\[zmc\]) implies the additional constraint $\bar \partial_A \psi=0$. The new ingredient now is a more general background for the adjoint scalar $\varphi$. Concretely, = m\_1\^2 z\_1 Q\_1 + m\_2\^2 z\_2 Q\_2 \[phibg\] where the $z_i$ are local coordinates, and $\varphi=\varphi_{12}$. Each $Q_i$ is the generator of a $U(1)_i$ inside $G_p$. The $m_i$ are some mass parameters expected to be related to the F-theory supergravity scale $M_*$. In what follows we will take $m_1=m_2=m$.
As discussed in section \[ss:sigma\], when $\langle \varphi \rangle \propto z_1$, the adjoint vev resolves the singularity on the curve $\Sigma_a$ characterized by $z_1=0$ [@kv]. Now the more general adjoint background resolves the $G_p$ singularity where three curves intersect. When $(z_1,z_2) \not= (0,0)$, the group is broken to $G_S$ but at the intersection $(z_1,z_2) = (0,0)$ the group is enhanced to $G_p$. Furthermore, at the curves $\Sigma_I$ the group is enhanced to $G_{\Sigma_I} \supset G_S \times U(1)$. For example, when $(G_p, G_S)=(E_7, SO(10))$, the group is enhanced to $E_6$ at $\Sigma_a$ and $\Sigma_b$ defined respectively by the loci $z_1=0$ and $z_2=0$, whereas it is enhanced to $SO(12)$ at $\Sigma_c$ defined by $z_1+z_2=0$. In the case $(G_p, G_S)=(E_8, E_6)$, the group is enhanced to $E_7$ at each curve.
At each curve $\Sigma_I$ there are matter fermions that correspond to open strings stretching between 7-branes wrapping $S$ and $S^\prime_I$. The $U(1)_i$ charges of these fermions, denoted $(q_1,q_2)$, depend on the curve as shown in table \[sigmaq\]. In this table we also indicate how the fermions transform under the gauge group in the examples $G_S=E_6$ and $G_S=SO(10)$ in which the group $G_p$ is respectively $E_8$ and $E_7$. For $G_S=SU(5)$ the rank two higher $G_p$ can be either $E_6$ or $SO(12)$. We have introduced parameters $(e_1,e_2)$ to take into account normalization of the charges.
curve $(q_1,q_2)$ locus $E_6$ $SO(10)$ $SU(5)$ $SU(5)$
------------ --------------- ------------- ---------- ---------- ---------- --------------
$\Sigma_a$ $(e_1,0)$ $z_1=0$ $\r{27}$ $\r{16}$ $\r{10}$ $\r{10}$
$\Sigma_b$ $(0,e_2)$ $z_2=0$ $\r{27}$ $\r{16}$ $\r{10}$ $\r{\bar 5}$
$\Sigma_c$ $(-e_1,-e_2)$ $z_1+z_2=0$ $\r{27}$ $\r{10}$ $\r{5}$ $\r{\bar 5}$
: Curves and charges[]{data-label="sigmaq"}
Zero modes in the absence of fluxes {#ss:noflux}
-----------------------------------
We will first solve the zero mode equations without turning on a background gauge field but with scalar vev $\langle \varphi\rangle$ given in (\[phibg\]). The fundamental form $\omega$ is assumed to have the standard local form (\[oms\]). Recall that $\psi=\psi_{\bar \imath} d\bz^i$ and $\chi=\chi_{12}dz^1 \wedge dz^2$ are forms on $S$. Substituting in the master equations (\[zm1\]) yields \_2 \_[|2]{} + \_1 \_[|1]{} - m\^2(\_1 q\_1 + \_2 q\_2) & = & 0\
|\_[|1]{} - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|1]{} & = & 0 \[noflux\]\
|\_[|2]{} - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|2]{} & = & 0 where now $\chi=\chi_{12}$. The constants $(q_1,q_2)$ are the $U(1)_i$’s charges of the fermions that belong to a representation $(\RR, q_1, q_2)$ of $G_S \times U(1)_1 \times U(1)_2$. In the following we will analyze the different possibilities for the fermions with charges and corresponding curves shown in table \[sigmaq\]. Notice that the condition $\bar\partial_{\bar 2} \psi_{\bar 1}= \bar\partial_{\bar 1} \psi_{\bar 2}$, implied by the additional constraint $\bar \partial \psi=0$, is automatic by virtue of the last two equations above.
After substituting the charges in (\[noflux\]) we obtain the solutions \_[|2]{} = 0 ; =f(z\_2) e\^[-\_1 |z\_1|\^2]{} ; \_[|1]{} = - \[s1nf\] where $f(z_2)$ is a holomorphic function of $z_2$. The equations (\[noflux\]) require the constant $\lambda_1$ to satisfy \_1\^2 = e\_1\^2 m\^4 \[l1sol\] We see that there are solutions localized at $z_1=0$ provided that we take the positive root $\lambda_1=e_1 m^2$. We then have two zero modes $\psi_{\bar 1}$ and $\chi$ which correspond to massless fermions of $D=6$ massless hypermultiplets living on $\Sigma_a$. In the presence of magnetic fluxes through $\Sigma_a$, chiral four dimensional fermions will appear coming from $\psi_{\bar 1}$ or/and $\chi $ as dictated by index theorems.
The characteristic width of the Gaussian wave functions is $v=1/(e_1m^2)$. We will assume that $v=1/M_*^2$, where $M_*$ is the F-theory mass scale. For sufficiently large compactification radius $R$ this width becomes negligibly small.
In this case the solutions of (\[noflux\]) turn out to be \_[|1]{} = 0 ; =g(z\_1) e\^[-\_2 |z\_2|\^2]{} ; \_[|2]{} = - \[s2nf\] with $g(z_1)$ a holomorphic function of the longitudinal coordinate $z_1$. The constant $\lambda_2$ now satisfies \_2\^2 = e\_2\^2 m\^4 \[l2sol\] As in the previous situation, having solutions localized at $z_2=0$ requires $\lambda_2=e_2 m^2$.
To treat this curve it is convenient to introduce new variables and fields, and to simplify by setting $e_1=e_2=e$. Consider then the definitions w=z\_1 + z\_2 & ; & \_[|w]{} = (\_[|1]{} + \_[|2]{})\
u=z\_1 - z\_2 & ; & \_[|u]{} = (\_[|1]{} - \_[|2]{}) \[newdefs\] The zero mode equations then become 2(\_w \_[|w]{} + \_u \_[|u]{}) + em\^2|w & = & 0\
|\_[|w]{} + em\^2 w \_[|w]{} & = & 0 \[noflux3\]\
|\_[|u]{} + em\^2 w \_[|u]{} & = & 0 Now there are localized solutions at $w=0$, namely \_[|u]{} = 0 ; =h(u) e\^[-\_3 |w|\^2]{} ; \_[|w]{} = \[s3nf\] with $h(u)$ a holomorphic function of the coordinate $u$ along $\Sigma_c$, and $\lambda_3=em^2/\sqrt2$. Notice that $\psi_{\bar u}=0$ implies \_[|1]{} = \_[|2]{} = h(u) e\^[-\_3 |w|\^2]{} \[p12s3nf\] These agree with results in [@tatar2].
Zero modes with fluxes {#ss:withflux}
----------------------
We now want to solve the zero mode equations with a background flux, still keeping the adjoint vev $\langle \varphi \rangle$ given in (\[phibg\]). We already know that without flux each curve $\Sigma_I$ supports localized modes with $U(1)_i$ charges given in table \[sigmaq\]. The fermions on each curve will now feel a total flux $\cf$ that includes various contributions. There is a bulk $U(1)$ flux $F$ in $G_S$ with generator $Q$ (for example, hypercharge or $Q_{B-L}$). There are also fluxes $F^{(i)}$ along the $U(1)_i$ inside $G_p$ with generators $Q_i$. The total flux can then be written as = F Q + F\^[(1)]{} Q\_1 + F\^[(2)]{} Q\_2 \[tflux\] The corresponding gauge potentials will be denoted $\ca$, $A$ and $A^{(i)}$, with the total potential $\ca$ decomposed like the total flux. We will use conventions in which the covariant derivative $\partial_A$ of a field of charge $q$ is defined as \_A = - iq A \[covdef\] All field strengths and gauge potentials are taken to be real.
The fermions $\chi$ and $\psi$ have $U(1)_i$ charges $(q_1,q_2)$ and transform in some representation $\RR$ of $G_S$. The bulk flux break $G_S$ to $\Gamma_S \times U(1)$ and $\RR$ decomposes into a direct sum of irreducible representations that can be labelled by $(\rr, q, q_1, q_2)$, where $q$ is the bulk $U(1)$ charge. The zero mode equations for the charged fermions then become (\_2 - i \_2) \_[|2]{} + (\_1 -i\_1)\_[|1]{} - m\^2(\_1 q\_1 + \_2 q\_2) & = & 0\
(|\_[|1]{}- i\_[|1]{}) - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|1]{} & = & 0 \[withflux\]\
(|\_[|2]{} -i\_[|2]{}) - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|2]{} & = & 0 Clearly, the total gauge potential that appears depends on the charges. It is explicitly given by = q A + q\_1 A\^[(1)]{} + q\_2 A\^[(2)]{} \[tpot\] The task is to solve the above equations for particular fluxes.
The 8-dimensional equations of motion further require the vanishing of the $(2,0)$ and $(0,2)$ components of the field strengths. We will only consider diagonal components $\cf_{1\bar 1}$ and $\cf_{2\bar 2}$, even though $\cf_{1\bar 2}$ and $\cf_{1 \bar 2}$ are also allowed. Using local coordinates the bulk flux takes the form F = F\_[1|1]{} dz\_1 d\_1 + F\_[2|2]{} dz\_2 d\_2 \[bfluxgen\] For the $U(1)_i$ fluxes we instead take F\^[(1)]{} = F\^[(1)]{}\_[2|2]{} dz\_2 d\_2 ; F\^[(2)]{} = F\^[(2)]{}\_[1|1]{} dz\_1 d\_1 \[cfluxgen\] The rationale is that, say $F^{(1)}$, is the flux along the curve $\Sigma_a$ that is defined by $z_1=0$ and has coordinate $z_2$.
We will start by analyzing constant field strengths in section \[sss:cflux\]. In this case it is possible to solve the zero mode equations exactly. We will then study variable fluxes that turn out to be necessary to generate corrections to Yukawa couplings [@hv3].
### Zero modes with constant flux {#sss:cflux}
In the case of constant field strengths the bulk flux can be written as F = 2i M dz\_1 d\_1 + 2i N dz\_2 d\_2 \[bflux\] where $M$ and $N$ are real constants. As explained before, the $U(1)_i$ fluxes have components only along the curves. They are then given by F\^[(1)]{} = 2i N\^[(1)]{} dz\_2 d\_2 ; F\^[(2)]{} = 2i M\^[(2)]{} dz\_1 d\_1 \[cflux\] with $N^{(1)}$ and $M^{(2)}$ some real constants.
For the gauge potentials we take the following gauge A & = & iM (z\_1 d\_1 - \_1 dz\_1) + iN (z\_2 d\_2 - \_2 dz\_2)\
A\^[(1)]{} & = & iN\^[(1)]{} (z\_2 d\_2 - \_2 dz\_2) \[sympots\]\
A\^[(2)]{} & = & iM\^[(2)]{} (z\_1 d\_1 - \_1 dz\_1) Notice that the total gauge potential defined in (\[tpot\]) can be cast as = i (z\_1 d\_1 - \_1 dz\_1) + i (z\_2 d\_2 - \_2 dz\_2) \[tpotc\] where the total flux coefficients are given by =(q M + q\_2 M\^[(2)]{}) ; =(q N + q\_1 N\^[(1)]{}) \[tfluxcoef\] where $q$ and $q_i$ are the bulk and $U(1)_i$ charges respectively. In appendix A we give the exact solution of the zero mode equations (\[withflux\]) with this total gauge potential for the three matter curves $\Sigma_a$, $\Sigma_b$ and $\Sigma_c$. Using these results we can then describe the localized wave functions at each curve.
As explained in appendix A, it is convenient to perform a gauge transformation such that $\hat \ca_{\bar 1}=\hat \ca_{\bar 2}=0$, and then work with the potential $\hat \ca$. We will refer to this choice as the holomorphic gauge. The wave functions in this gauge, denoted $\hat \chi$ and $\hat \psi_{\bar \imath}$, take a simpler form and are better suited to compute gauge invariant quantities such as Yukawa couplings.
In the case of $\Sigma_a$ we find wave functions = f(z\_2) e\^[-\_1 |z\_1|\^2]{} ; \_[|1]{} = - ; \_[|2]{} = 0 \[casosigma1\] where \_1 = -+ e\_1m\^2 = -+ e\_1m\^2 + 12 + \[ellambda1\] which reduces to $\lambda_1=e_1 m^2$ when $\cam =0$. For future purposes we record the expansion of the zero modes to first order in $\cam$, namely = \^[(0)]{} { 1 + |z\_1|\^2 + } ; \_[|1]{} = \_[|1]{}\^[(0)]{} { 1 - v + |z\_1|\^2 + } \[chipsi10\] where $v=1/(e_1m^2)$. Clearly, $\hat \chi^{(0)}= - \hat \psi_{\bar 1}^{(0)} =f(z_2)\, e^{-|z_1|^2/v}$ is the solution for $\cam=0$.
Notice that as expected the flux has the effect of deforming the wave function. In the holomorphic gauge defined above the wave functions depend on fluxes only through $\cam$. Since the matter fields in the curve $\Sigma_a$ have $q_2=0$ the wave function depends only on the flux in the bulk (e.g. from hypercharge in $SU(5)$ or $U(1)_{B-L}$ in $SO(10)$). Concretely, we must replace $\cam$ above by \_a = q\_a M \[ema\] where $q_a$ is the bulk charge and $M$ comes from the bulk flux. This is relevant later when extracting the $U(1)$ charge dependence of the Yukawa couplings.
Analogous results are obtained for the $\Sigma_b$ matter curve with the obvious replacements $\cam \rightarrow \cn$ and $e_1\rightarrow e_2$. In the holomorphic gauge the $\Sigma_b$ wave function depends only on the bulk flux. This means that $\cn$ must be replaced by $\cn=q_b N$.
For the $\Sigma_c$ curve the wave functions are found to be = h(u+w) e\^[-\_3 |w|\^2]{} e\^[w |u]{} ; \_[|w]{} = ; \_[|u]{} = - \[casosigma3\] where $h(u+\gamma w)$ is an holomorphic function of its argument and = ; = \[xiepstexto\] where $\Delta=(\cam +\cn )/2$ and $\lambda_3$ is given in appendix A. In the absence of fluxes one has $\xi=\gamma=0$, and $\lambda_3=em^2/\sqrt{2}$, recovering the fluxless result. Note that now it is the linear combination $\xi {\hat \psi_{\bar w}}
+\lambda_3\hat \psi_{\bar u}$ which vanishes. On the curve $\Sigma_c$ the matter fields have $U(1)_i$ charges $(q_1,q_2)=(-e_1,-e_2)$. Hence, $\cam$ and $\cn$ in this case are explicitly given by \_c = q\_c M - e\_2 M\^[(2)]{} ; \_c = q\_c N - e\_1 N\^[(1)]{} \[emenc\] We see that the wave function depends on both bulk and brane fluxes.
Zero modes with variable fluxes {#sss:vflux}
-------------------------------
In general it is quite complicated to obtain the exact wave functions for non-constant field strengths. In [@hv3] an adiabatic hypothesis is assumed whereby the wave functions basically follow from those of constant fluxes by replacing the constant coefficients $\cf_{i\bar j}$ by their variable counterparts. An expansion in powers of the $z_i$’s is then performed. In this article our approach will be to consider variable fluxes expanded in powers of the local coordinates from the beginning, and then solve the differential equations for the zero modes.
We will first expand the fields strengths up to second order in the local coordinates. We again turn on only components $\cf_{1\bar 1}$ and $\cf_{2 \bar 2}$. Specifically, we take \_[1|1]{} & = & 2i+ 4i(\_1 z\_1 + |\_1 \_1) + 6i(\_1 z\_1\^2 + |\_1 \_1\^2)\
\_[2|2]{} & = & 2i + 4i(\_2 z\_2 + |\_2 \_2) + 6i(\_2 z\_2\^2 + |\_2 \_2\^2) \[vflux\] where the flux coefficients $\a_i$ and $\b_i$ are complex constants while $\cam$ and $\cn$ are real. In practice the expansion parameter is $z_i/R$, where $R$ is the compactification radius (see section \[ss:eflux\]). We have neglected quadratic terms proportional to $(z_i\bz_i)$ because they do not give any new effects concerning Yukawa couplings. The total flux coefficients can be split into bulk and curve contributions in analogy to (\[tfluxcoef\]).
In our gauge choice the vector potential has components \_1 & = & -i\_1 - i(|\_1\_1\^2 + 2\_1z\_1\_1) - i(|\_1 \_1\^3 + 3\_1\_1 z\_1\^2)\
\_2 & = & -i\_2 - i(|\_2\_2\^2 + 2\_2z\_2\_2) - i(|\_2 \_2\^3 + 3\_2\_2 z\_2\^2) \[vpot\] whereas $\ca_{\bar 1}=\ca_1^*$, and $\ca_{\bar 2}=\ca_2^*$. In appendix A we discuss the solutions of the zero mode equations (\[withflux\]) with this total gauge potential.
We have not solved the zero mode equations exactly. Instead we found solutions in a perturbative expansion in the flux parameters $(\cam,\cn, \a_i, \b_i)$. We first go to the holomorphic gauge with $\hat \ca_{\bar 1}=\hat \ca_{\bar 2}=0$, and then iterate to obtain $\hat\chi = \sum_{I=0} \hat \chi^{(I)}$, where $\hat \chi^{(I)}$ is of order $I$ in the flux coefficients. The zeroth order wave function $\hat \chi^{(0)}$ is the fluxless solution derived in section \[ss:noflux\]. Once $\hat \chi$ is determined it is straightforward to deduce the $\hat \psi_{\bar \imath}$. For example, in $\Sigma_a$, $\hat \psi_{\bar 1}=\bar\partial_{\bar 1}\hat\chi/(e_1 m^2 z_1)$, and $\hat \psi_{\bar 2}=0$.
The iteration can be carried out to any desired order, but the number of terms will clearly be increasingly larger. In appendix A we only display results at most up to second order in the flux parameters. Already at first order there is an interesting feature that deserves further elaboration. To simplify the argument we set $\b_i=0$. Then, the wave functions in the curve $\Sigma_a$ are found to be & = & \^[(0)]{} { 1 + 43 v\_1z\_1 + |z\_1|\^2 +23|z\_1|\^2 (|\_1 \_1 + 2\_1 z\_1) + }\
\_[|1]{} & = & \_[|1]{}\^[(0)]{} { 1 - v - 43 v[|\_1]{}\_1 + |z\_1|\^2 +23|z\_1|\^2 (|\_1 \_1 + 2\_1 z\_1) + } \[chipsi11\] where $v=1/(e_1m^2)$. For constant $\cf_{i\bar\jmath}$ we have derived the exact solutions whose expansion to first order in $\cam$ agrees with the above results setting $\a_i=0$.
One point we wish to make is that in the presence of variable field strength $\cf$ the solution is not merely obtained by adiabatically including the coordinate dependence in $\cf$. In our case this would correspond to substituting $\cam$ by the effective value \_[eff]{}= + 2(|\_1 \_1 + \_1 z\_1) \[adiabsubs\] Indeed, once we replace $\cam$ by $\cam_{eff}$ in the solutions (\[chipsi10\]) for constant field strength, we reproduce some terms in the expansions (\[chipsi11\]). However, in $\hat \chi$ there is an additional piece linear in $z_1$ which cannot arise in the adiabatic approximation. In the expansion of $\hat \psi_{\bar 1}$ the term linear in $\bz_1$ is expected because in the exact solution there is actually a linear term in the constant $\cam$.
Evaluating the fluxes {#ss:eflux}
---------------------
Before going to the explicit computation of the Yukawa couplings let us evaluate the size of the expected $U(1)$ fluxes in F-theory grand unification schemes. Flux quantization demands \_[S]{} F = [m]{} ; \_[\_a]{} F\^[(1)]{} = n\^[(1)]{} ; \_[\_b]{} F\^[(2)]{} = m\^[(2)]{} \[flujillos\] where $\tilde m$, $\tilde n^{(1)}$ and $\tilde m^{(2)}$ generically denote flux quanta for the bulk and $U(1)_i$ fluxes. On the other hand, the GUT gauge coupling constant is given by = M\_\*\^4 \_S =(S) M\_\*\^4 = R\^4 M\_\*\^4 . \[alphagut\] where $R$ is the overall radius of the manifold $S$. We then estimate for the fluxes F = 2 M\_\*\^2 [m]{} ; F\^[(1)]{} = 2 M\_\*\^2 [n\^[(1)]{}]{} ; F\^[(2)]{} = 2 M\_\*\^2 [m\^[(2)]{}]{} . \[flujetes\] Here we have assumed that the volume of each matter curve is controlled by the overall size $R$, since they are embedded in $S$. Recall that standard MSSM gauge coupling unification gives $\alpha_G\simeq 1/24$, for the conventional gauge group normalization $\Tr T^2=1/2$, with generators $T$ in the fundamental of $SU(K)$. Thus, the compactification scale is only slightly below the F-theory scale $M_*$.
Equipped with the above estimates we can characterize more precisely the parametrization of the field strengths. For instance, we conclude that the total constant coefficients are generically given by = 2(q m + q\_2 m\^[(2)]{})M\_\*\^2 ;= 2 (q n + q\_1 n\^[(1)]{})M\_\*\^2 \[estfluxm\] where $q$ and $q_i$ are respectively the bulk and $U(1)_i$ charges. Similarly, for the total linear coefficients we can write \_1 = 2(q \_[1B]{} + q\_2 \_1\^[(2)]{}) ; \_2 = 2 (q \_[2B]{} + q\_1 \_2\^[(1)]{}) \[estfluxa\] where $\tilde \a_{iB}$ and $\tilde\a_i^{(j)}$ are adimensional constants that come respectively from bulk and $U(1)_i$ fluxes.
Recall that on the curves $\Sigma_a$ and $\Sigma_b$ the effective wave functions, in the holomorphic gauge, depend only on parameters given by bulk quantities. Specifically they are functions of \_a = 2q\_a m M\_\*\^2 & ; & \_[1a]{} \_a = 2q\_a \_[1B]{} \[fluxsa\]\
\_b = 2q\_b n M\_\*\^2 & ; & \_[2b]{} \_b = 2q\_b \_[2B]{} \[fluxsb\] Other coefficients such as say, $\cam_b = 2\pi \sqrt{\alpha_G}(q_b \tilde m + e_2 \tilde m^{(2)}) M_*^2$, do not appear in the wave functions in the holomorphic gauge $\hat{\ca}$. On the other hand, the parameters for the curve $\Sigma_c$ depend on bulk and $U(1)_i$ fluxes according to (see appendix A) \_c = 2(q\_c m - e\_2 m\^[(2)]{}) M\_\*\^2 & ; & \_c = 2(q\_c n - e\_1 n\^[(1)]{}) M\_\*\^2 \[fluxsc\]\
\_[1c]{} = 2(q\_c \_[1B]{} -e\_2 \_1\^[(2)]{}) & ; & \_[2c]{} = 2(q\_c \_[2B]{} -e\_1 \_2\^[(1)]{}) In the following we will use $\Delta=(\cam_c+\cn_c)/2$ instead of $\cn_c$, and $\delta=(\a_{1c}+\a_{2c})/2$ in place of $\a_{2c}$, and we will denote $\a_{1c} \equiv \a_c$. The decomposition of the quadratic and higher order coefficients of $\cf$ is completely analogous. Observe that gauge invariance imposes constraints such as $\cam_a + \cam_b + \cam_c =0$.
Note that the bulk and $U(1)_i$ charges, $q$, $q_1$, and $q_2$, depend on the normalization of the gauge coupling constants. Consider for example the case of a bulk hypercharge $U(1)_Y$ with $q_Y$ integer normalization such that . Then, , evaluating at a $SU(5)$ 5-plet. In order to get the standard $SU(5)$ normalization with $\Tr T^2=1/2$, a normalization factor $e_Y=1/\sqrt{60}\simeq 0.13$ is needed. The same exercise for $U(1)_{B-L}$ in $SO(10)$ yields a factor $e_{B_L}=1/\sqrt{24}$, with assignments .
The normalization factors for $q_{1,2}$ are found in an analogous way, taking into account the enhanced gauge symmetry at each matter curve. Consider for example matter curves at which an $SU(5)$ symmetry is enhanced to $SO(10)$ or $SU(6)$. This means that there are branchings $SO(10)=SU(5)\times U(1)_{1,2}$ or $SU(6)=SU(5)\times U(1)_{1,2}$. One finds normalization constants $1/\sqrt{40}$ and $1/\sqrt{60}$ respectively, with matter fields having charges $\pm 1$. In the case of $SO(10)$ with matter curves enhancing to $E_6$ or $SO(12)$ one finds $1/\sqrt{20}$ and $1/\sqrt{8}$ respectively. These factors must be taken into account in the explicit computation of coupling constants.
There is an additional constraint on the fluxes in the bulk coming from the BPS condition in eq.(\[bpscond\]) which now reduces to $\omega \wedge F =0$. In particular, locally this condition implies $M+N = 0$ for constant $F$. Nevertheless, in what follows we will not impose this constraint so that we can keep track of the effect of all flux parameters.
![Triple intersection of three matter curves.[]{data-label="calvicie"}](yukawaR.eps)
Yukawa couplings
================
Computing Yukawa couplings {#ss:yukint}
--------------------------
We are interested in evaluating the Yukawa coupling of three chiral fields coming from three intersecting matter curves locally described by $z_1=0$, $z_2=0$ and $z_1+z_2=0$ in the surface $S$. The piece of the superpotential relevant for Yukawa couplings will be W\_Y = M\_\*\^4\_[ S]{} ( [A]{} ) \[superpot1\] where $M_*$ is the typical mass scale characteristic of the supergravity limit of F-theory. The Yukawa couplings are obtained as overlap integrals over $S$ of the three wave functions involved. In principle such a computation requires a knowledge of the wave functions over the whole complex surface $S$. On the other hand, we know that the wave functions are peaked around the local curves $z_1=0$, $z_2=0$ and $z_1+z_2=0$ so that the coupling is dominated by the region around the origin, $z_1=z_2=0$, where the three curves meet. If this is correct, a local knowledge of the wave functions of the type discussed in the previous sections will be sufficient to evaluate the Yukawa couplings. We will thus be interested on overlap integrals of the form[^1] Y = M\_\*\^4\_S d\^2z\_1 d\^2z\_2 \_[|1]{} \_[|2]{} \_[12]{} \[yuk1\] involving zero modes of the curves $z_1=0$, $z_2=0$, and $z_1+z_2=0$ respectively, $z_i$ being the local coordinates. Given this structure it is natural to assign the physical, e.g. quark/lepton, fields to $\psi_{\bar \imath}$ zero modes and $\varphi_{12}$ to the Higgs boson and we will assume this in what follows.
We will then take $\psi_{\bar 1}$, $\psi_{\bar 2}$, and $\varphi_{12}$ to be the zero modes localized at the curves $\Sigma_a$, $\Sigma_b$ and $\Sigma_c$ respectively. By supersymmetry the wave function of $\varphi_{12}$ is equal to that of $\chi_c$. We have seen in the previous sections that, in the holomorphic gauge, the relevant wave functions in the presence of variable fluxes take the general form \_[|1]{} & =& - f(z\_2)e\^[-e\_1m\^2|z\_1|\^2]{} G\_a(z\_1,[|z]{}\_1; q)\
\_[|2]{} & =& - g(z\_1)e\^[-e\_2m\^2|z\_2|\^2]{} G\_b(z\_2,[|z]{}\_2; q) \[fdo1\]\
\_[12]{} & =& e\^[- |z\_1+z\_2|\^2]{} G\_c(z\_1, \_1, z\_2, \_2;q,q\_1,q\_2) \[rayos\] where $G_I$, $I=a,b,c$, are functions which can be computed to any desired order both in fluxes and derivatives of fluxes. Recall that $G_a$ and $G_b$ depend only on bulk charges whereas $G_c$ depends on all charges. In absence of fluxes one simply has $G_a=G_b=G_c=1$. Here $f(z_2)$ and $g(z_1)$ are holomorphic functions. As in [@hv3] we will choose a basis in which they are given by $f_k=(z_2/R)^{3-k}$ and $g_\ell=(z_1/R)^{3-\ell}$, $k,\ell=1,2,3$, corresponding to the three generations of quarks and leptons. Since there is only one Higgs field we can take the corresponding holomorphic function to be a constant. We then have to perform the integral Y\_[k]{} = M\_\*\^4\_S d\^2z\_1 d\^2z\_2 e\^[-em\^2 (|z\_1|\^2+ |z\_2|\^2+ |z\_1+z\_2|\^2)]{} f\_k(z\_2) g\_(z\_1) G(z\_1,\_1, z\_2, \_2) , \[yuk2\] where $G=G_a G_b G_c$. To simplify the analysis we have set $e_1=e_2=e$.
An important point to remark is that turning on fluxes does not induce mixing in the wave functions among different flavors. Indeed, as seen e.g. in eq.(4.3), the flux corrections in $G_a$($G_b$) do not introduce additional holomorphic dependence on $z_2$($z_1$) which would signal generation mixing in the wave functions. This is an important simplification because otherwise we would need an additional diagonalization of wave functions in order to extract the physical couplings from eq.(\[yuk1\]).
The measure in the Yukawa integral can be thought to be d= d\^2z\_1 d\^2z\_2 e\^[-em\^2(|z\_1|\^2 + |z\_2|\^2 + |z\_1+z\_2|\^2 )]{} \[measure1\] Clearly, the third exponential, due to the zero mode on $\Sigma_c$, breaks the symmetry under separate $U(1)$ rotations $z_i\rightarrow e^{i\theta_i}z_i$. Instead, there is only invariance under the diagonal $U(1)$. This is enough to show that without non-constant fluxes, in which case $G$ cannot depend separately on the antiholomorphic variables, the only non-vanishing Yukawa coupling is $Y_{33}$ because $f_3=g_3=1$. Thus, the heaviest third generation of quarks and leptons will acquire masses through $Y_{33}$.
As pointed out originally in [@hv3], to generate non-vanishing Yukawa couplings for all families it is necessary to turn on non-constant background fluxes. To see this it is useful to rewrite the measure as d= 14 d\^2u d\^2w e\^[-em\^2(12|u|\^2 +|w|\^2)]{} \[measure2\] where $s={\sqrt2}-1$. As before, $w=z_1+z_2$ and $u=z_1 - z_2$. In presence of variable fluxes the function $G$ can furnish adequate powers of $\bar w$ and $\bar u$ so that the integrand becomes invariant under separate phase rotations of $w$ and $u$. The couplings $Y_{k\ell}$ will thus be non-zero and the light generations will gain masses and mixings. We have introduced the parameter $s$ in order to study also the case $s=1$, which corresponds to ignoring the zero mode exponential from $\Sigma_c$. In this situation the measure becomes invariant under separate $U(1)$ rotations $z_i\rightarrow e^{i\theta_i}z_i$ and there will be additional cancellations when computing the integrals.
In performing the integration we will assume that the width of the matter curves is determined by the F-theory scale $M_*$, this means $v=1/(em^2)=1/M_*^2$. Consistency of the local analysis requires the matter curves to be well localized within $S$. This amounts to the condition $v/R^2 \ll 1$, which is approximately valid for $v=1/M_*^2$ because $1/R^2 = \sqrt{\alpha_G} M_*^2$, and $\sqrt{\alpha_G} \simeq 0.2$. In practice we will evaluate the integrals over $S$, with the above measure $d\mu$, by extending $|w|$ and $|u|$ to infinite radius. The main contribution to the integrand still comes from the region near the origin because the measure is sufficiently peaked. The upshot is that in the end all integrals can be done exactly.
Without varying fluxes there is only one non-vanishing Yukawa coupling $Y_{33}$ for the third generation which may be explicitly estimated as Y\_[33]{}\^[(0)]{} = M\_\*\^4\_S d\^2z\_1 d\^2z\_2 e\^[-M\_\*\^2(|z\_1|\^2+|z\_2|\^2+|z\_1+z\_2|\^2)]{} = . \[yuk3\] To get the physical Yukawa coupling we really need to work with wave functions normalized to unity, but to actually normalize our wave functions we would need a global knowledge of them over all $S$. We can however make an estimate by neglecting the effect of fluxes and computing the norm of the $\psi_{\bar \imath}$ wave functions from M\_\*\^4\_S e\^[-2M\_\*\^2|z\_i|\^2]{} = M\_\*\^2R\^2 \[normfdo\] Thus, the normalized $\psi_{\bar \imath}$ wave functions are obtained multiplying our wave functions by the normalization factor $\cc^{-1/2}$. Similarly computed, the normalization for $\chi_{12}$, arising in the curve $\Sigma_c$, is found to be $\cc^\prime=\cc/\sqrt2$. We then obtain the normalized third generation Yukawa coupling Y\_[33]{}\^[norm]{} ()\^[ 3/2]{} = \_G\^[3/4]{} 0.23 \[yuktop\] where we have taken the $SU(5)$ value $\alpha_G=1/24$. The $\alpha_G^{3/4}$ dependence in eq.(\[yuktop\]) was previously noted in [@bhv2].
The $Y_{33}$ Yukawa just computed is given at the unification scale. Taking into account QCD renormalization effects down to the weak scale there is an extra factor $\simeq 3$ in the case of quarks so that one obtains m\_t 0.23 3 H\_u = 0.69 170 117 [GeV]{} \[eltop\] where in the last step we have assumed a large value for $\tan\beta=\langle H_u/H_d \rangle$. This is in reasonable agreement with experiment, given the uncertainties. A large value for $\tan\beta $ is required to understand within this scheme the relative smallness of the masses of b-quark and $\tau$ lepton compared to the top. For them one finds m\_b ; m\_ \[bytau\] which gives reasonable agreement for $\tan\beta \simeq 35-45 $. Note that the tau lepton is lighter by a factor $\simeq 3$ due to the absence of QCD renormalization.
There are also subleading contributions to $Y_{33}$ from flux corrections which appear even for constant flux (see appendix B.1). We will eventually neglect all subleading corrections so for consistency we will only keep the leading term in $Y_{33}$. When we calculate the rest of the Yukawa couplings we will then normalize them relative to the 3rd generation Yukawa in eq.(\[yuk3\]).
The case of a constant $\chi_c$ wave function {#ss:chic}
---------------------------------------------
We study first this simple case because it has some interesting features by itself. Furthermore, a constant wave function is unlocalized and hence could serve to give an idea of the results to be expected for Yukawa couplings in which the third particle, presumably the Higgs field, lives in the bulk rather than in a localized matter curve. Such type of couplings do appear in type IIB and F-theory models in which the base $S$ is not del Pezzo.
When $\chi_c$ is a constant, taken equal to one, the Yukawa couplings are determined by Y\_[k]{}|\_[\_c=1]{}. = M\_\*\^4\_S d\^2z\_1 d\^2z\_2 e\^[-em\^2(|z\_1|\^2+ |z\_2|\^2)]{} f\_k(z\_2) g\_(z\_1) G\_a(z\_1,\_1;q)G\_b(z\_2, \_2; q) \[yuknoc\] where $q$ denotes the bulk charges. Substituting the expressions for $G_a$ and $G_b$, which may be extracted from the wave functions in appendix \[a:vflux\], leads to Y\_[k]{}|\_[\_c=1]{}. = \^2 \_[k-3,-3]{} \[cancelara\] Hence, the flux-induced distortion of the wave functions does not give rise to any new couplings, only the coupling $Y_{33}$ which is there already for constant fluxes is non-vanishing. This is true for any order in the flux expansion. In the next section and in appendix \[b:yukresu\] we will provide some examples of the cancellation in the expansion of the $Y_{k\ell}$. The result can also be proven analytically. In fact, notice that in (\[yuknoc\]) the integrals in $z_1$ and $z_2$ decouple so that it suffices to show that $I_\ell=\int \! d^2z_1 e^{-em^2|z_1|^2} z_1^{3-\ell} G_a$ vanishes when $\ell=1,2$. The key point is that $ e^{-em^2|z_1|^2} G_a$ can be written as $\frac{1}{z_1}\bar\partial_{\bar 1}F_a$, as explained in appendix A.2. The function $F_a$ can be extracted explicitly, in particular it goes to zero when $|z_1| \to \infty$ and to 1 when $|z_1| \to 0$. It is then easy to show that $I_\ell=0$ for $\ell=1,2$.
The main conclusion is that in order to get non-trivial fermion mass hierarchies one needs all three wave functions to be localized on matter curves. We then proceed to this most interesting case of three overlapping localized wave functions.
Yukawa matrices {#ss:yukmat}
----------------
The physical Yukawas are obtained evaluating the overlapping integral in eq.(\[yuk2\]) which is dominated by the region close to the intersection point. The heavy task is to compute the function $G$ by substituting the wave functions found in the previous sections expanded in powers both of the flux and derivatives of the flux. In the end each Yukawa coupling reduces to a sum of Gaussian integrals that can evaluated analytically. As expected, $Y_{ij}$ and $Y_{ji}$ are related by an appropriate exchange of flux parameters.
To begin we have considered the simplest case in which the field strengths are expanded only to linear order. This means that we only take into account the first derivative of the fluxes (i.e. the $\alpha_I$ and $\delta$ parameters) and neglect the effect of higher derivatives. The integrals can be determined exactly. For example, the coupling $Y_{23}$ is found to be Y\_[23]{} = \[y23exact\] where $s=\sqrt2-1$ is the parameter appearing in the measure (\[measure2\]). This coupling is normalized with respect to $Y_{33}^{(0)}=\pi^2 s$. This exact expression also shows that when $s=1$ the terms that depend purely on $\a_a$ and $\a_b$ completely drop out. In all couplings it happens that for $s=1$ all pieces involving only parameters of the curves $\Sigma_a$ and $\Sigma_b$ do cancel out. This implies that when $\chi_c=1$, the only coupling that survives is $Y_{33}$.
In appendix \[b:f1\] we display the leading terms in the expansion in $\a$-fluxes for each entry of the Yukawa matrix, normalized with respect to $Y_{33}^{(0)}$. Some of the elements $Y_{ij}$ have complicated expressions in terms of the flux parameters but the pattern behind can be easily understood. Schematically, the couplings turn out to be Y\_[ij]{} \~()\^[3-i]{} ()\^[3-j]{} \[yijalpha\] As explained in section \[ss:eflux\], we have for instance $\a_a=2\pi\sqrt{\a_G}q_a \tilde\a_{1B} M_*^2/R$. Therefore, we find $v^2\a_a/R=2\pi \a_G q_a \tilde \a_{1B}$, because $v=1/M_*^2$ and $M_*^2 R^2 = 1/\sqrt{\a_G}$.
More generally, the corrections to the Yukawa couplings due to first derivatives of the fluxes have the general form Y\_[ij]{} \_[ij]{} (2\_G(aq + a’q\^))\^[(3-i)]{} (2\_G(aq+a’q\^))\^[(3-j)]{} \[yukflux1\] Here we have simply replaced the $\tilde \a_{iB}$ and the $\tilde \a_i^{(j)}$ of section \[ss:eflux\] by generic constants $a$ and $a^\prime$ in order to get an idea of the structure. The $\xi_{ij}$ are numerical coefficients appearing upon integration which are typically in the range $0.1-10$, as may be seen in appendix \[b:f1\]. The constants $q$ and $q^\prime$ are the bulk and matter curve $U(1)$ charges respectively. Recall that for the fields in matter curves $\Sigma_a$ and $\Sigma_b$, which include quarks and leptons, one has $q^\prime=0$ and the corresponding $\alpha_a$ and $\alpha_b$ parameters only depend on the bulk $U(1)$ charges. This is not the case for the matter curve $\Sigma_c$, the parameters $\alpha_c$ and $\delta$ do depend on the matter curve charges. Note that the normalization of the $U(1)$ charges is relevant here. As we explained, for the $SU(5)$ case for integer hypercharge there is a normalization factor $1/\sqrt{60}$ and the $U(1)$’s on the matter curves containing $10$’s and ${\bar{5}}$’s have normalization $1/\sqrt{40}$ and $1/\sqrt{60}$ respectively.
We have just discussed the general form of each of the terms in the Yukawa couplings shown in appendix \[b:f1\]. To get more accurate results we would need to specify the different flux parameters for the three matter curves involved. In particular, we would need a precise knowledge of how the $U(1)$ field strengths vary in the vicinity of the intersecting points. In principle, given a set of assumptions about the derivatives of fluxes on the different matter curves in a concrete model, the formulas in appendix B will allow us to compute the different Yukawa couplings.
It is already quite encouraging that a hierarchical structure of fermion masses seems to be built in. Using eq.(\[yukflux1\]) we can further estimate the Yukawa couplings by taking into account the normalization of the $U(1)$ charges explained in section \[ss:eflux\]. To this end we will write the bulk charges as $q=\hat q e_B$, where $\hat q$ is an integer and $e_B$ is the bulk charge normalization. We will similarly write $q^\prime=\hat q^\prime e$, where $e$ is the normalization of the appropriate $U(1)$ on the matter curve, and reabsorb the ratio $e/e_B$ into the coefficient $a^\prime$. It is also convenient to introduce $\alpha_{U(1)}= e_B^2 \alpha_G$, which corresponds to the $U(1)$ fine structure constant normalized for integer charges of massless fields. We then conclude that the Yukawa matrix has the form Y\^[(1)]{} \~(
[ccc]{} \^4\^4 & \^3\^3 & \^2\^2\
\^3\^3 & \^2\^2 &\
\^2\^2 & & 1
) ;=2 ;= \[flx2\] In the terminology of [@hv3], we can say that the physical parameter $\e$, which is tied to the $\a_I$ coefficients, controls the flux expansion. The parameter $\eta$ is related to powers of the width $v$ and the overall radius $R$ that appear in the couplings and controls instead the derivative expansion. Taking $\alpha_G=1/24$ and $e_B=e_Y=1/\sqrt{60}$ gives $\e = 0.16$, $\eta =0.20$ so that $\e\eta \simeq 1/31$. We then find Y\_[ij]{} \_[ij]{} ( )\^[(3-i)]{} ( )\^[(3-j)]{} \[yukflux2\] Therefore, the fermion hierarchies are roughly of the type (m\_3:m\_2:m\_1) (1:10\^[-3]{}(aq+a’q\^)\^2 : 10\^[-6]{}(aq+a’q\^)\^4) \[jerarquias0\] in qualitative agreement with the observed spectra of quarks and leptons. In the next chapter we discuss in slightly more detail to what extent this structure may be successful in describing the pattern of quark and lepton masses.
Let us now see what happens if further terms in the derivative expansion of the fluxes were non-negligible. In particular, we have studied the corrections to the Yukawa couplings when the second derivative flux parameters $\beta_I$ and $\rho$ are non-zero. We found that the couplings $Y_{13}$ and $Y_{22}$ receive leading contributions linear in $\b_I$ or $\rho$. They also have quadratic corrections, proportional to the constant coefficients $\cam$ and $\cn$ of the various curves times the $\b_I$ or $\rho$, that are subleading and can be neglected. The leading linear terms are Y\_[13]{} & = & \[y13exact\]\
Y\_[22]{} & = & \[y22exact\] where $s=\sqrt2-1$ as before. Here we notice again that when $\chi_c=1$ the couplings will vanish identically because in this case $s=1$ while $\b_c=0$ and $\rho=0$. The couplings $Y_{12}$ and $Y_{11}$ have leading corrections typically proportional to $\a_I \b_J$ and $\b_I \b_J$ respectively, but the exact expressions are too long to display. In appendix \[b:f2\] we show the numeric results for the extra leading contributions
To figure out the size of the corrections due to the second derivative flux parameters we will estimate them for the case of the $Y_{11}$ and $Y_{22}$ Yukawa couplings. We have Y\_[22]{} & \~ & = 2\_G (bq+b’q\^)\
Y\_[11]{} & \~ & = (2)\^2\_G\^2\_[U(1)]{} (bq+b’q\^)\^2 \[yukder2\] These terms would contribute to the hierarchy of fermion masses as (m\_3:m\_2:m\_1) (1:(2)\_G: (2)\^2\_G\^2\_[U(1)]{}) . \[jerarquias\] We can evaluate the remaining couplings in the same way (the first order in fluxes $Y_{23}$ and $Y_{32}$ as well). Including the zeroth order $Y_{33}$ we obtain the structure Y\^[(2)]{} \~(
[ccc]{} \^2\^4 & \^2\^3 & \^2\
\^2 \^3 & \^2 &\
\^2 & & 1
) \[der22\] Since $\e \simeq \eta $, there are hierarchies $(1: \e^{3}(b\hat q+b'\hat q^\prime):\e^{6}(b\hat q+b'\hat q^\prime)^2)$. We see that for coefficients of order one, these corrections will generically dominate over the corresponding terms in the flux expansion with only first derivatives of fluxes.
We can go one step beyond and consider also the effect of terms of order three and four in the derivative expansion of the fluxes. In this case, to leading order there appear contributions to $Y_{12}$, $Y_{21}$ and $Y_{11}$. The results are given in \[b:f34\] where we also explain the notation. The additional corrections may be approximated by Y\_[11]{} & \~& = 2\_G\^2 (dq+d’q\^) \[yukder411\]\
Y\_[12]{} & \~& Y\_[21]{} \~ = = 2\_G\^[3/2]{} (cq+c’q\^) \[yukder4\] In this case there are new contributions to the Yukawa couplings of the form Y\^[(3,4)]{} \~(
[ccc]{} \^4 & \^3 & 0\
\^3 & 0 & 0\
0 & 0 & 0
) \[der44\] Thus, the first generation Yukawa has corrections of order $Y_{11}\simeq \e \eta^4$.
The contributions leading to the terms captured by $Y^{(1)}$ are an explicit evaluation of the flux expansion of the authors of [@hv3]. On the other hand, their derivative expansion would correspond to taking the terms linear in $\e$ in $Y^{(1)}$, $Y^{(2)}$ and $Y^{(3,4)}$ above. Thus, this derivative expansion has the structure Y\^[DER]{} \~(
[ccc]{} \^4 & \^3 & \^2\
\^3 & \^2 &\
\^2 & & 1
) \[DER\] Note however that, for instance in $Y^{(2)}$, we also find corrections which do not correspond to either of both expansions.
As a general conclusion, one observes that, for a given Yukawa matrix element, the correction due to a higher order term in the derivative expansion will always dominate over the flux expansion.
Fermion Yukawa couplings in F-theory GUT’s
==========================================
![Intersecting matter curves and Yukawa couplings.[]{data-label="losacoplos"}](yft3n.eps)
In the previous sections we have studied the zero modes of the 8-dimensional quasi-topological field theory as well as the computation of the Yukawa couplings using these zero modes, without specifying any particular geometry nor identifying the nature of the three particles involved in the couplings. We did neither specify the bulk $U(1)$ to be considered, which could be hypercharge in $SU(5)$, or other in $SO(10)$. In a GUT model, the quark, lepton and Higgs superfields will be localized in matter curves like those we have described (see figure \[losacoplos\]). We will have Yukawa couplings from a superpotential of the form W\_[Yuk]{} = Y\^U\_[ij]{}Q\^iU\^jH\_u + Y\^D\_[ij]{}Q\^iD\^jH\_d +Y\^L\_[ij]{} L\^iE\^jH\_d \[supyuk\] in an obvious notation. In principle the intersection points of the different matter curves will be different and, correspondingly the flux parameters $\alpha_I$, $\beta_I$, etc., will also be different at each intersection. The geometry of each given model may constrain the possibilities though. For example, in an $SU(5)$ GUT the left-handed leptons $L^i$ and the right-handed $D$-quarks $D^j$ live in the same matter curve. In other settings that need not be the case. For example, in a flipped SU(5) setting, $D$, $U$ and $L$ masses come from independent couplings $10\times 10\times 5_H$, $10\times {\bar 5}\times {\bar 5}_H$ and $1\times {\bar 5}\times { 5}_H$.
Another point to emphasize is that in previous chapters we have made use of the possibility of choosing a local basis of holomorphic wave functions of the canonical form $1,z_i,z_i^2$ at the intersection point. Note however that in the case of quarks we cannot make use of the freedom to choose that basis both at the intersection point leading to $U$-quark masses and that giving rise to $D$-quark masses. On the other hand, if the holomorphic basis at both points are very different, one expects very large CKM mixing angles. This may be an indication that both points must be quite close in $S$ in order for the basis to be aligned to give reduced mixing, as required phenomenologically [@hv3]. That points towards further unification into at least $E_7$ at the F-theory level.
![Matter curves intersecting to provide $U$-quark Yukawas.[]{data-label="lastres"}](yft2n.eps)
An important issue in F-theory grand unification is the generation of appropriate Yukawa couplings for the $U$-quarks. After all, one of the main motivations for going to F-theory GUT’s instead of perturbative IIB orientifolds is that in the former case these couplings are allowed, while they are perturbatively forbidden in type IIB orientifolds. In $SU(5)$ the $U$-quark Yukawas come from couplings $10^i\times 10^j\times 5_H$, and if such coupling comes from three distinct matter curves, there can be no diagonal $U$-quark couplings. This implies that the trace of $Y^U$ vanishes which makes impossible a hierarchy of $U$-quark masses. In [@bhv1; @bhv2; @hv3] it was suggested that the matter curve associated to the $10$’s in $SU(5)$ could [*self-pinch*]{} as in figure , allowing for diagonal entries. It was noted though [@tatar2] that in such configuration the two branches of the wave functions of the $10$ are independent so that there would be two independent rank one contributions to $Y^U$. This would then lead to a rank two Yukawa matrix, with no automatic hierarchical structure. In [@tatar2] (see also [@hv3]) it was suggested that in fact the two independent branches of the wave functions could be identified by some symmetry in the geometry (figure ). In such a case the two rank one contributions would be identical and rank one (before the addition of flux effects). It was also argued that these symmetries are ubiquitous in F-theory and correspond to non-trivial monodromies. Another alternative in order to obtain diagonal entries in $U$-quark Yukawa couplings was also suggested in [@fi]. It is more easily described in $SO(10)$ but it also applies to $SU(5)$. In the context of $SO(10)$ the Yukawa couplings come from terms $16^i\times 16^j\times 10_H$. If one associates both $16$’s in the coupling to two matter curves $\Sigma_a$ and $\Sigma_b$, and one allows for appropriate flux with opposite restriction on the curves, the massless spectrum splits as in figure . One curve has matter content $2Q + U$, and the other $2U+Q$, and this splitting allows for diagonal couplings. In fact, as already noted in [@fi], both matter curves could be local branches of some self-pinched matter curve, as in figures and . An interesting feature of this possibility is that the mixing of the first generation with the other two is expected to be suppressed. In what follows we will not specify the particular scheme for the understanding of the $U$-quark Yukawa couplings. We will just consider that the approximate rank one structure already assumed in the previous sections does apply.
In this section we make a preliminary analysis of the application of our previous results to the description of quark and lepton spectra. We would like to see to what extent flux distortion may explain the data. Let us first consider for definiteness the case of a $SU(5)$ GUT broken down to the SM by fluxes along the hypercharge direction. Let us first see whether the flux-induced distortion of wave functions due to first derivatives of fluxes is enough to describe the observed structure of quarks and leptons. In order to get manageable results we will first assume for simplicity that the fluxes going through the third matter curve are approximately constant, i.e. $\alpha_c=\beta_c=\delta=\rho=0$ (we will also denote the subscripts $(a,b)$ as $(1,2)$ hereafter). Under these circumstances the formulas in appendix \[b:f1\] substantially simplify. In particular, the diagonal entries reduce to Y\_[22]{} & & \^2\^2( 0.067(Y\_R\^2[|a]{}\_1\^2+Y\_L\^2[|a]{}\_2\^2)+0.11 Y\_RY\_L[|a]{}\_1[|a]{}\_2) \[fluxsm1\]\
Y\_[11]{} & & \^4\^4( 0.10(Y\_R\^4[|a]{}\_1\^4+Y\_L\^4[|a]{}\_2\^4)+0.09Y\_RY\_L[|a]{}\_1[|a]{}\_2(Y\_R\^2 [|a]{}\_1\^2+Y\_L\^2[|a]{}\_2\^2)+0.12Y\_L\^2Y\_R\^2[|a]{}\_1\^2[|a]{}\_2\^2 ) where $Y_R,Y_L$ are the (integer) hypercharges of right-handed fermions (on matter curve $\Sigma_a$) and left-handed ones (on matter curve $\Sigma_b$). Recall that = 2 = 0.16 ; = 0.20 where we have taken into account the hypercharge normalization factor $e_Y=1/ \sqrt{60}$. Note that, as we mentioned before, the wave functions in matter curves $\Sigma_a$ and $\Sigma_b$ in the holomorphic gauge have no dependence on the $U(1)_i$ fluxes, they only depend on the bulk fluxes which go in this case along hypercharge. Here $a_{1,2}$ are the adimensional constants parametrizing the variation of the bulk flux close to the intersection point. Note that in principle these parameters may be different for the three different Yukawa couplings, i.e. there are $a^{U,D,L}_{1,2}$. To get an idea of the size of the Yukawas let us for the moment assume that $a^{U.D,L}_1\simeq a^{U,D,L}_2\simeq a^{U,D,L}$. Note that in this case that we neglect the flux variation for the Higgs matter curve, the Yukawa matrix is strongly dependent on the hypercharge of the quarks and leptons involved in the couplings. Since the maximum value of the quark and lepton hypercharges is $|Y_{max}|=6,4,2$ respectively for leptons and $U$ and $D$-quarks, one expects larger effects for leptons, $U$-quarks and $D$-quarks in that order.
Inserting the values of the hypercharges for the different particles involved in each Yukawa coupling leads to the results in table \[jerarsm1\] for the diagonal Yukawas.
Yukawa $Y_{33}$ $Y_{22}$ $Y_{11}$
------------- ---------- -------------------------------- -------------------------------
$ Y^U $ 1 $0.7 (a^U)^2 \times 10^{-3}$ $2.2 (a^U)^4\times 10^{-5}$
$Y^U$(exp) 1 $(3-4)\times 10^{-3}$ $(0.5-1.6)\times 10^{-5}$
$Y^L$ 1 $1.1 (a^L)^2 \times 10^{-3}$ $1.1 (a^L)^4 \times 10^{-4}$
$Y^L$(exp) 1 $5.9\times 10^{-2}$ $2.8\times 10^{-4}$
$Y^D$ 1 $0.6 (a^D)^2 \times 10^{-3}$ $ 3.2 (a^D)^4\times 10^{-6}$
$Y^D$ (exp) 1 $(1-3)\times 10^{-2}$ $(0.6-1.8)\times 10^{-3} $
: Hierarchies of fermion masses from the flux expansion. First order in derivatives.[]{data-label="jerarsm1"}
The couplings are normalized to the one of the corresponding third generation particle. We also show for comparison experimental results for that hierarchy evaluated at the electroweak scale from [@fritzsch]. The results for the $U$-quarks hierarchies are encouraging, for values $a^U,a^L \simeq 1$ one can describe reasonably well the observed pattern. For the case of charged leptons the mass of the electron is again well described for $a^L\simeq 1$. However, the mass of the muon would turn out too light unless $a^L\simeq 7.3$, which would be quite large and incompatible with the $a^L\simeq 1$ required for the electron. Thus, the correct numerical description would require some further contribution for the muon. Alternatively, it could be that for charged leptons neglecting the flux variation coming from the Higgs matter curve is not the correct assumption. For the case of the the $D$-quark hierarchies one would need slightly large values $a^D\simeq 5.8$ and $a^D\simeq 4$ for $Y_{22}$ and $Y_{11}$ respectively.
Let us now explore what would be the effect of higher order terms in the derivatives of the fluxes. If we consider second order in derivatives there are extra corrections which may be extracted from appendix \[b:f2\]. We will again set to zero all flux parameters from the curve $\Sigma_c$. The corresponding diagonal terms are found to be Y\_[22]{} & & \^2 0.18( Y\_R[|b]{}\_1+ Y\_L[|b]{}\_2) \[fluxsm2\]\
Y\_[11]{} & & \^2\^4 (-0.23 ( Y\_R\^2 [|b]{}\_1\^2+Y\_L\^2[|b]{}\_2\^2) +0.29 Y\_LY\_R[|b]{}\_1[|b]{}\_2 ) Again taking $b^{U,D,L}_1\simeq b^{U,D,L}_2\simeq b^{U,D,L}$ yields contributions as in table \[jerarsm2\].
Yukawa $Y_{33}$ $Y_{22}$ $Y_{11}$
--------- ---------- ---------------------------- ------------------------------
$ Y^U $ 1 $ 3.4 (b^U)\times 10^{-3}$ $2.1 (b^U)^2 \times 10^{-4}$
$Y^L$ 1 $3.4 (b^L) \times 10^{-3}$ $6.4(b^L)^2 \times 10^{-4}$
$Y^D$ 1 $3.4 (b^D) \times 10^{-3}$ $2.3(b^D)^2 \times 10^{-5}$
: Hierarchies of fermion masses from the flux expansion. Second order in derivatives.[]{data-label="jerarsm2"}
Note that here the $Y_{22}$ entries have the same structure because in the three cases $Y_R+Y_L=\pm3$ (we are ignoring the overall sign of the contribution which is not relevant for this estimate). As expected, the corrections to the Yukawa couplings are always higher than those coming from only first derivatives. This is true even for the leptons, which have the highest maximal hypercharge and hence get the largest contribution to first order in derivatives. In fact, to avoid too large $Y_{11}^{U,L}$ values one rather needs $b^U,b^L < 1$. On the other hand, the contribution to $Y_{22}^L$ is still too small. The same happens with the $D$-quarks, one would need $b^D\simeq 5.6$ to reproduce the observed $D$-quark mass hierarchies, so that strong variation is again required for $D$-quarks. Terms of order 3 and 4 in flux derivatives could also add to the relatively large values of the $D$-quarks. Equation (\[yukder411\]) shows that the expected contribution is of order $Y_{11}\simeq \e\eta^4(d\hat q+d'\hat q') \simeq 2.6 d \times 10^{-4}$, which reproduces the $D$-quark mass result for a flux parameter $d\simeq 2.4$. Hence, if we do not want to rely on relatively large flux parameters the case of $D$-quarks requires substantial input from higher orders in the derivative expansion, up to order four.
In [@hv3] it was pointed out that the flux expansion to first order in derivatives gives a good explanation of the hierarchies observed for leptons and $U$-quarks but terms coming from the higher derivative flux expansion were needed in order to describe the hierarchies for $D$-quarks. It was also suggested that a possible reason for this different behavior could arise from the fact that leptons and $U$-quarks have higher maximal hypercharge than the $D$-quarks. We indeed find that the hierarchies for $U$-quarks may be quite well described by first order flux variations of order one. The resulting electron mass is also of the correct order. However, the dependence on hypercharge does not seem to explain the different behavior of $L$ and $U$ compared to $D$ fermions. In particular, higher derivative terms always generically dominate over the first order terms, even taking into account the hypercharge dependence. The milder behavior of the $D$-quark hierarchies can be understood either by assuming a relatively strong first/second order flux variation (i.e. $a^D\simeq 5.8$ or $b^D\simeq 5.6$) or larger 4th order contributions with $d\simeq 2.4$. The muon has the tendency to come out too light which may indicate that neglecting flux variation in the Higgs matter curve could perhaps be inappropriate for the leptons and possibly for the $D$-quark matter curves.
If we assume that $U$-quarks get their Yukawas already at first order in derivatives (eq.(\[flx2\])) and on the contrary the $D$-quarks need a dominant contribution at order two or higher (eq.(\[der22\]) or eq.(\[DER\]), it does not matter for this approximation), we can also give an estimate of the CKM mixing matrix [@hv3]. Indeed in this case the respective mass squared matrices will be proportional to Y\^[U]{}(Y\^U)\^ \~(
[ccc]{} \^4\^4 & \^3\^3 & \^2 \^2\
\^3 \^3 & \^2 \^2 &\
\^2 \^2 & & 1
) ; Y\^[D]{}(Y\^D)\^ \~(
[ccc]{} \^2\^4 & \^2\^3 & \^2\
\^2 \^3 & \^2 \^2 &\
\^2 & & 1
) \[upydown\] Then, as in [@fn], one can estimate the matrices $V^{U,D}$ which diagonalize each of them V\^U \~(
[ccc]{} 1 & & \^2 \^2\
& 1 &\
\^2 \^2 & & 1
) ; V\^D \~(
[ccc]{} 1 & & \^2\
& 1 &\
\^2 & & 1
) The CKM matrix, $V^{CKM} \simeq V^U(V^D)^{\dagger}$, then turns out to be V\^[CKM]{} & & (
[ccc]{} 1 & & \^2\
& 1 &\
\^2 & & 1
) (
[ccc]{} 1 & \_G\^[1/2]{} & 2\_Y\^[1/2]{}\_G\
\_G\^[1/2]{} & 1 & 2\_Y\^[1/2]{} \_G\^[1/2]{}\
2\_Y\^[1/2]{}\_G & 2\_Y\^[1/2]{}\_G\^[1/2]{} & 1
)\
& & (
[ccc]{} 1 & 0.20 & 0.006\
0.20 & 1 & 0.03\
0.006 & 0.03 & 1
) \[CKM\] which is in reasonable agreement with experiment. This structure is similar to that found in [@hv3], although in comparison, in the above formula the separate dependence on the hypercharge flux is explicit and the 3rd generation mixing is slightly smaller.
As a general conclusion, in this simplified scheme in which we have set the flux variation in the third curve to zero, one can reproduce the general pattern of quark and lepton hierarchies as well as quark mixing, for reasonable choices of flux variation parameters. This is particularly the case for the $U$-quarks and the electron. Nevertheless, a more complete numerical study, not neglecting flux parameters of the Higgs matter curve, may be required to get full agreement. The order of magnitude estimates for the CKM matrix are on the other hand quite promising. We leave a more detailed phenomenological analysis of this framework for future work.
Final comments
==============
In this paper we have studied the local structure of zero mode wave functions of chiral matter fields in F-theory compactifications. We have solved the relevant differential equations for the zero modes which were derived from local Higgssing in the world-volume effective action of the F-theory 7-branes [@bhv1]. These wave functions have a Gaussian profile centered on the matter curves and become distorted in the presence of $U(1)$ fluxes both on the bulk and on the matter curves themselves. In our approach we first write the fluxes in a power series of the local coordinates and then make a perturbative expansion of the wave functions in powers of the flux coefficients. In this way we obtain expressions which may then be applied to compute physical quantities of interest. In this paper we have concentrated on the calculation of Yukawa couplings but the wave functions could also help to examine other problems. For instance, they could be used to explore the effects of closed string fluxes and warping on the effective action, which could prove important in relation to compactifications with broken supersymmetry.
With the wave functions at our disposal we have computed Yukawa couplings by performing explicitly the overlap integrals of the three wave functions linked to fermions and the Higgs field. By choosing an appropriate gauge, the wave functions of quark or lepton generations are shown to depend only on the bulk fluxes but not on the extra $U(1)$’s associated to the unfolding of the singularities. For example, in the case of a $SU(5)$ F-theory GUT broken to the SM by hypercharge flux, the effective distortion of the wave function depends on the hypercharge of the specific particle considered. The Yukawa integrals can be done analytically and in appendix B we provide the leading terms in the flux expansion. One interesting fact we find is that for a constant non-localized Higgs wave function, presumably corresponding to a Higgs field living on the bulk of the base $S$, the flux distortion cancels in such a way that the possible Yukawa matrices remain of rank one. On the other hand, when the three wave functions are localized, corresponding to three intersecting matter curves, a non-constant $U(1)$ flux gives rise naturally to a hierarchy of Yukawa couplings as first pointed out in [@hv3].
We have applied our findings to the understanding of the observed hierarchies of quark and lepton masses and mixings. In a simplified situation in which the flux variation in the Higgs matter curve is negligible we obtain explicit compact formulas for Yukawa couplings as a function of flux parameters and the charges of the bulk $U(1)$. In a $SU(5)$ setting broken to the SM by hypercharge flux, the resulting Yukawa couplings depend on different powers of the hypercharge of each quark and lepton. It turns out that reasonable values of flux parameters, involving only a first derivative expansion of the fluxes, can account for the hierarchical structure of the masses of $U$-quarks and the electron. The explanation of $D$-quark hierarchies seems to require larger contributions from the higher order terms in the flux derivative expansion. A reasonable semiquantitative understanding of the CKM matrix is then obtained somewhat analogous to the results in [@hv3].
The natural appearance of hierarchies for masses and mixings looks quite promising. However, a full explanation of the data would require a more detailed phenomenological analysis. In particular in the numerical estimations we assumed weakly varying fluxes in the Higgs matter curve, which needs not necessarily be the case. Furthermore, we also took flux variations of the same order for the matter curves corresponding to left- and right-handed fermions, which again is suggestive but not generally true. We think that our explicit formulas are a good starting point for a more thorough investigation which we plan to carry out elsewhere [@afi].
Another interesting topic to address is the origin and structure of neutrino masses, which seem to follow a pattern quite distinct from that of quarks. Here the crucial point is the nature and origin of the mass of right-handed neutrinos. We think that our results will also be useful in this case. More generally, $U(1)$ fluxes may have meaningful implications for other physical issues such as supersymmetry breaking. As an example, in [@aci] it was proposed that in F-theory or type IIB orientifolds, local volume modulus dominance of supersymmetry breaking gives rise to a very predictive pattern of soft terms consistent with radiative electroweak symmetry breaking. It was also pointed out that the presence of $U(1)$ fluxes affects in a small but significant way the values of the soft terms and that these flux contributions could be needed in fact in order to obtain the proper amount of neutralino dark matter. Corrections coming from hypercharge fluxes could also play an important role in the detailed understanding of gauge coupling unification [@dw1; @blumen]. It thus appears that the distortion caused by fluxes could be indeed important in several physical issues in F-theory unification.
[**Acknowledgments**]{}\
We thank F. Marchesano, A. Uranga and S. Theisen for useful advice. A.F. acknowledges a research grant No. PI-03-007127-2008 from CDCH-UCV, as well as hospitality and support from the Instituto de Física Teórica UAM/CSIC, and the Max-Planck-Institut für Gravitationsphysik, during completion of this paper. L.E.I. thanks the PH-TH Division at CERN for hospitality while writing up this paper. This work has been supported by the CICYT (Spain) under project FPA2006-01105, the Comunidad de Madrid under project HEPHACOS P-ESP-00346 and the Ingenio 2010 CONSOLIDER program CPAN.
[**Note added**]{}
The Yukawa couplings among fields on curves $\Sigma_a$, $\Sigma_b$ and $\Sigma_c$ arise from the superpotential term W\_Y = M\_\*\^4\_[ S]{} ( [A\_a]{} ) + [cyclic permutations]{} \[superpot2\] where $\pmb A$ and $\pmb \Phi$ are chiral superfields given in (\[superfi\]). It is enough to focus on $\theta\theta$ terms involving two fermions and one scalar. The three families of quark and leptons are taken to reside in curves $\Sigma_a$ and $\Sigma_b$ while the Higgs lives on $\Sigma_c$. Then, neglecting an overall constant, the coupling is given by Y\_[ij]{} = \_S Note that the Yukawa computations in the main text of the paper involve only the contribution from the first two terms. On the other hand, in a fully symmetric local interaction the additional four terms from cyclic permutations should also be included. This has been recently addressed in refs.[@cchv] and [@cp]. In [@cchv] it has been shown that $\tilde Y_{ij}$ does not receive corrections when fluxes are turned on. We wish to stress that there is a delicate cancellation among the six contributions in eq.(\[fullyuk\]), each term being in general flux dependent. This happens independently of whether or not the field strengths satisfy the BPS condition $\omega \wedge F =0$.
It is instructive to consider the example of constant fluxes. In this case it can be exactly shown that each non-trivial term in (\[fullyuk\]) separately gives a flux dependent contribution to the third generation coupling $\tilde Y_{33}$, but the full coupling is flux independent. As in [@cchv], using our notation, we turn a gauge field along the $Q_1$ and $Q_2$ directions given by $A=A_a Q_1 + A_b Q_2$. Furthermore, we choose A\_a=A\_b=-iM |z\_1 dz\_1 - iN |z\_2 dz\_2 + [c.c.]{} \[cfluxab\] Notice that the gauge field acting on $\Sigma_c$ is $A_c=-(A_a+A_b)$. The resulting field strength satisfies the BPS condition provided $M+N=0$.
The zero modes on each curve follow from the results in appendix A. We find \_a & : & \_a = f\_a(z\_2) e\^[-\_a |z\_1|\^2]{} ; \_a=-M + 1[v]{}\
\_b & : & \_b= f\_b(z\_1) e\^[-\_b |z\_2|\^2]{} ; \_b=-N + 1[v]{} \[zmcf\]\
\_c & : & \_c = e\^[-\_c |w|\^2]{} e\^[w|u]{} ; where $\xi= \frac{\lambda_c(N-M)}{(\lambda_c-M-N)}$. Here we have already set $f_c=1$ in the curve $\Sigma_c$ that is taken to host the Higgs. Also, we will take $f_a(z_2)=z_2^{3-i}$ and $f_b(z_1)=z_1^{3-j}$. Since we are working in the holomorphic gauge, from the zero mode equations (\[hatflux\]), we further have $\psi_{a\bar\jmath}=v\bar\partial_{\bar\jmath}\varphi_a/z_1$, $\psi_{b\bar\jmath}=v\bar\partial_{\bar\jmath}\varphi_b/z_2$ and $\psi_{c\bar\jmath}=-v\bar\partial_{\bar\jmath}\varphi_c/w$, where we have dropped the family index to ease notation. Observe that in the example of constant fluxes these expressions lead to simple results such as $\psi_{a\bar 1}=-v\lambda_a \varphi_a$, $\psi_{a\bar 2}=0$, and so on, so that only three of the terms in (\[fullyuk\]) are not zero. It is straightforward to show that the coupling vanishes except when $i=j=3$, and that Y\_[33]{}= v\^2\_S d\^2z\_1 d\^2 z\_2 \_a \_b \_c \[ty33\] where the $\varphi_I$ are given in (\[zmcf\]). Evaluation of the Gaussian integral yields \_S d\^2z\_1 d\^2 z\_2 \_a \_b \_c = \[iy33\] Therefore, $\tilde Y_{33}=\pi^2v^2$, independent of fluxes. However, notice that each separate term in (\[ty33\]) depends on fluxes even if the BPS condition $M+N=0$ is satisfied.
In the example of [@cchv], in which the BPS condition $\omega \wedge F=0$ is satisfied, it also happens that the flux effects on the couplings only cancel when all terms in (\[ty33\]) are included. On the other hand, in the setup of this article, in which $\omega \wedge F=0$ is not enforced, nonetheless it can be checked that when all terms in (\[ty33\]) are added only the coupling $\tilde Y_{33}$ survives and is flux independent. In [@cp] the sum of all contributions to the couplings has also been taken into account.
In [@cchv] it was proved that the cancellation of flux effects in the full coupling $\tilde Y_{ij}$ follows from an exact residue formula. For a pedestrian derivation of this formula we start from (\[fullyuk\]) and manipulate the integrand to write it as a sum of total derivatives. To this purpose, following [@cchv], we write the zero modes $\psi_{I\bar \jmath}$ and $\varphi_I$, which satisfy the last two equations in (\[hatflux\]), as $\psi_{I\bar \jmath}=\bar\partial_{\bar \jmath}\, \xi_I$, together with \_a = \_a + h\_a ; \_b = \_b + h\_b ; \_c = - \_c + h\_c \[phixi\] The functions $h_I$ are holomorphic and correspond to $\varphi_I\big|_{\Sigma_I}$. An elementary calculation, dropping family indices to simplify, then shows that Y & = & \_S d\^2z\_1 d\^2 z\_2 { |\_[|1]{}- |\_[|2]{}}\
& + & \_S d\^2z\_1 d\^2 z\_2 { |\_[|1]{}( h\_c \_a |\_[|2]{} \_b ) - |\_[|2]{}(h\_c \_a |\_[|1]{} \_b ) } \[presi\] The integrals in the first line can be evaluated by parts, and then the boundary terms are seen to vanish because the zero modes $\varphi_a$ and $\varphi_b$ are localized. In the second line, integrating by parts twice, using $\xi_a=v(\varphi_a - h_a)/z_1$, similarly for $\xi_b$, and invoking localization, gives the final residue formula $\tilde Y \sim {\rm Res}\left(\frac{h_a h_b h_c}{z_1z_2}\right)$ [@cchv].
The computation of Yukawa couplings just described is purely local. If the symmetry among the cyclic permutations in eq.(\[fullyuk\]) still remains after a global completion of the theory, only one generation acquires a Yukawa coupling. In this case the observed hierarchy of fermion masses cannot be generated just by turning on magnetic fluxes, some additional ingredient, e.g. non-perturbative effects, should also be at work to produce these mass hierarchies.
Fluxed zero modes and wave functions {#a:gensols}
====================================
In this appendix we study the solutions of the zero mode equations (\[withflux\]) both for constant and variable field strengths. We will explicitly consider the curves $\Sigma_a$ and $\Sigma_c$. As in the fluxless case, the results for $\Sigma_a$ and $\Sigma_b$ are completely similar, but the curve $\Sigma_c$ must be treated separately.
We find it convenient to rewrite the total gauge potential as = + d\[gtrans1\] in such a way that $\hat \ca_{\bar 1}=\hat \ca_{\bar 2}=0$. We can then work in this ‘holomorphic’ gauge where the potential is just $\hat \ca$ and the corresponding fermions are denoted $\hat \chi$ and $\hat \psi$. The advantage is that the equations reduce to (\_2 - i\_2) \_[|2]{} + (\_1 - i\_1)\_[|1]{} - m\^2(\_1 q\_1 + \_2 q\_2) & = & 0\
|\_[|1]{} - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|1]{} & = & 0 \[hatflux\]\
|\_[|2]{} - m\^2(z\_1 q\_1 + z\_2 q\_2) \_[|2]{} & = & 0 and the gauge fields do not appear in the last two equations. The further constraint $\bar\partial_{\ca} \psi=0$ becomes $\bar\partial \hat\psi=0$ and is automatically verified on account of the last two equations above.
The solutions for the original flux are recovered by performing a gauge transformation, namely = e\^[i]{}; \_[|1]{} = e\^[i]{}\_[|1]{} ; \_[|2]{} = e\^[i]{}\_[|2]{} \[osols\] To compute Yukawa couplings it suffices to work with the hatted fields because the couplings are gauge invariant.
Constant flux {#a:cflux}
-------------
From the total gauge potential given in (\[tpotc\]) it follows that the transformed potential and gauge function are = -2i\_1 dz\_1 - 2i\_2 dz\_2 ; =i(|z\_1|\^2 + |z\_2|\^2) \[tpotcnew\] We then need to find the solutions of (\[hatflux\]) when $\hat \ca_1=-2i\cam \bz_1$ and $\hat \ca_2=-2i\cn \bz_2$. The charges $(q_1, q_2)$ that must also be specified depend on the curve.
Notice that in this case $\cam=qM$ and $\cn=qN+e_1N^{(1)}$, where $(M,N)$ come from the bulk flux and $N^{(1)}$ from the flux along the curve. As in the fluxless case, we find that $\hat \psi_{\bar 2}=0$, which then implies $\bar \partial_{\bar 2} \hat \chi=0$. We make the Ansatz = f(z\_2) e\^[-\_1 |z\_1|\^2]{} \[hatchi1\] The equation $\bar \partial_{\bar 1}\hat \chi=z_1\hat \psi_{\bar 1}$ then fixes \_[|1]{} = - \[hatpsi1\] There is still an equation that requires $\lambda_1$ to satisfy \_1\^2 + 2\_1 - e\_1\^2m\^4 = 0 \[eql1\] To have localized solutions we choose the root \_1 = -+ e\_1m\^2 \[l1exp\] which reduces to $\lambda_1=e_1 m^2$ when $\cam =0$. Inserting in (\[hatchi1\]) and (\[hatpsi1\]) gives the solutions found in [@hv3] in a different gauge.
In the fluxless case we saw that to solve the equations it is convenient to set $e_1=e_2=e$, and to use the variables $w=(z_1+z_2)$ and $u=(z_1-z_2)$, together with the redefined fermions $\psi_{\bar w}=(\psi_{\bar 1} + \psi_{\bar 2})/2$, and $\psi_{\bar u}=(\psi_{\bar 1} - \psi_{\bar 2})/2$.
The gauge potential is still formally given by (\[tpotcnew\]) but now $\cam=(qM -e_2M^{(2)})$ and $\cn=(qN-e_1N^{(1)})$. In the new variables the non-vanishing components of $\hat \ca$ are \_w & = & - i |w - i(-) |u\
\_u & = & - i |u - i(-) |w \[hpot3\] where $\Delta=(\cam+\cn)/2$. In the gauge $\hat \ca$, the zero mode equations imply that the $\hat \psi$ fermions neatly depend on $\hat \chi$ as \_[|w]{} = - 1[em\^2w]{} |\_[|w]{} ; \_[|u]{} = - 1[em\^2w]{} |\_[|u]{} \[psichi\] In turn $\hat \chi$ can be determined from the remaining equation (\_w - i\_w) \_[|w]{} + (\_u - i\_u)\_[|u]{} + 12 e m\^2 |w = 0 \[hatchieq\] To solve we make the Ansatz = h(u+w) e\^[-\_3 |w|\^2]{} e\^[w |u]{} \[chiansatz\] where $h(u+\gamma w)$ is a holomorphic function of its argument. It then follows \_[|w]{} = ; \_[|u]{} = - \[psichi2\] Substituting in (\[hatchieq\]) determines the unknown constants. We find = ; = \[xieps\] Finally, $\lambda_3$ is a positive root of the cubic equation \_3(\_3 + )(\_3 + 2-) - 12 e\^2 m\^4 (\_3 + ) = 0 \[eql3\] When $\cam=\cn=0$ we recover the fluxless solution with $\xi=\gamma=0$, and $\lambda_3=em^2/\sqrt{2}$. In the special cases $\Delta=0$ ($\cn=-\cam$) and $\Delta=\cam$ ($\cn=\cam$) the cubic becomes quadratic and the positive root is easily identified.
Variable flux {#a:vflux}
-------------
We consider the quadratic flux given in (\[vflux\]). The corresponding transformed potential and the gauge function turn out to be \_1 & = & -2i\_1 - 2i(|\_1\_1\^2 + 2\_1z\_1\_1) - 2i(|\_1 \_1\^3 + 3\_1\_1 z\_1\^2)\
\_2 & = & -2i\_2 - 2i(|\_2\_2\^2 + 2\_2z\_2\_2) - 2i(|\_2 \_2\^3 + 3\_2\_2 z\_2\^2) \[vpothat\]\
& = & i|z\_1|\^2\
& + & i|z\_2|\^2 As described below for particular curves, we have only been able to obtain zero mode solutions in a perturbative expansion in the flux parameters.
As in the constant flux case we find $\hat \psi_{\bar 2}=0$ which implies $\bar \partial_{\bar 2} \hat \chi=0$. On the other hand, $\hat \psi_{\bar 1}=\frac{v}{z_1} \bar\partial_{\bar 1}\hat\chi$, where $v=1/e_1m^2$. There is still an equation (\_1 - i\_1) \_[|1]{} - e\_1 m\^2 |z\_1 = 0 \[hatchieq2\] with $\hat \ca_1$ given in (\[vpothat\]). We have found a solution $\hat\chi = \sum_{I=0} \hat \chi^{(I)}$, where $\hat \chi^{(I)}$ is of order $I$ in the flux coefficients. There is a corresponding expansion for $\hat \psi_{\bar 1}$ with $\hat \psi_{\bar 1}^{(I)}=\frac{v}{z_1} \bar\partial_{\bar 1}\hat\chi^{(I)}$.
The zeroth order solutions are the fluxless ones presented in section \[ss:noflux\]. They are \^[(0)]{} = f(z\_2) e\^[-e\_1 m\^2|z\_1|\^2]{} ; \_[|1]{}\^[(0)]{} = - f(z\_2) e\^[-e\_1 m\^2|z\_1|\^2]{} \[s1zeroth\] The expansion of $\hat\chi$ to second order turns out to be = \^[(0)]{} { 1 + v\^3 H\_[23]{} + v\^2 H\_[22]{} + v (H\_[21]{} + H\_[11]{}) + H\_[10]{} + 12 H\_[10]{}\^2 + } \[chi12\] where $v=1/e_1m^2$ is the volume defined before. The auxiliary functions are given by H\_[10]{} & = & z\_1 \_1 \[h10def\]\
H\_[11]{} & = & 43 \_1 z\_1 + 32\_1 z\_1\^2 \[h11def\]\
H\_[21]{} & = & -z\_1\_1\[h21def\]\
H\_[22]{} & = & z\_1\
& & + 2[15]{}z\_1\_1 (2|\_1\_1 z\_1 - \_1|\_1 \_1 ) \[h22def\]\
H\_[23]{} & = & 4[15]{}|\_1\_1 z\_1 \[h23def\] It can be checked that when $\a_1=\b_1=0$, the results match those of section \[a:cflux\] to second order in $\cam$.
The expansion of the wave function $\hat \psi_{\bar 1}$ needed to compute Yukawa couplings follows from $\hat \psi_{\bar 1}=\frac{v}{z_1} \bar\partial_{\bar 1}\hat\chi$. We obtain \_[|1]{} = \_[|1]{}\^[(0)]{} { 1 + v\^3 H\_[23]{}\^\* + v\^2 K\_[22]{} - v(+ H\_[11]{}\^\* + K\_[21]{}) + H\_[10]{} + 12 H\_[10]{}\^2 + } \[psi12\] with the additional definitions K\_[22]{} & = & 12\^2 + \_1\
&& + 2[15]{}z\_1\_1 (2\_1|\_1 \_1 - |\_1 \_1 z\_1 ) \[k22def\]\
K\_[21]{} & = & z\_1\_1
We need to solve the zero mode equations (\[psichi\]) and (\[hatchieq\]). The gauge potential components $\hat \ca_w$ and $\hat \ca_u$ can be easily found changing to coordinates $w=(z_1+z_2)$ and $u=(z_1-z_2)$ starting from (\[vpothat\]). As before we define $\Delta=(\cam+\cn)/2$. In analogy we also introduce =12(\_1+\_2) ; =12(\_1+\_2) \[delabdefs\] and the corresponding $\bar\delta = \delta^*$ and $\bar\rho=\rho^*$.
To iterate we begin with the zeroth order solutions presented in section \[ss:noflux\], taking $h=1$. They are \^[(0)]{} = e\^[-em\^2|w|\^2/]{} ; \_[|w]{}\^[(0)]{} = e\^[-em\^2|w|\^2/]{} ; \_[|u]{}\^[(0)]{} = 0 \[s3zeroth\] To higher orders we will only report the wave function $\hat\chi$ that enters in Yukawa couplings. To first order we find \^[(1)]{} = \^[(0)]{} { D\_[11]{} + D\_[10]{} } \[chi13\] with functional coefficients given by D\_[10]{} & = & w \[d10def\]\
D\_[11]{} & = & w\[d11def\] We have also computed the second order correction to $\hat \chi$. We refrain from presenting it because it involves too many terms.
Yukawa couplings {#b:yukresu}
================
The purpose of this appendix is to provide the explicit expressions for the Yukawa couplings $Y_{ij}$ obtained upon performing the overlap integral of the localized wave functions on the curves $\Sigma_I$, $I=a,b,c$. The procedure is to determine the integrand $z_2^{3-i} z_1^{3-j} G_a G_b G_c$, where the $G_I$ are the corrections of the wave functions due to fluxes that were derived in appendix A. The integral with measure (\[measure2\]) is evaluated assuming that the size of the compact manifold $S$ is much larger than the width $v$ of the matter curves. The piece of the integrand that can contribute is a sum of terms $|w|^{2m}|u|^{2n}$ and the integral is easily computed.
Flux expansion, first order in derivatives {#b:f1}
------------------------------------------
To proceed systematically, to begin we consider the field strengths expanded up to linear order. In this way we obtain Yukawa couplings depending only on the parameters $\a_I$ and $\d$ that characterize the first derivative of the total flux acting on the matter curves $\Sigma_I$. We focus on the leading terms for each entry. There are corrections proportional to powers of the zeroth order coefficients, $\cam$ and $\cn$, times powers of the $\a_I$, which are always subleading, i.e. higher order in $\e$. We have normalized with respect to the zeroth order third generation Yukawa coupling $Y_{33}^{(0)}=\pi^2 s$, where $s=\sqrt2-1$. The results are as follows: Y\_[23]{} & = & \[y23a\]\
Y\_[22]{} & = & \[y22a\]\
Y\_[13]{} & = & \[y13a\] Y\_[12]{} & = & \[y12a\]\
Y\_[11]{} & = & \[y11a\] The couplings $Y_{ij}$ satisfy the property Y\_[ij]{}(|\_a, |\_b, |\_c, |) = Y\_[ji]{} (|\_b, |\_a, 2|- |\_c, |) \[dualp\] Then, the $Y_{ij}$ for $i > j$ can be easily found from the above results.
We want to stress that just as $Y_{23}$ given in (\[y23exact\]), all couplings can be computed exactly. The results are given numerically only for ease of presentation. For example, $Y_{22}$ is found to be Y\_[22]{} & = & where $s=\sqrt2-1$ is the parameter in the measure (\[measure2\]).
For completeness we also provide the expansion of $Y_{33}$ to first order order in fluxes, namely Y\_[33]{} = 1 + v \[y33exact\] with $s=\sqrt2-1$. This is the simplest example showing that the corrections vanish when $\chi_c=1$ which implies $s=1$ and $\Delta=0$.
Flux expansion, second order in derivatives {#b:f2}
-------------------------------------------
To second oder in derivatives of the fluxes there are further contributions to the Yukawas with leading terms as follows: Y\_[13]{} & = & \[y13b\]\
Y\_[22]{} & = & \[y22b\]\
Y\_[12]{} & = & \[y12b\]\
Y\_[11]{} & = & \[y11b\] The coupling $Y_{31}$ follows from $Y_{13}$ by exchanging $\ba \leftrightarrow \bb$ and $\bc \leftrightarrow (2\rb - \bc)$. A similar remark applies to $Y_{21}$.
Flux expansion, third and fourth orders in derivatives {#b:f34}
------------------------------------------------------
In section \[ss:yukmat\] we also discuss the effects of third and fourth order derivatives in the fluxes. The modified wave functions needed to calculate the couplings are obtained as explained in \[a:vflux\] but with new terms in the gauge potential because now the components of the total field strength have the additional pieces \_[i|]{}\^[extra]{} = 8i(C\_i z\_i\^3 + |C\_i \_i\^3) + 10i(D\_i z\_i\^4 + |D\_i \_i\^4) \[vfluxcd\] To third order the effective flux acting on the $\Sigma_I$ is characterized by parameters $C_I$ and $\Delta_c$ as discussed in section \[ss:eflux\]. The notation at fourth order is analogous. The couplings that receive new corrections are Y\_[11]{} & = & \[y11exact\]\
Y\_[12]{} & = & \[y12exact\]\
Y\_[21]{} & = & \[y21exact\] where $s=\sqrt2-1$. Observe again that these couplings vanish when $\chi_c=1$. Numeric evaluation gives Y\_[11]{} & = & \[y11c\]\
Y\_[12]{} & = & \[y12c\]\
Y\_[21]{} & = & \[y21c\]
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A. Collinucci, “New F-theory lifts II: Permutation orientifolds and enhanced singularities,” arXiv:0906.0003 \[hep-th\].
J. Marsano, N. Saulina and S. Schafer-Nameki, “Monodromies, Fluxes, and Compact Three-Generation F-theory GUTs,” arXiv:0906.4672 \[hep-th\].
S. H. Katz and C. Vafa, “Matter from geometry,” Nucl. Phys. B [**497**]{} (1997) 146 \[arXiv:hep-th/9606086\].
S. Katz and D. R. Morrison, “Gorenstein Threefold Singularities with Small Resolutions via Invariant Theory for Weyl Groups,” J. Alg. Geom. [**1**]{} (1992) 449, arXiv:alg-geom/9202002.
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L. Aparicio, A. Font and L.E. Ibáñez, in progress.
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[^1]: See however the note added at the end of the paper.
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---
author:
- Arnau Pujol
- Jerome Bobin
- Florent Sureau
- Axel Guinot
- 'Martin Kilbinger\'
bibliography:
- 'biblist.bib'
date: 'Received date / Accepted date'
title: ' Shear measurement bias II: a fast machine learning calibration method '
---
Acknowledgements {#acknowledgements .unnumbered}
================
AP, FS and JB acknowledge support from a European Research Council Starting Grant (LENA-678282). This work has been supported by MINECO grants AYA2015-71825, PGC2018-102021-B-100 and Juan de la Cierva fellowship, LACEGAL and EWC Marie Sklodowska-Curie grant No 734374 and no.776247 with ERDF funds from the EU Horizon 2020 Programme. IEEC is partially funded by the CERCA program of the Generalitat de Catalunya.
|
---
author:
- Nigel Hitchin
title: Higgs bundles and diffeomorphism groups
---
Introduction
============
The prime motivation for this paper is an attempt to find geometric structures on a compact surface $\Sigma$ of genus $g>1$ whose moduli space is described by the higher Teichmüller spaces introduced in [@Hit1]. These are distinguished components of the space of flat $SL(n,{\mathbf{R}})$-connections on $\Sigma$. Each one is diffeomorphic to ${\mathbf{R}}^N$ where $N=2(n^2-1)(g-1)$ and for $n=2$ it is concretely identifiable as Teichmüller space, the moduli space of hyperbolic structures on $\Sigma$. In some ways this goal has already been achieved by others [@GW],[@Lab] but in a language far removed from the differential geometry of metrics of constant curvature. We shall here produce a conjectural solution to this question, not for $SL(n,{\mathbf{R}})$ for finite $n$, but for what can formally be considered as $SL(\infty,{\mathbf{R}})$. The finite rank cases may then be thought of as quantizations of a classical piece of geometry.
The original discovery of the higher Teichmüller spaces used the theory of Higgs bundles, which requires the introduction of a complex structure on $\Sigma$, and an $SO(n)$-bundle with connection $A$. The associated rank $n$ vector bundle $E$ has a holomorphic orthogonal structure and we also require a Higgs field $\Phi$, a holomorphic section of the bundle $\End E\otimes K$, where $K$ is the canonical bundle, which is symmetric with respect to the orthogonal structure. A solution to the equations $F_A+[\Phi,\Phi^*]=0$ defines a flat connection $\nabla_A+\Phi+\Phi^*$ whose holonomy is in $SL(n,{\mathbf{R}})$ and a special choice of the holomorphic data $(E,\Phi)$ produces the higher Teichmüller space. Our approach is to use this theory replacing gauge groups by groups of diffeomorphisms.
Within this context, we may not know what meaning to attach to $SL(\infty,{\mathbf{R}})$, but the compact group $SU(n)$ has a well-studied infinite-dimensional version $SU(\infty)$ which is defined to be the symplectic diffeomorphisms of the 2-sphere. We can then introduce $SO(\infty)$ as the subgroup which commutes with reflection in an equator and use Higgs bundle data without worrying about what a flat $SL(\infty,{\mathbf{R}})$-connection means.
Taking the action of $SO(\infty)$ on the sphere, we replace the principal bundle by an $S^2$-bundle over $\Sigma$ and with this formalism a solution to the Higgs bundle equations analogous to those used to define the higher Teichmüller spaces defines an incomplete 4-dimensional hyperkähler metric on a disc bundle inside the sphere bundle. It acquires a singularity, a [*fold*]{}, on the boundary circle bundle, and extends as a negative-definite hyperkähler metric on the complementary disc bundle, just like the folded Kähler metrics in [@Bay]. In fact in our case an involution interchanges the two disc bundles. The three hyperkähler forms extend as smooth closed 2-forms to the whole sphere bundle.
Each point of the traditional Teichmüller space, in its realization as an $SU(2)$-Higgs bundle, defines such a hyperkähler metric via the symplectic action of $SU(2)$ on the sphere, and in particular the point corresponding to the given complex structure on $\Sigma$. This is called the canonical Higgs bundle and we call its associated hyperkähler metric the canonical model. It has appeared in the mathematics and physics literature several times over the past 30 years, though its global structure has been largely ignored. The sphere bundle in this case is ${{\mathbf {\rm P}}}(1\oplus K)$ and the fold can be identified with the unit circle bundle in the cotangent bundle. We conjecture that there exist deformations of this model, where the boundary of the disc bundle embeds nonquadratically in the cotangent bundle.
There are two existence theorems for the standard Higgs bundle equations: one starts with the holomorphic data of a stable Higgs bundle $(E,\Phi)$, the other starts with an irreducible flat connection. For the higher Teichmüller spaces the first point of view yields a description of the moduli space as a vector space of holomorphic differentials of different degrees on $\Sigma$. The second characterises the holonomy of the flat connection as positive hyperbolic representations of $\pi_1(\Sigma)$ in $SL(n,{\mathbf{R}})$ [@FG]. We expect the existence of infinite-dimensional versions of these for the case of $SU(\infty)$. For the first approach, we may observe that given a folded hyperkähler metric on the disc bundle, then if $\theta$ is the tautological (complex) 1-form on the cotangent bundle, integrating $\theta^m$ against the symplectic form along the fibres yields a holomorphic section of $K^m$ over $\Sigma$. These sections should determine the metric uniquely. For the second approach, we expect the boundary data on the fold, embedded in the cotangent bundle, to play the role of the holonomy of the flat connection.
We produce three pieces of evidence for this conjecture. For a circle bundle in the cotangent bundle, the real and imaginary parts of the holomorphic symplectic form $d\theta$ restrict to the data of a generic pair of closed 2-forms on a 3-manifold. The first result, an idea of O.Biquard, says that if these forms are real analytic, then they determine a [*local*]{} folded hyperkähler extension. The proof uses twistor theory techniques. The challenge then is to find global boundary conditions which will produce an extension to the whole disc bundle.
The second piece of evidence is to consider the examples given by the Higgs bundle version of Teichmüller space itself. We show that the fold in this case is the unit cotangent bundle of the hyperbolic metric defined by the quadratic differential $q$, and not just for the canonical model $q=0$. We then describe the infinitesimal deformation of the corresponding hyperkähler metric obtained by varying $q$ and generalise the resulting formula to an arbitrary differential of degree $m$. The problem, from the first point of view, is then to extend the first order deformations for $m>2$ to genuine ones.
The final evidence is a global one, and based on the observation that for the higher Teichmüller spaces expressed as a sum of vector spaces of differentials the origin is the unique fixed point under the action $\Phi\mapsto e^{i\phi}\Phi$ and this is a standard embedding of the canonical rank 2 Higgs bundle. We prove here analogously that the canonical model is unique among $S^1$-invariant hyperkähler metrics satisfying the folded boundary condition.The proof uses a geometrical reformulation of what has long been known as the $SU(\infty)$-Toda equation, describing 4-dimensional hyperkähler manifolds with this symmetry.
The author wishes to thank Olivier Biquard and Robert Bryant for useful exchanges of ideas; the Academia Sinica in Taipei and the Mathematical Sciences Center, Tsinghua University, Beijing for their hospitality while part of this paper was being written, and ICMAT Madrid and QGM Aarhus for support.
$SU(\infty)$ and $SO(\infty)$
=============================
We first explain how the group $\SDiff(S^2)$ of symplectic diffeomorphisms of the standard $2$-form $\omega$ on the 2-sphere may be described as $SU(\infty)$. Its Lie algebra consists of $C^{\infty}(S^2)$ modulo the constant functions – the Hamiltonian functions for the symplectic vector fields. Equivalently we may consider functions whose integral against $\omega$ is zero. The group $SU(2)$ acts via its quotient $SO(3)$ and breaks up (the $L^2$-completion of) the algebra into an orthogonal sum of irreducible representations ${\bf 3}+{\bf 5}+{\bf 7}+\dots$. Each irreducible representation of $SO(3)$ occurs with multiplicity one.
Now consider the irreducible representation ${\bf n}$ of $SU(2)$. This is a homomorphism $SU(2)\rightarrow SU(n)$ and the Lie algebra of $SU(n)$ under the restricted adjoint action breaks up as ${\bf 3}+{\bf 5}+\dots + {\bf (2n-1)}$. The analogy with $\SDiff(S^2)$ is clear and justifies the notation $SU(\infty)$, although ${\mathfrak{su}}(n)$ does not embed as a Lie subalgebra in ${\mathfrak{su}}(\infty)$: the Poisson bracket does not restrict to the Lie bracket except on the 3-dimensional component ${\mathfrak{su}}(2).$
The image of this homomorphism is the distinguished [*principal*]{} 3-dimensional subgroup for $SU(n)$. Higher Teichmüller spaces exist for all split real forms $G^r\subset G^c$ [@Hit1] and the principal 3-dimensional subgroup for any simple Lie group plays a fundamental role in this construction. Moreover it is the homomorphism of the split groups $SL(2,{\mathbf{R}})\rightarrow G^r$ and the corresponding associated flat connections which maps ordinary Teichmüller space into its higher version.
The group $SU(\infty)$ has several formal attributes in common with a compact Lie group, in particular there are invariant polynomials on the Lie algebra which generalize $p_m(A)=\tr A^m$, namely $$p_m(f)=\int_{S^2}f^m\omega$$ and for $m=2$ this defines a bi-invariant positive definite metric.
The real form $SL(n,{\mathbf{R}})$ is the fixed point set of an involution on $SL(n,{\mathbf{C}})$ and restricting to the maximal compact subgroup $SU(n)$, it fixes the subgroup $O(n)$, with identity component $SO(n)$. Writing the sphere as $x_1^2+x_2^2+x_3^2=1$, and the symplectic form $\omega=2dx_1\wedge dx_2/x_3$ (the induced area form) we define $SO(\infty)$ to be the subgroup which commutes with the involution $\tau(x_1,x_2,x_3)=(x_1,x_2,-x_3)$ and acts on the equatorial circle $x_3=0$ by an orientation-preserving diffeomorphism. The Lie algebra of $SO(\infty)$ is then formally the functions $f$ which are odd with respect to the involution: since $\tau^*\omega=-\omega$ the Hamiltonian vector fields defined by $i_X\omega=df$ are then even and commute with $\tau$. Note that the induced action on the equator gives a homomorphism $SO(\infty)\rightarrow \Diff(S^1)$.
Higgs bundles
=============
We recall here the main features of Higgs bundles for a finite-dimensional Lie group and in particular the choice which yields the higher Teichmüller spaces [@Hit0],[@Hit1],[@Sim],[@Cor]. Given a compact Riemann surface $\Sigma$ and a compact Lie group $G$ with complexification $G^c$, we take a principal $G$-bundle with connection $A$. To this we can associate a principal $G^c$-bundle and the $(0,1)$ component of the connection defines on it a holomorphic structure. A Higgs field is a holomorphic section $\Phi$ of ${{\mathfrak{g}}}\otimes K$ where ${{\mathfrak{g}}}$ denotes the associated Lie algebra bundle and $K$ the canonical line bundle of $\Sigma$. The Higgs bundle equations are $F_A+[\Phi,\Phi^*]=0$ where $F_A$ is the curvature of the connection $A$ and $x\mapsto -x^*$ is the involution on ${{\mathfrak{g}}}\otimes K$ induced by the reduction to the compact real form $G$ of $G^c$.
Given a solution of these equations, the $G^c$-connection with covariant derivative $\nabla_A+\Phi+\Phi^*$ is flat. Conversely, given a flat connection with holonomy a reductive representation $\pi_1(\Sigma)\rightarrow G^c$ there is a reduction of the structure group of the flat principal $G^c$-bundle to $G$ which is defined by a $\pi_1(\Sigma)$-equivariant harmonic map from the universal covering $\tilde \Sigma$ to the symmetric space $G^c/G$. Using this reduction, the flat connection may be written in the above form for a solution of the Higgs bundle equations.
If we want a flat connection which corresponds to a representation into a real form $G^r$ of $G^c$, we must take the $G$-connection to reduce to a maximal compact subgroup $H\subset G^r$ and the Higgs field to lie in ${{\mathfrak{m}}}\otimes K$ where ${{\mathfrak{g}}}={\mathfrak{h}}\oplus{\mathfrak{m}}$. Thus, for $G^r=SL(n,{\mathbf{R}})$ we need an $SO(n)$ connection $A$, or equivalently a rank $n$ vector bundle $E$ with a holomorphic orthogonal structure and $\Lambda^nE$ trivial together with a Higgs field which is symmetric with respect to this structure.
Uniformization of a Riemann surface gives a representation $\pi_1(\Sigma)\rightarrow PSL(2,{\mathbf{R}})$ and a choice of holomorphic square root $K^{1/2}$ of the canonical bundle defines a lift to $SL(2,{\mathbf{R}})$. The corresponding Higgs bundle consists of $E=K^{-1/2}\oplus K^{1/2}$ with the canonical pairing defining the orthogonal structure: $((u,v),(u,v))=\langle u,v\rangle$, and the nilpotent Higgs field $\Phi(u,v)=(v,0)$ is then symmetric. The $SO(2)=U(1)$-connection $A$ is a connection on $K^{1/2}$ which defines a Levi-Civita connection and the Higgs bundle equation $F_A+[\Phi,\Phi^*]=0$ says that the Gaussian curvature is $-4$.
This canonical example can be modified by taking a holomorphic section $q$ of $K^2$ and defining $\Phi(u,v)=(v, qu)$. The Higgs bundle equations are again for a Hermitian metric $h$ but can be interpreted as saying that the metric $$\hat h=q+\left(h+\frac{q\bar q}{h}\right)+\bar q
\label{hyp}$$ has curvature $-4$ [@Hit0]. Then the $3g-3$-dimensional space of quadratic differentials defines Teichmüller space from the Higgs bundle point of view.
For the higher Teichmüller spaces of representations into $SL(n,{\mathbf{R}})$ we take for $n=2m+1$ the vector bundle $$E=K^{-m}\oplus K^{1-m}\oplus \dots \oplus K^{m}$$ and for $n=2m$ $$E=K^{-(2m-1)/2}\oplus K^{1-(2m-1)/2}\oplus \dots \oplus K^{(2m-1)/2}.$$ The pairing of $K^{\pm \ell}$ or $K^{\pm \ell/2}$ defines an orthogonal structure on $E$ and $\Lambda^nE$ is trivial so it has structure group $SO(n,{\mathbf{C}})$.
The Higgs field must be symmetric with respect to this orthogonal structure. We set:
$$\Phi=\begin{pmatrix}0 & 1 & 0 &\dots & & 0\\
a_2 & 0 & 1 & \dots & & 0\\
a_3 & a_2 & 0 & 1& \dots & 0\\
\vdots & & &\ddots & &\vdots \\
a_{n-1}& & & & \ddots & 1\\
a_n& a_{n-1} & \dots & a_3 & a_2& 0
\end{pmatrix}
\label{sym}$$
where $a_i\in H^0(\Sigma, K^i)$.
Then the higher Teichmüller space is the space $H^0(\Sigma,K^2)\oplus H^0(\Sigma,K^3)\oplus \cdots \oplus H^0(\Sigma, K^n)$ of differentials of degree $2$ to $n$. The invariants $\tr \Phi^m\in H^0(\Sigma,K^m)$ are universal polynomials in the entries $a_i$ so that given these differentials we have a natural way of defining a Higgs bundle and correspondingly a flat $SL(n,{\mathbf{R}})$-connection which fills out a connected component in the moduli space. Setting $a_i=0$ for $i>2$ gives the embedding of Teichmüller space.
Fock and Goncharov [@FG] identified this connected component, from the flat connection point of view, with the moduli space of positive hyperbolic representations of $\pi_1(\Sigma)$ in $SL(n,{\mathbf{R}})$ (and did so for all split real forms): each homotopy class is mapped to a matrix with $n$ positive eigenvalues. Their methods are quite distinct from the Higgs bundles which lay behind the original discovery.
Higgs bundles for $SU(\infty)$
==============================
If we replace the compact group $G$ in a Higgs bundle by $SU(\infty)$ then it is equivalent to consider instead of a principal bundle with structure group $\SDiff(S^2)$, its associated 2-sphere bundle $p:M^4\rightarrow \Sigma$. A connection is defined by a horizontal distribution – a rank 2 subbundle $H\subset TM$ which is transverse to the tangent bundle along the fibres $T_F$. Then a horizontal lift of a vector field on $\Sigma$ is a vector field $X$ which integrates to a diffeomorphism taking fibres to fibres. We want this to preserve a symplectic form along the fibres which is a section $\omega_F$ of $\Lambda^2T^*_F$ and so we require ${\mathcal L}_X\omega_F=0$. If $M$ is a symplectic manifold and the fibres are symplectic submanifolds then the symplectic orthogonal to $T_F$ is an example of a horizontal distribution which defines such an $\SDiff(S^2)$-connection. Any $SU(2)$-Higgs bundle defines this structure by virtue of its symplectic action on the sphere.
For a reduction to $SO(\infty)$ we need an involution $\tau$ on $M$ which acts trivially on the base $\Sigma$ and on each fibre is equivalent by a symplectic diffeomorphism to the standard reflection $(x_1,x_2,x_3)\mapsto (x_1,x_2,-x_3)$. It must preserve the horizontal distribution. The fixed point set is a circle bundle $p:N^3\rightarrow \Sigma$ with a horizontal distribution which we can think of as the connection associated to the homomorphism $SO(\infty)\rightarrow \Diff (S^1)$.
Consider the Higgs field for $SU(\infty)$ first as a real object: the Lie algebra bundle ${{\mathfrak{s}}u(\infty)}$ consists of the bundle of smooth functions along the fibres of integral zero. So if $z=x+iy$ is a local coordinate on $\Sigma$ the Higgs field is $\phi_1dx+\phi_2dy$ where $\phi_1,\phi_2$ are local smooth functions on $M$. Then $\Phi=(\phi_1dx+\phi_2dy)^{1,0}$ is to be interpreted as a global section of $p^*K$. If we want an analogue of the Higgs field for a higher Teichmüller space then we want $\phi_1,\phi_2$ to be invariant under the involution, or $\tau^*\Phi= \Phi$.
We now need to interpret the Higgs bundle equations in terms of the geometry of this data. A parallel situation was considered many years ago [@AJS] considering Nahm’s equations for volume-preserving diffeomorphisms of a 3-manifold.
Consider first the connection – the horizontal lift of a vector field on $\Sigma$. Locally $\partial/\partial x, \partial/\partial y$ lift to vector fields $$\frac{\partial}{\partial x}+A_1,\quad \frac{\partial}{\partial y}+A_2$$ where $A_1,A_2$ are Hamiltonian vector fields along the fibres, depending smoothly on $x,y$. The notation is intended to suggest connection forms in a local trivialization. Replacing the vector fields by Hamiltonian functions $a_1,a_2$ and using the Poisson bracket the equation $F_A+[\Phi,\Phi^*]=0$ reads $$(a_2)_x-(a_1)_y+\{a_1,a_2\}+\{\phi_1,\phi_2\}=0.
\label{higgs1}$$ The Cauchy-Riemann equation $\bar\partial_A\Phi=0$ is then $$2(\phi_1+i\phi_2)_{\bar z}+\{a_1+ia_2,\phi_1+i\phi_2\}=0.
\label{higgs2}$$
The horizontal distribution splits the tangent bundle of $M$ as $TM\cong T_F\oplus p^*T\Sigma$ and so the relative symplectic form $\omega_F$, a section of $\Lambda^2T^*_F$, defines a genuine 2-form whose restriction to a fibre is $\omega_F$. Using the local formula above for horizontal vector fields this may be written, with $\omega$ the standard symplectic form on $S^2$ and $x_1,x_2$ local coordinates $$\omega+\{a_1,a_2\}dx\wedge dy-d_Fa_1\wedge dx-d_Fa_2\wedge dy$$ where $d_Fa=a_{x_1}dx_1+a_{x_2}dx_2$. Substituting from the Higgs bundle equation (\[higgs1\]) gives $$\omega-\{\phi_1,\phi_2\}dx\wedge dy-da_1\wedge dx-da_2\wedge dy.$$ Now $\{\phi_1,\phi_2\}dx\wedge dy$ is a well-defined section of $p^*\Lambda^2T^*\Sigma$ and so we may add it to the above form and define on $M$ the closed 2-form locally expressed as $$\omega_1= \omega-da_1\wedge dx-da_2\wedge dy.
\label{w1}$$ Then $$\omega_1^2=-2\omega\wedge((a_2)_x- (a_1)_y) dx\wedge dy-2 d_Fa_1\wedge d_Fa_2\wedge dx\wedge dy$$ and from (\[higgs1\]) this is equal to $2\{\phi_1,\phi_2\}\omega_F\wedge dx\wedge dy$. Hence it is non-degenerate so long as $\{\phi_1,\phi_2\}$ is non-zero. Moreover, in that case, by construction the symplectic orthogonal of $T_F$ is the horizontal distribution.
Now consider the complex 1-form, the Higgs field $\Phi=(\phi_1+i\phi_2)dz$ and its exterior derivative $$\omega^c=d(\phi_1+i\phi_2)\wedge dz.
\label{wc}$$ Then $d(\phi_1+i\phi_2)$ and $dz$ span the $(1,0)$-forms for a complex structure so long as $\omega^c\wedge\bar \omega^c\ne 0$. But $$\omega^c\wedge\bar \omega^c=d_F(\phi_1+i\phi_2)\wedge d_F(\phi_1-i\phi_2) \wedge d\bar z\wedge dz=4\{\phi_1,\phi_2\}\omega_F\wedge dx\wedge dy$$ and so once more, if $\{\phi_1,\phi_2\}$ is non-zero we obtain a complex structure and a closed holomorphic 2-form $\omega^c$.
Finally consider $$\omega_1\wedge \omega^c=\omega\wedge (\phi_1+i\phi_2)_{\bar z}d\bar z\wedge dz-(da_1\wedge dx+da_2\wedge dy)\wedge d(\phi_1+i\phi_2)\wedge dz.$$
From (\[higgs2\]) the first term is $-\omega_F\wedge \{a_1+ia_2,\phi_1+i\phi_2\}idx\wedge dy$. The second term is $$(id_Fa_1\wedge d_F(\phi_1+i\phi_2)-d_Fa_2\wedge d_F(\phi_1+i\phi_2))\wedge dx\wedge dy$$ which can be written as $\{ia_1-a_2,\phi_1+i\phi_2\}\wedge \omega_F\wedge dx\wedge dy$ and both terms together yield $\omega_1\wedge \omega^c=0$.
Writing $\omega^c=\omega_2+i\omega_3$ from $(\omega^c)^2=0$ we have $\omega_2^2=\omega_3^2$ and $\omega_2\wedge \omega_3=0$. From $\omega_1\wedge \omega^c=0$ we have $\omega_1\wedge\omega_2=0=\omega_1\wedge\omega_3$ and with the other calculations we have $$\omega_1^2=\omega_2^2=\omega_3^2=2\{\phi_1,\phi_2\}\omega_F\wedge dx\wedge dy \qquad \omega_1\wedge \omega_2=\omega_2\wedge \omega_3=\omega_3\wedge \omega_1=0.$$ It is a standard fact that, given these equations for closed forms, raising and lowering indices with the symplectic forms and their inverses one can recapture the metric up to a sign and integrable complex structures $I,J,K$ which define a quaternionic structure on the tangent bundle. This means that we have a [*hyperkähler metric*]{} on the open set $\{\phi_1,\phi_2\}\ne 0$.
Any Higgs bundle for the group $SU(2)$ or $SO(3)$ defines such a hyperkähler metric. The Poisson bracket $\{\phi_1,\phi_2\}$ is then the moment map for a section of the Lie algebra bundle and since moment maps for $SO(3)$ are height functions on the 2-sphere the degeneracy locus of the hyperkähler metric contains a metric circle bundle, though there may be other points where $[\Phi,\Phi^*]$ itself vanishes.
Locally, there is nothing exceptional about these metrics. Given a 4-dimensional hyperkähler metric, $\omega^c=\omega_2+i\omega_3$ is a holomorphic 2-form with respect to the complex structure $I$ and one may find local holomorphic coordinates such that $\omega^c=dw\wedge dz$. Then regard $z$ as a projection to a local Riemann surface. The Higgs field $\Phi$ is then $wdz$, and the Kähler form $\omega_1$ for complex structure $I$ defines the $SU(\infty)$-connection by the symplectic orthogonal to the tangent bundle of the fibres.
Folding
=======
Within symplectic geometry there exists a notion of [*folded*]{} structure, based on the geometric notion of folding a piece of paper along the $y$-axis: the smooth map $f:{\mathbf{R}}^2\rightarrow {\mathbf{R}}^2$ defined by $f(x,y)=(x^2,y)$. The standard symplectic form $\omega=dx\wedge dy$ then pulls back to the degenerate form $2xdx\wedge dy$ which is said to be folded along the line $x=0$. In general, a folded symplectic manifold $M^{2m}$ is defined by a closed 2-form $\omega$ such that $\omega^m$ vanishes transversally on a hypersurface $N$ and $\omega$ restricted as a form to $N$ is of maximal rank $2m-2$. There is a Darboux theorem [@CGW] which states that locally around the fold $N$ there are coordinates such that $$\omega= xdx\wedge dy+\sum_1^{m-1}dx_i\wedge dy_i.$$ There is also a Kähler version of this and in [@Bay] Baykur proves the remarkable result that any compact smooth 4-manifold has a folded Kähler structure, the two components of $M\setminus N$ being Stein manifolds. However, the metric changes signature from positive-definite to negative-definite on crossing the fold.
For our 4-manifold above, a 2-sphere bundle over the surface $\Sigma$, we have closed 2-forms $\omega_1,\omega_2,\omega_3$ such that $\omega_1^2=\omega_2^2=\omega_3^2$ and this vanishes when the section of $p^*\Lambda^2T^*\Sigma$ defined by $\{\phi_1,\phi_2\}dx\wedge dy$ vanishes. In the case that the Higgs field is invariant by an involution $\tau$ (the $SO(\infty)$-connection case), the Poisson bracket $\{\phi_1,\phi_2\}$ is anti-invariant and so each term $\omega_i^2$ vanishes on the hypersurface which is the fixed-point set – a circle bundle over $\Sigma$. We could proceed to define a folded hyperkähler 4-manifold by using the same definition as in the symplectic case, but there is an issue.
When $\omega_1^2=0$ at a point $x\in M$ we have the algebraic equations in $\Lambda^2T^*_xM$ $$\omega_1^2=\omega_2^2=\omega_3^2=0 \qquad \omega_1\wedge \omega_2=\omega_2\wedge \omega_3=\omega_3\wedge \omega_1=0.$$ If we assume that the $\omega_i$ are linearly independent at $x$ then they span a 3-dimensional subspace on which the quadratic form given by the exterior product is zero. Now $\omega^2=0$ is the condition for $\omega$ to be decomposable – geometrically it defines a point in the 4-dimensional Klein quadric parametrizing lines in ${{\mathbf {\rm P}}}(T^*_x)$ – and so we have a plane in the quadric. But there are two types: an $\alpha$-plane consists of the lines through a point (and so $\omega_i=\beta_i\wedge\varphi$) or a $\beta$-plane consists of lines in a plane (or $i_X\omega_i=0$ for some tangent vector $X\in T_xM$).
If all three closed forms are folded in the usual sense then we would have $\omega_i=xdx\wedge\alpha_i+\beta_i\wedge \gamma_i$ with $\beta_i\wedge\gamma_i$ non-vanishing when restricted as a form to the fold $x=0$. From the normal form, at $x=0$ we have $i_{\partial/\partial x}(\beta_i\wedge \gamma_i)=0$ and hence we have a $\beta$-plane.
This is not, however the case we are interested in – where the connection is preserved by the involution and the Higgs field is anti-invariant. In this case from (\[w1\]) we have $\tau^*\omega_1=-\omega_1$ and from (\[wc\]), $\tau^*\omega_2=\omega_2, \tau^*\omega_3=\omega_3$. In particular $\omega_1$ must vanish as a form when restricted to the fold and thus is not a folded symplectic structure according to the above definition.
So take local coordinates such that $x=\{\phi_1,\phi_2\}$ then because $\omega_2,\omega_3$ are even we must have to order $x$ $$\omega_2=xdx\wedge\alpha_2+\beta_2\wedge\gamma_2,\quad \omega_3=xdx\wedge\alpha_3+\beta_3\wedge\gamma_3$$ and $$\omega_1=dx\wedge \alpha_1+x\beta_1\wedge \gamma_1$$ where $i_{\partial/\partial x} (\alpha_i,\beta_i,\gamma_i)=0.$ From $\omega_1\wedge \omega_2=0=\omega_1\wedge \omega_3$ we have $\alpha_1\wedge\beta_2\wedge\gamma_2=0=\alpha_1\wedge\beta_3\wedge\gamma_3$ and so we can take $\gamma_2=\gamma_3=\alpha_1=\varphi$. Then at $x=0$ the $\omega_i$ are all divisible by $\varphi$ and we have explicitly an $\alpha$-plane.
Also, $d\omega_1=0$ and so $\omega_1=dx\wedge \varphi+xd\varphi$ to order $x$. Furthermore, since $\omega_1^2=2xdx\wedge \varphi\wedge d\varphi$ and vanishes transversally on $x=0$ we have $\varphi\wedge d\varphi\ne 0$ and so $\varphi$ defines a contact structure on the fold $x=0$.
Relabelling $\beta_2,\beta_3$ as $\eta_1,\eta_2$, the restriction of $\omega_2$ and $\omega_3$ to the fold $N$ gives two closed forms $\eta_1\wedge\varphi,\eta_2\wedge\varphi$ where $\varphi$ is a contact form and $\eta_1\wedge\eta_2\wedge\varphi\ne 0$. This latter condition is equivalent to the linear independence of the 2-forms on $M$ at $x=0$.
Hyperkähler metrics with $\alpha$-type folds have been considered in the physics literature. The simplest uses the Gibbons-Hawking Ansatz $$g= V\sum_{i=1}^3dx_i^2+V^{-1}(d\phi+\alpha)^2$$ where $V(x_1,x_2,x_3)$ and $\alpha=a_1dx_1+a_2dx_2+a_3dx_3$ satisfy $dV=\ast d\alpha$. If we take $V=1/\vert {\mathbf x}+{\mathbf a}\vert + 1/\vert {\mathbf x}-{\mathbf a}\vert $ we get the complete Eguchi-Hanson metric. With a minus sign $V=1/\vert {\mathbf x}+{\mathbf a}\vert - 1/\vert {\mathbf x}-{\mathbf a}\vert $ there is a fold when $V=0$, a 3-manifold given by a circle bundle over the plane through the origin orthogonal to ${\mathbf a}$. There is an involution here covering ${\mathbf x}\mapsto -{\mathbf x}$.
The more general case where $V=\sum_{i=1}^m \pm 1/\vert {\mathbf x}-{\mathbf a_i}\vert$ is folded but the Kähler form which vanishes on the fold varies from point to point. This is like the general $SU(2)$-Higgs bundle rather than those which yield flat $SL(2,{\mathbf{R}})$-connections.
Metrics like these can be utilized to construct non-singular Lorentzian signature 5-manifolds which are asymptotically flat and satisfy the equations of 5-dimensional supergravity [@GWar].
Teichmüller space
==================
The canonical model
-------------------
The example which we propose to generalize is the canonical $SU(2)$-Higgs bundle described in Section 3 which gives the hyperbolic metric of curvature $-4$ with the given conformal structure on $\Sigma$:
$$E=K^{-1/2}\oplus K^{1/2}\qquad \Phi= \begin{pmatrix}0 & 1 \\
0& 0
\end{pmatrix}.$$
From the inclusion $SO(3)\subset SU(\infty)$ this defines a folded hyperkähler metric on the 2-sphere bundle $M={{\mathbf {\rm P}}}(K^{-1/2}\oplus K^{1/2})$. Moreover, since the $SU(2)$-connection reduces to $SO(2)$ and the Higgs field lies in ${{\mathfrak{m}}}\otimes K$ we have the involution $\tau: M\rightarrow M$ defined by $\tau([u,v])=[\bar v h^{-1/2}, \bar u h^{1/2}]$ where $h$, a section of $K\bar K$, is the hyperbolic metric. If we write $M={{\mathbf {\rm P}}}(1\oplus K)$ and $w=v/u$ then the fixed point set is $h^{-1}w\bar w=1$, the unit circle bundle in the cotangent bundle. We have $[\Phi,\Phi^*]=\diag(h,-h)$ and $\{\phi_1,\phi_2\}$ is the moment map for $h(i,-i)$ which vanishes only on the circle bundle which is the fixed point set of $\tau$. The corresponding positive-definite hyperkähler metric is thus defined on the unit disc bundle in the cotangent bundle, and has an $\alpha$-fold on the unit circle bundle boundary.
As we described in Section 4, the Higgs field $\Phi$ can, in the $SU(\infty)$-interpretation, be thought of as a section of $p^*K$ or equivalently as a map $f:M\rightarrow T^*\Sigma$ to the total space of the cotangent bundle in which case we can take $\Phi=f^*\theta$ for $\theta=wdz$ the canonical holomorphic 1-form. In our case $f$ is a diffeomorphism restricted to the open disc bundle.
The canonical Higgs bundle has the property that the map $\Phi\mapsto e^{i\theta}\Phi$, which preserves the equations, takes the solution to a gauge-equivalent one. It follows that the hyperkähler metric is invariant under the action $w\mapsto e^{i\theta}w$, scalar multiplication by a unit complex number on the fibres of $T^*\Sigma$. This metric then fits into the more general result of Feix and Kaledin [@Feix],[@Kal] that a real analytic Kähler metric has a unique $S^1$-invariant hyperkähler extension to a neighbourhood of the zero section of the cotangent bundle, where $\omega_2+i\omega_3$ is the canonical holomorphic symplectic form on the cotangent bundle. This is a local result. The canonical model is thus the unique hyperkähler extension of a metric on $\Sigma$, and because the universal covering is invariant under $PSL(2,{\mathbf{R}})$ this must be a hyperbolic metric.
The corresponding $SO(3)$-invariant metric for the extension of the round metric on $S^2$ is the Eguchi-Hanson metric. This is complete and extends to the whole cotangent bundle. The hyperbolic analogue, which we are considering here, is less well-known but certainly appears in the physics literature in [@Geg],[@Ped]. It was also used by Donaldson [@Don] to investigate the hyperkähler extension of the Weil-Petersson metric on Teichmüller space.
We describe this metric first following [@Ped], but with different coordinates. It will reappear in several forms later on. We use the upper half-plane model with coordinate $z=x+iy$ and the metric $(dx^2+dy^2)/y^2$ (for convenience of formulae we take constant curvature $-1$ not $-4$) and $w=u_1+iu_2$ the fibre coordinate on the cotangent bundle, with $\theta=wdz$ the canonical 1-form. Then $$\omega_1= d(u_3(dx-yd\phi))
\label{om1}$$ where $\phi=\arg w$ and $y^2(u_1^2+u_2^2+u_3^2)=1$. This latter expression (put $x_i=yu_i$) is the equation for the $S^2$-fibre, and if we put $\varphi=dx-yd\phi$ we see that $\omega_1=du_3\wedge\varphi+u_3d\varphi$ making $u_3=0$ the fold. We then have $$\omega_2+i\omega_3=dw\wedge dz=w(dx+idy)\wedge \left(\frac{1}{y(1-y^2u_3^2)}(dy+y^3u_3du_3)-id\phi\right)$$ and to first order in $u_3$ this is $$-wy^2u_3du_3\wedge dz+w\left(\frac{dy}{y}-id\phi\right)\wedge (dx-yd\phi)$$ and we observe that $\varphi=dx-yd\phi$ is a contact form on the unit circle bundle. In fact $d(dx/y)=dx\wedge dy/y^2$ so $d\phi-dx/y$ is a connection form for the circle bundle as a principal $S^1$-bundle.
Restricted as forms to $u_3=0$ we have $\omega_1=0$ and $$\omega_2=\frac{1}{y^2}(\sin \phi dx+\cos \phi dy)\wedge (dx-yd\phi),\quad
\omega_3=\frac{1}{y^2}(-\cos \phi dx+\sin \phi dy)\wedge (dx-yd\phi).$$ The null foliation of $\omega_2$ (which is the real canonical 2-form on $T^*\Sigma$) is tangential to the geodesic flow: in fact explicitly the two equations $\sin \phi \,dx+\cos \phi \,dy=0, dx-yd\phi=0$ give the hyperbolic geodesics $y=c_1\cos\phi, x=c_1\sin\phi+c_2$. We therefore encounter the hyperbolic metric from the data on the fold through its geodesics.
Note that the forms $\omega_2$ and $\omega_3$ are folded in the usual sense, and $\omega_2$ is the Kähler form for the complex structure $J$. With a circle action as above the moment map for $\omega_1$ (which from the above calculation is $-u_3y=-x_3$) is a Kähler potential for $\omega_2$, and it follows that the complex structure $J$ is Stein and fits in with Baykur’s results [@Bay]. By contrast the complex structure $I$ admits the compact holomorphic curve given by the zero section of $K$ and is not Stein.
There is a more general picture which we address next.
Quadratic differentials
-----------------------
The description of Teichmüller space in the theory of Higgs bundles uses the following pair: $$E=K^{-1/2}\oplus K^{1/2}\qquad \Phi= \begin{pmatrix}0 & 1 \\
q& 0
\end{pmatrix}$$ where $q$ is a holomorphic section of $K^2$. The canonical model is the case $q=0$. The metric given by solving the Higgs bundle equations is again a Hermitian metric $h$ on $\Sigma$, respecting the holomorphic orthogonal structure on $E$ and the Higgs field is symmetric, thus from the inclusion $SO(3)\subset SU(\infty)$ this defines a folded hyperkähler metric on the same $S^2$-bundle over $\Sigma$. The actual Higgs bundle equation is $$F=2(1-\vert q\vert^2)\omega
\label{qhiggs}$$ where $\omega$ is the volume form of the metric and $F$ the curvature of $K$. We shall investigate the geometry of the fold in this case.
Locally, we write $q=a(z)dz^2$ and the metric as $h=kdzd\bar z$. Then $(k^{-1/4}dz^{-1/2},k^{1/4}dz^{1/2})$ is a unitary basis and relative to this basis $$\Phi= \begin{pmatrix}0 & k^{1/2} \\
ak^{-1/2}& 0
\end{pmatrix}dz=k^{1/2}\begin{pmatrix}0 & 1 \\
0& 0
\end{pmatrix}dz+ak^{-1/2}\begin{pmatrix}0 & 0 \\
1& 0
\end{pmatrix}dz.$$ and then, as a 1-form with values in Hamiltonian functions we have $$\Phi=(\phi_1+i\phi_2)dz=\frac{1}{2}\left(k^{1/2}(x_1-ix_2)dz+ak^{-1/2}(x_1+ix_2)dz\right)
\label{phiham}$$ where $x_1,x_2$ are standard height functions on the unit sphere with $\{x_1,x_2\}=2x_3$.
If we regard $\Phi$ as a map to $T^*\Sigma$ coordinatized by $(w,z)\mapsto wdz$ then $$x_1-ix_2=2\frac{k^{1/2}w-ak^{-1/2}\bar w}{k-\vert a\vert^2k^{-1}}.$$ The fold $x_3=0$ is then where $x_1^2+x_2^2=1$ which gives the ellipse in the fibre of $T^*\Sigma$ $$\frac{4}{(k-\vert a\vert^2k^{-1})^2} (\bar aw^2+a\bar w^2 +(k+k^{-1}\vert a\vert^2)w\bar w)=-1.$$ Inverting the matrix of coefficients of $w^2,\bar w^2, w\bar w$ this is the unit circle bundle in the cotangent bundle for the metric $$\hat h=adz^2+(k+\vert a\vert^2k^{-1})dzd\bar z+\bar ad\bar z^2.$$ But, as in (\[hyp\]) the Higgs bundle equations imply that this is a metric of constant curvature $-4$. Hence the data on the fold describes the geodesic flow for the hyperbolic metric determined by the quadratic differential $q=adz^2$.
Local existence
===============
We have seen in Section 5 that an $\alpha$-folded hyperkähler 4-manifold with involution induces a contact structure on the fold $N^3$ defined by a form $\varphi$ and two closed 2-forms $\eta_1\wedge\varphi,\eta_2\wedge\varphi$ with $\eta_1\wedge\eta_2\wedge\varphi\ne 0$. (Note that there is an ambiguity $\varphi\mapsto f\varphi, \eta_1\mapsto f^{-1}\eta_1+g_1\varphi, \eta_2\mapsto f^{-1}\eta_2+g_2\varphi$ in the definition of the 1-forms). In fact, for real analytic data this is sufficient to find a local folded hyperkähler metric on $N\times (-\epsilon,\epsilon)$ for some interval $(-\epsilon,\epsilon)$. I owe the idea below to Olivier Biquard who dealt with a similar issue, with CR boundary data, in [@Biq].
To find a hyperkähler metric we shall use the twistor construction [@HKLR]. This means finding a complex 3-manifold $p:Z\rightarrow {{\mathbf {\rm P}}}^1$ fibring over the projective line together with a holomorphic section $\varpi$ of $\Lambda^2T_F^*(2)$, where $T_F$ is the tangent bundle along the fibres and the factor 2 indicates the tensor product with $p^*{\mathcal O}(2)$, the line bundle of degree $2$ on ${{\mathbf {\rm P}}}^1$. We also need an antiholomorphic involution $\sigma$, which covers the antipodal map $\zeta\mapsto -1/\bar\zeta$ on ${{\mathbf {\rm P}}}^1$. This defines a real structure on $Z$ and associated geometrical objects. Then on a family of real sections with normal bundle ${\mathcal O}(1)\oplus {\mathcal O}(1)$ there exists a hyperkähler metric.
Let $X$ be a real analytic 3-manifold with two analytic closed forms $\eta_1\wedge\varphi, \eta_2\wedge\varphi$ such that the annihilator of $\varphi$ is a contact distribution and $\eta_1,\eta_2$ satisfy $\eta_1\wedge\eta_2\wedge \varphi\ne 0$. Then this data defines naturally an $\alpha$-folded hyperkähler metric with involution $(x,t)\mapsto (x, -t)$ on $X\times (-\epsilon,\epsilon)$.
From the real analyticity we complexify locally to a complex manifold $X^c$.
1\. Consider $\eta=(\eta_1+i\eta_2)-\zeta^2(\eta_1-i\eta_2)$ where $\zeta\in {\mathbf{C}}$. On ${{\mathbf {\rm P}}}^1\times X^c$ we can extend this to a section of $T^*X^c(2)$ by defining $\tilde \eta =-(\eta_1-i\eta_2)+\tilde\zeta^{2}(\eta_1+i\eta_2)$ for $\tilde\zeta=\zeta^{-1}$. Then $\tilde\eta=\zeta^2\eta$ and the pair define a global section. This is moreover real under the conjugation map $(\eta,\zeta)\mapsto (-\bar\eta,-1/\bar\zeta)$. Since $\eta$ is even in $\zeta$ it is also invariant under the holomorphic involution $\tau$ on ${{\mathbf {\rm P}}}^1\times X^c$ defined by $(\zeta,x)\mapsto(-\zeta,x)$.
Now $\eta\wedge\varphi$ cannot be zero for taking the exterior product with $\eta_1-i\eta_2$ we have $$(\eta_1+i\eta_2)\wedge (\eta_1-i\eta_2)\wedge \varphi=-2i\eta_1\wedge\eta_2\wedge\varphi\ne 0.$$ It is moreover real since $\varphi$ is real. Its annihilator is a holomorphic 1-dimensional subbundle of $TX^c$ over ${{\mathbf {\rm P}}}^1\times X^c$ and hence describes a foliation by curves.
Locally there is a well-defined quotient space of the foliation which is a complex 3-manifold $Z$ with a projection to ${{\mathbf {\rm P}}}^1$ induced from the first factor in ${{\mathbf {\rm P}}}^1\times X^c$. It has a real structure and a holomorphic involution induced by $\tau$ whose fixed point set consists of the fibres over $\zeta=0,\infty$.
Since the form $\eta\wedge\varphi$ is closed on the three-dimensional manifold $X^c$, this is the quotient by its degeneracy foliation and thus each fibre has an induced symplectic structure. We therefore have a holomorphic section $\varpi$ of $\Lambda^2T_F^*(2)$ on $Z$. There are distinguished holomorphic sections of the fibration, the images of ${{\mathbf {\rm P}}}^1\times \{x\}$. These form a 3-dimensional family but what we need is a 4-dimensional family with normal bundle ${\mathcal O}(1)\oplus {\mathcal O}(1)$.
2\. On ${{\mathbf {\rm P}}}^1\times X^c$ the normal bundle of a section ${{\mathbf {\rm P}}}^1\times \{x\}$ is trivial and since the subbundle defined by the foliation is isomorphic to ${\mathcal O}(-2)$ the normal bundle $N$ in $Z$ fits in an exact sequence $$0\rightarrow {\mathcal O}(-2)\stackrel{\alpha}\rightarrow {\mathcal O}^3\rightarrow N\rightarrow 0.$$ But $\varphi$ defines a trivial bundle which annihilates the foliation and is thus a trivial subbundle of $N^*$. The map $\alpha$ is defined by three sections $(s_1,s_2,s_3)$ of ${\mathcal O}(2)$ but the trivial annihilator means that there is a linear relation. With a change of basis the map is $(t_1,t_2,0)$ and $N$ is ${\mathcal O}\oplus L$ where $L$ is the quotient of ${\mathcal O}^2$ by ${\mathcal O}(-2)$ – in other words ${\mathcal O}(2)$. Thus $N\cong {\mathcal O}(2)\oplus {\mathcal O}$. In fact each of these sections is preserved by the involution $\tau$ so we could take $t_1=1,t_2=\zeta^2$.
This is not the right normal bundle for a hyperkähler metric but in any case we are expecting the metric to be singular for these sections. However, Kodaira’s deformation theory tells us that, given one curve ${{\mathbf {\rm P}}}^1\subset Z$, as long as $H^1({{\mathbf {\rm P}}}^1,N)=0$ there is a smooth family of deformations $M^c$ whose tangent space at a curve is isomorphic to the space of holomorphic sections of the normal bundle $N$. For $N={\mathcal O}(2)\oplus {\mathcal O}$, $H^1({{\mathbf {\rm P}}}^1,N)$ indeed vanishes and $\dim H^0({{\mathbf {\rm P}}}^1,N)=1+3=4$ so we have a 4-manifold $M^c$. The real members of this family which have normal bundle ${\mathcal O}(1)\oplus {\mathcal O}(1)$ will define a hyperkähler manifold. Note that the 3-dimensional family $X^c$ has tangent space $(a_0, a_1\zeta^2-a_2)\in H^0({{\mathbf {\rm P}}}^1,{\mathcal O}(2)\oplus {\mathcal O})$, the fixed point subspace under $\tau$. The involution induces one on $M^c$ and the fixed point set is $X^c$.
The condition for a rank 2 holomorphic vector bundle $E$ over ${{\mathbf {\rm P}}}^1$ with $c_1(E)=0$ to be trivial is $H^0({{\mathbf {\rm P}}}^1,E(-1))=0$. In our case, since the curves are sections of $Z\rightarrow {{\mathbf {\rm P}}}^1$, the normal bundle is the tangent bundle along the fibres $T_F$ and $c_1(T_F)\cong p^*{\mathcal O}(2)$ so the condition for the holomorphic structure to be ${\mathcal O}(1)\oplus {\mathcal O}(1)$ is $H^0({{\mathbf {\rm P}}}^1,T^*_F)=0$. There is a determinant line bundle over $M^c$ for the $\bar\partial$-operators on $T^*_F$ with a determinant section which vanishes when $H^0({{\mathbf {\rm P}}}^1,T^*_F)\ne 0$. Unless the determinant is identically zero this is a divisor and the smooth 3-manifold $X^c$ is already contained in it, so to show that points in $M^c\backslash X^c$ sufficiently close to $X^c$ have normal bundle ${\mathcal O}(1)\oplus {\mathcal O}(1)$ we have to show that ${\mathcal O}(2)\oplus {\mathcal O}$ is not the generic case. We need the contact condition to do this.
3\. So suppose for a contradiction that all deformations of the lines parametrized by $X^c$ have normal bundle ${\mathcal O}(2)\oplus {\mathcal O}$. The ${\mathcal O}(2)$ factor is canonically determined as the maximal destabilizing subbundle. Then the sections of ${\mathcal O}(2)$ define a rank 3 distribution on the 4-manifold $M^c$. Take one line $L$ of the family and a one-parameter family $L_t$ passing through a point $z\in L\subset Z$. Then because $z$ is fixed, the section of the normal bundle $N$ associated to the infinitesimal variation vanishes at $z$ (for all $L_t$) and so lies in the ${\mathcal O}(2)$ component. Hence the whole curve in $M^c$ is tangential to the distribution. Now consider all lines through $z$. This is a 2-dimensional family whose tangent space at $L$ consists of the sections of ${\mathcal O}(2)$ which vanish at $z$ – the quadratic polynomials with factor $(x-z)$. This surface is everywhere tangential to the distribution. It is locally defined by a 1-form $\gamma$, so take two vector fields $U,V$ in the surface so that $i_U\gamma=i_V\gamma=0$, then $i_{[U,V]}\gamma=0$ and $d\gamma$ vanishes as a form on the surface.
Now the 2-dimensional subspaces of $H^0({{\mathbf {\rm P}}}^1,{\mathcal O}(2))$ consisting of sections which vanish at points $z\in {{\mathbf {\rm P}}}^1$ give elements which span $\Lambda^2(H^0({{\mathbf {\rm P}}}^1,{\mathcal O}(2)))^*$. Indeed under the $PSL(2, {\mathbf{C}})$-invariant isomorphism with $H^0({{\mathbf {\rm P}}}^1,{\mathcal O}(2))$ these are polynomials of the form $a(x-z)^2$ and any quadratic is a sum of squares. It follows that $d\gamma$ vanishes on the whole distribution so that $\gamma\wedge d\gamma=0$ and the distribution is integrable – a foliation.
The tangent space of $X^c$ is, as remarked above, $(a_0, a_1\zeta^2-a_2)\in H^0({{\mathbf {\rm P}}}^1,{\mathcal O}\oplus {\mathcal O}(2))$. So the leaves of this foliation intersect $X^c$ tangential to the distribution $a_0=0$ defined by $\varphi$. But $\varphi\wedge d\varphi\ne 0$ so we have a contradiction to integrability. Thus a twistor line given by a point in the family near $X^c$ must have normal bundle ${\mathcal O}(1)\oplus {\mathcal O}(1)$.
4\. The three hyperkähler forms may be considered as the coefficients of a quadratic polynomial $\omega_{\zeta}=(\omega_2+i\omega_3)+2\zeta\omega_1-\zeta^2(\omega_2-i\omega_3)$ and in the twistor approach this is obtained as follows. Since the normal bundle $T_F$ to a twistor line $L_m$ is ${\mathcal O}(1)\oplus {\mathcal O}(1)$ then the natural map $$H^0(L_m,T_F(-1))\otimes H^0(L_m,p^*{\mathcal O}(1))\rightarrow H^0(L_m,T_F)= T_mM^c
\label{tensor}$$ is an isomorphism. A tangent vector can then be considered as a linear function in $\zeta$ with coefficients sections of $T_F(-1)$. The section $\varpi$ of $\Lambda^2T^*_F(2)$ is a skew form on the first factor and so evaluating on a pair of tangent vectors gives a quadratic polynomial in $\zeta$. This defines $\omega_{\zeta}$. When the normal bundle jumps, the homomorphism above is no longer an isomorphism, but we can still define $\omega_{\zeta}$ as we shall see next.
5\. The argument in paragraph 2 of this proof concerning normal bundles only uses the condition $\varphi\wedge d\varphi\ne 0$ and not the full integrability of the leaves of the foliation. It is just the 1-jet of $\varphi$ which is relevant. This means that the section of $T^*_F$ on a line $L_x$ for $x\in X^c$ does not extend to the first order neighbourhood. Equivalently, the determinant section vanishes on $X^c$ with multiplicity one. On ${{\mathbf {\rm P}}}^1$ a local model for the minimal jump in holomorphic structure for a rank 2 bundle $E$ of degree $0$ is given by the one-parameter family of extensions in $H^1({{\mathbf {\rm P}}}^1,{\mathcal O}(-2))\cong {\mathbf{C}}$. If $[e]$ is a generator then the extension $t[e]$ defines a vector bundle $E$: $$0\rightarrow {\mathcal O}(-1)\rightarrow E\rightarrow {\mathcal O}(1)\rightarrow 0$$ which is trivial if $t\ne 0$ and ${\mathcal O}(-1)\oplus {\mathcal O}(1)$ if $t=0$.
We can describe such an extension by a transition matrix defined on ${\mathbf{C}}^*$ by $$g_{10}(t,z)= \begin{pmatrix}\zeta & t \\
0& \zeta^{-1}
\end{pmatrix}$$ and a global section is given by holomorphic vector-valued functions $v_0(\zeta), v_1(\tilde\zeta)$ satisfying $v_1(\tilde\zeta)=g_{10}(t,z)v_0(\zeta)$ where $\tilde\zeta=\zeta^{-1}$. The 2-dimensional space of such sections is defined by $v_0(\zeta)=(-ta_1, a_0+a_1\zeta)$. Since $\det g_{10}=1$ it preserves the standard skew form $\epsilon$ on ${\mathbf{C}}^2$ and taking a basis $s_1=(t,-\zeta), s_2=(0,1)$ this gives $\epsilon(s_1,s_2)=t$. In a similar fashion, the 4-dimensional space of global sections of $T_F$, with transition matrix $\zeta^{-1}g_{10}(t,z)$ is given by $v_0(\zeta)=(b-tc_3\zeta, c_0+c_1\zeta+c_2\zeta^2)$.
Applying this to the bundle $T_F(-1)$ together with the skew form defined by $\varpi$. we can implement the isomorphism (\[tensor\]) to obtain the $\zeta$-dependent skew form on $T_m$ $$\omega_{\zeta}=db\wedge dc_0+(db\wedge dc_1+tdc_0\wedge dc_2)\zeta+(db+tdc_1)\wedge dc_2\zeta^2.$$ When $t=0$ it is well defined and all coefficients are divisible by $db$ – the characteristic property of an $\alpha$-fold. More importantly, identifying this with $(\omega_2+i\omega_3)+2\zeta\omega_1-\zeta^2(\omega_2-i\omega_3)$ we see that $2\omega_2^2=(\omega_2+i\omega_3)\wedge(\omega_2-i\omega_3)=tdb\wedge dc_0\wedge dc_1\wedge dc_2$ and so vanishes transversally on $X^c$.
6\. Consider now the real structure, given by the antiholomorphic involution $\sigma$, and the real lines parametrised by $M\subset M^c$ . Then $\omega_{\zeta}$ is real and the reality condition on the coefficients means that the $\omega_i$ are real forms on $M$. As we have seen, $\omega_i^2$ vanishes transversally on $X\subset M$. Moreover, since $\omega_{\zeta}$ is preserved by the holomorphic involution $\tau$ which covers $\zeta\mapsto -\zeta$, we have $\tau^*\omega_1=-\omega_1$ and $\tau^*(\omega_2+i\omega_3)=(\omega_2+i\omega_3)$. We therefore have an $\alpha$-fold $X$ on $M$, which is the fixed point set of an involution.
1\. Consider the real twistor lines given by $M\backslash X$. The standard twistor approach gives that the intersection with any fibre $Z_{\xi}$ of $Z\rightarrow {{\mathbf {\rm P}}}^1$ is a local diffeomorphism, and we obtain a description of the hyperkähler metric as a $C^{\infty}$ product ${{\mathbf {\rm P}}}^1\times M$. The twisted holomorphic 2-form is then written as $(\omega_2+i\omega_3)+2\zeta\omega_1-\zeta^2(\omega_2-i\omega_3)$.
Take $\xi=0$. This fibre is fixed by $\tau$ so $L_m$ and $\tau(L_{m})=L_{\tau(m)}$ meet it at the same point. The restriction map $M\rightarrow Z_0$ therefore factors through the involution $\tau$. Pulling back $\varpi$ gives a closed complex 2-form $\omega_2+i\omega_3$ with $\tau^*(\omega_2+i\omega_3)=(\omega_2+i\omega_3)$ and by construction its restriction to $X$ is $(\eta_1+i\eta_2)\wedge\varphi$ which is rank 2. So this is folding in a quite concrete sense: the map from $M$ to $Z_0$ is a folding map, and its image is a manifold with boundary diffeomorphic to $X$.
2\. It is natural to ask about the geometry on $Z_0$ on the other side of the hypersurface. In fact, composing the real structure on $Z$ with $\tau$ gives a new real structure covering $\zeta\mapsto 1/\bar \zeta$ on ${{\mathbf {\rm P}}}^1$, the reflection in an equator. With this structure $\omega_1$ is imaginary and we have the relation $\omega_2^2=\omega_3^2=-\omega_1^2$ for the three closed forms. The twistor lines which are real for this structure define a 4-manifold with hypersymplectic structure (see e.g.[@Hit01],[@DS]). In four dimensions this is a Ricci-flat anti-self-dual Einstein metric of signature $(2,2)$. The analytic continuation of the canonical model to the [*exterior*]{} of the disc bundle is an example. Inside $M^c$ we thus have two real submanifolds, intersecting in $X$ but mapping to opposite sides of the corresponding hypersurface in the fibre $Z_0$. This is analogous to the real and imaginary axes in ${\mathbf{C}}$ and the map $z\mapsto z^2$ to ${\mathbf{R}}$.
3\. To reverse the construction, if a twistor section has normal bundle $T_F\cong {\mathcal O}(2)\oplus{\mathcal O}$ then the distinguished subbundle ${\mathcal O}(2)$ lifts it canonically to the 4-manifold ${{\mathbf {\rm P}}}(T_F)\rightarrow Z$. It has trivial normal bundle there and so locally this space is a product $ {{\mathbf {\rm P}}}^1\times X^c$.
4\. The intrinsic differential geometry of a 3-manifold with the boundary data of the theorem has been studied by R.Bryant [@Bry1] who proved elsewhere a hyperkähler extension theorem analogous to the above but for $\beta$-folds [@Bry2].
Global invariants
=================
Topological invariants
----------------------
Our conjectural folded hyperkähler metrics are deformations of the canonical model and so are defined on the 4-manifold $M$ which is the $S^2$-bundle ${{\mathbf {\rm P}}}(1\oplus K)$ over the surface $\Sigma$. The second homology is generated by the classes of a fibre and the zero section in $K\subset {{\mathbf {\rm P}}}(1\oplus K)$. We have three closed 2-forms $\omega_1,\omega_2,\omega_3$ which define de Rham classes in $H^2(M,{\mathbf{R}})$. But we also have the involution $\tau$ and in particular $\omega_2,\omega_3$ are invariant by $\tau$. Since $\tau$ changes the orientation of each fibre it follows that the classes $[\omega_2],[\omega_3]$ evaluated on a fibre are zero. But also, $\omega_2+i\omega_3$ restricted to $K$ is holomorphic and so vanishes on the zero section. Thus $[\omega_2]=0=[\omega_3]$.
From the $SU(\infty)$ point of view $\omega_1$ restricts to the standard symplectic form on a fibre and evaluates to $4\pi$. From the explicit formula (\[om1\]) for the canonical model $\omega_1=d(u_3(dx-yd\phi))$ and so on the zero section $x_3=yu_3=1$, $\omega_1=dx\wedge dy/y^2$ and evaluating the integral gives the area of the hyperbolic metric of curvature $-1$ which is $4\pi(g-1)$.
Invariant polynomials
---------------------
One of the fundamental features of the moduli space of Higgs bundles is the associated integrable system, based on evaluating an invariant polynomial of degree $m$ on $\Phi$ to give a holomorphic section of $K^m$ on $\Sigma$. As we remarked above, we can replace the polynomial $\tr A^m$ on ${{\mathfrak{s}}u}(n)$ by $$p_m(f)=\int_{S^2}f^m\omega
\label{inv}$$ for $SU(\infty)$, and this we can do for the Lie algebra and its complexification.
Then for the Higgs field $\Phi$, a section of $p^*K$ on the sphere bundle $M^4$, integration over the fibres of $\Phi^m$ defines likewise a section $\alpha_m$ of $K^m$ on $\Sigma$. This is again holomorphic, for from (\[higgs2\]) we have $2(\phi_1+i\phi_2)_{\bar z}+\{a_1+ia_2,\phi_1+i\phi_2\}=0$. Then writing $\psi=\phi_1+i\phi_2, a=a_1+ia_2$ we have $$\frac{\partial}{\partial \bar z}\left(\int_{M/\Sigma}\psi^m\omega\right) dz^m=-\frac{1}{2}\left(\int_{M/\Sigma}m\{a,\psi \}\psi^{m-1}\omega\right) dz^m=-\frac{1}{2}\left(\int_{M/\Sigma}{\mathcal L}_{X_a}(\psi^{m}\omega)\right) dz^m.$$ where $X_a$ is the Hamiltonian vector field of $a$. But this integral is zero by Stokes’s theorem as ${\mathcal L}_{X_a}(\psi^{m}\omega)= d(i_{X_a}(\psi^{m}\omega))$. This holds for any $SU(\infty)$-Higgs bundle. For the ones we are considering where the connection lies in $SO(\infty)$ and the Higgs field is symmetric under the involution, the differential $\alpha_m$ is twice the integral over the disc bundle. When $\Phi$ maps the closed disc bundle diffeomorphically to a submanifold $D$ of the cotangent bundle then, using standard local coordinates $(z,w)\mapsto wdz$ this integral is obtained from the canonical holomorphic 1-form $\theta=wdz$: $$\alpha_m=2\left(\int_{D/\Sigma}w^m\Phi_*\omega\right) dz^m=2\left(\int_{D/\Sigma}w^m\omega_1\right) dz^m$$ using the Kähler form $\omega_1$.
In the example relating to Teichmüller space above we can perform this calculation using (\[phiham\]) to get $$\alpha_m=2\int_{0\le \theta\le 2\pi}\int_{0\le r \le1}\frac{1}{2^m}(k^{1/2}re^{-i\theta} +ak^{-1/2}re^{i\theta})^m\frac{rdrd\theta}{2\sqrt{1-r^2}} dz^m$$ which is zero if $m$ is odd and if $m=2\ell$ is $$\left(\frac{\pi}{2^{2\ell}}{2\ell \choose \ell}\right)^2a^{\ell}dz^{2\ell}$$ and we recover a power of the quadratic differential $q=adz^2$ used in the definition of the Higgs bundle. For the canonical model $q=0$ and all these invariants therefore vanish.
One may ask what happened to $\alpha_0$ and $\alpha_1$. But $\alpha_0$ is just the area of the $2$-sphere which is the standard $4\pi$. As for $\alpha_1$, in the finite-dimensional case this is set to zero because we are considering the group $SU(n)$ and the trace of the Higgs field is zero. For our case, a translation $wdz\mapsto wdz+a(z)dz$ by a holomorphic section of $K$ preserves the holomorphic symplectic form and takes one solution to another. Setting $\alpha_1=0$ removes this trivial deformation.
First order deformations
========================
If a generalization of the higher Teichmüller spaces exists then the holomorphic differentials $\alpha_m$ should uniquely determine a folded hyperkähler metric. It makes sense then to look for deformations, and initially infinitesimal deformations, of the canonical model to be determined by a holomorphic differential. The Teichmüller example offers a test: we have a genuine deformation in the direction of a quadratic differential.
Differentiating equation \[qhiggs\] for the quadratic differential $tq$ with respect to $t$ and setting $t=0$ we get $\dot F=2\dot\omega$. Now the conformal structure is unchanged so $\dot\omega=f\omega$ for some function $f$ But then $\dot F= dd^cf$ and and so $dd^cf=f\omega$ and this implies $f=0$ since $dd^c$ is a negative operator. The infinitesimal variation of the metric is therefore zero and so $\dot\omega_1=0$ for the hyperkähler metric. From (\[phiham\]) $$\dot \Phi=\dot w dz= ay^2\bar wdz$$ using the hyperbolic metric $k=y^{-2}$ and $\dot k=0$. Then $$\dot \omega_2+i\dot\omega_3=d(\dot w dz)=d(ay^2\bar w dz).
\label{Lomega}$$ Introduce the complex vector field (tangential to the fibres) $$X^c=ay^2\bar w\frac{\partial}{\partial w}$$ and let $X$ be the real part, then we can write (\[Lomega\]) as $$\dot \omega_2+i\dot\omega_3=d{\mathcal L}_X wdz={\mathcal L}_X(\omega_2+i\omega_3).$$
We can generalise this infinitesimal deformation by taking a holomorphic section $\alpha_m$ of $K^m$ and writing it in local coordinates as $\alpha_m=a(z)dz^m$. Then define a complex vector field $$X^c=ay^{2m-2}\bar w^{m-1}\frac{\partial}{\partial w}.$$ This is globally well-defined because we have the hermitian form $h=dzd\bar z/y^2$, the canonical 1-form $\theta=wdz$ and the canonical holomorphic Poisson tensor $\pi=\partial/\partial w\wedge \partial/\partial z$. The vector field is then $X^c=\pi(\alpha h^{-(m-1)}\bar\theta^{m-1}).$
If $X$ is the real part of $X^c$, then the three closed 2-forms ${\mathcal L}_X\omega_i$ are anti-self-dual.
The Kähler forms $\omega_i$ span the space of self-dual 2-forms at each point and so we need to prove that ${\mathcal L}_X\omega_i\wedge\omega_j=0$ for all $i,j$.
First consider ${\mathcal L}_X(\omega_2+i\omega_3)={\mathcal L}_Xd(wdz)$. This is $d(a\bar w^{m-1}y^{2m-2}dz)$ and since $a(z)$ is holomorphic and $y=(z-\bar z)/2i$ we obtain $$(m-1)ay^{2m-3}\bar w^{m-2}(yd\bar w\wedge dz+i\bar wd\bar z\wedge dz).
\label{lie}$$ So clearly $${\mathcal L}_X(\omega_2+i\omega_3)\wedge dw\wedge dz=0={\mathcal L}_X(\omega_2+i\omega_3)\wedge d\bar w\wedge d\bar z$$ and we have ${\mathcal L}_X\omega_2\wedge\omega_2=0={\mathcal L}_X\omega_3\wedge\omega_2={\mathcal L}_X\omega_2\wedge\omega_3={\mathcal L}_X\omega_3\wedge\omega_3.$ In particular ${\mathcal L}_X(\omega_2^2)=0$ but since $\omega_2^2=\omega_1^2$ we also have ${\mathcal L}_X\omega_1\wedge\omega_1=0$.
It remains to prove that ${\mathcal L}_X(\omega_2+i\omega_3)\wedge\omega_1=0$ for then we shall have ${\mathcal L}_X\omega_2\wedge \omega_i=0$ for $i=1,2,3$ and similarly for $\omega_3$. Taking the Lie derivative of $\omega_3\wedge \omega_1=0=\omega_2\wedge \omega_1$ we shall then have the same result for ${\mathcal L}_X\omega_1$.
Recall then that $$\omega_1=d(u_3(dx-yd\phi))=du_3\wedge (dx-yd\phi)-u_3dy\wedge d\phi$$ and $u_3y=\sqrt{1-y^2\vert w\vert^2}, e^{2i\phi}=w/\bar w.$ Since from (\[lie\]) $dz=d(x+iy)$ is a factor of ${\mathcal L}_X(dw\wedge dz)$ we calculate first $$\omega_1\wedge(dx+idy)=idu_3\wedge dx\wedge dy-u_3d\phi\wedge dx\wedge dy-ydu_3\wedge d\phi\wedge dx-iydu_3\wedge d\phi\wedge dy
\label{wedge}$$ and now, from the form of ${\mathcal L}_X(dw\wedge dz)$ in (\[lie\]), take the exterior product with $yd\bar w+i\bar w d\bar z$.
Using $$2id\phi=\frac{dw}{w}-\frac{d\bar w}{\bar w},\quad du_3=-(u_3^2+\vert w\vert^2)\frac{dy}{yu_3}-\frac{1}{2u_3}(\bar wdw+wd\bar w)$$ in the expression (\[wedge\]) we get zero.
Setting $\dot\omega_1=0, \dot\omega_2= {\mathcal L}_X\omega_2$ and $\dot\omega_3= {\mathcal L}_X\omega_3$ from the proposition we certainly have a first order deformation of the algebraic equations $$\omega_1^2=\omega_2^2=\omega_3^2=0 \qquad \omega_1\wedge \omega_2=\omega_2\wedge \omega_3=\omega_3\wedge \omega_1=0$$ by closed forms. We also have a deformation which is orthogonal to just rotating the forms $\omega_i$.
Infinitesimal deformations of Ricci-flat metrics define elements in the null space of the Lichnerowicz Laplacian acting on trace-free symmetric tensors. On a hyperkähler 4-manifold these are constructed from tensor products of the covariant constant self-dual 2-forms $\omega_i$ and closed anti-self dual 2-forms, thus the proposition gives us potential deformations of hyperkähler metrics (and in the compact case would give actual deformations).
We can calculate the infinitesimal variation in the global invariants using the above. Since $\dot\omega_1=0$ the cohomology class is unchanged. Since $\omega_1^2$ is fixed to first order then so is the fold, so we only have to vary $w$ which is the vector field $X$ applied to $w$. The variation in the moment of $w^k$ is therefore given by
$$\int_{y^2\vert w\vert^2<1}kw^{k-1}a\bar w^{m-1}y^{2m-2}y^2\frac{dwd\bar w}{\sqrt{1-y^2\vert w\vert^2}}.$$ The measure is $S^1$-invariant and so the multiple integral vanishes unless $m=k$. In this case put $u=yw$ and we get $$a\int_{\vert u\vert<1}\vert u\vert^{2k-2}\frac{dud\bar u}{\sqrt{1-\vert u\vert^2}}=2\pi a\int_0^1r^{2k-2}\frac{rdr}{\sqrt{1-r^2}}=\pi a \frac{4^k (k!)^2}{(2k)!}.$$ This is consistent with our conjecture that the differentials determine the metrics.
$S^1$-invariance
================
The usual Teichmüller space is embedded in its higher analogue by the homomorphism $SL(2,{\mathbf{C}})\rightarrow SL(n,{\mathbf{C}})$ given by the irreducible $n$-dimensional representation. Restricted to the compact real form we take the $SU(n)$-Higgs bundle induced from the $SU(2)$-Higgs bundled discussed in Section 6.2 and restricted to the split form we take the positive hyperbolic $SL(n,{\mathbf{R}})$-representation coming from the uniformizing representations. The scalar $S^1$-action $\Phi\mapsto e^{i\phi}\Phi$ preserves the higher Teichmüller spaces but acts non-trivially on all differentials except zero and so the only solution fixed by this action is the canonical solution. If our conjecture holds, then the canonical model should be the only $S^1$-invariant folded hyperkähler metric on a disc bundle in the cotangent bundle. Here, where we regard the Higgs field as the canonical 1-form, the action is just scalar multiplication by a unit complex number in the fibre of $T^*\Sigma$.
This circle action acts on $\omega_2+i\omega_3=d(wdz)$ by $e^{i\phi}$ and preserves $\omega_1$ and for this type of action in four dimensions there is a well-known differential equation called either the Boyer-Finley equation [@BF] or the $SU(\infty)$-Toda equation [@Tod].
Locally the Ansatz for the metric is $$u_t(e^u(dx^2+dy^2)+dt^2)+u_t^{-1}(d\tau+u_ydx-u_xdy)^2
\label{Boy}$$ and for the Kähler form $$\omega_1= u_te^udx\wedge dy+dt\wedge (d\tau+u_ydx-u_xdy).
\label{kform}$$ Requiring this to be closed gives the Boyer-Finley equation $$u_{xx}+u_{yy}+(e^u)_{tt}=0.
\label{toda}$$ The holomorphic symplectic form is $\omega_2+i\omega_3=d(wdz)$ where $w=\sqrt{2} e^{u/2+i\tau}$.
For the canonical model $u=\log(1-t^2)-2\log y$, giving $u_x=0$ and $u_y=-2/y$. Hence $u_{yy}=2/y^2$ and $(e^u)_{tt}=-2/y^2$. Then $t=0$ is the fold and at $t=-1$, $u_te^u=2/y^2$ which gives the hyperbolic metric $2y^{-2}(dx^2+dy^2)$ on the zero section of the cotangent bundle.
One form of the 2-dimensional Toda lattice is the differential-difference equation $$v_{xx}+v_{yy}+e^{v_{n+1}}-2e^{v_n}+e^{v_{n-1}}=0.$$ The limit where the difference operator becomes the second derivative is the reason for the $SU(\infty)$ terminology and is consistent with our use in this paper.
We need to give the local formulae above a more geometric interpretation. The Killing field is $\partial/\partial\tau$ so from (\[kform\]) the moment map with respect to $\omega_1$ is $-t$. Setting $t$ to be a constant and taking the quotient by the circle action gives the Kähler quotient as the metric $4u_te^u(dx^2+dy^2)$ so $z=x+iy$ is a local complex coordinate on the quotient. The function $u$ itself is defined by $w\bar w=2e^u$.
If we focus now on the $t$-dependent metric $g=w\bar w dzd\bar z=2e^u(dx^2+dy^2)$ on a surface then its Gaussian curvature $\kappa=-e^{-u}(u_{xx}+u_{yy})/4$ and then the Toda equation (\[toda\]) may be written in the more geometrical form $$g_{tt}=4\kappa g
\label{toda1}$$ for a $t$-dependent family of metrics on a compact surface.
In our setting $wdz$ is the canonical 1-form on $T^*\Sigma$ restricted to a disc bundle. The $S^1$-invariance $w\mapsto e^{i\phi}w$ means our boundary value data for the fold is the unit cotangent bundle for a hermitian metric .
In the Higgs bundle picture $\omega_1$ restricted to the fibre is the restriction of the standard symplectic form on $S^2$ to the disc $x_3\ge 0$ and $x_3$ itself is the moment map for the circle action so $t=-x_3$ and with this normalisation the fold is $t=0$ and the zero section $t=-1$ as in the canonical model. Note from formula (\[kform\]) that integrating $\omega_1$ over each disc is integrating $dt\wedge d\tau$ for $-1\le t\le 0,0\le \tau\le 2\pi$ which gives $2\pi$, the correct normalisation.
For regularity we assume smoothness of the forms $\omega_i$ on the $S^2$-bundle ${{\mathbf {\rm P}}}(1\oplus K)$ and on this space the Higgs field is described locally as $\phi_1dx+\phi_2dy$ where $\phi_1,\phi_2$ are even functions. The metric $g=(\phi_1^2+\phi_2^2)(dx^2+dy^2)$ and since the $\phi_i$ are even $\partial \phi_i/\partial x_3=0$ on the fold. Hence $g_t=0$ at the fold. We now have boundary conditions $g_t(0)=0$ and $g(-1)=0$ for the equation (\[toda1\]). In fact, we are interested in a solution over the whole sphere bundle, invariant under an involution, so we could take equivalently the boundary conditions $g(-1)=0=g(1)$.
The only $S^1$-invariant hyperkähler metric on ${{\mathbf {\rm P}}}(1\oplus K)\rightarrow \Sigma$ which has an $\alpha$-fold on a circle bundle is the canonical model for a hyperbolic metric on $\Sigma$.
1\. Note that since $g=w\bar wdzd\bar z$ the conformal structure is independent of $t$, so if $g=fh$ for some fixed metric $h$, the area form $v_g=fv_h$. Now integrate (\[toda1\]) against $v_h$ and use Gauss-Bonnet. If $A(t)$ is the area of the metric $g(t)$ on $\Sigma$ we get $$A_{tt}=16\pi(1-p)$$ where $p$ is the genus, and hence $A(t)=(a+bt+ct^2)$ for constants $a,b,c$. But the metric vanishes at $t=\pm1$ so $b=0=a+c$ and we have $$A(t)=8\pi (t^2-1)(1-p).$$ and since $-1\le t\le 1$ the genus must be greater than 1.
2\. Consider the metric $h=g/2(1-t^2)$, then the area is fixed and the metric is regular at $t=\pm 1$. The right hand side of the equation (\[toda1\]) is the Gauss-Bonnet integrand which is unchanged under constant rescaling so the equation for $h$ is $$(1-t^2)h_{tt}-4th_t-2h=4\kappa h$$ and clearly $h$ of constant curvature $-1/2$ satisfies this. These metrics are all conformally related so we can write $h=fg_{H}$ where $g_{H}$ is the hyperbolic metric and then we get $(1-t^2)f_{tt}-4tf_t=2\Delta_H\log f $ where $\Delta_H$ is the Laplace-Beltrami operator for the hyperbolic metric.
Rewrite this as $((1-t^2)^2f_t)_t=2(1-t^2)(\Delta_H\log f). $ Integrating by parts and using the fact that $f$ is finite at $t=\pm1$, we have $$2\int_{-1}^1(1-t^2)f\Delta_H\log f dt =\int_{-1}^1((1-t^2)^2f_t)_tfdt=-\int_{-1}^1(1-t^2)^2f^2_tdt.$$ But $f\Delta_H\log f=\Delta_Hf+f^{-1}(df,df)$ so integrating over $[-1,1]\times \Sigma$ we get $$-\int_{-1}^1(1-t^2)^2\int_{\Sigma}f^2_tdt\,v_H=2\int_{-1}^1(1-t^2)\int_{\Sigma}(\Delta_Hf+f^{-1}(df,df))dt\,v_H$$ which, since the integral of $\Delta_Hf$ over $\Sigma$ vanishes, is a contradiction unless $f$ is a constant on $[-1,1]\times \Sigma$.
1\. There do exist $S^1$-invariant solutions to the Higgs bundle equations other than the canonical one. The holomorphic Higgs bundle is of the form $$E=L\oplus L^* \qquad \Phi= \begin{pmatrix}0 & s \\
0& 0
\end{pmatrix}$$ where $s$ is a holomorphic section of $L^2K$. But when $L\ne K^{-1/2}$, $s$ has zeros and then the Higgs field itself vanishes, which gives a rather different singularity for the hyperkähler metric, apart from the circle bundle fold.
2\. The uniqueness result above, coupled with the local existence theorem of Section 7, shows that the unit cotangent bundle of a Riemannian metric which has a global folded hyperkähler extension is a hyperbolic metric. If our conjecture holds, then the general situation must involve a geometry of Finsler type.
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abstract: 'A big challenge in solar and stellar physics in the coming years will be to decipher the magnetism of the solar outer atmosphere (chromosphere and corona) along with its dynamic coupling with the magnetic fields of the underlying photosphere. To this end, it is important to develop rigorous diagnostic tools for the physical interpretation of spectropolarimetric observations in suitably chosen spectral lines. Here we present a computer program for the synthesis and inversion of Stokes profiles caused by the joint action of atomic level polarization and the Hanle and Zeeman effects in some spectral lines of diagnostic interest, such as those of the He [i]{} 10830 Å and 5876 Å (or D$_3$) multiplets. It is based on the quantum theory of spectral line polarization, which takes into account in a rigorous way all the relevant physical mechanisms and ingredients (optical pumping, atomic level polarization, level crossings and repulsions, Zeeman, Paschen-Back and Hanle effects). The influence of radiative transfer on the emergent spectral line radiation is taken into account through a suitable slab model. The user can either calculate the emergent intensity and polarization for any given magnetic field vector or infer the dynamical and magnetic properties from the observed Stokes profiles via an efficient inversion algorithm based on global optimization methods. The reliability of the forward modeling and inversion code presented here is demonstrated through several applications, which range from the inference of the magnetic field vector in solar active regions to determining whether or not it is canopy-like in quiet chromospheric regions. This user-friendly diagnostic tool called “HAZEL" (from HAnle and ZEeman Light) is offered to the astrophysical community, with the hope that it will facilitate new advances in solar and stellar physics.'
author:
- 'A. Asensio Ramos, J. Trujillo Bueno'
- 'E. Landi Degl’Innocenti'
title: Advanced Forward Modeling and Inversion of Stokes Profiles Resulting from the Joint Action of the Hanle and Zeeman Effects
---
Introduction
============
The present paper describes a computer program for the synthesis and inversion of Stokes profiles resulting from the joint action of the Hanle and Zeeman effects in some spectral lines of diagnostic interest, such as those pertaining to the 10830 Å and 5876 Å (or D$_3$) multiplets. The effects of radiative transfer on the emergent spectral line radiation are taken into account through a suitable slab model. Our aim is to provide the solar and stellar physics communities with a robust but user-friendly tool for understanding and interpreting spectropolarimetric observations, with the hope that this will facilitate new advances in solar and stellar physics.
In particular, the lines of neutral helium at 10830 Å are of great interest for empirical investigations of the dynamic and magnetic properties of plasma structures in the solar chromosphere and corona, such as active regions [e.g., @harvey_hall71; @ruedi96; @Lagg04; @centeno06], filaments [e.g., @lin98; @trujillo_nature02], prominences [e.g., @merenda06] and spicules [e.g., @trujillo_merenda05; @socas_elmore05]. The same applies to the lines of the D$_3$ multiplet at 5876 Å which have been used for investigating the magnetic field vector in solar prominences and spicules [e.g., @landi_d3_82; @querfeld85; @bommier94; @casini03; @lopezariste_casini05; @ramelli06_2; @ramelli06]
Such helium lines result from transitions between terms of the triplet system of helium (ortho-helium), whose respective $J$-levels (with $J$ the level’s total angular momentum) are far less populated than the ground level of helium (that is, than the singlet level $^1$S$_0$), except perhaps in flaring regions. On the other hand, the lower term (2s$^3$S$_1$) of the 10830 Å multiplet is the ground level of ortho-helium, while its upper term (2p$^3$P$_{2,1,0}$) is the lower one of 5876 Å (whose upper term is 3d$^3$D$_{3,2,1}$). Therefore, the significant difference in the ensuing optical thicknesses of the observed solar plasma structure implies that when the radiation in these spectral lines is observed on the solar disk it is much easier to see structures in 10830 Å than in 5876 Å, while both lines are clearly seen in emission when observing off-limb structures such as prominences and spicules. The additional fact that the Hanle effect in forward scattering creates measurable linear polarization signals in the lines of the 10830 Å multiplet when the magnetic field is inclined with respect to the local solar vertical direction [@trujillo_nature02], and that there is a nearby photospheric line of Si [i]{}, makes the 10830 Å spectral region very suitable for investigating the coupling between the photosphere and the corona.
While the Stokes $I$ profiles of the 10830 Å and 5876 Å helium lines depend mainly on the distribution of the populations of their respective upper ($J_u$) and lower ($J_l$) levels along the line-of-sight (LOS), their Stokes $Q$, $U$ and $V$ profiles depend on the strengths and wavelength positions of the $\pi$ ($\Delta{M}=M_u-M_l=0$), $\sigma_{\rm blue}$ ($\Delta{M}=+1$) and $\sigma_{\rm
red}$ ($\Delta{M}=-1$) transitions, which can only be calculated correctly within the framework of the Paschen-Back effect theory. Moreover, the $Q$, $U$ and $V$ profiles are also affected by the atomic level polarization induced by anisotropic pumping processes [e.g., @landi_landolfi04]. An atomic level of total angular momentum $J$ is said to be polarized when its magnetic sublevels are unequally populated and/or when there are quantum coherences between them. The radiative transitions induced by the anisotropic illumination of the helium atoms in the solar atmosphere are able to create a significant amount of atomic polarization in the helium levels, even in the metastable lower level of the 10830 Å multiplet [@trujillo_nature02]. If the net circular polarization of the incident radiation at the wavelengths of the helium transitions is negligible, as it uses to be the case, the radiatively induced atomic level polarization is such that the populations of substates with different values of $|M|$ are different (non-zero atomic alignment), while substates with magnetic quantum numbers $M$ and $-M$ are equally populated (zero atomic orientation). On the other hand, elastic collisions with the neutral hydrogen atoms of the solar chromospheric and coronal structures are unable to destroy the atomic polarization of the He [i]{} levels. As a result, even in the absence of magnetic fields, linearly polarized spectral line radiation would be produced, simply because the population imbalances among the magnetic sublevels imply more or fewer $\pi$-transitions, per unit volume and time, than $\sigma$ transitions. The atomic polarization of the upper level of the line transition under consideration is thus responsible of a [*selective emission*]{} of polarization components, while that of the lower level may give rise to a [*selective absorption*]{} of polarization components (“zero-field” dichroism). In order for this type of dichroism to produce a measurable contribution to the emergent linear polarization it is necessary to have a substantial line-center optical thickness along the LOS or, assuming a small but non-negligible optical thickness, that the plasma structure under consideration is observed against the bright background of the solar disk [@trujillo_nature02; @trujillo_asensio07]. Therefore, the observable effects of dichroism are easier to detect in 10830 Å than in 5876 Å.
In the presence of a magnetic field the emergent polarization changes because of the following two reasons. First, because a magnetic field modifies the atomic level polarization, not only by producing the Hanle-effect relaxation of the quantum coherences pertaining to each individual $J$-level, but also through possible interferences between the magnetic sublevels pertaining to different $J$-levels, which give rise to a variety of remarkable effects such as the transfer of atomic alignment to atomic orientation in the $J$-levels of the upper term of the D$_3$ multiplet [@landi_d3_82] or the enhancement of the scattering polarization in the D$_2$ line of by a vertical magnetic field [@trujillo_casini02]. Second, because the magnetic splitting of the atomic energy levels give rise to significant wavelength shifts between the $\pi$ and $\sigma$ transitions (as compared with the spectral line width) and, consequently, to the generation of measurable linear and/or circular polarization (i.e., the familiar transverse and longitudinal Zeeman effects, respectively). Obviously, a correct modeling of the spectral line polarization that results from the joint action of the Hanle and Zeeman effects requires the application of the quantum theory of spectral line polarization [see the monograph by @landi_landolfi04], as done by several researchers for interpreting spectropolarimetric observations of solar plasma structures in the D$_3$ multiplet [e.g., @landi_d3_82; @bommier94; @casini03; @lopezariste_casini05] and in the 10830 Å triplet [e.g., @trujillo_nature02; @trujillo_merenda05; @merenda06].
Over the last few years new computer programs for the synthesis and inversion of Stokes profiles induced by the joint action of atomic level polarization and the Hanle and Zeeman effects have been developed and applied to the interpretation of spectropolarimetric observations. For example, [@landi_landolfi04] have developed some forward modeling codes with which they have calculated several Hanle effect diagrams and theoretical Stokes profiles of lines from complex atomic models, while the computer program of [@casini_manso05] can treat even the case of a hyperfine structured multiplet taking into account the quantum interferences between the $F$ levels belonging to the $J$ levels of different terms. Concerning Stokes inversion techniques we should mention that [@casini03] applied an inversion code based on principal component analysis [@arturo_casini02] to spectropolarimetric observations of solar prominences in the D$_3$ multiplet, providing two-dimensional maps of the magnetic field vector and further evidence for strengths significantly larger than average. On the other hand, [@merenda06] opted for an inversion strategy for the 10830 Å multiplet in which the longitudinal component of the magnetic field vector is obtained from the measured Stokes $V$ profiles (which are dominated by the Zeeman effect for the lines of the 10830 Å multiplet), and its orientation from the observed Stokes $Q$ and $U$ profiles (which in solar prominences are due to the presence of atomic level polarization). In the just mentioned computer programs and applications the optically thin assumption was used, which was a suitable approximation for the particular prominences observed, but an unreliable one in general (and especially for the interpretation of observations in the 10830 Å multiplet).
A simple but suitable model for taking into account radiative transfer effects is the constant-property slab model used by [@trujillo_nature02; @trujillo_merenda05] for the interpretation of spectropolarimetric observations in solar filaments and spicules, which has been recently extended by [@trujillo_asensio07] to include magneto-optical effects. These authors have applied this “cloud" model for the interpretation of spectropolarimetric observations in order to point out that the atomic polarization of the helium levels may have an important impact on the emergent linear polarization of the 10830 Å multiplet, even for magnetic field strengths as large as 1000 G. Therefore, inversion codes that neglect the influence of atomic level polarization, such as the Milne-Eddington codes of [@Lagg04] and [@socas_trujillo04], should ideally be used only for the inversion of Stokes profiles emerging from strongly magnetized regions (with $B>1000$ G), or when the observed Stokes $Q$ and $U$ profiles turn out to be dominated by the transverse Zeeman effect [e.g., as it happens with some active regions filaments as a result of the particular illumination conditions explained in @trujillo_asensio07].
It is also necessary to mention that [@manso_trujillo03a] developed a general radiative transfer computer program for solving the so-called non-LTE problem of the 2nd kind –that is, the multilevel scattering polarization problem including the Hanle effect of a weak magnetic field. These authors considered the multilevel atom approximation (which neglects quantum interferences between the sublevels of [*different*]{} $J$-levels), but took fully into account the effects of radiative transfer in realistic atmospheric models and the role of elastic and inelastic collisions in addition to all the relevant optical pumping mechanisms. An interesting application using a semi-empirical model of the solar atmosphere can be found in [@manso_trujillo03b]. The recent advances in the development of efficient numerical methods for the solution of non-LTE polarization transfer problems [e.g., the review by @trujillo03] and in computer technology make now possible even the performing of three-dimensional radiative transfer simulations of the Hanle effect in convective atmospheres [e.g., @trujillo_shchukina07].
The previous introductory paragraphs strongly suggest that it would be of great interest to develop a robust but user-friendly computer program for the synthesis and inversion of Stokes profiles resulting from the joint action of atomic level polarization and the Hanle and Zeeman effects. We have done this by implementing an efficient global optimization method for the solution of the inversion problem and by calculating at each iterative step the emergent spectral line polarization through the solution of the Stokes-vector transfer equation in a slab of constant physical properties in which the radiatively-induced atomic level polarization is assumed to be dominated by the photospheric continuum radiation. At each point of the observed field of view the slab’s optical thickness is chosen to fit the Stokes $I$ profile, which is a strategy that accounts implicitly for the true physical mechanisms that populate the triplet levels of [e.g., the photoionization-recombination mechanism, as shown by @centeno08b]. The observed Stokes $Q$, $U$ and $V$ profiles are then used to infer the magnetic field vector along with a few extra physical quantities.
The outline of this paper is as follows. The formulation of the problem is presented in §2, where we review the relevant equations within the framework of the quantum theory of spectral line polarization. The forward modelling option of our computer program is described in §3, including some interesting examples of possible applications. §4 deals with a detailed description of the Stokes inversion option, while §5 considers a variety of applications aimed at an in-depth testing of this diagnostic tool. The important issue of the possible ambiguities and degeneracies is addressed in §6, with emphasis on the Van Vleck ambiguity and on the possibility of inferring the atmospheric height at which the observed on-disk plasma structure is located. Finally, in §7 we summarize the main conclusions and comment on some ongoing developments.
Formulation of the problem
==========================
We consider a constant-property slab of atoms, located at a height $h$ above the visible solar “surface", in the presence of a deterministic magnetic field of arbitrary strength $B$, inclination $\theta_B$ and azimuth $\chi_B$ (see Fig. 1). The slab’s optical thickness at the wavelength and line of sight under consideration is $\tau$. We assume that all the atoms inside this slab are illuminated from below by the photospheric solar continuum radiation field, whose center-to-limb variation has been tabulated by [@pierce00]. The ensuing anisotropic radiation pumping produces population imbalances and quantum coherences between pairs of magnetic sublevels, even among those pertaining to the different $J$-levels of the adopted atomic model. This atomic level polarization and the Zeeman-induced wavelength shifts between the $\pi$ ($\Delta{M}=M_u-M_l=0$), $\sigma_{\rm blue}$ ($\Delta{M}=+1$) and $\sigma_{\rm red}$ ($\Delta{M}=-1$) transitions produce polarization in the emergent spectral line radiation.
In order to facilitate the reading and understanding of this paper, in the following subsections we summarize the basic equations which allow us to calculate the spectral line polarization taking rigorously into account the joint action of the Hanle and Zeeman effects. To this end, we have applied the quantum theory of spectral line polarization, which is described in great detail in the monograph by [@landi_landolfi04]. We have also applied several methods of solution of the Stokes-vector transfer equation, some of which can be considered as particular cases of the two general methods explained in §6 of [@trujillo03].
The radiative transfer approach {#sec:radiative_transfer}
-------------------------------
The emergent Stokes vector $\mathbf{I}(\nu,\mathbf{\Omega})=(I,Q,U,V)^{\dag}$ (with $\dag$=transpose, $\nu$ the frequency and $\mathbf{\Omega}$ the unit vector indicating the direction of propagation of the ray) is obtained by solving the radiative transfer equation
$$\frac{d}{ds}\mathbf{I}(\nu,\mathbf{\Omega}) =
\bm{\epsilon}(\nu,\mathbf{\Omega}) - \mathbf{K}(\nu,\mathbf{\Omega})
\mathbf{I}(\nu,\mathbf{\Omega}),
\label{eq:rad_transfer}$$
where $s$ is the geometrical distance along the ray under consideration, $\bm{\epsilon}(\nu,\mathbf{\Omega})=({\epsilon}_I,{\epsilon}_Q,{\epsilon
}_U,{\epsilon}_V)^{\dag}$ is the emission vector and $$\mathbf{K} = \left( \begin{array}{cccc}
\eta_I & \eta_Q & \eta_U & \eta_V \\
\eta_Q & \eta_I & \rho_V & -\rho_U \\
\eta_U & -\rho_V & \eta_I & \rho_Q \\
\eta_V & \rho_U & -\rho_Q & \eta_I
\end{array} \right)
\label{eq:propagation}$$ is the propagation matrix. Alternatively, introducing the optical distance along the ray, ${\rm d}{\tau}=-{\eta_I}{\rm d}s$, one can write the Stokes-vector transfer Eq. (\[eq:rad\_transfer\]) in the following two ways:
- The first one, whose formal solution requires the use of the evolution operator introduced by [@landi_landi_85], is $${{d}\over{d{\tau}}}{\bf I}\,=\,{\bf K}^{*}
{\bf I}\,-\,{\bf S},
\label{eq:rad_transfer_peo}$$ where ${\bf K}^{*}={\bf K}/{\eta_I}$ and ${\bf S}=\bm{\epsilon}/{\eta_I}$. The formal solution of this equation can be seen in eq. (23) of [@trujillo03].
- The second one, whose formal solution does not require the use of the above-mentioned evolution operator is [e.g., @rees89] $${{d}\over{d{\tau}}}{\bf I}\,=\,{\bf I}\,-\,{\bf S}_{\rm eff},
\label{eq:rad_transfer_delo}$$ where the effective source-function vector $\,{\bf S}_{\rm eff}\,=\,{\bf S}\,-\,
{\bf K}^{'}{\bf I},\,\,\,$ being $\,{\bf K}^{'}={\bf K}^{*}-{\bf 1}$ (with $\bf 1$ the unit matrix). The formal solution of this equation can be seen in eq. (26) of [@trujillo03].
Once the coefficients $\epsilon_I$ and $\epsilon_X$ (with $X=Q,U,V$) of the emission vector and the coefficients $\eta_I$, $\eta_X$, and $\rho_X$ of the $4\times4$ propagation matrix are known at each point within the medium it is possible to solve formally Eq. (\[eq:rad\_transfer\_peo\]) or Eq. (\[eq:rad\_transfer\_delo\]) for obtaining the emergent Stokes profiles for any desired line of sight. Our computer program considers the following levels of sophistication for the solution of the radiative transfer equation:
- [*Numerical Solutions*]{}. The most general case, where the properties of the slab vary along the ray path, has to be solved numerically. To this end, two efficient and accurate methods of solution of the Stokes-vector transfer equation are those proposed by [@trujillo03] (see his eqs. (24) and (27), respectively). The starting points for the development of these two numerical methods were Eq. (\[eq:rad\_transfer\_peo\]) and Eq. (\[eq:rad\_transfer\_delo\]), respectively. Both methods can be considered as generalizations, to the Stokes-vector transfer case, of the well-known short characteristics method for the solution of the standard (scalar) transfer equation.
- [*Exact analytical solution of the problem of a constant-property slab including the magneto-optical terms of the propagation matrix*]{}. For the general case of a constant-property slab of arbitrary optical thickness we actually have the following analytical solution, which can be easily obtained as a particular case of eq. (24) of [@trujillo03]:
$${\bf I}={\rm e}^{-{\mathbf{K}^{*}}\tau}\,{\bf I}_{\rm sun}\,+\,\left[{\mathbf{K}^{*}}\right]^{-1}\,
\left( \mathbf{1} - {\rm e}^{-{\mathbf{K}^{*}}\tau} \right) \,\mathbf{S},
\label{eq:slab_peo}$$
where $\mathbf{I}_{\rm sun}$ is the Stokes vector that illuminates the slab’s boundary that is most distant from the observer. We point out that the exponential of the propagation matrix ${\mathbf{K}^{*}}$ has an analytical expression similar to eq. (8.23) in [@landi_landolfi04].
- [*Approximate analytical solution of the problem of a constant-property slab including the magneto-optical terms of the propagation matrix*]{}. An approximate analytical solution to the constant-property slab problem can be easily obtained as a particular case of eq. (27) of [@trujillo03]:
$$\mathbf{I} = \left[ \mathbf{1}+\Psi_0 \mathbf{K}' \right]^{-1} \left[ \left(
e^{-\tau} \mathbf{1} - \Psi_M \mathbf{K}' \right) \mathbf{I}_{\rm sun} +
(\Psi_M+\Psi_0) \mathbf{S} \right],
\label{eq:slab_delo}$$
where the coefficients $\Psi_M$ and $\Psi_0$ depend only on the optical thickness of the slab at the frequency and line-of-sight under consideration, since their expressions are: $$\begin{aligned}
\Psi_M&=& \frac{1-e^{-\tau}}{\tau} - e^{-\tau},\nonumber \\
\Psi_0 &=&1-\frac{1-e^{-\tau}}{\tau}.\end{aligned}$$
Note that Eq. (\[eq:slab\_delo\]) for the emergent Stokes vector is the one used by [@trujillo_asensio07] for investigating the impact of atomic level polarization on the Stokes profiles of the He [i]{} 10830 Å multiplet. We point out that, strictly speaking, it can be considered only as the exact analytical solution of the optically-thin constant-property slab problem[^1]. The reason why Eq. (\[eq:slab\_delo\]) is, in general, an approximate expression for calculating the emergent Stokes vector is because its derivation assumes that the Stokes vector within the slab varies linearly with the optical distance. However, it provides a fairly good approximation to the emergent Stokes profiles (at least for all the problems we have investigated in this paper). Moreover, the results of fig. 2 of [@trujillo_asensio07] remain also virtually the same when using instead the exact Eq. (\[eq:slab\_peo\]), which from a computational viewpoint is significantly less efficient than the approximate Eq. (\[eq:slab\_delo\]).
- [*Exact analytical solution of the problem of a constant-property slab when neglecting the second-order terms of the Stokes-vector transfer equation*]{}. Simplified expressions for the emergent Stokes vector can be obtained when $\epsilon_I{\gg}\epsilon_X$ and $\eta_I{\gg}(\eta_X,\rho_X)$, which justifies to neglect the second-order terms of Eq. (\[eq:rad\_transfer\]). The resulting approximate formulae for the emergent Stokes parameters are given by eqs. (9) and (10) of [@trujillo_asensio07], which are identical to those used by [@trujillo_merenda05] for modeling the Stokes profiles observed in solar chromospheric spicules. We point out that there is a typing error in the sentence that introduces such eqs. (9) and (10) in [@trujillo_asensio07], since they are obtained only when the above-mentioned second-order terms are neglected in Eq. (\[eq:rad\_transfer\]), although it is true that there are no magneto-optical terms in the resulting equations.
- [*Optically thin limit*]{}. Finally, the most simple solution is obtained when taking the optically thin limit ($\tau{\ll}1$) in the equations reported in the previous point, which lead to the equations (11) and (12) of [@trujillo_asensio07]. Note that if $\mathbf{I}_{\rm sun}=0$ (i.e., $I_0=X_0=0$), then such optically thin equations imply that ${X/I}\,{\approx}\,{\epsilon_X}/{\epsilon_I}$.
The coefficients of the emission vector and of the propagation matrix depend on the multipolar components, $\rho^K_Q(J,J^{'})$, of the atomic density matrix. Let us recall now the meaning of these physical quantities and how to calculate them in the presence of an arbitrary magnetic field under given illumination conditions.
The multipolar components of the atomic density matrix
------------------------------------------------------
We quantify the atomic polarization of the levels using the multipolar components of the atomic density matrix. The atom can be correctly described under the framework of the $L$-$S$ coupling [e.g., @condon_shortley35]. As illustrated in Fig. 2, the different $J$-levels are grouped in terms with well defined values of the electronic angular momentum $L$ and the spin $S$. Since the $^4$He atoms are devoid of nuclear angular momentum, we do not have to consider hyperfine structure[^2]. The energy separation between the $J$-levels pertaining to each term is very small in comparison with the energy difference between different terms. For example, the energy separation between the $J=3$ and $J=2$ levels of the term 3d$^3$D (the upper term of the D$_3$ multiplet) is of the order of $0.0025\,{\rm cm}^{-1}$, which is $\sim$2$\times$10$^{5}$ times smaller than the separation between the 3d$^3$D and 3p$^3$P terms. On the other hand, although the energy separations between the $J$-levels of the upper terms of the 10830 Å and D$_3$ multiplets are much larger than their natural widths, such $J$-levels suffer crossings and repulsions for the typical magnetic strengths encountered in the solar atmospheric plasma (e.g., the $J=2$ and $J=1$ levels of the upper term of the 10830 Å multiplet cross for magnetic strengths between 400 G and 1600 G, while the $J=3$ and $J=2$ levels of the upper term of the D$_3$ multiplet cross for strengths between 10 G and 100 G, approximately). This can be seen clearly in Fig. \[fig:splitting\]. Therefore, it turns out to be fundamental to allow for coherences between different $J$-levels pertaining to the same term but not between the $J$-levels pertaining to different terms. As a result, we can represent the atom under the formalism of the multi-term atom discussed by [@landi_landolfi04].
In the absence of magnetic fields the energy eigenvectors can be written using Dirac’s notation as $|\beta L S J M\rangle$, where $\beta$ indicates a set of inner quantum numbers specifying the electronic configuration. In general, if a magnetic field of arbitrary strength is present, the vectors $|\beta L S J M\rangle$ are no longer eigenfunctions of the total Hamiltonian and $J$ is no longer a good quantum number. In this case, the eigenfunctions of the full Hamiltonian can be written as the following linear combination: $$\label{eq:eigenfunctions_total_hamiltonian}
|\beta L S j M\rangle = \sum_J C_J^j(\beta L S, M) |\beta L S J M\rangle,$$ where $j$ is a pseudo-quantum number which is used for labeling the energy eigenstates belonging to the subspace corresponding to assigned values of the quantum numbers $\beta$, $L$, $S$, and $M$, and where the coefficients $C_J^j$ can be chosen to be real.
In the presence of a magnetic field sufficiently weak so that the magnetic energy is much smaller than the energy intervals between the $J$-levels, the energy eigenvectors are still of the form $|\beta L S J M\rangle$ ($C_J^j(\beta L S, M) \approx \delta_{Jj}$), and the splitting of the magnetic sublevels pertaining to each $J$-level is linear with the magnetic field strength. For stronger magnetic fields, we enter the incomplete Paschen-Back effect regime in which the energy eigenvectors are of the general form given by Eq. (\[eq:eigenfunctions\_total\_hamiltonian\]), and the splitting among the various $M$-sublevels is no longer linear with the magnetic strength. This regime is reached for magnetic strengths of the order of 10 G for the D$_3$ multiplet and of the order of 100 G for the 10830 Å multiplet (see Fig. \[fig:splitting\]). If the magnetic field strength is further increased we eventually reach the so-called complete Paschen-Back effect regime, where the energy eigenvectors are of the form $|L S M_L M_S\rangle$ and each $L$-$S$ term splits into a number of components, each of which corresponding to particular values of ($M_L+2M_S$). Within the framework of the multi-term atom model the atomic polarization of the energy levels is described with the aid of the density matrix elements $$\rho^{\beta L S}(jM,j'M') = \langle \beta L S j M | \rho | \beta L S j' M'\rangle,$$ where $\rho$ is the atomic density matrix operator. Using the expression of the eigenfunctions of the total Hamiltonian given by Eq. (\[eq:eigenfunctions\_total\_hamiltonian\]), the density matrix elements can be rewritten as: $$\rho^{\beta L S}(jM,j'M') = \sum_{JJ'} C_J^j(\beta L S, M) C_{J'}^{j'}(\beta L
S, M') \rho^{\beta L S}(JM,J'M'),$$ where $\rho^{\beta L S}(JM,J'M')$ are the density matrix elements on the basis of the eigenvectors $| \beta L S J M\rangle$.
Following [@landi_landolfi04], it is helpful to use the spherical statistical tensor representation, which is related to the previous one by the following linear combination: $$\begin{aligned}
{^{\beta LS}\rho^K_Q(J,J')} &=& \sum_{jj'MM'} C_J^j(\beta L S, M)
C_{J'}^{j'}(\beta L S, M') \nonumber \\
&\times& (-1)^{J-M} \sqrt{2K+1} { \left(\begin{array}{ccc}
J&J'&K\\
M&-M'&-Q
\end{array}\right) }
\rho^{\beta L S}(jM,j'M'),\end{aligned}$$ where the 3-j symbol is defined as indicated by any suitable textbook on Racah algebra. This alternative representation has some advantages. Firstly, the well-known results obtained when atomic polarization effects are disregarded are easily recovered by considering only the elements of the density matrix with $K=0$ and $Q=0$. Secondly, the transformation law under rotations is much simpler because it involves only one rotation matrix. Finally, each $\rho^K_Q$ element has a clear physical interpretation: the $\rho^2_Q$ elements (with $Q=0,{\pm}1, {\pm}2$) are called the atomic alignment components, while the $\rho^1_Q$ elements (with $Q=0,{\pm}1$) are the atomic orientation components. The ${^{\beta LS}\rho^K_Q(J,J')}$ elements are, in general, complex quantities (except for the elements with $Q=0$ and $J=J'$, that are real quantities), so that, taking into account that the density matrix is an hermitian operator, the number of real quantities required to describe the atomic polarization properties of a given $L$-$S$ term is $(2S+1)^2(2L+1)^2$. This makes a total of 405 real quantities to describe the atomic polarization of the 5-term model atom shown in Fig. \[fig:helium\_atom\].
Statistical equilibrium equations
---------------------------------
In order to obtain the ${^{\beta LS}\rho^K_Q(J,J')}$ elements we have to solve the statistical equilibrium equations. These equations, written in a reference system in which the quantization axis ($Z$) is directed along the magnetic field vector and neglecting the influence of collisions, can be written as [@landi_landolfi04]: $$\begin{aligned}
\frac{d}{dt} {^{\beta LS}\rho^K_Q(J,J')} &=& -2\pi \mathrm{i} \sum_{K' Q'}
\sum_{J'' J'''} N_{\beta LS}(KQJJ',K'Q'J''J''') {^{\beta LS}\rho^{K'}_{Q'}(J'',J''')}
\nonumber \\
&+& \sum_{\beta_\ell L_\ell K_\ell Q_\ell J_\ell J_\ell'} {^{\beta_\ell L_\ell
S}\rho^{K_\ell}_{Q_\ell}(J_\ell,J_\ell')}
\mathbb{T}_A(\beta L S K Q J J', \beta_\ell L_\ell S K_\ell Q_\ell J_\ell
J_\ell') \nonumber \\
&+& \sum_{\beta_u L_u K_u Q_u J_u J_u'} {^{\beta_u L_u
S}\rho^{K_u}_{Q_u}(J_u,J_u')}
\Big[ \mathbb{T}_E(\beta L S K Q J J', \beta_u L_u S K_u Q_u J_u J_u') \nonumber \\
& &\qquad \qquad \qquad \qquad \qquad + \mathbb{T}_S(\beta L S K Q
J J', \beta_u L_u S K_u Q_u J_u J_u') \Big] \nonumber \\
&-& \sum_{K' Q' J'' J'''} {^{\beta L S}\rho^{K'}_{Q'}(J'',J''') } \Big[
\mathbb{R}_A(\beta L S K Q J J' K' Q' J'' J''') \nonumber \\
& & + \mathbb{R}_E(\beta L S K Q J J' K'
Q' J'' J''') + \mathbb{R}_S(\beta L S K Q J J' K' Q' J'' J''') \Big].
\label{eq:see}\end{aligned}$$ The first term in the right hand side of Eq. (\[eq:see\]) takes into account the influence of the magnetic field on the atomic level polarization. This term has its simplest expression in the chosen magnetic field reference frame [see eq. 7.41 of @landi_landolfi04]. In any other reference system, a more complicated expression arises. The second, third and fourth terms account, respectively, for coherence transfer due to absorption from lower levels ($\mathbb{T}_A$), spontaneous emission from upper levels ($\mathbb{T}_E$) and stimulated emission from upper levels ($\mathbb{T}_S$). The remaining terms account for the relaxation of coherences due to absorption to upper levels ($\mathbb{R}_A$), spontaneous emission to lower levels ($\mathbb{R}_E$) and stimulated emission to lower levels ($\mathbb{R}_S$), respectively.
The stimulated emission and absorption transfer and relaxation rates depend explicitly on the radiation field properties [see eqs. 7.45 and 7.46 of @landi_landolfi04]. The symmetry properties of the radiation field are accounted for by the spherical components of the radiation field tensor:
$$J^K_Q(\nu) = \oint \frac{d\Omega}{4\pi} \sum_{i=0}^3
\mathcal{T}^K_Q(i,\mathbf{\Omega}) S_i(\nu,\mathbf{\Omega}).
\label{eq:jkq}$$
The quantities $\mathcal{T}^K_Q(i,\mathbf{\Omega})$ are spherical tensors that depend on the reference frame and on the ray direction $\mathbf{\Omega}$. They are given by $$\mathcal{T}^K_Q(i,\mathbf{\Omega}) = \sum_P t^K_P(i) \mathcal{D}^K_{PQ}(R'),
\label{eq:tkq}$$ where $R'$ is the rotation that carries the reference system defined by the line-of-sight $\mathbf{\Omega}$ and by the polarization unit vectors $\mathbf{e}_1$ and $\mathbf{e}_2$ into the reference system of the magnetic field, while $\mathcal{D}^K_{PQ}(R')$ is the usual rotation matrix [e.g., @edmonds60]. Table 5.6 in [@landi_landolfi04] gives the $\mathcal{T}^K_Q(i,\mathbf{\Omega})$ values for each Stokes parameter $S_i$ (with $S_0=I$, $S_1=Q$, $S_2=U$ and $S_3=V$).
Emission and absorption coefficients
------------------------------------
Once the multipolar components ${^{\beta L S}\rho^{K}_{Q}(J,J') }$ are known, the coefficients $\epsilon_I$ and $\epsilon_X$ (with $X=Q,U,V$) of the emission vector and the coefficients $\eta_I$, $\eta_X$, and $\rho_X$ of the propagation matrix for a given transition between an upper term $(\beta L_u S)$ and an lower term $(\beta L_\ell S)$ can be calculated with the expressions of §7.6.b in [@landi_landolfi04]. These radiative transfer coefficients are proportional to the number density of atoms, $\mathcal{N}$. Their defining expressions contain also the Voigt profile and the Faraday-Voigt profile [see §5.4 in @landi_landolfi04], which involve the following parameters: $a$ (i.e., the reduced damping constant), $v_\mathrm{th}$ (i.e., the velocity that characterizes the thermal motions, which broaden the line profiles), and $v_\mathrm{mac}$ (i.e., the velocity of possible bulk motions in the plasma, which produce a Doppler shift).
It is important to emphasize that the expressions for the emission and absorption coefficients and those of the statistical equilibrium equations are written in the reference system whose quantization axis is parallel to the magnetic field. The following equation indicates how to obtain the density matrix elements in a new reference system: $$\left[ {^{\beta L S}\rho^{K}_{Q}(J,J') } \right]_\mathrm{new} = \sum_{Q'} \left[
{^{\beta L S}\rho^{K}_{Q'}(J,J') } \right]_\mathrm{old}
\mathcal{D}^K_{Q' Q}(R)^*,$$ where $\mathcal{D}^K_{Q' Q}(R)^*$ is the complex conjugate of the rotation matrix for the rotation $R$ that carries the old reference system into the new one.
The atomic model
----------------
The atomic model we have used in our calculations includes the following five terms of the triplet system of neutral helium: 2s$^3$S, 3s$^3$S, 2p$^3$P, 3s$^3$P and 3d$^3$D (see Fig. \[fig:helium\_atom\]). It has been concluded that the inclusion of these five terms is sufficient for a reliable calculation of the atomic polarization that the anisotropic pumping of the photospheric continuum radiation produces in the lower and upper $J$-levels of the D$_3$ multiplet [@bommier80]. Since the only level (with $J=1$) of the lower term 2s$^3$S is metastable, the adopted atomic model should be also satisfactory for the 10830 Å multiplet. In order to check this we have compared the results obtained using two different atomic models. The first one is that of Fig. \[fig:helium\_atom\]. The second one is a simplified version in which only the 2s$^3$S, 2p$^3$P and 3d$^3$D terms are taken into account. We have verified, for a large number of possible configurations, that differences in the resulting values of the multipolar components of the atomic density matrix are never larger than $\sim$5% for the terms involved in the 10830 Å transitions. Although the selection rules allow six transitions among the 5 terms, we have only included the four transitions indicated in Fig. \[fig:helium\_atom\] (i.e., we have neglected the influence of the infrared transitions 3d$^3$D-3p$^3$P and 3p$^3$P-3s$^3$S). The Einstein coefficients for the four included transitions shown in Table \[tab:tab\_einstein\] and the energy of the levels shown in Fig. \[fig:helium\_atom\] have been obtained from the NIST database[^3] [see also @wiese_nist66; @drake_helium98]. Table \[tab:tab\_einstein\] also indicates the value of the critical magnetic field strength for the operation of the upper-level Hanle effect, which results from equating the Zeeman splitting of the level with its natural width:
$$B_\mathrm{critical} \approx 1.137 \times 10^{-7} / (t_\mathrm{life} g_L),
\label{eq:critical_field}$$
where $t_\mathrm{life}$ is the level’s lifetime in seconds, $g_L$ is its Landé factor and $B_\mathrm{critical}$ is given in gauss. Note that for obtaining the $B_\mathrm{critical}^\mathrm{upper}$ values of Table \[tab:tab\_einstein\] we have used $t_\mathrm{life}{\approx}1/A_{ul}$, while an estimation of the critical magnetic field strength for the operation of the lower-level Hanle effect requires using $t_\mathrm{life} \approx 1/(B_{lu}J^0_0)$ (which for the metastable lower-level of the He [i]{} 10830 Å multiplet gives $B_\mathrm{critical}^\mathrm{lower} \approx 0.1$ G).
We point out that in our modeling we are not explicitly taking into account the radiative mechanism that is thought to be responsible of the overpopulation of the triplet levels of required to produce the absorption or emission features observed in the spectral lines of such two multiplets –that is, ionizations from the singlet states of caused by EUV ionizing radiation coming downwards from the corona followed by recombinations towards both the singlet and triplet states [e.g., @avrett94]. The fact that most of the ionizations take place from the singlet states suggests that the atomic polarization of the triplet states should be little affected by such EUV coronal irradiation. This expectation is reinforced by the fact that the number of photoionizations per unit volume and time from the triplet levels of is way smaller than the number of bound-bound transitions [see Fig. 7 in @centeno08b], which suggests that the atomic polarization of the $J$-levels of the 10830 Å and D$_3$ multiplets is indeed dominated by optical pumping in the line transitions themselves. Following our approach, the key mechanism responsible of the absorption or emission observed in such lines of neutral helium, be it the above-mentioned photoionization-recombination mechanism and/or collisional excitation in regions with T$>20000$ K [cf., @andretta_jones97], is unimportant in our forward modeling and inversion approach. The reason is that they are implicitly accounted for via the definition of the optical thickness of the slab which, being a free parameter in our modeling, is used to fit the observed intensity profiles. This point has been clarified by [@centeno08b].
The incident radiation field
----------------------------
As mentioned before, we consider a slab of atoms anisotropically illuminated from below by the photospheric continuum radiation, which we assume to have axial symmetry around the solar local vertical direction. Since the illumination conditions are assumed to be known a priori, the radiation field tensors can be calculated directly from the given incident radiation.
For symmetry reasons, it is advantageous to calculate the statistical tensors of the radiation field in a reference frame in which the $Z$-axis is along the vertical of the atmosphere. Since the incoming radiation is assumed to be unpolarized, the only non-zero spherical tensor components of the radiation field are $J^0_0$ and $J^2_0$. They quantify the mean intensity and the “degree of anisotropy” of the radiation field, respectively. For the case of our plane-parallel slab model, their expressions reduce to: $$\begin{aligned}
\label{eq:j00_j20}
J^0_0 &=& \frac{1}{2} \int_{-1}^{1} I(\mu) \mathrm{d}\mu \\
J^2_0 &=& \frac{1}{2\sqrt{2}} \int_{-1}^{1} (3\mu^2-1) I(\mu) \mathrm{d}\mu,\end{aligned}$$ where $\mu=\cos \theta$ is the cosine of the heliocentric angle $\theta$.
It is convenient to parameterize the radiation field in terms of the number of photons per mode $\bar n$ and the anisotropy factor $w$: $$\bar n = \frac{c^2}{2 h \nu^3} J^0_0, \qquad w = \sqrt{2} \frac{J^2_0}{J^0_0},
\label{eq:nbar_omega}$$ where $c$ is the light speed, $h$ is the Planck constant and $\nu$ is the frequency of the transition. The anisotropy factor fulfills $-1/2 \leq w \leq 1$. It reaches $w=1$ when the radiation is unidirectional along the polar axis of the reference system, while $w=-1/2$ when the radiation field is azimuth-independent and confined to the plane perpendicular to the quantization axis. We have calculated the tensors of the radiation field by using the center-to-limb variation and the wavelength dependence of the solar continuum radiation field tabulated by [@pierce00]. Since the lines originate in the outer regions of the solar/stellar atmosphere, it is necessary to take into account the geometrical effect produced by the non-negligible height $h$ above the surface of the star. To this end, we have applied the strategy outlined in §12.3 of [@landi_landolfi04]. Figure \[fig:nbar\_omega\] shows the sensitivity of $w$ and $\bar n$ to the height in the solar atmosphere at which our slab is assumed to be located. The number of photons per mode decreases, while the anisotropy factor rapidly increases. This effect is of a purely geometrical nature.
It is important to note that the theory of spectral line polarization presented in the monograph of [@landi_landolfi04] is only valid if the radiation field illuminating the atomic system is spectrally flat (independent of frequency) over a frequency interval larger than the Bohr frequencies connecting the levels that present quantum coherences. Fortunately, this is the case for all the lines included in the atomic model of Fig. \[fig:helium\_atom\], except for the 3p$^3$P-2s$^3$S transition at 3888.6 Å which is situated in a quite crowded region of the spectrum. Fortunately, this line is of secondary importance in setting the statistical equilibrium, and according to [@landi_landolfi04] the statistical tensors can be calculated by simply reducing the photospheric continuum radiation intensity at that wavelength by a factor 5. In any case, its inclusion has a rather negligible impact on the final results.
The forward modeling code
=========================
By forward modeling we mean the calculation of the emergent Stokes profiles for given values of the height $h$ at which the slab is located above the visible solar “surface", of the slab’s optical thickness and of the magnetic field vector. We have written this option of our computer program in a way such that it performs the calculation at various levels of realism. We point out that similar type of forward-modeling calculations can be carried out also with some of the computer programs mentioned in §1, which are likewise based on the density-matrix theory of spectral line polarization. The main difference with ours is that we have taken into account radiative transfer effects (without neglecting the magneto-optical terms of the Stokes-vector transfer equation) and that we have developed a user-friendly interface to facilitate the performing of numerical experiments (see Fig. \[fig:front-end\]). In what follows we first describe the various options of our computer program and then show some illustrative examples of possible applications.
Description of the computing options
------------------------------------
The solution of the statistical equilibrium equations, using the radiation field tensors calculated from the given illumination conditions, provides the mutipolar components of the atomic density matrix. The computer program calculates such $\rho^K_Q(J,J^{'})$ elements in both the magnetic field reference frame (if a deterministic magnetic field is present) and in a reference system where the quantization axis lies along the solar local vertical direction (hereafter, “vertical frame"). This option of the forward modeling code helps to understand what’s going on at the atomic level when an atomic system is subjected to anisotropic radiative pumping processes in the absence and in the presence of a magnetic field. Some illustrative examples for the $J$-levels involved in the 10830 Å and D$_3$ transitions are shown in §3.2 below.
The values of the multipolar components of the density matrix is all we need for calculating the coefficients of the emission vector and of the propagation matrix, which enter the Stokes vector transfer equation (\[eq:rad\_transfer\]). Our forward modeling code can solve this equation at the various levels of approximation explained in §\[sec:radiative\_transfer\]. Moreover, it can also compute the emergent Stokes profiles calculating the wavelength positions of the $\pi$ and $\sigma$ components assuming the linear Zeeman-effect regime (instead of using the general Paschen-Back effect theory), incorporating or discarding the influence of atomic level polarization.
Atomic level polarization in two reference systems
--------------------------------------------------
Figure \[fig:coherences\] shows an illustrative example of the population imbalances \[$\rho^2_0(J)$ and $\rho^1_0(J)$\] and quantum coherences \[$\rho^2_Q(J)$ and $\rho^1_Q(J)$, with $Q{\ne}0$\] induced by optical pumping processes in the levels of the 10830 Åmultiplet that can carry atomic polarization (i.e., the lower level, with $J_{l}=1$, and the upper levels with $J_{u}=2$ and $J_{u}=1$). Like in [@trujillo_asensio07], we assume a slab with $\Delta{\tau}_{\rm red}=1$, where the label “red” indicates that the optical thickness of the slab along its normal direction is measured at the frequency where the peak of the red blended component of the 10830 Å multiplet is located. The slab is assumed to be at a height of only 3 arcseconds and in the presence of a horizontal magnetic field whose strength we can vary at will.
Consider first the results for $J_{l}=1$ and $B=0$ G in a reference system with the quantization axis along the local vertical. Since the pumping radiation is assumed to be unpolarized and with axial symmetry around the vertical, for $B \approx 0$ G we see only population imbalances of the form $\rho^2_0(J)$. Note that in the absence of magnetic fields the $\rho^1_0(J_l) $ atomic orientation value is zero because there is no net circular polarization in the incident radiation field. The bottom right panel of Fig. \[fig:coherences\] shows that, for $B=0$ G, we have lower-level quantum coherences of the form $\rho^2_Q(J)$ (with $Q{\ne}0$) in the magnetic reference system, whose $Z$-axis is inclined with respect to the symmetry axis of the pumping radiation field. In fact, for given illumination conditions in the absence of a magnetic field, whether or not we have such quantum coherences depends only on the reference system. In each of the two bottom panels it is shown what happens with the population imbalances and the coherences of the $J_{l}=1$ lower level as the strength of the assumed horizontal field is increased. Consider, for instance, the right panel results corresponding to the magnetic field reference frame. Note that the lower-level Hanle effect starts to operate for field strengths as low as 0.01 G, and that for field intensities of the order or larger than 1 G all the lower-level coherences have been relaxed. This happens because such a lower level is metastable, which implies that its critical Hanle field is only $0.1$ G (see Eq. \[eq:critical\_field\]). It is also of interest to point out that if the $J$-levels of our multiterm atomic model were isolated levels then the $\rho^2_0(J)$ population imbalances would be constant in the magnetic field reference frame, and the $\rho^1_0(J)$ atomic orientation value would be zero. However, since the $J$-levels suffer crossings and repulsions non-zero $\rho^1_0(J)$ values appear through the alignment-to-orientation conversion mechanism [cf. @landi_d3_82], which for the 10830 Å triplet has however a negligible impact on the emergent Stokes $V$ profiles. In addition, as shown in the bottom right panel, the $\rho^2_0(J_l)$ alignment coefficient itself starts to be modified as soon as the field strength is sensibly larger than 100 G. Although this lower level does not suffer any crossings with the other $J$-levels of the model atom of Fig. \[fig:helium\_atom\], its atomic polarization is modified because it sensitively depends on that of the upper levels of the 10830 Å multiplet.
Consider now the results for the upper levels with $J_{u}=1$ (central panels of Fig. \[fig:coherences\]) and $J_{u}=2$ (top panels). Obviously, we only see population imbalances of the form $\rho^2_0(J)$ at zero field in the vertical frame. Around $B=10^{-2}$ G, the density matrix elements start to be affected by the magnetic field. This modification is due to the feedback effect that the alteration of the lower-level polarization has on the upper levels. It is also possible to note in the central and top panels of Fig. \[fig:coherences\] the action of the upper level Hanle effect on the $J_{u}=1$ and $J_{u}=2$ levels. In fact, there is a hint of a small plateau just above 0.1 G, as expected from the fact that the critical upper-level Hanle field value is 0.8 G for the 10830 Å multiplet (see Table \[tab:tab\_einstein\]). Similar features can be seen in the corresponding right panels of Fig. \[fig:coherences\]. Probably, the most notable conclusion to highlight from this figure is that between approximately 10 and 100 G we only have population imbalances of the form $\rho^2_0(J)$ in the magnetic field reference frame (i.e., we can consider that between 10 and 100 G the He [i]{} 10830 Å multiplet is in the saturation regime of the Hanle effect, where the coherences are negligible and the atomic alignment values of the lower and upper levels are insensitive to the strength of the magnetic field).
Finally, in Fig. \[fig:coherencesD3\] we show similar results but for the upper levels of the He [i]{} D$_3$ multiplet. The situation now is much more complicated, as evidenced by the fact that it is impossible to find a range of solar magnetic field strength values between which the coherences are zero in the magnetic field reference frame and, at the same time, the population imbalances are insensitive to the magnetic strength.
How to investigate empirically the possibility of magnetic canopies in the quiet solar chromosphere? {#sec:canopies}
----------------------------------------------------------------------------------------------------
In the presence of an *inclined* magnetic field forward scattering processes can produce linear polarization signatures in spectral lines [e.g., the review by @trujillo01]. In this geometry, the polarization signal is *created* by the Hanle effect, a physical phenomenon that has been clearly demonstrated via spectropolarimetry of solar coronal filaments in the 10830 Å multiplet [@trujillo_nature02].
Fig. \[fig:canopies\] shows theoretical examples of the emergent fractional linear polarization in the lines of the 10830 Å multiplet assuming a constant-property slab of helium atoms permeated by a horizontal magnetic field of 10 G. As expected, the smaller the optical thickness of the assumed plasma structure the smaller the fractional polarization amplitude. In principle, the Tenerife Infrared Polarimeter [TIP; @martinez_pillet99; @collados_tipII07] mounted on the Vacuum Tower Telescope (VTT) of the Observatorio del Teide allows the detection of very low amplitude polarization signals, such as those corresponding to the $\Delta{\tau}_{\rm red}=0.1$ case of Fig. \[fig:canopies\]. However, in order to be able to achieve this goal without having to sacrifice the spatial and/or temporal resolution we need a larger aperture solar telescope.
We consider now the question of whether it is safer to interpret disk-center observations or to opt for a different scattering geometry. To this end, we have investigated how is the variation of $Q/I$ and $U/I$ at the central wavelengths of the red and blue components of the 10830 Å multiplet for different inclinations ($\theta_B$) of the magnetic field vector and for different line-of-sights, assuming $\Delta{\tau}_{\rm red}=0.1$ and $B=10$ G (which implies that we are very close to the saturation regime of the upper level Hanle effect). Fig. \[fig:canopies\_qi\_peak\] shows only the case of magnetic field vectors with azimuth $\chi_B=90^{\circ}$ (i.e., contained in the Z-Y plane of Fig. \[fig:geometry\]) and for line-of-sights with $\chi=0$ (i.e., contained in the Z-X plane of Fig. \[fig:geometry\]). It is interesting to note that for the case of a horizontal magnetic field (i.e., $\theta_B=90^{\circ}$, which implies that the magnetic field vector forms always an angle of $90^{\circ}$ with respect to any of the line-of-sights of Fig. \[fig:canopies\_qi\_peak\]) Stokes $U=0$ and the Stokes $Q$ amplitudes of the emergent spectral line radiation are identical for all such line-of-sights. This is easy to understand by using the following approximate expressions for $\epsilon_Q/\epsilon_I$ and $\eta_Q/\eta_I$, which for the case of a deterministic magnetic field provide a suitable approximation if we are in the saturation regime of the upper-level Hanle effect [@trujillo03]:
$${\frac{\epsilon_Q}{\epsilon_I}}\,{\approx}\,{\frac{3}{2\sqrt{2}}}\,({\mu}_B^2 -
1)\,{\cal W}\,[\sigma^2_0(J_u)]_B\, ,
\label{eq:QIa}$$
$${\frac{\eta_Q}{\eta_I}}\,{\approx}\,{\frac{3}{2\sqrt{2}}}\,({\mu}_B^2 -
1)\,{\cal Z}\,[\sigma^2_0(J_l)]_B\, ,
\label{eq:QIb}$$
where $[\sigma^2_0]_B=[\rho^2_0]_B/\rho^0_0$ quantifies the fractional atomic alignment in the magnetic field reference frame, while ${\cal W}$ and ${\cal Z}$ are numerical coefficients that depend on the $J_l$ and $J_u$ values[^4]. It is very important to note that in Eqs. (\[eq:QIa\]) and (\[eq:QIb\]) ${\mu}_B$ is [*the cosine of the angle between the magnetic field vector and the line of sight*]{}. Note also that in these expressions for $\epsilon_Q/\epsilon_I$ and $\eta_Q/\eta_I$, that are valid in the magnetic field reference frame, we have chosen the positive reference direction for Stokes $Q$ parallel to the projection of the magnetic field vector onto the plane perpendicular to the LOS, while in the similar Eqs. (16) and (17) of [@trujillo_asensio07] we chose it along the perpendicular direction. In both cases, we have $\epsilon_U=\eta_U=0$ (if we are really in the above-mentioned Hanle-effect saturation regime). Obviously, the observed $Q/I$ amplitude depends on the fractional atomic alignment of the upper and lower levels of the 10830 Å line transitions calculated in the magnetic field reference frame, but their values are independent of the LOS. Actually, in the weak anisotropy limit they are given by
$$[\sigma^2_0]_{B}=[\sigma^2_0]_{V}{\,} \frac{1}{2} (3 \cos^2{\theta_B}-1),$$
where $[\sigma^2_0]_{V}$ is the fractional atomic alignment in the vertical frame for the zero field case. Therefore, the dependency of the emergent Stokes $Q$ signal on the scattering angle is only through the factor $({\mu_B}^2 - 1)$, with ${\mu_B}=0$ for all the line-of-sights of Fig. \[fig:canopies\_qi\_peak\] if $\theta_B=90^{\circ}$.
As seen in Fig. \[fig:canopies\_qi\_peak\], for non-horizontal fields there are notable differences between the curves corresponding to each LOS, mainly for the cases with an inclination ($\theta_B$) of the magnetic field smaller than the Van-Vleck angle ($\theta_{VV}=54.73^{\circ}$, which corresponds to ${\rm
cos}^{2}(\theta_{VV})=1/3$). For the forward-scattering case of a disk-center observation (the $\mu=1$ curves of Fig. \[fig:canopies\_qi\_peak\]) Stokes $U \approx 0$ and Stokes $Q$ admits only one solution for $\theta_B > \theta_{VV}$ (with the exception of the well-known $180^{\circ}$ ambiguity of the Hanle effect)[^5]. However, for $\theta_B < \theta_{VV}$ we may have two different magnetic field inclinations producing the same Stokes $Q$ values, each of them having its corresponding $180^{\circ}$ ambiguity of the Hanle effect. As seen in the figure, the range of $\theta_B$ values where two such solutions for Stokes $Q$ are possible depends on the $\mu$-value of the LOS. Stokes $U$ is clearly non-zero for $\mu < 1$, but the fact that $U \approx 0$ for the case of magnetic field with fixed inclination but with random-azimuth within the spatio-temporal resolution element of the observation, leads us to conclude that the best one can do for a reliable empirical investigation of the possibility of canopy-like fields in the quiet solar chromosphere is to interpret spectropolarimetric observations at the solar disk center, such as those considered in §\[sec:internetwork\_regions\].
The Inversion Code {#sec:inversion}
==================
Our inversion strategy is based on the minimization of a merit function that quantifies how well the Stokes profiles calculated in our atmospheric model reproduce the observed Stokes profiles. To this end, we have chosen the standard $\chi^2$–function, defined as: $$\chi^2 = \frac{1}{4N_\lambda} \sum_{i=1}^4 \sum_{j=1}^{N_\lambda}
\frac{\left[S_i^\mathrm{syn}(\lambda_j)-S_i^\mathrm{obs}(\lambda_j) \right]^2}{
\sigma_i^2(\lambda_j)} ,$$ where $N_\lambda$ is the number of wavelength points and $\sigma_i^2(\lambda_j)$ is the variance associated to the $j$-th wavelength point of the $i$-th Stokes profiles. The minimization algorithm tries to find the value of the parameters of our model that lead to synthetic Stokes profiles $S_i^\mathrm{syn}$ with the best possible fit to the observations. For our slab model, the number of parameters (number of dimensions of the $\chi^2$ hypersurface) lies between 5 and 7, the maximum value corresponding to the optically thick case (see Table \[tab:parameters\]). The magnetic field vector ($B$, $\theta_B$ and $\chi_B$), the thermal velocity ($v_\mathrm{th}$) and the macroscopic velocity ($v_\mathrm{mac}$) are always required. This set of parameters is enough for the case of an optically thin slab. In order to account for radiative transfer effects, we need to define the optical depth of the slab along its normal direction and at a suitable reference wavelength (e.g., the central wavelength of the red blended component for the 10830 Å multiplet). In addition, we may additionally need to include the damping parameter ($a$) of the Voigt profile if the wings of the observed Stokes profiles cannot be fitted using Gaussian line profiles.
A critical problem in any inversion code is to identify possible degeneracies among different parameters of the model. When two or more parameters produce similar effects on the emergent Stokes profiles, the inversion algorithm is unable to distinguish between them. As a result, the emergent Stokes profiles corresponding to different combinations of the model parameters are indistinguishable within the noise level. Concerning this critical point, we investigate in detail in §\[sec:invert\_height\] the possibility of using the observed Stokes profiles of the 10830 Å triplet to obtain the value of the height $h$ at which the observed plasma structure is located.
Ambiguities in the determination of the model’s parameters can also result from the presence of degeneracies. However, this type of ambiguities occur only for a finite number of combinations of some parameters. Although the problem is complicated, it is possible to develop techniques that can help selecting one of the combinations as the most plausible. The well-known 180$^\circ$ ambiguity of the Hanle effect adds to the unfamiliar Van Vleck ambiguity [@house77; @casini_judge99; @casini05; @merenda06]. The role of these ambiguities in the inferred model’s parameters will be explored in §\[sec:van\_vleck\], although some hints have been already given in §\[sec:canopies\].
It is also important to point out the interest of developing methods capable of providing reliable confidence intervals for all the inferred parameters. In a future development, we plan to implement Bayesian inference techniques [@asensio_martinez_rubino07].
Levenberg-Marquardt
-------------------
Probably, the most well-known procedure for the minimization of the $\chi^2$-function is the Levenberg-Marquardt (LM) method [e.g., @numerical_recipes86]. The minimization strategy uses the Hessian method when the parameters are close to the minimum of the $\chi^2$-function (a quadratic form approximately describes this function around the minimum) and the steepest descent method when the parameters are far from the minimum. The transition between both methods is done in an adaptive manner. Its main drawback (which applies also to the majority of the standard numerical methods of function minimization) is that it can easily get trapped in local minima of the $\chi^2$-function. Some alternatives are available to overcome this difficulty. The most straightforward but time consuming one is to restart the minimization process at different values of the initial parameters. If the obtained minimum is systematically the same, the probability that this is the global optimum is high. However, when the $\chi^2$ parameter hypersurface is complicated, this technique does not give any confidence on the validity of the global minimum. Other possibilities rely on the application of some kind of “inertia” to the method, so that the LM method can overcome such a local minimum problem when moving on the parameter hypersurface. Again, these methods do not guarantee the success of getting the global minimum of the function. On the contrary, the LM method turns out to be one of the fastest and simplest options when the initial estimate of the parameters is close to the absolute minimum.
Global Optimization techniques
------------------------------
In order to avoid the possibility of getting trapped in a local minimum of the $\chi^2$ hypersurface, global optimization methods have to be used. Several optimization methods have been developed to obtain the global minimum of a function [e.g., @global_optimization95]. The majority of them are based on stochastic optimization techniques. The essential philosophy of these methods is to sample efficiently the whole space of parameters to find the global minimum of a given function. One of the most promising methods is genetic optimization (inspired by the fact of biological evolution), in which the parameters of the merit function are encoded in a gene. Although no mathematical proof of the convergence properties of these algorithms exists, recent advances suggest that the probability of convergence is very high [@gutowsky04]. Actually, they perform quite well in practice for the optimization of very hard problems. In solar spectropolarimetry, genetic optimization methods have been recently applied by [@Lagg04] to the inversion of Stokes profiles induced by the Zeeman effect in the 10830 Å triplet, neglecting the influence of atomic level polarization. The main disadvantage is that the computing time needed to reach convergence increases dramatically (by a factor $\sim$10 with respect to standard methods based on the gradient descent like LM). Another different approach is based on deterministic algorithms [@global_optimization95]. Typically, these algorithms rely on a strong mathematical basis, so that their convergence properties are well known. We have chosen the DIRECT algorithm [@Jones_DIRECT93], whose name derives from one of its main features: *di*viding *rect*angles. The idea is to recursively sample parts of the space of parameters, improving in each iteration the location of the part of the space where the global minimum is potentially located. The decision algorithm is based on the assumption that the function is Lipschitz continuous [see @Jones_DIRECT93 for details]. The method works very well in practice and can indeed find the minimum in functions that do not fulfill the condition of Lipschitz continuity. The reason is that the DIRECT algorithm does not require the explicit calculation of the Lipschitz constant but it uses all possible values of such a constant to determine if a region of the parameter space should be broken into subregions because of its potential interest [see @Jones_DIRECT93 for details]. A schematic illustration of the subdivision process for a function of two parameters is shown in Fig. \[fig:direct\_method\].
Convergence {#sec:convergence}
-----------
Taking into account that the dimension of our space of parameters is between 5 and 7, it seems unreasonable to use an algorithm like DIRECT to obtain a precise determination of the values of the model’s parameters at the global minimum. The reason is that the precision in the values of the parameters decreases with the size of the hyperrectangles. Therefore, we would need to perform many divisions to end up with a reasonable precision. What we do is to let the DIRECT algorithm locate the global minimum in a region whose hypervolume is $V$. This hypervolume is obtained as the product of the length $d_i$ of each dimension associated with each of the $N$ parameters: $$V = \prod_i^N d_i.$$ When the hypervolume decreases by a factor $f$ after the DIRECT algorithm has discarded some of the hyperrectangles, its size along each dimension is approximately decreased by a factor $f^{1/N}$. In order to end up with a small region where the global minimum is located, many subdivisions are necessary, thus requiring many function evaluations. For this reason, it has been observed that although the DIRECT algorithm rapidly finds the region where the global minimum is located, its local convergence properties are rather poor [see, e.g., @cox01; @bartholomew02; @ljungberg04 for applications in the extremely hard problems of the design of high-speed civil transport, aircrafts and bioinformatics]. In summary, the DIRECT method is an ideal candidate for its application as an estimator of the region where the global minimum is located, but not for determining it.
The most time consuming part of any optimization procedure is the evaluation of the merit function. The DIRECT algorithm needs only a reduced number of evaluations of the merit function to find the region where the global minimum is located. For this reason, we have chosen it as the initialization part of the LM method. Since the initialization point is close to the global minimum, the LM method, thanks to its quadratic behavior, rapidly converges to the minimum.
Stopping criterium
------------------
A critical and fundamental problem in the optimization of functions (either local or global) is to identify when the method has converged to the solution. We have used two stopping criteria for the DIRECT algorithm. The first one is stopping when the ratio between the hypervolume where the global minimum is located and the original hypervolume is smaller than a given threshold. This method has been chosen when using the DIRECT algorithm as an initialization for the LM method, giving very good results. The other good option, suggested by [@Jones_DIRECT93], is to stop after a fixed number of evaluations of the merit function.
Since the intensity profile is not very sensitive to the presence of a magnetic field (at least for magnetic field strengths of the order of or smaller than 1000 G), we have decided to estimate the optical thickness of the slab, the thermal and the macroscopic velocity of the plasma and the damping constant by using only the Stokes $I$ profile, and then to determine the magnetic field vector by using the polarization profiles. The full inversion scheme, shown schematically in Table \[tab:inversion\], begins by applying the DIRECT method to obtain a first estimation of the indicated four parameters by using only Stokes $I$. Afterwards, some LM iterations are carried out to refine the initial values of the model’s parameters obtained in the previous step. Once the LM method has converged, the inferred values of $v_\mathrm{th}$, $v_\mathrm{mac}$ (together with $a$ and $\Delta \tau$, when these are parameters of the model) are kept fixed in the next steps, in which the DIRECT method is used again for obtaining an initial approximation of the magnetic field vector ($B$,$\theta_B$,$\chi_B$). According to our experience, the first estimate of the magnetic field vector given by the DIRECT algorithm is typically very close to the final solution. Nevertheless, some iterations of the LM method are performed to refine the value of the magnetic field strength, inclination and azimuth. In any case, although we have found very good results with this procedure, the specific inversion scheme is fully configurable and can be tuned for specific problems.
Applications
============
The main aim of this section is to illustrate the application of our inversion code to some selected spectropolarimetric observations in the 10830 Å multiplet, showing that it gives results that are in agreement with the published ones. In addition, in §\[sec:internetwork\_regions\] we show a new application aimed at determining the strength and inclination of the magnetic field vector in the chromosphere above an internetwork region observed at solar disk center. Note that in the following applications $\Delta{\tau}_{\rm red}$ will continue denoting the optical thickness of the constant-property slab, along its normal direction, measured at the center of the red blended component of the 10830 Å multiplet.
Prominences
-----------
The first application is for the case of solar prominences, which are relatively cool and dense plasma structures embedded in the $T\sim 10^6$ K solar corona. In these objects the observed Stokes $Q$ and $U$ profiles of the 10830 Å multiplet are dominated by the presence of atomic level polarization, while the Stokes $V$ profile is dominated by the Zeeman effect. Recently, [@merenda06] have shown how to infer the magnetic field that confines the plasma of solar prominences via the inversion of the Stokes profiles induced by scattering processes and the Hanle and Zeeman effects in the lines of the 10830 Å multiplet. They analyzed in detail spectropolarimetric observations of the 10830 Å multiplet in a polar crown prominence and concluded that if the observed prominence was located in the plane of the sky the magnetic field had to be relatively strong ($B \approx 30 $ G) and inclined by only $25^{\circ}$ with respect to the local vertical.
We have applied our inversion code to the spectropolarimetric observations shown in Fig. \[fig:prominence\], taken from Fig. 9 of [@merenda06]. We have assumed that the observed plasma structure was optically thin and that the prominence is located in the plane of the sky. The inversion code was used to infer the value of the thermal velocity $v_\mathrm{th}$, the macroscopic velocity shift $v_\mathrm{mac}$ (to allow for a shift in the wavelength calibration) and the magnetic field vector $(B,\theta_B,\chi_B)$. The atmospheric height was fixed to $h=20"$, the same value used by [@merenda06]. After the four steps summarized in Table \[tab:inversion\], we end up with a thermal velocity $v_\mathrm{th}=7.97$ km s$^{-1}$, a bulk velocity that is compatible with zero and a magnetic field vector characterized by $B=26.8$ G, $\theta_B=25.5^\circ$ and $\chi_B=161.0^\circ$. These values are in very good agreement with the results of [@merenda06], namely $B=26$ G, $\theta_B=25^\circ$ and $\chi_B=160.5^\circ$. Note that since the prominence plasma was assumed to lie in the plane of the sky, the following solutions are also valid: $\theta_B^{*}=180^\circ-\theta_B$ and $\chi_B^{*}=-\chi_B$ (i.e., the well-known $180^{\circ}$ ambiguity of the Hanle effect).
We point out that the total number of evaluations of the merit function was 132. For the inversion of the Stokes profiles corresponding to other points of the field of view, one can initialize the inversion using the model’s parameters corresponding to the previous point. Using a few LM iterations, one should be able to reach the global minimum. In case this procedure does not work properly, one should return to the four-steps inversion scheme already presented in Table \[tab:inversion\].
Spicules {#sec:spicules}
--------
Another interesting problem is the determination of the magnetic field vector in solar chromospheric spicules. [@trujillo_merenda05] interpreted spectropolarimetric observations of spicules in the 10830 Å multiplet and concluded that the magnetic field of spicules in quiet regions of the solar chromosphere has a strength of the order of 10 G and is inclined by about 35$^\circ$ with respect to the local vertical. Their conclusion that the typical magnetic field strength is $\sim 10$ G required to obtain the longitudinal component of the magnetic field vector via some careful measurements of the Stokes $V$ profiles, such as that shown in Fig. 13 of [@trujillo_esa05]. This was needed because for field strengths larger than a few gauss the 10830 Å multiplet enters the saturation regime of the upper-level Hanle effect and the observed Stokes $Q$ and $U$ profiles provide only information on the orientation of the magnetic field vector. In fact, the application of our inversion code to the observed Stokes profiles shown in Fig. \[fig:spicules\] (where the Stokes $V$ profile is at the noise level), assuming that the spicular material is located in the plane of the sky, provides several different magnetic field vectors that lead to equally reliable fits. One of them, given by $B=10$ G, $\theta_B=37^\circ$ and $\chi_B=172^\circ$ is similar to the one chosen by [@trujillo_merenda05]. Another possible fit is the one illustrated in Fig. \[fig:spicules\], which corresponds to $B=2.6$ G, $\theta_B=37^\circ$ and $\chi_B=35^\circ$.
Fig. \[fig:spicules\_chi2\] gives the values of the $\chi^2$ function for all possible magnetic field inclinations and azimuths corresponding to the cases $B=10$ G (left panel) and $B=2.6$ G (right panel). In each of these panels we have indicated with white dots and numbers the four solutions that correspond to equally good best fits to the observed Stokes profiles of Fig. \[fig:spicules\]. The pair of solutions $1$ and $2$ correspond to the Van-Vleck ambiguity[^6]. The same applies to the $1'$ and $2'$ solutions. On the contrary, the pair of solutions $1$ and $1'$ or the $2$ and $2'$ are not strictly equivalent. The inversion code considers such pairs of solutions as equivalent because the observed Stokes $V$ profile is at the noise level and it is not able to differentiate between the two cases. Note that these pairs of solutions give exactly the same Stokes $Q$ and $U$ profiles, but their corresponding Stokes $V$ profiles have opposite signs. Concerning each pair of solutions in Fig. \[fig:spicules\_chi2\], it is possible to verify that the projections on the plane of the sky of the magnetic fields corresponding to solutions $1$ and $2$ form an angle close to to 90$^\circ$, which is typical of the Van-Vleck ambiguity. The same happens for the magnetic fields corresponding to solutions $1'$ and $2'$. This holds for both cases, $B=10$ G and $B=2.6$ G. As pointed out above, when the observed Stokes $V$ signal is very small, it is very hard (or impossible) to differentiate between the two possibilities having azimuths $\chi_B$ and $180^\circ-\chi_B$. The magnetic field vectors $1$ and $1'$ (or those corresponding to the $2$ and $2'$ solutions of Fig. \[fig:spicules\_chi2\]) have the same projection on the line of sight, except for a sign change. Therefore, the detection of Stokes $V$ turns out to be fundamental to determine which is the correct one [@trujillo_merenda05; @merenda06]. Apart from the considered solutions, which are restricted to the interval $0^\circ < \theta_B < 180^\circ$, one has also to take into account the well-known ambiguity of the Hanle effect, which applies when the emitting plasma is located in the plane of the sky. In this case, we have to add to the possible set of solutions all the combinations fulfilling $\theta_B^{*}=180^\circ-\theta_B$ and $\chi_B^{*}=-\chi_B$ since both pairs produce the same Stokes profiles.[^7].
Filaments
---------
We have also considered the inversion of the Stokes profiles presented in [@trujillo_nature02], which were observed in a solar coronal filament at the solar disk center. Such profiles, which are reproduced in Fig. \[fig:filament\_observation\], were used by those authors to demonstrate the presence of atomic polarization in the lower level of the 10830 Å multiplet and that the Hanle effect due to an inclined magnetic field creates linearly polarized radiation in forward scattering geometry. Note that the Stokes $Q$, $U$ and $V$ profiles are normalized to the maximum depression in Stokes $I$ (which is $0.4\,I_c$, approximately). The application of our inversion code using $h=40"$ confirms the conclusions of [@trujillo_nature02], yielding the following values for the model’s parameters: $\Delta \tau=0.86$, $v_\mathrm{th}=6.6$ km s$^{-1}$, $a=0.19$ and a magnetic field vector characterized by $B=18$ G and $\theta_B=105^\circ$.
Inter-network regions {#sec:internetwork_regions}
---------------------
As discussed in §\[sec:canopies\], the investigation of the possibility of horizontal magnetic canopies in the quiet solar chromosphere above internetwork regions is feasible with TIP, especially when interpreting measurements of the polarization of the 10830 Å multiplet in forward scattering at the solar disk center[^8]. We present in Fig. \[fig:canopy\_observation\] an observation carried out with TIP very close to the solar disk center ($\mu=0.98$). The slit was crossing an enhanced network region of circular shape. A time series of 50 steps with an integration time of 3 seconds was performed. The resulting polarimetric sensitivity after averaging over the 50 time steps and along a small spatial interval within the observed internetwork region is close to 6$\times$10$^{-5}$ in units of the continuum intensity. According to the results of the right panel of Fig. \[fig:canopies\], this is sufficient for detecting the linear polarization signal of a horizontal magnetic field provided the optical depth at the wavelength of the red component of the 10830 Å multiplet is larger than $\sim 0.01$. The reference system has been rotated until Stokes $U$ is minimized. Since the inferred magnetic field strength implies that the 10830 Å is in the saturation regime of the upper-level Hanle effect, the resulting reference direction for Stokes $Q$ lies either along the projection of the magnetic field vector on the solar disk, or along the direction perpendicular to such a projection. We have applied our inversion code to the above-mentioned observed profiles assuming a slab located at a height of 3 arcsec and we have obtained the following results: $\Delta \tau=0.19$, $v_\mathrm{th}=9.2$ km s$^{-1}$, $a=0.62$ and a magnetic field vector characterized by $B=35$ G, $\theta_B=21^\circ$ and $\chi_B=0^\circ$. However, other possible solutions can be found with a similar goodness of the fit (e.g., $B=47$ G, $\theta_B=47^\circ$ and $\chi_B=0^\circ$). These results obtained from a solar disk center observation suggest the presence of magnetic fields inclined by no more than $50^{\circ}$ in the observed quiet Sun chromospheric region.
Emerging magnetic flux regions
------------------------------
As pointed out by @trujillo_asensio07, the modeling of the emergent Stokes $Q$ and $U$ profiles of the 10830 Å multiplet should be done by taking into account the possible presence of atomic level polarization, even for magnetic field strengths as large as 1000 G. An example of a spectropolarimetric observation of an emerging magnetic flux region is shown by the circles of Fig. \[fig:lagg\_emerging\]. The solid lines show the best theoretical fit to these observations of [@Lagg04]. Here, in addition to the Zeeman effect, we took into account the influence of atomic level polarization. The dotted lines neglect the atomic level polarization that is induced by anisotropic radiation pumping in the solar atmosphere. Our results indicate the presence of atomic level polarization in a relatively strong field region (${\sim}$1000 G). However, it may be tranquilizing to point out that both inversions of the observed profiles yield, at least for this case, a similar magnetic field vector, in spite of the fact that the corresponding theoretical fit is much better for the case that includes atomic level polarization.
Ambiguity and degeneracies
==========================
In the previous subsections we have shown how our inversion code can be used for recovering the parameters of the assumed slab atmospheric model from the Stokes profiles observed in different solar plasma structures. The aim of this section is to discuss the Van Vleck ambiguity and to investigate whether we can infer the height of the observed plasma structure directly through the inversion approach.
Van Vleck Ambiguity {#sec:van_vleck}
-------------------
In general, the solution to any inversion problem is not unique –that is, it is often possible to detect several solutions which are compatible with the observations [e.g., @asensio_martinez_rubino07]. Some of the unicity problems are associated with the physics of the polarization phenomena (e.g., the 180$^\circ$ ambiguity of the Hanle effect for plasma structures located in the plane of the sky or the Van Vleck ambiguity). However, as seen in §\[sec:spicules\], in addition to this type of ambiguities, other degeneracies can appear because of the presence of noise in the observed Stokes profiles.
The Van Vleck ambiguity occurs only for some combinations of the inclinations and azimuths. Moreover, it occurs mainly in the saturation regime of the Hanle effect. For example, Fig. 6 of [@merenda06] shows the region of parameters for which the Van Vleck ambiguity occurs in the Hanle-effect saturation regime of the 10830 Å triplet. Since two different magnetic field vectors give rise to exactly the same emergent Stokes profiles, it is impossible to distinguish between them using only the 10830 Å multiplet (or four solutions, if the 180$^\circ$ ambiguity of the Hanle effect also applies). However, if more information is introduced in the inversion procedure (for instance, by using simultaneous observations in the 10830 Å and D$_3$ multiplets), it might be possible to distinguish between the two possible solutions.
Unfortunately, it is not easy to determine the range of parameters in which we may have the Van Vleck ambiguity. One possibility [used by @merenda06] is to consider the theoretical Hanle diagram of the red line of the He [i]{} 10830 Å multiplet and detect if the observed profiles fall in the ambiguity region. We propose another method based on the DIRECT algorithm implemented in our inversion code. The DIRECT algorithm can rapidly detect regions of the space of parameters where the global minimum may be located. Therefore, we can take advantage of this property to identify the two (or more) points in the space of parameters $(\theta_B,\chi_B)$ that produce the same emergent Stokes profiles for a given magnetic field strength.
To this end, we have calculated the synthetic emergent Stokes profiles of the 10830 Å line from an optically thin prominence, located in the plane of the sky, with $v_\mathrm{th}=8$ km s$^{-1}$, $h=20"$, $B=25$ G, $\theta_B=40^\circ$ and $\chi_B=19^\circ$. According to the Hanle diagram shown by [@merenda06], these profiles are indistinguishable from the ones given by the combination $B=22$ G, $\theta_B=100^\circ$ and $\chi_B=46^\circ$. To these two combinations, we have to add those corresponding to the 180$^\circ$ ambiguity: ($B=25$ G, $\theta_B=140^\circ$, $\chi_B=-19^\circ$) and ($B=22$ G, $\theta_B=80^\circ$, $\chi_B=-46^\circ$). Using the standard four steps inversion procedure explained in §\[sec:inversion\], the global minimum is rapidly located at position $B=22$ G, $\theta_B=100^\circ$ and $\chi_B=46^\circ$. Keeping fixed the value of all the thermodynamical properties and the field strength, the DIRECT algorithm is used to recover the inclination and azimuth of the magnetic field vector. We show in Fig \[fig:vanvleck\_ambiguity\] the position in the $(\theta_B,\chi_B)$ space of parameters of the $N$ evaluations performed by the DIRECT method. It has been possible to detect the two combinations of parameters that give the same emergent Stokes profiles, as stated above. With only $N=100$ evaluations of the merit function, the DIRECT algorithm has located and refined the position of the global minimum. It has also identified the second possible solution. When the number of function evaluations increases (even with only $N=200$), the DIRECT algorithm rapidly locates the two minima. For $N>200$, we face a degradation in the convergence rate as discussed in §\[sec:convergence\].
As already discussed, an interesting property of the DIRECT method is that no hyper-rectangle is ever discarded from the search. Therefore, a rectangle that in one iteration is not considered to be potentially interesting, can be chosen for division in a later iteration[^9]. This behavior is shown in Fig. \[fig:vanvleck\_ambiguity\]. When $N=100$, only a part of the space of parameters has been sampled, with clear gaps for inclinations above 110$^\circ$. In spite of these gaps, the two global minima have been already found. However, when the number of function evaluation is increased, the numerical scheme finally evaluates the function in those regions with the aim of discarding the presence of additional global minima.
Can we infer the height of the observed plasma structure? {#sec:invert_height}
---------------------------------------------------------
In this section we briefly discuss the possibility of determining the height at which the atoms are located by only using the information contained in the Stokes profiles of the 10830 Å multiplet. For simplicity, we consider first the optically thin limit, the case of off-limb observations (i.e., $90^{\circ}$ scattering geometry) and a magnetic field with a fixed azimuth and strength. The synthetic profiles correspond to the case $v_\mathrm{th}=8$ km s$^{-1}$ and $h=20"$, with the magnetic field vector given by $B=25$ G, $\theta_B=40^\circ$ and $\chi_B=19^\circ$. The aim of this experiment is to infer the inclination $\theta_B$ and height $h$ from synthetic Stokes profiles without noise. The positions where the merit function has been evaluated by the DIRECT algorithm are presented in the upper panels and in the bottom left panel of Fig. \[fig:height\_degeneration\]. The $\chi^2$ surface is shown in the bottom right panel of Fig. \[fig:height\_degeneration\]. The presence of the vertical strip where the minimum is located makes it very difficult to converge to the minimum using gradient-based methods like the LM method. The derivatives cannot be correctly approximated when the $\chi^2$ function has a large variation in such a small region of the space of parameters. This shallow strip is produced by the quasi-degeneracy of the problem in both parameters. An infinity of combinations of both parameters give Stokes profiles that can approximately reproduce the observations. The difference in the $\chi^2$ merit function between these spurious cases and the exact one is very small. Two reasons produce such a behavior. On the one hand, the linear polarization signal is enhanced when the height is increased because the anisotropy of the radiation field increases (see Fig. \[fig:nbar\_omega\], right panel). On the other hand, the Hanle effect turns out to be particularly efficient in reducing the atomic polarization when the magnetic field is significantly inclined with respect to the symmetry axis of the radiation field (the vertical direction). In a realistic case, the problem is much more complicated due to the presence of other additional parameters and the noise contamination.
When the observed structure is off the limb, imaging techniques can be used to estimate $h$. On the contrary, the case of on-disk observations is much more complicated since no straightforward technique for estimating the height is available. One possibility is to follow the observed active region until it approaches the limb. The height can then be estimated if we assume that the height of the plasma structure has not changed between both observations. An even less precise procedure is to assume a given $h$ value based on the typical height of the solar structure type under study. Obviously, the ideal situation would be the one where $h$ could be inferred directly from the observed Stokes profiles. In order to investigate this possibility, we have performed an experiment in which the DIRECT method is used with disk-center ($\theta=0^\circ$) synthetic Stokes profiles. The emergent profiles have been calculated with $v_\mathrm{th}=8$ km s$^{-1}$, $\Delta \tau=0.8$, $h=20"$, $B=25$ G, $\theta_B=40^\circ$ and $\chi_B=19^\circ$, taking into account the effects of radiative transfer in the slab. We keep fixed all the parameters except for the inclination of the magnetic field $\theta_B$ and the height $h$. The upper panels and the bottom left panel of Figure \[fig:height\_diskcenter\] present the points at which the DIRECT algorithm has evaluated the merit function, showing that it is possible to infer the height of the observation by only using the Stokes profiles. The shape of the $\chi^2$ surface is shown in the bottom right panel of Fig. \[fig:height\_diskcenter\]. In comparison with the off-limb case shown in Fig. \[fig:height\_degeneration\], the minimum is located in a much less complicated region of the $\chi^2$ surface. The quasi-degeneracy present in the off-limb case is not present in the on-disk case. This is associated with the fact that the blue component gives no signal in the optically thin limit, while it does if an inclined field is present for the disk center case [@trujillo_nature02].
Interestingly, if one wants to infer the magnetic field vector and the height simultaneously from the observations, the code is unable to get a suitable global minimum, even in the noiseless case. However, an easily accessible global minimum exists when one of the parameters is kept fixed, thus inferring only the following combinations of parameters: ($B$, $\theta_B$, $h$), ($B$, $\chi_B$, $h$) and ($\theta_B$, $\chi_B$, $h$).
Conclusions {#sec:conclusions}
===========
The physical interpretation of spectropolarimetric observations of lines of neutral helium, such as those of the 10830 Å and D$_3$ multiplets, represents a very important diagnostic window for investigating the dynamical behavior and the magnetic field of plasma structures in the solar chromosphere and corona, such as spicules, filaments, regions of emerging magnetic flux, network and internetwork regions, sunspots, flaring regions, etc. In order to facilitate this type of investigations we have developed a powerful forward modeling and inversion code that permits either to calculate the emergent spectral line intensity and polarization for any given magnetic field vector or to infer the dynamical and magnetic properties from the observed Stokes profiles. This diagnostic tool is based on the quantum theory of spectral line polarization [see @landi_landolfi04], which self-consistently accounts for the presence of atomic level polarization and the Hanle and Zeeman effects in the most general situation of the incomplete Paschen-Back effect regime. It is also of interest to mention that the same computer program can be easily applied to other chemical species apart from (e.g., in order to investigate the magnetic sensitivity of the polarization caused by the joint action of the Hanle and Zeeman effects in many other spectral lines of diagnostic interest, both in the solar atmosphere and in other astrophysical plasmas).
The influence of radiative transfer on the emergent spectral line radiation is taken into account by solving the Stokes-vector transfer equation in a slab of constant physical properties, including the magneto-optical terms of the propagation matrix. Although this “cloud" model for the interpretation of polarimetric observations in such lines of is suitable for inferring the magnetic field vector of plasma structures embedded in the solar chromosphere and corona, there are several interesting improvements and generalizations on which we are presently working on. The first one will be useful for improving the modeling of the Stokes profiles observed in low-lying optically-thick plasma structures embedded in the solar chromosphere, such as those of active region filaments. It consists in taking into account that in optically-thick plasma structures located at low atmospheric heights, the atomic level polarization is not going to be necessarily dominated by the anisotropic continuum radiation coming from the underlying solar photosphere (as we have assumed here), given that the radiation field generated by the optically-thick structure itself will tend to reduce the anisotropy factor of the true radiation field that pumps the helium atoms of the plasma structure under consideration [see @trujillo_asensio07]. The second additional development consists in considering a Milne-Eddington atmospheric model, but determining consistently the height-dependent atomic level polarization induced by the anisotropic radiation field within the atmosphere model that provides the best fit to the observed Stokes profiles. Since the anisotropy factor is very sensitive to the source-function gradient [e.g., Fig. 4 in @trujillo01] the solution of these type of problems in stratified model atmospheres may be facilitated by the application of efficient iterative schemes, such as those used by [@manso_trujillo03a; @manso_trujillo03b] for developing a general multilevel radiative transfer program for modeling scattering line polarization and the Hanle effect in weakly magnetized stellar atmospheres.
For the solution of the Stokes inversion problem we have applied an efficient algorithm based on global optimization methods, which permits a fast and reliable determination of the global minimum and facilitates the determination of the solutions corresponding to the unfamiliar Van-Vleck ambiguity. Our inversion approach is based on the application of the Levenberg-Marquardt (LM) method for locating the minimum of the merit function that quantifies the goodness of the fit between the observed and synthetic Stokes profiles. However, gradient-based methods suffer from convergence problems when the initial value of the parameters is not close to the minimum. In order to improve the convergence properties of the LM method, we have introduced a novel initialization technique. This method is based on the DIRECT algorithm, a deterministic global optimization scheme that performs very well. We have shown that a four-steps scheme using the DIRECT method to initialize the parameters and the LM method to refine the first estimation close to the minimum leads to a very robust technique.
Our computer program has been developed with the aim of being computationally efficient and user-friendly. The relevant equations of the problem result from a general and robust theory, so that it is straightforward to treat limiting cases and include or discard several physical effects in a very transparent way. It is appropriate for its application to a wide variety of problems, from simple Zeeman-dominated Stokes profiles to more complex situations in which the influence of atomic level polarization cannot be neglected. The code is written in FORTRAN 90, and incorporates a user-friendly front-end based on IDL[^10] which facilitates the execution and analysis of the synthesis and/or inversion calculations (see Fig. \[fig:front-end\]).
Obviously, our inversion strategy cannot compete in speed with algorithms based on look-up tables, like those applied by [@casini03] and [@merenda06]. At present, with a modern portable computer, we need of the order of 1 min. for the inversion of the Stokes profiles shown in Fig. \[fig:canopy\_observation\]. The strength of our approach is that it is very general and robust, and very suitable also to investigate the impact of the different physical mechanisms and parameters on the retrieved models. Concerning future improvements, we think that it would be worthwhile to treat the inversion problem within the framework of Bayesian inference techniques [see @asensio_martinez_rubino07 for a first application of such techniques to the inference of Milne-Eddington parameters from Stokes profiles induced by the Zeeman effect]. The aim is to sample the joint posterior probability distribution of the parameters of the model once the observation has been taken into account, and to carry out marginalizations to infer the probability distribution of each parameter. One of the main obstacles to overcome is to determine how to sample efficiently the full posterior probability distribution in the complex physical problem that we have investigated in this paper. A possible solution could be to rely on machine learning techniques for a fast solution of the forward problem, something that could be in perfect synergy with Markov Chain Montecarlo methods [@mackay03].
The reliability of the developments presented in this paper has been demonstrated through several model calculations and applications. Of particular interest is the investigation described in Section 3.4, which aimed at clarifying which is the optimum strategy for determining, from He [i]{} 10830 Å spectropolarimetric observations, whether or not we have magnetic canopies with horizontal fields in the quiet solar chromosphere. The results of an aplication to an observation of a disk-center internetwork region can be found in §5.4, which suggest the presence of magnetic fields inclined by no more than $50^{\circ}$ in the observed quiet chromospheric region.
We have also discussed the potential problems that one may encounter. For example, we have investigated the presence of degeneracies, paying particular attention to the possibility of determining the height of the observed plasma structure from the observed Stokes profiles themselves and to demonstrate that the DIRECT method is a very efficient technique for detecting the solutions associated to the Van Vleck ambiguity.
“HAZEL" (an acronym from HAnle and ZEeman Light) is the name we have given to our IAC computer program for the synthesis and inversion of Stokes profiles resulting from the joint action of the Hanle and Zeeman effects in slabs of finite optical thickness. HAZEL will be continuously improved over the years (e.g., with extensions to more complicated radiative transfer models), but is now ready for systematic applications to a variety of spectropolarimetric observations in the spectral lines of the 10830 Å and D$_3$ multiplets. We offer it to the astrophysical community with the hope that it will help researchers to achieve new breakthroughs in solar and stellar physics. To get a copy, it suffices with making an e-mail request to the authors of this paper.
We thank Roberto Casini (HAO) for carefully reviewing of our paper. Finantial support by the Spanish Ministry of Education and Science through project AYA2007-63881 and by the European Commission through the SOLAIRE network (MTRN-CT-2006-035484) is gratefully acknowledged.
{width="\columnwidth"}
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[^1]: More precisely, when the optical thickness of the slab is small in comparison with the eigenvalues of the matrix $\mathbf{K}'$.
[^2]: See [-@belluzzi07] for the formulation and solution of an interesting problem where hyperfine structure is important.
[^3]: `http://physics.nist.gov/PhysRefData/ASD/index.html`
[^4]: Actually, ${\cal W}=w^{(2)}_{J_uJ_l}$ and ${\cal
Z}=w^{(2)}_{J_lJ_u}$, with $w^{(2)}_{JJ^{'}}$ given by Eq. (10.12) of [@landi_landolfi04]. For instance, ${\cal W}=0$ and ${\cal Z}=1$ for a line transition with $J_l=1$ and $J_u=0$, such as that of the blue component of the 10830 Å triplet.
[^5]: The reason why Stokes $U$ in forward scattering geometry is not exactly zero for the red component is because for B=10 G some of the coherences are not completely insignificant.
[^6]: Information on this ambiguity, typical of the Hanle-effect saturation regime, can be found in [@casini05], in [@merenda06] and in §\[sec:van\_vleck\] below.
[^7]: We point out that this ambiguity of the Hanle effect applies only to some particular scattering geometries (i.e., to those of 90$^\circ$ and of forward scattering). If the observed plasma structure is not located in the plane of the sky (which implies scattering processes at an angle different from 90$^\circ$), or if it is not located at the solar disk center, then one has a sort of quasi-degeneracy which can disappear when the angle $\theta$ of Fig. 1 is considerably different from 90$^\circ$ or from 0$^\circ$. This fact has been exploited by [@landi_bommier93] to propose a method for removing the azimuth ambiguity intrinsically present in vector magnetograms.
[^8]: For a preliminary interpretation of an observation of a quiet region located at $\mu=0.5$ see [@Lagg07].
[^9]: This proves to be fundamental to demonstrate that the global minimum will always be found [@Jones_DIRECT93].
[^10]: `http://www.ittvis.com/idl`
|
---
abstract: |
Superscaling analyses of inclusive electron scattering from nuclei are extended from the quasielastic processes to the delta excitation region. The calculations of $(e,e^\prime)$ cross sections for the target nucleus $^{12}$C at various incident electron energies are performed using scaling functions $f(\psi^{\prime})$ obtained in approaches going beyond the mean-field approximation, such as the coherent density fluctuation model (CDFM) and the one based on the light-front dynamics (LFD) method. The results are compared with those obtained using the relativistic Fermi gas (RFG) model and the extended RFG model (ERFG). Our method utilizes in an equivalent way both basic nuclear quantities, density and momentum distributions, showing their role for the scaling and superscaling phenomena. The approach is extended to consider scaling function for medium and heavy nuclei with $Z\neq N$ for which the proton and neutron densities are not similar. The asymmetry of the CDFM quasielastic scaling function is introduced, simulating in a phenomenological way the effects which violate the symmetry for $\psi^{\prime}\geq
0$ including the role of the final-state interaction (FSI). The superscaling properties of the electron scattering are used to predict charge-changing neutrino-nucleus cross sections at energies from 1 to 2 GeV. A comparison with the results of the ERFG model is made. The analyses make it possible to gain information about the nucleon correlation effects on both local density and nucleon momentum distributions.
author:
- 'A. N. Antonov'
- 'M. V. Ivanov'
- 'M. K. Gaidarov'
- 'E. Moya de Guerra'
- 'J. A. Caballero'
- 'M. B. Barbaro'
- 'J. M. Udias'
- 'P. Sarriguren'
title: 'Superscaling analysis of inclusive electron scattering and its extension to charge-changing neutrino-nucleus cross sections beyond the relativistic Fermi gas approach'
---
Introduction
============
Over the past four decades electron scattering has provided a wealth of information on nuclear structure and dynamics. Form factors and charge distributions have been extracted from elastic scattering data, while inelastic measurements have allowed for a systematic study of the dynamic response over a broad range of momentum and energy transfer. The nuclear $y$-scaling analysis of inclusive electron scattering from a large variety of nuclei (e.g. [@West75; @Sick80; @Day90; @Cio87; @Cio90; @Cio96; @AMD+88; @CW97; @CW99; @FCW00]) showed the existence of high-momentum components in the nucleon momentum distributions $n(k)$ at momenta $k>2$ fm$^{-1}$ due to the presence of nucleon-nucleon (NN) correlations. It was shown (see, e.g. [@AGK+04; @AGK+05; @AIG+06; @DS99l; @DS99]) that this specific feature of $n(k)$, which is similar for all nuclei, is a physical reason for the scaling and superscaling phenomena in nuclei.
The concepts of scaling [@West75; @Sick80; @Day90; @Cio87; @Cio90; @Cio96; @CW97; @CW99; @FCW00] and superscaling [@AMD+88; @AGK+05; @AIG+06; @DS99l; @AGK+04; @DS99; @BCD+98] have been explored in [@DS99; @MDS02] for extensive analyses of the $(e,e^{\prime})$ world data (see also [@Benhar2006]). Scaling of the first kind (no dependence on the momentum transfer) is reasonably good as expected, at excitation energies below the quasielastic (QE) peak, whereas scaling of second kind (no dependence on the mass number) is excellent in the same region. When both types of scaling behavior occur one says that superscaling takes place. At energies above the QE peak both scaling of the first and, to a lesser extent, of the second kind are shown to be violated because of important contributions introduced by effects beyond the impulse approximation, namely, inelastic scattering [@BCD+04; @AlvRuso03] together with correlation contributions and meson exchange currents [@Amaro2002; @Pace03].
The superscaling analyses of inclusive electron scattering from nuclei for relatively high energies (several hundred MeV to a few GeV) have recently been extended to include not only quasielastic processes, but also the region where $\Delta$-excitation dominates [@Amaro2005]. A good representation of the electromagnetic response in both quasielastic and $\Delta$ regions has been obtained using the scaling ideas, importantly, with an asymmetric QE scaling function $f^{QE}(\psi^{\prime})$ ($\psi^{\prime}$ is the scaling variable in the QE region) and a scaling function $f^{\Delta}(\psi^{\prime}_{\Delta})$ in the region up to inelasticities where the $\Delta$ contribution reaches its maximum. Both functions were deduced from phenomenological fits to electron scattering data. Particularly, for the scaling function in the quasielastic region it has been shown in Ref. [@Amaro2005] that, in contrast to the relativistic Fermi gas model scaling function, which is symmetric, limited strictly to the region $-1 \leq \psi^{\prime} \leq +1$, and with a maximum value 3/4, the empirically determined $f^{QE}(\psi^{\prime})$ has a somewhat asymmetric shape with a tail that extends towards positive values of $\psi^{\prime}$ and its maximum is only about 0.6. Of course, the specific features of the scaling function should be accounted for by reliable microscopic calculations that take FSI into account. In particular, the asymmetric shape of $f^{QE}$ tested in Refs. [@Caballero2005; @Caballero2006] by using a relativistic mean field (RMF) for the final states shows a very good agreement with the behavior presented by the experimental scaling function.
The superscaling analyses and the present knowledge of inclusive electron scattering allowed one to start studies of neutrino scattering off nuclei on the same basis. The reactions of incident neutrino beams interacting with a complex nucleus have offered unique opportunities for exploring fundamental questions in different domains in physics. Recently, positive signals of neutrino oscillations confirmed the hypothesis of non-zero neutrino masses and triggered much interest on this issue [@Fukuda]. To better analyze the next generation of high-precision neutrino oscillation experiments and to reduce their systematic uncertainty both neutral- (e.g. [@Meucci20041; @Amaro2006; @Martinez2006; @Barbaro2006; @Nieves2005]) and charged-current (e.g. [@Amaro2005; @Caballero2005; @Martinez2006; @Barbaro2006; @Barbaro2005; @Maieron2003; @Meucci20042; @Benhar20041; @Benhar2004; @Benhar2005]) neutrino-nucleus scattering have stimulated detailed investigations.
The neutrino-nucleus interactions have been studied within several approaches investigating a variety of effects. Using the superscaling analysis of few-GeV inclusive electron-scattering data, a method was proposed in Ref. [@Amaro2005] to predict the inclusive $\nu A$ and $\overline\nu A$ cross sections for the case of $^{12}$C in the nuclear resonance region, thereby effectively including delta isobar degrees of freedom. It was shown in Refs. [@Meucci20041; @Meucci20042] that the important final-state interaction effects arising from the use of relativistic optical potentials within a relativistic Green’s function approach lower the cross section by at least a 14% factor at incoming neutrino energies of 1 GeV. A similar result has been obtained in Refs. [@Co2006; @Botrugno2005] where the use of Random Phase Approximation (RPA) to predict the neutrino-nucleus cross section was discussed. Apart from relativistic dynamics and FSI, other effects may influence the neutrino-nucleus reactions. The role of Pauli blocking and FSI in charged-current neutrino induced reactions is analyzed in Refs. [@Benhar20041; @Benhar2004; @Benhar2005].
In this article we follow our method presented in Refs. [@AGK+04; @AGK+05; @AIG+06] to calculate the scaling function in finite nuclei firstly within the coherent density fluctuation model (e.g., Refs. [@AHP88; @AHP93; @ANP79+; @A+89+]). This approach, which is a natural extension of the RFG model, has shown how both basic quantities, density and momentum distributions, are responsible for the scaling and superscaling phenomena in various nuclei. Although the scaling function obtained in [@AGK+04] is symmetrical around $\psi^{\prime}=0$, the results agree with the available experimental data at different transferred momenta and energies below the quasielastic peak position, showing superscaling for $\psi^{\prime}<0$ including $\psi^{\prime}<-1$, whereas in the RFG model $f(\psi^{\prime})=0$ for $\psi^{\prime} \leq -1$. It was shown in [@AGK+05] that the QE scaling function can be obtained within the CDFM in two equivalent ways, on the basis of the local density distribution, as well as of the nucleon momentum distribution. As pointed out in [@AGK+05], the nucleon momentum distributions $n(k)$ for various nuclei obtained in [@AGI+02] within a parameter-free theoretical approach based on the light-front dynamics method (e.g., [@CK95; @CDK98] and references therein) can also be used to describe both $y$- and $\psi^{\prime}$-scaling data. So, in our present work we explore both methods, CDFM and LFD, to investigate further the scaling functions and their applications to analyses of electron- and neutrino scattering off nuclei.
Our work is motivated by the fact that different models of the nuclear dynamics (including those with RMF dynamics and with RPA-type correlations accounted for) describe with different success the basic size and shape of the cross sections in studies of high-energy inclusive lepton scattering used so far. For this reason we extend further our consideration and calculate within the CDFM and LFD the scaling functions in the kinematical regions of the QE and $\Delta$ peak on the basis of momentum and density distributions of finite nuclear systems in which nucleon correlations are included. This can be done either by using available empirical data for these quantities or theoretical calculations in which correlations are included to some extent. Then, the obtained scaling functions are applied to calculate electron-nucleus cross sections in QE and $\Delta$ regions in the energy range from 500 MeV to 2 GeV for the target nucleus $^{12}$C and to predict charge-changing neutrino and antineutrino reaction cross sections from the scaling region to the QE peak at energies of few GeV. We also make comparisons of the results obtained using our methods with those obtained using the RFG model and other theoretical schemes.
The paper is organized in the following way. In Sec. II we present the formalism needed in studies of scaling functions in the quasielastic region and validate the superscaling within the CDFM and LFD for a variety of nuclei with $Z=N$ and $Z\neq N$. Then, we consider the nucleon momentum distributions and their applications in both approaches showing the sensitivity of the calculated scaling functions to the peculiarities of $n(k)$ in different regions of momenta. Section III contains the CDFM and LFD methods to build up the scaling function in the $\Delta$ region. The formalism involved in obtaining the electron-nucleus cross sections in QE and $\Delta$ kinematical regions and the results of the numerical calculations are presented in Sec. IV A. In Sec. IV B we give our theoretical predictions for cross sections of quasielastic charge-changing neutrino reactions. Finally, in Sec. V we summarize the results of our work.
Scaling function in the quasielastic region
===========================================
QE scaling function in the CDFM
-------------------------------
As already mentioned in the Introduction, the superscaling behavior was firstly considered within the framework of the RFG model [@AMD+88; @BCD+98; @DS99l; @DS99; @MDS02; @BCD+04] where a properly defined function of the $\psi^{\prime}$-variable was introduced. As pointed out in [@DS99], however, the actual nuclear dynamical content of the superscaling is more complex than that provided by the RFG model. It was observed that the experimental data have a superscaling behavior in the low-$\omega$ side ($\omega$ being the transfer energy) of the quasielastic peak for large negative values of $\psi^{\prime}$ (up to $\psi^{\prime}\approx -2$), while the predictions of the RFG model are $f(\psi^{\prime})=0$ for $\psi^{\prime}\leq -1$. This imposes the consideration of the superscaling in realistic finite systems. One of the approaches to do this was developed [@AGK+04; @AGK+05] in the CDFM [@AHP88; @AHP93; @ANP79+; @A+89+] which is related to the $\delta$-function limit of the generator coordinate method [@AGK+04; @Grif57]. It was shown in [@AGK+04; @AGK+05; @AIG+06] that the superscaling in nuclei can be explained quantitatively on the basis of the similar behavior of the high-momentum components of the nucleon momentum distribution in light, medium and heavy nuclei. As already mentioned, the latter is related to the effects of the NN correlations in nuclei (see, e.g. [@AHP88; @AHP93]).
The scaling function in the CDFM was obtained starting from that in the RFG model [@AMD+88; @BCD+98; @DS99l; @DS99] in two equivalent ways, on the basis of the local density distribution $\rho(r)$ and of the nucleon momentum distribution $n(k)$. This allows one to study simultaneously the role of the NN correlations included in $\rho(r)$ and $n(k)$ in the case of the superscaling phenomenon. To explore these properties the scaling function $f(\psi^{\prime})$ has been derived in two ways in CDFM in [@AGK+05]. Firstly, by means of the density distribution $\rho(r)$, it leads to $$f^{QE}(\psi')= \int_{0}^{\alpha/(k_{F}|\psi'|)}dR |F(R)|^{2}
f_{RFG}^{QE}(\psi'(R)),
\label{eq:1}$$ with a weight function of the form $$|F(R)|^{2}=-\frac{1}{\rho_{0}(R)} \left. \frac{d\rho(r)}{dr}\right
|_{r=R},
\label{eq:2}$$ where $$\rho_{0}(R)=\frac{3A}{4\pi R^{3}}.
\label{eq:3}$$ $f_{RFG}^{QE}(\psi^{\prime}(R))$ with $\psi^{\prime}(R)=k_{F}R\psi^{\prime}/\alpha$ is the scaling function related to the RFG model $$\begin{aligned}
f_{RFG}^{QE}(\psi'(R))& =& \displaystyle \frac{3}{4} \left[
1-\left( \frac{k_FR|\psi'|}{\alpha} \right)^{2}\right] \left\{ 1+
\left( \frac{Rm_N}{\alpha}\right)^2 \left(
\frac{k_FR|\psi'|}{\alpha}
\right)^2 \right. \nonumber\\
&& \times \displaystyle \left. \left[2+ \left( \frac{\alpha}{Rm_N}
\right)^2- 2\sqrt{1+ \left( \frac{\alpha}{Rm_N} \right)^2}\right]
\right\},
\label{eq:4}\end{aligned}$$ $m_{N}$ being the nucleon mass and $\alpha=(9\pi A/8)^{1/3}\simeq
1.52A^{1/3}$. Secondly, by means of the momentum distribution $n(k)$, the scaling function is expressed by $$f^{QE}(\psi')= \int_{k_{F}|\psi'|}^{\infty} d\overline{k}_F
|G(\overline{k}_F)|^2 f_{RFG}^{QE}(\psi'(\overline{k}_F)),
\label{eq:5}$$ where $\psi'(\overline{k}_F)=k_{F}\psi'/\overline{k}_F$ and the weight function is $$|G(\overline{k}_F)|^2=- \frac{1}{n_0(\overline{k}_F)}\left.
\frac{dn(k)}{dk}\right |_{k=\overline{k}_F}
\label{eq:6}$$ with $$n_0(\overline{k}_F)= \frac{3A}{4\pi {\overline{k}_F}^3}.
\label{eq:7}$$ In Eq. (\[eq:5\]) the RFG scaling function $f_{RFG}^{QE}(\psi'(\overline{k}_F))$ can be obtained from $f_{RFG}^{QE}(\psi'(R))$ \[Eq. (\[eq:4\])\] by changing there $\alpha/R$ by $\overline{k}_F$. In Eqs. (\[eq:1\]), (\[eq:4\]) and (\[eq:5\]) the Fermi momentum $k_{F}$ is not a free parameter for different nuclei as it is in the RFG model, but $k_{F}$ is calculated within the CDFM for each nucleus using the corresponding expressions: $$k_F= \int_{0}^{\infty} dR k_{F}(R)|F(R)|^2= \alpha \int_{0}^{\infty}
dR \frac{1}{R}|F(R)|^{2}= \frac{4\pi(9\pi)^{1/3}}{3A^{2/3}}
\int_{0}^{\infty} dR \rho(R) R
\label{eq:8}$$ when the condition $$\lim_{R\rightarrow \infty} \left[ \rho(R)R^2 \right]=0
\label{eq:9}$$ is fulfilled and $$k_F= \frac{16\pi}{3A} \int_0^\infty d\overline{k}_F n(\overline{k}_F
) {\overline{k}_F}^3
\label{eq:10}$$ when the condition $$\lim_{\overline{k}_F\rightarrow \infty}\left[
n(\overline{k}_F){\overline{k}_F}^4 \right]=0
\label{eq:11}$$ is fulfilled.
As shown in [@AGK+05], the integration in Eqs. (\[eq:1\]) and (\[eq:5\]), using Eqs. (\[eq:2\]) and (\[eq:6\]), leads to the explicit relationships of the scaling functions with the density and momentum distributions: $$f^{QE}(\psi')= \frac{4\pi}{A}\int_{0}^{\alpha/(k_{F}|\psi'|)}dR
\rho(R) \left[ R^2 f_{RFG}^{QE}(\psi'(R))+ \frac{R^3}{3}
\frac{\partial f_{RFG}^{QE}(\psi'(R))}{\partial R} \right]
\label{eq:12}$$ and $$f^{QE}(\psi')= \frac{4\pi}{A} \int_{k_{F}|\psi'|}^{\infty}
d\overline{k}_F n(\overline{k}_F) \left[ {\overline{k}_F}^2
f_{RFG}^{QE}(\psi'(\overline{k}_F))+ \frac{{\overline{k}_F}^3}{3}
\frac{\partial f_{RFG}^{QE}(\psi'(\overline{k}_F))}{\partial
\overline{k}_F} \right],
\label{eq:13}$$ the latter at $$\lim_{\overline{k}_F\rightarrow \infty}\left[
n(\overline{k}_F){\overline{k}_F}^3 \right]=0.
\label{eq:14}$$ One can see the symmetry in both Eqs. (\[eq:12\]) and (\[eq:13\]) written in $r$- and $k$-space. We also note that in the consideration up to here the CDFM scaling function $f^{QE}(\psi^{\prime})$ is symmetric under the change of $\psi^{\prime}$ by -$\psi^{\prime}$.
In Refs. [@AGK+04; @AGK+05] we used the charge density distributions to determine the weight function $|F(R)|^{2}$ and $f^{QE}(\psi^{\prime})$ in Eqs. (\[eq:1\]), (\[eq:2\]) and (\[eq:8\]) for the cases of $^{4}$He, $^{12}$C, $^{27}$Al, $^{56}$Fe and $^{197}$Au. The results for the scaling function $f^{QE}(\psi^{\prime})$ agree well with the available data from the inclusive quasielastic electron scattering for $^{4}$He, $^{12}$C, $^{27}$Al, $^{56}$Fe and only approximately for $^{197}$Au for various values of the transfer momentum $q=500,
1000, 1650$ MeV/c [@AGK+04] and $q=1560$ MeV/c [@AGK+05], showing superscaling for negative values of $\psi^{\prime}$ including also those smaller than -1, whereas in the RFG model $f(\psi^{\prime})=0$ for $\psi^{\prime} \leq -1$. One can see this in Fig. \[fig01\] for $^{4}$He, $^{12}$C and $^{27}$Al at $q=1000$ MeV/c. At the same time, however, in [@AGK+04; @AGK+05] we encountered some difficulties to describe the superscaling in $^{197}$Au which was the heaviest nucleus considered. We related this in [@AGK+04; @AGK+05] to the particular A-dependence of $n(k)$ in the model that does not lead to realistic high-momentum components of $n(k)$ in the heaviest nuclei. We followed in Refs. [@AGK+04; @AGK+05] an artificial way to “improve” the high-momentum tail of $n(k)$ in $^{197}$Au by taking the value of the diffuseness parameter $b$ in the Fermi-type charge density distribution of this nucleus to be $b$=1 fm instead of the value $b$=0.449 fm (as obtained from electron elastic scattering experiments, see e.g. [@PP03]). In this way the high-momentum tail of $n(k)$ for $^{197}$Au in CDFM becomes similar to those of $^{4}$He, $^{12}$C, $^{27}$Al, and $^{56}$Fe and this leads to a good agreement of the scaling function $f^{QE}(\psi^{\prime})$ with the data also for $^{197}$Au. Still in [@AGK+04] we pointed out, however, that all the nucleons (not just the protons) may contribute to $f^{QE}(\psi^{\prime})$ for the transverse electron scattering and this could be simulated by increasing of the diffuseness of the matter density with respect to that of the charge density for a nucleus like $^{197}$Au that has much larger number of neutron than of protons.
![The quasielastic scaling function $f^{QE}(\psi^{\prime})$ at $q=1000$ MeV/c for $^{4}$He, $^{12}$C, $^{27}$Al, $^{82}$Kr and $^{197}$Au calculated in CDFM. Dotted line: RFG model result. The curves for $^{4}$He, $^{12}$C and $^{27}$Al nuclei almost coincide. Grey area: experimental data [@DS99l; @DS99]. \[fig01\]](fig01.eps){width="10cm"}
In [@AIG+06] we assumed that the reason why the CDFM does not work properly in the case of $^{197}$Au is that we had used in [@AGK+04; @AGK+05] only the phenomenological charge density, while this nucleus has many more neutrons than protons ($N$=118 and $Z$=79) and therefore, proton and neutron densities may differ considerably. In the case when $Z\neq N$ and the proton and neutron densities are not similar, the total scaling function will be expressed by the sum of the proton $f_{p}^{QE}(\psi^{\prime})$ and neutron $f_{n}^{QE}(\psi^{\prime})$ scaling functions which are determined by the proton and neutron densities $\rho_{p}(r)$ and $\rho_{n}(r)$, respectively: $$f_{p(n)}^{QE}(\psi^{\prime})=\!\!\!\!\int\limits_{0}^{\alpha_{p(n)}/(k^{p(n)}_{F}
|\psi^{\prime}|)}\!\!\!\!dR
|F_{p(n)}(R)|^{2}f_{RFG}^{p(n)}(\psi'(R)).
\label{eq:15}$$ In Eq. (\[eq:15\]) the proton and neutron weight functions are obtained from the corresponding proton and neutron densities $$\left|F_{p(n)}(R)\right|^2=-\dfrac{4\pi
R^3}{3Z(N)}\left.\dfrac{d\rho_{p(n)}(r)}{dr}\right|_{r=R},
\label{eq:16}$$ $$\alpha_{p(n)}=\left(\dfrac{9\pi Z(N)}{4}\right)^{1/3},
\label{eq:17}$$ $$\int\limits_{0}^{\infty}\rho_{p(n)}(\vec{r})d\vec{r}=Z(N),
\label{eq:18}$$ and the Fermi momentum for the protons and neutrons is given by $$k_{F}^{p(n)}=\alpha_{p(n)}\int\limits_{0}^{\infty}dR
\frac{1}{R}|{F}_{p(n)}(R)|^{2}.
\label{eq:19}$$ The RFG proton and neutron scaling functions $f_{RFG}^{p(n)}(\psi'(R))$ have the form of Eq. (\[eq:4\]), where $\alpha$ and $k_{F}$ stand for $\alpha_{p(n)}$ from Eq. (\[eq:17\]) and $k_{F}^{p(n)}$ from Eq. (\[eq:19\]), respectively. The functions are normalized as follows: $$\int\limits_{0}^{\infty}|F_{p(n)}(R)|^{2}dR=1,
\label{eq:20}$$ $$\int\limits_{-\infty}^{\infty}f_{p(n)}^{QE}(\psi^{\prime})d\psi^{\prime}=1.
\label{eq:21}$$ Then the total scaling function can be expressed by means of both proton and neutron scaling functions: $$f^{QE}(\psi^{\prime})=\dfrac{1}{A}[Zf_p^{QE}(\psi^{\prime})+Nf_n^{QE}(\psi^{\prime})]
\label{eq:22}$$ and is normalized to unity.
The same consideration can be performed equivalently on the basis of the nucleon momentum distributions for protons $n^{p}(k)$ and neutrons $n^{n}(k)$ presenting $f^{QE}(\psi^{\prime})$ by the sum of proton and neutron scaling functions (\[eq:22\]) calculated similarly to Eqs. (\[eq:15\])-(\[eq:22\]) (and to Eqs. (\[eq:5\]), (\[eq:6\]), (\[eq:10\]) and (\[eq:11\])): $$f_{p(n)}^{QE}(\psi')=\!\!
\int\limits_{k^{p(n)}_{F}|\psi'|}^{\infty}\!\! d\overline{k}_F
|G_{p(n)}(\overline{k}_F)|^2
f^{p(n)}_{RFG}(\psi'(\overline{k}_F)),
\label{eq:23}$$ where $$|G_{p(n)}(\overline{k}_F)|^2=- \frac{4\pi
{\overline{k}_F}^3}{3Z(N)}\left. \frac{dn^{p(n)}(k)}{dk}\right
|_{k=\overline{k}_F}
\label{eq:24}$$ with $f_{RFG}^{p(n)}(\psi'(\overline{k}_F))$ containing $\alpha_{p(n)}$ from Eq. (\[eq:17\]) and $k_F^{p(n)}$ calculated as $$k^{p(n)}_F= \int\limits_0^\infty d\overline{k}_F \overline{k}_F
|G_{p(n)}(\overline{k}_F)|^2.
\label{eq:25}$$ The scaling functions for several examples, such as the medium stable nuclei $^{62}$Ni and $^{82}$Kr and the heavy nuclei $^{118}$Sn and $^{197}$Au are calculated following Eqs. (\[eq:15\])-(\[eq:22\]) using the corresponding proton and neutron densities obtained in deformed self-consistent mean-field (HF+BCS) calculations with density-dependent Skyrme effective interaction (SG2) and a large harmonic-oscillator basis with 11 major shells [@Sarriguren99; @Vautherin73]. In Fig. \[fig01\] we give the results for the $^{82}$Kr and $^{197}$Au nuclei in which $Z\neq N$ and compare them with the results for $^{4}$He, $^{12}$C ($Z=N$) and $^{27}$Al ($Z\simeq
N$), as well as with the experimental data (presented by a grey area and taken from [@DS99]) obtained for $^{4}$He, $^{12}$C, $^{27}$Al, $^{56}$Fe, and $^{197}$Au. The scaling functions are in a reasonable agreement with the data, which was not the case for $^{197}$Au calculated in [@AGK+04] by using only the Fermi-type charge density with phenomenological parameter values $b=0.449$ fm and $R=6.419$ fm from [@PP03]. At the same time we note also the improvement in comparison with the RFG model result in which $f^{QE}(\psi^{\prime})=0$ for $\psi^{\prime}\leq
-1$. Thus, it can be concluded that the scaling function $f^{QE}(\psi^{\prime})$ for nuclei with $Z\neq N$ for which the proton and neutron densities are not similar has to be expressed by the sum of the proton and neutron scaling functions. The latter can be calculated by means of theoretically and/or experimentally obtained proton and neutron local density distributions or momentum distributions.
As known (e.g. [@Amaro2005; @DS99]), the total inclusive electron scattering response is assumed to be composed of several contributions: i) the entire longitudinal contribution which superscales and is represented by the QE scaling function $f^{QE}(\psi^{\prime})$; ii) a part of the transverse response, which arises from the quasielastic knockout of nucleons and is also driven by the scaling function $f^{QE}(\psi^{\prime})$, and iii) the additional contribution of the transverse response from MEC effects and from inelastic single-nucleon processes including the excitation of the $\Delta$ isobar. The effects of point iii) break the scaling. In [@Amaro2005] an universal scaling function $f^{QE}(\psi^{\prime})$ has been determined by reliable separations of the empirical data into their longitudinal and transverse contributions for $A>4$. Such separations are available only for a few nuclei [@Jordan96]. All of these response functions have been used to extract the “universal” QE response function $f^{QE}(\psi^{\prime})$ (see Fig. 1 of [@Amaro2005]) which is parametrized by a simple function. This function has a somewhat asymmetric shape. Its left tail $(\psi^{\prime}<0)$ passes through the grey area of Fig. \[fig01\]. The right tail $(\psi^{\prime}>0)$ extends larger towards positive values of $\psi^{\prime}$. In contrast, the RFG scaling function is symmetric. The CDFM scaling function discussed so far, which is based on the RFG one, is also symmetric. As mentioned, the maximum value of $f^{QE}(\psi^{\prime})$ in RFG (and in CDFM) is 3/4, while the empirical scaling function extracted in Ref. [@Amaro2005] reaches about 0.6.
As mentioned in [@Amaro2005], if FSI are neglected, the RMF theory [@Kim95; @Alberico97; @Maieron2003] and relativized shell-model studies [@Amaro96] provide rather modest differences from the RFG predictions. Another possible reason for the differences between the RFG (or mean-field results) and the empirically determined scaling function arises from high-momentum components in realistic wave functions which may be large enough. In [@Amaro2005] the scaling function was taken from the experiment. In the present work we also limit our approach to phenomenology when considering the asymmetric shape and the maximum value of the quasielastic scaling function. In order to simulate the role of all the effects which lead to asymmetry, we impose the latter on the RFG scaling function (and, correspondingly, on the CDFM one) by introducing a parameter which gives the correct maximum value of the scaling function ($c_{1}$ in our expressions given below) and also an asymmetry in $f^{QE}(\psi^{\prime})$ for $\psi^{\prime}\geq 0$. We consider the main parts of the RFG scaling function for $\psi^{\prime}\leq 0$ and $\psi^{\prime}\geq 0$ in the following forms, keeping the parabolic dependence on $\psi^{\prime}$ as required in Ref. [@AMD+88]: $$f_{RFG,1}^{QE}(\psi^{\prime})=c_{1}(1-\psi^{\prime
2})\Theta(1-\psi^{\prime 2}), \;\;\;\;\; \psi^{\prime}\leq 0,
\label{eq:26}$$ $$f_{RFG,2}^{QE}(\psi^{\prime})=c_{1}\left[1-\left(\frac{\psi^{\prime}}{c_{2}}\right
)^{2}\right ]\Theta\left[1-\left(\frac{\psi^{\prime}}{c_{2}}\right
)^{2}\right ], \;\;\;\;\; \psi^{\prime}\geq 0.
\label{eq:27}$$ The total RFG scaling function is normalized to unity: $$\int_{-\infty}^{\infty}
f^{QE}_{RFG}(\psi^{\prime})d\psi^{\prime}=\int_{-\infty}^{\infty}
[f_{RFG,1}^{QE}(\psi^{\prime})+f_{RFG,2}^{QE}(\psi^{\prime})]
d\psi^{\prime}=1.
\label{eq:28}$$ If the normalization of the scaling function for negative values of $\psi^{\prime}$ is equal to $$a=\int_{-\infty}^{0} d\psi^{\prime}
f_{RFG,1}^{QE}(\psi^{\prime})=\frac{2}{3}c_{1},
\label{eq:29}$$ then, in order to keep the total normalization \[Eq. (\[eq:28\])\], the normalization for positive $\psi^{\prime}$ has to be: $$1-a=\int_{0}^{\infty} d\psi^{\prime}
f_{RFG,2}^{QE}(\psi^{\prime})=\frac{2}{3}c_{1}c_{2}.
\label{eq:30}$$ From Eqs. (\[eq:29\]) and (\[eq:30\]) we get the relationship between $c_{2}$ and $c_{1}$: $$c_{2}=\frac{3}{2c_{1}}-1.
\label{eq:31}$$ In the RFG $c_{1}=3/4$ and, correspondingly, $c_{2}=1$. In the CDFM the QE scaling function will be: $$f^{QE}(\psi^{\prime})=f_{1}^{QE}(\psi^{\prime})+f_{2}^{QE}(\psi^{\prime}),
\label{eq:32}$$ where $$f_{1}^{QE}(\psi^{\prime})\cong
\int_{0}^{\alpha/k_{F}|\psi^{\prime}|} dR |F(R)|^{2}
c_{1}\left[1-\left(\frac{k_{F}R|\psi^{\prime}|}{\alpha}\right
)^{2}\right ], \psi^{\prime}\leq 0,
\label{eq:33}$$ $$f_{2}^{QE}(\psi^{\prime})\cong
\int_{0}^{c_{2}\alpha/k_{F}|\psi^{\prime}|} dR |F(R)|^{2}
c_{1}\left[1-\left(\frac{k_{F}R|\psi^{\prime}|}{c_{2}\alpha}\right
)^{2}\right ], \psi^{\prime}\geq 0.
\label{eq:34}$$ In this approach, parametrizing the RFG scaling function by the coefficient $c_{1}$ we account for the experimental fact that $c_{1}\neq 3/4$ and take this value in accordance with the empirical data. Then from the normalization \[Eqs. (\[eq:28\])-(\[eq:30\])\] we determine the corresponding value of $c_{2}$ using Eq. (\[eq:31\]). As in [@AGK+04; @AGK+05], the CDFM scaling function is obtained \[Eqs. (\[eq:32\])-(\[eq:34\])\] by averaging the RFG scaling function. As an example, we give in Fig. \[fig02\] the CDFM QE scaling function for different values of $c_{1}$ (0.75, 0.72, 0.60 and 0.50) in comparison with the empirical data and the phenomenological fit. We also include for reference the scaling function obtained from calculations for $(e,e^{\prime})$ reaction based on the relativistic impulse approximation (RIA) with FSI described using the RMF potential (see [@Caballero2005; @Caballero2006] for details). In this way we simulate in a phenomenological way the role of the effects which violate the symmetry for positive values of $\psi^{\prime}$ of the QE scaling function, which in the RMF approximation are seen to be due to the FSI.
![The quasielastic scaling function $f^{QE}(\psi^{\prime})$ for $^{12}$C calculated in CDFM in comparison with the experimental data (black squares) [@Amaro2005]. The CDFM results for different values of $c_{1}$ are presented by solid lines. Also shown for comparison is the phenomenological curve which fits the data (dash-two dots), as well as the curve (dash-dot line) corresponding to the result for $(e,e^{\prime})$ obtained within the relativistic impulse approximation and FSI using the relativistic mean field (see Refs. [@Caballero2005; @Caballero2006]). \[fig02\]](fig02.eps){width="10cm"}
QE scaling function in the LFD method
-------------------------------------
In this Subsection we will obtain the QE scaling function on the basis of calculations of nucleon momentum distribution (using Eqs. (5)–(7) or Eq. (13)) obtained within a modification of the approach from [@AGI+02]. The latter is based on the momentum distribution in the deuteron from the light-front dynamics (LFD) method (e.g., [@CK95; @CDK98] and references therein). Using the natural-orbital representation of the one-body density matrix [@Low55], $n(k)$ was written as a sum of contributions from hole-states $[n^{\text{h}}(k)]$ and particle-states $[n^{\text{p}}(k)]$ (see also [@AGK+05]) $$n_A(k)=N_A\left[ n^{\text{h}}(k)+ n^{\text{p}}(k)\right] .
\label{eq:35}$$ In (\[eq:35\]) $$n^{\text{h}}(k)= C(k)\sum_{nlj}^{\text{F.L.}} 2(2j+1)
\lambda_{nlj}
|R_{nlj}(k)|^2 ,
\label{eq:36}$$ where F.L. denotes the Fermi level, and $$C(k)= \frac{m_N}{(2\pi)^3\sqrt{k^2+m_N^2}} ,
\label{eq:37}$$ $m_N$ being the nucleon mass. To a good approximation for the hole states, the natural occupation numbers $\lambda_{nlj}$ are close to unity in [@AGI+02] and the natural orbitals $R_{nlj}(k)$ are replaced by single-particle wave functions from the self-consistent mean-field approximation. In [@AGI+02] Woods-Saxon single-particle wave functions were used for protons and neutrons. $N_A$ is the normalization factor. Concerning the particle-state $[n^{\text{p}}(k)]$ contribution in (\[eq:35\]), we used in [@AGI+02] and [@AGK+05] the well-known facts that: (i) the high-momentum components of $n(k)$ caused by short-range and tensor correlations are almost completely determined by the contributions of the particle-state natural orbitals (e.g. [@SAD93]), and (ii) the high-momentum tails of $n_{A}(k)/A$ are approximately equal for all nuclei and are a rescaled version of the nucleon momentum distribution in the deuteron $n_d(k)$ [@FCW00], $$n_A(k)\simeq \alpha_A n_d(k),
\label{eq:38}$$ where $\alpha_A$ is a constant. These facts made it possible to assume in [@AGI+02] and [@AGK+05] that $n^{\text{p}}(k)$ is related to the high-momentum components $n_5(k)$ of the deuteron, that is, $$n^{\text{p}}(k)= \frac{A}{2} n_5(k).
\label{eq:39}$$ In (\[eq:39\]) $n_5(k)$ is expressed by an angle-averaged function [@AGI+02] as $$n_5(k)= C(k) \overline{(1-z^2) f_5^2(k)}.
\label{eq:40}$$ In (\[eq:40\]) $z=\cos(\widehat{\vec{k},\vec{n}})$, $\vec{n}$ being a unit vector along the three vector ($\vec{\omega}$) component of the four-vector $\omega$ which determines the position of the light-front surface [@CK95; @CDK98]. The function $f_5(k)$ is one of the six scalar functions $f_{1-6}(k^2,\vec{n}\cdot\vec{k})$ which are the components of the deuteron total wave function $\Psi(\vec{k},\vec{n})$. It was shown [@CK95] that $f_5$ largely exceeds other $f$-components for $k\geq$ 2.0–2.5 fm$^{-1}$ and is the main contribution to the high-momentum component of $n_d(k)$, incorporating the main part of the short-range properties of the nucleon-nucleon interaction.
It was shown in Fig. 2 of [@AGK+05] that the calculated LFD $n(k)$’s are in good agreement with the “$y$-scaling data” for $^4$He, $^{12}$C and $^{56}$Fe from [@Cio90] and also with the $y_{\text{CW}}$ analysis [@CW99; @CW97] up to $k\lesssim
2.8$ fm$^{-1}$. For larger $k$ the momentum distributions from LFD exceeds that obtained from $y_{\text{CW}}$ analysis. We should note also that the calculated scaling function $f^{QE}(\psi')$ using the approximate relationship (see Eq. (75) and Fig. 4 of [@AGK+05]) $$f^{QE}(\psi')\simeq 3\pi \int_{|y|}^{\infty} d\overline{k}_F \,
n(\overline{k}_F) {\overline{k}_F}^2, \;\;\;
|y|=\frac{1-\sqrt{1-4ck_F |\psi'|}}{2c}, \;\;\; c\equiv
\frac{\sqrt{1+m_{N}^{2}/q^{2}}}{2m_{N}},
\label{eq:41}$$ for $^{56}$Fe at $q=1000$ MeV/c is in agreement with the data for $-0.5\lesssim\psi'\leq0$, while in the region $-1.1\leq\psi'\leq-0.5$ it shows a dip in the interval $-0.9\leq\psi'\leq-0.6$. This difference is due to the particular form of $n(k)$ from LFD shown in Fig. 2 of [@AGK+05] (a dip around $k\approx1.7$ fm$^{-1}$ and a very high-momentum tail at $k\gtrsim2.8$ fm$^{-1}$). This result showed that the assumption (\[eq:39\]) for the particle-state contribution is a rather rough one. In this paper we consider a modification of the approach in which we include partially in the particle-state part $n^{p}(k)$ not only $n_5(k)$ but also $n_2(k)$ which is related to the angle-averaged function $f_2(k)$: $$n_2(k)= C(k) \overline{f_2^2(k)}.
\label{eq:42}$$ Then the particle-state part can be written in the form $$n^{\text{p}}(k)=\beta\big[n_2(k)+n_5(k)\big],
\label{eq:43}$$ where $\beta $ is a parameter. Then the LFD nucleon momentum distribution for the nucleus with $A$ nucleons will be: $$n_{\text{LFD}}(k)=N_A\big[n^{\text{h}}(k)+\beta\big(n_2(k)+n_5(k)\big)\big],
\label{eq:44}$$ with $n^{\text{h}}(k)$ from Eq. (\[eq:36\]) and $$N_A= \left\{ 4\pi \int_0^\infty dq \, q^2 \left[
\sum_{nlj}^{\text{F.L.}} 2(2j+1) \lambda_{nlj} C(q) |R_{nlj}(q)|^2
+ \beta \big(n_2(q)+n_5(q)\big)\right] \right\}^{-1} .
\label{eq:45}$$ In Fig. \[fig03\] we present the nucleon momentum distribution for $^{12}$C calculated within the LFD method using Eqs. (\[eq:35\])–(\[eq:37\]), (\[eq:40\]), (\[eq:42\])–(\[eq:45\]) with the parameter value $\beta=0.80$. It is compared with the band of CDFM momentum distributions for $^4$He, $^{12}$C, $^{27}$Al, $^{56}$Fe, $^{197}$Au (grey area), with $n_{\text{CW}}(k)$ from the $y_{\text{CW}}$ analysis [@CW99; @CW97] and with the $y$-scaling data [@Cio90] for $^4$He, $^{12}$C, and $^{56}$Fe. It can be seen that up to $k\simeq2.8$ fm$^{-1}$ $n_{\text{LFD}}$ curve is close to the results of [@Cio90; @CW99; @CW97]. For the region $1\leq k\leq2.5$ fm$^{-1}$ it is between them and for $k\geqslant2.8$ fm$^{-1}$ it is close to $n_{\text{CW}}(k)$, in contrast to our previous results in Fig. 2 of Ref. [@AGK+05] (see also [@AGI+02]) which were based on Eq. (\[eq:39\]) and which are also shown for comparison in Fig. \[fig03\]. This behavior of $n_{\text{LFD}}(k)$ reflects in the result of the calculation of the QE scaling function using Eq. (\[eq:41\]) given in Fig. \[fig04\]. It can be seen that both momentum distributions $n_{\text{CW}}$ [@CW97] and $n_{\text{LFD}}(k)$ \[Eq. (\[eq:44\])\] give a good agreement with the experimental data for the QE scaling function at least up to $\psi'\simeq-1.2$. This result is an improvement of that for LFD shown in Fig. 4 of [@AGK+05], where only the contribution $n_5$ was used in the calculation of $n^{\text{p}}(k)$ (\[eq:39\]) and $n_{\text{LFD}}(k)$.
![The nucleon momentum distribution $n(k)$. Grey area: CDFM combined results for $^{4}$He, $^{12}$C, $^{27}$Al, $^{56}$Fe and $^{197}$Au. Solid line: result of the present work for $^{12}$C using the modified LFD approach with $\beta=0.80$. Dashed line: $y_{CW}$-scaling result [@CW99; @CW97]. Dash-dotted line: result of LFD for $^{12}$C from [@AGI+02]. Dotted line: the mean-field result using Wood-Saxon single-particle wave functions for $^{56}$Fe. Open squares, circles and triangles are $y$-scaling data [@Cio90] for $^{4}$He, $^{12}$C and $^{56}$Fe, respectively. The normalization is: $\int n(k)d^{3}{\bf k}=1$. \[fig03\]](fig03.eps){width="10cm"}
![The quasielastic scaling function $f^{QE}(\psi^{\prime})$ calculated using Eq. (\[eq:41\]) at $q=1000$ MeV/c with $n_{CW}(k)$ from the $y_{CW}$-scaling analysis [@CW99; @CW97] for $^{56}$Fe (solid line) and $n_{LFD}(k)$ from modified LFD approach \[Eq. (\[eq:44\])\] for $^{12}$C (dashed line). \[fig04\]](fig04.eps){width="10cm"}
Scaling function in the quasielastic delta region
=================================================
In this Section we will extend our analysis within both CDFM and LFD to the $\Delta$-peak region, which is not too far above the QE peak region and is the main contribution to the inelastic scattering. Dividing the cross section by the appropriate single-nucleon cross section, now for $N\rightarrow\Delta$ transition, and displaying the results versus a new scaling variable ($\psi_\Delta'$) (in which the kinematics of resonance electro-production is accounted for) it is obtained in [@Amaro2005] that the results scale quite well. This is considered as an indication that the procedure has identified the dominant contributions not only in the QE region, but also in the $\Delta$-region.
The shifted dimensionless scaling variable in the $\Delta$-region $\psi_\Delta'$ is introduced (see, e.g. [@Amaro2005]) by the expression: $$\label{eq:46}
\psi _{\Delta}'\equiv \left[ \frac{1}{\xi _{F}}\left( \kappa
\sqrt{{{\rho}_{\Delta}'}^{2}+1/\tau' }-\lambda'
{\rho}_{\Delta}'-1\right) \right] ^{1/2}\times \left\{
\begin{array}{cc}
+1, & \lambda' \geq {\lambda'}_{\Delta}^{0} \\
-1, & \lambda' \leq {\lambda'}_{\Delta}^{0}
\end{array}
\right. ,$$ where $$\label{eq:47}
\xi_F\equiv \sqrt{1+\eta_F^2}-1,\qquad \eta_F \equiv
\dfrac{k_F}{m_N},$$ $$\label{eq:48}
\lambda'=
\lambda-\dfrac{E_{shift}}{2m_N},\qquad\tau'=\kappa^2-\lambda'^2,$$ $$\label{eq:49}
\lambda =\dfrac{\omega}{2m_{N}},\qquad\kappa =
\dfrac{q}{2m_{N}},\qquad\tau =\kappa ^{2}-\lambda ^{2},$$ $$\label{eq:50}
{\lambda'}_{\Delta}^{0}=\lambda
_{\Delta}^{0}-\frac{E_{shift}}{2m_N},\qquad \lambda
_{\Delta}^{0}=\frac{1}{2}\left[ \sqrt{\mu _{\Delta}^{2}+4\kappa
^{2}}-1\right] ,$$ $$\label{eq:51}
\qquad \mu _{\Delta}=m_{\Delta }/m_{N},$$ $$\label{eq:52}
\rho_{\Delta} =1+\dfrac{\beta _{\Delta}}{\tau},\qquad
{\rho}_{\Delta}' =1+\dfrac{\beta _{\Delta}}{\tau'},$$ $$\label{eq:53}
\beta _{\Delta} =\dfrac{1}{4}\left( \mu _{\Delta}^{2}-1\right).$$
The relativistic Fermi gas superscaling function in the $\Delta$ domain is given by [@Amaro2005]: $$\label{eq:54}
f_{RFG}^{\Delta}(\psi'_{\Delta})=\dfrac{3}{4}(1-{\psi'_{\Delta}}^2)\Theta(1-
{\psi'_{\Delta}}^2).$$
Following the CDFM application to the scaling phenomenon, the $\Delta$-scaling function in the model will be: $$\label{eq:55}
f^{\Delta}(\psi'_{\Delta})=\int_{0}^{\infty}dR|F_\Delta(R)|^2f_{RFG}^{\Delta}(\psi'_{\Delta}(R)).$$ In Eq. (\[eq:55\]): $$\label{eq:56}
{\psi'_{\Delta}}^2(R)=\dfrac{1}{\left[\sqrt{1+\dfrac{k_F^2(R)}{m_N^2}}-1\right]}
\left[ \kappa \sqrt{{{\rho}_{\Delta}'}^{2}+\dfrac{1}{\tau'}
}-\lambda' {\rho}_{\Delta}'-1\right]\equiv t(R).{\psi_\Delta'}^2,$$ where $$\label{eq:57}
t(R)\equiv\dfrac{\left[\sqrt{1+\dfrac{k_F^2}{m_N^2}}-1\right]}{\left[\sqrt{1+\dfrac{k_F^2(R)}
{m_N^2}}-1\right]}\quad \text{and}\quad k_F(R)=\dfrac{\alpha}{R}.$$
In the CDFM $k_F$ can be calculated using the density distribution (Eqs. (\[eq:8\]), (\[eq:9\]) or (\[eq:19\]) and (\[eq:16\])) or the momentum distribution (Eqs. (\[eq:10\]), (\[eq:11\]) or (\[eq:25\]) and (\[eq:24\])). The weight function $|F_{\Delta}(R)|^2$ is related to the density distributions (Eqs. (\[eq:2\]) or (\[eq:16\])). In the equivalent form of the CDFM, the scaling function can be written in the form: $$\label{eq:58}
f^{\Delta}(\psi'_{\Delta})=\int_{0}^{\infty} d\overline{k}_F
|G_\Delta(\overline{k}_F)|^2
f_{RFG}^{\Delta}(\psi_\Delta'(\overline{k}_F)),$$ where $G_\Delta(\overline{k}_F)$ is determined by means of the momentum distribution (Eqs. (\[eq:6\]) or (\[eq:24\])) and $$\label{eq:59}
\psi_\Delta'^{2}(\overline{k}_F)\equiv\widetilde{t}(\overline{k}_F).\psi_\Delta'^{2}$$ with $$\label{eq:60a}
\widetilde{t}(\overline{k}_F)\equiv
\dfrac{\left[\sqrt{1+\dfrac{k_F^2}{m_N^2}}-1\right]}
{\left[\sqrt{1+\dfrac{\overline{k}_F^2}{m_N^2}}-1\right]}.$$ Here we would like to note that though the functional forms of $f^{\Delta}(\psi'_\Delta)$ \[Eq. (\[eq:55\])\] and the weight function $|F_\Delta(R)|^2$ (Eqs. (\[eq:2\]) or (\[eq:16\])) are like before, i.e. as in the case of the QE region, the parameters of the densities (e.g. the half-radius $R_\Delta$ and the diffuseness $b_\Delta$ when Fermi-type forms have been used) may be different from those ($R$ and $b$) in the QE case. Along this line, we calculated firstly the scaling function $f^{\Delta}(\psi_\Delta')$ by means of Eqs. (\[eq:55\])–(\[eq:57\]) using the Fermi-type density for $^{12}$C. We found the values of $R_\Delta$ and $b_\Delta$ fitting the scaling data at the $\Delta$ peak extracted from the high-quality world data for inclusive electron scattering (given in Fig. 2 of [@Amaro2005]). Our results are presented in Fig. \[fig05\]. As mentioned already in the QE case, the empirical data require to use a value of the coefficient in the right-hand side of Eq. (\[eq:54\]) for the RFG scaling functions $f_{RFG}^{\Delta}(\psi_\Delta')$ different from $3/4$. In our calculations in the $\Delta$-region we use the value $0.54$. We found that reasonable agreement with the data can be achieved using the parameter values $R_\Delta=1.565$ fm and $b_\Delta=0.420$ fm (the Fermi-momentum value is taken to be $k_F=1.20$ fm$^{-1}$ and this choice leads to normalization to unity of $f_{RFG}^{\Delta}(\psi_\Delta')$). The value of $R_\Delta$ is smaller than that used in the description of the QE superscaling function for $^{12}$C [@AGK+04; @AGK+05] ($R=2.470$ fm) while the value of $b_\Delta$ is the same as $b$ in the QE case. Secondly, we calculated $f^{\Delta}(\psi_\Delta')$ using Eqs. (\[eq:58\]), (\[eq:59\]) and (\[eq:60a\]). In Eq. (\[eq:58\]) the weight function $|G_\Delta(\overline{k}_F)|^2$ was determined by means of Eq. (\[eq:6\]) and the nucleon momentum distribution $n_\text{LFD}$ (Eqs. (\[eq:44\]) and (\[eq:45\])) calculated with the parameter value $\beta=0.80$ (shown in Fig. \[fig03\]). We note that the use of $n_\text{LFD}(k)$ with this value of $\beta$ gives simultaneously a reasonable agreement both with the results for the momentum distribution from the $y$-scaling data shown in Fig. \[fig03\], as well as with the QE scaling function shown in Fig. \[fig04\].
![The $f^{\Delta}(\psi_{\Delta}^{\prime})$ scaling function for $^{12}$C in the $\Delta$-region. Dashed line: CDFM result (with $R_{\Delta}=1.565$ fm, $b_{\Delta}=0.420$ fm, $k_{F}=1.20$ fm$^{-1}$). Solid line: result of modified LFD approach ($\beta=0.80$, $k_{F}=1.20$ fm$^{-1}$). The coefficient $c_{1}=0.54$ in both CDFM and LFD cases. Averaged experimental values of $f^{\Delta}(\psi_{\Delta}^{\prime})$ are taken from [@Amaro2005]. \[fig05\]](fig05.eps){width="10cm"}
Scaling functions and inclusive lepton scattering
=================================================
Scaling functions and $(e,e^{\prime})$ reaction cross sections
--------------------------------------------------------------
In the beginning of this Subsection we will give some basic relationships concerning inclusive electron scattering from nuclei. An electron with four-momentum $k^{\mu}=(\epsilon,{\bf
k})$ is scattered through an angle $\theta$ to four-momentum $k^{\prime\mu}=(\epsilon^{\prime},{\bf k^{\prime}})$. The four-momentum transfer is then $$Q^{\mu}=(k-k^{\prime})^{\mu}=(\omega,{\bf q}),
\label{eq:60}$$ where $\omega=\epsilon-\epsilon^{\prime}$, $q=|{\bf q}|=|{\bf
k}-{\bf k^{\prime}}|$ and $$Q^{2}=\omega^{2}-q^{2}\leq 0.
\label{eq:61}$$ In the one-photon-exchange approximation, the double-differential cross section in the laboratory system can be written in the form (e.g. [@AMD+88]): $$\frac{d^{2}\sigma}{d\Omega_{k^{\prime}}d\epsilon^{\prime}}=\sigma_{M}\left
[\left (\frac{Q^{2}}{q^{2}}\right )^{2}R_{L}(q,\omega)+\left
(\frac{1}{2}\left|\frac{Q^{2}}{q^{2}}\right
|+\tan^{2}\frac{\theta}{2}\right )R_{T}(q,\omega)\right],
\label{eq:62}$$ where $$\sigma_{M}=\left [\frac{\alpha \cos (\theta/2)}{2\epsilon
\sin^{2}(\theta/2)}\right ]^{2}
\label{eq:63}$$ is the Mott cross section and $\alpha$ is the fine structure constant. In Eq. (\[eq:62\]) $R_{L}$ and $R_{T}$ are the longitudinal and transverse response functions which contain all the information on the distribution of the nuclear electromagnetic charge and current densities, being projections (with respect to the momentum transfer direction) of the nuclear currents. They can be separated experimentally by plotting the cross section against $\tan^{2}(\theta/2)$ at fixed $(q,\omega)$ (the so-called “Rosenbluth plot”). These functions can be evaluated as components of the nuclear tensor $W_{\mu\nu}$. In [@AMD+88] this tensor is computed in the framework of the RFG model and $R_{L(T)}$ for the QE electron scattering are expressed by means of the RFG scaling function (Eq. (\[eq:9\]) of Ref. [@AMD+88]).
At leading-order in the parameter $k_{F}/m_{N}$ the QE responses have the form [@Amaro2005]: $$R_{L}^{QE}(\kappa,\lambda)=\Lambda_{0}\frac{\kappa^{2}}{\tau}[(1+\tau)W_{2}(\tau)-
W_{1}(\tau)]\times f_{RFG}^{QE}(\psi^{\prime}),
\label{eq:64}$$ $$R_{T}^{QE}(\kappa,\lambda)=\Lambda_{0}[2W_{1}(\tau)]\times
f_{RFG}^{QE}(\psi^{\prime}),
\label{eq:65}$$ with $$\Lambda_{0}\equiv \frac{{\cal N}\xi_{F}}{m_{N}\kappa\eta_{F}^{3}},
\label{eq:66}$$ where ${\cal N}=Z$ or $N$ and $W_{1}$, $W_{2}$ are the structure functions for elastic scattering which are linked to the Sachs form factors $$(1+\tau)W_{2}(\tau)-W_{1}(\tau)=G_{E}^{2}(\tau),
\label{eq:67}$$ $$2W_{1}(\tau)=2\tau G_{M}^{2}(\tau).
\label{eq:68}$$ In [@Amaro99; @Amaro2005] the electro-production of the $\Delta$-resonance is considered computing the nuclear tensor also within the RFG model and analytical expressions for the response functions are obtained. The latter contain the RFG $\Delta$-peak scaling function (\[eq:54\]) and read [@Amaro99]: $$R_{L}(\kappa,\lambda)=\frac{3{\cal
N}\xi_{F}}{2m_{N}\eta_{F}^{3}\kappa}\frac{\kappa^{2}}{\tau}[(1+\tau\rho^{2})w_{2}(\tau)-
w_{1}(\tau)+w_{2}(\tau)D(\kappa,\lambda)]\times
f_{RFG}^{\Delta}(\psi_{\Delta}^{\prime}),
\label{eq:69}$$ $$R_{T}(\kappa,\lambda)=\frac{3{\cal
N}\xi_{F}}{2m_{N}\eta_{F}^{3}\kappa}[2w_{1}(\tau)+w_{2}(\tau)D(\kappa,\lambda)]\times
f_{RFG}^{\Delta}(\psi_{\Delta}^{\prime}),
\label{eq:70}$$ where ${\cal N}=Z$ or $N$, $$\begin{aligned}
D(\kappa,\lambda)& \equiv &
\frac{\tau}{\kappa^{2}}[(\lambda\rho+1)^{2}+(\lambda\rho+1)(1+\psi_{\Delta}^
{\prime 2})\xi_{F}\\ \nonumber &+&
\frac{1}{3}(1+\psi_{\Delta}^{\prime 2}+\psi_{\Delta}^{\prime
4})\xi_{F}^{2}]-(1+\tau\rho^{2}).
\label{eq:71}\end{aligned}$$ The single-baryon structure functions can be expressed by means of the electric $(G_{E})$, magnetic $(G_{M})$ and Coulomb $(G_{C})$ delta form factors [@Amaro99]: $$w_{1}(\tau)=\frac{1}{2}(\mu_{\Delta}+1)^{2}(2\tau\rho+1-\mu_{\Delta})
(G_{M}^{2}+3G_{E}^{2}),
\label{eq:72}$$ $$w_{2}(\tau)=\frac{1}{2}(\mu_{\Delta}+1)^{2}\frac{2\tau\rho+1-\mu_{\Delta}}{1+\tau\rho^{2}}
\left
(G_{M}^{2}+3G_{E}^{2}+4\frac{\tau}{\mu_{\Delta}^{2}}G_{C}^{2}
\right ).
\label{eq:73}$$ These form factors are parametrized as follows [@Amaro99]: $$G_{M}(Q^{2})=2.97f(Q^{2}),
\label{eq:74}$$ $$G_{E}(Q^{2})=-0.03f(Q^{2}),
\label{eq:75}$$ $$G_{C}(Q^{2})=-0.15G_{M}(Q^{2}),
\label{eq:76}$$ where $$f(Q^{2})=G_{E}^{P}(Q^{2})\frac{1}{\left
[1-\frac{Q^{2}}{3.5(GeV/c)^{2}}\right]^{1/2}}
\label{eq:77}$$ with $$G_{E}^{P}=\frac{1}{(1+4.97\tau)^{2}}
\label{eq:78}$$ being the Galster parametrization [@Galster71] of the electric form factor.
In the CDFM the longitudinal and transverse response functions can be obtained by averaging the RFG response functions in the QE region \[Eqs. (\[eq:64\]) and (\[eq:65\])\] and $\Delta$-region \[Eqs. (\[eq:69\]) and (\[eq:70\])\] by means of the weight functions in $r$-space $|F(R)|^{2}$ and $k$-space $|G(\overline{k}_{F})|^{2}$, similarly as in the case of the QE- and $\Delta$-scaling functions (Eqs. (\[eq:1\]), (\[eq:5\]), (\[eq:15\]), (\[eq:23\]) and (\[eq:55\]), (\[eq:58\]), respectively). As a result, accounting for the different behavior of the RFG scaling functions and terms containing $\eta_{F}(R)=k_{F}(R)/m_{N}$ as functions of $R$ or $\overline{k}_{F}=\alpha/R$ in (\[eq:64\]), (\[eq:65\]), (\[eq:69\]) and (\[eq:70\]), the CDFM response functions $R_{L(T)}$ in QE- and $\Delta$-regions have approximately the same forms as in the equations just mentioned, in which, however, the RFG scaling functions are changed by the CDFM scaling functions obtained in Sections 2 and 3.
In Figs. \[fig06\]-\[fig15\] we give results of calculations within the CDFM of inclusive electron scattering on $^{12}$C at different incident energies and angles. The QE-contribution is calculated using the Fermi-type density distribution of $^{12}$C with the same values of the parameters as in [@AGK+04; @AGK+05]: $R=2.47$ fm and $b=0.42$ fm (which lead to a charge rms radius equal to the experimental one) and Fermi momentum $k_{F}=1.156$ fm$^{-1}$. The delta-contribution is calculated using the necessary changes of the parameter values of the Fermi-type density (used in Fig. \[fig05\]): $R_{\Delta}=1.565$ fm, $b_{\Delta}=0.42$ fm and $k_{F}=1.20$ fm$^{-1}$. The coefficient $c_{1}$ used in the $\Delta$-region scaling function is fixed to be equal to 0.54 so that the maximum of the scaling function to be in agreement with the data. The scaling function $f^{\Delta}(\psi_{\Delta}^{\prime})$ is symmetric, its maximum is chosen to be 0.54 (but not 0.75) and it is normalized to unity by means of the fixed value of $k_{F}=1.20$ fm$^{-1}$. The inclusive electron-$^{12}$C scattering cross sections shown in Figs. \[fig06\]-\[fig15\] are the sum of the QE and $\Delta$-contribution. The results of the CDFM calculations are presented for two values of the coefficient $c_{1}$ in the QE case (noted further by $c_{1}^{QE}$), namely for $c_{1}^{QE}\simeq
0.72$ and $c_{1}^{QE}=0.63$. This is related to two types of experimental data. In the first one the transferred momentum in the position of the maximum of the QE peak extracted from data ($\omega_{exp}^{QE}$) is $q_{exp}^{QE}\geq 450$ MeV/c $\approx
2k_{F}$, roughly corresponding to the domain where scaling is fulfilled [@BCD+04; @Amaro2005]. Such cases are presented in Figs. \[fig06\]-\[fig12\]. In these cases we found by fitting to the maximum of the QE peak the value of $c_{1}^{QE}$ to be 0.72–0.73, i.e. it is not the same as in the RFG model case (case of symmetry of the RFG and of the CDFM scaling functions with $c_{1}^{QE}=0.75$), but is slightly lower. This leads to a weak asymmetry of the CDFM scaling function for cases in which $q_{exp}^{QE}\geq 450$ MeV/c. In the second type of experimental data $q_{exp}^{QE}$ is not in the scaling region ($q_{exp}^{QE}<450$ MeV/c). Such cases are given in Figs. \[fig13\]-\[fig15\]. For them we found by fitting to the maximum of the QE peak extracted from data the value of $c_{1}^{QE}$ to be 0.63. For these cases the CDFM scaling function is definitely asymmetric. So, the results in Figs. \[fig06\]-\[fig15\] are presented for both almost symmetric ($c_{1}^{QE}\simeq 0.72$) and asymmetric ($c_{1}^{QE}=0.63$) CDFM scaling functions. One can see that the results for the almost symmetric CDFM scaling function agree with the electron data in the region close to the QE peak in cases where $q_{exp}^{QE}\geq 450$ MeV/c and overestimate the data for cases where approximately $q_{exp}^{QE}<450$ MeV/c. The results with asymmetric CDFM scaling function agree with the data in cases where $q_{exp}^{QE}<450$ MeV/c and underestimate the data in cases where $q_{exp}^{QE}\geq 450$ MeV/c. Here we would like to emphasize that, in our opinion, the usage of asymmetric CDFM scaling function is preferable, though the results in some cases can underestimate the empirical data, because other additional effects, apart from QE and $\Delta$-resonance (e.g. meson exchange currents effects) could give important contributions to the cross section for some specific kinematics and minor for others. A similar situation occurs for the results obtained within the RMF approach [@Caballero2006] particularly when the CC2 current operator is selected.
In Table \[table1\] we list the energies, the angles, the values of $c_{1}^{QE}$ obtained by fitting the magnitude of the QE peak, and the energy shifts in the QE and $\Delta$-case, as well as the approximate values of the transfer momentum $q_{exp}^{QE}$ in the position of the maximum of the QE peak ($\omega_{exp}^{QE}$) for different cases. The values of the energy shifts $\epsilon_{shift}^{QE(\Delta)}$ for the QE- and $\Delta$-regions are generally between 20 and 30 MeV. In the Figures we also present the QE-contribution (as well as $\Delta$-contribution) for the value of $c_{1}^{QE}$ which fits approximately the magnitude of the QE peak.
![Inclusive electron scattering on $^{12}$C at $\epsilon=1299$ MeV and $\theta=37.5^{\circ}$ ($q_{exp}^{QE}=792$ MeV/c $>2k_{F}$). The results obtained using $c_{1}^{QE}=0.72$ in the CDFM scaling function for the QE cross section and the total result are given by dashed and thick solid line, respectively. Dotted line: using CDFM $\Delta$-scaling function; thin solid line: total CDFM result with $c_{1}^{QE}=0.63$. Dash-dotted line: result of ERFG method [@BCD+04; @Amaro2005]. The experimental data are taken from [@Sealock89]. \[fig06\]](fig06.eps){width="10cm"}
![Inclusive electron scattering on $^{12}$C at $\epsilon=2020$ MeV and $\theta=20.02^{\circ}$ ($q_{exp}^{QE}=703$ MeV/c $>2k_{F}$). The results obtained using $c_{1}^{QE}=0.73$ in the CDFM scaling function for the QE cross section and the total result are given by dashed and thick solid line, respectively. Dotted line: using CDFM $\Delta$-scaling function. Thin solid line: total CDFM result with $c_{1}^{QE}=0.63$. The experimental data are taken from [@Day93]. \[fig07\]](fig07.eps){width="10cm"}
![The same as in Fig. \[fig07\] for $\epsilon=1108$ MeV and $\theta=37.5^{\circ}$ ($q_{exp}^{QE}=675$ MeV/c $>2k_{F}$). Dot-dashed line: using QE- and $\Delta$-scaling functions obtained in the LFD approach. The experimental data are taken from [@Sealock89]. \[fig08\]](fig08.eps){width="10cm"}
![The same as in Fig. \[fig07\] for $\epsilon=620$ MeV and $\theta=60^{\circ}$ ($q_{exp}^{QE}=552$ MeV/c $>2k_{F}$). The experimental data are taken from [@Barr83]. \[fig09\]](fig09.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=2020$ MeV and $\theta=15.02^{\circ}$ ($q_{exp}^{QE}=530$ MeV/c $>2k_{F}$) for the CDFM results. The experimental data are taken from [@Day93]. \[fig10\]](fig10.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=500$ MeV and $\theta=60^{\circ}$ ($q_{exp}^{QE}=450$ MeV/c $\geq 2k_{F}$). Here the dot-dashed line shows the result using QE- and $\Delta$-scaling functions obtained in the LFD approach. The experimental data are taken from [@Whitney74]. \[fig11\]](fig11.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=730$ MeV and $\theta=37.1^{\circ}$ ($q_{exp}^{QE}=442$ MeV/c $\leq 2k_{F}$) for the CDFM results. The experimental data are taken from [@Oconnell87]. \[fig12\]](fig12.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=1650$ MeV and $\theta=13.5^{\circ}$ ($q_{exp}^{QE}=390$ MeV/c $\leq 2k_{F}$) for the CDFM results. The experimental data are taken from [@Baran88]. \[fig13\]](fig13.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=1500$ MeV and $\theta=13.5^{\circ}$ ($q_{exp}^{QE}=352$ MeV/c $\leq
2k_{F}$). The experimental data are taken from [@Baran88]. \[fig14\]](fig14.eps){width="10cm"}
![The same as in Fig. \[fig06\] for $\epsilon=537$ MeV and $\theta=37.1^{\circ}$ ($q_{exp}^{QE}=326$ MeV/c $\leq 2k_{F}$) for the CDFM results. The experimental data are taken from [@Oconnell87]. \[fig15\]](fig15.eps){width="10cm"}
Figure $\epsilon$ $\theta$ $c_{1}^{QE}$ $\epsilon_{shift}^{QE}$ $\epsilon_{shift}^{\Delta}$ $\approx q_{exp}^{QE}$
-------- -- ------------ -- ---------- -- -------------- -- ------------------------- -- ----------------------------- -- ------------------------ --
6 1299 37.5 0.72 30 30 792
7 2020 20.02 0.73 25 20 703
8 1108 37.5 0.73 30 30 675
9 620 60 0.73 20 0 552
10 2020 15.02 0.72 20 30 530
11 500 60 0.72 30 0 450
12 730 37.1 0.72 20 20 442 $\simeq 2k_{F}$
13 1650 13.5 0.63 20 30 390
14 1500 13.5 0.63 20 20 352
15 537 37.1 0.63 20 20 326
: Values of energies $\epsilon$, angles $\theta$, the coefficient $c_{1}^{QE}$ obtained by fitting the magnitude of the QE peak, energy shifts $\epsilon_{shift}^{QE}$ and $\epsilon_{shift}^{\Delta}$, and transferred momenta $q_{exp}^{QE}$ for the cases of inclusive electron scattering cross sections considered. Energies are in MeV, angles are in degrees and momenta are in MeV/c.
\[table1\]
In Figs. \[fig08\] and \[fig11\] we present also the calculations of the electron cross sections using QE- and $\Delta$-scaling functions obtained by using the nucleon momentum distributions obtained in the LFD approach (Section 3) which give a reasonable agreement with the empirical electron scattering data. In Figs. \[fig06\] and \[fig14\] we also give for comparison the results of the cross sections obtained within the ERFG method [@BCD+04; @Amaro2005]. In this method the response functions and differential cross sections are calculated using the scaling function fitted to the data.
It is interesting to note that for those kinematics where the overlap between the QE and $\Delta$ peaks is bigger (Figs. \[fig06\], \[fig07\] and \[fig08\]), the asymmetric CDFM model ($c_{1}^{QE}=0.63$) gives rise to an excess of strength in the transition region. This makes a difference with the ERFG model (see Fig. \[fig06\]) which fits nicely the data in that region. This discrepancy between the two models, asymmetric CDFM and ERFG, can be explained by noting the different behavior presented by the two scaling functions in the region of $\psi^{\prime}$ between 0.5 and 1.5, being the asymmetric CDFM one significantly larger.
Note on the other hand, that in the cases where the overlap between the QE and $\Delta$ peaks is weaker (Figs. \[fig09\]-\[fig12\]), the asymmetric CDFM model, compared to the almost symmetric CDFM one, reproduces better the data in the transition region although it underpredicts importantly the maximum of the QE peak. Concerning results in Figs. \[fig13\]-\[fig15\] (it can be also applied to Figs. \[fig11\] and \[fig12\]), one observes that both CDFM approaches do not reproduce the strength of data located in the region between the QE and delta peaks. This is not the case for the ERFG model (see Fig. \[fig14\]) which fits nicely the experiment for $\omega\geq 180$ MeV. This result is connected with the much bigger strength of the scaling function provided by the ERFG model for larger values of the scaling variable, $\psi^{\prime}\geq 2$ (see Fig. \[fig02\]).
From this whole analysis, one may conclude that the phenomenological procedure introduced in the CDFM model to get an asymmetric scaling function, gives rise to an excess of strength in the region $0.5\leq\psi^{\prime}\leq 1.5$, whereas the model lacks strength for larger $\psi^{\prime}$-values, $\psi^{\prime}\geq 2$.
Scaling functions and charge-changing neutrino-nucleus reaction cross sections
------------------------------------------------------------------------------
In this Subsection we will present applications of the CDFM and LFD scaling functions to calculations of charge-changing neutrino-nucleus reaction cross sections. We follow the description of the formalism given in [@Amaro2005]. The charge-changing neutrino cross section in the target laboratory frame is given in the form $$\left [ \frac{d^{2}\sigma}{d\Omega dk^{\prime}}\right
]_{\chi}\equiv \sigma_{0}{\cal F}_{\chi}^{2},
\label{eq:79}$$ where $\chi=+$ for neutrino-induced reaction (e.g., $\nu_{\ell}+n\rightarrow \ell^{-}+p$, where $\ell=e, \mu, \tau$) and $\chi=-$ for antineutrino-induced reactions (e.g., $\overline{\nu}_{\ell}+p\rightarrow \ell^{+}+n$), $$\sigma_{0}\equiv
\frac{(G\cos\theta_{c})^{2}}{2\pi^{2}}[k^{\prime}\cos\tilde{\theta}/2]^{2},
\label{eq:80}$$ $G=1.16639\times 10^{-5}$ GeV$^{-2}$ being the Fermi constant, $\theta_{c}$ being Cabibbo angle $(\cos\theta_{c}=0.9741)$, $$\tan^{2}\tilde{\theta}/2\equiv \frac{|Q|^{2}}{v_{0}},
\label{eq:81}$$ $$v_{0}\equiv (\epsilon +
\epsilon^{\prime})^{2}-q^{2}=4\epsilon\epsilon^{\prime}-|Q|^{2}.
\label{eq:82}$$ The quantity ${\cal F}_{\chi}^{2}$ which depends on the nuclear structure is written in [@Amaro2005] as a generalized Rosenbluth decomposition having charge-charge, charge-longitudinal, longitudinal-longitudinal and two types of transverse responses. The nuclear response functions are expressed in terms of the nuclear tensor $W^{\mu\nu}$ in both QE and $\Delta$-regions using its relationships with the RFG model scaling functions. Following [@Amaro2005], in the calculations of the neutrino-nucleus cross sections the Höhler parametrization 8.2 [@Hohler76] of the form factors in the vector sector was used, while in the axial-vector sector the form factors given in [@Amaro2005] were used.
In our work, instead of the RFG scaling functions in QE- and $\Delta$-regions, we use those obtained in the CDFM and LFD approach (Sections 2 and 3). In Fig. \[fig16\] we give the results of calculations for cross sections of QE neutrino $(\nu_{\mu},\mu^{-})$ scattering (Figs. \[fig16\]a, c, d, e, f) on $^{12}$C and also antineutrino $(\overline{\nu}_{\mu},\mu^{+})$ scattering (Fig. \[fig16\]b) for energies of neutrino $\epsilon_{\nu}=1, 1.5$ and 2 GeV and of antineutrino $\epsilon_{\overline{\nu}}=1$ GeV. The presented cross sections are functions of muon kinetic energy. The energy shift is equal to 20 MeV. The calculations of the neutrino-nucleus cross sections in the $\Delta$-region will be a subject of a future work.
We give the results of our calculations using the CDFM scaling function which is almost symmetric (with $c_{1}=0.72$), as well as the asymmetric CDFM scaling function (with $c_{1}=0.63$). These values of $c_{1}$ correspond to the cases of inclusive electron scattering considered. As can be seen the results obtained by using the almost symmetric CDFM scaling function are close to the RFG model results. On the other hand, the results obtained with the use of asymmetric CDFM and LFD scaling functions are quite different from those in the RFG model, but are close to the predictions of the ERFG model [@BCD+04; @Amaro2005]. The basic difference from the ERFG model result is observed in the tail extended to small muon energy values, where the ERFG model gives more strength.
![The cross section of quasielastic charge-changing $(\nu_{\mu},\mu^{-})$ reaction \[(a), (c)-(f)\] and of $(\overline{\nu}_{\mu},\mu^{+})$ reaction (b) on $^{12}$C for $\epsilon=1, 1.5$ and 2 GeV using QE-scaling functions in CDFM (thin solid line: with $c_{1}=0.63$; thin dashed line: with $c_{1}=0.72$). The results using QE-scaling functions in LFD (thick solid line: with $c_{1}=0.63$; thick dashed line: with $c_{1}=0.72$) are presented in (b) and (f). The RFG model result and ERFG result [@BCD+04; @Amaro2005] are shown by dotted and dash-dotted lines, respectively. \[fig16\]](fig16.eps){width="15cm"}
Conclusions
===========
The results of the present work can be summarized as follows:
i\) In Ref. [@AGK+05] we extended the CDFM description of the quasielastic $\psi^{\prime}$-scaling function from [@AGK+04] by expressing it explicitly and equivalently by means of both density and nucleon momentum distributions. In [@AGK+04; @AGK+05] our results on $f^{QE}(\psi^{\prime})$ were obtained on the basis of the experimental data on the charge densities for a wide range of nuclei. In the present work we extended our approach to consider the scaling function $f^{QE}(\psi^{\prime})$ for medium and heavy nuclei with $Z\neq N$ for which the proton and neutron densities are not similar. In this case $f^{QE}(\psi^{\prime})$ is a sum of the proton and neutron scaling functions calculated by means of the proton and neutron densities obtained from nonrelativistic self-consistent mean-field calculations. This concerns calculations, as examples, of nuclei like $^{197}$Au, $^{82}$Kr, as well as $^{62}$Ni and $^{118}$Sn [@AIG+06]. The comparison with the data from [@DS99l; @DS99] shows superscaling for negative values of the QE $\psi^{\prime}$ including $\psi^{\prime}<-1$, whereas in the RFG model $f(\psi^{\prime})=0$ for $\psi^{\prime}\leq -1$ (see Fig. \[fig01\]).
ii\) We introduce the asymmetry in the CDFM QE scaling function using the fact that the maximum value of $f(\psi^{\prime})$ in RFG model is 3/4 while the empirical scaling function reaches values smaller than 0.6. In relation with this and the normalization, we parametrize the RFG scaling function for $\psi^{\prime}\geq 0$, thus simulating the role of all the effects which lead to asymmetry and imposing this to the CDFM QE scaling function. In this way, simulating phenomenologically the effects which violate the symmetry of $f^{QE}(\psi^{\prime})$ for $\psi^{\prime}\geq 0$ including the role of the FSI, one can obtain in the CDFM a reasonable agreement of $f^{QE}(\psi^{\prime})$ with the empirical data also for positive values of $\psi^{\prime}$ (Fig. \[fig02\]).
iii\) We obtain the QE scaling function also on the basis of calculations of nucleon momentum distribution $n(k)$ within an approach based on the light-front dynamics method [@AGI+02; @CK95; @CDK98] which improves that used in Ref. [@AGK+05]. Here we include in the particle-state part of $n(k)$ not only a contribution of the function $f_{5}$ (as in [@AGI+02] and [@AGK+05]) but also a contribution of the function $f_{2}$. $f_{5}$ and $f_{2}$ are two of the six scalar components of the deuteron total wave function in the LFD [@CK95; @CDK98] and are the main contributions to the tail of $n_{d}(k)$. It can be seen in Fig. \[fig03\] the reasonable agreement of $n(k)$ in LFD with the $y_{CW}$-scaling data [@CW99; @CW97]. This result made it possible to obtain a good description of the experimental QE scaling function (Fig. \[fig04\]) at least up to $\psi^{\prime}\simeq -1.2$.
iv\) We extend our analysis within the CDFM and LFD to the $\Delta$-peak region which is the main contribution to the inelastic scattering. Here we emphasize that reasonable agreement with the experimental data (Fig. \[fig05\]) was obtained using the empirical value of the coefficient in front of the RFG scaling function (0.54 instead of 0.75) in both CDFM and LFD. Also, the parameter $R_{\Delta}$ used in the Fermi-type density for $^{12}$C (necessary to calculate the weight function $|F_{\Delta}(R)|^{2}$ and thus the scaling function $f^{\Delta}
(\psi_{\Delta}^{\prime})$) has a smaller value (1.565 fm) than that ($R$=2.42 fm) in the QE case, while the value of the diffuseness parameter $b_{\Delta}$ remains the same as $b$ in the QE case. We note that the use of $n_{LFD}(k)$ with the same values of $\beta$ and of $k_{F}$ ($\beta=0.80$, $k_{F}=1.20$ fm$^{-1}$) gives a reasonable agreement with results for both QE- and $\Delta$-region scaling functions (Figs. \[fig04\] and \[fig05\]).
v\) The QE- and $\Delta$-region scaling functions obtained in the CDFM and in the LFD approach are applied to description of experimental data on differential cross sections of inclusive electron scattering by $^{12}$C at large energies and transferred momenta (Figs. \[fig06\]-\[fig15\]). The CDFM results are presented for both almost symmetric ($c_{1}^{QE}\simeq 0.72$) and asymmetric ($c_{1}^{QE}=0.63$) scaling functions. We observe that there are two regions of the value of $q_{exp}^{QE}$ in different experiments at which the above mentioned (almost symmetric and asymmetric) scaling functions work better. The almost symmetric scaling function leads to results in agreement with the data in the region of the QE peak in cases when the transferred momentum $(q_{exp}^{QE})$ in the position of maximum of the QE peak $(\omega_{exp}^{QE})$ is in the scaling region ($q_{exp}^{QE}\geq
450$ MeV/c $\approx 2k_{F}$), while the data are overestimated in cases where $q_{exp}^{QE}<450$ MeV/c. The results obtained when asymmetric scaling function ($c_{1}^{QE}=0.63$) is used agree with the data in cases when $q_{exp}^{QE}< 450$ MeV/c and underestimate them when $q_{exp}^{QE}\geq 450$ MeV/c in the region close to the QE peak, but differences emerge in the transition region. In our opinion, the latter case is preferable because additional effects (apart from QE and $\Delta$-resonance), e.g. of the meson exchange currents could give additional important contributions to the inclusive electron cross sections for some specific kinematics and minor for others.
vi\) The CDFM and LFD scaling functions are applied to calculations of QE charge-changing neutrino-nuclei reaction cross sections. We present in Fig. \[fig16\] the predicted cross sections for the reactions $(\nu_{\mu},\mu^{-})$ and $(\overline{\nu}_{\mu},\mu^{+})$ on the $^{12}$C nucleus for energies of the incident particles from 1 to 2 GeV. Our results are compared with those from the RFG model and from the ERFG model [@BCD+04; @Amaro2005]. The results obtained by using the asymmetric CDFM scaling function are close to those of ERFG and are quite different from the RFG results while the almost symmetric CDFM scaling function leads to cross sections which are similar to the results of the RFG model.
Four of the authors (A.N.A., M.V.I., M.K.G. and M.B.B.) are grateful to C. Giusti and A. Meucci for the discussion. This work was partly supported by the Bulgarian National Science Foundation under Contracts No.$\Phi$-1416 and $\Phi$-1501 and by funds provided by DGI of MCyT (Spain) under Contract Nos. FIS 2005-00640, BFM 2003-04147-C02-01, INTAS-03-54-6545, FPA 2005-04460, and FIS 2005-01105.
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---
abstract: 'We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic $0$, we then apply the Eisenbud-Harris theory of limit linear series to deduce a new proof of the Gieseker-Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost-transverse flags.'
author:
- Melody Chan
- Brian Osserman
- Nathan Pflueger
bibliography:
- 'gen.bib'
title: 'The Gieseker-Petri theorem and imposed ramification'
---
[^1]
Introduction
============
The classical Brill-Noether theorem states that if we are given $g,r,d \geq 0$, a general curve $X$ of genus $g$ carries a linear series $(\sL,V)$ of projective dimension $r$ and degree $d$ if and only if the quantity $$\rho(g,r,d):=g-(r+1)(r+g-d)$$ is nonnegative [@g-h1]. Moreover, in this case the moduli space $G^r_d(X)$ of such linear series has pure dimension $\rho$. This statement was generalized by Eisenbud and Harris to allow for imposed ramification: given marked points $P_1,\dots,P_n \in X$, and sequences $0\le a^i_0<\dots<a^i_r \leq d$ for $i=1,\dots,n$, consider the moduli space $G^r_d(X,(P_1,a^1_{\bullet}),\dots,(P_n,a^n_{\bullet})) \subseteq
G^r_d(X)$ parametrizing linear series with vanishing sequence at least $a^i_{\bullet}$ at each of the $P_i$. Then Eisenbud and Harris used their theory of limit linear series to show in [@e-h1] that in characteristic $0$, if $(X,P_1,\dots,P_n)$ is a general $n$-marked curve of genus $g$, the dimension of $G^r_d(X,(P_1,a^1_{\bullet}),\dots,(P_n,a^n_{\bullet}))$—if it is nonempty—is given by the generalized formula $$\rho(g,r,d,a^1_\bullet,\ldots,a^n_\bullet):=g-(r+1)(r+g-d)-\sum_{i=1}^n \sum_{j=0}^r (a^i_j-j).$$ The condition for nonemptiness is still combinatorial, but becomes more complicated in this context.
This theorem fails in positive characteristic for $n \geq 3$, but is still true if $n \leq 2$. In this case, we also have a simple criterion for nonemptiness. To state it, we shift notation, supposing we have marked points $P,Q\in X$, and sequences $a_{\bullet}$, $b_{\bullet}$. We then introduce the following notation: $$\widehat{\rho}(g,r,d,a_\bullet,b_\bullet) :=
g-\sum_{j:a_j+b_{r-j} > d-g} a_j+b_{r-j}-(d-g).$$
We summarize what was previously known about the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$, as follows.
\[thm:bg\] Given $(g,r,d)$ nonnegative integers, and sequences $0 \leq a_0<a_1 <\dots<a_r \leq d$, $0 \leq b_0<b_1 <\dots<b_r \leq d$, let $(X,P,Q)$ be a twice-marked smooth projective curve of genus $g$ over a field of any characteristic. Set $\rho=\rho(g,r,d,a_{\bullet},b_{\bullet})$ and set $\widehat{\rho}=\widehat{\rho}(g,r,d,a_{\bullet},b_{\bullet})$.
Suppose that $X$ and $P,Q$ are general. Then $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is nonempty if and only if $\widehat{\rho} \geq 0$, and if nonempty, has pure dimension $\rho$. Furthermore, it is reduced and Cohen-Macaulay, and if $\widehat{\rho} \geq 1$, it is connected.
For the nonemptiness and dimension statements, see [@os18]; for reducedness and connectedness, see [@os26]. The Cohen-Macaulayness statement follows from the construction of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ (see for instance the proof of Proposition \[prop:non-smooth\] below) together with the Cohen-Macaulayness of relative Schubert cycles.
What has remained open until now is the question of the singularities of the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$. In the absence of marked points, Gieseker in 1982 used degenerations to prove a conjecture of Petri that if $X$ is general, then the space $G^r_d(X)$ is also smooth [@gi1]. This proof was later simplified by Eisenbud and Harris [@e-h2] and Welters [@we4] using ideas closely related to the theory of limit linear series. These proofs all relied on proving injectivity of the Petri map, by taking a hypothetical nonzero element of the kernel, and carrying out a careful analysis of how it would behave under degeneration.
In this paper, we give a new proof of the Gieseker-Petri theorem, and generalize it to the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$, proving that the singular locus of this space consists precisely of linear series with a certain type of excess vanishing. Our Gieseker-Petri theorem with imposed ramification, Theorem \[thm:main\] below, generalizes two statements.
1. In the absence of marked points, it reduces to the Gieseker Petri theorem, which holds for curves of any genus.
2. With marked points allowed, but in genus $0$, it reduces to the well-known characterization of the singular loci of Schubert varieties and Richardson varieties.
Indeed, in the case $g=0$, a single ramification condition corresponds to a Schubert cycle in the Grassmannian $\mathrm{Gr}(r+1,\mathcal{O}_{\mathbb{P}^1}\!(d))$, while a pair of ramification conditions similarly corresponds to a Richardson variety. These spaces are singular, and their singularities can be characterized precisely as loci with a specific type of excess vanishing. Our main theorem extends this characterization to all genera, and also deduces additional consequences on the geometry of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$. To state it, the following preliminary notation will be helpful.
If $(\sL,V)$ is a $\fg^r_d$ on a smooth projective curve $X$, and $D$ is an effective divisor on $X$, write $$V(-D)=V \cap \Gamma(X,\sL(-D)) \subseteq \Gamma(X,\sL).$$
Thus, to say that $(\sL,V)$ has vanishing sequence at least $a_{\bullet}$ at $P$ is equivalent to saying that $$\label{eq:van-ineq}
\dim V(-a_j P) \geq r+1-j$$ for $j=0,\dots,r$.
We then make the following definition:
\[defn:good-open\] In the situation of Theorem \[thm:bg\], let $$G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \subseteq G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$$ be the open subset consisting of $(\sL,V)$ such that holds with equality for all $j>0$ such that $a_j>a_{j-1}+1$, and the analogous condition holds for $(Q,b_\bullet)$.
We see that $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ contains all linear series with precisely the prescribed vanishing at $P$ and $Q$, but it also contains many linear series with more than the prescribed vanishing. For instance, if $a_{\bullet} = b_{\bullet} = (0,1,\ldots,r)$ are both minimal, so that $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))=G^r_d(X)$, then we also have $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))=G^r_d(X)$. Our main theorem is then the following.
\[thm:main\] In the situation of Theorem \[thm:bg\], suppose further that we are in characteristic $0$. Then the smooth locus of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is precisely equal to $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$.
Furthermore, the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ has singularities in codimension at least $3$, is normal, and when $\widehat{\rho} \geq 1$ is irreducible.
Thus, we are in particular giving a new proof of the Gieseker-Petri theorem (in characteristic $0$). As an immediate consequence of Theorem \[thm:main\], the twice-pointed Brill-Noether curves studied in [@c-l-p-t1], as well as twice-pointed Brill-Noether surfaces [@c-p3; @a-c-t1] are smooth.
Our proof proceeds by degenerating to a chain of elliptic curves, and studying the geometry of the corresponding moduli space of Eisenbud-Harris limit linear series. The key idea in this step is that although the space of limit linear series will be singular in codimension $1$, after base change and blowup one can ensure that any given point of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ on the generic fiber will specialize to a smooth point of the limit linear series space of a chain of curves of genus 0 or 1. This is where the characteristic-$0$ hypothesis comes in. The case of genus 0 is well-known, so our main calculation is the following result, which does not depend on characteristic, concerning the case $g=1$.
\[thm:genus-1\] In the situation of Theorem \[thm:bg\], suppose that $g=1$, and make the generality condition explicit as follows: $X$ is arbitrary, and $P,Q$ are such that $P-Q$ is not a torsion point of $\Pic^0(X)$ of order less than or equal to $d$. Then the space $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth.
The proof of Theorem \[thm:genus-1\] proceeds by consideration of the morphism $$\label{eq:to-pic}
G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \Pic^d(X).$$ The main subtlety that needs to be addressed is that the map is not smooth. The fibers are each described as an intersection of a pair of Schubert cycles in a Grassmannian. But in finitely many fibers, namely the ones above line bundles of the form $\mathcal{O}_X(aP+(d\!-\!a)Q)$ for $0<a<d$, the pairs of flags defining the Schubert cycles are not transverse, but only almost-transverse (see Definition \[def:a-t\]). We prove Theorem \[thm:genus-1\] by first showing that in fibers, the tangent spaces have dimension at most $1$ greater than expected, and then showing that at the points where the tangent space dimension jumps in the fiber, there cannot be any horizontal tangent vectors.
The statement on tangent spaces in fibers, which is Corollary \[cor:at\] below, takes place entirely inside the Grassmannian, and may be of independent interest. Indeed, § \[sec:a-t\] is a study of tangent spaces of intersections of pairs of Schubert cycles, and we address the case of arbitrary pairs of flags in Theorem \[thm:workhorse\] and Remark \[rem:arbitrary-flags\]. We then prove Theorems \[thm:main\] and \[thm:genus-1\] in §\[sec:proofs\].
Almost-transverse intersections of Schubert cycles {#sec:a-t}
==================================================
It is well known that the intersection of two Schubert varieties associated to transverse flags—commonly called a Richardson variety—is smooth on the open subset of points which are smooth in both Schubert varieties. In this section we consider intersections of pairs of Schubert varieties associated to not necessarily transverse flags. Our analysis recovers the usual smoothness statement in the transverse case, but our main purpose is to analyze the almost-transverse case in Corollary \[cor:at\], where we characterize the smooth points and show that the dimension of the tangent space jumps only by $1$ at the non-smooth points. While Schubert intersections and non-transverse flags have been studied by Vakil [@va7] and Coskun [@co3], those situations involved studying the flat limits of transverse intersections, rather than the direct analysis of the non-transverse intersections required in the present work.[^2] We fix $k$ to be an algebraically closed field of any characteristic. Throughout this section, we will work entirely with $k$-valued (equivalently, closed) points. We index our complete flags by codimension, so that for a complete flag $P^\bullet$ in a $d$-dimensional vector space $H$, $$\begin{aligned}
0 &= P^d \subset \cdots \subset P^1 \subset P^0 = H.\end{aligned}$$ We fix further notation as follows.
\[defn:schubert\] Given a $k$-vector space $H$ of finite dimension $d$ and a complete flag $P^{\bullet}$ in $H$, if we are given also $a_{\bullet} = (a_0,\ldots,a_r)\in \ZZ^{r+1}$ with $$0 \le a_0 < \cdots < a_r < d,$$ we let $\Sigma_{P^{\bullet}, a_{\bullet}}$ be the **Schubert variety** defined as the closed subscheme of $\Gr(r+1,H)$ given by the set of $\Lambda \in \Gr(r+1,H)$ such that $$\label{eq:one-flag}
\dim (\Lambda\cap P^{a_i})
\ge r+1-i$$ for $i=0,\ldots,r$.
More precisely, the conditions in are determinantal, yielding a scheme structure on $\Sigma_{P^{\bullet},a_{\bullet}}$ (which turns out to be reduced). In our notation, the codimension of $\Sigma_{P^{\bullet},a_{\bullet}}$ is given by $\sum_{i=0}^r (a_i\!-\!i)$.
\[defn:active\] With $a_{\bullet} = (a_0,\ldots , a_r)$ an increasing sequence as above, say that an index $i$ with $0\le i\le r$ is **active** in $a_{\bullet}$ if $i>0$ and $a_i > a_{i-1} + 1$, or $i=0$ and $a_0 >0.$
\[defn:sigma-circ\] Let ${\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$ be the open subscheme of $\Sigma_{P^{\bullet},a_{\bullet}}$ consisting of subspaces $\Lambda$ for which for every active index $i$, the inequality in is an equality.
Note that is automatically an equality when $i=0$, so in Definition \[defn:sigma-circ\] we can restrict to positive active choices of $i$.
We fix the following situation throughout this section.
\[sit:flags\] Let $H$ be a finite-dimensional $k$-vector space, and write $d:=\dim H$. Fix complete flags $P^{\bullet}$, $Q^{\bullet}$ in $H$, and sequences $a_{\bullet},b_{\bullet}\in \ZZ^{r+1}$ with $0 \le a_0 < \cdots < a_r < d$ and $0 \le b_0 < \cdots < b_r < d.$
Recall that we are indexing by codimension; thus $\codim P^i=\codim Q^i=i$. Note that for any $\Lambda \in {\Sigma}_{P^{\bullet},a_{\bullet}}$, the distinct subspaces in the collection $\Lambda\cap P^{j}$ form a complete flag in $\Lambda$; we denote the flag $\Lambda\cap P^{\bullet}$ by abuse of notation.
We have the following description of the tangent space at any point in ${\Sigma}_{P^{\bullet},a_{\bullet}}$. Tangent spaces to Schubert varieties are well understood [@b-l2], but for the sake of completeness, we provide a description in the particular case that we need of Grassmannian Schubert varieties.
\[prop:tangent-schubert\]
1. Given $\Lambda \in {\Sigma}_{P^{\bullet},a_{\bullet}}$, let $S$ be the set of active indices $i$ such that $\dim \Lambda\cap P^{a_i} = r+1-i$. Then there is a canonical isomorphism of vector spaces $$T_{\Lambda}\Sigma_{P^{\bullet},a_{\bullet}}\cong
\left\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i})
\subseteq (P^{a_i}+\Lambda)/\Lambda \text{ for } i\in S\right\}.$$
2. In particular, the smooth locus of $\Sigma_{P^{\bullet},a_{\bullet}}$ is precisely ${\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$.
By definition, $\Sigma_{P^{\bullet},a_{\bullet}}$ is the scheme-theoretic intersection of the following subschemes of $\Gr(r+1,H)$ (for $i=0,1,\cdots,r$): $$\Sigma_i = \{ \Lambda \in \Gr(r+1,H)\colon \dim(\Lambda \cap P^{a_i}) \geq
r+1-i \}.$$ Define $\Sigma_i^\circ$ to be the open subscheme of $\Sigma_i$ where equality holds. Note also that in fact $\Sigma_{P^{\bullet},a_{\bullet}}$ can be cut out as the intersection of the $\Sigma_i$ over all active indices $i$: this is immediate set-theoretically, and is also true scheme-theoretically because whenever $a_{i+1}=a_i+1$, the condition for $a_{i+1}$ is obtained from that of $a_i$ by adding a single row to the local matrix expression, and considering minors of size one larger. Thus, every minor occuring in the $a_{i+1}$ condition can be expanded in terms of minors occuring in the $a_i$ condition.
The first statement of the proposition then follows immediately from the following claim. For a fixed index $i$, $$T_{\Lambda} \Sigma_i =
\begin{cases}
\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i}) \subseteq
(P^{a_i}+\Lambda)/\Lambda \} & \mbox{ if $\Lambda \in \Sigma^\circ_i$}\\
T_{\Lambda} \Gr(r+1,H) & \mbox{ otherwise,}
\end{cases}$$ where we identify $T_\Lambda \Gr(r+1,H)$ with $\Hom(\Lambda,H/\Lambda)$ as usual.
To prove this claim, one may work on an affine open subset of $\Gr(r+1,H)$, as follows. Choose a basis of $H$ extending a basis of $\Lambda$; then an affine neighborhood of $\Lambda$ is given by the set of $(r+1) \times d$ matrices whose first $(r+1)$ columns form the identity matrix (where the point in $\Gr(r+1,H)$ is given by taking the span of the rows). More precisely, for any $k$-algebra $R$, we may identify the $R$-points of this open subscheme with $R$-valued matrices whose first $r+1$ columns form the identity matrix. In particular, taking $R = k[\epsilon]$, the tangent space $T_{\Lambda}
\Gr(r+1,H)$ is identified with matrices in block form $( I\ \epsilon M)$, where $M$ is a matrix of values of $k$; the matrix $M$ then determines an element of $\Hom(\Lambda,H / \Lambda)$. Now, we may further assume that the chosen basis of $H$ also includes a basis of $P^{a_i}$ as a subset. Then the $R$-points of $\Sigma_i$ consist of those matrices such that the submatrix consisting of all columns not corresponding to the basis of $P^{a_i}$ has rank at most $i$. Assuming that we order our basis of $\Lambda$ so that a basis of $\Lambda
\cap P^{a_i}$ comes at the end, the submatrix in question has the form $$\begin{pmatrix} I & A \\ 0 & B \end{pmatrix},$$ where the size of the identity matrix in the upper left is $\dim (\Lambda /
(\Lambda \cap P^{a_i}))$. Therefore, lying in $\Sigma_i$ corresponds to the condition that $$\rk(B) \leq i - (r+1) + \dim (\Lambda \cap P^{a_i}).$$ Now, specialize to the case $R = k[\epsilon]$, and consider a tangent vector to $\Gr(r+1,H)$ at $\Lambda$. The submatrix $B$ is a multiple of $\epsilon$. Therefore all $2 \times 2$ and larger minors of $B$ are guaranteed to vanish. Thus in the case $\dim (\Lambda \cap P^{a_i}) > r+1-i$ (i.e. $\Lambda \not\in
\Sigma_i^\circ$), all tangent vectors to $\Gr(r+1,H)$ at $\Lambda$ are also tangent vectors to $\Sigma_i$ at $\Lambda$. On the other hand, when $\Lambda
\in \Sigma_i^\circ$, a tangent vector to $\Gr(r+1,H)$ at $\Lambda$ is a tangent vector to $\Sigma_i$ if and only if the matrix $B$ vanishes entirely. This condition can be made intrinsic by observing that, if $\phi\colon \Lambda
\rightarrow H / \Lambda$ is the linear map encoding a tangent vector, then $B$ is a matrix representation for the linear map $\Lambda \cap P^{a_i} \to H /
(P^{a_i} + \Lambda)$ induced by $\phi$. Therefore it follows that, in the case $\Lambda \in \Sigma_i^\circ$, $\phi$ described a tangent vector to $\Sigma_i$ if and only if $\phi(\Lambda \cap P^{a_i}) \subseteq (P^{a_i} +
\Lambda)/\Lambda$. This proves the claim, and the first statement of the proposition.
The second statement follows by direct computation of the codimension imposed by the conditions on the tangent space in the first part. If we have $i \in S$, let $i_n$ denote the next (greater) element of $S$, setting $i_n=r+1$ if $i$ is maximal in $S$. By starting from the condition imposed at the maximal element of $S$, and inductively working downwards, one computes that the codimension of the tangent space is given by $$\sum_{i \in S} (i_n-i)(a_i-i).$$ Each term of this sum is always less than or equal to $\sum_{j=i}^{i_n-1} (a_j-j)$, with equality if and only if there are no actives indices strictly between $i$ and $i_n$. The proposition follows.
\[cor:tangent-schubert-circ\] Given $\Lambda \in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}}$, there is a canonical isomorphism of vector spaces $$T_{\Lambda}\Sigma_{P^{\bullet},a_{\bullet}}\cong
\left\{\phi\colon \Lambda\to H/\Lambda: \phi(\Lambda \cap P^{a_i})
\subseteq (P^{a_i}+\Lambda)/\Lambda \text{ for active } i=0,\ldots,r\right\}.$$
Following Definition 4.1 of [@c-p3], we define:
\[def:a-t\] Two complete flags $P^{\bullet}$ and $Q^{\bullet}$ are called **almost-transverse** if there exists an index $t\in\{1,\ldots,d-1\}$ such that $$\dim P^i \cap Q^{d-i}
= \begin{cases} 0 & \text{ if $i\ne t$,}\\ 1 & \text{ if }i=t.\end{cases}$$
More generally, we have the following statement, which is easy to check:
\[prop:sigma\] There is a unique permutation $\sigma \in S_d$ associated to the flags $P^{\bullet}$ and $Q^{\bullet}$ with the property that there exists a basis $e_1,\ldots,e_d$ for $H$ satisfying $$e_i \in P^{i-1}\smallsetminus P^i, \quad \text{and} \quad
e_i \in Q^{\sigma(i)-1} \smallsetminus Q^{\sigma(i)}.$$
Such a basis can also be characterized by the property that for all indices $i$ and $j$, $P^i\cap Q^j$ is spanned by $\{e_1,\ldots,e_d\}\cap P^i\cap Q^j$. In particular, if $\dim P^i\cap Q^j = 1$ then $P^i\cap Q^j$ contains one of the $e_{\ell}$.
We refer to a basis as in Proposition \[prop:sigma\] as a $(P^{\bullet},Q^{\bullet})$-**basis**.
Thus, following the notation of Proposition \[prop:sigma\], we have that $P^{\bullet}=Q^{\bullet}$ if and only if $\sigma=\mathrm{id}$, $P^{\bullet}$ and $Q^{\bullet}$ are transverse if and only if $\sigma = \omega := (d,d-1,\ldots,1)$, and $P^{\bullet}$ and $Q^{\bullet}$ are almost-transverse if and only if $\sigma$ is the composition of $\omega$ with an adjacent transposition.
We set $\rho = \rho(1,r,d,a_{\bullet},b_{\bullet})$, so that $$\rho-1=(r+1)(d-r-1)-\sum_{i=0}^r (a_i-i)-\sum_{i=0}^r (b_i-i)
= \rho(0,r,d-1,a_{\bullet},b_{\bullet})$$ is precisely the expected dimension of $\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}}$. Also recall the definition of the complete flags $\Lambda\cap P^\bullet$ and $\Lambda\cap Q^\bullet$ from Situation \[sit:flags\].
\[thm:workhorse\] Given $\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$, let $\sigma \in S_{r+1}$ denote the permutation associated to $\Lambda\cap P^{\bullet}$ and $\Lambda\cap Q^{\bullet}$ in $\Lambda$ by Proposition \[prop:sigma\]. Given any $j\in \{0,\ldots,r\}$, let $$m(j) = \max\{a_i: i\text{ is active in $a_{\bullet}$ and $i\le j$}\},$$ setting $m(j) = 0$ if no such $a_i$ exists. Similarly, let $$n(j) = \max\{b_i: i\text{ is active in $b_{\bullet}$ and $i\le \sigma(j)$}\},$$ or $n(j) = 0$ if no such $b_j$ exists. Then $$\label{eq:the-answer}
\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
= \rho-1 + \sum_{j=0}^r \codim_H (P^{m(j)} + Q^{n(j)} + \Lambda).$$
Let $\lambda_0,\ldots,\lambda_r$ be a $(\Lambda\cap P^{\bullet},\Lambda\cap Q^{\bullet})$-basis for $\Lambda$. Then for any $i$ active in $a_{\bullet}$, respectively $b_{\bullet}$, have $$\label{eq:lambda-basis}
\Lambda \cap P^{a_i} =\langle \lambda_i,\ldots,\lambda_r\rangle,\qquad \Lambda\cap Q^{b_i} = \langle \lambda_{\sigma^{-1}(i)}, \ldots,\lambda_{\sigma^{-1}(r)}\rangle.$$ In other words, given any $j$, and any $i$ that is active in $a_{\bullet}$, we have $\lambda_j\in \Lambda\cap P^{a_i}$ if and only if $i\le j$. Similarly, for any $i$ that is active in $b_{\bullet}$, we have $\lambda_j\in \Lambda\cap Q^{b_i}$ if and only if $i\le\sigma( j)$. By Corollary \[cor:tangent-schubert-circ\], we have isomorphisms $$\begin{aligned}
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
&\cong& \{\phi\colon \Lambda\to H/\Lambda ~:~ \phi(\lambda_j) \in (P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda.\}\\
&\cong& \bigoplus_{j=0}^r \mathrm{Hom} \left(\langle \lambda_j\rangle, (P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda\right).\end{aligned}$$ We are thus reduced to computing the dimensions $(P^{m(j)}+\Lambda)/\Lambda \cap (Q^{n(j)}+\Lambda)/\Lambda$, which are equal to $$\dim (P^{m(j)}+\Lambda)+ \dim (Q^{n(j)}+\Lambda)
- \dim (P^{m(j)}+ Q^{n(j)}+\Lambda)-\dim \Lambda.$$ Moreover, the first two terms are determined by the fact that $\dim P^{m(j)}\cap \Lambda = r+1-m(j)$ and $\dim Q^{n(j)} \cap \Lambda = r+1-n(j)$, by assumption that $m(j)$ and $n(j)$ are active or are equal to $0$. A straightforward calculation produces .
We observe that the well-known case of transverse flags follows immediately from Theorem \[thm:workhorse\].
\[cor:transverse\] If $P^{\bullet}$ and $Q^{\bullet}$ are transverse, then $\dim \, T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
= \rho-1 $ for all $\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$.
Following the notation of the proof of Theorem \[thm:workhorse\], for each $j$ we have $\lambda_j \in P^{m(j)} \cap Q^{n(j)} $ by construction. Since $P^{\bullet}$ and $Q^{\bullet}$ are transverse, it follows that $m(j) + n(j) < d$, so $P^{m(j)} + Q^{n(j)} + \Lambda = P^{m(j)} + Q^{n(j)} = H$.
More importantly, we can also deduce the desired statement in the almost-transverse case.
\[cor:at\] Given $\Lambda
\in {\Sigma}^\circ_{P^{\bullet},a_{\bullet}} \cap {\Sigma}^\circ_{Q^{\bullet},b_{\bullet}}$, suppose $P^{\bullet}$ and $Q^{\bullet}$ are almost-transverse, with $t+t'=d$ such that $\dim P^t \cap Q^{t'} =1$.
Suppose first that $t=a_i$ for $i$ active in $a_{\bullet}$, that $t'=b_{i'}$ for $i'$ active in $b_{\bullet}$, and that $$P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}.$$ Then $$\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho.$$ If those conditions do not all hold, then $$\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho-1.$$
Let $\lambda_0,\ldots,\lambda_r$ be a $(\Lambda\cap P^{\bullet},\Lambda\cap Q^{\bullet})$-basis for $\Lambda$. First suppose that $t=a_i$ for $i$ active in $a_{\bullet}$, and $t'=b_{i'}$ for $i'$ active in $b_{\bullet}$, and that $P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}.$ We will deduce that $\dim \,
T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
=\rho.$
We have that $\Lambda\cap P^t$ and $\Lambda\cap Q^{t'}$ are elements in the flags $\Lambda\cap P^\bullet$ and $\Lambda\cap Q^\bullet$ respectively with intersection $\Lambda\cap P^t \cap Q^{t'}$ of dimension 1. Proposition \[prop:sigma\] implies that $P^t \cap Q^{t'} = \langle \lambda_j \rangle$ for a unique $j\in\{0,\ldots,r\}$. By Theorem \[thm:workhorse\] it is enough to show that for each $j'\in\{0,\ldots,r\}$, $$\codim_H (P^{m(j')} + Q^{n(j')} + \Lambda) = \begin{cases}
1 &\text{ if }j'=j,\\
0&\text{ if }j'\ne j.
\end{cases}$$
Now, for $j'\ne j$, the fact that $\lambda_{j'}\in P^{m(j')}\cap Q^{n(j')}$ and $P^\bullet$ and $Q^\bullet$ are almost-transverse implies that either $m(j')+n(j')<d$, or that $m(j') = t$ and $n(j') = t'$. But the latter case cannot be, since then both $\lambda_j, \lambda_{j'}\in P^{t}\cap Q^{t'}$, contradicting that $\dim P^{t}\cap Q^{t'}=1$. Therefore $m(j')+n(j')<d$ and $$P^{m(j')}+ Q^{n(j')} = P^{m(j')}+ Q^{n(j')}+\Lambda = H,$$ as desired.
Next, to show that $\codim_H (P^{m(j)} + Q^{n(j)}+\Lambda )=1$, we claim that $m(j) = t$ and $n(j) = t'$. Recall that $a_i = t$ and $a_{i'} =t'$. By assumption, $i$ is active in $a_{\bullet}$ and $\lambda_j\in\Lambda\cap Q^{a_i}$, so $i\le j$ by . We want to show that $i$ is the largest active index in $a_{\bullet}$ with $i\le j$. Indeed, if $l$ is active in $a_{\bullet}$ with $i<l\le j$, then $\lambda_j\in P^{a_l}\cap Q^{b_{i'}}$. But now $a_l > a_i$, so $a_l + b_{i'} > a_i + b_{i'} = t+t'=d$. Therefore $P^{a_l}\cap Q^{b_{i'}}=0$, contradiction. A similar argument shows $n(j) = t'$. Therefore, $$\codim_H (P^{m(j)} + Q^{n(j)} + \Lambda) = \codim_H (P^{t} + Q^{t'} + \Lambda)=1,$$ since $P^{t} + Q^{t'} $ is a hyperplane in $H$, and $\Lambda$ is contained in it by assumption.
It remains to show that if the conditions in the statement of Corollary \[cor:at\] do not all hold, then $\dim T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})= \rho-1.$ We prove the contrapositive. Suppose that $\dim T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})>\rho-1.$ By Theorem \[thm:workhorse\], there is an index $j$ such that $\codim_H (P^{m(j)} + Q^{n(j)} + \Lambda) >0$. Again, given that $\lambda_j \in P^{m(j)} \cap Q^{n(j)}$ and that $P^\bullet$ and $Q^\bullet$ are almost-transverse, it follows that either $m(j)+n(j)<d$ or that $m(j)=t$ and $n(j)=t'$. But $m(j)+n(j)<d$ would imply $P^{m(j)} + Q^{n(j)} = H$, contradicting the codimension statement. So $m(j)=t$ and $n(j)=t'$, implying that $t=a_i$ and $t'=b_{i'}$ for active indices $i$ and $i'$ in $a_{\bullet}$ and $b_{\bullet}$ respectively. (It is not possible that $m(j) = 0$ or $n(j) = 0$, since $\codim_H P^{m(j)}+Q^{n(j)}+\Lambda>0$.) Furthermore, $$\langle \lambda_j \rangle = P^t\cap Q^{t'} \subseteq \Lambda \subseteq P^t + Q^{t'}$$ where the last containment holds again by the codimension assumption.
Summarizing, we have shown that the [*only*]{} way that $\dim T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})> \rho-1$ is for all the conditions in the statement of Corollary \[cor:at\] to hold, in which case we have already proved that the dimension is exactly $\rho$.
\[rem:arbitrary-flags\]
For arbitrary flags $P^{\bullet}$ and $Q^{\bullet}$ and $\Lambda\in\Sigma^\circ_{P^\bullet,a_{\bullet}}\cap \Sigma^\circ_{Q^\bullet,b_{\bullet}}$, let $\tau\in S_d$ be the associated permutation from Proposition \[prop:sigma\] (maintaining other notation as in Theorem \[thm:workhorse\]). Then the extent to which the dimension of the tangent space at $\Lambda$ of $\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}}$ exceeds $\rho-1$ can be bounded in terms of $\tau$ as follows. We have: $$\label{eq:coxeter}
\dim T_\Lambda (\Sigma_{P^{\bullet},a_{\bullet}} \cap \Sigma_{Q^{\bullet},b_{\bullet}})
\leq (\rho-1) + \inv( \omega \tau ).$$ Here $\omega$ denotes the decreasing permutation $(d,d-1,\cdots,1)$, and $\inv(\omega\tau)$ denotes the inversion number of $\omega\tau$, i.e. the number of $i<j$ with $\omega\tau(i)>\omega\tau(j)$.[^3] We briefly sketch a proof of this more general inequality.
Using the second part of Proposition \[prop:sigma\], it follows that for each $i,j$, we have $$\dim P^i \cap Q^j = \# \{ i' \geq i:\ \tau(i') \geq j \}.$$ From this it follows that $\dim P^i \cap Q^j > \dim P^{i+1} \cap Q^j$ if and only if $\tau(i) \geq j$. Now, since we have $P^{a_i} \cap Q^{b_{\sigma(i)}} \neq P^{a_i+1} \cap
Q^{b_{\sigma(i)}}$, we find that $b_{\sigma(i)} \leq \tau(a_i)$. Note also that for all $j$, $m(j) \leq a_j$ and $n(j) \leq b_{\sigma(j)}$. Then:
$$\begin{aligned}
\sum_{j=0}^r \codim ( \Lambda + P^{m(j)} + Q^{n(j)} ) &\leq&
\sum_{j=0}^r \codim(P^{m(j)} + Q^{n(j)})\\
&\leq& \sum_{j=0}^r \codim(P^{a_j} + Q^{b_{\sigma(j)}})\\
&\leq& \sum_{j=0}^r \codim( P^{a_j} + Q^{\tau(a_j)} )\\
&\leq& \sum_{j=0}^{d-1} \codim( P^j + Q^{\tau(j)}).\end{aligned}$$
Using that $\dim(P^j \cap Q^{\tau(j)})=\#\{j' \geq j: \tau(j') \geq \tau(j)\}$, we compute that $\displaystyle \sum_{j=0}^{d-1} \codim( P^j
+ Q^{\tau(j)}) = \inv(\omega \tau)$, and the inequality follows from . When $P^{\bullet}$ and $Q^{\bullet}$ are almost-transverse, $\inv(\omega
\tau) = 1$, and the precise statement in Corollary \[cor:at\] can be deduced from characterizing the equality cases of the four inequalities above in the case where $\omega \tau$ is equal to an adjacent transposition.
Linear series in positive genera {#sec:proofs}
================================
We begin with a proposition that will show that our smoothness result, Theorem \[thm:main\] to be proved below, is sharp.
\[prop:non-smooth\] In the situation of Theorem \[thm:bg\], every point of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ in the complement of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is singular.
This is a consequence of the standard construction of the space $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$: we let $\wt{\sL}$ be a Poincaré line bundle on $X \times \Pic^d(X)$. Take a sufficiently ample[^4] effective divisor $D$ on $X$ with support disjoint from $P$ and $Q$, and write $\wt{D}=D \times \Pic^d(X)$. Writing $p\colon X\times \Pic^d(X)\to\Pic^d(X)$ for projection, let $G$ be the relative Grassmannian $\Gr(r+1,p_{*} \wt{\sL}(\wt{D}))$, equipped with structure map $\pi\colon G\to \Pic^d(X)$. Let $\wt{\sV} \hookrightarrow \pi^* p_{*} \wt{\sL}(\wt{D})$ denote the universal subbundle. Then $G^r_d(X)$ is cut out in $G$ by the condition that the induced map $$\wt{\sV} \hookrightarrow \pi^*p_{*}\left( \wt{\sL}(\wt{D})|_{\wt{D}}\right)$$ vanishes identically. Because we have chosen $D$ to have support disjoint from $P$ and to be sufficiently ample, the space $G^r_d(X,(P,a_{\bullet}))$ is cut out by imposing the additional Schubert condition that the maps $$\wt{\sV} \hookrightarrow \pi^* p_{*}\left( \wt{\sL}(\wt{D})|_{a_j P}\right)$$ have rank at most $j$ for each $j$. Imposing the analogous condition at $Q$, we obtain $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ as an intersection of three conditions: a determinantal condition (in fact a complete intersection), and two relative Schubert cycles. It is routine to check that for $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ to have dimension $\rho(g,r,d,a_{\bullet},b_{\bullet})$, as asserted by Theorem \[thm:bg\], these three conditions must intersect in the maximal codimension. Given that we know from Theorem \[thm:bg\] that $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ does in fact have dimension $\rho(g,r,d,a_{\bullet},b_{\bullet})$, it then further follows that in order for $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ to be smooth at any point, that point must lie in the smooth locus of each of the three conditions, and in particular of the two Schubert cycles.
But we claim that $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ consists precisely of the points of $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ which lie in the smooth locus of both relative Schubert cycles. Indeed, each relative Schubert cycle is nothing but a locally constant family of Schubert varieties over the base $G$, so we are done by the standard characterization of the smooth locus of a Schubert variety (see the second part of Proposition \[prop:tangent-schubert\]).
We now use our calculations in Grassmannians in §2 to complete the proof of our main theorem, beginning with the case of genus $1$ in Theorem \[thm:genus-1\].
Set $\rho=\rho(1,r,d,a_{\bullet},b_{\bullet})$. We may assume $d>0$, as otherwise the result is trivial. Thus, $G^r_d(X)$ is a Grassmannian bundle over $\Pic^d(X)$, with the fiber over a line bundle $\sL$ being canonically identified with $\Gr(r+1,\Gamma(X,\sL))\cong \Gr(r+1,d)$. The condition imposed by requiring vanishing sequence at least $a_{\bullet}$ at $P$ then gives a Schubert cycle in each fiber, corresponding to the complete flag determined by vanishing order at $P$. The codimension of spaces in the flag corresponds precisely to vanishing order except over the point $\sL \cong \sO_X(dP)$, where no sections vanish to order precisely $d-1$, and vanishing to order $d$ imposes codimension only $d-1$. Consequently, there are two possibilities for $G^r_d(X,(P,a_{\bullet}))$. First, if $a_r<d$, it is a relative Schubert cycle of codimension $\sum_j (a_j-j)$ in $G^r_d(X)$, Cohen-Macaulay and flat over $\Pic^d(X)$. Or, if $a_r=d$, it is supported entirely over $\sL \cong \sO_X(dP)$ (even scheme-theoretically), and is still a Schubert cycle, but of codimension $(\sum_j (a_j-j))-1$. The same analysis applies to $G^r_d(X,(Q,b_{\bullet}))$, so we find that every fiber of $$\label{eq:str-map}
G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \Pic^d(X)$$ is an intersection of a pair of Schubert cycles. The basic properties of the map are analyzed for instance in Lemma 2.1 of [@os18] and Proposition 2.1 of [@os26]; we review the main points of this analysis in order to carry out the necessary tangent space analysis.
First, we see that in most fibers of , the relevant Schubert cycles are associated to transverse flags: the only way in which the flags fail to be transverse is if $\sL \cong \sO_X(aP+(d-a)Q)$ for some $a\in\{1,\ldots,d-1\}$, which is unique by genericity of $P$ and $Q$; then the conditions of vanishing to order $a$ at $P$ and $d-a$ at $Q$ intersect in dimension $1$ instead of dimension $0$. Thus, on fibers of over points not of the form $\sO_X(aP+(d-a)Q)$ for $0 \leq a \leq d$, we have that the Schubert indexing matches vanishing sequences, and the flags are transverse, so the standard theory (see for instance Corollary \[cor:transverse\]) gives us that (on these fibers) the space $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth (of relative dimension $(r+1)(d-r-1)-\sum_j (a_j-j)-\sum_j (b_j-j) = \rho-1$) over $\Pic^d(X)$, and hence smooth of relative dimension $\rho$ over $\Spec k$. Similarly, it is easily verified that we still obtain Richardson varieties over $\sO_X(aP+(d-a)Q)$ for $0<a<d$ unless $a$ occurs in $a_{\bullet}$ and $d-a$ occurs in $b_{\bullet}$. Thus, we obtain the desired statement in these cases. On the other hand, if $a=0$ or $a=d$, we have transverse intersection of flags, but a potential difference in indexing. In the case $a=d$, the difference in indexing arises only in that imposing vanishing order $d$ at $P$ is a codimension $d-1$ condition. Thus, this can only affect the final term in the vanishing sequence, which is irrelevant for determining membership in $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$. We conclude that—whether or not $a_r=d$—the fiber of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ over $\sO_X(dP)$ precisely corresponds to the open subset addressed in Corollary \[cor:transverse\], and hence has the desired smoothness property. The case that $a=0$ is the same, with $Q$ in place of $P$.
It thus remains to analyze the fibers with $\sL=\sO_X(aP+(d-a)Q)$, for $0<a<d$, and with $a$ occurring in $a_{\bullet}$ and $d-a$ occurring in $b_{\bullet}$. Our hypothesis on $P-Q$ implies that $\sO_X(aP+(d-a)Q) \not \cong \sO_X(a'P+(d-a')Q)$ for any $a \neq a'$, so even in these cases, our flags in $\Gamma(X,\sL)$ are almost-transverse. In the case that $a=a_j$ and $d-a=b_{r-j}$ for some $j$, then $G^r_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is supported (scheme-theoretically) in the given fiber, and one checks (see the proof of Proposition 2.1 of [@os26]) that the nonempty fiber can still be described as a Richardson variety, by replacing $a_j$ with $a-1$, and changing the choice of codimension-$a$ subspace in the first flag. Because of this modification to the flag, we will still have that $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ corresponds precisely to the open subset treated in Corollary \[cor:transverse\], and is hence smooth (but this time of dimension $\rho$).
Finally, we consider the case that $a_j+b_{r-j}<d$ for all $j$, but we have $\sL=\sO_X(aP+(d-a)Q)$, and $a=a_j$ and $d-a=b_{j'}$ for some $j,j'$ with $j+j'>r$; in particular, we must have $0<a<d$. See Example \[ex:0202\] for what is essentially the smallest nontrivial example of this case. In this situation, the given fiber of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ over $\Pic^d(X)$ may be singular or even reducible, but at least it is pure of dimension $\rho-1$: see the proof of Proposition 2.1 of [@os26]. Moreover, as we have observed, the fiber is an intersection of Schubert cycles associated to almost-transverse flags, so we can invoke Corollary \[cor:at\] to conclude that the fiber of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ has tangent space dimension equal to $\rho-1$ or $\rho$ everywhere. Moreover, the latter occurs precisely at linear series $(\sL,V)$ satisfying the following conditions:
1. $V$ contains a section $s$ vanishing to order $a$ at $P$ and $d-a$ at $Q$;
2. $V$ is contained in the linear span of the spaces of sections vanishing to order $a$ at $P$ and order $d-a$ at $Q$;
3. there is some $j$ with $a=a_j$ and $j$ active in $a_{\bullet}$ in the sense of Definition \[defn:active\];
4. there is some $j'$ with $d-a=b_{j'}$ and $j'$ active in $b_{\bullet}$.
Thus, in order to complete the proof of the theorem, we will prove that the space $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is smooth of dimension $\rho$ at every point of the given fiber by showing that if $(\sL,V)$ is a point at which the tangent space of the fiber has dimension $\rho$, then every tangent vector of the total space at $(\sL,V)$ is in fact vertical. Accordingly, given $(\sL,V)$ satisfying the four conditions above, suppose $(\wt{\sL},\wt{V})$ is a first-order deformation of $(\sL,V)$, and let $s \in V$ be a section vanishing to order $a$ at $P$ and $d-a$ at $Q$. We claim that $s$ has a lift $\tilde{s} \in \wt{V}$ which vanishes (scheme-theoretically) to order $a$ at $P$. Indeed, in the notation of the proof of Proposition \[prop:non-smooth\], recall that on $G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ we have a map of vector bundles $$\label{eq:rank-at-most-j}
\Phi\colon\tilde{\sV}\hookrightarrow\pi^*p_*\left(\tilde{\sL}(\tilde{D})|_{a_jP}\right)$$ which has rank at most $j$. The assumption that $(\sL,V)$ lies in the open subset $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$, together with the fact that $j$ is active in $a_{\bullet}$, says that on every point in an open neighborhood of $(\sL,V)$, the map has rank [*exactly*]{} $j$. In this situation, a standard argument shows that $\operatorname{ker}\Phi$ is locally free, and that the restriction map $(\operatorname{ker}\Phi)|_{(\tilde{\sL},\tilde{V})} \rightarrow (\operatorname{ker}\Phi)|_{(\sL,V)}$ is surjective. A sketch of this standard argument is as follows. The cokernel of $\Phi$ must be locally free (see e.g. [@ei1 §16.7]), and hence also the image and kernel. Therefore, the short exact sequences $0\to\operatorname{ker}\Phi\to \tilde{\sV} \to \operatorname{im}\Phi\to0$ and $0\to\mathrm{im}\Phi\to\pi^*p_*\left(\tilde{\sL}(\tilde{D})|_{a_jP}\right) \to\operatorname{cok}\Phi\to 0$ remain exact after base change, and hence taking kernels commutes with base change.
From surjectivity of the restriction map above, we deduce that our section $s$ admits a lift $\tilde{s}$ that also vanishes (scheme-theoretically) to order $a$ at $P$. Similarly, $s$ must have another lift which vanishes (scheme-theoretically) to order $d-a$ at $Q$; since it is another lift of $s$, it can be expressed as $\tilde{s}+\epsilon v$ for some $v \in V$. Now, recall our hypothesis that $V$ is contained in the span of sections vanishing to order at least $a$ at $P$ and at least $d-a$ at $Q$. Write $v=v_1 + v_2$, where $v_1$ vanishes to order at least $a$ at $P$ and $v_2$ vanishes to order at least $d-a$ at $Q$. But then $\tilde{s}+ \epsilon v_1$ still vanishes to order $d-a$ at $Q$, and also vanishes to order $a$ at $P$. This forces $\wt{\sL}$ to be the trivial deformation of $\sL$, yielding the desired verticality assertion and the theorem.
\[rem:reduced-fibers\] Although it was shown in [@os26] that $G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is reduced, we are not aware of a proof in the literature that every fiber of $G^{r}_d(X,(P,a_{\bullet}),(Q,b_{\bullet})) \to \Pic^d(X)$ is reduced, even for genus $1$. However, this genus-$1$ case follows from the proof of Theorem \[thm:genus-1\]. Indeed, the fibers can be expressed as intersections of a pair of Schubert varieties having the expected dimension, and they are therefore Cohen-Macaulay. Furthermore, our proof produces dense open subsets of each fiber which are smooth, so we conclude reducedness.
To conclude the proof of our main theorem, we need to make use of the Eisenbud-Harris theory of limit linear series. We first set up notation for our reducible curves, and recall the relevant definitions.
\[sit:chain\] Fix $g,d,n$. Let $Z_1,\dots,Z_n$ be smooth projective curves, with (distinct) points $P_i, Q_i$ on $Z_i$ for each $i$, and let $X_0$ be the nodal curve obtained by gluing $Q_i$ to $P_{i+1}$ for $i=1,\dots,n-1$.
\[defn:lls\] Given $r,d$ a **limit linear series** of dimension $r$ and degree $d$ on $X_0$ consists of a tuple $(\sL^i,V^i)$ of linear series of dimension $r$ and degree $d$ on the $Z_i$, satisfying the following condition: if $a^i_{\bullet},b^i_{\bullet}$ are the vanishing sequences of $(\sL^i,V^i)$ at $P_i$ and $Q_i$ respectively, then we require $$\label{eq:eh-ineq}
b^i_j+a^{i+1}_{r-j} \geq d$$ for all $i=1,\dots,n-1$ and $j=0,\dots,r$. If is an equality for all $i,j$, we say that the limit linear series is **refined**.
The space of all such limit linear series on $X_0$ is denoted by $G^r_d(X_0)$. If we have sequences $a_{\bullet}$ and $b_{\bullet}$, we also have the closed subscheme $G^r_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet})) \subseteq G^r_d(X_0)$ consisting of limit linear series such that, following the above notation, we have $a^1_{\bullet} \geq a_{\bullet}$ and $b^n_{\bullet} \geq b_{\bullet}$. Finally, denote by $G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet})) \subseteq
G^r_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ the open subscheme consisting of refined limit linear series which further satisfy $$\begin{aligned}
\dim V^1(-a_j P_1) =r+1-j &\text{for $j>0$ active in $a$,}\\
\dim V^n(-b_j Q_n) =r+1-j &\text{for $j>0$ active in $b$.}\end{aligned}$$ We comment that these last two conditions can be re-expressed purely in terms of $a^1_\bullet$ and $b^n_\bullet$ as follows: for all $j>0$ active in $a$, we require $\#\{j':a^1_{j'} \ge a_j\} = r+1-j$, and similarly for $b$.
We are now ready to prove our main smoothness result.
In Situation \[sit:chain\], observe that we can decompose the limit linear series space $G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ into disjoint open subsets according to the vanishing sequences at each node, i.e. according to the possible values in the left hand side of the equalities . Then each such open subset is almost a product over $i$ of spaces of the form $G^{r,\circ}_d(Z_i,(P_i,a^i_{\bullet}),(Q_i,b^i_{\bullet}))$. In fact, it is an open subset of this product, since the refinedness condition completely fixes the vanishing sequences at the nodes. If further each $Z_i$ has genus $0$ or $1$, and for the $Z_i$ of genus $1$ we suppose that $P_i-Q_i$ is not $m$-torsion for any $m \leq d$, then we know that each $G^{r,\circ}_d(Z_i,(P_i,a^i_{\bullet}),(Q_i,b^i_{\bullet}))$ is smooth. Indeed, the genus-$1$ case is Theorem \[thm:genus-1\], while the genus-$0$ case is well known, but follows in particular immediately from Corollary \[cor:transverse\] taking into account that $\dim \Gamma(\PP^1,\sO(d))=d+1$, so there is a shift of $1$ in the value of $d$. We thus conclude that $G^{r,\circ}_d(X_0,(P_1,a_{\bullet}),(Q_n,b_{\bullet}))$ is also smooth, of dimension $\rho(g,r,d,a_{\bullet},b_{\bullet})$. Now, fix $n=g$ and suppose each $Z_i$ has genus $1$. Let $B$ be the spectrum of a discrete valuation ring, and $\pi:X \to B$ be a flat, proper family family of curves of genus $g$, with $X$ regular, the generic fiber $X_{\eta}$ smooth, and the special fiber isomorphic to $X_0$. Further assume that $\pi$ has sections $P$, $Q$, specializing to $P_1$ and $Q_n$ respectively on $X_0$.
Suppose that we have a closed point of $G^{r,\circ}_d(X_{\eta},(P_{\eta},a_{\bullet}),(Q_{\eta},b_{\bullet}))$. Extend the base so that the corresponding linear series is defined on $X_{\eta}$, and then extend further so that all ramification points are also rational over the base field. Blow up the nodes in $X_0$ as necessary to resolve any resulting singularities,[^5] and finally, blow up $P_1$ and $Q_n$ as necessary so that no generic ramification point distinct from $P$ or $Q$ limits to $P_1$ or $Q_n$ in the special fiber. Denote the resulting family by $\pi':X' \to B'$, and the special fiber by $X'_0$, and write $P'$ and $Q'$ (respectively, $P'_1$ and $Q'_n$) for the resulting sections of $\pi'$ and their restrictions to $X'_0$. Then $X'_0$ is obtained by $X_0$ by base extension and insertion of chains of genus-$0$ curves at the nodes and at $P_1$ and $Q_n$. By construction, none of the ramification points on $X_\eta$ can specialize to nodes of $X'_0$, so by Proposition 2.5 of [@e-h1] (and using the characteristic $0$ hypothesis), the extension of the given linear series is a refined limit linear series on $X'_0$. Moreover, by the same argument, the ramification at $P'_1$ and at $Q'_n$ must be precisely equal to the ramification at $P_{\eta}$ and $Q_{\eta}$, so that the induced limit linear series lies in $G^{r,\circ}_d(X'_0,(P'_1,a_{\bullet}),(Q'_n,b_{\bullet}))$. But as we have discussed above, this space is smooth. Moreover, by [@o-m1] (see also Theorem 3.4 of [@os26] for the situation with imposed ramification) there is a flat relative moduli space recovering linear series on the generic fiber and limit linear series on the special fiber.[^6] It follows that the original point of $G^{r,\circ}_d(X_{\eta},(P_{\eta},a_{\bullet}),(Q_{\eta},b_{\bullet}))$ must have been smooth as well.
Now, since the spaces we are considering are in general not proper, the condition that $G^{r,\circ}_d$ is smooth is not open in families. However, the condition does define a constructible subset of $\cM_{g,2}$, and the generic fibers of the possible families $\pi$ as above correspond to a Zariski-dense subset, so we conclude the main smoothness statement of the theorem. The fact that the remaining points are not smooth is Proposition \[prop:non-smooth\].
The statement on codimension of singularities follows from the observation that a point in the complement of $G^{r,\circ}_d(X,(P,a_{\bullet}),(Q,b_{\bullet}))$ is a union of closed subvarieties of the form $G^{r}_d(X,(P,a'_{\bullet}),(Q,b'_{\bullet}))$, where $a'_\bullet \ge a$ and $b'_\bullet \ge b$ are sequences such that for some $j>0$ active in $a_\bullet$, we have $a'_{j-1} \ge a_j$, implying that $a'_{j-1}\ge a_{j-1}+2$ and $a_j' \ge a_j+1$ and hence $\sum a'_\bullet \ge \sum a_\bullet+3$; or analogously for $b_\bullet$. We then conclude normality from the Cohen-Macaulayness and Serre’s criterion, and the irreducibility statement follows immediately from the connectedness in the case $\widehat{\rho} \geq 1$.
\[ex:0202\] We provide here essentially the smallest interesting example in the $g=1$ case, exhibiting a fiber of $G^r_d(X, (P, a_\bullet), (Q,b_\bullet))\to \Pic^d(X)$ that is not a Richardson variety, and is in fact reducible. Let $a_\bullet = b_\bullet = (0,2)$, let $\sL = \sO_X(2P + 2Q)$, and consider the fiber of $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ over $\sL$. This fiber has two irreducible components $Z_1$ and $Z_2$, each isomorphic to $\mathbb{P}^2$, meeting along a $\mathbb{P}^1$. It may be described as the variety of lines in $\mathbb{P}^3$ that meet two fixed lines that themselves intersect at a point.
In this situation, the vertical tangent space at a point in $Z_1\cap Z_2\cong \mathbb{P}^1$ has dimension jumping up to $3$. Now, the fiber of $G^{1,\circ}_4(X, (P, a_\bullet), (Q,b_\bullet))$ over $\sL$ is obtained from that of $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ by removing two points of $Z_1\cap Z_2$. Those two points correspond to the space of sections of $\sL$ vanishing to order at least $2$ at $P$, respectively the space of sections of $\sL$ vanishing to order at least $2$ at $Q$. Then Theorem \[thm:genus-1\] asserts that on $Z_1\cap Z_2$, except for at those two points, $G^1_4(X, (P, a_\bullet), (Q,b_\bullet))$ has no horizontal tangent vectors.
[^1]: The first author is supported by the Henry Merritt Wriston Fellowship and by NSF DMS-1701924. The second author is partially supported by a grant from the Simons Foundation \#279151.
[^2]: More precisely, they work with closures of loci with prescribed behavior with respect to both flags; these are in particular irreducible, so are not the same thing as the intersection of Schubert cycles which we consider.
[^3]: If one views $S_d$ as a Coxeter group with reflections being the adjacent transpositions, then $\inv(\omega \tau)$ is also the Coxeter length of $\omega \tau$.
[^4]: Precisely, of degree strictly greater than $2g-2$
[^5]: The preceding constitutes an alternative for the argument of Theorem 2.6 of [@e-h1], avoiding invocation of the stable reduction theorem.
[^6]: In fact, since we only need refined limit linear series for our specialization argument, it is likely possible to make a flatness argument using only the original Eisenbud-Harris construction of [@e-h1], rather than appealing to the general results of [@o-m1; @os26]. But we are not aware of a reference for the more restrictive statement.
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abstract: 'Intelligent reflecting surface (IRS) is a revolutionary and transformative technology for achieving spectrum and energy efficient wireless communication cost-effectively in the future. Specifically, an IRS consists of a large number of low-cost passive elements each being able to reflect the incident signal independently with an adjustable phase shift so as to collaboratively achieve three-dimensional (3D) passive beamforming without the need of any transmit radio-frequency (RF) chains. [In this paper, we study an IRS-aided single-cell wireless system where one IRS is deployed to assist in the communications between a multi-antenna access point (AP) and multiple single-antenna users.]{} We formulate and solve new problems to minimize the total transmit power at the AP by jointly optimizing the transmit beamforming by active antenna array at the AP and reflect beamforming by passive phase shifters at the IRS, subject to users’ individual signal-to-interference-plus-noise ratio (SINR) constraints. [Moreover, we analyze the asymptotic performance of IRS’s passive beamforming with infinitely large number of reflecting elements and compare it to that of the traditional active beamforming/relaying. Simulation results demonstrate the significant performance gain achieved by the proposed scheme with IRS over a benchmark massive MIMO system without using IRS. We also draw useful insights into optimally deploying IRS in future wireless systems.]{}'
author:
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bibliography:
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- 'mybib.bib'
title: Intelligent Reflecting Surface Enhanced Wireless Network via Joint Active and Passive Beamforming
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Intelligent reflecting surface, joint active and passive beamforming, phase shift optimization.
Introduction
============
To achieve 1,000-fold network capacity increase and ubiquitous wireless connectivity for at least 100 billion devices in the forthcoming fifth-generation (5G) networks, a variety of wireless technologies have been proposed and thoroughly investigated in the last decade, including most prominently the ultra-dense network (UDN), massive multiple-input multiple-output (MIMO), and millimeter wave (mmWave) communication [@boccardi2014five]. However, the network energy consumption and hardware cost still remain critical issues in practical systems [@zhang2016fundamental]. For example, UDNs almost linearly scale up the circuit and cooling energy consumption with the number of deployed base stations (BSs), while costly radio frequency (RF) chains and complex signal processing are needed for achieving high-performance communication at mmWave frequencies, especially when massive MIMO is employed to exploit the small wavelengths. Moreover, adding an excessively large number of active components such as small-cell BSs/relays/remote radio heads (RRHs) in wireless networks also causes a more aggravated interference issue. As such, innovative research on finding both spectrum and energy efficient techniques with low hardware cost is still imperative for realizing a sustainable wireless network evolution with scalable cost in the future [@wu2016overview].
In this paper, intelligent reflecting surface (IRS) is proposed as a promising new solution to achieve the above goal [@JR:wu2019IRSmaga; @JR:wu2019discreteIRS; @JR:wu2019SWIPT:IRS; @JR:xinrong:IRS]. Specifically, IRS is a planar array consisting of a large number of reconfigurable passive elements (e.g., low-cost printed dipoles), where each of the elements is able to induce a certain phase shift (controlled by an attached smart controller) independently on the incident signal, thus collaboratively changing the reflected signal propagation. Although passive reflecting surfaces have found a variety of applications in radar systems, remote sensing, and satellite/deep-space communications, they were rarely used in mobile wireless communication. This is mainly because traditional reflecting surfaces only have fixed phase shifters once fabricated, which are unable to cater to the dynamic wireless channels arising from user mobility. However, recent advances in RF micro electromechanical systems (MEMS) and metamaterial (e.g., metasurface) have made the reconfigurability of reflecting surfaces possible, even by controlling the phase shifters in real time [@cui2014coding]. By smartly adjusting the phase shifts of all passive elements at the IRS, the reflected signals can add coherently with the signals from other paths at the desired receiver to boost the received signal power or destructively at non-intended receivers to suppress interference as well as enhancing security/privacy [@JR:wu2019IRSmaga; @JR:xinrong:IRS].
[|m[2.25cm]{}|m[2.3cm]{}|m[1.6cm]{}|m[2.0cm]{}|m[1.4cm]{}|m[2.0cm]{}|m[1.7cm]{}|]{} &[Operating mechanism]{}& [Duplex]{} & No. of transmit RF chains needed &[Hardware cost]{} &[Energy]{} & [Role]{}\
&Passive, reflect & [Full duplex]{} & $0$ & Low & Low & Helper\
& Passive, reflect & [Full duplex]{} & $0$ &Very low& Very low & Source\
& Active, receive and transmit & [Half/full duplex]{} & $N$ &High & High & Helper\
& Active, transmit/receive& [Half/full duplex]{} & $N$ &Very high& Very high & [Source/ Destination]{}\
[It is worth noting that the proposed IRS differs significantly from other related existing technologies such as amplify-and-forward (AF) relay, backscatter communication, and active intelligent surface based massive MIMO [@JR:wu2019IRSmaga]. First, compared to the AF relay that assists in source-destination transmission by amplifying and regenerating signals, IRS does not use a transmitter module but only reflects the received signals as a passive array, which thus incurs no transmit power consumption.[^1] Furthermore, active AF relay usually operates in half-duplex (HD) mode and thus is less spectrally efficient than the proposed IRS operating in full-duplex (FD) mode. Although AF relay can also work in FD, it inevitably suffers from the severe self-interference, which needs effective interference cancellation techniques. Second, different from the traditional backscatter communication of the radio frequency identification (RFID) tag that communicates with the receiver by reflecting the signal sent from a reader, IRS is utilized mainly to enhance the existing communication link performance instead of delivering its own information by reflection. As such, the direct-path signal (from reader to receiver) in backscatter communication is undesired interference and hence needs to be canceled/suppressed at the receiver. However, in IRS-enhanced communication, both the direct-path and reflect-path signals carry the same useful information and thus can be coherently added at the receiver to maximize the total received power. Third, IRS is also different from the active intelligent surface based massive MIMO [@hu2017beyond] due to their different array architectures (passive versus active) and operating mechanisms (reflect versus transmit). [A more detailed comparison between the above technologies and IRS is summarized in Table \[table1\], where $N$ denotes the number of active antennas in massive MIMO or MIMO relay.]{} ]{}
On the other hand, from the implementation perspective, IRSs possess appealing advantages such as low profile, lightweight, and conformal geometry, which enable them to be easily attached/removed to/from the wall or ceiling, thus providing high flexibility for their practical deployment [@subrt2012intelligent]. For example, by installing IRSs on the walls/ceilings which are in line-of-sight (LoS) with an access point (AP)/BS, the signal strength in the vicinity of each IRS can be significantly improved. In addition, integrating IRSs into the existing networks (such as cellular or WiFi) can be made transparent to the users without the need of any change in the hardware and software of their devices. All the above features make IRS a compelling new technology for future wireless networks, particularly in indoor applications with high density of users (such as stadium, shopping mall, exhibition center, airport, etc.). To validate the feasibility of IRS, an experimental testbed for a two-user setup was developed [@tan2016increasing], where the spectral efficiency is shown to be greatly improved by using the IRS. However, the research on IRS design as well as the performance analysis and optimization for IRS-aided wireless communication systems is still in its infancy, which thus motivates this work.
[In this paper, we consider an IRS-aided multiuser multiple-input single-output (MISO) communication system in a single cell as shown in Fig. \[system:model\], where a multi-antenna AP serves multiple single-antenna users with the help of an IRS.]{} Since each user in general receives the superposed (desired as well as interference) signals from both the AP-user (direct) link and AP-IRS-user (reflected) link, we jointly optimize the (active) transmit beamforming at the AP and (passive) reflect beamforming by the phase shifters at the IRS to minimize the total transmit power at the AP, under a given set of signal-to-interference-plus-noise ratio (SINR) constraints at the user receivers. For the special case of single-user transmission without any interference, it is intuitive that the AP should beam toward the user directly if the channel of the AP-user link is much stronger than that of the AP-IRS link; while in the opposite case, especially when the AP-user link is severally blocked by obstacles (e.g., thick walls in indoor applications), the AP ought to adjust its beam toward the IRS to maximally leverage its reflected signal to serve the user (i.e., by creating a virtual LoS link with the user to bypass the obstacle). In this case, a large number of reflecting elements with adjustable phases at the IRS can focus the signal into a sharp beam toward the user, thus achieving a high beamforming gain similarly as by the conventional massive MIMO [@Hien2013], but only via a passive array with significantly reduced energy consumption and hardware cost.
Moreover, under the general multiuser setup, an IRS-aided system will
![An IRS-aided multiuser communication system. []{data-label="system:model"}](Visio-IRS_model8_multiuser "fig:"){width="85.00000%"}
benefit from two main aspects: the beamforming of desired signal as in the single-user case as well as the spatial interference suppression among the users. Specifically, a user near the IRS is expected to be able to tolerate more interference from the AP as compared to the user farther away from the IRS, because the phase shifts of the IRS can be tuned such that the interference reflected by the IRS can add destructively with that from the AP-user link at the near user to suppress its overall received interference. This thus provides more flexibility for designing the transmit beamforming at the AP for serving the other users outside the IRS’s covered region, so as to improve the SINR performance of all users in the system. Therefore, the transmit beamforming at the AP needs to be jointly designed with the phase shifts at the IRS based on all the AP-IRS, IRS-users, and AP-users channels in order to fully reap the network beamforming gain. However, this design problem is difficult to be solved optimally in general, due to the non-convex SINR constraints as well as the signal unit-modulus constraints imposed by passive phase shifters. Although beamforming optimization under unit-modulus constraints has been studied in the research on constant-envelope precoding [@mohammed2012single; @zhang2018constant] as well as hybrid digital/analog processing [@el2014spatially; @foad16jstsp], such designs are mainly restricted to either the transmitter or the receiver side, which are not applicable to our considered joint active and passive beamforming optimization at both the AP and IRS.
To tackle this new problem, we first consider a single-user setup and apply the semidefinite relaxation (SDR) technique to obtain a high-quality approximate solution as well as a lower bound of the optimal value to evaluate the tightness of approximate solutions. To reduce the computational complexity, we further propose an efficient algorithm based on the alternating optimization of the phase shifts and transmit beamforming vector in an iterative manner, where their optimal solutions are derived in closed-form with the other being fixed. Then, we extend our designs for the single-user case to the general multiuser setting, and propose two algorithms to obtain suboptimal solutions that also offer different tradeoffs between performance and complexity. Numerical results demonstrate that the required transmit power at the AP to meet users’ SINR targets can be considerably reduced by deploying the IRS as compared to the conventional setup without using IRS for both single-user and multiuser setups. In particular, for serving a single-user in the vicinity of the IRS, it is shown that the AP’s transmit power decreases with the number of reflecting elements $N$ at the IRS in the order of $N^2$ when $N$ is sufficiently large, which is consistent with the performance scaling law derived analytically. [Note that in [@huangachievable], the authors also considered the use of passive intelligent mirror (analogous to IRS) to enhance the sum-rate in a multiuser system. This paper differs from [@huangachievable] in the following two main aspects. First, to simplify the system model and algorithm design, [@huangachievable] ignored the direct channels from the AP to users, while this paper considers the more general setting with the AP-user direct channels considered. Second, [@huangachievable] adopted the suboptimal zero-forcing (ZF) based precoding at the AP to simplify the optimization of passive phase shifters, while in this paper we optimize AP transmit precoding jointly with IRS’s phase shifts. As such, the algorithm proposed in [@huangachievable] is not applicable to solving the formulated problems in this paper. ]{}
The rest of this paper is organized as follows. Section II introduces the system model and the problem formulation for designing the IRS-aided wireless network. In Sections III and IV, we propose efficient algorithms to solve the formulated problems in the single-user and multiuser cases, respectively. Section V presents numerical results to evaluate the performance of the proposed designs. Finally, we conclude the paper in Section VI.
*Notations:* Scalars are denoted by italic letters, vectors and matrices are denoted by bold-face lower-case and upper-case letters, respectively. $\mathbb{C}^{x\times y}$ denotes the space of $x\times y$ complex-valued matrices. For a complex-valued vector $\bm{x}$, $\|\bm{x}\|$ denotes its Euclidean norm, $\arg(\bm{x})$ denotes a vector with each element being the phase of the corresponding element in $\bm{x}$, and $\text{diag}(\bm{x})$ denotes a diagonal matrix with each diagonal element being the corresponding element in $\bm{x}$. The distribution of a circularly symmetric complex Gaussian (CSCG) random vector with mean vector $\bm{x}$ and covariance matrix ${\bm \Sigma}$ is denoted by $\mathcal{CN}(\bm{x},{\bm \Sigma})$; and $\sim$ stands for “distributed as”. For a square matrix ${\bm S}$, ${\rm{tr}}({\bm S})$ and ${\bm S}^{-1}$ denote its trace and inverse, respectively, while ${\bm S}\succeq \bm{0}$ means that ${\bm S}$ is positive semi-definite. For any general matrix ${\bm{M}}$, ${\bm{M}}^H$, ${\rm{rank}}({\bm{M}})$, and ${\bm{M}}_{i,j}$ denote its conjugate transpose, rank, and $(i,j)$th element, respectively. ${\bm{I}}$ and $\bm{0}$ denote an identity matrix and an all-zero matrix, respectively, with appropriate dimensions. $\mathbb{E}(\cdot)$ denotes the statistical expectation. $ \mathrm{Re}\{\cdot\}$ denotes the real part of a complex number.
System Model and Problem Formulation
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System Model
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[As shown in Fig. \[system:model\], we consider the IRS-aided downlink communications in a single-cell network where an IRS is deployed to assist in the communications from a multi-antenna AP to $K$ single-antenna users over a given frequency band.]{} The set of the users is denoted by $\mathcal{K}$. The number of transmit antennas at the AP and that of reflecting units at the IRS are denoted by $M$ and $N$, respectively. The IRS is equipped with a controller that coordinates its switching between two working modes, i.e., receiving mode for channel estimation and reflecting mode for data transmission [@JR:wu2019IRSmaga; @subrt2012intelligent]. Due to the high path loss, it is assumed that the power of the signals that are reflected by the IRS two or more times is negligible and thus ignored [@JR:wu2019IRSmaga]. To characterize the theoretical performance gain brought by the IRS, we assume that the channel state information (CSI) of all channels involved is perfectly known at the AP. In addition, the quasi-static flat-fading model is adopted for all channels. Since the IRS is a passive reflecting device, we consider a time-division duplexing (TDD) protocol for uplink and downlink transmissions and assume channel reciprocity for the CSI acquisition in the downlink based on the uplink training.
[The baseband equivalent channels from the AP to IRS, from the IRS to user $k$, and from the AP to user $k$ are denoted by $\bm{G}\in \mathbb{C}^{N\times M}$, $\bm{h}^H_{r,k}\in \mathbb{C}^{1\times N}$, and $\bm{h}^H_{d,k}\in \mathbb{C}^{1\times M}$, respectively, with $k = 1, \cdots,K$.]{} [ It is worth noting that the reflected channel from the AP to each user via the IRS is usually referred to as a dyadic backscatter channel in RFID communications [@JR:wu2019IRSmaga], which behaves different from the AP-user direct channel. Specifically, each element of the IRS receives the superposed multi-path signals from the transmitter, and then scatters the combined signal with adjustable amplitude and/or phase as if from a single point source [@JR:wu2019IRSmaga].]{} Let $\bm{\theta}= [\theta_1, \cdots, \theta_N]$ and define a diagonal matrix ${\mathbf \Theta}= \text{diag} (\beta_1 e^{j\theta_1}, \cdots, \beta_N e^{j\theta_N})$ (with $j$ denoting the imaginary unit) as the reflection-coefficients matrix of the IRS, where $\theta_n\in [0, 2\pi)$ and $\beta_n \in [0, 1]$ denote the phase shift[^2] and the amplitude reflection coefficient[^3] of the $n$th element of the IRS, respectively [@JR:wu2019IRSmaga]. The composite AP-IRS-user channel is thus modeled as a concatenation of three components, namely, the AP-IRS link, IRS reflection with phase shifts, and IRS-user link. In this paper, we consider linear transmit precoding at the AP where each user is assigned with one dedicated beamforming vector. Hence, the complex baseband transmitted signal at the AP can be expressed as $\bm{x}= \sum_{k=1}^K\bm{w}_ks_k$, where $s_k$ denotes the transmitted data for user $k$ and $\bm{w}_k\in \mathbb{C}^{M\times 1}$ is the corresponding beamforming vector. It is assumed that $s_k$, $k = 1, \cdots,K$, are independent random variables with zero mean and unit variance (normalized power). The signal received at user $k$ from both the AP-user and AP-IRS-user channels is then expressed as $$\begin{aligned}
y_k= ( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G} + \bm{h}^H_{d,k}) \sum_{j=1}^K\bm{w}_js_j + n_k, k=1, \cdots, K,\end{aligned}$$ where $n_k \sim \mathcal{CN}(0, \sigma^2_k)$ denotes the additive white Gaussian noise (AWGN) at the user $k$’s receiver. Accordingly, the SINR of user $k$ is given by $$\begin{aligned}
\label{eq:SINR}
\text{SINR}_k = \frac{|( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_k |^2}{\sum_{j\neq k}^{K}|( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_j |^2 + \sigma^2_k}, k=1, \cdots, K.\end{aligned}$$
Problem Formulation
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Let ${\bm W}= [{\bm w}_1, \cdots,{\bm w}_K]\in \mathbb{C}^{M\times K}$, ${\bm H}_r = [{\bm h}_{r,1}, \cdots,{\bm h}_{r,K}]\in \mathbb{C}^{N\times K}$, and ${\bm H}_d = [{\bm h}_{d,1}, \cdots,{\bm h}_{d,K}]\in \mathbb{C}^{M\times K}$. In this paper, we aim to minimize the total transmit power at the AP by jointly optimizing the transmit beamforming at the AP and reflect beamforming at the IRS, subject to individual SINR constraints at all users. Accordingly, the problem is formulated as $$\begin{aligned}
\text{(P1)}: ~~\min_{{\bm W}, \bm{\theta}} ~~~&\sum_{k=1}^{K}\|\bm{w}_k\|^2 \\
\mathrm{s.t.}~~~~&\frac{|( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_k |^2}{\sum_{j\neq k}^{K}|( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_j |^2 + \sigma^2_k}\geq \gamma_k, \forall k, \label{eq:coupling}\\
& 0\leq \theta_n \leq 2\pi, n=1,\cdots, N, \label{eq:modulus}\end{aligned}$$ where $\gamma_k>0$ is the minimum SINR requirement of user $k$. Although the objective function of (P1) and constraints in are convex, it is challenging to solve (P1) due to the non-convex constraints in where the transmit beamforming and phase shifts are coupled. In general, there is no standard method for solving such non-convex optimization problems optimally. Nevertheless, in the next section, we apply the SDR and alternating optimization techniques, respectively, to solve (P1) approximately for the single-user case, which are then generalized to the multiuser case.
Prior to solving problem (P1), we present a sufficient condition for its feasibility as follows. Let ${\bm H}= [{\bm h}_1, \cdots,{\bm h}_K]\in \mathbb{C}^{M\times K}$ where ${\bm h}^H_k = \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k}$, $\forall k$.
\[feasibility:condition\] Problem (P1) is feasible for any finite user SINR targets $\gamma_k$’s if ${\rm{rank}}({\bm G}^H{\bm H}_r + {\bm H}_d)=K$.
If ${\rm{rank}}({\bm G}^H{\bm H}_r + {\bm H}_d)=K$, the (right) pseudo inverse of ${\bm H}^H ={\bm H}_{r}^{H}{\mathbf \Theta}{\bm G}+ {\bm H}_{d}^{H}$ exists with ${\mathbf \Theta}= {\bm{I}}$ and the precoding matrix ${\bm W}$ at the AP can be set as $$\begin{aligned}
{\bm W}= {\bm H}( {\bm H}^H {\bm H})^{-1} \text{diag}(\gamma_1\sigma_1^2,\cdots, \gamma_k\sigma_k^2)^{\frac{1}{2}}.\end{aligned}$$ It is easy to verify that the above solution allows all users to achieve their corresponding $\gamma_k$’s and thus (P1) is feasible.
[ Thanks to the additional AP-IRS-user link, the rank condition in Proposition \[feasibility:condition\] is practically easier to be satisfied in an IRS-aided system, as compared to that in the case without the IRS, i.e., ${\rm{rank}}({\bm H}_d)=K$. For instance, if the AP-user direct channels of two users lie in the same direction, then ${\rm{rank}}({\bm H}_d)=K$ does not hold. While the rank condition in an IRS-aided system may still hold since the combined AP-user channels (including both the AP-user direct and AP-IRS-user reflected links) of these two users are unlikely to be aligned too, due to the additional IRS reflected paths.]{}
Single-User System {#sec:III}
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In this section, we consider the single-user setup, i.e., $K=1$, to draw important insights into the optimal joint beamforming design. In this case, no inter-user interference is present, and thus (P1) is simplified to (by dropping the user index) $$\begin{aligned}
\text{(P2)}: ~~\min_{\bm{w}, \bm{\theta}} ~~~& \|\bm{w}\|^2 \label{eq:obj}\\
\mathrm{s.t.}~~~~&| (\bm{h}^H_r{\mathbf \Theta}\bm{G}+\bm{h}^H_d )\bm{w}|^2 \geq \gamma \sigma^2, \label{SINR:constraints} \\
& 0\leq \theta_n \leq 2\pi, n=1,\cdots, N. \label{phase:constraints}\end{aligned}$$ Although much simplified, problem (P2) is still a non-convex optimization problem since the left-hand-side (LHS) of is not jointly concave with respect to $\bm{w}$ and $\bm{\theta}$. In the next two subsections, we solve (P2) by applying the SDR and alternating optimization techniques, respectively, which will be extended to the general multiuser system in the next section.
SDR
---
We first apply SDR to solve problem (P2), which also helps obtain a lower bound of the optimal value of (P2) for evaluating the performance gaps from other suboptimal solutions. For any given phase shift $\bm{\theta}$, it is known that the maximum-ratio transmission (MRT) is the optimal transmit beamforming solution to problem (P2) [@tse2005fundamentals], i.e., $\bm w^* = \sqrt{ P} \frac{(\bm{h}^H_r\Theta \bm{G}+\bm{h}^H_d )^H}{\|\bm{h}^H_r{\mathbf \Theta}\bm{G} +\bm{h}^H_d \|}$, where $P$ denotes the transmit power of the AP. Substituting $\bm w^*$ to problem (P2) yields the following problem $$\begin{aligned}
\label{secIII:p2}
\min_{P, \bm{\theta}} ~~~&p \\
\mathrm{s.t.}~~~~&p\|\bm{h}^H_r{\mathbf \Theta}\bm{G}+ \bm{h}^H_d\|^2\geq \gamma\sigma^2, \label{SecIII:SNRconstraint0}\\
&0\leq \theta_n \leq 2\pi, \forall n. \label{SecIII:phaseconstraint0}\end{aligned}$$ It is not difficult to verify that the optimal transmit power satisfies $P^* = \frac{\gamma\sigma^2}{\|\bm{h}^H_r{\mathbf \Theta}\bm{G}+ \bm{h}^H_d\|^2}$. As such, minimizing the transmit power is equivalent to maximizing the channel power gain of the combined channel, i.e., $$\begin{aligned}
\label{secIII:p3}
\max_{\bm{\theta}} ~~~&\|\bm{h}^H_r{\mathbf \Theta}\bm{G}+ \bm{h}^H_d\|^2\\
\mathrm{s.t.}~~~~&0\leq \theta_n \leq 2\pi, \forall n. \label{SecIII:phaseconstraint}\end{aligned}$$ Let $\bm{v} = [v_1, \cdots, v_N]^H$ where $v_n = e^{j\theta_n}$, $\forall n$. Then, constraints in are equivalent to the unit-modulus constraints: $|v_n|^2=1, \forall n$. By applying the change of variables $\bm{h}^H_r{\mathbf \Theta}\bm{G} =\bm{v}^H\bm{\Phi} $ where $\bm{\Phi}=\text{diag}(\bm{h}^H_r)\bm{G} \in \mathbb{C}^{N \times M}$, we have $\|\bm{h}^H_r{\mathbf \Theta}\bm{G} + \bm{h}^H_d\|^2 =\|\bm{v}^H\bm{\Phi}+ \bm{h}^H_d\|^2 $. Thus, problem is equivalent to $$\begin{aligned}
\label{secIII:p4}
\max_{\bm{v}} ~~~&\bm{v}^H\bm{\Phi}\bm{\Phi}^H\bm{v} + \bm{v}^H\bm{\Phi}\bm{h}_d+\bm{h}^H_d\bm{\Phi}^H \bm{v} + \|\bm{h}^H_d\|^2\\
\mathrm{s.t.}~~~~& |v_n|^2=1, \forall n. \label{P4:C9}\end{aligned}$$ Note that problem is a non-convex quadratically constrained quadratic program (QCQP), which can be reformulated as a homogeneous QCQP [@so2007approximating]. Specifically, by introducing an auxiliary variable $t$, problem is equivalently written as $$\begin{aligned}
\label{secIII:p5}
\max_{\bm{\bar{v}}} ~~~&\bm{\bar{v}}^H\bm{R}\bm{\bar{v}} + \|\bm{h}^H_d\|^2 \\
\mathrm{s.t.}~~~~& |\bar{v}_n|^2=1, n=1,\cdots, N+1, \label{P4:C9}\end{aligned}$$ where $$\bm{R}=\begin{bmatrix}
\bm{\Phi}\bm{\Phi}^H & \bm{\Phi}\bm{h}_d \\
\bm{h}^H_d\bm{\Phi}^H & 0 \\
\end{bmatrix},~~
\bm{\bar{v}}=\begin{bmatrix}
\bm{v} \\
t \\
\end{bmatrix}.$$ However, problem is still non-convex in general [@so2007approximating]. Note that $\bm{\bar{v}}^H\bm{R}\bm{\bar{v}}={\rm{tr}}(\bm{R}\bm{\bar{v}}\bm{\bar{v}}^H) $. Define $\bm{V}=\bm{\bar{v}}\bm{\bar{v}}^H$, which needs to satisfy $\bm{V}\succeq \bm{0}$ and ${\rm{rank}}(\bm{V})=1$. Since the rank-one constraint is non-convex, we apply SDR to relax this constraint. As a result, problem is reduced to $$\begin{aligned}
\label{secIII:p6}
\max_{\bm{V}} ~~~&{\rm{tr}}(\bm{RV}) + \|\bm{h}^H_d\|^2 \\
\mathrm{s.t.}~~~~& \bm{V}_{n,n} = 1, n=1,\cdots, N+1, \label{P6:C9} \\
~~~~&\bm{V} \succeq 0. \label{P6:C9}\end{aligned}$$ As problem is a convex semidefinite program (SDP), it can be optimally solved by existing convex optimization solvers such as CVX [@cvx]. Generally, the relaxed problem may not lead to a rank-one solution, i.e., ${\rm{rank}}(\bm{V})\neq1$, which implies that the optimal objective value of problem only serves an upper bound of problem . Thus, additional steps are needed to construct a rank-one solution from the obtained higher-rank solution to problem , while the details can be found in [@wu2018IRS] and thus are omitted here. It has been shown that such an SDR approach followed by a sufficiently large number of randomizations guarantees at least a $\frac{\pi}{4}$-approximation of the optimal objective value of problem [@so2007approximating].
Alternating Optimization
------------------------
To achieve lower complexity than the SDR-based solution presented in the preceding subsection, we propose an alternative suboptimal algorithm in this subsection based on alternating optimization. Specifically, the transmit beamforming direction and transmit power at the AP are optimized iteratively with the phase shifts at the IRS in an alternating manner, until the convergence is achieved.
Let $\bm{w}=\sqrt{P}\bm{\bar w}$ where $\bm{\bar w}$ denotes the transmit beamforming direction and $P$ is the transmit power. [For fixed transmit beamforming direction $\bm{\bar w}$, (P2) is reduced to a joint transmit power and phase shifts optimization problem which can be formulated as (similar to and ),]{} $$\begin{aligned}
\label{secIII2:p3}
\max_{\bm{\theta}} ~~~&|(\bm{h}^H_r{\mathbf \Theta}\bm{G}+ \bm{h}^H_d){\bm{\bar w}}|^2\\
\mathrm{s.t.}~~~~&0\leq \theta_n \leq 2\pi, n=1,\cdots, N. \label{SecIII2:phaseconstraint}\end{aligned}$$ Although being non-convex, the above problem admits a closed-form solution by exploiting the special structure of its objective function. Specifically, we have the following inequality: $$\begin{aligned}
\label{SecIV:distributed}
| (\bm{h}^H_r{\mathbf \Theta}\bm{G}+\bm{h}^H_d )\bm{\bar w}| &=| \bm{h}^H_r{\mathbf \Theta}\bm{G}\bm{\bar w}+ \bm{h}^H_d\bm{\bar w}| \overset{(a)} \leq | \bm{h}^H_r{\mathbf \Theta}\bm{G}\bm{\bar w}| + | \bm{h}^H_d\bm{\bar w}|,\end{aligned}$$ where $(a)$ is due to the triangle inequality and the equality holds if and only if $\mathrm{arg}(\bm{h}^H_r{\mathbf \Theta}\bm{G}\bm{\bar w})=\mathrm{arg}( \bm{h}^H_d\bm{\bar w} )\triangleq \varphi_0$. Next, we show that there always exists a solution $\bm{\theta}$ that satisfies $(a)$ with equality as well as the phase shift constraints in . Let $\bm{h}^H_r{\mathbf \Theta}\bm{G}\bm{w} =\bm{v}^H\bm{a}$ where $\bm{v} = [e^{j\theta_1}, \cdots, e^{j\theta_N}]^H$ and $\bm{a}=\text{diag}(\bm{h}^H_r)\bm{G}\bm{\bar w}$. With , problem is equivalent to $$\begin{aligned}
\max_{\bm{v}} ~~~&|\bm{v}^H\bm{a}|^2\\
\mathrm{s.t.}~~~~& |v_n|=1, \forall n=1,\cdots, N,\\
& \mathrm{arg}(\bm{v}^H\bm{a})= \varphi_0. $$ It is not difficult to show that the optimal solution to the above problem is given by $\bm{v}^* = e^{j (\varphi_0 - \arg({\bm{a}}) )}=e^{j ( \varphi_0 - \arg( \text{diag}(\bm{h}^H_r)\bm{G}\bm{\bar w}) )}$. Thus, the $n$th phase shift at the IRS is given by $$\begin{aligned}
\label{phase:sub}
\theta^*_n &=\varphi_0 - \arg({h}^H_{n,r}\bm{g}^H_{n}\bm{\bar w})= \varphi_0 - \arg({h}^H_{n,r})- \arg(\bm{g}^H_{n}\bm{\bar w}),\end{aligned}$$ where ${h}^H_{n,r}$ is the $n$th element of $\bm{h}^H_r$ and $\bm{g}^H_{n}$ is the $n$th row vector of $\bm{G}$. Note that $\bm{g}^H_{n}\bm{\bar w}$ combines the transmit beamforming and the AP-IRS channel, which can be regarded as the effective channel perceived by the $n$th reflecting element at the IRS. Therefore, suggests that the $n$th phase shift should be tuned such that the phase of the signal that passes through the AP-IRS and IRS-user links is aligned with that of the signal over the AP-user direct link to achieve coherent signal combining at the user. Furthermore, it is interesting to note that the obtained phase $\theta^*_n $ is independent of the amplitude of ${h}_{n,r}$. As a result, the optimal transmit power is given by $P^*=\frac{\gamma \sigma^2}{\|(\bm{h}^H_r{\mathbf \Theta}\bm{G}+\bm{h}^H_d){\bm{\bar w}} \|^2}$ from (P2). Next, we optimize the transmit beamforming direction for given $\bm{\theta}$ in . As in Section III-A, the combined AP-user channel is given by $\bm{h}^H_r{\mathbf \Theta}\bm{G} + \bm{h}^H_d$ and hence MRT is optimal, i.e., $\bm{\bar w}^{*} = \frac{(\bm{h}^H_r{\mathbf \Theta}\bm{G}+\bm{h}^H_d )^H}{\|\bm{h}^H_r{\mathbf \Theta}\bm{G} +\bm{h}^H_d \|}$. [The above alternating optimization approach is practically appealing since both the transmit beamforming and phase shifts are obtained in closed-form expressions, without invoking the SDP solver. Its convergence is guaranteed by the following two facts. First, for each subproblem, the optimal solution is obtained which ensures that the objective value of (P2) is non-increasing over iterations. Second, the optimal value of (P2) is bounded from below due to the SNR constraint. Thus, the proposed algorithm is guaranteed to converge. ]{}
[The power scaling law with the optimal IRS phase design in Proposition \[scaling:law\] is highly promising since it implies that by using a large number of reflecting units at the IRS, we can scale down the transmit power of the AP by a factor of $1/N^2$ without compromising the user received SNR. The fundamental reason behind such a “squared gain” is that the IRS not only achieves the transmit beamforming gain of order $N$ in the IRS-user link as in the conventional massive MIMO [@Hien2013], but also captures an inherent aperture gain of order $N$ by collecting more signal power in the AP-IRS link, which, however, cannot be achieved by scaling up the number of transmit antennas in massive MIMO due to the fixed total transmit power. Moreover, for the two benchmark cases with unit and random phase shifts at the IRS, a received power gain of order $N$ is also achieved. This shows the practical usefulness of the IRS, even without requiring any channel knowledge for optimally setting the phase shifts. Note that the received noise power in the IRS-aided system remains constant as $N$ increases and thus the corresponding user receive SNR also has the same squared gain as the received signal power with increasing $N$. ]{}
Next, we show the performance scaling law of an FD AF relay aided system under the same setup as the above IRS-aided system. The relay is equipped with $N$ transmit and $N$ receive antennas and the direct channel from the AP to the user can be similarly ignored when $N$ is asymptotically large. We assume that the relay adopts linear receive and transmit beamforming vectors, denoted by ${ \bm{x} }^H_r$ and ${ \bm{x} }_t$, respectively. In addition, perfect self-interference cancellation (SIC) is assumed at the relay so that the obtained performance serves as an upper bound for the practical case with imperfect SIC. In this case, the user receive SNR can be expressed as $$\begin{aligned}
\label{SNR:AF}
\gamma_{FD} =\frac{P P_r\| { \bm{x} }_r^H{\bm g}\|^2 \| {\bm h}^H_{r}{ \bm{x} }_t \|^2}{P_r \sigma^2_r\| { \bm{x} }_r \|^2\| {\bm h}^H_{r}{ \bm{x} }_t \|^2 + P\sigma^2\| { \bm{x} }_t \|^2\|{ \bm{x} }^H_r{\bm g}\|^2+\sigma^2_r\sigma^2\| { \bm{x} }_t \|^2 \|{ \bm{x} }_r \|^2},\end{aligned}$$ where $P_r$ and $\sigma^2_r$ denote the transmit power and the noise power at the relay, respectively. It is not difficult to show that the optimal solution maximizing $\gamma_{FD}$ satisfies ${ \bm{x} }^*_t = \frac{{\bm h}_{r}}{\| {\bm h}_{r}\|}$ and ${ \bm{x} }^*_r = \frac{{\bm g}}{\| {\bm g}\| }$. Substituting ${ \bm{x} }^*_t$ and ${ \bm{x} }^*_r$ into , we have $\gamma_{FD} = \frac{PP_r\|{\bm g}\|^2 \|{\bm h}_{r}\|^2 }{ P_r \sigma^2_r \|{\bm h}_{r}\|^2 + P \sigma^2 \|{\bm g}\|^2 +\sigma^2_r\sigma^2}.$ Then, we have the following proposition.
\[scaling:law:AF\] Assume $\bm{h}^H_{r} \sim \mathcal{CN}(\bm{0},\varrho^2_h{\bm{I}})$ and $\bm{g} \sim \mathcal{CN}(\bm{0},\varrho^2_g{\bm{I}})$. As $N\rightarrow \infty$, it holds that $$\begin{aligned}
\gamma_{FD} \rightarrow \frac{PP_r \varrho^2_{g}\varrho^2_{h}N }{ P_r \sigma^2_r \varrho^2_{h} + P \sigma^2\varrho^2_{g} }. \label{SNR2}
$$
Since $\frac{\|{\bm g}\|^2}{N}\rightarrow \varrho^2_{g}$ and $\frac{\|{\bm h}_{r}\|^2}{N}\rightarrow \varrho^2_{h}$ as $N\rightarrow \infty$ [@Hien2013], it follows that $$\begin{aligned}
\label{SNR:3}
\gamma_{FD} \rightarrow \frac{P P_r \varrho^2_{g}\varrho^2_{h} N^2 }{ P_r \sigma^2_r \varrho^2_{h}N + P\sigma^2 \varrho^2_{g}N +\sigma^2_r\sigma^2}\approx \frac{PP_r \varrho^2_{g}\varrho^2_{h}N }{ P_r \sigma^2_r \varrho^2_{h} + P \sigma^2\varrho^2_{g} }.\end{aligned}$$ This thus completes the proof.
[Proposition \[scaling:law:AF\] shows that even with perfect SIC, the receive SNR by using the FD AF relay increases only linearly with $N$ when $N$ is asymptotically large. This is fundamentally due to the noise effect at the AF relay. To be specific, although the signal power in the FD AF relay system scales in the order of $N^2$ same as that in the IRS-aided system, its effective noise power at the receiver also scales linearly with $N$ (see ) in contrast to the constant noise power $\sigma^2$ in the IRS-aided system, thus resulting in a lower SNR gain order with $N$. Last, it is worth mentioning that for the HD AF relay system, its receive SNR scaling order with $N$ can be shown to be identical to that of the FD AF relay system given in Proposition 3.]{}
Multiuser System {#multiuser:sec}
================
In this section, we consider the general multiuser setup. Specifically, we propose two efficient algorithms to solve (P1) suboptimally by generalizing the two approaches in the single-user case.
Alternating Optimization Algorithm
-----------------------------------
This algorithm leverages the alternating optimization similarly as in the single-user case, while the transmit beamforming at the AP is designed by applying the well-known minimum mean squared error (MMSE) criterion to cope with the multiuser interference instead of using MRT in the single-user case without interference. For given phase shift ${\bm \theta}$, the combined channel from the AP to user $k$ is given by ${\bm{h}}^H_k= \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k}$. Thus, problem (P1) is reduced to $$\begin{aligned}
\text{(P3)}: ~~\min_{{\bm W}} ~~~&\sum_{k=1}^{K}\|\bm{w}_k\|^2 \\
\mathrm{s.t.}~~~~&\frac{|{\bm{h}}^H_k\bm{w}_k |^2}{\sum_{j\neq k}^{K}|{\bm{h}}^H_k\bm{w}_j |^2 + \sigma^2_k}\geq \gamma_k, \forall k.\label{P2:SINR}\end{aligned}$$ Note that (P3) is the conventional power minimization problem in the multiuser MISO downlink broadcast channel, which can be efficiently solved by using second-order cone program (SOCP) [@wiesel2006linear], SDP [@bengtsson2001handbook], or a fixed-point iteration algorithm based on the uplink-downlink duality [@schubert2004solution; @luo2006introduction]. In addition, it is easy to verify that at the optimal solution to problem (P3), all the SINR constraints in are met with equalities.
On the other hand, for given transmit beamforming ${\bm W}$, problem (P1) is reduced to a feasibility-check problem. Let $\bm{h}^H_{d,k}\bm{w}_j ={b}_{k,j}$ and $v_n=e^{j\theta_n}$, $n=1,\cdots, N$. By applying the change of variables $\bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}\bm{w}_j =\bm{v}^H\bm{a}_{k,j}$ where $\bm{v} = [e^{j\theta_1}, \cdots, e^{j\theta_N}]^H$ and $\bm{a}_{k,j}=\text{diag}(\bm{h}^H_{r,k})\bm{G}\bm{w}_j$, problem (P1) is reduced to $$\begin{aligned}
\label{secIV:p3}
\text{Find}~~ &~~ \bm{v} \\
\mathrm{s.t.}~~&\frac{|\bm{v}^H\bm{a}_{k,k} + b_{k,k} |^2}{\sum_{j\neq k}^{K}|\bm{v}^H\bm{a}_{k,j} + b_{k,j} |^2 + \sigma^2_k}\geq \gamma_k, \forall k, \label{eq:coupling2}\\
& |v_n|=1, n=1,\cdots, N. \label{eq:modulus1}\end{aligned}$$ [While the above problem appears similar to the relay beamforming optimization problem for multi-antenna relay broadcast channel [@zhang2009joint], it cannot be directly transformed into an SOCP optimization problem because the phase rotation of the common vector ${\bm v}$ may not render $\bm{v}^H\bm{a}_{k,k} + b_{k,k}$’s in to be real numbers for all users. Moreover, it has non-convex unit-modulus constraints in . However, by observing that constraints and can be transformed into quadratic constraints, we apply the SDR technique to approximately solve problem efficiently.]{}
Specifically, by introducing an auxiliary variable $t$, can be equivalently written as $$\begin{aligned}
\label{secIV:p4}
\text{Find} ~~~~&\bm{v}\\
\mathrm{s.t.}~~~~& \bm{\bar v}^H{\bm R}_{k,k}{\bm {\bar v}} + |b_{k,k}|^2 \geq \gamma_k\sum_{j\neq k}^{K} \bm{\bar v}^H{\bm R}_{k,j}{\bm {\bar v}} + \gamma_k(\sum_{j\neq k}^{K} |b_{k,j}|^2 + \sigma^2_k), \forall k,\label{P10:qos}\\
& |v_n|^2=1, n=1,\cdots, N+1,\end{aligned}$$ where $$\bm{R}_{k,j}=\begin{bmatrix}
\bm{a}_{k,j}\bm{a}_{k,j}^H & \bm{a}_{k,j}b^H_{k,j} \\
\bm{a}_{k,j}^Hb_{k,j} & 0 \\
\end{bmatrix},~~
\bm{\bar{v}}=\begin{bmatrix}
\bm{v} \\
t \\
\end{bmatrix}.$$ Note that $\bm{\bar{v}}^H\bm{R}_{k,j}\bm{\bar{v}}={\rm{tr}}(\bm{R}_{k,j}\bm{\bar{v}}\bm{\bar{v}}^H) $. Define $\bm{V}=\bm{\bar{v}}\bm{\bar{v}}^H$, which needs to satisfy $\bm{V}\succeq \bm{0}$ and ${\rm{rank}}(\bm{V})=1$. Since the rank-one constraint is non-convex, we relax this constraint and problem is then transformed to $$\begin{aligned}
\text{(P4)}: ~~\text{Find} ~~~~&\bm{V}\\
\mathrm{s.t.}~~~~&{\rm{tr}}( {\bm{R}_{k,k}}{\bm V}) + |b_{k,k}|^2 \geq \gamma_k\sum_{j\neq k}^{K} {\rm{tr}}( {\bm{R}_{k,j}}{\bm V}) + \gamma_k(\sum_{j\neq k}^{K} |b_{k,j}|^2 + \sigma^2_k), \forall k,\label{P6:SINR:39}\\
~~~~& \bm{V}_{n,n} = 1, n=1,\cdots, N+1, \label{P6:C9} \\
~~~~&\bm{V} \succeq 0. \label{P6:C10}\end{aligned}$$ It is not difficult to observe that problem (P4) is an SDP and hence it can be optimally solved by existing convex optimization solvers such as CVX [@cvx]. While the SDR may not be tight for problem , the Gaussian randomization can be similarly used to obtain a feasible solution to problem based on the higher-rank solution obtained by solving (P4). In addition, it is worth pointing out that the SINR constraints in are not necessarily to be met with equality for a feasible solution of (P4), due to the common phase shifting matrix (${\bm V}$) for all users.
In the proposed alternating optimization algorithm, we solve problem (P1) by solving problems (P3) and (P4) alternately in an iterative manner, where the solution obtained in each iteration is used as the initial point of the next iteration. The details of the proposed algorithm are summarized in Algorithm \[Alg:MMSE\]. In particular, the algorithm starts with solving problem (P3) for given $\bm{\theta}$ instead of solving (P4) for given ${\bm W}$. This is deliberately designed since (P3) is always feasible for any arbitrary $\bm{\theta}$, provided that ${\rm{rank}}({\bm G}^H{\mathbf \Theta}{\bm H}_r + {\bm H}_d)=K$, while this may not be true for (P4) with arbitrary ${\bm W}$. On the other hand, as solving (P4) only attains a feasible solution, it remains unknown whether the objective value of (P3) will monotonically decrease or not over iterations in Algorithm \[Alg:MMSE\]. Intuitively, if the feasible solution obtained by solving (P4) achieves a strictly larger user SINR than the corresponding SINR target $\gamma_k$ for user $k$, then the transmit power of user $k$ and hence the total transmit power in problem (P3) can be properly reduced without violating all the SINR constraints. More rigorously, the convergence of Algorithm 1 is ensured by the following proposition.
The objective value of (P3) is non-increasing over the iterations by applying Algorithm \[Alg:MMSE\].
Denote the objective value of (P3) based on a feasible solution $({\bm{\theta}}, {\bm W})$ as $f({\bm{\theta}}, {\bm W})$. As shown in step 4 of Algorithm 1, if there exists a feasible solution to problem (P4), i.e., $({\bm{\theta}}^{r+1}, {\bm W}^{r})$ exists, it is also feasible to problem (P3). As such, $({\bm{\theta}}^r, {\bm W}^{r})$ and $ ({\bm{\theta}}^{r+1}, {\bm W}^{r+1})$ in step 3 are the feasible solutions to (P3) in the $r$th and $(r+1)$th iterations, respectively. It then follows that $f({\bm{\theta}}^{r+1}, {\bm W}^{r+1})\overset{(a)}{\geq} f({\bm{\theta}}^{r+1}, {\bm W}^r)\overset{(b)}{=} f({\bm{\theta}}^{r}, {\bm W}^r)$, where $(a)$ holds since for given ${\bm{\theta}}^{r+1}$ in step 3 of Algorithm 1, ${\bm W}^{r+1}$ is the optimal solution to problem (P3); and $(b)$ holds because the objective function of (P3) is regardless of ${\bm{\theta}}$ and only depends on ${\bm W}$.
Initialize the phase shifts ${ \bm{\theta}}={\bm{\theta}}^1$ and set the iteration number $r=1$. Solve problem (P3) for given ${\bm{\theta}}^r$, and denote the optimal solution as ${\bm W}^{r}$. Update $r=r+1$.
To achieve better converged solution, we further transform problem (P4) into an optimization problem with an explicit objective to obtain a generally more efficient phase shift solution to reduce the transmit power. The rationale is that for the transmit beamforming optimization problem, i.e., (P3), all the SINR constraints are active at the optimal solution. As such, optimizing the phase shift to enforce the user achievable SINR to be larger than the SINR target in (P4) directly leads to the transmit power reduction in (P3) (e.g., by simply scaling down the power of transmit beamforming). To this end, problem (P4) is transformed into the following problem $$\begin{aligned}
\text{(P4')}: ~\max_{{\bm V}, \{\alpha_k\}, } ~&\sum_{k=1}^{K} \alpha_k \\
\mathrm{s.t.}~~&{\rm{tr}}( {\bm{R}_{k,k}}{\bm V}) + |b_{k,k}|^2 \geq \gamma_k\sum_{j\neq k}^{K} {\rm{tr}}( {\bm{R}_{k,j}}{\bm V}) + \gamma_k(\sum_{j\neq k}^{K} |b_{k,j}|^2 + \sigma^2_k) + \alpha_k, \forall k,\label{P6:SINR}\\
~~& \bm{V}_{n,n} = 1, n=1,\cdots, N+1, \label{P6:C9} \\
~~&\bm{V} \succeq 0, \alpha_k \geq 0, \forall k, \label{P6:C10}\end{aligned}$$ where the slack variable $\alpha_k$ can be interpreted as the “SINR residual” of user $k$ in phase shift optimization. Note that (P4) and (P4’) have the same set of feasible ${\bm V}$, while (P4’) is more efficient than (P4) in terms of the converged solution, as will be verified in Section V-B by simulation.
Two-Stage Algorithm
--------------------
Inspired by the combined channel gain maximization problem in the single-user case, we next propose a two-stage algorithm with lower complexity compared to the alternating optimization algorithm by decoupling the joint beamforming design problem (P1) into two beamforming subproblems, for optimizing the phase shifts and transmit beamforming, respectively. Specifically, the phase shifts at the IRS are optimized in the first stage by solving a weighted effective channel gain maximization problem. This aims to align with the phases of different user channels so as to maximize the beamforming gain of the IRS, especially for the users near to the IRS. In the second stage, we solve problem (P3) to obtain the optimal MMSE-based transmit beamforming with given phase shifts $ \bm{\theta}$.
Let $\bm{v} = [e^{j\theta_1}, \cdots, e^{j\theta_N}]^H \in \mathbb{C}^{N \times 1}$ and $\bm{\Phi}_k = \text{diag}(\bm{h}^H_{r,k})\bm{G} \in \mathbb{C}^{N \times M}$, $\forall k$. The weighted sum of the combined channel gain of all users is expressed as $$\begin{aligned}
\label{eq:obj}
\sum_{k=1}^{K}t_k\| \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k} \|^2= &\sum_{k=1}^{K}t_k\|{\bm v}^H \text{diag}(\bm{h}^H_{r,k})\bm{G} + \bm{h}^H_{d,k} \|^2 = \sum_{k=1}^{K}t_k\|{\bm v}^H \bm{\Phi}_k + \bm{h}^H_{d,k} \|^2, \end{aligned}$$ where we set the weights to be $t_k = \frac{1}{ \gamma_k\sigma^2_k}, k=1,...,K$, motivated by constraint . Based on , the phase shifts can be obtained by solving the following problem $$\begin{aligned}
\text{(P5)}: ~~\max_{{\bm v}} ~~~&\sum_{k=1}^{K}t_k\|{\bm v}^H \bm{\Phi}_k + \bm{h}^H_{d,k} \|^2 \\
\mathrm{s.t.}~~~~& |v_n|=1, n=1,\cdots, N. \label{eq:modulus2}\end{aligned}$$ Note that for $K=1$, (P5) is equivalent to problem for the single-user case in Section III-A. However, in the multiuser case, due to the same set of phase shifts applied for all users with different channels, the combined channel power gains of different users cannot be maximized at the same time in general, which thus need to be balanced for optimally solving (P5). Nevertheless, since problem (P5) is a non-convex QCQP, it can be similarly reformulated as a homogeneous QCQP as in Section III-A and then solved by applying the SDR and Gaussian randomization techniques. The details are omitted here for brevity. With the phase shifts obtained from (P5), the MMSE-based transmit bemaforming is then obtained by solving (P3). [Compared to the alternating optimization based algorithm proposed in Section \[multiuser:sec\]-A, the two-stage algorithm has lower computational complexity as (P5) and (P3) only need to be respectively solved for one time, but may suffer from certain performance loss, which will be evaluated in the next section.]{}
Simulation Results {#simulation:sec}
==================
We consider a three-dimensional (3D) coordinate system where a uniform linear array (ULA) at the AP and a uniform rectangular array (URA) at the IRS are located in $x$-axis and $x$-$z$ plane, respectively. The antenna spacing is half wavelength and the reference (center) antennas at the AP and IRS are respectively located at $(0, 0, 0)$ and $(0, d_0, 0)$, where $d_0>0$ is the distance between them. For the IRS, we set $N=N_{x}N_{z}$ where $N_{x}$ and $N_{z}$ denote the numbers of reflecting elements along the $x$-axis and $z$-axis, respectively. For the purpose of exposition, we fix $N_x=5$ and increase $N_z$ linearly with $N$. The distance-dependent path loss model is given by $$\begin{aligned}
\label{pathloss}
L(d) = C_0\left( \frac{d}{D_0} \right)^{-\alpha},\end{aligned}$$ where $C_0$ is the path loss at the reference distance $D_0=1$ meter (m), $d$ denotes the individual link distance, and $\alpha$ denotes the path loss exponent. To account for small-scale fading, we assume the Rician fading channel model for all channels involved. Thus, the AP-IRS channel ${\bm G}$ is given by $$\begin{aligned}
{\bm G}= \sqrt{\frac{ \beta_{\rm AI} }{1+ \beta_{\rm AI} }}{\bm G}^{\rm LoS} + \sqrt{\frac{1}{1+\beta_{\rm AI}}}{\bm G}^{\rm NLoS},
\end{aligned}$$ where $\beta_{\rm AI}$ is the Rician factor, and ${\bm G}^{\rm LoS} $ and ${\bm G}^{\rm NLoS}$ represent the deterministic LoS (specular) and Rayleigh fading components, respectively. In particular, the above model is reduced to the LoS channel when $ \beta_{\rm AI} \rightarrow \infty$ or Rayleigh fading channel when $ \beta_{\rm AI} = 0$. The elements in $ {\bm G}$ are then multiplied by the square root of the distance-dependent path loss in with the path loss exponent denoted by $\alpha_{\rm AI}$. The AP-user and IRS-user channels are also generated by following the similar procedure. The path loss exponents of the AP-user and IRS-user links are denoted by $\alpha_{\rm Au}$ and $\alpha_{\rm Iu}$, respectively, and the Rician factors of the two links are denoted by $\beta_{\rm Au}$ and $\beta_{\rm Iu}$, respectively. Due to the relatively large distance and random scattering of the AP-user channel, we set $\alpha_{\rm Au}=3.5$ and $\beta_{\rm Au}=0$ while their counterparts for AP-IRS and IRS-user channels will be specified later to study their effects on the system performance. Without loss of generality, we assume that all users have the same SINR target, i.e., $\gamma_k=\gamma$, $\forall k$. [The number of random vectors used for the Gaussian randomization is set to be 1000 and the stopping threshold for the alternating optimization algorithms is set as $\epsilon=10^{-4}$.]{} Other system parameters are set as follows: $C_0= -30$ dB, $\sigma_k^2=-80$dBm, $\forall k$, and $d_0= 51$ m (if not specified otherwise).
Single-User System {#single-user-system}
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First, we consider a single-user system with the user SNR target $\gamma=10$ dB and $M=4$. As shown in Fig. \[simulation:setup\], the user lies on a horizontal line that is in parallel to the one that connects the reference antennas of AP and IRS, with the vertical distance between these two lines being $d_v = 2$ m. Denote the horizontal distance between the AP and user by $d$ m. Accordingly, the AP-user and IRS-user link distances are given by $d_1= \sqrt{d^2+d_v^2}$ and $d_2= \sqrt{(d_0-d)^2+d_v^2}$, respectively. By varying the value of $d$, we can study the transmit power required for serving the user located between the AP and IRS, under the given SNR target. The path loss exponents and Rician factors are set as $\alpha_{\rm AI} =2$, $\alpha_{\rm Iu}=2.8$, $\beta_{\rm Iu}=0$, and $\beta_{\rm AI} = \infty$, respectively, where ${\bm G}$ is of rank one, i.e., an LoS channel between the AP and IRS. We compare the following schemes: 1) Lower bound: the minimum transmit power based on the optimal solution of the SDP problem ; 2) SDR: the solution obtained by applying SDR and Gaussian randomization techniques in Section III-A; 3) Alternating optimization: the solution proposed in Section III-B; 4) AP-user MRT: we set $\bm{\bar w} = {{\bm{h}_d}}/{\|{\bm{h}_d}\|}$ to achieve MRT based on the AP-user direct channel; 5) AP-IRS MRT: we set $\bm{\bar w} ={{\bm{g}}}/{\|\bm{g}\|}$ to achieve MRT based on the AP-IRS rank-one channel, with $\bm{g}^H$ denoting any row in ${\bm G}$; 6) Random phase shift: we set the elements in $\bm{\theta}$ randomly in $[0, 2\pi]$ and then perform MRT at the AP based on the combined channel; 7) Benchmark scheme without the IRS by setting $\bm{w} = \sqrt{\gamma \sigma^2}{\bm{h}_d}/{\|\bm{h}_d\|^2}$. Note that for scheme 3), the transmit beamforming is initialized by using the AP-user MRT, and for schemes 4) and 5) with given $\bm{\bar w}$, the transmit power and phase shifts are optimized by using the results in Section III-B.
### AP Transmit Power versus AP-User Distance
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In Fig. \[simulation:distance\], we compare the transmit power required by all schemes versus the horizontal distance between the AP and user, $d$. First, it is observed that the proposed two schemes both achieve near-optimal transmit power as compared to the transmit power lower bound, and also significantly outperform other benchmark schemes. Second, for the scheme without the IRS, one can observe that the user farther away from the AP requires higher transmit power at the AP due to the larger signal attenuation. However, this problem is alleviated by deploying an IRS, which implies that a larger AP-user distance does not necessarily lead to a higher transmit power in IRS-aided wireless networks. This is because the user farther away from the AP may be closer to the IRS and thus it is able to receive stronger reflected signal from the IRS. As a result, the user near either the AP (e.g., $d=23$ m) or IRS (e.g., $d=47$ m) requires lower transmit power than a user far away from both of them (e.g., $d=40$ m). This phenomenon also suggests that the signal coverage can be effectively extended by deploying only a passive IRS rather than installing an additional AP or active relay. For example, for the same transmit power about 13 dBm, the coverage of the network without the IRS is about 33 m whereas this value is improved to be beyond 50 m by applying the proposed joint beamforming designs with an IRS. On the other hand, it is observed from Fig. \[simulation:distance\] that the AP-user MRT scheme performs close to optimal when the user is nearer to the AP, while it incurs considerably higher transmit power when the user is nearer to the IRS. This is expected since in the former case, the user received signal is dominated by the AP-user direct link whereas the IRS-user link is dominant in the latter case. Moreover, it can be observed that the AP-IRS MRT scheme behaves oppositely as the user moves away from the AP toward IRS. Finally, Fig. \[simulation:distance\] also shows that if the transmit beamforming is not designed properly, the performance achieved by using the IRS may be even worse than that of the case without the IRS, e.g., with the AP-IRS MRT scheme for $d\leq 35$ m. This further demonstrates that the proposed joint beamforming designs can dynamically adjust the AP’s beamforming to strike an optimal balance between the signal power transmitted directly to the user and that to the IRS, to achieve the maximum received power at the user.
### AP Transmit Power versus Number of Reflecting Elements
In Fig. \[simulation:N\], we compare the AP’s transmit power of all the above schemes versus the number of reflecting elements at the IRS when $d=50$, $41$, and $15$ m, respectively. From Fig. \[simulation:N\] (a), it is observed that for the case of $d=50$ m (i.e., the user is very close to the IRS), the AP-IRS MRT scheme achieves near-optimal transmit power since the signal reflected by the IRS is much stronger than that directly from the AP at the user. Furthermore, it is interesting to note that the transmit power required by the proposed schemes scales down with the number of reflecting elements $N$ approximately in the order of $N^2$ in this case, which is in accordance with Proposition \[scaling:law\] (even when the AP-IRS channel is LoS rather than Rayleigh fading). For example, for the same user SNR, a transmit power of 2 dBm is required at the AP when $N=30$ while this value is reduced to $-4$ dBm when $N=60$, which suggests an around 6 dB gain by doubling the number of reflecting elements. In contrast, the transmit power required by using the random phase shift decreases with increasing $N$ in a much slower rate, because without reflect beamforming the average signal power of the reflected signal is comparable to that of the signal from the AP-user direct link in this case. Finally, it is observed that the above gains diminish as the user moves away from the IRS. For example, for the case of $d=15$ m shown in Fig. \[simulation:N\] (c) where the AP-user direct link signal is much stronger than that of the IRS-user link, the required transmit power is insensitive to the number of reflecting elements. For the case of $d=41$ m shown in Fig. \[simulation:N\] (b) when the user is neither close to the AP nor close to the IRS, it is observed that the transmit power gain of the proposed schemes is generally lower than $N^2$. This is because in this case the signal power received at the IRS is compromised as the AP transmit beamforming is steered to strike a balance between the AP-IRS link and the AP-user direct link. In practice, the number of reflecting elements can be properly selected depending on the IRS’s location as well as the target user SNR/AP coverage range.
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### Comparison with AF Relay
[Next, we compare the achievable rates of the IRS versus the FD AF relay based on the results derived in Section III. We consider the setup in Fig. \[simulation:setup\] with $d_0=d=100$ m, $d_v=1$ m, ${\sigma}^2_r=-80$ dBm, $\alpha_{\rm AI} =3.2$, $\alpha_{\rm Iu}=2$, $\beta_{\rm AI} = 0$, $\beta_{\rm Iu}=\infty$, and $M=1$. To focus on the comparison with large $N$, the direct link from the AP to the user is ignored, and perfect SIC is assumed for the FD AF relay. As such, the SNRs and the corresponding achievable rates for the IRS-aided and the FD AF relay-aided systems can be obtained based on and , respectively. For a fair comparison, we assume that both systems have the same total transmit power budget $P=5$ mW (for single link only). Since the IRS is passive, all the transmit power is used at the AP, whereas since the AF relay is active like the AP, an optimal power allocation between them is required which can be obtained by exhaustive search. ]{}
[Under the above setup, we plot the achievable rate in bits/second/Hertz (bps/Hz) versus $N$ in Fig. \[simulation:AFrelay\], where the HD AF relay is also considered as a benchmark. It is observed that when $N$ is small, the IRS-aided system is able to achieve the same rate as the FD/HD AF relay-aided system by using more reflecting elements. However, since the IRS’s elements are passive, no transmit RF chains are needed for them and thus the cost is much lower as compared with that of active antennas for the AF relay requiring transmit RF chains. Furthermore, one can observe that by doubling $N$ from 400 to 800, the achievable rate of using IRS increases about 2 bps/Hz whereas that of using the FD AF relay only increases about 1 bps/Hz. This is due to their different SNR gains ($N^2$ versus $N$) with increasing $N$ as revealed in Propositions 2 and 3. As a result, it is expected that the IRS-aided system will eventually outperform the FD/HD AF relay-aided system when $N$ is sufficiently large, as shown in Fig. \[simulation:AFrelay\].]{}
Multiuser System {#multiuser-system}
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Next, we consider a multiuser system with eight users, denoted by $U_k$, $k=1,\cdots, 8$, and their locations are shown in Fig. \[simulation:MU\]. Specifically, $U_k$’s, $k=2,4, 6, 8$, lie evenly on a half circle centered at the reference antenna of the IRS with radius $d_2=3$ m, which are usually considered as “cell-edge” users, as compared to $U_k$’s, $k=1,3, 5, 7$, which lie evenly on a circle centered at the reference antenna of the AP with radius $d_1=20$ m. Since the IRS can be practically deployed in LoS with the AP and “cell-edge” users, we set $\alpha_{\rm AI} =\alpha_{\rm Iu}=2.8$, $\beta_{\rm AI} =\beta_{\rm Iu}=3$ dB, respectively. We compare our proposed two algorithms (named as Alternating optimization w/ IRS and Two-stage algorithm w/ IRS, respectively) in Section IV with the two conventional designs in the case without the IRS, i.e., MMSE and zero-forcing (ZF) based beamforming [@bengtsson2001handbook; @yoo2006optimality]. Specifically, the transmit power of the MMSE-based scheme without the IRS is obtained by solving (P3) with ${\mathbf \Theta}= {\bm 0}$, while that of the ZF-based scheme without the IRS is given by ${\rm{tr}}({\bm P}( {\bm H}_d^H {\bm H}_d)^{-1})$ where ${\bm P}= \text{diag}( \sigma^2_1\gamma_1,\cdots, \sigma^2_K\gamma_K)$. The transmit power required by using the random phase shift at the IRS and MMSE beamforming at the AP is also plotted as a benchmark. Before comparing their performances, we first show the convergence behaviour of the proposed Algorithm \[Alg:MMSE\] in Fig. \[convergence\] by setting $M=4$ and considering that only $U_k$, $k=1,2,3,4$, are active (need to be served) with $\gamma=20$ dB. The phase shifts are initialized using the two-stage algorithm. It is observed that the transmit power required by the proposed algorithm decreases quickly with the number of iterations and solving (P4’) instead of (P4) in Algorithm \[Alg:MMSE\] achieves lower converged power. Thus, in the following simulations, we use (P4’) instead of (P4) in Algorithm 1.
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### AP Transmit Power versus User SINR Target
In Fig. \[transmit:pow\], we show the transmit power at the AP versus the SINR target by considering that only two users, namely $U_k$’s, $k=1,2$, are active, which are far from and near the IRS, respectively. It is observed that by adding the IRS, the transmit power required by applying the proposed algorithms is significantly reduced, as compared to the case without the IRS. In addition, one can observe that the transmit power of the proposed two-stage algorithm asymptotically approaches that of the alternating optimization algorithm as $\gamma$ increases. Next, we show how the AP and IRS serve the two users by collaboratively steering the transmit beamforming and adjusting the phase shifts based on the proposed Algorithm \[Alg:MMSE\] (i.e., Alternating optimization w/ IRS). To visualize the transmit beamforming direction at the AP, we use the effective angle between the transmit beamforming (i.e., $\bm{w}_k$) and the AP-user direct channel (i.e., $\bm{h}^H_{d,k}$) for user $k$, which is defined as [@yoo2006optimality; @tse2005fundamentals] $\rho_k\triangleq \mathbb{E} \left( \frac{| \bm{h}^H_{d,k}\bm{w}_k|}{\|\bm{h}^H_{d,k}\|\|\bm{w}_k\|}\right), \forall k.$ In particular, $\rho_k=1$ when the AP steers $\bm{w}_k$ to align with $\bm{h}^H_{d,k}$ perfectly, i.e., MRT transmission, whereas in general $\rho_k<1$ when the AP sets $\bm{w}_k$ orthogonal to $\bm{h}^H_{d,j}$, $j\neq k$, i.e., ZF transmission. Generally, a higher $\rho_k$ implies that the transmit beamforming direction is closer to that of the AP-user direct channel. Figs. \[simulation:pow2\] (a) and (b) show both the desired signal power and interference power of the two users, respectively. For each user $k$, we plot the following four power terms: 1) Combined desired signal power, i.e., $|( \bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_k |^2$; 2) Desired signal power from AP-user link, i.e., $|\bm{h}^H_{d,k}\bm{w}_k |^2$; 3) Combined interference power, i.e., $|\bm{h}^H_{r,k}{\mathbf \Theta}\bm{G}+\bm{h}^H_{d,k})\bm{w}_j|^2$, $j\neq k$; 4) Interference power from AP-user link, i.e., $ |\bm{h}^H_{d,k}\bm{w}_j|^2$, $j\neq k$. For the purpose of exposition, both the desired signal and interference powers are normalized by the noise power. It is observed from Fig. \[simulation:pow2\] (a) that for user 1 (far from IRS), the desired signal and interference powers received from the combined channel are almost the same as those from the AP-user direct link. This is expected since user 1 is far away from the IRS and hence the signal (both desired and interference) from the IRS is negligible. However, the case of user 2 (near IRS) is quite different from that of user 1. As shown in Fig. \[simulation:pow2\] (b), the desired signal power from the combined channel is remarkably higher than that from the AP-user direct link, due to the reflect beamforming gain by the IRS. Furthermore, the IRS also helps align the interference from the AP-IRS-user reflect link oppositely with that from the AP-user direct link to suppress the interference at user 2. This is shown by observing that the interference power from the AP-user direct link monotonically increases with the SINR target whereas that from the combined channel decreases as SINR target increases.
In Figs. \[simulation:pow2\] (c) and (d), we plot $\rho_1$ and $ \rho_2$, respectively, for both the cases with and without the IRS. For user 1, it is observed in Fig. \[simulation:pow2\] (c) that for low SINR (noise-limited) regime, the AP in the case with the IRS steers its beam direction toward the AP-user direct channel as in the case without the IRS (i.e., MRT beamforming), while for high SINR (interference-limited) regime where the beam direction in the latter case is adjusted to null out the interference to user 2 (i.e, ZF beamforming), the former case still keeps a high correlation between user 1’s beam direction and the corresponding AP-user direct channel. This is expected since the IRS is not able to enhance the desired signal for user 1 (see Fig. \[simulation:pow2\] (a)), thus keeping a high $\rho_1$ helps reduce the transmit power for serving user 1. However, user 2 inevitably suffers more interference from user 1 in the AP-user direct link (see Fig. \[simulation:pow2\] (b)), which, nevertheless, is significantly suppressed at user 2, thanks to the IRS-assisted interference cancellation. In contrast, from Fig. \[simulation:pow2\] (d) it is observed that the user 2’s beam direction in the case with the IRS is steered toward the AP-user combined channel rather than the AP-user direct channel at low SINR regime, and then converges to the same ZF beamforming as in the case without the IRS at high SINR regime. This is expected since the IRS is not able to help cancel the AP-user direct interference at user 1, thus nulling out the interference caused by user 2 is the most effective way for meeting the high SINR target. From the above, it is concluded that the transmit beamforming directions for the users with the IRS are drastically different from those in the case without the IRS, depending on their different distances with the IRS. The above results further demonstrate the necessity of jointly optimizing the transmit beamforming and phase shifts in IRS-aided multiuser systems.
![AP’s transmit power versus the Rician factor of the AP-IRS channel ${\bm G}$.[]{data-label="Figures:K"}](pow_rician_factor){width="55.00000%"}
### AP Transmit Power versus Rician Factor of ${\bm G}$
In Fig. \[Figures:K\], we plot the AP’s transmit power required by the proposed algorithms with IRS against the benchmark schemes without IRS versus the Rician factor of the AP-IRS channel by assuming all eight users are active and setting $M=8$, $N=40$, and $\gamma = 10$ dB. One can observe that when the average power of the LoS component is comparable to that of the fading component (e.g., $\beta_{\rm AI} \geq -10$ dB), the transmit power of both proposed algorithms increases with the increasing Rician factor, for meeting the same set of SINR targets. This is because for users near the IRS, the reflected signals generally dominate the signals from AP-user direct links. As such, a higher Rician factor of ${\bm G}$ results in higher correlation among the combined channels of these users, which is detrimental to the spatial multiplexing gain due to the more severe multiuser interference. The implication of this result is that (somehow surprisingly) it is practically favorable to deploy the IRS in a relatively rich scattering environment to avoid strong LoS (low-rank ${\bm G}$) with the AP, so as to serve multiple users by the IRS which requires sufficient spatial degrees of freedom from the AP to the IRS.
### [Comparison with Massive MIMO]{}
[To compare with the existing TDD-based massive MIMO system (without the use of IRS), we consider there are sixteen users with eight users randomly distributed within $60$ m from the AP and the other eight users randomly distributed within $6$ m from the IRS. Other channel parameters are set to be the same as those for Fig. \[transmit:pow\]. To facilitate the channel estimation, we assume that the IRS is equipped with receive RF chains. The transmission protocol for the IRS-aided wireless system is described as follows: 1) all the users send orthogonal pilot signals concurrently as in the TDD-based massive MIMO system; 2) the AP and the IRS estimate the AP-user and IRS-user channels, respectively[^4]; 3) the AP starts to transmit data to the users and in the meanwhile sends its estimated AP-user channels to the IRS controller via a separate control link (see Fig. \[system:model\]), so that the IRS can jointly optimize the AP transmit beamforming vectors and its phase shifts by using the proposed algorithms in this paper; 4) the IRS controller sends optimized transmit beamforming vectors to the AP and sets its phase shifts accordingly; and 5) the AP and IRS start to transmit data to the users collaboratively. As such, different from the massive MIMO system, the IRS-aided system generally incurs additional delay in steps 3) and 4) due to information exchange and algorithm computation. We denote the channel coherence time as $T_c$ and the total delay caused by 3) and 4) as $\tau$, where we assume that $\tau<T_c$. The delay ratio is thus given by $\rho=\frac{\tau}{T_c}$. Since steps 1) and 2) are required in both the IRS-aided system and massive MIMO system, the time overhead for channel training is omitted for both schemes for a fair comparison. Note that users are served by the AP only over $\tau$ with the achievable SINR of user $k$, $k\in \mathcal{K}$, denoted by $\text{SINR}^1_k$, while when they are served by both the AP and IRS during the remaining time of $T_c-\tau$, the SINR of user $k$ is denoted by $\text{SINR}^2_k$. Accordingly, the achievable rates of user $k$ in the above two phases can be expressed as $R^1_{k}=\log_2( 1+ \text{SINR}^1_k )$ and $R^2_{k}=\log_2( 1+ \text{SINR}^2_k )$ in bps/Hz, respectively. The average achievable rate of user $k$ over $T_c$ is thus given by $r_k=\frac{ \tau}{T_c} R^1_{k} + (1-\frac{\tau}{T_c}) R^2_{k}= \rho R^1_{k} + (1-\rho) R^2_{k}$. Given the transmit power constraint at the AP, we aim to maximize the minimum achievable rate of $r_k$ among all the users, which is a dual problem of (P1) and thus can be solved by using the proposed alternating optimization algorithm together with an efficient bisection search over the AP transmit power [@wiesel2006linear]. For the case of massive MIMO, we adopt the optimal MMSE-based transmit beamforming at the AP. ]{}
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[In Fig. \[rate:power\], we show the achievable max-min rate versus the transmit power at the AP for the ideal case with negligible delay (i.e., $\rho=0$) in the IRS-aided system, which provides a throughput upper bound for practical implementation. From Fig. \[rate:power\], it is observed that a hybrid deployment of an AP with $M=20$ active antennas and an IRS with $N=80$ passive reflecting elements achieves nearly the same performance as deploying only an AP with $M=50$ but without using the IRS, which implies that the IRS is indeed effective in reducing the number of active antennas required in conventional massive MIMO. To take into account the practical delay due to IRS-AP coordination, we show in Fig. \[rate:delay:tradeoff\] the achievable max-min rate versus the delay ratio $\rho $ by fixing the transmit power at the AP as 15 dBm. It is observed that as the delay ratio increases, the achievable rate of the IRS-aided system decreases since users are served by a relatively small MIMO system (with $M=20$) for more time before the IRS is activated for signal reflection. However, it is also observed that even with a large delay of $0.18T_c$, the IRS-aided system with $M=20$ and $N=80$ can still achieve better performance than the AP-only system with $M=40$. This validates the practical throughput gain of IRS even by taking into account a moderate delay for its coordination with the AP. ]{}
Conclusions
===========
In this paper, we proposed a novel approach to enhance the spectrum and energy efficiency as well as reducing the implementation cost of future wireless communication systems by leveraging the passive IRS via smartly adjusting its signal reflection. Specifically, given the user SINR constraints, the active transmit beamforming at the AP and passive reflect beamforming at the IRS were jointly optimized to minimize the transmit power in an IRS-aided multiuser system. By applying the SDR and alternating optimization techniques, efficient algorithms were proposed to trade off between the system performance and computational complexity. It was shown for the single-user system that the receive SNR increases quadratically with the number of reflecting elements of the IRS, which is more efficient than the conventional massive MIMO or multi-antenna AF relay. While for the multiuser system, it was shown that IRS-enabled interference suppression can be jointly designed with the AP transmit beamforming to improve the performance of all users in the system, even for those that are far away from the IRS. Extensive simulation results under various practical setups demonstrated that by deploying the IRS and jointly optimizing its reflection with the AP transmission, the wireless network performance can be significantly improved in terms of energy saving, coverage extension as well as achievable rate, as compared to the conventional setup without using the IRS such as the massive MIMO. Useful insights on optimally deploying the IRS and its delay-performance trade-off were also drawn to provide useful guidance for practical design and implementation.
[In practice, if IRS is equipped with receive RF chains, the commonly used pilot-assisted channel estimation methods can be similarly applied to the IRS as shown in Section V-B; otherwise, it is infeasible for the IRS to directly estimate the channels with its associated AP/users. For the latter (more challenging) case, a viable approach may be to design the IRS’s passive beamforming based on the feedback from the AP/users that receive the signals reflected by the IRS, which is worth investigating in the future work. In addition, after this paper was submitted, we became aware of another parallel work [@huang2018large], which shows that the IRS-aided wireless system is more energy-efficient than the conventional multi-antenna AF relay system with HD operation. Although the spectrum efficiency can be further improved by using the FD AF relay, effective SIC is required, which incurs additional energy consumption. As such, it is worthy of further comparing the energy efficiency of IRS with the FD AF relaying in future work.]{}
[^1]: [Although using devices like MEMs as mentioned previously to adjust the phase shifts at the IRS requires some power consumption, it is practically negligible as compared to the much higher transmit power of active communication devices.]{}
[^2]: To characterize the fundamental performance limits of IRS, we assume that the phase shifts can be continuously varied in $[0, 2\pi)$, while in practice they are usually selected from a finite number of discrete values from 0 to $2\pi$ for the ease of circuit implementation. The design of IRS with discrete phase shifts is left for our future work [@JR:wu2019discreteIRS; @wu2018IRS_discrete].
[^3]: In practice, each element of the IRS is usually designed to maximize the signal reflection. Thus, we set $\beta_n=1, \forall n$, in the sequel of this paper for simplicity. Note that this scenario is different from the traditional backscatter communication where the RFID tags usually need to harvest a certain amount of energy from the incident signals for powering their circuit operation and thus a much smaller amplitude reflection coefficient than unity is resulted in general.
[^4]: Since in practice the AP and IRS are deployed at fixed locations, we assume for simplicity that the channel ${\bm G}$ between them is quasi-static and changes much slower as compared to the AP-user and IRS-user channels, and thus is constant and known for the considered communication period of interest.
|
---
abstract: 'Although the parent iron-based pnictides and chalcogenides are itinerant antiferromagnets, the use of local moment picture to understand their magnetic properties is still widespread. We study magnetic Raman scattering from a local moment perspective for various quantum spin models proposed for this new class of superconductors. These models vary greatly in the level of magnetic frustration and show a vastly different two-magnon Raman response. Light scattering by two-magnon excitations thus provides a robust and independent measure of the underlying spin interactions. In accord with other recent experiments, our results indicate that the amount of magnetic frustration in these systems may be small.'
address:
- '$^1$Stanford Institute for Materials and Energy Science, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA'
- '$^2$Department of Physics, Stanford University, Stanford, California 94305, USA'
- '$^3$Department of Applied Physics, Stanford University, Stanford, California 94305, USA'
- '$^4$Department of Physics, University of California, Davis, California 95616, USA'
- '$^5$Geballe Laboratory for Advanced Materials, Stanford University, California 94305, USA'
author:
- 'C.-C. Chen$^{1,2}$'
- 'C. J. Jia$^{1,3}$'
- 'A. F. Kemper$^{1}$'
- 'R. R. P. Singh$^4$'
- 'T. P. Devereaux$^{1,5}$'
title: 'Theory of two-magnon Raman scattering in iron pnictides and chalcogenides'
---
Since their discovery in 2008 [@Hosono; @MKWu], iron-based superconductors have led to extensive studies in the condensed matter field. Like the high-$T_c$ cuprate superconductors, the iron pnictides and chalcogenides are quasi-two-dimensional materials with a layered structure. Both the copper and iron based superconductors exhibit antiferromagnetic (AF) order in the parent phases, and superconductivity emerges by suppressing magnetism upon doping. However, there are also clear differences. At stoichiometry, the cuprates are Mott insulators and can be described by an effective single-band model. On the other hand, the iron-based pnictide and chalcogenide parent compounds are multi-orbital systems and remain metallic even in the AF state. The description of magnetism and its interplay with superconductivity in this new class of superconductors remain an active area of research.
There is growing evidence that weak coupling theories with varying degrees of sophistication and input from density functional theory (DFT) calculations can quantitatively describe several experimental findings [@Mazin; @Kuroki; @Kemper]. However, many aspects of the magnetic properties in these materials are also remarkably captured by a local moment perspective. These include the phase diagram with orthorhombic distortion and antiferromagnetic order, temperature dependence of uniform susceptibility, and neutron studies of spin-wave dispersion throughout the Brillouin zone (BZ). It also has been shown that a local moment model can reproduce the essence of magnetism from DFT calculations [@Antropov]. There are various reasons why a strong coupling perspective is still relevant. It provides the potential for a unified framework for understanding high-$T_c$ superconductivity derived from electron-electron interactions. More importantly, *Local moment models are being widely used in experiments* to explain magnetic properties in these systems [@Dai_Ca122; @Ba122_Ewings; @Sr122_Ewings; @Sugai_122_1; @Sugai_122_2; @Neutron_Dai_FeTe; @Sugai_FeTe].
In the pnictides, proposed theories for their collinear AF phase based on localized moments include the spatially anisotropic $J_{1a}$-$J_{1b}$-$J_2$ model [@J1abJ2_1; @J1abJ2_2], where the coupling is ferromagnetic (FM) in one direction and AF in the other, and the strongly frustrated $J_1$-$J_2$ model [@J1J2_1; @J1J2_2; @J1J2_3; @J1J2_4], where collinear antiferromagnetism arises via order by disorder (see Fig. 1). Similarly, the diagonal double stripe AF order in iron chalcogenides can be obtained by either invoking strong frustration in a $J_1$-$J_2$-$J_3$ model [@Hu_J1J2J3] or utilizing a model with strong spatial anisotropy stemming from orbital order [@Ashvin_SO]. These Hamiltonians can lead to ground states of the same broken symmetry, but they vary greatly in the degree of magnetic frustration. It is necessary to distinguish these scenarios and narrow the possible models for the iron-based superconductors.
The preceding difficulty in some cases could be resolved by inelastic neutron scattering (INS) experiments. Indeed, the INS spin wave spectrum of CaFe$_2$As$_2$ [@Dai_Ca122] has a local maximum at momentum $(\pi,\pi)$ (in the single iron BZ), strongly favoring the $J_{1a}$-$J_{1b}$-$J_2$ model. However, as high energy spin wave signals are strongly damped, interpretations of the neutron results remain disputed [@Neutron_itinerant]. Moreover, there are also cases where different Hamiltonians can lead to virtually indistinguishable spin wave spectra. Therefore, further studies based on different measurements appear crucial.
One way to achieve the above goal is by studying magnetic Raman scattering [@RMP]. Light scattering by two-magnon flips is dominated by short-range excitations and sensitive to details of the exchange couplings. This probe was instrumental in the first accurate determination of the exchange constants in cuprate superconductors [@Raman_cuprates].
![ (Color online) Schematics of quantum spin models that lead to (a) the $(\pi,0)$ collinear AF order in iron pnictides, and (b) the $(\pi/2,\pi/2)$ diagonal double stripe AF order in iron chalcogenides. []{data-label="fig:Cartoon"}](Cartoon.eps){width="\columnwidth"}
In the present study, following the Fleury-Loudon (FL) formalism [@FL_formalism; @SS_formalism], we investigate the two-magnon Raman spectra for strongly frustrated models and spatially anisotropic models that are unfrustrated. Using exact diagonalization (ED) and spin wave theory, we find that in strongly frustrated models the two-magnon peak appears at relatively low energies. Strong frustration implies abundant low-energy configurations, which thereby provides symmetry-dependent pathways for low-energy two-magnon flips. In contrast, in unfrustrated models the two-magnon peak occurs at a higher energy, closer to twice the single-magnon bandwidth. Magnetic Raman scattering thus can be a robust and independent measure of the underlying spin interactions.
We study quantum spin models on a square lattice: $$\label{eq:Model}
H=\sum_{ij}\frac{J_{ij}}{2}\mathbf{S}_i\cdot\mathbf{S}_{j},$$ where $J_{ij}$ are short range exchange couplings up to the third nearest-neighbor (NN). Depending on the interaction parameters, the model can support ground states of different broken symmetries, such as the $(\pi,0)$ collinear AF order or the $(\pi/2,\pi/2)$ diagonal double stripe order. The FL Raman scattering operator $\hat{\cal{O}}$ is given by [@RMP] $$\label{eq:FL_operator}
\hat{\cal{O}}=\sum_{i,j} \eta_{ij} (\hat{\mathbf{e}}_{in}\cdot \hat{\mathbf{d}}_{ij})(\hat{\mathbf{e}}_{out}\cdot \hat{\mathbf{d}}_{ij})\mathbf{S}_i\cdot\mathbf{S}_j,$$ where the coupling strengths $\eta_{ij}$ are proportional to the exchange interactions $J_{ij}$, $\hat{\mathbf{d}}_{ij}$ is the unit vector that connects lattice sites $i$ and $j$, and $\hat{\mathbf{e}}_{in}$ and $\hat{\mathbf{e}}_{out}$ are the incident and scattered photon polarizations, respectively. Specifically, we will consider the following light-polarization geometries: $$\begin{aligned}
B_{1g}:&&\hat{\mathbf{e}}_{in}=\frac{1}{\sqrt{2}}(\hat{x}+\hat{y}),\hat{\mathbf{e}}_{out}=\frac{1}{\sqrt{2}}(\hat{x}-\hat{y}),
\nonumber\\
A_{1g}^{\prime}:&&\hat{\mathbf{e}}_{in}=\frac{1}{\sqrt{2}}(\hat{x}+\hat{y}),\hat{\mathbf{e}}_{out}=\frac{1}{\sqrt{2}}(\hat{x}+\hat{y}),\end{aligned}$$ where we note that $A^{\prime}_{1g}$ transforms as $A_{1g}\oplus B_{2g}$ [@RMP]. With a given scattering operator $\hat{\cal{O}}$, the two-magnon Raman cross-section is given by $R(\omega)=-(1/\pi)\textrm{Im}[I(\omega)]$, where $$\label{eq:Raman}
I(\omega)\equiv \langle \Psi_0 \vert \hat{\cal{O}}^\dagger \frac{1}{\omega+E_0+i\epsilon -H} \hat{\cal{O}}\vert\Psi_0\rangle.$$ Here $\vert \Psi_0\rangle$ is the ground state with energy $E_0$. Below we perform spin one-half ($S=1/2$) calculations in systems with a $(\pi,0)$ collinear AF order, and spin one ($S=1$) calculations in systems with a $(\pi/2,\pi/2)$ diagonal double stripe order. These calculations are relevant since in the pnictides the Fe moment is unexpectedly small (ranging from 0.3 to 0.9 $\mu_B$), while in the chalcogenides the moments ($\sim 2\mu_B$) are much larger [@David_Review]. For the $S=1/2$ case, we perform ED calculations on square lattices of 16, 32, and 36 sites. For the $S=1$ case, we focus on a 16-site cluster as the Hilbert space is substantially larger. We further compare the calculations with spin wave theory and comment on the effects of $S$.
We first discuss the case of ($\pi,0$) AF order. We consider (i) a frustrated $J_1-J_2$ model with $J_1=J_2$ and (ii) the spatially anisotropic $J_{1a}$-$J_{1b}$-$J_2$ model, as depicted in Fig. \[fig:Cartoon\](a). For the latter we take $J_{1b}=-0.1J_{1a}$ and $J_2=0.4J_{1a}$ as reported by INS measurements on CaFe$_2$As$_2$ [@Dai_Ca122]. In this case, even though $(J_{1a}+J_{1b})$ is comparable to $2J_2$, the system remains unfrustrated.
Figure \[fig:Raman\_CAF\] shows the two-magnon Raman spectra. In the 36-site ED calculations, the $J_1$-$J_2$ model has a $B_{1g}$ peak at $\sim SJ_1$, while in the $A_{1g}^{\prime}$ channel it is at $\sim 6SJ_1$. On the other hand, for the $J_{1a}$-$J_{1b}$-$J_2$ model the two-magnon excitation appears at $\sim 7SJ_{1a}$ in both polarizations. We note that while finite-size effects give small corrections to the Raman resonance energy, the clear qualitative difference in the $B_{1g}$ channel between the two models is robust, depending weakly on cluster sizes.
The sharp distinction in the $B_{1g}$ two-magnon response is related to the difference in the amount of magnetic frustration. In both models, the single-magnon bandwidth is roughly the same [@spin-orbital]; however, their magnon spectral weights are rather distinct. In the $J_{1a}$-$J_{1b}$-$J_2$ model, the magnon density of states (DOS) diverges at the top of the single-magnon band, similar to that in the Heisenberg model. On the other hand, in the frustrated $J_1$-$J_2$ model a large spectral weight is distributed at lower energies [@impurity]. In general, a substantial low-energy magnon DOS as implied by frustration provides pathways for low-energy two-magnon flips [@triangular_1; @triangular_2].
The low-energy $B_{1g}$ mode of the $J_1$-$J_2$ model can be related to the proximity of the system to a disordered phase characterized by either a quantum spin liquid or a valence bond solid (VBS). At a transition to a phase lacking magnetic order, one expects singlet modes with appropriate quantum numbers to soften to zero energy [@Lhuillier]. Close to such a critical point low energy excitations would appear. For a quantum spin liquid, low energy singlets may appear in all light polarizations, while for a VBS phase low energy singlets may appear only in specific channels. Our finding of a low-energy excitation in $B_{1g}$ and not in $A^{\prime}_{1g}$ implies a nearby VBS phase with columnar dimers, since this state has the right symmetry [@Dimer; @ED_36].
![ (Color online) The Fleury-Loudon two-magnon Raman cross-sections calculated with $S=1/2$ exact diagonalization. In the frustrated $J_1$-$J_2$ model, the $B_{1g}$ two-magnon excitation appears at $\sim SJ_1$, an energy much lower than twice the single-magnon bandwidth. []{data-label="fig:Raman_CAF"}](Raman_CAF.eps){width="\columnwidth"}
To compare the calculations to experiments, we convert our results into physically measurable units. By fitting to the $J_{1a}$-$J_{1b}$-$J_2$ model, an $SJ_{1a}\sim50$ meV has been reported by neutron scattering experiments on CaFe$_2$As$_2$ [@Dai_Ca122]. Therefore, the 36-site ED calculation indicates a $B_{1g}$ two-magnon peak at $7.4J_{1a}\sim3000$ cm$^{-1}$. Anisotropic exchange couplings are also found in BaFe$_2$As$_2$ [@Ba122_Ewings] and SrFe$_2$As$_2$ [@Sr122_Ewings], with a slightly different energy scale. Even with a material dependence and possible corrections due to finite-size calculations, the two-magnon peak in an unfrustrated $J_{1a}$-$J_{1b}$-$J_2$ model is clearly at an energy several times higher than $SJ_{1a}$. On the other hand, for a $J_1$-$J_2$ model with the same exchange energy scale, a $B_{1g}$ two-magnon peak occurs around a few hundred wavenumbers. As in the cuprates [@J1J2Raman_1990], the absence of a low-energy $B_{1g}$ two-magnon peak in recent experiments implies that magnetic frustration in the pnictides might also be small [@Sugai_122_1; @Sugai_122_2].
To further understand the Raman cross-sections, we discuss results based on spin wave theory. In the AF Heisenberg model, linear spin-wave theory indicates a $B_{1g}$ two-magnon peak located at $8SJ$, twice the energy of the single-magnon bandwidth. This value is appropriate only in the classical $S=\infty$ limit, where magnon-magnon interactions can be neglected. In systems with a finite $S$, two magnons are bound to each other locally in space, thereby shifting the two-magnon energy to a lower value. When interaction effects are treated at the mean-field level, the two-magnon Raman response takes the following RPA form: $R(\omega) \sim \textrm{Im} \left[I(\omega)/(1+\frac{\bar{J}}{S}I(\omega))\right]$, where $I(\omega)$ \[defined in Eq. (\[eq:Raman\])\] is calculated within the linear spin wave approximation, and $\bar{J}$ is the mean-field interaction characteristic to the problem. In the $S=1/2$ AF Heisenberg model, the $B_{1g}$ two-magnon energy is renormalized to $\sim 6SJ$. Similarly, in the $S=1/2$ ED calculations for the $J_{1a}$-$J_{1b}$-$J_2$ model, the two-magnon excitation in the $B_{1g}$ channel occurs at $\sim 7SJ_{1a}$, again understood by an RPA renormalization of its classical $B_{1g}$ two-magnon peak at $8SJ_{1a}$. In frustrated models, however, the correspondence between the classical $B_{1g}$ two-magnon energy and twice the single-magnon bandwidth fails to describe the Raman spectra. Based on linear spin wave theory, while the top of the single magnon band is roughly $6SJ_1$ in the $J_1$-$J_2$ model, its classical $B_{1g}$ two-magnon energy occurs at zero energy. This zero energy two-magnon resonance is closely related to the zero mode in the spin wave spectrum at momentum $(\pi,\pi)$, which is known to be shifted to a finite energy by quantum fluctuations. The spin-wave theory calculation agrees well with the ED results shown in Fig. \[fig:Raman\_CAF\], where the $B_{1g}$ two-magnon energy $\sim SJ_1$ is much closer to zero, rather than twice the single-magnon bandwidth. Moreover, our $S=1$ 16-site ED calculation indicates that the $B_{1g}$ two-magnon peak moves further down to $\sim0.7SJ_1$. The distinction in the two-magnon response thus serves as a clear benchmark to distinguish the $J_1$-$J_2$ and the $J_{1a}$-$J_{1b}$-$J_2$ models.
We next focus on situations where the model Hamiltonian \[Eq. (\[eq:Model\])\] has a $(\pi/2,\pi/2)$ AF ground state as observed in iron chalcogenides [@p2p2_1; @p2p2_2]. Like the case of the pnictides, below we discuss two models that have a similar single-magnon bandwidth but vary in the amount of magnetic frustration.
In the first case we consider a system which is spatially anisotropic but unfrustrated. We use parameters from DFT calculations [@J1abJ2_2] (in units of meV): $SJ_{1a}=-7.6$, $SJ_{1b}=-26.5$, $SJ_{2a}=46.5$, and $SJ_{2b}=-34.9$ \[see Fig. \[fig:Cartoon\](b)\]. In the second frustrated case, we use parameters from Fe$_{1.05}$Te INS measurements [@Neutron_Dai_FeTe] (in units of meV): $SJ_{1a}=-17.5$, $SJ_{1b}=-51.0$, $SJ_{2a}=21.7$, $SJ_{2b}=21.7$, and $SJ_3=6.8$. In the latter case a strong frustration is present, as the NN interactions in both crystal-axes are FM, and the second NN exchanges are all AF, independent of spin sub-lattice. In both cases, $SJ$ below is referred to as the energy scale of the dominant coupling.
Figure \[fig:Raman\_FeTe\] shows the two-magnon Raman spectra from 16-site $S=1$ ED calculations. In the unfrustrated system, there is no low-energy two-magnon excitation. The two-magnon Raman spectra start to show spectral features between $6\sim 8 SJ$, and the cross-sections are dominated by peaks located at $\sim 11 SJ$ and $10.5 SJ $ respectively in the $B_{1g}$ and $A^{\prime}_{1g}$ polarizations \[Fig. \[fig:Raman\_FeTe\](a) and (b)\]. In this unfrustrated case, the two-magnon peak is expected to occur at an energy close to twice the single-magnon bandwidth \[see Fig. \[fig:Spinwave\_FeTe\]\]. On the other hand, when the system is frustrated there is a substantial low-energy two-magnon weight between $1\sim3 SJ$ \[Fig. \[fig:Raman\_FeTe\](c) and (d)\] These results show a clear difference in the two-magnon Raman response between the two models.
![ (Color online) Two-magnon Raman spectra for models with a $(\pi/2,\pi/2)$ diagonal double stripe AF order. In both cases $SJ$ is the energy of the dominant exchange coupling. We note the plots have a different energy scale. []{data-label="fig:Raman_FeTe"}](Raman_FeTe.eps){width="\columnwidth"}
As mentioned previously, the distinction in the two-magnon Raman spectra is not caused by a difference in the single-magnon bandwidth. As shown in Fig. \[fig:Spinwave\_FeTe\], the magnon dispersion from spin wave theory [@Hu_J1J2J3; @Neutron_Dai_FeTe] and the dynamical structure factor $S^{zz}(\mathbf{q},\omega)$ from ED calculations both indicate a similar magnon bandwidth of the two models. However, in a frustrated model the magnon DOS is no longer dominated by zone boundary magnons, and an estimate based on twice the magnon bandwidth fails to describe the two-magnon Raman profile.
We last note that recent Raman scattering experiments do not suggest the existence of low-energy two-magnon excitations in the chalcogenides [@Sugai_FeTe]. This implies that weakly frustrated but spatially anisotropic models based on orbital order [@Ashvin_SO] might be relevant. When there is an ambiguity or freedom in fitting the neutron scattering spin wave spectra to localized model models, additional experiments such as Raman scattering can be helpful for pinning down the interaction parameters.
In summary, we have calculated the two-magnon Raman spectra for various spin models proposed for the iron-based superconductors. We have shown that a distinct two-magnon Raman response can result between models that vary in the level of magnetic frustration. Complementary to neutron scattering, magnetic Raman scattering detects short-wavelength spin fluctuations and can serve as an independent measure of the underlying spin interactions. Together with recent experiments, our results favor spatially anisotropic models, implying that the amount of magnetic frustration in the pnictides and chalcogenides is small.
![ (Color online) $S^{zz}(\mathbf{q},\omega)$ \[blue peaks\] from ED and the magnon dispersion spectra \[red curves\] from spin wave theory. Here the momentum points are (in the single iron BZ): $\Gamma =(0,0)$, $X=(\pi,0)$, and $M=(\pi,\pi)$. []{data-label="fig:Spinwave_FeTe"}](Spinwave_FeTe.eps){width="\columnwidth"}
The authors acknowledge discussions with M. Bernhard, R. Hackl, M. Gingras, J.-H. Chu, and P. Hubbard. CCC, CJJ, AFK, and TPD are supported by the U.S. DOE under Contract No. DE-AC02-76SF00515. CJJ is also supported by the Stanford Graduate Fellowships Program. RRPS is supported by NSF Grant No. DMR-1004231. This research used resources of NERSC, supported by U.S. DOE under Contract No. DE-AC02-05CH11231.
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abstract: 'Ferromagnetic proximity effect is studied in InAs nanowire (NW) based quantum dots (QDs) strongly coupled to a ferromagnetic (F) and a superconducting (S) lead. The influence of the F lead is detected through the splitting of the spin-1/2 Kondo resonance. We show that the F lead induces a local exchange field on the QD, which has varying amplitude and sign depending on the charge states. The interplay of the F and S correlations generates an exchange field related subgap feature.'
author:
- 'L. Hofstetter$^{1,*}$, S. Csonka$^{1,2,*}$, A. Geresdi$^{2}$, M. Aagesen$^{3}$, J. Nyg[å]{}rd$^{3}$ and C. Schönenberger$^{1}$'
title: 'Ferromagnetic proximity effect in F-QDot-S devices'
---
Superconductor-ferromagnet (S/F) heterostructures have been investigated extensively in the past and they have potential impact on the field of spintronics and quantum computation. Due to the different spin ordering of textcolor[red]{}[a]{} s-wave superconductor and a ferromagnet, a large variety of interesting phenomena can be studied at a S/F interface: probing the spin polarization by Andreev reflection [@Soulen_1998], $\pi$-junction behaviour [@Kontos_2001; @Zareyan_2001], ferromagnetically induced triplet superconductivity [@Keizer_2006] or ferromagnetically assisted Cooper pair splitting [@Beckmann_2004].
Here we focus on a novel S/F hybrid device where a quantum dot (QD) is incorporated between a S and a F lead. Recently, the modification of the Andreev reflection of such hybrid sytems has been calculated [@Feng2003; @Cao2004]. Also configurations, where a QD is strongly coupled to two F leads (F-QDot-F) attracted increasing theoretical [@Martinek2003; @Choi2004; @Cottet2006a; @Cottet2006b] and experimental [@Sahoo2005; @Pasupathy2004; @Hamaya2007; @Hauptmann2008] attention. It has been verified [@Pasupathy2004; @Hamaya2007; @Hauptmann2008], that in this situation the spin-$\uparrow$ and $\downarrow$ energy levels of the QD can split by an exchange energy, $E_{ex}$ due to the hybridization between the QD states and the F lead. This proximity ferromagnetism has the same effect on the QD as the presence of an external magnetic field, $B$. Thus, the exchange splitting is often characterized by the so-called local magnetic exchange field ($B_{ex}$) in the literature, where $E_{ex} = g\mu_B B_{ex}$. As has been proposed by Martinek $et \, al.$ [@Martinek2005], $B_{ex}$ can be gate dependent and even change sign. Hence, electric controlled reversal of the spin occupation of the QD can be achieved. This has very recently been shown in carbon nanotubes with two F contacts [@Hauptmann2008].
In this work, we show the influence of F and S correlations on the transport properties of F-QD-S devices. We demonstrate that already a single F lead introduces a local exchange field which strongly varies for different QD levels. Furthermore, it even can change its sign within the same charge state. On the other hand, the presence of the S contact serves as a spectroscopic tool. Concentrating on the subgap transport, a novel mini-gap feature is observed, which is related to $B_{ex}$.\
A typical device is shown in Fig. 1a. We use high-quality molecular beam-epitaxy grown InAs nanowires (NWs). NWs are dissolved in IPA and deposited on doped Si substrates with $400$ nm insulating SiO$_2$. Afterwards, ohmic contacts at spacings of $300-500$ nm are fabricated. The F contacts consist of a Ni/Co/Pd trilayer (15 nm/80 nm/10 nm), whereas the S ones consist of a Ti/Al bilayer (10 nm/110 nm). Prior to metal evaporation, argon gun sputtering is used to remove the native oxide layer from the nanowires. In agreement with our previous measurements, QDs form between these contacts, which can be tuned by the voltage applied on the backgate, $V_{BG}$ [@Csonka2008; @Jespersen2006]. Measurements are performed at a temperature of $25$ mK. A $B$ field is applied parallel to the easy axis of the F contact which is defined by shape anisotropy (see Fig. 1a). The $B$ field allows to switch the orientation of the F stripe and to control the size of the superconducting gap. To magnetize the F lead the external field is ramped to $B=+300$ mT and then back to $B=0$ mT before measuring.
In Fig. 1b the energy diagram of our devices is shown schematically. The F lead induces an asymmetry of the tunnel coupling ($\Gamma$) of spin-$\uparrow$ and spin-$\downarrow$ electrons to the QD, described by the tunneling spin polarization $P=(\Gamma_{\uparrow}-\Gamma_{\downarrow})/(\Gamma_{\downarrow}+\Gamma_{\uparrow})$. This asymmetry is caused by the difference of the spin-$\uparrow$ and spin-$\downarrow$ electron density of the F lead at the Fermi energy and by the tunneling matrix elements of these electrons. E.g. in Ni at $E_F$ two bands couple with opposite spin-imbalance and very different tunneling matrix elements to the QD [@Mazin1999]. By hybridization, the spin dependent tunnel coupling generates a spin imbalance on the QD, described as an exchange field. The S lead is represented by the BCS density of states (DOS) with its energy gap, $\Delta$.
![(Color online)(a) A SEM picture of a device. The right lead is a Ti/Al bilayer superconductor, while the left longer one is a Ni/Co/Pd trilayer ferromagnet. The $B$ field is applied parallel to easy axis of the F lead. (b) Schematic view of the F-QDot-S system. The spin degeneracy on the QD is lifted by an exchange splitting induced by the ferromagnetic proximity effect. (c) Due to the spin polarized charge fluctuations, the spin ground state of the QD is opposite when the QD occupation fluctuates between 1 and 0 ($\epsilon_d \approx 0$) or 1 and 2 ($\epsilon_d \approx -
U$). []{data-label="Figure1"}](Figure1.eps)
In order to investigate the ferromagnetic proximity effect, the QD is operated in the strongly coupled regime, which allows the study of the cotunneling induced manybody spin-1/2 Kondo resonance [@Goldhaber-Gordon1998] at odd occupation number of the QD. Kondo resonances split into a doublet in a $B$ field according to the Zeeman energy $E_{\downarrow/\uparrow} = \mp 1/2 g\mu_B B$, thus being also a sensitive tool to visualize $B_{ex}$. Probing the exchange splitting by the Kondo effect has been demonstrated in C60-molecules [@Pasupathy2004] and carbon nanotubes based QDs [@Hauptmann2008]. InAs NW QDs have the advantage that the g-factor can be comparable to the bulk value of g$\approx$15 [@Csonka2008], making them particularly sensitive for studying the level splitting created by $B_{ex}$. Much smaller external fields are needed to access the regime, where the exchange energy and the Zeeman splitting are comparable.
The ferromagnetic proximity effect and the therewith connected ground state transition on the QD can be described in a simple model. Using perturbative scaling analysis for a flatband bandstructure with spin dependent tunneling rates and including finite Stoner splitting in the leads an analytical formula for the energy splitting of the spin-$\uparrow$ and the spin-$\downarrow$ is found [@Martinek2005]: $$eV_{sd} = g\mu_B B + \Delta_0 + (P\Gamma / \pi)ln(|\epsilon_d| / |U + \epsilon_d |).$$ Here $P$ is as defined earlier, $\Gamma$ is the coupling to the F lead ($\Gamma=\Gamma_\uparrow+\Gamma_\downarrow$), $U$ the charging energy of the QD and $\epsilon_d$ the level position of the QD, tunable by $V_{BG}$. $g \mu_B B$ is the Zeeman splitting due to an external magnetic field, $\Delta_0$ is a Stoner splitting induced shift. Elaborated NRG calculations also support the result of Eq. 1 [@Martinek2005]. Interestingly, based on Eq. 1, the spin ground state of the QD is different for $\epsilon_d$ close to $0$ than for $\epsilon_d$ close to $-U$, if $\Delta_0$ or the Zeeman term are not too big. This change in the ground state of the QD can be explained by the charge fluctuations between the QD and the F lead (see Fig. 1c). Electrons with majority tunneling spin orientation dominate the charge fluctuations. Thus, when the QD occupation fluctuates between $1$ and $0$ ($\epsilon_d \approx 0$), the majority tunneling spin occupies the QD preferably. However, when the occupation fluctuates between 1 and 2 ($\epsilon_d \approx - U$), the remaining (non fluctuating) spin on the QD has the minority spin orientation of the tunneling electrons [@Martinek2005; @Hauptmann2008]. The transition between these opposite ground states is described by the sign change of the exchange field in Eq. 1. However, if $\Delta_0$ is the dominant term, a roughly constant splitting of the Kondo resonance is expected within a charge state.
![(Color online)(a) Differential conductance as a function of $V_{BG}$ and $V_{sd}$ of a F-QD-S device at $B=0$T. $B_{ex}$ modifies the Kondo resonances (odd numbered states) differently. The S lead induces peaks in the conductance at $V_{sd}=\pm \Delta$. $B$ dependence of different charge states: (b) with no signature of $B_{ex}$ (state 7 in Fig. 2a), (c) with $B_{ex} < 0$, which is compensated by external $B$ (state 3 in Fig. 2a), (d) with $B_{ex} > 0$, which is enhanced by $B$. Panel d is measured in a charge state at $V_{BG}=2.31$ V.[]{data-label="Figure2"}](Figure2.eps)
In Fig. 2a the differential conductance as a function of $V_{BG}$ and source drain voltage $V_{sd}$ , $G(V_{BG},V_{sd})$, of several charge states of a studied F-QD-S device is presented. Measurements were performed with standard lock-in technique with an *ac* excitation of $4 \mu$eV. In accordance with the spin-1/2 Kondo effect, pronounced conductance is seen around ($V_{sd} = 0$ V) in every odd charge state. However, in these states (labeled with numbers 1,3,5,7) the Kondo resonance shows different signatures due to correlations induced by the F lead. State 7 demonstrates a spin-1/2 Kondo situation with a single resonance line at $V_{sd}=0$ V. As it is shown in Fig. 2b this zero bias Kondo resonance splits up linearly with $B$. In contrast, state 3 exhibits clear signature of F correlations, i.e. the Kondo resonance has a finite and roughly constant splitting at $B=0$ mT (see black cross-section). Fig. 2c shows, that this splitting is compensated by $B\approx 64$ mT and split again at higher $B$ fields. Thus $B_{ex}$ has an opposite sign as the $B$ field. Another type of $B$ dependence of the Kondo ridge is presented in Fig. 2d (measured at $V_{BG}=2.31$ V), where the zero field splitting of the resonance is further increased by an applied $B$ field. It means, that $B_{ex}$ is parallel to $B$ for this state. The three markedly different $B$ field behaviors (Fig. 2b-d) demonstrate that $B_{ex}$ strongly depends on the QD level. Even in a small backgate range the amplitude and the sign of $B_{ex}$ varies. This observation highlights the particular importance of the coupling of the QD state to the F lead for the charge fluctuations induced local exchange field [@Martinek2005]. Note, the different $B$ dependencies also prove that the observed effect can not solely be described by stray field, since the stray field cannot depend on $V_{BG}$.
The most interesting class among the different exchange field manifestations is the one of state 1 (see Fig. 2a). The Kondo resonance is also split for this state. However the size of the splitting strongly varies with $V_{BG}$. (The resonance lines are highlighted with dashed lines in Fig. 2a). Interestingly, the split Kondo resonance lines also cross the $V_{sd}=0$ mV value inside the charge state. This dependence suggests that within this charge state $B_{ex}$ first gets smaller as $V_{BG}$ is increased. With further increasing $V_{BG}$ it changes its sign and increases further in the opposite direction. This situation corresponds to the electrically controlled ground state transition on the QD described by Eq. 1 and Fig. 1c.
![(Color online) Measurements on state 1. (a) Differential conductance as a function of $V_{BG}$ and $B$ measured at $V_{sd}=0$ mV. The two horizontal ridges (dotted lines) are the charge state boundaries. The high conductance ridge is the restored Kondo resonance, where $B=-B_{ex}(V_{BG})$. This ridge separates the regions where spin-$\uparrow$ or spin-$\downarrow$ is the ground state. The white line is a plot based on Eq. 1 using the parameters obtained from the fits shown in (c). The white arrow shows the sweeping direction of $B$. The black arrow points out the position where the magnetization of the F lead changes sign. (b) Colorscale plots at different magnetic fields. The dashed line highlights the position of the tilted Kondo resonance at $B=0.2$ T. (c) The evolution of the Kondo resonance is presented for different $B$ fields by symbols. The lines are fits using Eq. 1. []{data-label="Figure3"}](Figure3.eps)
Since $B_{ex}$ in situations like state 1 of Fig. 2a is strongly gate dependent, a measure for the exchange splitting at different gate voltages can be obtained by measuring the compensation field for which the Kondo peak is restored. Thus, if one measures $G(V_{BG},B)$ at $V_{sd}=0$ mV, at each $V_{BG}$ signatures of high conductance are expected if $B=-B_{ex}(V_{BG})$ due to the restored Kondo resonance. Such a plot is presented in Fig. 3a. The lower/upper ridge (see dashed lines) defines the resonance positions when the occupation of the QD level changes between $0\leftrightarrow 1$ /$1\leftrightarrow 2$. The white arrow indicates the sweep direction of $B$. The high conductance lines with finite slope show the evolution of the Kondo resonance (marked with a white line). At each $B$ field the gate position of the restored Kondo resonance defines the border between the spin-$\uparrow$ and the spin-$\downarrow$ ground states. At high external magnetic fields (i.e. $B \geq 0.3$ T) the Zeeman term dominates (see Eq. 1). Therefore, considering that the g-factor of InAs is negative, the spin-$\uparrow$ is the ground state for all gate values in high field and the Kondo resonance coincides with the border at $\epsilon_d=0$. As the $B$ field decreases the restored Kondo peak moves towards $\epsilon_d=-U$, opening backgate regions where spin-$\downarrow$ is the ground state. At $B=0$ mT the spin-$\downarrow$ state dominates, and spin-$\uparrow$ remains only in a small gate region around $\epsilon_d=-U$ the preferable spin orientation. The gate voltage value as a function of $B$, where the spin ground state change takes place, can be expressed with Eq. 1 using the condition of $eV_{sd}=0$. The white line in Fig. 3a shows such a curve, giving a reasonable agreement with the measurement using the parameters of $|g|=12.3$, $\Delta_0 = -90$ mT and $P\Gamma =
0.22$ meV.
As the polarization of the F lead is switched to the opposite direction by a $B$ field larger than the coercive field of the contact, $B_{ex}$ also changes its sign. This appears as a step in the $G(V_{BG},B)$ measurement (see Fig. 3a) since the splitting turns from $g \mu_B (B+B_{ex})$ to $g \mu_B (B-B_{ex})$. Due to the high g-factor of the InAs QD state, this is clearly seen in Fig. 3a at the position of the black arrow. Note, that in the vicinity of the $0\leftrightarrow 1$ border at $B=0$ mT, the ground state is spin-$\downarrow$. At this border the preferable spin orientation on the QD is the majority tunneling spin orientation (see Fig. 1c). [@Martinek2005; @Hauptmann2008] Since the F lead has been previously polarized into spin-$\downarrow$ state by a positive $B$ field, we can conclude, that the polarization of the F lead and the polarization of tunneling electrons are the same.
Eq. 1 describes the energy difference between the spin-$\uparrow$ and the spin-$\downarrow$ states as a function of $V_{BG}$. This is equivalent with the energy ($eV_{sd}$), where the Kondo ridge appears. $G(V_{BG},V_{sd})$ of state 1 is also measured in different magnetic fields. As it is seen in Fig. 3b, the position of the more pronounced split Kondo line moves to higher source drain voltage values as $B$ is increased. The $(V_{BG}$,$V_{sd})$ coordinates of this line is read out from the measurements, see e.g. the dotted line for $B=0.2$ T. Fig. 3c summarizes the measured position of the Kondo ridge by symbols at different $B$ fields. The theory described by Eq. 1 nicely fits the experiment (see lines) with the parameters used for the plot in Fig. 3a.
![(Color online) (a) $G(B,V_{sd})$ measurement: A subgap feature appears (horizontal dashed lines) at the energy scale of the exchange field. The feature is suppressed above the critical field of the superconductor. (b) $G(V_{BG},V_{sd})$ at $B=0$ mT showing the relation of the new subgap structure to $B_{ex}$. Inside the superconducting gap additional resonance lines appear in even charge states (see dotted lines). These resonances change their position with gate voltage and merge into the exchange split Kondo resonance lines at the border of the odd charge state. The line graphs show $G(V_{sd})$ cuts in the middle of the three charge states.[]{data-label="Figure4"}](Figure4.eps)
Since our device has a novel hybrid configuration, i.e. a QD connected to one F and one S lead, the interplay of S and F correlations can be studied. Up to this point, charge states with Kondo physics were investigated, where the Kondo resonance dominates the low energy behavior also for the S state. For such states the S lead induces conductance maxima at $V_{sd}=\pm\Delta$ related to the singularities of the S DOS, which diminish with increasing B field. However, in several charge states of different devices, where spin-1/2 Kondo resonances are not present (neither in the S nor in the normal state), an interesting subgap feature appears. This is shown in Fig. 4a, where the $G(B,V_{sd})$ measurement of such a state is presented. The conductance shows in a bias window of $\approx 50~\mu$V significantly smaller values than in the rest of the superconducting gap (see horizontal dashed line). This mini-gap is clearly connected to superconductivity since it is suppressed above the critical field of the S lead. On the other hand the energy scale of the novel subgap feature coincides with the exchange energy observed in the split Kondo states (see e.g. Fig. 2d). It suggests that the subgap is also exchange field related. Further evidence for such an relation is shown in the $G(V_{BG},V_{sd})$ measurement at $B=0$ mT on an other device (see Fig. 4b). The middle charge state with odd electron filling ($o$) shows a slightly split Kondo resonance, while the two even states ($e$) demonstrate the new subgap feature. As it is shown by the dotted lines, the mini-gap lines of the even charge states merge into the split Kondo resonance at the border between the even and the odd states. Thus, superconductivity induced transport processes in charge states with an even electron number are presented, which seem to be correlated with the exchange splitting.
Concluding, our measurements demonstrate that ferromagnetic proximity effect is indeed present in InAs NW based F-QD-S devices. A single F lead induces a local exchange field on the QD, which is strongly level dependent: it even changes sign for a single charge state. Demonstration of controlling the spin ground state of the QD by electric means makes the F-InAs QD-S system a promising building block for spin correlation studies, if implemented into a Cooper pair splitter device [@Hofstetter2009; @Herrmann2010].
We thank L. Borda, V. Koerting for fruitful discussions and C.B. Soerensen, III-V Nanolab, Niels Bohr Institute for MBE growth. This work has been supported by the Swiss NSF, the NCCR on Nanoscale Science, the Danish Natural Science Research Council, the OTKA-Norwegian Financial Mechanism NNF 78842, OTKA 72916 and the EU M.C. 41139 projects. S. C. is a grantee of the Bolyai Janos Scholarship.
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abstract: 'We study the generation of planar quantum squeezed (PQS) states by quantum non-demolition (QND) measurement of a cold ensemble of $^{87}$Rb atoms. Precise calibration of the QND measurement allows us to infer the conditional covariance matrix describing the ${F_y}$ and ${F_z}$ components of the PQS, revealing the dual squeezing characteristic of PQS. PQS states have been proposed for single-shot phase estimation without prior knowledge of the likely values of the phase. We show that for an *arbitrary* phase, the generated PQS gives a metrological advantage of at least $\unit{3.1}{dB}$ relative to classical states. The PQS also beats traditional squeezed states generated with the same QND resources, except for a narrow range of phase values. Using spin squeezing inequalities, we show that spin-spin entanglement is responsible for the metrological advantage.'
author:
- 'G. Colangelo'
- 'F. Martin Ciurana'
- 'G. Puentes'
- 'M. W. Mitchell'
- 'R. J. Sewell'
title: 'Entanglement-enhanced phase estimation without prior phase information'
---
Estimation of interferometric phases is at the heart of precision sensing, and is ultimately limited by quantum statistical effects [@WisemanBook2010]. Entangled states can improve sensitivity beyond the “classical limits” that restrict sensing with independent particles, and a diversity of entangled states have been demonstrated for this task, including photonic squeezed states [@SlusherPRL1985; @WuPRL1986] and spin-squeezed states [@MeyerPRL2001]. These give improved sensitivity for a narrow range of phases, but worsened sensitivity for most phases. Optical “NOON” states [@MitchellN2004] give improved sensitivity over the whole phase range, but introduce additional phase ambiguity that increases with the size, and thus sensitivity advantage, of the NOON state. Recent proposals [@Toth2009; @HePRA2011; @HeNJP2012] suggest using *planar quantum squeezed* (PQS) states to obtain an entanglement-derived advantage for all phase angles, with no additional phase ambiguity. A natural application is in high-bandwidth atomic sensing [@ShahPRL2010; @VasilakisPRL2011; @SewellPRL2012], in which the precession angle may not be predictable in advance. PQS states may also be valuable for *ab initio* phase estimation using feedback [@XiangNPhot2010; @YonezawaScience2012; @BerniNatPhoton2015].
Discussion of such states under the name “intelligent spin states” [@AragoneJPA1974] predates modern squeezing terminology, and analogous states have been studied with optical polarization [@KorolkovaPRA2002; @SchnabelPRA2003; @Predojevic2008]. Generation of PQS states in material systems has been proposed using two-well Bose-Einstein condensates with tunable and attractive interactions [@HePRA2011; @HeNJP2012], and using quantum non-demolition (QND) measurements [@PuentesNJP2013]. Here we take the latter approach, using Faraday rotation QND measurements [@MitchellNJP2012; @SewellNP2013] applied to an ensemble of cold atomic spins with $f=1$. As the ensemble spin precesses about the $x$ axis in an external magnetic field [@BehboodPRL2013; @Behbood2013APL; @BehboodPRL2014], we measure the $y$ and $z$ spin components to generate measurement-induced squeezing in these two components, creating a PQS state. The resulting state has enhanced sensitivity to precession angle, i.e., to Zeeman-shift induced phase. The demonstrated PQS state beats the best possible classical state at any precession angle, and beats traditional spin-squeezed states when averaged over the possible angles. Spin-squeezing inequalities [@HePRA2011; @HeNJP2012; @VitaglianoPRL2011] detect spin entanglement in the PQS state, showing the sensing advantage is due to spin entanglement [@BeduiniPRL2015].
A spin $\mathbf{F}$ obeys the Robertson uncertainty relation $$\label{eq:Robertson}
\Delta {F_y}\Delta {F_z}\ge \frac{1}{2} | \langle [{F_y}, {F_z}] \rangle | = \frac{1}{2} | {\ensuremath{\langle{F_x}\rangle}} |.$$ Unlike the canonical Heisenberg uncertainly relation, the rhs of Eq. (\[eq:Robertson\]) may vanish, e.g. for ${\ensuremath{\langle{F_x}\rangle}}=0$, with the consequence that two spin components, e.g. ${F_y}$ and ${F_z}$, may be [*simultaneously*]{} squeezed, with the uncertainty absorbed by the third component, ${F_x}$. We refer to a state fulfilling this condition as a PQS state.
Following the approach of He [*et al.*]{} [@HePRA2011; @HeNJP2012], we adopt an operational definition planar squeezing. We take $\Delta^2 {F_y}= \Delta^2 {F_z}= {F_{||}}/2$ as the standard quantum limit, where ${F_{||}}\equiv \sqrt{{F_y}^2 + {F_z}^2}$, so that ${F_{||}}$ is the magnitude of the in-plane spin components. We define the planar variance $\Delta^2 {F_{||}}\equiv \Delta^2 {F_y}+ \Delta^2 {F_z}$, with standard quantum limit $\Delta^2 {F_{||}}= {F_{||}}$, and the planar squeezing parameter $$\label{eq:PQSParameter}
{\xi_{||}^2}\equiv \frac{\Delta^2 F_{||}}{{F_{||}}}.$$ A PQS state has ${\xi_{||}^2}< 1$, and has individual component variances below the standard quantum limit, i.e., [$ {\xi_{y}^2}< 1$, and $ {\xi_{z}^2}< 1$ ]{}, where $\xi_i^2 \equiv {2\Delta^2 F_{i}}/{{F_{||}}}$, so that ${\xi_{||}^2}= ({\xi_{y}^2}+ {\xi_{z}^2})/2$.
Entanglement is detected using the witness ${\xi_{\rm e}^2}\equiv\Delta^2 F_{||}/{\ensuremath{\langle\tilde{{N_{\rm A}}}\rangle}}$, derived in Ref. [@HePRA2011]; for $f=1$ atoms, entanglement is detected if ${\xi_{\rm e}^2}<7/16$. Here $\tilde{N_{\rm A}}\equiv({\eta_{\rm sc}}+p(1-{\eta_{\rm sc}})){N_{\rm A}}$ is the number of atoms remaining in the $f=1$ state after probing, ${\eta_{\rm sc}}$ accounts for off-resonant scattering of atoms, and $p$ is the fraction of scattered atoms that return to $f=1$ [@ColangeloNJP2013]. We also define a metrological squeezing parameter ${\xi_{\rm m}^2}\equiv F \Delta^2 F_{||}/{F_{||}}^2$, where $F\equiv{\ensuremath{\langle{N_{\rm A}}\rangle}}$ is the input spin coherence, similar to the Wineland criterion [@WinelandPRA1992; @WinelandPRA1994], in that it compares noise to the magnitude of the coherence ${F_{||}}$. A PQS with ${\xi_{\rm m}^2}<1$ gives enhanced metrological sensitivity to arbitrary phase shifts.
![ Experimental setup. A cloud of laser-cooled $^{87}$Rb atoms is held in a singe-beam optical dipole trap. The atoms precess in the $y$–$z$ plane due to an external magnetic field $B_x$. Off-resonant optical probe pulses experience Faraday rotation as they pass through the atoms by an angle $\varphi$ proportional to the collective on axis spin component ${F_z}$. Rotation of the optical polarization from ${S_x}$ into ${S_y}'$ is detected by a balanced polarimeter that consists in a wave plate (WP), a polarizing beam splitter (PBS), and photodiodes PD$_{2}$ and PD$_3$. The input ${S_x}$ polarization is recorded with a reference photodetector (PD$_1$). \[fig:exp\]](Fig1.png){width="\columnwidth"}
A PQS state may be used to measure arbitrary phase angles with quantum-enhanced precision. For example, we consider an ensemble of atomic spins precessing in the $y$–$z$ plane in an external magnetic field $B_x$. The spin projection onto the $z$-axis is given by ${F_z}(t)={F_z}\cos \phi -{F_y}\sin \phi$, where ${F_y}$ and ${F_z}$ are evaluated at $t=0$ and the phase $\phi={\omega_{\rm L}}t$ is proportional to the magnetic field. The uncertainty in estimating $\phi$ of the atomic precession is $$\begin{aligned}
\Delta^2 \phi &= \frac{\Delta^2 {F_z}(\phi)}{|d {\ensuremath{\langle{F_z}(\phi)\rangle}}/d \phi|^2} = \frac{\Delta^2 {F_z}(\phi)}{({\ensuremath{\langle{F_y}\rangle}}\cos\phi+{\ensuremath{\langle{F_z}\rangle}}\sin\phi)^2}
\label{eq:phase}\end{aligned}$$ where $\Delta^2 {F_z}(\phi)\equiv \Delta^2 {F_y}\sin^2\phi +\Delta^2 {F_z}\cos^2\phi + \mathrm{cov}({F_y},{F_z})\sin2\phi$, and $ {\mathrm{cov}}(A,B) \equiv \frac{1}{2} {\ensuremath{\langleA B + BA\rangle}} - {\ensuremath{\langleA\rangle}}{\ensuremath{\langleB\rangle}}$ is the covariance. The standard quantum limit is $\Delta^2\phi_{\rm SQL}=1/2 {F_{||}}$. We note that PQS states reduce the planar variance for arbitrary angles on a finite interval, except where the denominator in Eq. (\[eq:phase\]) is equal to zero. In contrast, squeezing a single spin component is only beneficial to refine the estimate of a phase over a limited range of angles, and requires prior knowledge of the phase, or adaptive procedures to determine the phase during the measurement [@HeNJP2012].
We work with an ensemble of up to $1.75\times 10^6$ laser-cooled [$^{87}\mathrm{Rb}\ $]{}atoms held in a single beam optical dipole trap [@KubasikPRA2009; @KoschorreckPRL2010a; @KoschorreckPRL2010b], as illustrated in Fig. \[fig:exp\]. The atoms are initially polarized via high efficiency ($\sim98\%$) stroboscopic optical pumping, in the presence of a small magnetic field applied along the $x$-axis, such that ${\ensuremath{\langle{F_y}\rangle}}\simeq{\ensuremath{\langle{N_{\rm A}}\rangle}}$. ${N_{\rm A}}$ is subject to Poissonian fluctuations because accumulation of independent atoms into the ensemble is a stochastic process limited by Poisson statistics $\Delta^2{N_{\rm A}}= {\ensuremath{\langle{N_{\rm A}}\rangle}}$. We refer to this kind of state as a [*Poissonian coherent spin state*]{} (PCSS), with variances $\Delta^2{F_x}= \Delta^2{F_z}= {\ensuremath{\langle{N_{\rm A}}\rangle}}/2$ and $\Delta^2{F_y}= {\ensuremath{\langle{N_{\rm A}}\rangle}}$. Generating sub-Poissonian atom number statistics, either via strong interaction among the atoms during accumulation [@SchlosserN2001; @SortaisPRA2012; @ChuuPRL2005; @ItahPRL2010; @SannerPRL2010; @WhitlockPRL2010; @HofmannPRL2013], or precise non-destructive measurement [@StocktonThesis2007; @TakanoPRL2009; @AppelPNAS2009; @SchleierSmithPRL2010; @HumePRL2013; @BeguinPRL2014; @BohnetNPhot2014; @GajdaczPRL2016; @HostenN2016; @ZhangPRL2012; @StroescuPRA2015], remains a significant experimental challenge.
![ Rotation angle $\varphi$ in the $y-z$ plane of a ${F_y}$-polarized state precessing under a magnetic field oriented in the $x$ direction. We use the measurement record to predict the ${F_z}$ and ${F_y}$ components at a time $t={t_{\rm e}}$ using two sequential measurements $M_1$ and $M_2$. \[fig:FID\] ](Fig2.png){width="\columnwidth"}
We probe the atomic spins via off-resonant paramagnetic Faraday-rotation. The effective atom-light interaction is given by the hamiltonian $$H_{\rm eff} = g {S_z}{F_z}\label{qnd}$$ Here, the atoms are described by the collective spin operators ${\mathbf{F}}\equiv \sum_{i} {\mathbf{f}}^{(i)}$, with ${\mathbf{f}}^{(i)}$ the spin orientation of individual atoms. The optical polarization of the probe pulses is described by the Stokes operators ${S_k}= \frac{1}{2}(a_L^\dagger, a_R^\dagger) \sigma_k (a_L, a_R)^T$, with Pauli matrices $\sigma_k$. The coupling constant $g$ depends on the detuning from the resonance of the probe beam, the atomic structure, the geometry of the atomic ensemble and probe beam [@KubasikPRA2009; @KoschorreckPRL2010a; @Deutsch2010OC; @KuzmichPRL2000; @AppelPNAS2009].
{width="\textwidth"}
Equation describes a quantum non-demolition measurement of the collective atomic spin ${F_z}$: an input ${S_x}$-polarized optical pulse interacting with the atoms experiences a rotation by an angle $\varphi = g {F_z}$. The transformation produced by the interaction is ${S_y}' = {S_y}\cos \varphi + {S_x}\sin \varphi $. In our experiment we measure ${S_x}$ at the input by picking off a fraction of the optical pulse and sending it to a reference detector, and ${S_y}'$ using a fast home-built balanced polarimeter [@MartinOL2016]. Both signals are recorded on a digital oscilloscope, from which we calculate $\hat{\varphi}=\arcsin \left({S_y}'/{S_x}\right)$, the estimator for $\varphi$. We correct for slow drifts in the polarimeter signal by subtracting a baseline from each pulse, estimated by repeating the measurement without atoms in the trap.
We probe the atoms using a train of $\tau=\unit{0.6}{\micro\second}$ duration pulses of linearly polarized light, with a detuning of to the red of the [$^{87}\mathrm{Rb}\ $]{}D$_2$ line, sent through the atomic cloud at intervals. The probe pulses are $V$-polarized, with on average ${n_{\rm l}}=2.74 \times 10^6$ photons. Between the probe pulses, we send $H$-polarized compensation pulses with on average ${n_{\rm l}^{(H)}}= 1.49 \times 10^6$ photons through the atomic cloud to compensate for tensor light shifts [@KoschorreckPRL2010b; @SewellNP2013; @ColangeloNJP2013]. During the measurement, an external magnetic field $B_x$ coherently rotates the atoms in the $y$–$z$ plane at the larmor frequency ${\omega_{\rm L}}$. The time taken to complete a single-pulse measurement is small compared to the Larmor precession period, i.e. $\tau \ll T_L$. Off-resonant scattering of probe photons during the measurement leads to decay of the atomic coherence at a rate $\eta=3\times10^{-10}$ per photon.
The measurable signal is described by the free induction decay model [@Behbood2013APL] $$\label{eq:FIDForm}
\varphi(t)=g \Big( {F_z}({t_{\rm e}}) \cos \phi -{F_y}({t_{\rm e}}) \sin \phi \Big) e^{-{t_{\rm r}}/T_2} + {\varphi_0}$$ where ${t_{\rm r}}\equiv t- {t_{\rm e}}$ and the phase $\phi={\omega_{\rm L}}{t_{\rm r}}$ is proportional to the magnetic field. We record a set of measurements $\varphi({t_k})$, and detect the PQS state at time ${t_{\rm e}}$. A typical free induction decay signal is illustrated in Fig. \[fig:FID\]. An independent measurement is used to calibrate $g$, while ${\omega_{\rm L}}$, $T_2$, and ${\varphi_0}$ are found by fitting the measured $\varphi({t_k})$ over all the data points. The model described in Eq. allows a simultaneous estimation of ${\mathbf{F}}_1 = ({F_y}^{(1)}, {F_z}^{(1)})$ at a time $t={t_{\rm e}}$ by fitting the the data using the measurements from an interval ${\Delta t}$ *prior* to ${t_{\rm e}}$ (labeled $M_1$ in Fig. \[fig:FID\]), producing a conditional PQS at time ${t_{\rm e}}$. We detect the PQS by comparing the first measurement outcome to a second estimate ${\mathbf{F}}_2 = ({F_y}^{(2)}, {F_z}^{(2)})$ using the measurements from an interval ${\Delta t}$ *after* to ${t_{\rm e}}$ (labeled $M_2$ in Fig. \[fig:FID\]). The classical parameters $g$, ${\omega_{\rm L}}$, $T_2$ and ${\varphi_0}$ are fixed beforehand. As a result, these are two linear, least-squares estimates of the vector ${\mathbf{F}}$ obtained from disjoint data sets [^1]. Statistics are gathered over 450 repetitions of the experiment, taking into account the inhomogeneous atom-light coupling [@AppelPNAS2009; @SchleierSmithPRL2010; @ColangeloNature2017].
The estimate of the state from the two independent measurements is subject to technical noise due to amplitude and phase fluctuations of the input state, and shot-to-shot variations of the magnetic field. In Fig. \[fig:PQS\] a) we plot the estimate of ${\mathbf{F}}_1$ at time ${t_{\rm e}}$ for an input state with ${\ensuremath{\langle{N_{\rm A}}\rangle}}=1.75\times10^6$ atoms. In contrast, the conditional uncertainty of ${\mathbf{F}}_2$ given ${\mathbf{F}}_1$ is limited mainly by the measurement read-out noise, as shown in Figs. \[fig:PQS\] b) and c).
From the measurement record we compute the conditional covariance matrix $\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1}=\Gamma_{{\mathbf{F}}_2}-\Gamma_{{\mathbf{F}}_{2} {\mathbf{F}}_1} \Gamma_{{\mathbf{F}}_1}^{-1}\Gamma_{{\mathbf{F}}_1 {\mathbf{F}}_2}$ which quantifies the error in the best linear prediction of ${\mathbf{F}}_2$ based on ${\mathbf{F}}_1$ [@BehboodPRL2014]. $\Gamma_{\bf v}$ indicates the covariance matrix for vector ${\bf v}$, and $\Gamma_{\bf uv}$ indicates the cross-covariance matrix for vectors ${\bf u}$ and ${\bf v}$. The difference between the best linear prediction of ${\mathbf{F}}$ using ${\mathbf{F}}_1$ and the confirming estimate ${\mathbf{F}}_2$ is visualized using the vector $\bm{\mathcal{F}} = \{ \mathcal{F}_{\rm y}, \mathcal{F}_{\rm z} \} = \tilde{{\mathbf{F}}}_2 - \Gamma_{{\mathbf{F}}_{2} {\mathbf{F}}_1} \Gamma_{{\mathbf{F}}_1}^{-1} \tilde{{\mathbf{F}}}_1$, where $\tilde{{\mathbf{F}}_i}={\mathbf{F}}_i-{\ensuremath{\langle{\mathbf{F}}_i\rangle}}$. Standard errors in the estimated conditional covariance matrix are calculated from the statistics of $\{\bm{\mathcal{F}}\}$ [@Kendall1979].
Empirically, we find ${\Delta t}= \unit{270}{\micro\second}$ minimizes the total variance ${\rm Tr}(\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1})$. This reflects a trade-off of photon shot noise versus scattering-induced decoherence and magnetic-field technical noise. At this point ${N_{\rm L}}= 2.47 \times 10^8$ photons have been used in the measurement and the atomic state coherence has decayed by a factor ${\eta_{\rm sc}}=0.89$ due to off-resonant scattering, and a factor ${\eta_{\rm dec}}=0.93$ due to dephasing induced by magnetic field gradients [@ColangeloNJP2013]. The resulting spin coherence of the PQS is ${F_{||}}= {\eta_{\rm dec}}{\eta_{\rm sc}}{N_{\rm A}}$ spins.
![ Semi-log plot of the planar squeezing parameter, ${\xi_{||}^2}$, as function of the in-plane coherence ${F_{||}}$ of the atomic ensemble. We vary ${F_{||}}$ by changing the number of atoms loaded in the optical dipole trap. A PQS is detected for ${\xi_{||}^2}<1$ (shaded region). Entanglement is detected for ${\xi_{\rm e}^2}= ({F_{||}}/{\ensuremath{\langle\tilde{{N_{\rm A}}}\rangle}}){\xi_{||}^2}<7/16$ (dashed line). Error bars represent $\pm1\sigma$ statistical errors. \[fig:PlotPQS\]](Fig4.png){width="\columnwidth"}
![ Estimated phase sensitivity of the PQS state as a function of the measurement phase $\phi$ (red solid line). The standard quantum limit $\Delta^2\phi_{\rm SQL}$ is indicated by the shaded region. For comparison, we plot the phase sensitivity of the input PCSS (blue solid line), and an ideal single-variable spin squeezed state (green solid line). We also show the metrologically significant *enhancement* in phase sensitivity relative to that of the PCSS, $\Delta^2\phi / \Delta^2\phi_{\rm PCSS}$, for both the PQS (red dashed line) and SSS (green dot-dashed line) states. \[fig:PLPHASEvsPCSS\] ](Fig5.png){width="\columnwidth"}
From $\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1}$ we estimate the planar squeezing parameter ${\xi_{||}^2}={\mathrm{Tr}}(\tilde{\Gamma}_{{\mathbf{F}}_2\mid {\mathbf{F}}_1})/{F_{||}}$, where $\tilde{\Gamma}_{{\mathbf{F}}_2\mid {\mathbf{F}}_1}=\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1}-\Gamma_{0}$ and ${F_{||}}$ is estimated at ${t_{\rm e}}$. $\Gamma_{0}$ is the read out noise, quantified by repeating the measurement without atoms in the trap. In Fig. \[fig:PlotPQS\] we show ${\xi_{||}^2}$ as function of the in-plane coherence ${F_{||}}$ of the atomic ensemble, which we vary by changing the number of atoms in the optical dipole trap. We detect a PQS for ${F_{||}}\ge4\times10^{5}$ spins. With the maximum coherence ${F_{||}}=1.45\times10^{6}$ spins, we observe ${\xi_{||}^2}=0.37\pm0.03 < 1$, detecting a PQS with $> 20 \sigma$ significance, with $\xi_y^2 = 0.32\pm0.03$ and $\xi_z^2 = 0.42\pm0.04$, and ${\xi_{\rm e}^2}= 0.32\pm0.02 < 7/16 $, detecting entanglement among the atomic spins with $>5\sigma$ significance [@HePRA2011]. The measured conditional covariance (in units of spins$^2$) is $$\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1} =
\left[
\left(
\begin{array}{cc}
2.32 & 0.64 \\
0.64 & 3.00 \\
\end{array}
\right)
\pm
\left(
\begin{array}{cc}
0.21 & 0.16 \\
0.16 & 0.28 \\
\end{array}
\right)
\right]
\times 10^5.
\label{eq:cov}$$ For comparison, the estimated read-out noise is $$\Gamma_{0} =
\left[
\left(
\begin{array}{cc}
1.02 & 0.14 \\
0.14 & 1.03 \\
\end{array}
\right)
\pm
\left(
\begin{array}{cc}
0.07 & 0.05 \\
0.05 & 0.07 \\
\end{array}
\right)
\right]
\times 10^5.$$
For this state, the observed metrological squeezing parameter is ${\xi_{\rm m}^2}= 0.45\pm0.03$, indicating that entanglement-enhanced phase sensitivity is achievable. To estimate the enhanced phase sensitivity provided by the PQS state, we evaluate Eq. (\[eq:phase\]) using the conditional covariance $\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1}$ and the measured coherences. The PQS state achieves a maximum sensitivity $\Delta^2\phi = 0.38\, \Delta^2\phi_{\rm SQL}$ ($\Delta\phi=3.6\times10^{-4}$ radians) at a phase $\phi=0.68\,\pi$ radians. Note that this phase is determined by the choice of measurement time ${t_{\rm e}}$.
In Fig. \[fig:PLPHASEvsPCSS\] we plot the estimated phase sensitivity $\Delta^2\phi$ of the observed PQS state (red solid line). For comparison purposes, we rotate the PQS so that the spin coherence is aligned along the $y$-axis, i.e. ${\mathbf{F}}\rightarrow R(\theta)\cdot{\mathbf{F}}$ and $\Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1} \rightarrow R(\theta)\cdot \Gamma_{{\mathbf{F}}_2\mid {\mathbf{F}}_1} \cdot R(\theta)^{T}$, where $\arctan\theta\equiv{F_y}/{F_z}$. We compare this with the sensitivity of a PCSS with input spin coherence ${\ensuremath{\langle{F_y}\rangle}}={N_{\rm A}}$ (blue dashed line), and an *ideal* single-variable spin squeezed state (SSS) that would be produced by a single instantaneous quantum non-demolition measurement with the same precision, i.e. with $\Delta^2{F_y}= {\ensuremath{\langle{N_{\rm A}}\rangle}}$, $\Delta^2{F_z}$ reduced by a factor $1/(1+g^2{N_{\rm L}}{N_{\rm A}}/2)$, and input coherence ${\ensuremath{\langle{F_y}\rangle}}={\eta_{\rm sc}}{N_{\rm A}}$ (green dot-dashed line).
We also plot the calculated *enhancement* in phase sensitivity $\Delta^2\phi$ of both the PQS and SSS states relative to the classical input PCSS. The measured PQS state achieves $\ge \unit{3.1}{dB}$ quantum-enhanced, metrologically-significant phase sensitivity with respect to the PCSS for all phases, with a maximum of , enabling quantum-enhanced measurement of an *arbitrary* phase shift. In contract, the SSS achieves 6.6 dB enhancement relative to the PCSS at $\phi=0$, but performs worse than the PQS state outside the range $-0.09\, \pi < \phi < 0.12\, \pi$ radians.
In contrast to the well known spin-squeezed states, planar quantum squeezed states enhance the precision of phase estimation without requiring *a priori* information about the phase. Here we have shown that QND measurement can efficiently produce such states, demonstrating more than of advantage relative to classical states over the full range of phase angles. We also detect spin-spin entanglement underlying the metrological advantage. Such states are attractive for high-bandwidth and high-sensitivity optical magnetometers [@KominisN2003; @ShahPRL2010] and other atomic sensing applications employing non-destructive spin detection [@LodewyckPRA2009; @ShengPRL2013; @HostenN2016].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Q. Y. He, M. Reid, P. Drummond, G. Vitagliano, G. Tóth, E. Distante, V.G. Lucivero, L. Bianchet, N. Behbood and M. Napolitano for helpful discussions. Work supported by MINECO/FEDER, MINECO projects MAQRO (Ref. FIS2015-68039-P), XPLICA (FIS2014-62181-EXP) and Severo Ochoa grant SEV-2015-0522, Catalan 2014-SGR-1295, by the European Union Project QUIC (grant agreement 641122), European Research Council project AQUMET (grant agreement 280169) and ERIDIAN (grant agreement 713682), and by Fundació Privada CELLEX. GP gratefully acknowledges funding from the Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), PICT2014-1543, PICT2015-0710, and UBACYT PDE 2017.
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[^1]: Further details of the fitting procedure are given in Ref. [@ColangeloNature2017]
|
---
abstract: |
We present global 3D MHD simulations of geometrically thin but unstratified accretion disks in which a near Keplerian disk rotates between two bounding regions with initial rotation profiles that are stable to the MRI. The inner region models the boundary layer between the disk and an assumed more slowly rotating central, non magnetic star. We investigate the dynamical evolution of this system in response to initial vertical and toroidal fields imposed in a variety of domains contained within the near Keplerian disk. Cases with both non zero and zero net magnetic flux are considered and sustained dynamo activity found in runs for up to fifty orbital periods at the outer boundary of the near Keplerian disk.
We find a progression of behavior regarding the turbulence resulting from the MRI and the evolving structure of the disk and boundary layer according to the initial field configuration. Simulations starting from fields with small radial scale and with zero net flux lead to the lowest levels of turbulence and smoothest variation of disk mean state variables. As found in local simulations, the final outcome is shown to be independent of the form of the imposed field. For our computational set up, average values of the Shakura & Sunyaev (1973) $\alpha$ parameter in the Keplerian disk are typically $0.004\pm 0.002.$ Magnetic field eventually always diffuses into the boundary layer resulting in the build up of toroidal field, inward angular momentum transport and the accretion of disk material. The mean radial velocity, while exhibiting large temporal fluctuations is always subsonic.
Simulations starting with net toroidal flux may yield an average $\alpha \sim 0.04.$ While being characterized by one order of magnitude larger average $\alpha$, simulations starting from vertical fields with large radial scale and net flux may lead to the formation of persistent non-homogeneous, non-axisymmetric magnetically dominated regions of very low density. In these gaps, angular momentum transport occurs through magnetic torques acting between regions on either side of the gap. Local turbulent transport occurs where the magnetic field is not dominant. These simulations are indicative of the behavior of the disk when threaded by magnetic flux originating from an external source. However, the influence of such presumed sources in determining the boundary conditions that should be applied to the disk remains to be investigated.
author:
- Adriane Steinacker
- 'John C.B. Papaloizou'
title: '3D-MHD simulations of an accretion disk with star-disk boundary layer'
---
Introduction {#S0}
============
The study of the boundary layer, the region where the angular velocity of an accretion disk drops to match the rotation velocity of the central object, is of great importance for the understanding of accreting objects, since up to half of the total accretion energy can be released there (Lynden-Bell $\&$ Pringle 1974) if the central object is a slow rotator. Up to now detailed studies of this region (eg. Papaloizou & Stanley 1986, hereafter PS, Kley 1989, Popham & Narayan 1992) have been based on the Navier-Stokes equations with a modified viscosity prescription involving an anomalous viscosity coefficient. This is presumed to contain the effects of any turbulence present reducing the problem to one of laminar flow.
The decrease of angular velocity in the boundary layer is associated with an increase in the thermal pressure, which replaces centrifugal support for the accreting matter. The large pressure gradients in turn, may be associated with supersonic radial infall velocities in this region, if a standard disk viscosity prescription is continued into the layer. However, it has been argued that if such a supersonic flow occurred, the star would loose its causal connection to the outer disk (Pringle 1977). In relation to this issue, several studies based on the Navier-Stokes equations were performed but with a modified viscosity prescription in the boundary layer corresponding to its much lower pressure scale height (PS, Popham & Narayan 1992). This modification alone does not eliminate supersonic flows under all conditions (e.g. for large values of $\alpha\simeq 1$). Popham & Narayan (1992) suggested reducing the viscosity coefficient to zero as the radial infall velocity approaches the sound speed. This causally limited viscosity (Narayan 1992) always leads to subsonic infall. A different approach, appreciating the short time required for matter to pass through the layer, allows the viscous stress components to relax towards their equilibrium values on a relaxation time scale (Kley & Papaloizou 1997) and so naturally incorporates causality. Studies of one-dimensional models led to the conclusion that the boundary layer must be characterized not only by the value of $\alpha$ in the outer disk, but also by the nature of the viscous relaxation process. Additional time dependent studies demonstrated that for low $\alpha\simeq 0.01$, the boundary layer adjusts to a steady state, while for large $\alpha=0.1$, significant disturbances occurred in the boundary layer and to the power output. Periodic oscillations were seen throughout the disk by PS, whenever $\alpha$ was large (close to 1). For small $\alpha$, the oscillations were localized near the outer disk boundary. These oscillations are caused by the viscous overstability found by Kato (1978) and Blumenthal et al. (1984) and it was suggested they may be important in explaining the time-dependent behavior in accreting objects such as CVs and protostars.
All of the above models were based on an ad-hoc anomalous viscosity prescription and did not consider further its origin. The discovery of the relevance of the magnetorotational instability (MRI) (Balbus & Hawley 1991) has opened up a new era in accretion disk astrophysics. The instability provides a robust and self-consistent mechanism for the production of turbulence and angular momentum transport in these objects if they are adequately ionized, thereby removing the need for ad-hoc prescriptions. The development of different numerical codes has enabled a detailed investigation of the nonlinear phase of the instability. First numerical studies were performed in a local shearing box approximation, (Hawley, Gammie & Balbus 1995, Brandenburg et al. 1995, Sano, Inutsuka & Miyama 1998, Fleming, Stone & Hawley 2000, Miller & Stone 2000). These studies show that the turbulent outcome of the MRI depends on the initial field configuration applied to the disk. Thus, local simulations with initial vertical fields with zero net flux field indicate an average $\alpha=0.001-0.006$, while vertical fields with non zero net flux result in larger values of $\alpha$ up to $0.3.$ The turbulent outcome of an initially unstable toroidal field can lead to intermediate values of $\alpha$ up to $ 0.04$ depending on the net flux.
Relatively recently studies of instabilities in global disk models have begun (Armitage 1998). Such studies are needed to see how an unstable disk modifies its underlying structure in response to globally varying levels of turbulence and whether the longer time scale evolution is in any way like that of standard $'\alpha'$ disk models. Recent studies have been made by Hawley (2000), who concentrated on the evolution of thick tori, and Hawley & Krolik (2001). The latter study focuses on the evolution of the inner region of a disk accreting onto a black hole modeled with a pseudo-Newtonian potential.
In this paper we study the interaction of an accretion disk with a boundary layer region located between it and the central star. This situation is the relevant one to consider for non relativistic accretion onto a non magnetic central star. The inner boundary layer region together with an exterior stable region also provide convenient, relatively inert regions in which to embed a near Keplerian disk with an unstable rotation profile. Instead of prescribing the viscosity in an ad-hoc fashion, as in previous studies of the boundary layer, we self-consistently incorporate the turbulence arising from the MRI as the source of viscosity and diffusion of magnetic field.
We assume the disk to have a small ratio of scale height to radius ( $H/R\simeq 0.1$ ). The gravity is assumed to be entirely due to the central object and for simplicity and in common with other global disk studies we neglect vertical stratification by adopting a cylindrically symmetric potential thus focusing the study on the radial structure of the disk. We study the dynamical evolution of the disk over a time span of up to one thousand rotation periods measured at the inner disk edge. We consider different initial magnetic field configurations (poloidal and toroidal) imposed in the main body of the disk. Cases with both non zero and zero net magnetic flux are considered.
Simulations starting with small scale initial fields with zero net flux exhibit the lowest Shakura & Sunyaev (1973) parameter $\alpha$ with a mean value averaged over the Keplerian domain of $\simeq 0.005$. In this case the simulations on average attain a final state characterized by the same mean $\alpha$ and magnetic energy independently of (within computationally defined limits) the initial field strength. There may also be a tendency for the mean value of $\alpha$ to increase with the extent of the vertical domain and the numerical resolution. This is in agreement with results of shearing box simulations (Hawley, Gammie & Balbus 1996, hereafter HGB96). On the other hand simulations with large net magnetic flux may evolve turbulence with a larger mean value of $\alpha$ of $\simeq 0.04$, when the initial field is toroidal. Models starting from initial vertical fields with large radial scale are such that $\alpha$ attains maximum values $>1$ in regions associated with prominent density depressions, while outside these gaps, $\alpha$ reaches values similar to the zero net flux models (as low as $0.005).$ The volume averaged $\alpha$ depends on the initial plasma beta in such cases.
We find that all simulations locally exhibit strong variations of the vertically and azimuthally averaged values of $\alpha$ in time and with radius. All of the models display oscillations of the vertically and azimuthally averaged radial Mach number. Even though the boundary layer region is stable to the MRI, magnetic field always diffuses into it. This is the case even when the initial field is non zero only well away from the layer. Toroidal field build up enables mass to accrete through it onto the central star through the operation of magnetic torques.
The paper is organized as follows: in §\[S1\] we present the basic model and computational set up. In §\[S2\] we describe the numerical procedure. In §\[S3\] we discuss azimuthal and vertical averaging together with the global transport of angular momentum and energy dissipation in the disk. §\[S4\] is devoted to the investigation of cases with initial fields with zero net magnetic flux. In §\[S5\] we present results when the initial magnetic field has net flux. Finally, we summarize our results in §\[S6\] .
Initial model setup {#S1}
===================
The simulations are performed within the framework of ideal MHD. The governing equations are: $$\frac{\partial \rho}{\partial t}+ \nabla \cdot {\rho\bf v}=0, \label{cont}$$ $$\rho \left(\frac{\partial {\bf v}}{\partial t}
+ {\bf v}\cdot\nabla{\bf v}\right)=
-\nabla p -\rho \nabla\Phi +
\frac{1}{4\pi}(\nabla \times {\bf B}) \times {\bf B}, \label{mot}$$ $$\frac{\partial {\bf B}}{\partial t}=\nabla \times ({\bf v} \times {\bf B}).
\label{induct}$$ where all the quantities have their usual meanings. While in some of our exploratory simulations we adopted an adiabatic equation of state (with heating due to artificial viscosity retained), the runs described in this paper are either based on using a locally isothermal equation of state, i.e. $$P(R)=\rho(R)\cdot c(R)^2,$$ where $c(R)$ denotes the sound speed in the disk specified as a fixed function of $R,$ or after using this initially, solving the energy equation assuming that heating due to artificial viscosity is removed by an exactly compensating cooling. Either of those procedures gave very similar results. We found them to lead to a more consistent procedure since the numerical scheme currently applied with zero resistivity does not allow for magnetic energy dissipated on the grid scale to be recovered as heat. Tests showed that when an adiabatic condition was used, such that kinetic energy dissipated through an artificial viscosity was recovered as heat, there was a noticeable change in the thermal energy content in the Keplerian disk during the course of a simulation.
The computations are carried out in cylindrical coordinates $(z,R,\phi)$. We assume that the gravitational potential is only dependent on the radial coordinate, $\Phi=-GM/R$, where $M$ is the mass of the central object and $G$ is the gravitational constant.
Our computational domain is divided into three distinct regions: 1) An extended active Keplerian domain, ranging from a radius $R_1$ to $R_2$, in which the growth of the MRI leads to turbulence (Region I); 2) An inner boundary layer ranging from $R_0$, the lower radial boundary of the computational domain, to $R_1$ (Region II), and 3) an outer region ranging from $R_2$ to the outer radial boundary, which we have added to avoid numerical problems with trying to apply a boundary condition directly at the outer edge of the active domain (Region III). The corresponding values of $R_1, R_2$ and $R_3$ are specified in table 1. The initial density and angular velocity profiles for these three regions are calculated consistently with hydrostatic equilibrium in the radial direction. We present these solutions below.
[**Region I: $R_1\le R < R_2$**]{}. In contrast to previously published cylindrical global accretion disk simulations (Armitage 1998, Hawley 2000 and Hawley & Krolik 2001) where constant density profiles were used, we adopt a density varying inversely with the radial distance $$\rho(R)=\rho_0\cdot R_0/R,$$ and a radial dependence of the sound speed given by $$c(R)=c_0 \sqrt{R_0/R} \label{sound}$$ where $\rho_0$ and $c_0$ are the density and sound speed corresponding to the radial position $R_0$. With the pressure determined by a locally isothermal equation of state, we calculate the rotation velocity $$v_{\phi}(R)=\sqrt{G M/R_0-2c_0^2}\cdot \sqrt{R_0/R}\label{velc}.$$ For a thin disk in which the sound speed at a given radius is much smaller than the rotation velocity, this profile is nearly Keplerian.
[**Region II: $R_0\le R \le R_1$.**]{} In this region we require the angular velocity $\Omega$ to drop from its near-Keplerian value at $R=R_1$ to a lower value $\Omega_\ast$ which can be considered as matching the angular velocity of a central stellar object. For the rotation velocity, setting $i=1,$ we adopt $$v_{\phi}(R)=v_{\phi i} \cdot (R/R_i)^{n_b},$$ and for the sound speed we take $$c(R)=c_i (R_i/R)^{n_c}.$$ These prescriptions lead to the following density profile: $$\rho(R)=\rho_i\cdot(R/R_i)^{2n_c}\cdot\exp
{\left[\frac{0.5\delta -1}{n_b+n_c}\cdot\left((R/R_i)^{2(n_b+n_c)}-1\right)-\delta\cdot f_{i}(R)
\right]},$$ where $f_{i}(R)=\ln(R/R_i)$, if $n_c=0.5$, and $f_{i}(R)=\left[(R/R_i)^{2n_c-1}-1\right]/(2n_c-1)$, otherwise, and $\delta=GM/(R_0 c_0^2)$. Here $ \rho_i, c_i$ and $v_{\phi i}$ are the density, sound speed, and rotation velocity, at the radius $R_i$. These quantities are given by eq. (5), (6) and (7), respectively. We have considered cases with $n_b=1$ (corresponding to uniform rotation), $n_b=2$ and $n_b=3.$ When $n_b=2,$ we took $n_c=0$. When $n_b=1,$ or $n_b=3$ we took $n_c=0.625$ in most of the cases. Situations with $n_c=0.5$ have also been considered. We note that tests showed that when this region was initiated with a density profile not too far out of hydrostatic equilibrium, the situation rapidly adjusted so that such an equilibrium was attained.
[**Region III: $R_2 < R \le R_3$.**]{} This region is added primarily for numerical stability. It provides a high inertia which prevents the magnetic field from diffusing into the outer radial boundary and producing a severe drop in the density and angular velocity of the boundary zones during the course of the simulations. This would ultimately contaminate the global energetics, leading to incorrect behavior of the magnetic energy and the $\alpha$ parameter. In this region, the solutions presented above for region II continue to be valid, but with the index $i=2$. In practice we adopted uniform initial rotation profiles that are stable with respect to the MRI ($n_b=1$) and $n_c=0.5.$ The assumption of a thin disk requires the sound speed to be smaller than the rotation velocity, which is equivalent to the pressure scale height $H=c/\Omega,$ the disk would have if stratified vertically, being significantly less than the current radius. We chose $c_0^2=0.01 GM/R_0$ for most models but also ran two (b4,b5) with $c_0^2=0.04 GM/R_0$ for comparison purposes.
We comment that due to the chosen radial dependence of the sound speed and rotational velocity in region I, the scale height $H$ is a linear function of $R.$ In most cases we chose the extent of the computational domain in $z$ to be larger than one scale height at the outer radial edge of the Keplerian domain. In few cases this was taken to be one half of the scale height at the outer radial boundary of the Keplerian domain, thus allowing us to test the effect of the extent of the $z$ domain on the results.
For most calculations with zero net flux fields, the radial domain was chosen such that $R_1=1.2, R_2 =3.7$ and $R_3=4.5.$ The $\phi$-domain extends from 0 to $\pi/2$ in most models. Recent work by Hawley (2001) suggests that, with the same resolution, the extent of the $\phi$ domain does not greatly affect the results. We have checked this by performing runs with the $\phi$ domain extending to $\pi$, and $\pi/3$. The boundary conditions in $z$ and $\phi$ are periodic. In $R$ we set the scalar quantities, magnetic field components and z- and $\phi$-components of the velocity to have zero gradient. The radial velocity components at the inner and outer radial boundaries were set to zero. This ensures mass conservation in the computational domain. We also note that the computational set up is such that the zero gradient condition on the azimuthal velocity applied at the radial boundaries only affects values at ghost zones that do not affect the flow elsewhere. Thus artificially imposed viscous boundary layer effects do not occur in our calculations.
We have performed simulations with:
1. Initial vertical fields with and without net flux,
2. Initial toroidal fields with and without net flux,
the magnetic field being initially defined in a restricted radial domain: $${\bf B_i}=B_0\sin\left(2 n_R\pi \frac{R-R_b}{R_{b1}-R_b}\right)\bf{e_i}, \label{Binit}$$ whith $n_R=0.5, 1.5$ for net field runs, and $n_R=3, 6, 9$ for runs with zero net flux. The index $i$ indicates either the vertical or the toroidal field component with the corresponding unit vector $\bf{e_i}.$ For toroidal fields $B_0$ is a constant while for vertical fields $B_0 \propto 1/R,$ with $R_b$ and $R_{b1}$ being the boundaries of the region where the field was applied. These will be specified for each run in turn. An overview of the performed simulations is given in table 1, which shows the magnetic field topology used, the mean plasma beta defined as the ratio of thermal to magnetic pressure with each of these averaged over the Keplerian domain, the numbers of computational grid points in each direction, the number of wavelengths of the mode of maximum growth for the MRI contained in the z-domain ($z_u$ and $z_l$ being the upper and lower boundaries) at the location of the innermost field maximum, the number of wavelengths in radial direction, $n_R,$ specified in order to define the net and zero net flux fields, respectively, the location of the boundaries to the three different regimes of the R-domain, $R_1,R_2$ and $R_3$, the extent of the domain in $\phi$ and z, and the duration of the run in units of the inverse Keplerian angular frequency at the inner boundary located at $R_0\equiv 1.$ Typically the runs last for about 20 orbits measured at the outer radius of the active Keplerian region located at $R \sim 4$, although we performed one long run for a larger disk with $R_2/R_0 = 7.2$ for a time exceeding 50 orbits at the outer boundary of the Keplerian domain in order to confirm the persistence of the turbulence up to such times. For computational purposes, the unit of length is taken to be the inner boundary radius, thus $R_0 = 1,$ the unit of mass is the central mass, and the unit of time is the inverse Keplerian angular frequency at the inner boundary radius, or the period there divided by $2\pi.$
Numerical procedure {#S2}
-------------------
The numerical procedure follows the method of characteristics constrained transport MOCCT as outlined in Hawley & Stone(1995) and implemented in the ZEUS code. Alfv[è]{}n wave characteristics are used to integrate the induction equation and to evaluate the Lorentz force. The evolution of the magnetic field, ${\bf B}$, is constrained to enforce $\nabla \cdot {\bf B} =0$ to machine accuracy. The code has been developed from a version of NIRVANA originally written by U. Ziegler (see Ziegler & Rüdiger 2000 and references therein), and has been previously used for 2D hydrodynamic simulations of viscous disks interacting with migrating Jovian mass planets. The results have been validated by detailed comparison with other codes and independently obtained results (Bryden et al. 1999, Nelson et al. 2000, Kley, Angelo & Henning 2001). Shearing box MHD simulations have also been performed by Ziegler & Rüdiger (2000).
The time step is limited by the Courant condition. In addition, we have implemented the numerical technique used by Miller & Stone (2000) to prevent the time step becoming too small by not allowing the Alfv[è]{}n speed to exceed a limiting value in regions of very low density. Tests have shown that this makes no significant difference to the results obtained, while circumventing the severe drop in the time step which would sometimes make practical continuation of simulations impossible. We also set a global floor for the density. On performing a series of exploratory runs with different values for the limiting speed and the density floor, we found that simultaneous application of the two methods leads to the best results in a most economical manner. However, as stated above, the results were unaffected by reasonable changes to the floor or the limiting velocity. For the runs listed in table 1, the density floor was $\rho_{fl}=10^{-3}\rho_0$ and the limiting velocity was $v_{lim}=0.3$.
----- ----------------- -------------------------- -------------------------------- --------------------------------- --------- ------- ------- ------- --------- ------------ ------
run B $\langle \beta \rangle $ $N_z\times N_R\times N_{\phi}$ $\frac{z_u-z_l}{\lambda_{max}}$ $n_{R}$ $R_1$ $R_2$ $R_3$ $\phi$ z time
b1 0-net $B_z$ 105 $36\times 334\times 108$ 2 6 1.2 3.7 4.5 $\pi/2$ $\pm$0.2 526
b2 — 316 $36\times 334\times 108$ 4 — — — — — — 592
b3 — 1005 $36\times 334\times 108$ 6 — — — — — — 697
b4 — 143 $36\times 167 \times 108$ 0.5 3 1.5 — — — — 710
b5 0-net$B_{\phi}$ 32 $36\times 167\times 108$ 0.3 — — — — — — 739
b6 0-net $B_z$ 568 $40\times 370\times 100$ 1 9 1.2 7.2 8.8 — — 6627
b7 — 563 $60\times 370\times 100$ 2 — — — — $\pi/3$ $\pm$ 0.3 2818
b8 — 316 $54\times 334\times 108$ 1 3 — 3.7 4.5 $\pi/2$ $\pm$0.2 740
b9 0-net$B_{\phi}$ 38 $56\times 334\times 108$ 2 — — — — — — 741
b10 — 40 $36\times 334\times 108$ 2 — — — — — — 750
b11 0-net $B_z$ 80 $34\times 132\times 34$ 3 — — 3 3.6 $\pi/3$ $\pm$0.18 1193
b12 — 438 $54\times 132\times 64$ 4 6 — — — — $\pm$0.2 1381
n1 net $B_z$ 120 $44\times 202\times 54$ 3 0.5 — 4 4.8 — $\pm$ 0.2 1444
n2 — 360 $34\times 177\times 72$ — — — 3.7 4.5 — $\pm$ 0.18 942
n3 — 120 $44\times 202\times 102$ — — — 4 4.8 $\pi$ $\pm$ 0.2 628
n4 net $B_{\phi}$ 7.6 $34\times 132\times 34$ 1 — — — — $\pi/3$ — 1444
----- ----------------- -------------------------- -------------------------------- --------------------------------- --------- ------- ------- ------- --------- ------------ ------
: \[table1\] Parameters associated with the simulations discussed in this paper. The first column gives the simulation label, the second the nature of the initial field, the third the initial value of $\langle \beta \rangle ={\int P d\tau \over \int {\bf B}^2/(8\pi) d\tau},$ with the integrals being taken over the Keplerian domain, the fourth the computational grid and the fifth gives the number of wavelengths in the vertical domain of the most unstable MRI mode calculated for the initial field using equation (\[lmax\]). The sixth column gives the number of radial wavelengths in the initial field, and the remaining columns give the boundaries and extents of the radial, azimuthal and vertical computational domains as well as the run time. Models b1, b2, b3, b8, b9, and b10 were all run with heating and cooling exactly in balance while the remainder had an isothermal equation of state with sound speed a fixed function of radius.
In all cases, the instability was initiated by applying a sinusoidal perturbation in the radial velocity with an amplitude of $0.01c_0.$ The number of wavelengths in the vertical domain of the most unstable MRI mode calculated for the initial field at the location of the innermost field maximum varies between 1 and 6 as specified in column 5 of table 1. The wavelength associated with the mode of maximum growth was calculated from $$\lambda_{max}=\sqrt{\frac{16\pi}{15}} \frac{B}{\sqrt{\rho} \Omega}.
\label{lmax}$$ We note that due to the radial dependences of the density, angular velocity and magnetic field, $ \lambda_{max}$ is a function of radius. If $ \lambda_{max}$ is smaller than the extent of the z domain, in the case of vertical fields, regions of stability may alternate with unstable regions.
Angular momentum transport {#S3}
==========================
In order to describe the behavior of the different models, it is helpful to use quantities that are vertically and azimuthally averaged over the $(\phi, z)$ domain. These are defined with an overbar such that for any quantity $Q$ $${\overline {Q(R,t)}} ={\int \rho Q dz d\phi \over \int \rho dz d\phi}.$$ Note that although the numerical simulations are done over a fraction of the full $\phi$ domain, because of periodicity they can be stacked end to end so that without loss of generality we can assume they occupy the full $2\pi$ when performing azimuthal averages.
We also introduce the surface density $$\Sigma = {1\over 2\pi}\int \rho dz d\phi .$$
Using the above quantities it is possible to describe the angular momentum transport in the disk using functions dependent only on $R.$ In this way a connection to classical viscous disk $'\alpha'$ theory can be made ( Balbus & Papaloizou 1999).
We monitor the vertically and azimuthally averaged Maxwell and Reynolds stresses, which are respectively defined as follows: $$T_M(R,t)=2\pi
\Sigma{\overline{\left({B_R(z,R,\phi,t) B_\phi(z,R,\phi,t) \over 4\pi\rho}\right)}}$$ and $$T_{Re}(R,t)=2\pi
\Sigma
{\overline{\delta v_R(z,R,\phi,t)\delta v_\phi(z,R,\phi,t)}}$$ The velocity fluctuations $\delta v_R$ and $\delta v_\phi$ are defined through, $$\delta v_R(z,R,\phi,t)=v_R(z,R,\phi,t)-{\overline{v_R}}(R,t),$$ $$\delta v_\phi(z,R,\phi,t)=v_\phi(z,R,\phi,t)- {\overline{v_{\phi}}}(R,t).$$ With these definitions, the Shakura & Sunyaev (1973) $\alpha$ parameter for the total stress is given by $$\alpha(R,t)=\frac{T_{Re}-T_M}{2\pi
\Sigma{\overline{ \left(P/\rho\right)}}},$$ where the pressure average is taken at the actual time rather than at $t=0$ as is sometimes found in the literature. With the above definitions the vertically and azimuthally averaged continuity equation (1) may be written $$\frac{\partial \Sigma}{\partial t}+\frac{1}{R}
\frac{\partial\left(R\Sigma\overline{v_R}\right)}{\partial R}=0 \label{cona}$$ and the vertically and azimuthally averaged azimuthal component of the equation of motion(2) can be written in the form $$\frac{\partial \left(\Sigma \overline{j}\right)}{\partial t}
+\frac{1}{R}\left(
\frac{\partial\left( R\Sigma\overline{v_R}\overline{j}\right)}{\partial R}
+\frac{\partial\left(\Sigma R^2\alpha\overline{P /\rho}\right)}{\partial R}
\right) =0.\label{mota}$$ Here $j=rv_{\phi}$ is the specific angular momentum. Equations (20) and (21) are identical to what is obtained in viscous $'\alpha'$ disk theory (Balbus & Papaloizou 1999). However, as noted by those authors significant differences may occur when the energy balance is considered. From equations (1) and (2) we may derive the rate of energy dissipation, $\epsilon_{\nu}$ and doing $PdV$ work per unit volume as $$\frac{\partial (\rho\epsilon)}{\partial t}+ \nabla \cdot {\bf F}
=-P\nabla\cdot{\bf v}-\epsilon_{\nu}=-Q_T,
\label{energ}$$ where, the energy per unit mass and flux are respectively $$\epsilon={1/2}{\bf{v}}^2 + \Phi + {\bf B}^2/(8\pi \rho)$$ and $${\bf F} =\rho
{\bf{v}}\left({1/2}{\bf{v}}^2 + \Phi +\frac{{\bf B}^2}{4\pi \rho} +{P\over\rho}\right)
-\frac{({\bf{v}}\cdot{\bf{B}}){\bf{B}}}{4\pi}.$$ To reorganize the above into a form more related to $'\alpha'$ disk theory we define an energy per unit mass $\epsilon_k(R)$ and specific angular momentum $j_k(R).$ These can be used to define an angular velocity $$\Omega_k=
\frac{{d\epsilon_k(R)\over dR}}{ {d j_{k}(R)\over dR}},$$ and may be chosen to correspond to a free particle in circular Keplerian orbit. The latter assumption is not necessary and any convenient values could be adopted in principle. Note that in the general non Keplerian case $\Omega_k \ne j_k/R^2=v_k/R.$ Performing an azimuthal and vertical average on equation(\[energ\]) and subtracting (\[mota\]) after multiplying by $\Omega_k,$ we may write the energy balance in an alternative form $$\frac{\partial(\Sigma \bar{\cal{E}})}{\partial t}+
\frac{1}{R} \frac{\partial(R \bar {\cal{F}})}{\partial R}
+\Sigma\left(\bar{v_R}(\bar{v_{\phi}}-v_k)
+\alpha(R){\overline{ P/\rho} }\right)R\frac{d\Omega_k}{dR} =
-{1\over 2\pi}\int Q_Tdzd\phi.
\label{energbal}$$ Here $${\cal{E}}= \epsilon-\epsilon_k -\Omega_k(\bar{j}-j_k)$$ and $${\cal{F}}=\Sigma v_R\left(\epsilon-\epsilon_k -\Omega_k(\bar{j}-j_k)
+{P\over\rho}+\frac{{\bf B}^2}{8\pi\rho}
- R\Omega_k{\bf{\hat{\phi}}}\cdot({\bf{v}}-\bar{{\bf{v}}})\right)$$ $$-\Sigma({\bf{v}}-R\Omega_k{\bf{\hat{\phi}}})
\cdot\frac{{\bf {B}}B_{R}}{4\pi\rho}$$ It can be argued (adopting Keplerian values for $\epsilon_k, j_k$) that in a thin disk that is in a time average near steady state the term proportional to $\alpha$ on the left hand side of (\[energbal\]) is second order in $c/v_{\phi},$ while the others are at least third order (Balbus & Papaloizou 1999). To do this one needs to extend the azimuthal and vertical average to incorporate an additional time average which is to be carried out over a time long compared to the orbital period but short compared to a supposed much longer evolutionary time scale. The relation of $\alpha$ to energy dissipation that holds in standard $'\alpha'$ disk theory is then recovered in the thin disk limit.
However, in some practical cases, including the calculations reported here, which have somewhat large values of $c/v_{\phi}\sim 0.1,$ important deviations may occur. For example the ratio of the contribution of the pressure flux term $\bar{ v}_R P$ to the term $\propto \alpha$ in (\[energbal\]) may be estimated as $\bar{v}_R/(\alpha v_{\phi}).$ In our calculations, at any particular time, this can be of order unity indicating that energy redistribution through pressure wave modes may be important and signaling departures from standard viscous disk theory as far as the energetics is concerned. Thus the adopted $'\alpha'$ parameterization may be useful only as far as angular momentum redistribution is concerned. In this context in distinction to standard viscous disk theory, there is no reason why $\alpha$ should be invariably positive. Positive definite dissipation may still occur provided the $\cal{F}$ flux terms provide a source (see also Balbus & Papaloizou 1999). In general the magnetic stresses always give a positive contribution to $\alpha.$ However, the contribution of the Reynolds stress can be strongly variable in space, time and sign.
We monitor the average time dependent (radial) Mach number, $$M_s(R,t)=\frac{\bar{v_R}}{ c(R)}.$$ In general $\bar{v_R}$ exceeds the expected mean viscous inflow velocity $$v_{\nu}=
\alpha c^2/\bar{v_{\phi}}\label{viscv}$$ appropriate to a Keplerian disk because there are fluctuations in the average radial velocity with near zero mean (compared to their amplitude).
Following Hawley(2000) we use the vertically and azimuthally averaged stresses as a measure of angular momentum transport through $\alpha(R,t)$ defined above. As indicated above one can incorporate a time average with a view to indicating the behavior of a disk evolving on a time scale long compared to the orbital period. The form of the averaged equations given above is unchanged but the mean radial inflow velocity is significantly reduced, becoming comparable to the viscous inflow speed given by equation (\[viscv\]).
In addition, we have considered mean values of $\alpha$ which are obtained from volume averaging over the Keplerian domain. Thus we denote the volume averaged value of $\alpha,$ as a function of $t$, through
$$\langle \alpha (t)\rangle =
{\int \limits^{R_2}_{R_1}\alpha(R,t)RdR\over (1/2)(R_2^2 - R_1^2)}.$$
In practice we find for runs with zero net flux which do not show extreme density contrasts that radial averaging of the total stress over the Keplerian domain (I) gives a value with much reduced temporal fluctuations compared to those occurring at a particular point once the turbulence is established (e.g. the volume average is never negative). Most of the fluctuations seen in our runs appear to be due to the Reynolds stress. Once a time average is performed the contribution of the latter to the total stress is relatively small $\sim 1/3$ in line with local simulations (e.g. HGB96).
Power spectra of the vertically averaged magnetic field components are calculated as the squared Fourier amplitude of an azimuthal mode $m$ as a function of radius as follows $$\mid a_{mbi}(R,t)\mid^2=\left| \int\limits^{z_u}_{z_l}\int\limits^{2\pi}_0B_i(z,R,\phi,t)
e^{-im\phi}d\phi dz\right|^2, \label{spectra}$$ where $i=z,R,\phi$.
Simulations with zero net flux fields {#S4}
=====================================
Provided there is zero field at the radial boundaries, in a global simulation of the type considered here, both the total vertical and the total toroidal flux threading the system are conserved. The latter is guaranteed by the periodic boundary conditions in the vertical direction. As long as there is no flux entry through the radial boundaries simulations can therefore be characterized by the amount of net flux they contain. In particular, if there are no other conserved quantities that can distinguish different simulations, one might expect that (allowing for numerical limitations) all simulations with zero initial net flux should approach the same turbulent state. This has been found to be the case for shearing box simulations (HGB96). However, the shearing box set up is special in that fully periodic boundary conditions in shearing coordinates guarantee no flux entry into the system for all time. In addition, one can search for a time averaged steady state over arbitrary time intervals. The set up of the global simulations considered here only allows for testing that simulations starting with zero net flux approach the same steady state in a more restricted sense. Although we expect magnetic energy to grow through the action of instabilities only in the Keplerian domain, we inevitably find diffusion of field into the inner boundary layer region as well as the outer stable region. When there is significant shear in the boundary layer region, we find the field can grow through the winding up of the poloidal field to produce a toroidal field with significant magnetic energy. However, this phenomenon may be considerably delayed by starting a simulation with initial field set to be non zero only at large distances from the boundary regions. In this respect we remark that toroidal fields are preferred to vertical fields and the absence of shear in the inner layer delays the build up of strong fields there. In any case there is no guarantee of conservation of magnetic flux in the Keplerian domain once there has been significant field diffusion out of that region. Accordingly, checking whether solutions approach the same turbulent state in the Keplerian domain has been limited to the situation before such diffusion has occurred. If convergence of the solutions occurs, then it should of course hold at a later stage when interaction with the boundary regions occurs. In this context we have found that different prescriptions for the inner boundary layer (i.e. $n_b=1$ or $n_b=3$) do not seem to affect the situation in the Keplerian domain even though the behavior in the inner boundary layer region may be significantly different. In this section we investigate the disk response to varying initial fields with zero net flux in order to establish that when these are of small enough radial scale and adequate amplitude essentially the same state results. Other effects such as those of the numerical resolution, disk aspect ratio and size are also explored. We investigate both initially prescribed vertical and toroidal magnetic fields. We find essentially the same final state when the field has a small scale compared to the current radius and a local quasi-steady turbulent state can be achieved on a time scale short compared to the global evolution time of the entire disk. Initial large scale fields with significant magnetic energy may not lead to the same state as small scale fields on a time scale short compared to the evolution time scale of the disk or on one that can be reasonably followed here.
Disks with initially vertical fields
-------------------------------------
We first consider the three runs b1, b2, b3 that start from an initial vertical field defined between $R_b = 4/3$ and $R_{b1} = 10/3.$ In all of these cases $n_b =3$ and $n_c=0.625$ in region II. The runs differ only in the amplitude of the initial vertical field such that the initial $\langle \beta \rangle$ varies by an order of magnitude ranging between $\sim 100$ and $\sim 1000$ (see table 1). Many features characteristic for simulations b1, b2, and b3 are also found in others described below. The magnetic energy in the Keplerian domain expressed in units of the volume integrated pressure (or $1/\langle \beta \rangle$) is plotted as a function of time for the three runs in figure \[fig1\].
As found with simulations performed in a shearing box (Hawley et al. 1995, Stone et al. 1996, Brandenburg et al. 1995, Ziegler & Rüdiger 2000) and existing global simulations (Hawley 2000, Krolik & Hawley 2000), the simulations with smaller initial $\langle \beta \rangle$ show the development and growth of channel solutions. This is manifest in the early evolution of b1 through the strong peak in $1/\langle \beta \rangle$ that occurs at $t \sim 30.$ Streams of fluid at different vertical levels, alternately moving in opposite radial directions, evolve and persist over a period of several orbits at the corresponding radius. This phase is associated with large values of $\alpha$ peaking at 0.3-0.6 at smaller radii, where the instability grows first due to the smaller rotation period. This behavior is barely noticeable in run b3. We comment that there is not yet a significant penetration of the magnetic field into the boundary regions during the time interval shown. In spite of very different initial behavior for $0 < t < 200$ among these cases they all approach $1/\langle \beta \rangle \sim 0.01$ (in an average sense) as in many other cases we have run (see below). After $ t \sim 200,$ turbulence is established and maintained throughout the disk. At this stage, it is found that $\alpha$ exhibits strong variations in time and with radial distance, sometimes up to one order of magnitude. The value of $\alpha$ volume averaged over the Keplerian domain $\langle \alpha \rangle$ is plotted as a function of time in the lower panel of figure \[fig1\] for the three runs. The volume averaged $\alpha$ shows strong fluctuations on all time scale, which are, however, less than those found at any particular radius. Closer inspection reveals that most of the fluctuation is due to the Reynolds stress which may contribute up to one half of the total. The Maxwell stress is found to be always positive and to vary less strongly.
Simulations b1, b2, b3 attain time averaged values of $\langle \alpha \rangle \sim 0.005 $ as in many other cases we have run (see below). This value is similar to what is seen in shearing box simulations starting with weak zero net flux fields (Brandenburg et al. 1995, Hawley et al. 1995). We also comment that we obtain a similar approximate correlation between $(R, \phi)$-stress and energy as found in the shearing box simulations of HGB96. This was verified in all simulations with zero net flux and can be expressed, after averaging out short term variations, as $\langle \alpha \rangle \sim 0.5/\langle \beta \rangle.$
We plot $\alpha(R,t)$ against radius for run $b3$ at time 643 in figure \[fig2\]. No significant field has yet penetrated into the boundary layer region as can be seen by the small values there. Note that at this time and for this simulation there is a strong peak at smaller radii. However, such a peak may occur in the center of the Keplerian domain at other times and in other simulations (see below). The vertically and azimuthally averaged density corresponding to figure \[fig2\] is plotted in figure \[fig3\] together with the initial profile. Comparison between the two curves shows evidence of accretion with material moving into the inner region from the outer parts of the disk.
Although it can be delayed by starting with initial data that is non zero far away from the inner boundary layer, the field inevitably diffuses into this region with the degree of penetration increasing with time. This is the case even though the shear indicates stability with respect to the MRI. The pattern of behavior is the same for all runs. For runs b1, b2 and b3, which have $n_b=3$, the strong shear causes the toroidal field to build up and the value of $\alpha(R,t)$ to become negative corresponding to inward angular momentum transport.
As an illustrative example to indicate these points, we plot $\alpha(R,t)$ against radius for run $b1$ at time 540 in figure \[fig4\]. The largest magnitudes of $\alpha \sim -0.05$ occur in this region which tends to expand outwards. Nonetheless, radial motions remain subsonic. Eventually magnetic contact with the inner boundary zones occurs causing a violation of the conditions required for conservation of flux in the computational domain. This effect may be delayed by choosing a more extended inner boundary region.
Disks with initially toroidal fields and the effect of vertical resolution
---------------------------------------------------------------------------
We now describe models b8, b9 and b10. We set $n_b=3, n_c=0.625$ in region II and defined the magnetic field between $R_{b}=7/3$ and $R_{b1}= 10/3$ for b8 and between $R_{b}=4/3$ and $R_{b1}= 10/3$ for b9 and b10. These runs were carried out to check that simulations beginning with small scale zero net toroidal fields led to the same state as those starting from vertical fields. For the purpose of comparing with a vertical field run, simulation b8 started with a poloidal field with the same initial magnetic energy as b2, but with higher vertical resolution. Simulation b10 started with a toroidal field and had the same resolution as b1, b2 and b3, while b9 had higher vertical resolution. Because of the very much weaker instability apparent in the cases with an initial toroidal field, these could be started with significantly larger magnetic energy. No early channel phase occurs. In fact in these cases the magnetic energy decreases due to reconnection of oppositely directed field lines in the initial phases of the MRI.
The magnetic energy in the Keplerian domain ($1/\langle \beta \rangle)$ expressed in units of the volume integrated pressure is plotted as a function of time for these three runs in the upper panel of figure \[fig5\]. There is no significant penetration of magnetic field into the boundary regions at this stage. In spite of the different initial conditions and the initially weaker instability in the models starting from a toroidal field, they all eventually attain $1/\langle \beta \rangle \sim 0.01$, as was found for simulations b1, b2 and b3. The increased vertical resolution of simulations b8 and b9 appears to have little influence on the results.
The value of $\alpha$ volume averaged over the Keplerian domain $\langle \alpha \rangle$ is plotted as a function of time for the simulations b8, b9 and b10 in the lower panel of figure \[fig5\]. These values behave similarly to those found for simulations b1,b2 and b3, and become indistinguishable at later times, with the time averaged value of $\langle \alpha \rangle = 0.004 \pm 0.002.$
For simulation b8 we present in figure \[fig6\] the vertically averaged density, azimuthal velocity and radial Mach number profiles near the end of the run. These are similar to the initial profiles.
In order to emphasize the similarity between simulations with initial vertical and toroidal fields with zero net flux in figure \[fig8\] we plot the mid plane density contours for simulations b9 and b10 near the end of these runs. Dark regions correspond to low density. Stochastic spiral patterns are visible. In addition, we present the azimuthal power spectra for the vertically averaged magnetic field components calculated according to eq. (\[spectra\]) at $R=2$ in figure \[fig9\]. In fact these are characteristic of the turbulent state and are very similar at all radii. There is a flat spectrum for small azimuthal wavenumbers $m<10$ and a sharp cut-off at $m=10-20$ This behavior is in agreement with the results previously obtained by other authors (e.g. Armitage 1998, Hawley et. al 1995).
Large disks and long runs
-------------------------
We also ran two simulations, b6, and b7, for which the disk was approximately twice as large as in the previous cases. This has the consequence that the evolutionary time scale for the whole disk is about three times longer. A larger disk enabled insertion of initial magnetic field data further from the inner boundary region with $R_b = 3.5$ and $R_{b1} = 6.5$. In both these models $n_b=3$ and $n_c=0.625$ in region II. The resolution in simulation b6 was very similar to the cases discussed above. However, the vertical domain extends over only one half the disk semi thickness in the outer part of the Keplerian domain. In simulation b7, the azimuthal domain was contracted to $\pi/3$ effectively increasing the azimuthal resolution by fifty percent. In addition, the extent of the vertical domain was increased by fifty percent with respect to b6, while maintaining the same resolution.
The magnetic energy in the Keplerian domain expressed in units of the volume integrated pressure is plotted as a function of time for simulations b6 and b7 in the upper panel of figure \[fig10\]. These both attain $1/(\langle \beta \rangle) \sim 0.01$ but with some indication of b7 tending to produce larger values than b6. A larger magnetic energy in the saturated turbulent state in b7 might be expected from shearing box simulations ( HGB96). These have indicated a dependence on numerical resolution (Brandenburg et al. 1996), as well as larger values for larger boxes. The effect of increasing the size of the vertical domain in b7 is similar to increasing the size of a shearing box. The value of $\alpha$ volume averaged over the Keplerian domain $\langle \alpha \rangle$ is plotted as a function of time for simulations b6 and b7 in the lower panel of figure \[fig10\]. These attain $\langle \alpha \rangle \sim 0.004 \pm 0.002$ but with b7 tending to give somewhat larger values in the mean. These results are also consistent with the expected correlation between $(R,\phi)$-stress and magnetic energy mentioned above in the form $\langle \alpha \rangle \sim 0.5/\langle \beta \rangle.$ Simulation b6 indicates survival of the saturated turbulent state for more than fifty orbital periods at the outer boundary of the Keplerian domain. At the end of simulation b7, significant boundary layer penetration occurred producing negative $\alpha$ there. The situation then was very similar to that illustrated in figure \[fig4\] for a smaller disk model. Taken together, these simulations indicate some possible dependence of $\alpha$ on numerical resolution and the extent of the vertical domain.
A thicker disk
--------------
Simulations b4 and b5 are characterized by larger semi thickness than the other runs, with $c_0^2=0.04GM/R_0$, and a wider uniformly rotating inner boundary layer ($n_b=1, n_c=0.625$ in region II and $R_1 = 1.5).$ The initial fields were applied between $R_b = 7/3,$ and $R_{b1}=10/3.$ These runs were initiated with a zero net poloidal field (b4) and a zero net toroidal field (b5). The initial magnetic energy is about three times larger in b5. These runs have lower absolute radial resolution than those discussed above but the same resolution per scale height. Maintaining the extent of the vertical domain as in the previous runs b1, b2, b3 means that in these two cases only one half of the scale height is contained within the z-domain at the outer edge of the Keplerian domain.
The magnetic energy in the Keplerian domain, $1/(\langle \beta \rangle)$, is plotted as a function of time in the upper panel of figure \[fig11\]. In spite of the very different initial conditions these simulations both approach a state with $1/(\langle \beta \rangle)$ on average somewhat less than $0.01.$ The value of $\alpha$ volume averaged over the Keplerian domain $\langle \alpha \rangle$ is plotted as a function of time in the lower panel figure \[fig11\]. Similar values to those obtained in the other simulations are found.
We also plot $\alpha(R,t)$ as a function of $R$ at time $t=590$ for simulation b4 in figure \[fig12\]. This indicates a peak in the center of the Keplerian domain. Even though there is some diffusion of magnetic energy into the boundary layer at about 25 percent of the that in the Keplerian domain. There is little activity in the boundary layer region while there is no contact with the inner boundary zones, because of the low shear. This has been found to be the case in these and other simulations. For the types of computational set up used and the times we have been able to run the simulations, we have found no evidence that the solution in the Keplerian domain is significantly affected by the shear profile adopted in the inner boundary layer. This is found to be the case (not illustrated here) even when runs are continued well beyond the point where magnetic contact with the inner boundary zones occurs.
Simulations with initial fields with large radial scale of variation
--------------------------------------------------------------------
We here describe simulation b11 which had an initial vertical field with zero net flux with a larger scale of radial variation (see table \[table1\]) and $n_b=2$. The field was initially applied between $R_b=1.32$ and $R_{b1}=2.76$. This run was carried out with lower resolution than the other simulations but in many ways led to similar results. The large scale motion associated with the channel solutions is completed after $t \sim 240.$ After this time, as in other simulations, turbulence is established and maintained throughout the disk. The largest values of $\alpha(R,t) \sim 0.03 $ are typically obtained close to the boundary layer and at the outer edge of the Keplerian domain, while in the middle of the active domain, $\alpha(R,t) $ decreases to a mean value of 0.002. The large values of $\alpha$ are connected with vertically and azimuthally averaged density minima there, with a density contrast of up to one order of magnitude with respect to the surroundings. With this type of model, unlike the previously described, convergence of different solutions, if it should occur, is difficult to attain on a reasonable time scale.
As in other simulations, when the magnetic field penetrates the boundary layer region producing an interaction between the boundary layer and the inner regions of the Keplerian disk, the boundary layer moves to a somewhat larger radius (from $R=R_1=1.2$ initially to $ R \sim 1.36$). The state reached after time 628 is therefore somewhat different from the initial state, with surface density depressions near the boundary layer and the outer stable region. This behavior is typical for initial perturbations with only one to a few wavelengths in the vertical domain for the most unstable MRI mode and an initial field with one to three maxima.
Oscillations
------------
The vertically and azimuthally averaged radial Mach number remains subsonic throughout the simulations, with the typical radial velocity being $\sim 10$ times larger than the viscous inflow velocity given by equation (\[viscv\]). It displays oscillatory behavior close to the outer stable boundary region. Such oscillations, which occur in all models, were indicated previously in laminar viscous disk modeling, away from the boundary layer, as a consequence of the viscous overstability found by Kato (1978) (e.g. PS). Here they arise as residuals after averaging turbulent fluctuations vertically and azimuthally and have no obvious connection to the earlier theories because the simple modeling used there does not incorporate the complication that the time and length scales of the oscillations and turbulence are not clearly separable and so do not allow for a description using an anomalous viscosity coefficient. Such a description would only be expected to apply to phenomena on a global length scale averaged over a time scale long compared to that characteristic of the turbulent fluctuations.
In order to further illustrate these oscillations, in figure \[fig13\] we have represented the Mach number as a function of time for simulation b12 which had $n_b = 2 $ and as in all previously discussed simulations other than b11 was initiated with a small scale field. The superposed curves apply to locations in radius from $R=2.04$ to $R=3.04$ in steps of 0.2. The period of the oscillations can be estimated with some noise to be 3.5 expressed in units of the Keplerian period at the inner edge of the computational domain. The Keplerian period corresponding to this range of radii ranges from 2.9 to 5.3. The smaller spread seen in the oscillation periods suggests that they are not entirely purely local epicyclic oscillations but that propagation of information from one location to another occurs in the simulations. A time series of the vertically and azimuthally averaged density as a function of radius performed between two distinct times during the simulation and covering several hundred time units suggests the presence of outward propagating disturbances produced by the interaction between the boundary layer and the Keplerian disk. Some reflection from the boundary of region III may also occur. The phase speed of the waves can be estimated to be a typical sound speed in the Keplerian domain.
Simulations with initial vertical fields with net flux {#S5}
======================================================
The simulations described above were focussed on conditions starting from small scale fields at small amplitude which lead to a well defined turbulent state in the mean that was, within that framework, initial condition independent. Such a situation would apply to a non magnetic star interacting with a disk threaded by no external flux. However, there may be situations where the disk is threaded with a large scale vertical flux. In that case the disk is not isolated because conducting material external to the disk is implied as a source. The source could be the material supplied at large distances to the disk itself. Concentration of the flux towards the center of the disk may lead to a large scale poloidal field threading the disk that could be a source of a global outflow as has been considered by many authors (e.g. Blandford & Payne, 1982, Lubow, Papaloizou & Pringle, 1994, Shu et al. 1994, Spruit, Stehle, & Papaloizou, 1995, Konigl & Wardle, 1996). Simulations of disks with net flux may also be relevant to a situation in which the disk becomes dominated by an external magnetic field arising from the central star. The latter point of view was adopted by Miller & Stone (2000) who carried out simulations of a stratified disk in a shearing box. They found magnetically dominated solutions. In their case, vertical gravity was included and outflow boundary conditions were applied in the vertical direction, but Keplerian rotation was enforced by the boundary conditions. In our model this is not the case but there is no vertical stratification. However, the periodic boundary conditions in the vertical direction, neither take into account any constraints arising from the external material in which field lines might be embedded nor correctly match to an external vacuum field. We comment that despite the difference in the models, the outcome is very similar: magnetically dominated solutions.
We here describe simulation n1 with initial mean $\beta =120$ (see table \[table1\]). The magnetic field was applied between $R_b=1.32,$ $R_{b1}=3.72$ and $n_b =2.$ During the channel phase, the global magnetic energy increases by up to 2 orders of magnitude with respect to the initial value, and drops to a mean value of about one order of magnitude lower than this maximum. This model and other similar ones we ran are characterized by the tendency to produce regions with pronounced density minima or gaps in which the vertically averaged density is smaller by 1-2 orders of magnitude than the surroundings. Conditions in the gaps were strongly variable, with inhomogeneous variations in $z$ and $\phi$ of sometimes more than one order of magnitude. This inhomogeneity is evident in figure \[fig14\] where we show polar contours of the surface density (upper panel) along with a slice of the density in the R-z-plane ($\phi$=0) at time 1224. The gaps, one located next to the boundary layer, and a second near to the outer stable region, are visible as lumpy structures in azimuth, and radially elongated low-density filaments alternating with large density regions are visible in the R-z plane. This picture remains qualitatively unchanged throughout the simulation (1444 time units).
The formation and survival of the gaps appears to be a characteristic of initial vertical fields with large net flux, which in turn have strong initial channel solutions. After an associated reconnection, a net vertical magnetic flux is trapped in low density regions with a radial extent larger than one local scale height. A large scale vertical field persists in association with the gaps. In the bottom panel of figure \[fig14\] we show the magnetic field vectors in the R-z plane ($\phi=0$) at time 1224. Superimposed is a binary map of the critical wavelength [$\lambda_c$]{} (white represents stable regions). Note that the inner gap extends roughly from 1.2 to 2, while the outer gap is located from 3.2 to 4. When the critical wavelength exceeds the disk height, the angular momentum transport from the faster to the slower rotating regions is mediated by the torque exerted by the toroidal field that is built up from the poloidal field. The toroidal field is wound up until torque balance occurs maintaining the gap. This mechanism operates analagous to magnetospheric cavity formation (e.g. Ghosh & Lamb 1978). In that case, a poloidal field is wound up between the disk and the central star rotating at a different rate until a steady torque is transmitted between the two via a force-free field. At the edges of the gap, the axisymmetric MRI can always operate, as the condition for instability is fulfilled. Indeed, figure \[fig14\] shows that the field stretches out radially across the gaps until it reconnects. Filaments or blobs of magnetic flux are expelled from both sides of the gaps, and a concentration of vertical flux is left there. In this manner, the large-scale field is maintained in the gap by the MRI.
We find large values of $\alpha$ correlated with the low density within the gaps. The upper panel of figure \[fig15\] shows the radial dependence of $\alpha$ at time 1143, while the lower panel represents the time evolution of the volume averaged $\alpha$. $\alpha$ periodically becomes larger than unity in the regions corresponding to the gaps. In some models, values of $\alpha$ as large as 3 can be attained (this finding being in agreement with Hawley (2001)). The Maxwell stress dominates over the Reynolds stress there, while outside the gaps, the Reynolds stress becomes important. Outside the gaps, $\alpha$ may decrease to a mean of 0.005, similar to the zero net field models.
When gaps are formed next to the boundary layer, the large-scale poloidal magnetic field trapped in that region contributes to link the boundary layer to the outer disk. A periodic broadening and narrowing of the boundary layer becomes evident in this case. We attribute this oscillation to the interaction between the disk and the boundary layer via the field connecting these regions. A time sequence of azimuthally and vertically averaged density profiles (represented in figure \[fig16\]) shows that waves, most likely excited by this oscillation of the boundary layer propagate through the high density region exterior to the gap.
We performed a number of simulations with varying resolution (e.g. run n2) and computational domain (e.g. n3). All these runs show the formation of at least one prominent gap (located next to the boundary layer) and the presence of one or two smaller gaps which have the tendency to deepen with time and sometimes merge. Their position may vary slightly with resolution and extent of the radial or azimuthal domain. These findings are in agreement with Hawley (2001).
A simulation with a toroidal field with net flux
------------------------------------------------
Simulation n4 was performed with an initial toroidal field with net flux and with one maximum in R. The magnetic field was applied between $R_b=1.32$ and $R_{b1}=2.76$ and $n_b =2.$ As in the cases with zero net flux, the instability is weak when compared to models with an initial vertical field. The magnetic energy first shows growth after time 125.6 peaking after 289 then having grown by a factor of two. A quasi-steady turbulent state was reached after about 377 time units, with the global magnetic energy being 0.8 of the initial value at the end of the simulation (time=1444). The radial variations of the vertically and azimuthally averaged stress parameter $\alpha$ are relatively small, not exceeding a factor of 3. The lower panel of figure \[fig17\] shows the time variation of $\alpha$ at R=2, and indicates a mean value of 0.04. In the upper panel, we represent the radial dependence of $\alpha$ close to the end of the simulation (t=1388). The magnetic stress is always dominant, and the Reynolds stress can sometimes become negative as previously discussed and the boundary layer moves slightly inwards in the course of the simulation.
Discussion {#S6}
==========
In this paper we have studied the time dependent evolution of a near Keplerian accretion disk which rotates between two bounding regions with initial rotation profiles that are stable to the MRI. The inner region models the boundary layer between the disk and a central star. Because we self-consistently incorporate MRI-induced turbulence as the source of viscosity, the necessity of an ad-hoc viscosity prescription of the type used in earlier studies is removed.
For this first study we assumed the disk to be unstratified with aspect ratio $ H/R \simeq 0.1-0.2 $ by adopting a cylindrically symmetric potential assumed to be exclusively generated by the central object. For most of the cases considered, the dynamical evolution of the disk following from the imposition of different magnetic field configurations was followed over a time span between one hundred and two hundred rotation periods at the inner edge of the computational domain. However, simulations of radially more extended disks were followed up to one thousand orbital periods at the inner edge of the computational domain. For initial conditions both toroidal and poloidal magnetic fields with zero as well as net flux were applied in varying domains contained within the near Keplerian disk.
Simulations starting from toroidal and poloidal fields with zero net flux and a small scale of radial variation evolved to a state characterized by a smooth angular velocity and density profile similar to the initial one. This was independent of the type, and within numerically determined limits, the amplitude of the initial field. This result also holds in shearing box simulations (HGB96). But one must bear in mind that in numerical computations, the range of initial plasma betas to which one has reasonable access is restricted by the resolution. Typical values of $\langle \alpha \rangle$ representing a volume average over the Keplerian domain are 0.004$\pm$0.002. Moreover, runs with a radially extended disk showed that the saturated turbulent state is maintained over more than fifty orbital periods at the outer boundary of the Keplerian domain.
While the shearing box approach guarantees flux conservation for all time in a Keplerian domain, in global simulations this is not necessarily the case. This is because in large scale simulations the inevitable diffusion of the magnetic field out of the Keplerian domain and into the boundary domains can lead to the violation of flux conservation in the Keplerian domain. This happens even though the boundary domains are stable to the MRI. However, we found that different prescriptions of the inner boundary layer region do not affect the final state in the Keplerian domain even though the behavior of the boundary layer itself may vary significantly. Once significant field has leaked into the boundary layer, toroidal field is built up due to the shear and causes $\alpha$ to attain negative values corresponding to inward angular momentum transport and mass accretion. In such cases, the boundary layer expands outwards but radial motions remain subsonic.
We also find a similar approximate correlation between the Keplerian domain volume averaged ($R \phi$)-component of the stress and magnetic energy as found in local simulations: For simulations with zero net flux we find, after averaging out short term variations, $\langle \alpha \rangle\sim 0.5/\langle \beta \rangle$.
Models with an initial vertical field with zero-net flux and large scale of radial variation exhibit local minima in the density associated with maxima of the angular velocity. These density pockets can reach a contrast of one order of magnitude with respect to the surroundings. The turbulent state can nonetheless be characterized by an average $\alpha$ similar to the cases with small scale of radial variation. It may require a very long time scale for such states to relax to those found in the initially small scale field cases.
All models display large variations of $\alpha$ in time and radius including oscillations on a rotational time scale. Variations of one order of magnitude are typical.
Models staring with fields that have non zero net flux lead to a higher level of turbulence. Thus, model n4 that started with a toroidal field attains an average alpha of 0.04. Those starting with vertical fields with net flux such as n1 may display several gaps in density, with the radially innermost gap typically located next to the boundary layer. The density contrast in the most prominent gaps can reach up to 2 orders of magnitude (in an azimuthal and vertical average) with respect to their surroundings. In the gap-regions, $\alpha$ alternates between values $<1$ and $>1$, sometimes exceeding 3 and dropping to an average of 0.005 (comparable with the zero net flux simulations) in non gap regions. Values of $\alpha$ exceeding 1 indicate angular momentum transport by magnetic torques originating from fields that connect across the gap region.
Recognizing that there are issues to be resolved regarding the correct boundary conditions required to represent the effects of external conducting material, these solutions might be relevant when the disk becomes dominated by an external magnetic field. Such magnetic fields may affect a variety of processes that take place in accretion disks, from dust coagulation to the interaction of planets with the disks in which they have been formed.
Clearly there is much room for future improvements and developments. Convergence needs to be checked at much higher resolution than currently attainable. Studies of more extended inner MRI-stable boundary layers should be carried out, and vertical stratification should be included. In this connection the simple periodic boundary conditions used in the vertical direction leave the vertical flux relatively unconstrained and unconnected to any conductors external to the disk. Proper matching of boundary conditions to external fields is also an issue that may affect the behavior of the low density gap regions studied here and a subject for future study.
Acknowledgements
----------------
We would like to thank Richard Nelson for encouragement and support regarding computational matters and him together with Caroline Terquem and Greg Laughlin for valuable discussions. We acknowledge support from the UK Astrophysical Fluids Facility and the NASA Advanced Supercomputing Facility’s Information Power Grid Project’s Pool at NASA ames Research Center. AS thanks the Astronomy Unit at QMW for hospitality, the European Commission for support under contract number ERBFMRX-CT98-0195 (TMR network “Accretion onto black holes, compact stars and protostars”) and the NRC for a research fellowship.
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---
abstract: 'We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate $t^{-n/2}$ in space dimension $n$. We also compute the leading order term in the asymptotic expansion of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, at the expense of making the whole system quasilinear. Stability is then obtained by a global fixed point argument in temporally weighted Sobolev spaces.'
---
[Diffusive stability of oscillations in reaction-diffusion systems]{}\
[Thierry Gallay]{}\
Université de Grenoble I\
Institut Fourier, UMR CNRS 5582\
BP 74\
38402 Saint-Martin-d’Hères, France
[Arnd Scheel]{}\
University of Minnesota\
School of Mathematics\
206 Church St. S.E.\
Minneapolis, MN 55455, USA
[**Corresponding author:**]{} Arnd Scheel
[**Keywords:**]{} periodic solutions, diffusive stability, normal forms, quasilinear parabolic systems
Introduction and main results {#s:1}
=============================
Synchronization of spatially distributed dissipative oscillators has been observed in a wide variety of physical systems. We mention synchronization in yeast cell populations [@GCP], fireflies [@BB], coupled laser arrays [@RTF], and spatially homogeneous oscillations in reaction-diffusion systems such as the Belousov-Zhabotinsky reaction [@Win] and the NO+CO-reaction on a Pt(100) surface [@VMMI]. Synchronization strikes us most when the system size is large or the coupling strength is weak. Both situations relate in natural ways to the regime of large Reynolds number in fluid experiments, where one expects turbulent, incoherent rather than laminar, synchronized behavior. Still, one finds synchronization as a quite common, universal phenomenon, even in very large systems.
The aim of this article is to elucidate the robustness of spatially homogeneous temporal oscillations in spatially extended systems, under most general assumptions, without detailed knowledge of internal oscillator dynamics or coupling mechanisms. In fact, quantitative models are very rarely available for the systems mentioned above. Instead, we make phenomenological assumptions, related to the existence of oscillations and the absence of strongly unstable modes. These assumptions typically guarantee asymptotic stability of a spatially homogeneous oscillation in any *finite-size* system, when equipped with compatible (say, Neumann) boundary conditions. The results in this article are concerned with *infinite-size*, reaction-diffusion systems, $$\label{rdsystem}
u_t \,=\, D\Delta u + f(u),\qquad u = u(t,x) \in{\mathbb{R}}^N\,,
\quad x\in{\mathbb{R}}^n\,, \quad t \ge 0\,,$$ with positive coupling matrix $D \in \mathcal{M}_{N\times N}({\mathbb{R}})$, $D=D^T>0$, and smooth kinetics $f\in C^\infty({\mathbb{R}}^N,{\mathbb{R}}^N)$. In this spatially continuous setup, working in the whole space ${\mathbb{R}}^n$ is an idealization which corresponds to the limit of small coupling matrix and/or large domain size. We will briefly comment on the relation between our results in the whole space and the stability of temporal oscillations in finite domains, below.
To be specific, we make the following assumptions on the kinetics $f$ and the coupling matrix $D$.
\[h:1\] We suppose that the ODE $u_t = f(u)$ possesses a periodic solution $u_*(t) = u_*(t+T)$ with minimal period $T>0$.
In particular, to avoid trivial situations, we assume that the periodic orbit is not reduced to a single equilibrium. As is well-known, this is possible only if $N \ge 2$, i.e. if the system does not reduce to a scalar equation.
In addition to existence we will make a number of assumptions on the Floquet exponents of the linearized equation $$\label{e:lin}
u_t \,=\, D\Delta u + f'(u_*(t))u\,,$$ which is formally equivalent to the family of ordinary differential equations $$\label{e:linf}
u_t \,=\, -k^2D u + f'(u_*(t))u\,, \quad k\in{\mathbb{R}}^n\,.$$ For each fixed $k$ we denote by $F_k(t,s)$ the two-parameter evolution operator associated to the linear time-periodic system , so that $u(t) = F_k(t,s)u(s)$ for any $t \ge s$. The asymptotic behavior of the solutions of is well characterized by the Floquet multipliers of the system, that is, the eigenvalues of the period map $F_k(T,0)$. We shall rather work with the Floquet exponents $\lambda_1(k),\dots,
\lambda_N(k) \in {\mathbb{C}}/{\mathrm{i}}\omega{\mathbb{Z}}$, where $\omega = 2\pi/T$, which satisfy $$\mathrm{det}\,\Bigl(F_k(T,0) - {\mathrm{e}}^{\lambda_j(k)T}\Bigr)
\,=\,0\,, \quad j = 1,\dots,N\,.$$ Remark that any Floquet exponent which is simple (that is, of algebraic multiplicity one) depends smoothly on the parameter $k$. Also note that $\lambda_1 = 0$ is always a Floquet exponent for $k=0$. We refer to the set of Floquet exponents as the Floquet spectrum.
\[h:2\] We suppose that the Floquet spectrum in the closed half-space $\{\Re\lambda\ge 0\}$ is minimal. More precisely, we assume that\
(i) The Floquet spectrum in the closed half-space $\{\Re\lambda\ge
0\}$ is nonempty only for $k = 0$, in which\
case it consists of a simple Floquet exponent $\lambda_1 = 0$;\
(ii) Near $k = 0$, the neutral Floquet exponent continues as $\lambda_1(k) = -d_0 k^2+{\mathrm{O}}(k^4)$ for some $d_0>0$.
We emphasize that these assumptions are satisfied for an open class of reaction-diffusion systems. In particular the expansion (ii) with some $d_0\in{\mathbb{R}}$ is a consequence of the simplicity of the Floquet exponent $\lambda_1 = 0$ at $k=0$, and of the symmetry $k\mapsto -k$; assuming $d_0>0$ is therefore robust. In fact, it is not difficult to show that $$\label{e:d0def}
d_0 \,=\, \frac{\int_0^T (U_*(t),Du_*'(t))\,{\mathrm{d}}t}{\int_0^T
(U_*(t),u_*'(t))\,{\mathrm{d}}t}\,,$$ where $(\cdot,\cdot)$ denotes the usual scalar product in ${\mathbb{R}}^N$, and $U_*(t)$ is the (unique nontrivial) bounded solution of the adjoint equation $$\label{e:adj}
-U_t \,=\, f'(u_*(t))^T U\,.$$ Of course, a necessary condition for Hypothesis \[h:2\] to hold is that $u_*(t)$ be a [*stable*]{} periodic solution of the ODE $u_t = f(u)$, but this assumption alone is not sufficient in general, except if the diffusion matrix is a multiple of the identity. Indeed, even if $N = 2$, one can find examples of periodic solutions which are asymptotically stable for the ODE dynamics, but become unstable if a suitable diffusion is added [@Ri1; @Ri2; @Ri3]. One possible scenario, which is usually called [*phase instability*]{} or [*sideband instability*]{}, is that the coefficient $d_0$ be negative, in which case the periodic orbit is unstable with respect to long-wavelength perturbations. It may also happen that the Floquet spectrum is stable for $k$ in a neighborhood of the origin, but that there exists an unstable Floquet exponent for some $k_* \neq 0$, and therefore for all $k$ in a neighborhood of $k_*$. This mechanism is reminiscent of the [*Turing instability*]{} for spatially homogeneous equilibria. In Section \[s:ex\] below, we give an example of a simple 2-species system which exhibits both kinds of instabilities depending on the choice of the parameters.
In order to state our results, we introduce a function space which measures both the spatial localization and the amplitude of the perturbation to our spatially homogeneous profile $u_*$. We will consider initial perturbations in the space of functions $X =
L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$, with target space ${\mathbb{R}}^N$, equipped with the norm $$\|v\|_X \,=\, \int_{{\mathbb{R}}^n} |v(x)|\,{\mathrm{d}}x \,+\, \sup_{x \in
{\mathbb{R}}^n} |v(x)|\,,$$ where “sup” here refers to the essential supremum. We will measure decay in the space $L^\infty({\mathbb{R}}^n)$. Our first result is:
\[t:1\] Consider a reaction-diffusion system on ${\mathbb{R}}^n$ with oscillatory kinetics, Hypothesis \[h:1\], and marginally stable spectrum, Hypothesis \[h:2\]. Then there are $C,\delta>0$ such that for any initial data $u(0,x) = u_*(t_0) + v_0(x)$ with $t_0\in{\mathbb{R}}$ arbitrary and $\|v_0\|_X \le \delta$, there exists a unique, smooth global solution $u(t,x)$ of for $t \ge 0$. Moreover $u(t,x)$ converges to the periodic solution $u_*$ in the sense that $$\label{e:t1decay}
\sup_{x \in {\mathbb{R}}^n}\Big|u(t,x)-u_*(t_0+t)\Big| \,\le\, \frac{C
\|v_0\|_X}{(1+t)^{n/2}}\,, \quad \hbox{for all }t \ge 0\,.$$
We emphasize that the perturbations we consider are localized in space, and therefore do not alter the overall phase $t_0$ of the periodic solution. We refer however to Section \[s:ex\] for a discussion of possible stability results in more general situations. It is not difficult to verify that the decay rate in is optimal. In fact, under the assumptions of Theorem \[t:1\], one can even compute the leading term in the asymptotic expansion of the perturbation as $t \to +\infty$. Let $$\label{e:GGdef}
G(x) \,=\, \frac{1}{(4\pi d_0)^{n/2}}\,\exp\Bigl(
-\frac{|x|^2}{4d_0}\Bigr)\,, \quad x \in {\mathbb{R}}^n\,,$$ where $d_0 > 0$ is defined in . Our second result is:
\[t:2\] Under the assumptions of Theorem \[t:1\], the solution $u(t,x)$ of can be decomposed as $$\label{e:udecomp}
u(t,x) \,=\, u_*(t_0+t) + u_*'(t_0+t) \alpha(t,x) + \beta(t,x)\,,
\quad x \in {\mathbb{R}}^n\,, \quad t \ge 0\,,$$ where $\alpha : {\mathbb{R}}_+ \times {\mathbb{R}}^n \to {\mathbb{R}}$ and $\beta : {\mathbb{R}}_+ \times
{\mathbb{R}}^n \to {\mathbb{R}}^N$ satisfy $$\begin{aligned}
\label{e:betaest}
\|\beta(t,\cdot)\|_{L^1} + (1+t)^{n/2}\|\beta(t,\cdot)\|_{L^\infty}
& \xrightarrow[t\to +\infty]{}& 0\,, \\ \label{e:alphaest}
\|t^{n/2}\alpha(t,x\sqrt{t}) - \alpha_* G\|_{L^1 \cap
L^\infty} & \xrightarrow[t\to +\infty]{}& 0\,,\end{aligned}$$ for some $\alpha_* \in {\mathbb{R}}$. In addition, $$\label{e:alphastardef}
\alpha_* \,=\, \int_{{\mathbb{R}}^n} \frac{(U_*(t_0),v_0(x))}{(U_*(t_0),u_*'(t_0))}
\,{\mathrm{d}}x + {\mathrm{O}}(\delta^2)\,,$$ where $U_*(t)$ is the bounded solution of the adjoint equation .
In other words, the solution $u(t,x)$ of satisfies $$\begin{aligned}
u(t,x) &=& u_*(t_0+t) + u_*'(t_0+t)\,\frac{\alpha_*}{(4\pi d_0 t)^{n/2}}
\,{\mathrm{e}}^{-|x|^2/(4d_0t)} + {\mathrm{o}}(t^{-n/2}) \\
&=& u_*\Bigl(t_0+t + \frac{\alpha_*}{(4\pi d_0 t)^{n/2}}
\,{\mathrm{e}}^{-|x|^2/(4d_0t)}\Bigr) + {\mathrm{o}}(t^{-n/2})\,, \quad \hbox{as }
t \to +\infty\,.\end{aligned}$$ To leading order, the effect of the perturbation is thus a spatially localized modulation of the phase of the periodic solution. As is clear from the proof, the left-hand side of decays at least like $t^{-\gamma}$ as $t \to \infty$, for some $\gamma > 0$. However, to specify a convergence rate in , it is necessary to restrict ourselves to more localized perturbations. For instance, if we assume in addition that $(1+|x|)v_0 \in L^1({\mathbb{R}}^n)$, then we can prove that the left-hand side of is ${\mathrm{O}}(t^{-1/2})$ as $t \to \infty$.
Under Hypothesis \[h:2\], if we consider system in a large bounded domain (say $x \in \Omega/\varepsilon$ where $\Omega
\subset {\mathbb{R}}^n$ is bounded) with Neumann boundary conditions, the perturbations of the periodic solution $u_*(t)$ decay exponentially [@henry]: if $\|v_0\|_{L^\infty} \le \delta$ for some small $\delta > 0$, then $\|v(t)\|_{L^\infty} \le C \|v_0\|_{L^\infty}
\,{\mathrm{e}}^{-\mu t}$ for some $\mu > 0$. However, the relaxation rate $\mu$ and the size of admissible perturbations $\delta$ both depend on $\varepsilon$, with typical scalings $\mu,\delta =
{\mathrm{O}}(\varepsilon^2)$ predicted by the spectral gap of the Laplacian on the domain $\Omega/\varepsilon$. This gap vanishes in the limit $\varepsilon \to 0$, and Theorem \[t:1\] shows that exponential decay is replaced by diffusive decay. Nevertheless, we expect our results to give an accurate description of the intermediate asymptotics for large bounded domains, if the initial perturbations are sufficiently localized.
The type of diffusive decay that we establish in Theorems \[t:1\] and \[t:2\] has been observed in many other contexts. For instance, localized perturbations of spatially periodic, stationary patterns in the Ginzburg-Landau or Swift-Hohenberg equation exhibit a similar diffusive behavior [@BK92; @CE92; @CEE92; @Ka94; @Sc96; @Ue99]. At a technical level, the approach in [@BK92; @Sc96; @Ue99] is based on [*renormalization group*]{} theory, see for instance [@BK94]. Roughly speaking, the method relies on the fact that the time-$T$ map for the evolution of the perturbations becomes a contraction in a space of localized functions when composed with an appropriate rescaling, except for a neutral direction which specifies the profile of the self-similar solution describing the leading order asymptotics. A nice feature of renormalization theory is that it allows to determine easily which terms in the nonlinearity are “relevant” (that is, potentially dangerous) for the stability analysis. For example, if we consider the nonlinear heat equation $u_t = \Delta u +
|u|^p$ in ${\mathbb{R}}^n$, with small and localized initial data, it is well-known that the nonlinearity $|u|^p$ will not influence the decay predicted by the linear evolution if $p > 1+2/n$, whereas instabilities and even blow-up phenomena can occur if $p \le 1+2/n$ [@Fu66; @CEE92]. In particular, quadratic terms (which arise naturally in the Taylor expansion of any smooth function) are “irrelevant” if $n \ge 3$ and “relevant” if $n = 1,2$. For this reason, diffusive stability is often easier to establish in high space dimensions, when diffusion is strong enough to control all possible nonlinear terms, whereas serious problems can occur in low dimensions. This is the case in particular in the stability analysis of one-dimensional spatially periodic patterns [@Sc96; @EWW97], where a key step of the proof is to show that relevant “self-coupling” terms actually do not occur in the evolution equation for the neutral translational mode.
As one may expect from the discussion above, Theorem \[t:1\] is rather easy to prove when $n \ge 3$. For completeness, we first settle this case in Section \[s:high\] and then focus on the more interesting situation where $n = 1$ or $2$. Here the idea is to construct a [*normal form*]{} transformation for the ODE dynamics which removes all “relevant” terms in the nonlinear PDE satisfied by the perturbation. In Section \[s:nf\], we show that this is possible, at the expense of transforming the semilinear equation into a quasilinear parabolic system. The next important step is to obtain optimal decay estimates for the solutions of the linearized perturbation equation, including maximal regularity estimates, using the spectral assumptions in Hypothesis \[h:2\]. Since the perturbation equation is translation invariant in space and periodic in time, such bounds are relatively straightforward to obtain via Fourier analysis, see Section \[s:le\]. Using these linear estimates, we give in Section \[s:nl\] a proof of Theorem \[t:1\] which is valid for $n \le 3$, hence covering the missing cases $n =
1,2$. Instead of renormalization group theory, we prefer using a global fixed point argument in temporally weighted spaces, as in [@CE92; @CEE92; @Ka94]. After stability has been established, a rather classical procedure, which is recalled in Section \[s:asym\], allows to derive the first-order asymptotics and to prove Theorem \[t:2\] at least for $n \le 3$ (the higher dimensional case is again easier, and left to the reader). In the final Section \[s:ex\], we illustrate our spectral assumptions on a simple, explicit example, and we conclude with a short discussion including possible extensions of our results.
**Acknowledgments.** The authors thank Sylvie Monniaux for useful discussions on maximal regularity in parabolic equations. A.S. would like to thank the Université de Franche-Comté for generous support and hospitality during his stay, where part of this project was carried out. A.S also acknowledges partial support by the NSF through grant DMS-0504271.
Stability in high dimensions {#s:high}
============================
In this section we explore a straightforward and somewhat naive approach to the stability of the periodic orbit $u_*(t)$ as a solution of the reaction-diffusion system . This method gives a simple proof of Theorem \[t:1\] in the high-dimensional case $n
\ge 3$, the main ingredient of which is an $L^p$-$L^q$ estimate for the linearized evolution operator which will be established in Section \[s:le\]. Without loss of generality, we assume from now on that the parameter $t_0$ in Theorems \[t:1\] and \[t:2\] is equal to zero (this is just an appropriate choice of the origin of time).
Consider a solution $u(t,x) = u_*(t) + v(t,x)$ of . The perturbation $v$ satisfies the equation $$\label{e:pertv}
v_t \,=\, D\Delta v + f'(u_*(t))v + N(u_*(t),v)\,,$$ where $$N(u_*(t),v) \,=\, f(u_*(t)+v)-f(u_*(t))-f'(u_*(t))v\,.$$ The Cauchy problem for the semi-linear parabolic system is locally well-posed in the space $X = L^1({\mathbb{R}}^n)
\cap L^\infty({\mathbb{R}}^n)$, see e.g. [@henry; @pazy]. More precisely, for any $v_0 \in X$, there exists a time $\tilde T > 0$ (depending only on $\|v_0\|_X$) such that has a unique (mild) solution $v \in C^0([0,\tilde T],L^1({\mathbb{R}}^n)) \cap C^0_b((0,\tilde T],
L^\infty({\mathbb{R}}^n))$ satisfying $v(0) = v_0$.
\[r:H2data\] Due to parabolic regularization, the solution $v(t,x)$ of is smooth for $t > 0$. For instance, there exists $C > 0$ such that $\|v(t)\|_{H^2} \le C t^{-1}\|v_0\|_X$ for all $t \in (0,\tilde T]$. Therefore, in the proof of Theorem \[t:1\], we can assume without loss of generality that the initial perturbation is small in the space $X \cap
H^2({\mathbb{R}}^n)$.
To investigate the long-time behavior of the solutions of , we consider the corresponding integral equation $$\label{e:pertvint}
v(t) \,=\, \mathcal{F}(t,0)v_0 + \int_0^t \mathcal{F}(t,s)
N(u_*(s),v(s))\,{\mathrm{d}}s\,,$$ where $\mathcal{F}(t,s)$ is the two-parameter semigroup associated to the linearized equation . Due to our spectral assumptions (Hypothesis \[h:2\]), the operator $\mathcal{F}(t,s)$ satisfies the same $L^p$–$L^q$ estimates as the heat semigroup ${\mathrm{e}}^{(t-s)\Delta}$. More precisely, anticipating the results of Section \[s:le\], we have:
\[p:10\] There exists a positive constant $C$ such that, for all $t > s$ and all $1 \le p \le q \le \infty$, we have $$\label{e:newlplq}
\|\mathcal{F}(t,s)v\|_{L^q({\mathbb{R}}^n)} \,\le\, \frac{C}{(t-s)^{\frac{n}2
(\frac1p-\frac1q)}}\ \|v\|_{L^p({\mathbb{R}}^n)}\,.$$
The proof follows exactly the same lines as in Propositions \[p:1\], \[p:2\] and \[p:3\].
By construction, the nonlinearity $N(u_*,v)$ in is at least quadratic in $v$ in a neighborhood of the origin. More precisely, there exists a nondecreasing function $K : {\mathbb{R}}_+
\to {\mathbb{R}}_+$ such that, for all $t \in [0,T]$, $$|N(u_*(t),v)| \,\le\, K(R)|v|^2 \quad \hbox{whenever }
|v| \le R\,.$$ As was mentioned in the introduction, if the space dimension $n$ is greater or equal to $3$, the diffusive effect described in is strong enough to kill the potential instabilities due to the nonlinearity. In that case, nonlinear stability can therefore be established by a classical argument, which we briefly reproduce here for the reader’s convenience.
**Proof of Theorem \[t:1\]** ($n \ge 3$). Fix $v_0 \in X = L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$, and let $v \in
C^0([0,T_*),L^1({\mathbb{R}}^n)) \cap C^0((0,T_*),L^\infty({\mathbb{R}}^n))$ be the maximal solution of with initial data $v_0$. For $t \in [0,T_*)$ we denote $$\phi(t) \,=\, \sup_{0 \le s \le t}\|v(s)\|_{L^1} +
\sup_{0 \le s \le t}(1+s)^{n/2}\|v(s)\|_{L^\infty}\,.$$ Using the integral equation and the linear estimates , we easily find $$\begin{aligned}
\|v(t)\|_{L^1} &\le \|\mathcal{F}(t,0)v_0\|_{L^1} +
\int_0^t \|\mathcal{F}(t,s) N(u_*(s),v(s))\|_{L^1}\,{\mathrm{d}}s\\
&\le C\|v_0\|_{L^1} + CK(\phi(t))\int_0^t \|v(s)\|_{L^1}
\|v(s)\|_{L^\infty} \,{\mathrm{d}}s\\
&\le C\|v_0\|_{L^1} + CK(\phi(t))\phi(t)^2 \int_0^t
\frac{1}{(1+s)^{n/2}}\,{\mathrm{d}}s\,.\end{aligned}$$ Similarly, if $0 < t < 1$, we have $$\begin{aligned}
(1+t)^{n/2}\|v(t)\|_{L^\infty} &\le C\|v_0\|_{L^\infty}
+ C\int_0^t \|N(u_*(s),v(s))\|_{L^\infty}\,{\mathrm{d}}s\\
&\le C\|v_0\|_{L^\infty} + CK(\phi(t))\phi(t)^2 \int_0^t
\frac{1}{(1+s)^n}\,{\mathrm{d}}s\,,\end{aligned}$$ while for $t \ge 1$ we can bound $$\begin{aligned}
(1+t)^{n/2}\|v(t)\|_{L^\infty} &\le C\|v_0\|_{L^1} +
C(1+t)^{n/2}\int_0^{t/2}\frac{1}{(t-s)^{n/2}}\ \|N(u_*(s),
v(s))\|_{L^1}\,{\mathrm{d}}s\\
&\hspace{2cm} + C(1+t)^{n/2}\int_{t/2}^t \|N(u_*(s),v(s))\|_{L^\infty}
\,{\mathrm{d}}s\\
&\le C\|v_0\|_{L^1} + CK(\phi(t))\phi(t)^2\int_0^t\frac{1}{(1+s)^{n/2}}
\,{\mathrm{d}}s\,. \end{aligned}$$ Now, since $n \ge 3$, we have $\int_0^\infty (1+s)^{-n/2}\,{\mathrm{d}}s
< \infty$ and we see that there exist positive constants $C_1, C_2$ (independent of $T_*$) such that $$\label{e:phibd}
\phi(t) \,\le\, C_1 \|v_0\|_X + C_2 K(\phi(t))\phi(t)^2, \quad
\hbox{for all } t \in [0,T_*)\,.$$ So if we further assume that the initial perturbation $v_0 \in
X$ is small enough so that $$2 C_1 \|v_0\|_X \,<\, 1\,,\quad \hbox{and} \quad
4 C_1 C_2 K(1)\|v_0\|_X \,<\, 1\,,$$ then it follows from that $\phi(t) \le 2 C_1
\|v_0\|_X < 1$ for all $t \in [0,T_*)$. Since $[0,T_*)$ is the maximal existence interval, this bound implies that $T_* = +\infty$ and that the solution of satisfies $$\sup_{t\ge 0}\|v(t)\|_{L^1} + \sup_{t \ge 0}(1+t)^{n/2}
\|v(t)\|_{L^\infty} \,\le\, 2C_1 \|v_0\|_X\,.$$ This concludes the proof of Theorem \[t:1\] in the high-dimensional case $n \ge 3$. $\Box$
Reduction to a normal form {#s:nf}
==========================
In low space dimensions the argument presented in the previous section fails, and we must therefore have a closer look at the structure of the perturbation equation. The idea is to introduce a normal form transformation which simplifies the ODE dynamics in a neighborhood of the periodic orbit $u_*$. Applying this transformation to the reaction-diffusion equation , we obtain a quasilinear parabolic system which will be the starting point of our stability analysis in Sections \[s:le\] and \[s:nl\].
We thus consider the ordinary differential equation $$\label{e:ode}
u_t \,=\, f(u)\,, \quad u\in{\mathbb{R}}^N\,,$$ with smooth nonlinearity $f\in C^\infty({\mathbb{R}}^N,{\mathbb{R}}^N)$, and we assume the existence of a time-periodic solution $u_*(t)=u_*(t+T)$ with minimal period $T = 2\pi/\omega > 0$. As in Hypothesis \[h:2\], we suppose that $u_*$ is linearly asymptotically stable, in the sense that the Floquet exponents $\lambda_1,\dots,\lambda_N$ are all contained in the open left half-plane, except for $\lambda_1 = 0$ (which is therefore algebraically simple). We shall show that the dynamics of near the periodic orbit $u_*$ is conjugate to the dynamics of the following normal form $$\label{e:odenf}
\theta_t \,=\, \omega\,,\quad \tilde v_t \,=\, g(\theta,\tilde v)\,,
\quad \theta\in S^1 \cong {\mathbb{R}}/2\pi{\mathbb{Z}}\,,\quad \tilde v\in B_\epsilon
\subset {\mathbb{R}}^{N-1}\,,$$ where $B_\epsilon$ denotes the open ball of radius $\epsilon > 0$ centered at the origin in ${\mathbb{R}}^{N-1}$. Here the vector field $g$ has the expansion $$\label{e:gnfdef}
g(\theta,\tilde v) \,=\, L(\theta)\tilde v + g_2(\theta,\tilde v)
[\tilde v,\tilde v]\,,$$ where $L(\theta)$ is a real $(N-1)\times (N-1)$ matrix depending smoothly on $\theta$, and $g_2(\theta,\tilde v)$ is a symmetric bilinear form on ${\mathbb{R}}^{N-1}$ depending smoothly on $\theta,\tilde v$. In particular $g(\theta,0) = 0$, hence has a trivial solution $\theta(t) = \omega t$, $\tilde v(t) = 0$ which will correspond to the periodic solution $u_*(t)$ of . By construction, the Floquet exponents $\lambda_2,\dots,\lambda_N$ of the time-periodic linear operator $L(\omega t)$ are all contained in the open left half-plane.
In what follows we denote by $\Phi(t)$ the flow of in a neighborhood of $u_*$, and by $\Phi_\mathrm{nf}(t)$ the flow of in a neighborhood of $S^1 \times \{0\}$. These local flows are defined at least for $t \ge 0$.
\[p:fol\] Assume that the periodic solution $u_*$ is linearly asymptotically stable. Then there exist $\epsilon > \epsilon' > 0$ and a smooth diffeomorphism $\Psi$ from the solid torus $S^1\times B_\epsilon$ to a tubular neighborhood of the periodic orbit $u_*$ such that the local flow in $S^1\times {\mathbb{R}}^{N-1}$ defined on $S^1 \times B_{\epsilon'}$ by $$\Phi_\mathrm{nf}(t) \,=\, \Psi^{-1} \circ \Phi(t) \circ \Psi\,,
\quad t \ge 0\,,$$ is the flow induced by an ODE of the form , .
Since the periodic orbit $u_*$ is linearly asymptotically stable, we can find a tubular neighborhood which is smoothly foliated by strong stable fibers. Straightening out these fibers gives the desired representation of the flow. For completeness we construct this straightening change of coordinates in detail.
We start with the linearized equation at the periodic orbit, $u_t=f'(u_*(t))u$, which possesses a linear invariant smooth foliation: if we parametrize the periodic orbit $u_*(t)$ using $\theta
= \omega t \in S^1$, Floquet theory gives smooth families of complementary subspaces $E^\mathrm{ss}(\theta)$ and $E^\mathrm{c}
(\theta)$, such that $\mathrm{dim}\,E^\mathrm{ss}(\theta)=N-1$ and $E^\mathrm{c}(\theta) = \mathrm{span}\,(u_*'(\theta/\omega))$. The linearized evolution leaves these subspaces invariant: $u(t)\in E^\mathrm{ss}(\theta)$ implies $u(t+\tau)\in
E^\mathrm{ss}(\theta+\omega\tau)$, and the same holds for $E^\mathrm{c}(\theta)$. In particular, the family $\{E^\mathrm{ss}
(\theta)\}_{\theta \in S^1}$ forms a smooth normal bundle to the periodic orbit $u_*$ (which is an orientable manifold in ambient Euclidean space), and such a bundle is necessarily trivial. Thus, we can find smooth coordinates $(\theta,v)\in S^1\times
{\mathbb{R}}^{N-1}$ and a smooth map $\Psi_0 : S^1\times{\mathbb{R}}^{N-1} \to {\mathbb{R}}^N$ such that $\Psi_0(\theta,0) = u_*(\theta/\omega)$ and $\Psi(\theta,{\mathbb{R}}^{N-1})=u_*(\theta/\omega)+E^\mathrm{ss}
(\theta)$ for all $\theta \in S^1$.
On the other hand, for each $\theta \in S^1$, the strong stable manifold $W^\mathrm{ss}(\theta)$ of the nonlinear system is the graph of a local map $h_\theta : E^\mathrm{ss}
(\theta) \to E^\mathrm{c}(\theta)$, with $h_\theta(0)=0$ and $h_\theta'(0)=0$. The strong stable manifolds depend smoothly on the base point. In other words, $h_\theta$ depends smoothly on $\theta$, so that the map $$u_*(\theta/\omega)+v^\mathrm{ss} ~\mapsto~
\Psi_1(u_*(\theta/\omega)+v^\mathrm{ss}) \,:=\,
u_*(\theta/\omega) + v^\mathrm{ss}+h_\theta(v^\mathrm{ss})\,,$$ defines a smooth diffeomorphism in a tubular neighborhood $\mathcal{U}$ of the periodic solution. Thus, if $\epsilon > 0$ is sufficiently small, the map $\Psi := \Psi_1 \circ \Psi_0 :
S^1 \times B_\epsilon \to \mathcal{U}$ is also a smooth diffeomorphism onto its image, and $\Psi(\theta,B_\epsilon) \subset W^\mathrm{ss}
(\theta)$ for all $\theta \in S^1$. Since $$\label{e:fol}
\Phi(t )(W^\mathrm{ss}(\theta))\cap \mathcal{U} \,\subset\,
W^\mathrm{ss}(\theta+\omega t)\,,$$ we deduce that, if $\theta \in S^1$ and $\tilde v \in B_{\epsilon'}$ for some small $\epsilon'$, then $(\Phi(t)\circ\Psi)(\theta,\tilde v)$ belongs to the image of $\Psi$ for all $t \ge 0$, and $$\Phi_\mathrm{nf}(t)(\theta,\tilde v) \,:=\, (\Psi^{-1}\circ\Phi(t)
\circ\Psi)(\theta,\tilde v) \,=\, (\theta+\omega t,\hat v)\,,$$ for some $\hat v \in B_\epsilon$. This immediately implies the trivial form $\theta_t = \omega$ for the evolution equation associated to $\Phi_\mathrm{nf}$. Moreover, by construction, $\Phi_\mathrm{nf}
(\theta,0) = (\theta+\omega t,0)$ for all $t \ge 0$, hence the transverse variable $\tilde v$ evolves according to an ODE of the form , where the vector field satisfies $g(\theta,0) = 0$ and can therefore be expanded as in .
We conclude this section with the transformation of the full reaction-diffusion system. The pointwise change of coordinates $u=\Psi(v)=\Psi(\theta,\tilde v)$ yields $$v_t \,=\, \Psi'(v)^{-1}D\Delta \left(\Psi(v)\right)
+f_\mathrm{nf}(v)\,, \quad f_\mathrm{nf}(v) \,=\, (\omega, g(v))^T\,,$$ which can be expanded into $$\label{e:ednf}
v_t \,=\, \Psi'(v)^{-1}D\Psi'(v) \Delta v + \Psi'(v)^{-1} D\Psi''(v)
[\nabla v,\nabla v] + f_\mathrm{nf}(v)\,.$$ We are interested in the stability of the periodic orbit $v_*(t)=(\omega t,0)^T$, and therefore we set $v = v_*(t)+w(t,x)$, so that $w$ solves $$\label{e:ednfp}
w_t \,=\, \Psi'(v_*+w)^{-1}D\Psi'(v_*+w) \Delta w +
\Psi'(v_*+w)^{-1}D\Psi''(v_*+w)[\nabla w,\nabla w]
+f_\mathrm{nf}^0(v_*+w)\,,$$ where now $f_\mathrm{nf}^0(v)=(0,g(v))^T$. In what follows, the first component of the vector $w$ will play a distinguished role, as is clear from the expression of $f_\mathrm{nf}^0$. Thus we shall often write $w=(w_0,w_\mathrm{h})^T$, with $w_0 \in {\mathbb{R}}$ and $w_\mathrm{h}
\in {\mathbb{R}}^{N-1}$.
Linear evolution estimates {#s:le}
==========================
We consider the linearization of at $w = 0$, which reads $$\label{e:ednfl}
w_t \,=\, \Psi'(v_*)^{-1}D\Psi'(v_*) \Delta w
+f_\mathrm{nf}'(v_*)w\,.$$ To simplify the notations, we define $$A(t) \,=\, \Psi'(v_*(t))~, \quad \hbox{and} \quad
B(t) \,=\, \begin{pmatrix}0 & 0 \\ 0 & L(\omega t)
\end{pmatrix}~,$$ where $L(\theta)$ is the $(N-1) \times (N-1)$ matrix which appears in . Note that $A(t)$, $B(t)$ are $T$-periodic $N \times N$ matrices, and that $A(t)$ is invertible for all $t$. The linearization then becomes $$\label{e:ednfls}
w_t \,=\, A(t)^{-1}DA(t) \Delta w + B(t)w\,, \quad t \in {\mathbb{R}}\,.$$ By Fourier duality this system is equivalent to the family of ODEs $$\label{e:lf}
w_t \,=\, -k^2 A(t)^{-1}DA(t)w + B(t)w\,, \quad t \in {\mathbb{R}}\,,
\quad k \in {\mathbb{R}}^n\,.$$ Since is related to by the $T$-periodic linear transformation $u=\Psi'(v_*(t))w$, it is clear that the Floquet spectrum of is identical to that of and therefore satisfies Hypothesis \[h:2\]. Let $M(t,s;k)$ denote the evolution operator defined by , so that any solution of satisfies $w(t) = M(t,s;k)w(s)$ for $t \ge s$. As $w=(w_0,w_\mathrm{h})^T \in
{\mathbb{R}}\times{\mathbb{R}}^{N-1}$, it is natural to decompose the matrix $M$ in blocks as follows: $$M(t,s;k) \,=\, \begin{pmatrix} M_{00}(t,s;k) & M_{0\mathrm{h}}(t,s;k)
\\ M_{\mathrm{h}0}(t,s;k) & M_{\mathrm{hh}}(t,s;k) \end{pmatrix}\,,$$ where $M_{00}$, $M_{0\mathrm{h}}$, $M_{\mathrm{h}0}$, $M_{\mathrm{hh}}$ are matrices of size $1\times 1$, $1\times (N-1)$, $(N-1)\times 1$, $(N-1)\times (N-1)$, respectively. The main result of this section is the following pointwise estimate on $M(t,s;k)$:
\[p:1\] There exist constants $C,d>0$ such that, for all $t \ge s$ and all $k \in {\mathbb{R}}^n$, one has $$\begin{aligned}
|M_{00}(t,s;k)| &\le C\,{\mathrm{e}}^{-dk^2(t-s)}\,,\label{e:e1}\\
|M_{0\mathrm{h}}(t,s;k)|+|M_{\mathrm{h}0}(t,s;k)| &\le
\frac{C}{1+t-s} \,{\mathrm{e}}^{-dk^2(t-s)}\,,\label{e:e2}\\
|M_{\mathrm{hh}}(t,s;k)|&\le \frac{C}{(1+t-s)^2} \,{\mathrm{e}}^{-dk^2(t-s)}\,,
\label{e:e3}\end{aligned}$$ where the norms on the left-hand side are arbitrary, $k$-independent matrix norms.
Since the coefficients in are $T$-periodic, we have $M(t+T,s+T;k) = M(t,s;k)$ for all $t,s \in {\mathbb{R}}$ and all $k \in {\mathbb{R}}^n$. As a consequence, if $t \ge s$ and if $\tau_1,\tau_2 \in [0,T)$ are such that $t-\tau_1$ and $s+\tau_2$ are integer multiples of $T$, we have the identity $$\label{e:factor}
M(t,s;k) \,=\, M(t,t-\tau_1;k)\,M(k)^m\,M(s+\tau_2,s;k)\,,
\quad k \in {\mathbb{R}}^n\,,$$ where $M(k) = M(T,0;k)$ and $m \in {\mathbb{N}}$ is such that $t-s =
\tau_1 + mT + \tau_2$. To prove Proposition \[p:1\], it is therefore sufficient to estimate $M(k)^m$ and $M(t,s;k)$ for $0 \le t-s \le T$.
**Step 1:** Estimates on $M(k)^m$.\
The general strategy is to distinguish between various parameters regimes. For small and intermediate values of $k$, we essentially exploit Hypothesis \[h:2\], while for large $k$ it is sufficient to use the parabolicity of .
*Small $k$:* We solve the ODE perturbatively for $t \in [0,T]$ and obtain $$w(t) \,=\, U(t,0)w(0) - k^2 \int_0^t U(t,s)A(s)^{-1} D A(s)
U(s,0)w(0)\,{\mathrm{d}}s + {\mathrm{O}}(k^4)\,,$$ where $U(t,s) = M(t,s;0)$ is the evolution operator associated to the equation $w_t = B(t)w$. In particular, setting $t = T$, we find $$\begin{aligned}
\nonumber
M(k) \,&=\, U(T,0)\left({\mathrm{\,id}\,}- k^2 \int_0^T (A(t)U(t,0))^{-1}
D (A(t)U(t,0))\,{\mathrm{d}}t + {\mathrm{O}}(k^4)\right) \\ \label{e:Mkexp}
\,&=\, \begin{pmatrix}1 & 0 \\ 0 & V\end{pmatrix}
\begin{pmatrix}1 -d_0 T k^2 + {\mathrm{O}}(k^4) & {\mathrm{O}}(k^2)\\
{\mathrm{O}}(k^2) & {\mathrm{\,id}\,}+ {\mathrm{O}}(k^2)\end{pmatrix}\,,\end{aligned}$$ where $V$ is the $(N-1)\times(N-1)$ Floquet matrix associated to the $T$-periodic linear operator $L(\omega t)$, and $$\label{e:d0def2}
d_0 \,=\, \frac1T \int_0^T e_1^T (A(t)U(t,0))^{-1}
D (A(t)U(t,0))e_1\,{\mathrm{d}}t \,=\,\frac1T \int_0^T e_1^T A(t)^{-1}
D A(t)e_1\,{\mathrm{d}}t\,.$$ Here $e_1 = (1,0)^T$ is the first vector of the canonical basis in ${\mathbb{R}}^N$. As was already observed, all eigenvalues of $V$ are contained in the disk $\{z \in {\mathbb{C}}\,|\, |z| < e^{-\nu}\}$ for some $\nu > 0$, and it follows from that $M(k)$ has exactly $N-1$ eigenvalues in this disk if $k$ is sufficiently small. The remaining Floquet multiplier has the expansion $1 - d_0 T k^2 + {\mathrm{O}}(k^4)$, in agreement with Hypothesis \[h:2\]. Incidentally, we observe that is identical to . Indeed, since $u_*(t) = \Psi(v_*(t)) = \Psi(\omega t e_1)$, we have $u_*'(t) =
\omega\Psi'(v_*(t))e_1 = \omega A(t)e_1$, and it is also straightforward to verify that the bounded solution of the adjoint equation is $U_*(t) = (A(t)^{-1})^T e_1$. Thus can be written as $$d_0 \,=\, \frac{1}{\omega T} \int_0^T U_*(t)^T D u_*'(t)\,{\mathrm{d}}t
\,=\, \frac{\int_0^T U_*(t)^T D u_*'(t)\,{\mathrm{d}}t}
{\int_0^T U_*(t)^T u_*'(t)\,{\mathrm{d}}t}\,,$$ and Hypothesis \[h:2\] guarantees that $d_0 > 0$.
For $k \in {\mathbb{R}}^n$ sufficiently small, let $P(k)$ denote the spectral projection onto the one-dimensional eigenspace of $M(k)$ corresponding to the neutral Floquet exponent $\lambda_1(k) = -d_0 k^2 + {\mathrm{O}}(k^4)$. From it is easy to verify that $P(k)$ has the following form $$P(k) \,=\, \frac{1}{1+k^4 b^T a}\begin{pmatrix} 1 & k^2 b^T \\
k^2 a & k^4 a b^T\end{pmatrix}\,,$$ where $a(k), b(k)$ are $(N-1)$-dimensional vectors with $a(k),
b(k) = {\mathrm{O}}(1)$ as $k \to 0$. By construction, we have for any $m \in {\mathbb{N}}^*$: $$\label{e:Mkm}
M(k)^m \,=\, M(k)^m P(k) + M(k)^m ({\mathrm{\,id}\,}- P(k)) \,=\,
{\mathrm{e}}^{mT\lambda_1(k)}P(k) + (M(k) ({\mathrm{\,id}\,}- P(k)))^m\,.$$ Since $\lambda_1(k) \le -d_1 k^2$ for small $k$ if $0 < d_1 < d_0$, and since the spectral radius of $M(k)({\mathrm{\,id}\,}- P(k))$ is smaller than $e^{-\nu}$, we conclude that $$|M(k)^m| \,\le\, C\,{\mathrm{e}}^{-d_1 k^2 mT}\begin{pmatrix} 1 & k^2 \\
k^2 & k^4\end{pmatrix} + {\mathrm{O}}(e^{-\nu m}) \,\le\,
C\,{\mathrm{e}}^{-d_1 k^2 mT}\begin{pmatrix} 1 & (mT)^{-1} \\
(mT)^{-1} & (mT)^{-2}\end{pmatrix}\,,$$ for all $m \in {\mathbb{N}}^*$ if $|k| \le \kappa_0 \ll 1$. Here the matrix norm $|\cdot|$ is applied separately to each of the four blocks of $M(k)^m$, and for convenience the four upper bounds are collected in a $2\times 2$ matrix.
*Large $k$:* In this parameter regime, it is more convenient to set $v(t) = A(t)w(t)$ and to solve the $v$-equation corresponding to , namely $$\label{e:lff}
v_t \,=\, -k^2 D v + C(t)v\,, \quad \hbox{where} \quad
C(t) \,=\, A'(t)A(t)^{-1} + A(t)B(t)A(t)^{-1}\,.$$ The matrix $C(t)$ is $T$-periodic, hence uniformly bounded. A standard energy estimate yields $$\frac12\frac{{\mathrm{d}}}{{\mathrm{d}}t}\|v(t)\|_2^2 \,=\,
-k^2 (v(t),Dv(t)) + (v(t),C(t)v(t)) \,\le\, -d_2 k^2
\|v(t)\|_2^2 + K \|v(t)\|_2^2\,,$$ where $d_2 > 0$ is the smallest eigenvalue of the (symmetric and positive) matrix $D$, and $K = \sup_{t \in [0,T]}\|C(t)\|_2$. Thus any solution of satisfies $\|v(t)\|_2 \le
{\mathrm{e}}^{(-d_2 k^2 + K)t}\|v(0)\|_2$, and returning to the $w$-equation we obtain $$\label{e:largew}
\|w(t)\|_2 \,\le\, C\,{\mathrm{e}}^{(-d_2 k^2 + K)t}\,\|w(0)\|_2\,,
\quad t \ge 0\,,$$ for some $C > 0$. In particular, if we choose $t = mT$ and if we assume that $|k| \ge \kappa_1$ with $\kappa_1 = (2K/d_2)^{1/2}$, we arrive at $$|M(k)^m| \,\le\, C\,{\mathrm{e}}^{-d_3 k^2 mT}\,, \quad \hbox{where}
\quad d_3 \,=\, \frac{d_2}{2T}\,.$$
*Intermediate $k$:* By Hypothesis \[h:2\], if $\kappa_0
\le |k| \le \kappa_1$, the spectrum of $M(k)$ is entirely contained in the disk $\{z \in {\mathbb{C}}\,|\, |z| < {\mathrm{e}}^{-\mu}\}$ for some $\mu > 0$. Moreover, the resolvent matrix $(z - M(k))^{-1}$ is uniformly bounded for all $z$ on the circle $\{|z| = {\mathrm{e}}^{-\mu}\}$ and all $k$ in the annulus $\kappa_0 \le |k| \le \kappa_1$. If $0 < d_4 < \mu/(\kappa_1^2T)$, it follows that $$|M(k)^m| \,\le\, C\,{\mathrm{e}}^{-d_4 k^2 mT}\,, \quad m \in {\mathbb{N}}\,,$$ where the constant $C$ is independent of $k$.
Summarizing the results obtained so far, we proved that there exist $C > 0$ and $d > 0$ such that $$\label{e:step1}
|M(k)^m| \,\le\, C\,{\mathrm{e}}^{-d k^2 mT} \begin{pmatrix} 1 & (mT)^{-1} \\
(mT)^{-1} & (mT)^{-2}\end{pmatrix}\,,$$ for all $k \in {\mathbb{R}}^n$ and all $m \in {\mathbb{N}}^*$. This is the particular case of – when $s = 0$ and $t = mT$.
**Step 2:** Estimates on $M(t,s;k)$ for $0 \le t-s \le 2T$.\
Our goal is to show that $$\label{e:step2}
|M(t,s;k)| \,\le\, C\,{\mathrm{e}}^{-d k^2(t-s)} \begin{pmatrix} 1 &
k^2(t-s) \\ k^2(t-s) & 1\end{pmatrix}\,,$$ for some $C > 0$ and $d > 0$. Note that implies – if $0 \le t-s \le 2T$.
In the case where $k^2(t-s)$ is small, say $k^2(t-s) \le \kappa_2
\ll 1$, we can solve perturbatively and obtain as in $$\label{e:Mtsexp}
M(t,s;k) \,=\, \begin{pmatrix} 1 & 0 \\ 0 & V(t,s)\end{pmatrix}
\,\Bigl({\mathrm{\,id}\,}+ {\mathrm{O}}(k^2(t-s))\Bigr)\,,$$ from which follows (for any fixed $d > 0$). If $k^2(t-s) \ge \kappa_2$, then using we immediately find $$|M(t,s;k)| \,\le\, C\,{\mathrm{e}}^{(-d_2 k^2 + K)(t-s)} \,\le\,
C\,{\mathrm{e}}^{2KT}\,{\mathrm{e}}^{-d_2 k^2 (t-s)}\,,$$ which implies with $d = d_2$.
It is now straightforward to conclude the proof of Proposition \[p:1\]. In view of , it remains to prove – for $t-s \ge 2T$. We decompose $t-s = \tau_1 + mT + \tau_2$ with $\tau_1,\tau_2
\in [0,T)$ and $m \in {\mathbb{N}}^*$, and we factorize $M(t,s;k)$ as in . Using , , we find $$\begin{aligned}
|M(t,s;k)| \,&\le\, C\,{\mathrm{e}}^{-dk^2 (t-s)}
\begin{pmatrix} 1 & k^2 \tau_1 \\ k^2 \tau_1 & 1\end{pmatrix}
\begin{pmatrix} 1 & (mT)^{-1} \\ (mT)^{-1} & (mT)^{-2}\end{pmatrix}
\begin{pmatrix} 1 & k^2 \tau_2 \\ k^2 \tau_2 & 1\end{pmatrix} \\
\,&\le\, C\,{\mathrm{e}}^{-dk^2 (t-s)} \begin{pmatrix} 1 &
(1+t-s)^{-1} \\ (1+t-s)^{-1} & (1+t-s)^{-2}\end{pmatrix}\,,\end{aligned}$$ which is the desired estimate.
\[r:asym\] For all $k \in {\mathbb{R}}^n$ and all $s \in {\mathbb{R}}$, we have $$\lim_{t \to +\infty} M(t,s;kt^{-1/2}) \,=\, {\mathrm{e}}^{-d_0 k^2}
\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}\,,$$ where $d_0$ is as in Hypothesis \[h:2\]. This follows from the proof of Proposition \[p:1\], and in particular from , , .
\[p:2\] For any $\alpha \in {\mathbb{N}}^n$ there exists $C > 0$ such that, for all $t > s$ and all $k \in {\mathbb{R}}^n$, $$\label{e:derest}
|\partial_k^\alpha M(t,s;k)| \,\le\, C (t-s)^{|\alpha|/2}
\,{\mathrm{e}}^{-dk^2(t-s)}\,.$$ The same estimate holds for $M_{00}$, $(1+t-s)M_{0\mathrm{h}}$, $(1+t-s)M_{\mathrm{h}0}$, and $(1+t-s)^2 M_{\mathrm{hh}}$.
It is clear from that $M(t,s;k)$ depends on the parameter $k \in {\mathbb{R}}^n$ only through the scalar quantity $p = k^2$. In this proof, we set $M(t,s;k) = \tilde M(t,s;k^2)$ and we consider the derivatives of $\tilde M(t,s;p)$ with respect to $p$. Our goal is to prove the estimate $$\label{e:derest2}
|\partial_p^j \tilde M(t,s;p)| \,\le\, C_j (t-s)^j\,{\mathrm{e}}^{-dp(t-s)},
\quad j \in {\mathbb{N}}\,,$$ which implies immediately . Differentiating with respect to $p = k^2$, we find $$(\partial_p w)_t \,=\, -\mathcal{D}(t) w + (B(t) - p\mathcal{D}(t))
(\partial_p w)\,,$$ where $\mathcal{D}(t) = A(t)^{-1}DA(t)$. Since $\partial_p
M(s,s;p) = 0$, we deduce that $$\partial_p\tilde M(t,s;p) \,=\, -\int_s^t \tilde M(t,t_1;p)
\mathcal{D}(t_1)\tilde M(t_1,s;p)\,{\mathrm{d}}t_1\,.$$ Iterating this procedure, we obtain for any $j \in {\mathbb{N}}$ the representation formula $$\begin{aligned}
\partial_p^j\tilde M(t,s;p) &= (-1)^j j! \int_s^t \int_s^{t_1}
\dots \int_s^{t_{j-1}} \tilde M(t,t_1;p)\mathcal{D}(t_1) \times\\
&\quad \times M(t_1,t_2;p)\mathcal{D}(t_2) \dots
M(t_{j-1},t_j;p)\mathcal{D}(t_j)M(t_j,s;p)\,{\mathrm{d}}t_j \dots \,{\mathrm{d}}t_1\,. \end{aligned}$$ Using now the pointwise estimates established in Proposition \[p:1\], we easily obtain . Similar bounds can be proved for $\tilde M_{00}$, $(1+t-s)\tilde M_{0\mathrm{h}}$, $(1+t-s)\tilde
M_{\mathrm{h}0}$, and $(1+t-s)^2 \tilde M_{\mathrm{hh}}$ (we omit the details).
We now convert our estimates on the Fourier multipliers $M(t,s;k)$ into bounds on the linear evolution equation in various $L^p$ spaces. The two-parameter semigroup $\mathcal{M}(t,s)$ associated to is defined using Fourier transform by the relation $$\label{e:le}
\widehat{(\mathcal{M}(t,s)v)}(k) \,=\, M(t,s;k)\hat{v}(k)\,,
\quad k \in {\mathbb{R}}^n\,.$$ The following proposition contains the main estimates on $\mathcal{M}(t,s)$ which will be used in the nonlinear stability proof.
\[p:3\] There exist a positive constant $C$ such that, for all $t > s$ and all $1 \le p \le q \le \infty$, one has $$\label{e:lplq}
\|\mathcal{M}(t,s)v\|_{L^q({\mathbb{R}}^n)} \,\le\, \frac{C}{(t-s)^{\frac{n}2
(\frac1p-\frac1q)}}\ \|v\|_{L^p({\mathbb{R}}^n)}\,.$$ The same estimate holds for $\mathcal{M}_{00}$, $(1+t-s)
\mathcal{M}_{0\mathrm{h}}$, $(1+t-s)\mathcal{M}_{\mathrm{h}0}$, and $(1+t-s)^2\mathcal{M}_{\mathrm{hh}}$.
By construction $\mathcal{M}(t,s)$ is the convolution operator with the function $x \mapsto \mathcal{M}(t,s;x)$, which is just the inverse Fourier transform of $k \mapsto M(t,s;k)$. Thus using the pointwise estimate , we easily obtain $$\|\mathcal{M}(t,s;\cdot)\|_{L^\infty({\mathbb{R}}^n)} \,\le\, C
\|M(t,s;\cdot)\|_{L^1({\mathbb{R}}^n)} \le \frac{C}{(t-s)^{n/2}}\,.$$ To estimate the $L^1$ norm of $\mathcal{M}(t,s;\cdot)$, we use Sobolev embeddings. Let $m \in {\mathbb{N}}$ be the smallest integer such that $m > n/2$. Using the estimates of Proposition \[p:2\] together with Hölder’s inequality and Parseval’s identity, we find $$\begin{aligned}
&\|\mathcal{M}(t,s;\cdot)\|_{L^1({\mathbb{R}}^n)} \,=\,
\int_{{\mathbb{R}}^n} \Bigl(1 + \frac{|x|^2}{t-s}\Bigr)^{-m/2}
\Bigl(1 + \frac{|x|^2}{t-s}\Bigr)^{m/2} |\mathcal{M}(t,s;x)|\
{\mathrm{d}}x\\
&\qquad \,\le\, C(t-s)^{n/4}\left(\int_{{\mathbb{R}}^n}\Bigl(1 + \frac{|x|^2}{t-s}
\Bigr)^m |\mathcal{M}(t,s;x)|^2\,{\mathrm{d}}x\right)^{1/2}\\
&\qquad \,\le\, C(t-s)^{n/4} \left(\int_{{\mathbb{R}}^n}\sum_{|\alpha| \le m}
(t-s)^{-|\alpha|}|\partial_k^\alpha M(t,s;k)|^2 {\mathrm{d}}k\right)^{1/2}
\le C\,.\end{aligned}$$ Summarizing, we have shown that $$\|\mathcal{M}(t,s;\cdot)\|_{L^p({\mathbb{R}}^n)} \,\le\, \frac{C}{(t-s)^{
\frac{n}2(1-\frac1p)}}, \quad 1 \le p \le \infty\,,$$ and follows by Young’s inequality. The estimates for $\mathcal{M}_{00}$, $(1+t-s)\mathcal{M}_{0\mathrm{h}}$, $(1+t-s)\mathcal{M}_{\mathrm{h}0}$, and $(1+t-s)^2\mathcal{M}_{
\mathrm{hh}}$ are proved in exactly the same way.
\[r:1\] Under the assumptions of Proposition \[p:3\], we also have $$\|\partial_x^\alpha \mathcal{M}(t,s)v\|_{L^q({\mathbb{R}}^n)} \,\le\,
\frac{C_\alpha}{(t-s)^{\frac{n}2(\frac1p-\frac1q)+\frac{|\alpha|}{2}}}\,
\|v\|_{L^p({\mathbb{R}}^n)}\,, \quad \alpha \in {\mathbb{N}}^n\,,$$ and the same estimates hold for $\mathcal{M}_{00}$, $(1+t-s)
\mathcal{M}_{0\mathrm{h}}$, $(1+t-s)\mathcal{M}_{\mathrm{h}0}$, and $(1+t-s)^2\mathcal{M}_{\mathrm{hh}}$. This is obvious in view of Proposition \[p:1\], since the operator $\partial_x^\alpha
\mathcal{M}(t,s)$ is the Fourier multiplier associated to the function $(ik)^\alpha M(t,s;k)$.
\[r:2\] For later use, we also observe that, if $t > 0$ and $0 < s < t$, then $$\label{e:lplqdiff}
\|(\mathcal{M}_{00}(t,s) - \mathcal{M}_{00}(t,0))v\|_{L^q({\mathbb{R}}^n)}
\,\le\, \frac{C}{(t-s)^{\frac{n}2(\frac1p-\frac1q)}}\,\frac{s}{t}\,
\|v\|_{L^p({\mathbb{R}}^n)}\,,$$ for $1 \le p \le q \le \infty$. If $t/2 \le s \le t$, this bound follows immediately from and the triangle inequality. If $0 < s \le t/2$, we observe that $$M_{00}(s,0;k) \,=\, 1 + k^2 s R_0(s,k)\,, \quad
M_{\mathrm{h}0}(s,0;k) \,=\, k^2 s R_\mathrm{h}(s,k)\,,$$ where $R_0(s,k)$, $R_\mathrm{h}(s,k)$ are uniformly bounded for $s > 0$ and $k \in {\mathbb{R}}^n$, see . Since $$M_{00}(t,s;k) - M_{00}(t,0;k) \,=\, M_{00}(t,s;k)(1 - M_{00}(s,0;k))
- M_{0\mathrm{h}}(t,s;k)M_{\mathrm{h}0}(s,0;k)\,,$$ we obtain the pointwise bound $$|M_{00}(t,s;k) - M_{00}(t,0;k)| \,\le\, C k^2 s \,{\mathrm{e}}^{-k^2 d(t-s)}
\,\le\, C\,\frac{s}{t-s}\,{\mathrm{e}}^{-k^2 d(t-s)}\,,$$ which allows to establish the $L^p$-$L^q$ estimate using the same arguments as in the proof of Proposition \[p:3\].
Finally, to control the quasilinear terms in the perturbation equation we will use maximal regularity properties of the evolution semigroup $\mathcal{M}(t,s)$.
\[p:4\] For any $r \in (1,+\infty)$ and any $\tilde T > 0$, there exists $C > 0$ such that the following holds. If $v \in L^r((0,\tilde T),
L^2({\mathbb{R}}^n))$ and if $w$ satisfies $$\label{e:wvdef}
w(t) \,=\, \int_0^t \mathcal{M}(t,s)v(s)\,{\mathrm{d}}s\,, \quad t \in
[0,\tilde T]\,,$$ then $$\label{e:maxres}
\int_0^{\tilde T} \|\Delta w(t)\|_{L^2}^r \,{\mathrm{d}}t \,\le\, C
\int_0^{\tilde T} \|v(t)\|_{L^2}^r \,{\mathrm{d}}t\,.$$
For any $t \in {\mathbb{R}}$ the generator $\mathcal{L}(t) = A(t)^{-1}DA(t)
\Delta + B(t)$ is an elliptic operator in $L^2({\mathbb{R}}^n)$ with (time-independent) domain $H^2({\mathbb{R}}^n)$. Moreover $\mathcal{L}(t)$, considered as a bounded operator from $H^2({\mathbb{R}}^n)$ into $L^2({\mathbb{R}}^n)$, is a smooth function of $t$. Thus estimate is a particular case of the results established in [@HM00; @PS01].
We also state a corollary which will be useful in the next section.
\[c:max\] Fix $r \in (1,\infty)$ and $\alpha \in {\mathbb{R}}$. There exists $C > 0$ such that, if $$w(t) \,=\, \int_{t-T}^t \mathcal{M}(t,s)v(s)\,{\mathrm{d}}s\,, \quad
t \ge T\,,$$ then $$\int_T^{\infty} (1+t)^\alpha \|\Delta w(t)\|_{L^2}^r \,{\mathrm{d}}t
\,\le\, C \int_0^\infty (1+t)^\alpha \|v(t)\|_{L^2}^r \,{\mathrm{d}}t\,.$$
Fix $k \in {\mathbb{N}}^* := {\mathbb{N}}\setminus\{0\}$. If $t \in [kT,(k+1)T]$, where $T > 0$ is the period of $\mathcal{L}(t)$, we can write $$w(t) \,=\, \int_{(k-1)T}^t \mathcal{M}(t,s)v(s)\,{\mathrm{d}}s
- \int_{(k-1)T}^{t-T} \mathcal{M}(t,s)v(s)\,{\mathrm{d}}s\,.$$ Since $\mathcal{M}(t+T,s+T) = \mathcal{M}(t,s)$, and since $0 \le t - (k-1)T \le 2T$, we can use Proposition \[p:4\] (with $\tilde T = 2T$) to control both terms in the right-hand side. We obtain $$\int_{kT}^{(k+1)T} \|\Delta w(t)\|_{L^2}^r \,{\mathrm{d}}t \,\le\, C
\int_{(k-1)T}^{(k+1)T} \|v(t)\|_{L^2}^r \,{\mathrm{d}}t\,,$$ for some $C > 0$ independent of $k$. Multiplying both sides by $(kT)^\alpha \approx (1+t)^\alpha$ and summing over $k \in {\mathbb{N}}^*$, we obtain the desired result.
Nonlinear stability in low dimensions {#s:nl}
=====================================
This section is devoted to the proof of Theorem \[t:1\] in the case $n \le 3$. Smoothness of the change of coordinates $\Psi$ implies that it is sufficient to prove the decay estimate for the transformed equation , with smallness assumptions on the initial data $w_0$. To simplify the notations, we rewrite in the compact form $$\label{e:13}
w_t \,=\, \mathcal{L}(t) w + F(t,w,\Delta w) + G(t,w,\nabla w)+
H(t,w)\,,$$ where $\mathcal{L}(t)= A(t)^{-1}DA(t)\Delta + B(t)$ is the linear operator studied in Section \[s:le\] and $$\begin{aligned}
F(t,w,\Delta w)\,&=\,\left(\Psi'(v_*(t)+w)^{-1}-\Psi'(v_*(t))^{-1}\right)
D \left(\Psi'(v_*(t)+w)\right)\Delta w \\
&\quad + \Psi'(v_*(t))^{-1}D\left(\Psi'(v_*(t)+w)-\Psi'(v_*(t))\right)
\Delta w\,,\\
G(t,w,\nabla w) \,&=\, \Psi'(v_*(t)+w)^{-1}D\Psi''(v_*(t)+w)[\nabla w,
\nabla w]\,,\\
H(t,w) \,&=\, (0,\hat g_2(t,w))^T, \quad \hbox{where} \quad
\hat g_2(t,w) \,=\, g_2(v_*(t)+w)[w_\mathrm{h},w_\mathrm{h}]\,.\end{aligned}$$ Clearly, $F(t,w,\Delta w)={\mathrm{O}}(w\Delta w)$ and $G(t,w,\nabla w)=
{\mathrm{O}}(|\nabla w|^2)$. More precisely, if $\|w\|_{L^\infty}$ is sufficiently small, we have $$\begin{array}{l}
\|F(t,w,\Delta w)\|_{L^1} \,\le\, C\|w\|_{L^2}\|\Delta w\|_{L^2}\,,\\[1mm]
\|F(t,w,\Delta w)\|_{L^2} \,\le\, C\|w\|_{L^\infty}\|\Delta w\|_{L^2}\,,
\end{array} \qquad
\begin{array}{l}
\|G(t,w,\nabla w)\|_{L^1} \,\le\, C\|\nabla w\|^2_{L^2}\,,\\[1mm]
\|G(t,w,\nabla w)\|_{L^2} \,\le\, C\|\nabla w\|^2_{L^4}\,.
\end{array}$$ Since $$\|\nabla w\|^2_{L^2}\,\le\, C\|w\|_{L^2}\|\Delta w\|_{L^2}\quad
\hbox{and}\quad \|\nabla w\|^2_{L^4}\,\le\, C\|w\|_{L^\infty}
\|\Delta w\|_{L^2}\,,$$ we find $$\label{e:nlest}
\begin{array}{l}
\|F(t,w,\Delta w)\|_{L^1}+\|G(t,w,\nabla w)\|_{L^1} \,\le\,
C\|w\|_{L^2}\|\Delta w\|_{L^2}\,,\\[1mm]
\|F(t,w,\Delta w)\|_{L^2}+\|G(t,w,\nabla w)\|_{L^2} \,\le\,
C\|w\|_{L^\infty}\|\Delta w\|_{L^2}\,.
\end{array}$$ Under the same assumptions, we also have $$\|H(t,w)\|_{L^1}\,\le\, C \|w_\mathrm{h}\|_{L^2}^2\quad \hbox{and}\quad
\|H(t,w)\|_{L^2}\,\le\, C \|w_\mathrm{h}\|_{L^2}\|w_\mathrm{h}\|_{L^\infty}\,.
\label{e:nlesth}$$ Setting $K(t,w,\nabla w,\Delta w)=F(t,w,\Delta w)+G(t,w,\nabla w)$, we can write the integral equation associated with in the form, $$\label{e:mild}
w(t)\,=\,\mathcal{M}(t,0)w_0 + \int_0^t \mathcal{M}(t,s)
K(s,w,\nabla w,\Delta w) \,{\mathrm{d}}s + \int_0^t \mathcal{M}(t,s)
H(s,w) \,{\mathrm{d}}s\,.$$
We now describe the function space in which we shall look for solutions of . Assume that $n \le 3$ and choose $r \in (4,+\infty)$, so that $$\label{e:rineq}
\frac1r \,<\, \frac{n}4 \,<\, 1 - \frac1r\,.$$ We define the Banach space $$Y \,=\, \Bigl\{w \in C^0([0,+\infty),L^1({\mathbb{R}}^n)\cap
L^\infty({\mathbb{R}}^n)) \cap L^r((0,+\infty),\dot H^2({\mathbb{R}}^n)) \,\Big|\,
\|w\|_Y < \infty\Bigr\}\,,$$ where $$\label{e:norm}
\|w\|_Y \,=\, \sup_{t \ge 0} \|w(t)\|_{L^1} + \sup_{t \ge 0}
(1+t)^{n/2} \|w(t)\|_{L^\infty} + \left(\int_0^\infty
(1+t)^r \|\Delta w(t)\|^r_{L^2}\,{\mathrm{d}}t\right)^{1/r}\,.$$
If $w_0 \in L^1({\mathbb{R}}^n) \cap H^2({\mathbb{R}}^n)$, the linear solution $W_0(t) =
\mathcal{M}(t,0)w_0$ belongs to $Y$ and $\|W_0\|_Y \le C_1
(\|w_0\|_{L^1} + \|w_0\|_{H^2})$ for some $C_1 > 0$.
Since $n \le 3$, we have $H^2({\mathbb{R}}^n) \hookrightarrow L^\infty({\mathbb{R}}^n)$, hence $w_0 \in L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$. Thus, using the linear estimates established in Proposition \[p:3\], we obtain $$\sup_{t \ge 0} \|W_0(t)\|_{L^1} + \sup_{t \ge 0}(1+t)^{n/2}
\|W_0(t)\|_{L^\infty} \,\le\, C(\|w_0\|_{L^1} + \|w_0\|_{L^\infty})\,.$$ On the other hand, using Proposition \[p:3\] and Remark \[r:1\], we find $$\|\Delta W_0(t)\|_{L^2} \,\le\, C \|w_0\|_{H^2} \quad \hbox{for }
t \le 1\,, \qquad
\|\Delta W_0(t)\|_{L^2} \,\le\, \frac{C \|w_0\|_{L^1}}{t^{1+n/4}}
\quad \hbox{for } t \ge 1\,. \qquad$$ As $rn/4 > 1$ by , we conclude that $$\left(\int_0^\infty (1+t)^r \|\Delta W_0(t)\|^r_{L^2}\,{\mathrm{d}}t\right)^{1/r} \,\le\, C(\|w_0\|_{L^1} + \|w_0\|_{H^2})\,.$$
The next step consists in estimating the integral terms in , namely $$\mathcal{I}(t) \,=\, \int_0^t \mathcal{M}(t,s)K(s,w,\nabla w,\Delta w)
\,{\mathrm{d}}s\,, \quad \hbox{and} \quad \mathcal{J}(t) \,=\, \int_0^t
\mathcal{M}(t,s) H(s,w) \,{\mathrm{d}}s\,.$$
\[p:fp\] There exist $C_2 > 0$ and $\delta_2 > 0$ such that, for all $w \in Y$ with $\|w\|_Y \le \delta_2$, we have $$\|\mathcal{I}\|_Y + \|\mathcal{J}\|_Y \,\le\, C_2\|w\|_Y^2\,.$$
Throughout the proof $C$ denotes a constant that changes between estimates, but does not depend on $w$. Smallness of $w$ in $Y$ implies that estimates and hold for the nonlinearities.
**Estimate on $\|\mathcal{I}(t)\|_{L^1}$**\
Using with $p = q = 1$ and the first estimate in , we find $$\begin{aligned}
\nonumber
\|\mathcal{I}(t)\|_{L^1} &\le C\int_0^t \|w(s)\|_{L^2}\|\Delta
w(s)\|_{L^2}\,{\mathrm{d}}s\\ \label{e:IL1}
&\le C \|w\|_Y\int_0^t \frac{1}{(1+s)^{\frac{n}{4}+1}}(1+s)
\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\ \nonumber
&\le C \|w\|_Y \left(\int_0^t \frac{1}{(1+s)^{(\frac{n}{4}+1)
\frac{r}{r-1}}}\,{\mathrm{d}}s\right)^{1-1/r} \left(\int_0^t (1+s)^r
\|\Delta w(s)\|_{L^2}^r \,{\mathrm{d}}s\right)^{1/r}\,.\end{aligned}$$ In the second inequality we used the bound $\|w(s)\|_{L^2} \le
\|w(s)\|_{L^1}^{1/2}\|w(s)\|_{L^\infty}^{1/2} \le \|w\|_Y
(1+s)^{-n/4}$, and in the last line Hölder’s inequality. Taking the supremum over $t \ge 0$ and using , we conclude that $$\sup_{t \ge 0}\|\mathcal{I}(t)\|_{L^1} \,\le\, C \|w\|_Y^2\,.$$
**Estimate on $\|\mathcal{I}(t)\|_{L^\infty}$**\
For $t\le 1$, we use with $(p,q) = (2,\infty)$ and the second estimate in . We obtain $$\label{e:Ismallt}
\|\mathcal{I}(t)\|_{L^\infty} \le C\int_0^t
\frac{1}{(t-s)^{n/4}}\|w(s)\|_{L^\infty}\|\Delta w(s)\|_{L^2}
\,{\mathrm{d}}s \le C t^{1-\frac{n}4-\frac1r}\|w\|_Y^2\,,$$ where the last estimate is again a consequence of Hölder’s inequality. Note that $1-\frac{n}4-\frac1r > 0$ by . For $t\ge 1$ we split $$\begin{aligned}
\mathcal{I}(t) &= \int_0^{t/2}\mathcal{M}(t,s)
K(s,w,\nabla w,\Delta w)\,{\mathrm{d}}s +
\int_{t/2}^t\mathcal{M}(t,s)K(s,w,\nabla w,\Delta w)
\,{\mathrm{d}}s\\
\,&=:\, \mathcal{I}_1(t) + \mathcal{I}_2(t)\,.\end{aligned}$$ Using with $(p,q) = (1,\infty)$ and proceeding as in , we find $$\begin{aligned}
\nonumber
(1+t)^{n/2}\|\mathcal{I}_1(t)\|_{L^\infty}
&\le C(1+t)^{n/2} \int_0^{t/2} \frac{1}{(t-s)^{n/2}}
\|w(s)\|_{L^2}\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\ \label{e:IL2}
&\le C\int_0^{t/2} \|w(s)\|_{L^2}\|\Delta w(s)\|_{L^2}{\mathrm{d}}s
\le C \|w\|_Y^2\,.\end{aligned}$$ On the other hand, using with $(p,q) = (2,\infty)$ we obtain $$\begin{aligned}
\nonumber
(1+t)^{n/2}\|\mathcal{I}_2(t)\|_{L^\infty}
&\le C(1+t)^{n/2} \int_{t/2}^t \frac{1}{(t-s)^{n/4}}
\|w(s)\|_{L^\infty} \|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\ \label{e:IL3}
&\le C\|w\|_Y \frac{1}{(1+t)} \int_{t/2}^t
\frac{1}{(t-s)^{n/4}}(1+s)\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\ \nonumber
&\le C \|w\|_Y^2 \frac{1}{(1+t)^{\frac{n}4 +\frac1r}}\,,\end{aligned}$$ where in the last line we used Hölder’s inequality as in . This shows that $$\sup_{t\ge 0}(1+t)^{n/2}\|\mathcal{I}(t)\|_{L^\infty} \,\le\,
C \|w\|_Y^2\,.$$
**Estimate on $\|\Delta\mathcal{I}(t)\|_{L^2}$**\
The estimates for the second derivative require maximal regularity, Proposition \[p:4\]. We need to estimate $\int_0^\infty (1+t)^r\|\Delta\mathcal{I}(t)\|_{L^2}^r
\,{\mathrm{d}}t$. We therefore split the integral and first estimate $$\int_0^T (1+t)^r\|\Delta\mathcal{I}(t)\|_{L^2}^r \,{\mathrm{d}}t
\,\le\, C\int_0^T \|w(t)\|_{L^\infty}^r
\|\Delta w(t)\|_{L^2}^r {\mathrm{d}}t \le C\|w\|_Y^{2r}\,,$$ where we used Proposition \[p:4\] and estimate in the first inequality, and the definition of $\|w\|_Y$ in the second inequality.
We next derive estimates for $\Delta\mathcal{I}(t)$ at $t\ge T$. Here it is more convenient to decompose $$\begin{aligned}
\mathcal{I}(t) \,&=\, \int_0^{t-T}\mathcal{M}(t,s)
K(s,w,\nabla w,\Delta w)\,{\mathrm{d}}s +
\int_{t-T}^t\mathcal{M}(t,s)K(s,w,\nabla w,\Delta w)
\,{\mathrm{d}}s\\
\,&=:\, \mathcal{I}_3(t) + \mathcal{I}_4(t)\,.\end{aligned}$$ Using with $(p,q) = (1,2)$ and Remark \[r:1\] we control the first term as $$\begin{aligned}
(1+t)\|\Delta\mathcal{I}_3(t)\|_{L^2}
&\,\le\, C(1+t) \int_0^{t-T}\frac{1}{(t-s)^{1+\frac{n}{4}}}
\|w(s)\|_{L^2}\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\
&\,\le\, C\|w\|_Y (1+t) \int_0^{t-T} \frac{1}{(t-s)^{1+\frac{n}{4}}}
\frac{1}{(1+s)^{1+\frac{n}{4}}}(1+s)\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\
&\,\le\, \|w\|_Y^2 \frac{1}{(1+t)^{\frac{n}{4}}}\,,\end{aligned}$$ where in the last estimate we used Hölder’s inequality together with the fact that $$\label{e:auxbound}
\left\{\int_0^t \Bigl(\frac{1}{1+t-s}\frac{1}{(t-s)^{n/4}}
\frac{1}{(1+s)^{1+\frac{n}{4}}}\Bigr)^q \,{\mathrm{d}}s\right\}^{1/q}
\,\le\, \frac{C}{(1+t)^{1+\frac{n}{4}}}, \quad \hbox{for any }
q \ge 1\,.$$ Integrating over time and recalling that $nr/4 > 1$, we obtain the desired bound for $\mathcal{I}_3$: $$\int_T^\infty (1+t)^r \|\Delta\mathcal{I}_3(t)\|_{L^2}^r
\,{\mathrm{d}}t \,\le\, C\|w\|_Y^{2r}\,.$$ The other term $\mathcal{I}_4$ is estimated directly using and Corollary \[c:max\]: $$\int_T^\infty (1+t)^r \|\Delta\mathcal{I}_4(t)\|_{L^2}^r\,
{\mathrm{d}}t \,\le\, C\int_0^\infty (1+t)^r \|w(t)\|_{L^\infty}^r
\|\Delta w(t)\|_{L^2}^r \,{\mathrm{d}}t \,\le\, C\|w\|_Y^{2r}\,.$$
**Estimates on $\|\mathcal{J}(t)\|_{L^1}$ and $\|\mathcal{J}(t)\|_{L^\infty}$**\
In this term, the nonlinearity does not contain derivatives which would yield decay, but we can exploit the the stronger decay of the linear operator $\mathcal{M}(t,s)$ when acting on $H(s,w)$. Indeed, since $H(s,w) = (0,\hat g_2(s,w))^T$, we have $$\mathcal{M}(t,s) H(s,w) \,=\, \left(\begin{array}{l}
\mathcal{M}_{0\mathrm{h}}(t,s)\hat g_2(s,w) \\
\mathcal{M}_{\mathrm{hh}}(t,s)\hat g_2(s,w)\end{array}\right)
\,=:\, \mathcal{M}_{\cdot\mathrm{h}}(t,s)\hat g_2(s,w)\,,$$ and it follows from Proposition \[p:3\] that $$\label{e:MHest}
\|\mathcal{M}(t,s)H(s,w)\|_{L^q({\mathbb{R}}^n)} \,\le\, C\,\frac{1}{1+t-s}
\frac{1}{(t-s)^{\frac{n}2(\frac1p-\frac1q)}}\ \|H(s,w)\|_{L^p({\mathbb{R}}^n)}\,,$$ for $1 \le p \le q \le \infty$. Using with $p = q = 1$ and estimate , we thus find $$\label{e:Jest1}
\|\mathcal{J}(t)\|_{L^1} \,\le\, C\int_0^t \frac{1}{1+t-s}
\|w(s)\|_{L^2}^2\,{\mathrm{d}}s \,\le\, C\|w\|_Y^2 \int_0^t \frac{1}{1+t-s}
\frac{1}{(1+s)^{n/2}}\,{\mathrm{d}}s \,\le\, C \|w\|_Y^2\,.$$ In a similar way, using with $(p,q) = (2,\infty)$, we arrive at $$\begin{aligned}
\nonumber
(1+t)^{n/2}\|\mathcal{J}(t)\|_{L^\infty}
\,&\le\, C(1+t)^{n/2} \int_0^t \frac{1}{1+t-s}\frac{1}{(t-s)^{n/4}}
\|w(s)\|_{L^\infty} \|w(s)\|_{L^2}\,{\mathrm{d}}s\\ \label{e:Jest2}
\,&\le\, C \|w\|_Y^2 (1+t)^{n/2} \int_0^t \frac{1}{1+t-s}\frac{1}
{(t-s)^{n/4}} \frac{1}{(1+s)^{3n/4}}\,{\mathrm{d}}s\\ \nonumber
\,&\le\, C\|w\|_Y^2\,.\end{aligned}$$
**Estimate on $\|\Delta\mathcal{J}(t)\|_{L^2}$**\
We first observe that $$\Delta\mathcal{J}(t) \,=\, \int_0^t \mathcal{M}_{\cdot\mathrm{h}}
(t,s)\Delta \hat g_2(s,w)\,{\mathrm{d}}s\,.$$ Since $\hat g_2(t,w) = g_2(v_*(t)+w)[w_\mathrm{h},w_\mathrm{h}]$, it is clear that $|\Delta \hat g_2(t,w)| \le C (|w||\Delta w|
+|\nabla w|^2)$, so that $\Delta \hat g_2$ satisfies the same estimates as $F$ and $G$. Using the $L^1$–$L^2$ estimate for $\mathcal{M}_{\cdot\mathrm{h}}(t,s)$, we thus find $$\begin{aligned}
(1+t)\|\Delta \mathcal{J}(t)\|_{L^2}
\,&\le\, C(1+t) \int_0^t \frac{1}{1+t-s}\frac{1}{(t-s)^{n/4}}
\|w(s)\|_{L^2} \|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\
\,&\le\, C\|w\|_Y (1+t) \int_0^t \frac{1}{1+t-s}\frac{1}{(t-s)^{n/4}}
\frac{1}{(1+s)^{1+\frac{n}{4}}}(1+s)\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s\\
\,&\le\, C\|w\|_Y^2 \frac{1}{(1+t)^{n/4}}\,,\end{aligned}$$ where in the last line we used Hölder’s inequality and estimate . We deduce that $$\int_0^\infty (1+t)^r \|\Delta \mathcal{J}(t)\|_{L^2}^r
\,{\mathrm{d}}t \,\le\, C\|w\|_Y^{2r}\,,$$ and the proof of Proposition \[p:fp\] is complete.
**Proof of Theorem \[t:1\]** ($n \le 3$). As was observed in Section \[s:nf\], we can work with the transformed equation instead of the original perturbation equation . Also, we can assume without loss of generality that the initial perturbation $w_0$ satisfies $\|w_0\|_{L^1} +
\|w_0\|_{H^2} \le \delta_0$ for some small $\delta_0 > 0$. Under these assumptions, we can solve equation by a standard fixed point argument in the Banach space $Y$ defined by . Indeed, let $\mathcal{N}$ denote the right-hand side of , namely $$(\mathcal{N}w)(t) \,=\,\mathcal{M}(t,0)w_0 + \int_0^t \mathcal{M}(t,s)
K(s,w,\nabla w,\Delta w) \,{\mathrm{d}}s + \int_0^t \mathcal{M}(t,s)
H(s,w) \,{\mathrm{d}}s\,.$$ If $w \in Y$ satisfies $\|w\|_Y \le \delta_2$, where $\delta_2 > 0$ is as in Proposition \[p:fp\], we know that $\mathcal{N}w \in Y$ and that $$\label{e:NN1}
\|\mathcal{N}w\|_Y \,\le\, C_1 \delta_0 + C_2 \|w\|_Y^2\,.$$ Similar calculations show that $$\label{e:NN2}
\|\mathcal{N}w - \mathcal{N}\tilde w\|_Y \,\le\, C_2
(\|w\|_Y + \|\tilde w\|_Y)\|w - \tilde w\|_Y\,,$$ whenever $w, \tilde w \in Y$ with $\|w\|_Y \le \delta_2$, $\|\tilde
w\|_Y \le \delta_2$. Let $\mathcal{B} \subset Y$ denotes the ball of radius $R = \min(2C_1\delta_0,\delta_2)$ centered at the origin. If $\delta_0 > 0$ is small enough so that $2C_2 R < 1$, it follows easily from , that $\mathcal{N}(\mathcal{B})
\subset \mathcal{B}$ and that $\mathcal{N}$ is a strict contraction in $\mathcal{B}$. Let $w \in Y$ be the unique fixed point of $\mathcal{N}$ in $\mathcal{B}$. Then $w$ is a global solution of , and if we return to the original variables by setting $u(t,x) = \Psi(v_*(t) + w(t,x))$, we obtain a global solution of which satisfies the decay estimate (with $t_0 = 0$), because $$|u(t,x) - u_*(t)| \,=\, |\Psi(v_*(t) + w(t,x)) - \Psi(v_*(t))|
\,\le\, C\,\frac{\|w\|_Y}{(1+t)^{n/2}}\,,\quad t \ge 0\,.$$ This concludes the proof. $\Box$
\[r:3\] The limitation $n \le 3$ in the above proof is due to the fact that we use maximal regularity (MR) in the Hilbert space $L^2({\mathbb{R}}^n)$ only, see e.g. . This choice was made for simplicity, but for equation it is known that MR holds in all $L^p$ spaces with $1 < p < \infty$, see [@HM00; @PS01]. It is not difficult to verify that the argument above can be adapted to any space dimension $n$ if we use MR in $L^p({\mathbb{R}}^n)$ with $p$ sufficiently large, depending on $n$.
Asymptotic behavior {#s:asym}
===================
We know from Theorem \[t:1\] that small, localized perturbations of the periodic solution $u_*(t)$ converge to zero like $t^{-n/2}$ as $t \to +\infty$. This decay rate is optimal in general, and it is even possible to compute the leading term in the asymptotic expansion of the perturbation as $t \to +\infty$. In this section, we assume (for simplicity) that $1 \le n \le 3$ and we consider the solution $w(t,x)$ of with small initial data $w_0 \in L^1({\mathbb{R}}^n) \cap H^2({\mathbb{R}}^n)$. If we decompose this solution as $w(t,x) = (w_0(t,x),w_{\mathrm{h}}(t,x))^T$, we first observe that the hyperbolic part $w_{\mathrm{h}}(t,x) \in {\mathbb{R}}^{N-1}$ has a faster decay as $t \to \infty$.
\[p:faster\] If the initial data $w_0 \in L^1({\mathbb{R}}^n) \cap H^2({\mathbb{R}}^n)$ are sufficiently small, the hyperbolic component of the solution $w$ of satisfies $$\sup_{t \ge 0} (1+t)\|w_{\mathrm{h}}(t)\|_{L^1} + \sup_{t \ge 0}
(1+t)^{1+\frac{n}2} \|w_{\mathrm{h}}(t)\|_{L^\infty} \,\le\,
C(\|w_0\|_{L^1} + \|w_0\|_{H^2})\,.$$
Projecting the integral equation onto the hyperbolic component, we find $$\begin{aligned}
w_\mathrm{h}(t)\,&=\,\mathcal{M}_{\mathrm{h}\cdot}(t,0)w_0
+ \int_0^t \mathcal{M}_{\mathrm{h}\cdot}(t,s)K(s,w,
\nabla w,\Delta w)\,{\mathrm{d}}s +
\int_0^t \mathcal{M}_{\mathrm{h\cdot}}(t,s)H(s,w)\,{\mathrm{d}}s \\
\,&=:\,\mathcal{M}_{\mathrm{h}\cdot}(t,0)w_0 +
\mathcal{I}_{\mathrm{h}}(t) + \mathcal{J}_{\mathrm{h}}(t)\,,\end{aligned}$$ where $\mathcal{M}_{\mathrm{h\cdot}}(t,s) = (\mathcal{M}_{
\mathrm{h0}}(t,s),\mathcal{M}_{\mathrm{hh}}(t,s))$. We know from Proposition \[p:3\] that $$\label{e:MHest2}
\|\mathcal{M}_{\mathrm{h\cdot}}(t,s)w\|_{L^q({\mathbb{R}}^n)} \le C\,\frac{1}{1+t-s}
\frac{1}{(t-s)^{\frac{n}2(\frac1p-\frac1q)}}\ \|w\|_{L^p({\mathbb{R}}^n)}\,,$$ for $1 \le p \le q \le \infty$. In particular, we have $$\sup_{t \ge 0} (1+t)\|\mathcal{M}_{\mathrm{h}\cdot}(t,0)w_0\|_{L^1}
+ \sup_{t \ge 0} (1+t)^{1+\frac{n}2} \|\mathcal{M}_{\mathrm{h}\cdot}(t,0)
w_0 \|_{L^\infty} \,\le\, C(\|w_0\|_{L^1} + \|w_0\|_{L^\infty})\,.$$ Moreover, proceeding as in the proof of Proposition \[p:fp\] we obtain $$\|\mathcal{I}_{\mathrm{h}}(t)\|_{L^1} \,\le\, C\int_0^t
\frac{1}{1+t-s}\frac{\|w\|_Y}{(1+s)^{1+\frac{n}{4}}}
\,(1+s)\|\Delta w(s)\|_{L^2}\,{\mathrm{d}}s \,\le\, \frac{C}{1+t}
\,\|w\|_Y^2\,,$$ and the same result holds for $(1+t)^{n/2}\|\mathcal{I}_{
\mathrm{h}}(t)\|_{L^\infty}$. Finally, to bound the term $\mathcal{J}_{\mathrm{h}}$, we observe that $\mathcal{M}_{\mathrm{h\cdot}}(t,s)H(s,w) = \mathcal{M}_{
\mathrm{hh}}(t,s)\hat g_2(s,w)$ and we use the strong decay in time given by Proposition \[p:3\]. We thus find $$\begin{aligned}
\label{e:Jhest}
\|\mathcal{J}_{\mathrm{h}}(t)\|_{L^1} \,&\le\, C\int_0^t
\frac{1}{(1+t-s)^2}\,\|w_{\mathrm{h}}(s)\|_{L^1}
\|w_{\mathrm{h}}(s)\|_{L^\infty}\,{\mathrm{d}}s \\ \nonumber
\,&\le\, C\int_0^t \frac{1}{(1+t-s)^2}\frac{\|w\|_Y^2}{(1+s)^{n/2}}
\,{\mathrm{d}}s \,\le\, \frac{C}{(1+t)^{n/2}}\,\|w\|_Y^2\,,\end{aligned}$$ and the same result holds for $(1+t)^{n/2}\|\mathcal{J}_{
\mathrm{h}}(t)\|_{L^\infty}$. This gives the desired result if $n
\ge 2$. If $n = 1$, we only have $$\sup_{t \ge 0} (1+t)^{1/2}\|w_{\mathrm{h}}(t)\|_{L^1} + \sup_{t \ge 0}
(1+t) \|w_{\mathrm{h}}(t)\|_{L^\infty} \,\le\, C\|w_0\|_{L^1\cap H^2}\,,$$ but if we now return to we obtain the stronger estimate $$\|\mathcal{J}_{\mathrm{h}}(t)\|_{L^1} \,\le\, C\|w_0\|_{L^1\cap H^2}^2
\int_0^t \frac{1}{(1+t-s)^2}\,\frac{1}{(1+s)^{3/2}}\,{\mathrm{d}}s
\,\le\, C\,\frac{\|w_0\|_{L^1\cap H^2}^2}{(1+t)^{3/2}}\,,$$ which also holds for $(1+t)^{1/2}\|\mathcal{J}_{\mathrm{h}}(t)
\|_{L^\infty}$. This concludes the proof.
We next consider the central component $w_0(t,x) \in {\mathbb{R}}$, and prove that it behaves asymptotically like a solution of a linear equation with suitably modified initial data.
\[p:central\] If the initial data $w_0 \in L^1({\mathbb{R}}^n) \cap H^2({\mathbb{R}}^n)$ are sufficiently small, the central component of the solution $w$ of satisfies $$\|w_0(t) - \mathcal{M}_{00}(t,0)w_\infty\|_{L^1} + (1+t)^{n/2}
\|w_0(t) - \mathcal{M}_{00}(t,0)w_\infty\|_{L^\infty}
\,\le\, \frac{C}{(1+t)^\gamma}\,(\|w_0\|_{L^1} + \|w_0\|_{H^2})\,,$$ where $\gamma = \frac{n}{4} + \frac{1}{r} < 1$ and $w_\infty \in
L^1({\mathbb{R}}^n) \cap L^\infty({\mathbb{R}}^n)$ is defined by $$w_\infty \,=\, (w_0)_0 + \int_0^\infty K_0(s,w(s),\nabla w(s),
\Delta w(s))\,{\mathrm{d}}s\,.$$
Projecting onto the central component, we find $$\begin{aligned}
w_0(t)\,&=\,\mathcal{M}_{0\cdot}(t,0)w_0 + \int_0^t
\mathcal{M}_{0\cdot}(t,s)K(s,w,\nabla w,\Delta w)\,{\mathrm{d}}s +
\int_0^t \mathcal{M}_{0\cdot}(t,s)H(s,w)\,{\mathrm{d}}s \\
\,&=:\,\mathcal{M}_{0\cdot}(t,0)w_0 + \mathcal{I}_0(t) +
\mathcal{J}_0(t)\,,\end{aligned}$$ where $\mathcal{M}_{0\cdot}(t,s) = (\mathcal{M}_{00}(t,s),
\mathcal{M}_{0\mathrm{h}}(t,s))$. Our goal is to extract from $w_0(t)$ the leading contributions as $t \to +\infty$. The last term $\mathcal{J}_0(t)$ is clearly negligible in this limit. Indeed, using Proposition \[p:faster\] and proceeding as in , , we obtain $$\label{e:negl}
\|\mathcal{J}_0(t)\|_{L^1} + (1+t)^{n/2} \|\mathcal{J}_0(t)
\|_{L^\infty} \,\le\, \frac{C}{1+t}\,\|w_0\|_{L^1\cap H^2}\,.$$ The same estimate holds for the linear term $\mathcal{M}_{0\mathrm{h}}
(t,0)(w_0)_\mathrm{h}$, because $\mathcal{M}_{0\mathrm{h}}(t,0)$ decays as fast as $(1+t)^{-1}\mathcal{M}_{00}(t,0)$ by Proposition \[p:3\]. Using the same remark and proceeding as in , , , we see that the integral term $\mathcal{I}_{0\mathrm{h}}(t) := \int_0^t
\mathcal{M}_{0\mathrm{h}}(t,s)K_\mathrm{h}(s,w,\nabla w,\Delta w)
\,{\mathrm{d}}s$ also satisfies . So the only remaining terms are $\mathcal{M}_{00}(t,0)(w_0)_0$ and $$\begin{aligned}
\mathcal{I}_{00}(t) \,&=\, \int_0^t \mathcal{M}_{00}(t,s)
K_0(s,w,\nabla w,\Delta w)\,{\mathrm{d}}s \\
\,&=\, \int_{t/2}^t \mathcal{M}_{00}(t,s) K_0(s,w,\nabla w,
\Delta w)\,{\mathrm{d}}s
\,+\, \int_0^{t/2} (\mathcal{M}_{00}(t,s)-\mathcal{M}_{00}(t,0))
K_0(s,w,\nabla w,\Delta w)\,{\mathrm{d}}s \\
&\quad + \mathcal{M}_{00}(t,0)\int_0^\infty K_0(s,w,\nabla w,
\Delta w)\,{\mathrm{d}}s
\,-\, \mathcal{M}_{00}(t,0)\int_{t/2}^\infty K_0(s,w,\nabla w,
\Delta w)\,{\mathrm{d}}s \\
\,&=:\, \mathcal{I}_{01}(t) + \mathcal{I}_{02}(t) +
\mathcal{I}_{03}(t) + \mathcal{I}_{04}(t)\,.\end{aligned}$$ Proceeding as in , , it is straightforward to verify that $$\|\mathcal{I}_{01}(t)\|_{L^1} + (1+t)^{n/2} \|\mathcal{I}_{01}(t)
\|_{L^\infty} \,\le\, \frac{C}{(1+t)^\gamma}\,\|w\|_Y^2\,,
\quad t \ge 0\,,$$ where $\gamma = \frac{n}{4} + \frac{1}{r} < 1$, and the same estimate clearly holds for $\mathcal{I}_{04}(t)$ too. Finally, using Remark \[r:2\] to bound the difference $\mathcal{M}_{00}(t,s)-
\mathcal{M}_{00}(t,0)$, we obtain $$\|\mathcal{I}_{02}(t)\|_{L^1} + (1+t)^{n/2} \|\mathcal{I}_{02}(t)
\|_{L^\infty} \,\le\, C \int_0^{t/2} \frac{s}{t}\,
\|K_0(s,w,\nabla w,\Delta w)\|_{L^1}\,{\mathrm{d}}s \,\le\,
\frac{C}{(1+t)^\gamma}\,\|w\|_Y^2\,.$$ This concludes the proof of Proposition \[p:central\], because $\mathcal{M}_{00}(t,0)(w_0)_0 + \mathcal{I}_{03}(t) =
\mathcal{M}_{00}(t,0)w_\infty$.
It is now rather easy to prove Theorem \[t:2\] (in the case where $n \le 3$). Combining Propositions \[p:faster\] and \[p:central\], we find $$\label{e:fin1}
\|w(t) - e_1 W(t)\|_{L^1} + (1+t)^{n/2}\|w(t) - e_1
W(t)\|_{L^\infty} \,\le\, \frac{C}{(1+t)^\gamma}\,(\|w_0\|_{L^1}
+ \|w_0\|_{H^2})\,,$$ where $W(t) = \mathcal{M}_{00}(t,0)w_\infty$ and $e_1 = (1,0)^T$ is the first vector of the canonical basis in ${\mathbb{R}}^N$. Furthermore, we claim that $$\label{e:fin2}
\|t^{n/2}W(t,xt^{1/2}) - \tilde \alpha G\|_{L^1 \cap
L^\infty} ~\xrightarrow[t\to +\infty]{}~ 0\,,$$ where $G$ is defined in and $$\label{e:fin3}
\tilde \alpha \,=\, \int_{{\mathbb{R}}^n} w_\infty(x)\,{\mathrm{d}}x \,=\,
e_1^T \left(\int_{{\mathbb{R}}^n} w_0(x)\,{\mathrm{d}}x + \int_0^\infty \!\!\int_{{\mathbb{R}}^n}
K(t,w,\nabla w,\Delta w)\,{\mathrm{d}}x\,{\mathrm{d}}t\right)\,.$$ To prove the $L^\infty$ claim in , we use Fourier transforms and simply note that the quantity $$\|\hat W(t,kt^{-1/2}) - \tilde \alpha \hat{G}(k)\|_{L^1}
\,=\, \|M_{00}(t,0;kt^{-1/2})\hat w_\infty(k t^{-1/2}) - e^{-d_0 k^2}
\hat w_\infty(0)\|_{L^1}$$ converges to zero as $t \to \infty$ by Lebesgue’s dominated convergence theorem, in view of Proposition \[p:1\] and Remark \[r:asym\]. The $L^1$ claim can be established in a similar way, using the same ideas as in the proof of Proposition \[p:3\] (we omit the details).
We now return to the original variables. Since $u_*(t) =
\Psi(\omega t e_1)$, the solution of given by $u(t,x) = \Psi(v(t,x)) = \Psi(v_*(t) + w(t,x))$ can be decomposed as in , with $\alpha(t,x) =
\omega^{-1}W(t,x)$ and $$\beta(t,x) \,=\, \Psi(v_*(t) + w(t,x)) - \Psi(v_*(t)) -
\Psi'(v_*(t))w(t,x) + \Psi'(v_*(t))(w(t,x)-e_1 W(t,x))\,.$$ Estimates , immediately give , with $\alpha_* =
\omega^{-1}\tilde \alpha$. Finally, the formula for $\alpha_*$ follows from the expression of $\tilde
\alpha$ and the fact that $U_*(0) = (\Psi'(0)^{-1})^T e_1$, $u_*'(0) =
\omega \Psi'(0)e_1$. This concludes the proof of Theorem \[t:2\]. $\Box$
Examples and perspectives {#s:ex}
=========================
In this final section, we first give a simple example of a 2-species reaction-diffusion system with a periodic orbit $u_*(t)$ which is asymptotically stable for the ODE dynamics but does not satisfy Hypothesis \[h:2\]. We then discuss possible generalizations of the results of this paper.
Destabilization by diffusion: a simple example
----------------------------------------------
One may feel inclined to believe that ODE-stable periodic orbits tend to be stable for the PDE dynamics, that is, that our Hypothesis \[h:2\] is satisfied in most cases where the periodic orbit is stable for the ODE. Our example below shows that this is not the case, even for a simple reaction-diffusion system with only two species.
We consider the following $2$-species reaction-diffusion system $$\label{e:ex1}
u_t \,=\, D u_{xx} + Ju + (\epsilon^2-|u|^2)Ru \,,$$ where $u = (u_1,u_2)^T \in {\mathbb{R}}^2$ and $|u|^2 = u_1^2 + u_2^2$. Here $\epsilon > 0$ is a parameter, $D$ is a $2\times2$ real matrix with positive eigenvalues, and $$J \,=\, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\,, \qquad
R \,=\, \begin{pmatrix} \cos(\theta) & -\sin(\theta) \\
\sin(\theta) & \cos(\theta)\end{pmatrix}\,, \qquad \theta \in
(-\pi/2,\pi/2)\,.$$ The system has a $2\pi$-periodic solution $u_*(t) = \epsilon
\bar u(t)$, where $\bar u(t) = (\cos(t),\sin(t))^T$. Linearizing at $u_*(t)$ we obtain $$v_t \,=\, D v_{xx} + (J - 2\epsilon^2 R\,\bar u(t)\,\bar u(t)^T)v\,,$$ or equivalently $$\label{e:ex2}
v_t \,=\, -k^2 D v + (J - 2\epsilon^2 R\,\bar u(t)\,\bar u(t)^T)v\,,
\quad k \in {\mathbb{R}}\,.$$ Of course $v(t) = \bar u'(t)$ is a solution of for $k = 0$.
Let $F = F_0(2\pi,0)$ be the Floquet matrix associated to for $k = 0$. Then $$\Det(F) \,=\, \exp\Bigl(\Tr \int_0^{2\pi} (J - 2\epsilon^2
R\,\bar u(t)\,\bar u(t)^T)\,{\mathrm{d}}t\Bigr) \,=\, \exp(-2\pi
\epsilon^2\Tr(R))\,.$$ The Floquet exponents (for $k = 0$) are therefore $\lambda_1 = 0$ and $\lambda_2 = -\epsilon^2 \Tr(R)$. As $\Tr(R) = 2\cos(\theta) > 0$, it follows that $u_*(t)$ is a stable periodic orbit for the ODE dynamics associated to .
To compute the Floquet exponents for small $k$, we consider the adjoint ODE $$\label{e:ex3}
U_t \,=\, (J + 2 \epsilon^2 \bar u(t)\,\bar u(t)^T R^T)U\,.$$ As is easily verified, the unique nontrivial bounded solution of is $U_*(t) = R \bar u'(t)$. Using formula , we conclude that $$d_0 \,=\, \frac{\int_0^{2\pi} \bar u'(t)^T R^T D \,\bar u'(t)
\,{\mathrm{d}}t}{\int_0^{2\pi} \bar u'(t)^T R^T \bar u'(t)\,{\mathrm{d}}t}
\,=\, \frac{\Tr(R^T D)}{\Tr(R^T)} \,=\, \frac12 \Bigl(
\Tr(D) - \tan(\theta)\Tr(JD)\Bigr)\,.$$ If the diffusion matrix $D$ is symmetric, then $\Tr(D) > 0$ and $\Tr(JD) = 0$, hence necessarily $d_0 > 0$, which means that the periodic solution $u_*(t)$ is spectrally stable for long wave-length perturbations. But if $D$ is a nonsymmetric matrix, then $\Tr(JD) \neq
0$ and therefore we can choose $\theta \in (-\pi/2,\pi/2)$ in such a way that $d_0 < 0$. This gives an example of a periodic orbit exhibiting a [*sideband*]{} instability.
On the other hand, as $\epsilon \to 0$, the periodic orbit $u_*(t)$ reduces to the fixed point $u = 0$, for which it is easy to perform a stability analysis. For a fixed wavenumber $k \in {\mathbb{R}}$, we have to compute the eigenvalues $\lambda_1(k), \lambda_2(k)$ of the linearized operator $J-Dk^2$. By direct calculation, we find $$\Tr(J-Dk^2) \,=\, -\Tr(D)k^2 \,\le\, 0\,, \quad \hbox{and} \quad
\Det(J-Dk^2) \,=\, 1 + \Tr(JD)k^2 + \Det(D)k^4\,.$$ If we choose $D$ such that $\Tr(JD) + 2(\Det(D))^{1/2} < 0$, we see that there exists a nonempty open interval $I \subset (0,+\infty)$ such that $\Tr(J-Dk^2) < 0$ and $\Det(J-Dk^2) < 0$ if $k^2 \in I$. Thus, one of the eigenvalues $\lambda_i(k)$ is strictly positive, which means that the equilibrium $u = 0$ is unstable with respect to perturbations with wavenumbers $k$ such that $k^2 \in I$. By continuity, this Turing instability persists for the periodic orbit $u_*(t) = \epsilon
\bar u(t)$ if $\epsilon > 0$ is sufficiently small: for $k^2 \in I$, one of the Floquet exponents has positive real part.
Summarizing, the periodic solution $u_*(t) = \epsilon \bar u(t)$ of the reaction-diffusion system exhibits:
1. a [*sideband instability*]{}, if $\Tr(D) - \tan(\theta)
\Tr(JD) < 0$;
2. a [*Turing instability*]{}, if $\Tr(JD) + 2(\Det(D))^{1/2}
< 0$ and $\epsilon \ll 1$.
\
[**1.**]{} The instability criteria above are never satisfied if the matrix $D$ is symmetric. But if $D$ has eigenvalues $d_1 > d_2 > 0$, we can choose an invertible matrix $S$ so that $S^{-1}DS =
\mathcal{D} = \mathrm{diag}(d_1,d_2)$. Then setting $u = Sw$ we obtain the equivalent system $$\label{e:ex4}
w_t \,=\, \mathcal{D} w_{xx} + S^{-1}JSw + S^{-1}(1-|Sw|^2)RSw\,,$$ where now the diffusion matrix has the usual, diagonal form. The Floquet exponents characterizing the stability properties of the periodic orbit are of course unaffected by this linear transformation. Thus we can find 2-species systems of the form , with diagonal diffusion matrix, which exhibit either a sideband or a Turing instability.
[**2.**]{} Our example is clearly reminiscent of the complex Ginzburg-Landau equation (CGLE), $$\label{e:CGL}
u_t \,=\, (1+ia)\Delta u + u - (1-ic)|u|^2 u\,,$$ where $a,c$ are real parameters and $u : {\mathbb{R}}_+ \times {\mathbb{R}}^n \to {\mathbb{C}}$. The system possesses a homogeneous time-periodic solution of the form $u(t,x) = {\mathrm{e}}^{ict}$, which exhibits a sideband instability if $ac >
1$ ([*Benjamin-Feir criterion*]{}) and a Turing instability in other parameter regions, see e.g. [@AK02]. The complex Ginzburg-Landau equation arises as a modulation equation near Hopf bifurcations in reaction-diffusion systems (see, for example, [@sh; @Mi]). One therefore expects stability and instability properties of small-amplitude periodic solutions near Hopf bifurcation to be governed by those of the CGLE; see, for example, [@Ri2] for a result in this direction. Since the diffusion matrix of CGLE possesses complex eigenvalues, it cannot be cast as a real reaction-diffusion system with diagonal diffusion matrix. Our example and the remark above show that one can almost explicitly recover the properties of CGLE with diagonal diffusion matrices. Upon substituting the diffusion matrix of CGLE in our example, one would recover precisely the Benjamin-Feir criterion from the instability criterion $\Tr(D) - \tan(\theta)\Tr(JD) < 0$. In a different direction, one could also extend our results to diffusion matrices with $D+D^T>0$ without any additional difficulties, which would then include CGLE as a particular example. In the context of reaction-diffusion modeling, cross-diffusion phenomena that are associated with off-diagonal elements of $D$ are however quite uncommon.
Discussion and perspectives
---------------------------
We believe that the method presented here can be adapted to other situations. We mention the stability of wave trains, $u(kx-\omega t)$, with $u(\xi)=u(\xi+2\pi)$ and $\omega,k>0$, and Turing patterns $u(kx)=u(-kx)$, with $u(\xi)=u(\xi+2\pi)$ and $k>0$. In both cases, one finds continuous spectrum with diffusive decay properties for the linearization. In both cases, the absence of relevant, self-coupling terms in the neutral mode has been shown previously; see [@Sc96; @DSSS].
Interesting questions arise when one attempts to extend the class of allowed perturbations. One may for instance consider perturbations $v$ such that $\nabla v\in L^1$. In one space-dimension, this would correspond to perturbations with different phase shifts at $x=\pm\infty$. One would still expect the diffusive linear part to be dominant so that one would find error function asymptotics for the phase correction.
In fact, one would expect some type of stability for much more general perturbations. For instance, in one space-dimension, the homogeneous oscillation $u_*(t)$ is embedded in a family of wave train solutions $u(kx-\omega t;k)$, with $k\approx 0$ and $u\approx u_*$; [@ssdefect Section 3.3]. Under our stability assumptions, using Lyapunov-Schmidt reduction, one finds $\omega=\omega(k) = \omega_0 + \omega_2
k^2+{\mathrm{O}}(k^4)$; see for instance [@RS Lemma 2.1]. Of course, $u(kx-\omega t;k)$ is not close to any fixed homogeneous oscillation, but it is close to an appropriate phase shift of the oscillation in any finite region of space. All solutions in the basin of attraction of such wave trains then stay close to our homogeneous oscillation, orbitally, and pointwise in space. A natural question then asks for the asymptotics of initial conditions of the type $u(k(x)
x-\omega(k(x)) t;k(x))$, where $k(x)\to k_\pm$ for $x\to\pm\infty$. In the case of the real Ginzburg-Landau equation, which exhibits spatial oscillations $u(kx)$, this question was addressed in [@CEE92; @GM], showing that asymptotics are governed by a nonlinear diffusion equation $\theta_t=W(\theta_x)_x$, with locally uniform convergence to a fixed, *intermediate* wavenumber. Near temporal oscillations, one expects dynamics to be governed by a viscous conservation law $\theta_t=d(\theta_x)_x+j(\theta_x)$, so that solutions with asymptotically constant wavenumber would be expected to converge to viscous shocks [@DSSS Section 8], or to rarefaction waves.
More generally, one could ask about the orbital stability of a family of oscillations: starting with an initial condition $u(k(x)
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---
abstract: |
We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost.
For the TSP on graphic metrics ([graph-TSP]{}), the approach yields a $1.461$-approximation algorithm with respect to the Held-Karp lower bound. For [graph-TSP]{}restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio $4/3$. The framework allows for generalizations in a natural way and also leads to a $1.586$-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.
author:
- |
Tobias Mömke and Ola Svensson\
Royal Institute of Technology - KTH, Stockholm, Sweden\
[{moemke,osven}@kth.se ]{}
title: 'Approximating Graphic TSP by Matchings[^1]'
---
Introduction {#sec:intro}
============
The traveling salesman problem in metric graphs is one of most fundamental NP-hard optimization problems. In spite of a vast amount of research several important questions remain open. While the problem is known to be APX-hard and NP-hard to approximate with a ratio better than $220/219$ [@PV06], the best upper bound is still the 1.5-approximation algorithm obtained by Christofides [@Chr76] more than three decades ago. A promising direction to improve this approximation guarantee, has long been to understand the power of a linear program known as the Held-Karp relaxation [@HK70]. On the one hand, the best lower bound on its integrality gap (for the symmetric case) is $4/3$ and indeed conjectured to be tight [@Goe95]. On the other hand, the best known analysis [@SW90; @Wol80] is based on Christofides’ algorithm and gives an upper bound on the integrality gap of $1.5$.
In the light of this difficulty of even determining the integrality gap of the Held-Karp relaxation, a reasonable way to approach the metric TSP is to restrict the set of feasible inputs. One promising candidate is the *[graph-TSP]{},* that is, the traveling salesman problem where distances between cities are given by any graphic metric, i.e., the distance between two cities is the length of the shortest path in a given (unweighted) graph. Equivalently, [graph-TSP]{}can be formulated as the problem of finding an Eulerian multigraph within an unweighted input graph so as to minimize the number of edges. In contrast to TSP on Euclidean metrics that admits a PTAS [@Arora98; @Mitch99], the [graph-TSP]{} seems to capture the difficulty of the metric TSP in the sense that, as stated in [@GKP95], it is APX-hard and the lower bound 4/3 on the integrality gap of the Held-Karp relaxation is established using a [graph-TSP]{}instance.
The TSP on graphic metrics has recently drawn considerable attention. In 2005, Gamarnik et al. [@GLS05] showed that for cubic 3-edge-connected graphs, there is an approximation algorithm achieving an approximation ratio of $1.5-5/389$. This result was generalized to cubic graphs by Boyd et al. [@BSSS11], who obtained an improved performance guarantee of 4/3. For subcubic graphs, i.e., graphs of degree at most $3$, they also gave an 7/5-approximation algorithm with respect to the Held-Karp lower bound. In a major achievement, Gharan et al. [@GSS11] recently presented an approximation algorithm for [graph-TSP]{}with performance guarantee strictly better than 1.5. The approach in [@GSS11] is similar to that of Christofides in the sense that they start with a spanning tree and then add a perfect matching of those vertices of odd-degree to make the graph Eulerian. The main difference is that instead of starting with a minimum spanning tree, their approach uses the solution of the Held-Karp relaxation to sample a spanning tree. Although the proposed algorithm in [@GSS11] is surprisingly simple, the analysis is technically involved and several novel ideas are needed to obtain the improved performance guarantee $1.5-\epsilon$ for an $\epsilon$ of the order $10^{-12}$.
#### Our Results and Overview of Techniques.
We propose an alternative framework for approximating the metric TSP and use it to obtain an improved approximation algorithm for [graph-TSP]{}.
\[thm:approximationratio\] There is a polynomial time approximation algorithm for [graph-TSP]{}with performance guarantee $\frac{14\cdot ( \sqrt{2}-1)}{12\cdot
\sqrt{2}-13} < 1.461$.
The result implies an upper bound on the integrality gap of the Held-Karp relaxation for [graph-TSP]{}that matches the approximation ratio. For the restricted class of graphs, where each block (i.e., each maximally 2-vertex-connected subgraph) is either claw-free or of degree at most $3$, we use the framework to construct a polynomial time $4/3$-approximation algorithm showing that the conjectured integrality gap of the Held-Karp relaxation is tight for those graphs. In fact, the techniques allow us to prove the tight result that any $2$-vertex-connected graph of degree at most $3$ has a spanning Eulerian multigraph with at most $4n/3 - 2/3$ edges, which settles a conjecture of Boyd et al. [@BSSS11] affirmatively.
Our framework is based on earlier works by Frederickson & Ja’ja’ [@FJJ89] and Monma et al. [@MMP90], who related the cost of an optimal tour to the size of a minimum $2$-vertex-connected subgraph. More specifically, Monma et al. showed that a $2$-vertex-connected graph $G=(V,E)$ always has a spanning Eulerian multigraph with at most $\frac{4}{3} |E|$ edges, generalizing a previous result of Frederickson & Ja’ja’ who obtained the same result for the special case of planar $2$-vertex-connected graphs. One interpretation of their approaches is the following. Given a $2$-vertex-connected graph $G=(V,E)$, they show how to pick a random subset $M$ of edges satisfying: (i) an edge is in $M$ with probability $1/3$ and (ii) the multigraph $H$ with vertex set $V$ and edge set $E
\cup M$ is spanning and Eulerian. From property $(i)$ of $M$, the expected number of edges in $H$ is $\frac{4}{3} |E|$ yielding their result.
Although the factor $4/3$ is asymptotically tight for some classes of graphs (one example is the family of integrality gap instances for the Held-Karp relaxation described in Section \[sec:prelim\]), the bound rapidly gets worse for $2$-vertex-connected graphs with significantly more than $n$ edges. The novel idea to overcome this issue is the following. Instead of adding all the edges in $M$ to $G$, some of the edges in $M$ might instead be removed from $G$ to form $H$. As long as the removal of the edges does not disconnect the graph, this will again result in a spanning Eulerian multigraph $H$. To specify a subset $R$ of edges that safely may be removed we introduce, in Section \[sec:tspframework\], the notion of a “[$\mbox{removable pairing}$]{}”. The framework is then completed by Theorem \[thm:main\], where we show that a $2$-vertex-connected graph $G=(V,E)$ with a set $R$ of removable edges has a spanning Eulerian multigraph with at most $\frac{4}{3} |E| - \frac{2}{3} |R|$ edges.
In order to use the framework, one of the main challenges is to find a sufficiently large set of removable edges. In Section \[sec:circulation\], we show that this problem can be reduced to that of finding a min-cost circulation in a certain circulation network. To analyze the circulation network we then (in Section \[sec:algorithms\]) use several properties of an extreme point solution to the Held-Karp relaxation to obtain our main algorithmic result. The better approximation guarantees for special graph classes follows from that the circulation network has an easier structure in these cases, which in turn allows for a better analysis.
Finally, we note that the techniques generalize in a natural way. Our results can be adapted to the more general traveling salesman path problem ([graph-TSPP]{}) with prespecified start and end vertices to improve on the approximation ratio of $5/3$ by Hoogeveen [@Hoo91] when considering graphic metrics. More specifically, we obtain the following.
\[thm:approximationratiohpp\] For any $\varepsilon > 0$, there is a polynomial time approximation algorithm for [graph-TSPP]{}with performance guarantee $ 3-\sqrt{2} + \varepsilon < 1.586+\varepsilon$.
If furthermore each block of the given graph is degree three bounded, there is a polynomial time approximation algorithm for [graph-TSPP]{}with performance guarantee $1.5 + \varepsilon$, for any $\varepsilon>0$.
The generalization to the traveling salesman problem is presented in Section \[sec:tspp\].
Preliminaries {#sec:prelim}
=============
#### Held-Karp Relaxation.
The linear program known as the Held-Karp (or subtour elimination) relaxation is a well studied lower bound on the value of an optimal tour. It has a variable $x_{\{u,v\}}$ for each pair of vertices with the intuitive meaning that $x_{\{u,v\}}$ should take value $1$ if the edge $\{u,v\}$ is used in the tour and $0$ otherwise. Letting $G=(V,E)$ be the complete graph on the set of vertices and $c_{\{u,v\}}$ be the distance between vertices $u$ and $v$, the Held-Karp relaxation can then be formulated as the linear program where we wish to minimize $\sum_{e \in E} c_{e} x_{e}$ subject to $$x(\delta(v)) =
2 \mbox{ for }v\in V \mbox{,} \qquad x(\delta(S)) \geq 2 \mbox{ for } \emptyset \neq S \subset V, \qquad \mbox{ and $x\geq 0$},$$ where $\delta(S)$ denotes the set of edges crossing the cut $(S, \bar S)$ and $x(F) = \sum_{e\in F} x_e$ for any $F\subseteq E$.
Goemans & Bertsimas [@GB90] proved that for metric distances the above linear program has the same optimal value as the linear program obtained by dropping the equality constraints. Moreover, when considering a [graph-TSP]{}instance $G=(V,E)$ we only need to consider the variables $(x_{e})_{e\in E}$. Indeed, any solution $x$ to the Held-Karp relaxation without equality constraints such that $x_{\{u,v\}}>0$ for a pair of vertices $\{u,v\} \not \in E$ can be transformed into a solution $x'$ with no worse cost and $x'_{\{u,v\}} = 0$ by setting $x'_{e} = x_{e} + x_{\{u,v\}}$ for each edge on the shortest path between $u$ and $v$, and $x_e' = x_e$ for the other edges. The Held-Karp relaxation for [graph-TSP]{}on a graph $G=(V,E)$ can thus be formulated as follows: $$\min \sum_{e \in E} x_{e} \qquad \mbox{ subject to}
\qquad x(\delta(S)) \geq 2 \mbox{ for } \emptyset \neq S \subset V,\qquad \mbox{and } x\geq 0.$$ We shall refer to this linear program as [$LP(G)$]{} and denote the value of an optimal solution by [$OPT_{LP}(G)$]{}. Its integrality gap was previously known to be at most $3/2- \epsilon$ and at least $4/3$ for graphic instances. The lower bound is obtained by a claw-free graphic instance of degree at most $3$ that consists of three paths of equal length with endpoints $(s_1, t_1),
(s_2, t_2),$ and $(s_3,t_3)$ that are connected so as $\{s_1, s_2,
s_3\}$ and $\{t_1, t_2, t_3\}$ form two triangles (see Figure \[fig:intgap\]).
We end our discussion of [$LP(G)$]{} with a useful observation. When considering [graph-TSP]{}, it is intuitively clear that we can restrict ourselves to *$2$-vertex-connected* graphs, i.e., graphs that stay connected after deleting a single vertex. Indeed, if we consider a graph with a vertex $v$ whose removal results in components $C_1,
\ldots, C_\ell$ with $\ell >1$ then we can recursively solve the [graph-TSP]{} problem on the $\ell$ subgraphs $G_1, G_2, \dots, G_\ell$ induced by $C_1 \cup \{v\}, C_2 \cup \{v\}, \dots, C_\ell \cup
\{v\}$. The union of these solutions will then provide a solution to the original graph that preserves the approximation guarantee with respect to the linear programming relaxation since one can see that ${\ensuremath{OPT_{LP}(G)}} \geq \sum_{i=1}^\ell {\ensuremath{OPT_{LP}(G_i)}}$. We summarize this observation in the following lemma (see Appendix \[app:2connTSP\] for a fullproof).
\[lemma:2connTSP\] Let $G$ be a connected graph. If there is an $r$-approximation algorithm for [graph-TSP]{} on each $2$-vertex-connected subgraph $H$ of $G$ (with respect to [$OPT_{LP}(H)$]{}) then there is an $r$-approximation algorithm for [graph-TSP]{}on $G$ (with respect to [$OPT_{LP}(G)$]{}).
#### Matchings of Cubic $2$-Edge-Connected Graphs.
Edmonds [@Edmonds1965b] showed that the following set of equalities and inequalities on the variables $(x_e)_{e\in E}$ determines the perfect matching polytope (i.e., all extreme points of the polytope are integral and correspond to perfect matchings) of a given graph $G=(V, E)$: $$x(\delta(v)) = 1 \mbox{ for } v\in V,\qquad
x(\delta(S)) \geq 1 \mbox{ for } S \subseteq V\mbox{ with $|S|$ odd,}\qquad \mbox{ and }
x \geq 0.$$ The linear description is useful for understanding the structure of the perfect matchings. For example, Naddef and Pulleyblank [@NP81] proved that $x_e = 1/3$ defines a feasible solution when $G$ is *cubic* and *$2$-edge connected*, i.e., every vertex has degree $3$ and the graph stays connected after the removal of an edge. They used that result to deduce that such graphs always have a perfect matching of weight at least $1/3$ of the total weight of the edges.
Standard algorithmic versions of Carathéodory’s theorem (see e.g. Theorem $6.5.11$ in [@GLS1988]) say that, in polynomial time, we can decompose a feasible solution to the perfect matching polytope into a convex combination of polynomially many perfect matchings (see also [@Barahona04] for a combinatorial approach for the matching polytope). Combining these results leads to the following lemma (see [@BSSS11; @GLS05; @MMP90] for closely related variants that also have been useful for the [graph-TSP]{}problem).
\[lem:matching\] Given a cubic $2$-edge-connected graph $G$, we can in polynomial time find a distribution over polynomially many perfect matchings so that with probability $1/3$ an edge is in a perfect matching picked from this distribution.
Note that all $2$-vertex-connected graphs except the trivial graph on $2$ vertices are $2$-edge connected. We can therefore apply the above lemma to cubic $2$-vertex-connected graphs.
Approximation Framework {#sec:tspframework}
=======================
Lemma \[lemma:2connTSP\] says that the technical difficulty in approximating the [graph-TSP]{}problem lies in approximating those instances that are $2$-vertex connected. As alluded to in the introduction, we shall generalize previous results [@FJJ89; @MMP90] that relate the cost of an optimal tour to the size of a minimum $2$-vertex-connected subgraph. The main difference is the use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian whereas our framework offers a structured way to specify a set of edges that safely may be removed leading to a lower cost. To identify the set of edges that may be removed we use the following definition.
\[def:pairing\] Given a $2$-vertex-connected graph $G$ we call a tuple $(R,P)$ consisting of a subset $R$ of removable edges and a subset $P\subseteq R\times R$ of pairs of edges a *[$\mbox{removable pairing}$]{}* if
- an edge is in at most one pair;
- the edges in a pair are incident to a common vertex of degree at least $3$;
- any graph obtained by deleting removable edges so that at most one edge in each pair is deleted stays connected.
The following theorem generalizes the corresponding result of [@MMP90] (their result follows from the the special case of an empty removable pairing).
\[thm:main\] Given a $2$-vertex-connected graph $G=(V,E)$ with a [$\mbox{removable pairing}$]{} $(R,P)$, there is a polynomial time algorithm that returns a spanning Eulerian multigraph in $G$ with at most $\frac{4}{3}\cdot |E| - \frac{2}{3} \cdot |R|$ edges.
The proof of the theorem is presented after the following lemma on which it is based.
\[lemma:sample\] Given a $2$-vertex-connected graph $G=(V,E)$ with a [$\mbox{removable pairing}$]{} $(R,P)$, we can in polynomial time find a distribution over polynomially many subsets of edges such that a random subset $M$ from this distribution satisfies:
- each edge is in $M$ with probability $1/3$;
- at most one edge in each pair is in $M$; and
- each vertex has an even degree in the multigraph with edge set $E
\cup M$.
We shall use Lemma \[lem:matching\] and will therefore need a cubic $2$-edge-connected graph. In the spirit of [@FJJ89], we replace all vertices of $G$ that are not of degree three by gadgets to obtain a cubic graph $G'=(V',E')$ as follows (see also Figure \[fig:degreplace\]):
- A vertex $v$ of degree 2 with neighbors $u$ and $w$ is replaced by a cycle consisting of four vertices $v_N$, $v_W$, $v_S$, $v_E$ with the chord $\{v_W, v_E\}$. The gadget is then connected to the neighbours of $v$ by the the edges $\{u,v_N\}$ and $\{v_S,w\}$.
- A vertex $v$ with $d(v) > 3$ is replaced by a tree $T_v$ that has $\lfloor d(v)/2 \rfloor$ leaves, a binary root if $d(v)$ is odd, and otherwise only degree $3$ internal vertices. Each leaf is connected to two neighbours of $v$ such that the edges incident to $v$ that form a pair in $P$ are incident to the same leaf. If $d(v)$ is odd, one of the neighbors is left and connected to the binary root.
![Examples of the used gadgets to obtain a cubic graph.[]{data-label="fig:degreplace"}](degreereplace){width="14cm"}
The above gadgets guarantee that the graph $G'$ is cubic and it is $2$-vertex connected since $G$ was assumed to be $2$-vertex connected. We can therefore apply Lemma \[lem:matching\] in order to obtain a random perfect matching $M'$. Each edge of $G'$ is in $M'$ with a probability of exactly 1/3. Let $M$ be the set of edges obtained by restricting $M'$ to the edges of $G$ in the obvious way. Now $M$ contains each edge of $G$ with probability 1/3. We complete the proof by showing that $M$ also satisfies properties $(b)$ and $(c)$. As each pair of edges in $P$ is incident to a vertex of degree at least $3$, we have, by the construction of the gadgets, that they are incident to a common vertex in $G'$ and hence at most one edge of each pair is in $M$. Finally, property $(c)$ follows from that $E' \cup M'$ is clearly a spanning Eulerian multigraph of $G'$ and compressing a set of even-degree vertices results in one vertex of even degree.
Equipped with the above lemma we are now ready to prove the main result of this section.
[Theorem \[thm:main\]]{} Pick a random subset $M \subseteq E$ of edges that satisfies the properties of Lemma \[lemma:sample\]. Let $M_R$ be the set of those edges of $M$ that are removable and let $\bar M_R$ be the set of the remaining edges of $M$.
Consider the multigraph $H$ on vertex set $V$ and edge set $E
\setminus M_R \cup \bar M_R$. Observe that both adding an edge and removing an edge swaps the parity of the degree of an incident vertex. We have thus from property $(c)$ of Lemma \[lemma:sample\] that the degree of each vertex in $H$ is even. Moreover, as $(R,P)$ is a removable pairing, property $(b)$ of Lemma \[lemma:sample\] gives that $H$ is connected. Alltogether we have that $H$ is an Eulerian graph, i.e., a [graph-TSP]{} solution. We continue to calculate its expected number of edges, which is $$\label{eq:nredges}
\mathbb{E}[|E| + |\bar M_R| -
|M_R|].$$ Using that each edge is in $M$ with probability $1/3$, we have, by linearity of expectation, that equals $$|E| + \frac{1}{3} (|E| - |R|) - \frac{1}{3} |R| = \frac{4}{3} \cdot |E| - \frac{2}{3} \cdot |R|.$$
To conclude the proof, we note that the selection of $M$ can be derandomized since there are, by Lemma \[lemma:sample\], polynomially many edge subsets to choose from; taking the one that minimizes the number of edges of $H$ is sufficient.
Finding a Removable Pairing by Minimum Cost Circulation {#sec:circulation}
=======================================================
In order to use our framework, one of the main challenges is to find a [$\mbox{removable pairing}$]{} that is sufficiently large. In the following, we show how to obtain a useful [$\mbox{removable pairing}$]{} based on circulations.
Consider a $2$-vertex connected graph $G$ and let $T$ be a spanning tree of $G$ obtained by depth-first search (starting from some arbitrary root $r$). Then each edge in $G$ connects a vertex to either one of its predecessors or one of its successors. We call the edges in $T$ *tree-edges* and those in $G$ but not in $T$ *back-edges*.
We shall now define a circulation network $C(G,T)$. We start by introducing an orientation of $G$: all tree-edges become tree-arcs directed from the root to the leaves and all back-edges become back-arcs directed towards the root. To distinguish the circulation network and the original graphs, we use the names ${\ensuremath{\overrightarrow{G}}}$ and ${\ensuremath{\overrightarrow{T}}}$ for the network versions of $G$ and $T$. In order to ensure connectivity properties of subnetworks obtained from feasible circulations, we replace some of the vertices by gadgets.
For each vertex $v$ except the root that has $\ell$ children $w_1,
w_2, \ldots, w_\ell$ in the tree, we introduce $\ell$ new vertices $v_1$, $v_2$, $\ldots$, $v_\ell$ and replace the tree-arc $(v, w_j)$ by the tree-arcs $(v,v_j)$ and $(v_j, w_j)$ for $j= 1,2, \ldots, \ell$. Then we redirect all incoming back-arcs of $v$ from the subtree rooted by $w_j$ to $v_j$. For an illustration of the gadget see Figure \[fig:circreplace\] and for an example of a complete network see Figure \[fig:circreplace\_appendix\]. This way, all back-arcs start in old vertices and lead to new vertices or the root. In the following, we call the new vertices and the root *in-vertices* and the remaining old ones *out-vertices*. We also let $\mathcal{I}$ be the set of all in-vertices.
![The gadget that, for each child of $v$, introduces a new vertex (depicted in white) and redirects back-arcs.[]{data-label="fig:circreplace"}](circreplace2){width="10cm"}
We now specify a lower bound (demand) and an upper bound (capacity) on the circulation. For each arc $a$ in ${\ensuremath{\overrightarrow{T}}}$, we set the demand of $a$ to 1 and for all other arcs to 0. The capacity is $\infty$ for any arc. Finally, the cost of a circulation $f$ in $C(G,T)$ is the piecewise linear function $\sum_{v\in\mathcal{I}}
\max[f(B(v))-1, 0]$, where $B(v)$ is the set of incoming back-arcs of $v$. One can think of the cost as the total circulation on the back-arcs except that each in-vertex accepts a circulation of $1$ for free. Note that algorithmically there is no considerable difference whether we use our cost function or define a linear cost function on the arcs: for any in-vertex $v$ we can redirect all back-arcs of $v$ to a new vertex $v'$ and introduce two arcs $(v',v)$, one of cost $0$ and capacity $1$ and the other of cost $1$ and capacity $\infty$. All remaining arcs then have a cost of $0$.
The following lemma shows how to use a circulation in $C(G,T)$ to approximate [graph-TSP]{}.
\[lemma:costcirc\] Given a $2$-vertex connected graph $G$ and a depth first search tree $T$ of $G$ let $C^*$ be the minimum cost circulation to $C(G,T)$ of cost $c(C^*)$. Then there is a spanning Eulerian multigraph $G'$ in $G$ with at most $\frac{4}{3} n + \frac{2}{3} c(C^*) - 2/3$ edges.
We first note that, for any arc of $C(G,T)$, the demand and the capacity is integral. Therefore, applying Hoffman’s circulation theorem (see [@Sch03], Corollary 12.2a), we can assume the circulation $C^*$ to be integral. Let ${\ensuremath{{C^*(G,T)}}}$ be the support of $C^*$ in $C(G,T)$, i.e., the induced subgraph of the arcs with non-zero circulation in $C^*$, and let $G'$ be the subgraph of $G$ obtained from ${\ensuremath{{C^*(G,T)}}}$ by compressing the gadges of the circulation network in the obvious way.
To prove the lemma, we shall first prove that graph $G'$ is $2$-vertex connected and then define a removable pairing $(R,P)$ on $G'$ in order to apply Theorem \[thm:main\]. That $G'$ is $2$-vertex connected follows from flow conservation, that each arc $a$ in ${\ensuremath{\overrightarrow{T}}}$ has demand $1$, and the design of the gadgets. Indeed, if $G'$ would have a cut vertex $v$ with children $w_1, w_2, \ldots, w_\ell$ in $T$ then one of the subtrees, say the one rooted by $w_j$, has no back-edges to the ancestors of $v$ which in turn, by flow conservation, would contradict that the tree-arc $(v, v_j)$ in ${\ensuremath{\overrightarrow{T}}}$ carries a flow of at least $1$. (Recall that the edge $\{v, w_j\}$ in $T$ is replaced by tree-arcs $(v, v_j)$ and $(v_j, w_j)$ in ${\ensuremath{\overrightarrow{T}}}$.)
We now determine a [$\mbox{removable pairing}$]{}$(R,P)$ on $G'$. For ease of argumentation we shall first slightly abuse notation and define a [$\mbox{removable pairing}$]{}$(R_C, P_C)$ on ${\ensuremath{{C^*(G,T)}}}$. The set $P_C$ consists of all $(e,e')$ such that $e=(u,v)$ is a back-arc of cost zero in ${\ensuremath{{C^*(G,T)}}}$, $v$ has at least two incoming arcs, and $e'=(v,w)$ is a tree-arc. Note that each such $v$ is an in-vertex, the number of incoming back-arcs of cost zero is at most one, $e'$ is the unique outgoing tree-arc of $v$, and the only possible vertex $v$ with only one incoming back-arc and no other incoming arc is the root. The set $R_C$ contains all edges from $P_C$ and additionally all remaining back-arcs of ${\ensuremath{{C^*(G,T)}}}$. In other words, each edge of ${\ensuremath{{C^*(G,T)}}}$ that is neither in ${\ensuremath{\overrightarrow{T}}}$ nor in $P$ is a back-arc with integer non-zero cost in the circulation or a back-arc to the root. Hence, $|R_C| - 2|P_C| = c(C^*)$ if the root has more than one incoming back-arc and $|R_C|- 2|P_C| = c(C^*) +1$ otherwise.
The [$\mbox{removable pairing}$]{}$(R,P)$ on $G'$ is now obtained from $(R_C, P_C)$, by mereley compressing the gadgets used to form $C(G,T)$ and by dropping the orientations of the arcs. As all edges in $R_C$ are either back-arcs or they are tree-arcs starting from an in-vertex, no arc in $R_C$ is removed by the compression and thus $|R|=|R_C|$ and $|P| = |P_C|$. Moreover, $G'$ has $(n-1) + |R| - |P|$ edges and, assuming $(R,P)$ is a valid [$\mbox{removable pairing}$]{}, Theorem \[thm:main\] yields that $G'$ (and thus $G)$ has a spanning Eulerian multigraph with at most $ \frac{4}{3} ((n-1) + |R|-|P|) - \frac{2}{3} |R| =
\frac{4}{3} n + \frac{2}{3}( |R| - 2|P|) - \frac{4}{3} \leq
\frac{4}{3} n + \frac{2}{3} c(C^*) - \frac{2}{3} $ edges. The last inequality followed from that $|R| - 2|P|$ is at most $c(C^*) +1$.
Therefore, we can conclude the proof by showing that $(R,P)$ is a valid [$\mbox{removable pairing}$]{}. It is easy to verify that $(R,P)$ satisfies the first two conditions of Definition \[def:pairing\], that is, each edge is contained in at most one pair and the edges in each pair are incident to one common vertex of degree at least three. The third condition follows from that, for any vertex $v$ of $G'$, the vertices in the subtree $T_v$ of $T$ rooted by $v$ form a connected subgraph of $G'$ even after removing edges according to $(R,P)$. To see this we do a simple induction on the depth of $v$. In the base case, $v$ is a leaf and the statement is clearly true. For the inductive step, consider a vertex $v$ with $\ell$ children $w_1,
w_2, \dots, w_\ell$ in $T$. By the inductive hypothesis, the vertices in $T_{w_j}$ for $j=1,2, \dots, \ell$ stay connected after the removal of edges according to $(R,P)$. To complete the inductive step it is thus sufficient to verify that $v$ is connected to each $T_{w_j}$ after the removal of edges. If $\{v,w_j\}$ is not in $R$ this clearly holds. Otherwise if $e_j = \{v, w_j\} \in R$ then by the definition of $(R, P)$ there is an edge $e$ such that $(e, e_j)
\in P$ and $e$ is incident to $v$ and a vertex in $T_{w_j}$. Since at most one edge in each pair is removed we have that $v$ also stays connected to $T_{w_j}$ in this case, which completes the inductive step. We have thus proved that $(R,P)$ satisfies the properties of a [$\mbox{removable pairing}$]{}which completes the proof of the statement.
Improved Approximation Algorithms {#sec:algorithms}
=================================
We first show how to apply our framework to restricted graph classes for which we obtain a tight bound on the integrality gap of the Held-Karp relaxation. We then show how to use our framework to obtain an improved approximation algorithm for general graphs.
Bounded Degree and Claw-Free Graphs
-----------------------------------
We consider the class of graphs that have a degree bounded by three.
\[lem:boundeddeg\] Given a $2$-vertex-connected graph $G$ with $n$ vertices, there is a polynomial time algorithm that computes a spanning Eulerian multigraph $H$ in $G$ with at most $4n/3 - 2/3$ edges.
If $G$ has one or two vertices, we obtain an Eulerian multigraph of zero or two edges. Otherwise, we compute a depth-first search tree $T$ in $G$ and determine the circulation network $C(G,T)$. We now show that this network has a feasible circulation $f$ of cost at most one. Let us assign a circulation of one to each back-arc $e$ in $C(G,T)$ and push it through the path in ${\ensuremath{\overrightarrow{T}}}$ that is incident to both the start and end vertex of $e$. By the construction of $C(G,T)$ and from the assumption that $G$ is $2$-vertex connected, each tree-arc is in a directed cycle that contains exactly one back-arc. Therefore, all demand constraints are satisfied. Due to the degree-bounds, no vertex but the root has more than one incoming back-arc. The cost $\sum_{v\in\mathcal{I}} \max[f(B(v))-1, 0]$ of the circulation is therefore at most one and zero if the root has only one back-arc. If the circulation cost is zero, by Lemma \[lemma:costcirc\] we obtain a spanning Eulerian multigraph $H$ in $G$ with at most $4n/3-2/3$ edges. For those circulations where the cost is one, the proof of Lemma \[lemma:costcirc\] allows to save an additional constant of $2/3$ (since then the root has more than one incoming back-arc) and we obtain the same bound on the number of edges.
Note that it is sufficient to find a 2-vertex-connected degree three bounded spanning subgraph (a 3-trestle) and thus, using a result from [@KKN01], we can apply Lemma \[lem:boundeddeg\] also to claw-free graphs. Applying Lemma \[lemma:2connTSP\], we obtain an upper bound of 4/3 on the integrality gap for the Held-Karp relaxation for the considered class of graphs. In addition, along the lines of the proof of Lemma \[lemma:2connTSP\], one can see that the above arguments imply that any connected graph $G$ decomposed into $k$ blocks, i.e., maximal $2$-connected subgraphs, such that each block is either degree three bounded or claw-free, has a spanning Eulerian multigraph with at most $4n/3 + 2k/3- 4/3$ edges.
General Graphs
--------------
We now apply our framework to graphs without degree constraints. We start with an algorithm that achieves an approximation ratio better than $3/2$ for graphs for which the linear programming relaxation has a value close to $n$. Let $G= (V,E)$ be an $n$-vertex graph. The support $E' = \{e: x_e^* > 0\}$ of an extreme point $x^*$ of [$LP(G)$]{} is known to contain at most $2n-1$ edges (see Theorem $4.9$ in [@CFN85]). Moreover, if we let $x^*$ be an optimal solution, then any $r$-approximate solution to graph $G'=(V,E')$ with respect to [$OPT_{LP}(G')$]{} is an $r$-approximate solution to $G$ with respect to [$OPT_{LP}(G)$]{}, because $E' \subseteq E$ and ${\ensuremath{OPT_{LP}(G')}} = {\ensuremath{OPT_{LP}(G)}}$. We can thus restrict ourselves to $n$-vertex graphs with at most $2n-1$ edges and, by Lemma \[lemma:2connTSP\], we can further assume the graph to be $2$-vertex connected.
A 2-vertex-connected graph $G$ with $n$ vertices and at most $2n-1$ edges. Obtain an optimal solution $x^*$ to [$LP(G)$]{}. Obtain a depth-first-search tree $T$ of $G$ by starting at some root and in each iteration pick, among the possible edges, the edge $e$ with maximum $x_e^*$. Solve the min cost circulation problem $C(G,T)$ to obtain a circulation $C^*$ with cost $c(C^*)$.
Apply Lemma \[lemma:costcirc\] to find a spanning Eulerian multigraph with less than $\frac{4}{3} n + \frac{2}{3} c(C^*)$ edges.
To analyze the approximation ratio achieved by Algorithm \[alg:allgraphs\], we bound the cost of the circulation.
\[lemma:circcost\] We have $c(C^*) \leq 6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G)}}$.
For notational convenience, when considering an arc $a$ in the flow network we shall slightly abuse notation and use $x^*_a$ to denote the value of the corresponding edge in $G$ according to the optimal LP-solution $x^*$. We prove the statement by defining a fractional circulation $f$ of cost at most $6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G)}}$. The circulation $f$ will in turn be the sum of two circulations $f'$ and $f''$. We obtain the circulation $f'$ as follows: for each back-arc $a$ we push a flow of size $\min[x^*_a,1]$ along the cycle formed by $a$ and the tree-arcs in ${\ensuremath{\overrightarrow{T}}}$. We shall now define the circulation $f''$ so as to guarantee that $f$ forms a feasible circulation, i.e., one that satisfies the demands $f_a \geq 1$ for each $a\in
{\ensuremath{\overrightarrow{T}}}$. As out- and in-vertices are alternating in ${\ensuremath{\overrightarrow{T}}}$ and in-vertices have only one child in ${\ensuremath{\overrightarrow{T}}}$ and no outgoing back-edges, a sufficient condition for $f$ to be feasible can be seen to be $f_a \geq 1$ for each $a\in {\ensuremath{\overrightarrow{T}}}$ that is from an out-vertex to an in-vertex. To ensure this, we now define $f''$ as follows. For each vertex $v$ of $G$ that is replaced by a gadget consisting of an out-vertex $v$ and a set $\mathcal{I}_v$ of in-vertices, we push for each $w\in \mathcal{I}_v$ a flow of size $\max[1-f'_{(v,w)}, 0]$ along a cycle that includes the arc $(v,w)$ (and one back-arc). Note that such a cycle is guaranteed to exist since $G$ was assumed to be $2$-vertex connected. From the definition of $f''$, we have thus that $f=f'+f''$ defines a feasible circulation.
We proceed by analyzing the cost of $f$, i.e., $\sum_{v\in
\mathcal{I}} \max[f(B(v)) -1, 0]$, where $\mathcal{I}$ is the set of all in-vertices and $B(v)$ is the set of incoming back-arcs of $v\in \mathcal{I}$. Note that the cost is upper bounded by $\sum_{v\in \mathcal{I}} \max[f'(B(v)) -1, 0] + \sum_{v\in
\mathcal{I}} f''(B(v))$ and we can thus analyze these two terms separately. We start by bounding the second summation and then continue with the first one. If ${\ensuremath{OPT_{LP}(G)}} = n$ then one can see that $f'' = 0$. Moreover,
\[claim:firstcost\] We have $\sum_{v\in \mathcal{I}} f''(B(v)) \leq {\ensuremath{OPT_{LP}(G)}}-n$.
When considering a vertex $v$ as done above in the definition of $f''$, the flow pushed on back-arcs is $\sum_{w\in \mathcal{I}_v}
\max[1-f'_{(v,w)}, 0]$ which equals $\sum_{w\in \mathcal{I}'_v}
(1-f'_{(v,w)})$, where $\mathcal{I}'_v = \{w \in \mathcal{I}_v: f'_{(v,w)} < 1\}$. Letting $T_w$ be the set of vertices of $G$ in the subtree of the undirected tree $T$ rooted by the child of $w\in \mathcal{I}'_v$, we have, by the definition of $f'$, $$f'_{(v,w)} = \sum_{a\in \delta(T_w)\setminus \delta(v)} \min[x^*_a, 1]
= x^*(\delta(T_w) \setminus \delta(v)).$$ The second equality follows from that if $x^*_a > 1$ for some $a\in
\delta(T_w) \setminus \delta(v)$ then $f'_{(v,w)} \geq 1$ and hence $w\not \in \mathcal{I}'_v$. We have thus $
\sum_{w\in \mathcal{I}'_v} (1- f'_{(v,w)}) = |\mathcal{I}'_v| -
\sum_{w\in \mathcal{I}'_v} x^*(\delta(T_w) \setminus \delta(v)).
$ As we are considering a depth-first-search tree (see Figure \[fig:circcostOLA\]), $$\begin{aligned}
\label{eq:equalsums}
2 \sum_{w\in \mathcal{I}'_v} x^*(\delta(T_w) \setminus \delta(v)) &=
\sum_{w\in \mathcal{I}'_v} x^*(\delta(T_w)) +
x^*\left(\delta\left(\bigcup_{w\in \mathcal{I}'_v} T_w \cup
\{v\}\right)\right) - x^*(\delta(v)).\end{aligned}$$
![An illustration of Equality with $\mathcal{I}'_v = \{w_1, w_2, \dots, w_\ell\}$: both the left-hand-side and the right-hand-side of the equality express two times the value of the fat edges.[]{data-label="fig:circcostOLA"}](circcost){width="7cm"}
Since by the feasibility of $x^*$ each of the sets corresponds to a cut of fractional value at least $2$ we use $2\cdot (|\mathcal{I}'_v| + 1) - x^*(\delta(v))$ as a lower bound on .
Summarizing the above calculations yields $$\sum_{w\in \mathcal{I}'_v} \left(1- f'_{(v,w)}\right) = |\mathcal{I}'_v| -
\sum_{w\in \mathcal{I}'_v} x^*(\delta(T_w) \setminus \delta(v)) \leq \frac{x^*(\delta(v))}{2} - 1.$$
Repeating this argument for each $v$ we have $\sum_{v\in \mathcal{I}}
f''(B(v)) = \sum_{v \in V} \sum_{w \in \mathcal{I}'_v} \left(1-f'_{(v,w)}\right) \le
\sum_{v\in V} \left(\frac{x^*(\delta(v))}{2} -1\right)$, which equals ${\ensuremath{OPT_{LP}(G)}}-n$ since ${\ensuremath{OPT_{LP}(G)}} = \frac{1}{2}\sum_{v\in V}
x^*(\delta(v))$.
We proceed by bounding $\sum_{v\in \mathcal{I}} \max[f'(B(v)) -1, 0]$ from above.
\[claim:secondcost\] We have $\sum_{v\in \mathcal{I}} \max[f'(B(v)) -1, 0] \leq (7-6\sqrt{2})n + 4(\sqrt{2}-1){\ensuremath{OPT_{LP}(G)}}$
To analyze this expression we shall use two facts. First $G$ has at most $2n-1$ edges, and therefore the number of back-arcs is at most $2n-1 - (n-1) = n$. Second, as the depth-first-search chooses (among the available edges) the edge $a$ with maximum $x^*_a$ in each iteration, we have that $x_{a}^* \leq x_{t_v}^*$ for each $a \in
B(v)$ where $t_v$ is the outgoing tree-arc of $v\in
\mathcal{I}$. Moreover, as $f'_a = \min[x^*_{a}, 1 ]$ for each back-arc, the number of back-arcs in $B(v)$ is at least $\left\lceil
\frac{f'(B(v))}{\min[x^*(t_v),1]}\right\rceil$. Combining these two facts gives us that $$\label{eq:cap2}
\sum_{v\in \mathcal{I}}
\left\lceil \frac{f'(B(v))}{\min[x^*(t_v),1]}\right\rceil \leq n.$$ For $v\in \mathcal{I}$, we partition $f'(B(v))$ into $\ell_v =
\min[2-x^*(t_v), f'(B(v))]$ and $u_v = f'(B(v)) -
\ell_v$. Furthermore, let $u^* = \sum_{v\in \mathcal{I}} u_v$. With this notation we can upper bound $\sum_{v\in \mathcal{I}}
\max[f'(B(v)) -1, 0]$ by $$\label{eq:fcost}
\sum_{v\in \mathcal{I}} \max[\ell_v - 1, 0] + u^*$$ and relax Inequality to $$\label{eq:cap}
\sum_{v\in \mathcal{I}} \frac{\ell_v}{x^*(t_v)} \leq n- u^*.$$
The cost (where we ignore $u^*$) subject to can now be interpreted as a knapsack problem of capacity $n - u^*$ that is packed with an item of profit $ \max[\ell_v
-1, 0]$ and size $\ell_v/x^*(t_v)$ for each $v\in
\mathcal{I}$. Consequently, we can upper bound by considering the fractional knapsack problem with capacity $n-u^*$ and infinitely many items of a maximized profit to size ratio. Associating a variable $L$ with $\ell_v$ and $T$ with $x^*(t_v)$ this ratio is $
\max_{0\leq T \leq 1,0\leq L \leq 2 - T} \frac{L-1}{L}\cdot T.$ For any $T$ the ratio is maximized by letting $L=2-T$ and we can thus restrict our attention to items with profit to size ratio $\max_{0\leq T \leq 1} \frac{1-T}{2-T} \cdot T$. A simple analysis (see Appendix \[sec:maxsizratio\]) shows that the maximum is achieved when $T= 2-
\sqrt{2}$. Therefore, the profit is upper bounded by $$\frac{\sqrt{2} -1}{\sqrt{2}} \cdot(2- \sqrt{2})\cdot (n-u^*) +u^* = (\sqrt{2}-1)^2 \cdot (n-u^*) + u^*.$$ As the fractional degree of a vertex $v$ that is replaced by a gadget with a set $\mathcal{I}_v$ of in-vertices is at least $2+\sum_{w\in
\mathcal{I}_v} u_w$, we have $u^* \leq 2({\ensuremath{OPT_{LP}(G)}}-n)$. Hence, $$\eqref{eq:fcost} \leq (\sqrt{2}-1)^2 \cdot (n-2({\ensuremath{OPT_{LP}(G)}}-n)) + 2({\ensuremath{OPT_{LP}(G)}}-n),$$ which equals $(7-6\sqrt{2})n + 4(\sqrt{2}-1){\ensuremath{OPT_{LP}(G)}}$.
Finally, by summing up the bounds given by Claim \[claim:firstcost\] and Claim \[claim:secondcost\] we bound the cost of $f$ and hence $c(C^*)$ from above by $
{\ensuremath{OPT_{LP}(G)}}-n + n(7-6\sqrt{2}) + 4(\sqrt{2}-1){\ensuremath{OPT_{LP}(G)}},
$ which equals $6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G)}}$.
Having analyzed Algorithm \[alg:allgraphs\], we are ready to prove our main algorithmic result.
[**\[thm:approximationratio\]**]{} (Restated) *There is a polynomial time approximation algorithm for [graph-TSP]{}with performance guarantee $\frac{14\cdot( \sqrt{2}-1)}{12\cdot
\sqrt{2}-13} < 1.461$.*
By Lemma \[lemma:2connTSP\] and the discussion before Algorithm \[alg:allgraphs\], we can restrict ourselves to $n$-vertex graphs that are $2$-vertex connected and have at most $2n-1$ edges. The statement now follows by using Algorithm \[alg:allgraphs\] if ${\ensuremath{OPT_{LP}(G)}}$ is close to $n$ and otherwise by using Christofides’ algorithm.
On the one hand, since Christofides’ algorithm returns a solution with at most $n-1 + {\ensuremath{OPT_{LP}(G)}}/2$ edges (see [@SW90] for an analysis of Christofides’ algorithm in terms of [$OPT_{LP}(G)$]{}), it has an approximation guarantee of at most $$\frac{n + {\ensuremath{OPT_{LP}(G)}}/2}{{\ensuremath{OPT_{LP}(G)}}}.$$ On the other hand, by Lemma \[lemma:circcost\], the approximation guarantee of Algorithm \[alg:allgraphs\] is at most $$\frac{\frac{4}{3} n + \frac{2}{3} \left( 6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G)}} \right) }{{\ensuremath{OPT_{LP}(G)}}}.$$ In particular, the approximation guarantee of Algorithm \[alg:allgraphs\] for a graph $G$ with ${\ensuremath{OPT_{LP}(G)}}=n$ is $4/3 +
2/3\cdot (\sqrt{2}-1)^2 \approx 1.4477$ but deteriorates as [$OPT_{LP}(G)$]{} increases. The approximation guarantee of Christofides’ algorithm on the other hand is getting better and better as [$OPT_{LP}(G)$]{} increases.
![The approximation ratios of Algorithm \[alg:allgraphs\] and Christofides’ algorithm depending on the ratio ${\ensuremath{OPT_{LP}(G)}}/n$.[]{data-label="fig:ratios"}](approx.mps){width="10cm"}
Comparing these two ratios, one gets that the worst case happens when ${\ensuremath{OPT_{LP}(G)}} = \frac{24\sqrt{2}-26}{16\sqrt{2}-15} n$ (see Figure \[fig:ratios\]) and, by using simple arithmetics, the approximation guarantee can be seen to be $\frac{14(\sqrt{2}-1)}{12\cdot \sqrt{2}-13}$.
The Traveling Salesman Path Problem {#sec:tspp}
===================================
\[sec:tspp\] In this section, we describe a sequence of generalizations and modifications of the techniques that we previously presented for [graph-TSP]{}and conclude with improved approximation algorithms for the traveling salesman path problem on graphic metrics, [graph-TSPP]{}.
Using Held-Karp for Graph-TSPP {#sec:HKpath}
------------------------------
We can obtain a natural generalization of ${\ensuremath{LP(G)}}$ to [graph-TSPP]{}by distinguishing whether the end vertices $s$ and $t$ are in the same set of vertices. To this end, let $\Phi = \{S \subseteq V \mid \{s,t\}
\subseteq S \mbox{ or } S \cap \{s,t\} = \emptyset \}$. Then the relaxation can be written as $${\ensuremath{LP(G,s,t)}}: \qquad
\begin{aligned}[t]
\min & \sum_{e\in E} x_e \\[2mm]
x(\delta(S)) & \geq 2, & \emptyset \neq S \subset V, S \in \Phi\\[2mm]
x(\delta(S)) & \geq 1, & \emptyset \neq S \subset V, S \notin \Phi\\[2mm]
x & \geq 0.
\end{aligned}$$ We denote the optimum of this generalized linear program by [$OPT_{LP}(G,s,t)$]{}. It is not hard to see that ${\ensuremath{OPT_{LP}(G)}} = {\ensuremath{OPT_{LP}(G,s,s)}}$.
The graph on the right-hand-side in Figure \[fig:intgap\] has a fractional solution such that the integrality gap of ${\ensuremath{LP(G,s,t)}}$ is lower bounded by 1.5.
![Graphs for which the Held-Karp relaxation and the Held-Karp relaxation adapted to [graph-TSPP]{}have an integrality gap tending to $4/3$ and $1.5$, respectively.[]{data-label="fig:intgap"}](intgap){width="12cm"}
For a given graph $G=(V,E)$, let $G'=(V,E \cup \{e'\})$ be the graph obtained from $G$ by inserting $e'=\{s,t\}$. Note that, given any solution $x$ to ${\ensuremath{LP(G,s,t)}}$, we can obtain a feasible solution to ${\ensuremath{LP(G')}}$ by adding 1 to $x_{e'}$. This way, for each of the cuts where $S \notin \Phi$, we have $\delta(S) \ge 2$ and thus ${\ensuremath{OPT_{LP}(G')}} \le {\ensuremath{OPT_{LP}(G,s,t)}}+1$. In the following, we will generalize our results for [graph-TSP]{}by using ${\ensuremath{OPT_{LP}(G')}}-1$ as lower bound.
Similar to [graph-TSP]{}, we observe that the difficulty in approximating [graph-TSPP]{}lies in approximating those instances that are $2$-vertex connected. The proof of this lemma can be found in Appendix \[app:2connTSP\].
[**\[lemma:2connTSP\]**]{} (Generalized) *Let $G$ be a graph and let $\mathcal{A}$ be an algorithm that, given a $2$-vertex-connected subgraph $H$ of $G$ and $s,t \in V(H)$, returns a [graph-TSPP]{} solution to $(H,s,t)$ with cost at most $r \cdot
{\ensuremath{OPT_{LP}(H,s,t)}}$. Then there is an algorithm $\mathcal{A}'$ that returns a [graph-TSPP]{} solution to $(G,s,t)$ for any $s,t\in V(G)$ with cost at most $r \cdot {\ensuremath{OPT_{LP}(G,s,t)}}$. Furthermore, the running time of $\mathcal{A}'$ is a polynomial in the running time of $\mathcal{A}$.*
Generalization of the Approximation Framework to [graph-TSPP]{}
---------------------------------------------------------------
We generalize the framework to the problem [graph-TSPP]{}. We obtain an approximation ratio that depends on ${\ensuremath{dist(s,t)}}$, the distance of $s$ and $t$. Therefore we can see the variant of Theorem \[thm:main\] for [graph-TSP]{}as a special case where $s$ and $t$ have the distance 0.
[**\[thm:main\]**]{} (Generalized) *Given a $2$-vertex connected graph $G=(V,E)$ with a [$\mbox{removable pairing}$]{} $(R,P)$ and $s,t\in V$, there is a polynomial time algorithm that returns a spanning subgraph $H$ of $G$ with an Eulerian path between $s$ and $t$ with at most $\frac{4}{3}|E| - \frac{2}{3} |R| + {\ensuremath{dist(s,t)}}/3$ edges.*
A graph has an Eulerian path between $s$ and $t$ if and only if it is connected and the multigraph obtained by adding the edge $e'=\{s,t\}$ is a spanning Eulerian subgraph. Therefore, we basically want to apply (the original) Theorem \[thm:main\] and swap the degree of $s$ and $t$.
To this end we create the graph $G'=(V,E')$ from $G$ by adding the edge $e'$ to $E$ if it is not already present in $G$. Then we apply Theorem \[thm:main\] to $G'$ with the removable pairing $(R, P)$ to obtain the spanning Eulerian subgraph $\tilde{G}$.
If the Eulerian graph $\tilde{G}$ contains exactly one copy of $e'$, we simply remove it to obtain $H$. This case appears if and only if $e'$ was not chosen during the sampling, which happens with a probability of $2/3$. Note that the 2-edge-connectedness ensures that the removal does not disconnect $\tilde{G}$.
Otherwise, with probability $1/3$, $\tilde{G}$ contains either two copies of $e'$ if $e' \notin R$ or none if $e' \in R$. In either case we obtain $H$ from $\tilde{G}$ by removing all copies of $e'$ and adding a shortest path of length exactly ${\ensuremath{dist(s,t)}}$ to $\tilde{G}$. If $e' \in E$, we add a path with probability $1/3$ and apart from that we only remove edges; the claimed result follows immediately. If $e' \notin E$, it is also not in $R$ and thus the path is added if and only if two edges are removed. Furthermore, with probability $2/3$, one edge is removed. Then the expected number of edges in $H$ is $$\frac{4}{3} (|E|+1) - \frac{2}{3} |R| + \frac{{\ensuremath{dist(s,t)}}-2}{3} - 2/3 =
\frac{4}{3} |E| - \frac{2}{3} |R| + \frac{{\ensuremath{dist(s,t)}}}{3}.$$
Both the removal of $e'$ and adding the shortest path swaps the parities of $s$ and $t$, but of no other vertex.
By using the generalized Theorem \[thm:main\] within the proof of Lemma \[lemma:costcirc\], we obtain immediately the following generalization.
[**\[lemma:costcirc\]**]{} (Generalized) *Given a $2$-vertex connected graph $G$, two vertices $s,t$ in $G$, and a depth first search tree $T$ of $G$, let $C^*$ be the minimum cost circulation to $C(G,T)$ of cost $c(C^*)$. Then there is a spanning multigraph $H$ of $G$ that has an Eulerian path between $s$ and $t$ with at most $\frac{4}{3}n + \frac{2}{3} c(C^*) - 2/3 + {\ensuremath{dist(s,t)}}/3$ edges.*
Approximation Algorithms for Graph-TSPP
---------------------------------------
We are now equipped with the right tools to obtain algorithmic results for [graph-TSPP]{}.
[**\[thm:approximationratiohpp\]**]{} (Restated) *For any $\varepsilon > 0$, there is a polynomial time approximation algorithm for [graph-TSPP]{}with performance guarantee $3-\sqrt{2} + \varepsilon < 1.586 + \varepsilon.$*
*If furthermore each block of the given graph is degree three bounded, there is a polynomial time approximation algorithm for [graph-TSPP]{}with performance guarantee $1.5 + \varepsilon$, for any $\varepsilon>0$*.
By the generalized variant of Lemma \[lemma:2connTSP\], it is sufficient to show the theorem assuming that $G$ is 2-vertex connected.
If $G$ is degree three bounded, we apply Lemma \[lem:boundeddeg\] on $G$, but use the generalized version of Lemma \[lemma:costcirc\] to obtain a solution to [graph-TSPP]{}that has at most $4n/3 - 2/3 + {\ensuremath{dist(s,t)}}/3$ edges. Additionally we may replace ${\ensuremath{dist(s,t)}}$ by $n/2$, since in 2-vertex-connected graphs with more than two vertices there are two vertex-disjoint paths between $s$ and $t$.
To obtain the claimed approximation ratio, we use the trivial lower bound $n-1$ of ${\ensuremath{OPT_{LP}(G,s,t)}}$. For any $\varepsilon$, we determine a constant $n_0$ such that, for all $n \ge
n_0$, the approximation ratio is bounded from above by $1.5+\varepsilon$. If the graph has fewer than $n_0$ vertices, we compute an optimal solution in constant time.
We continue with the case of general unweighted graphs. As in the previous subsections, $e'=\{s,t\}$. We apply Algorithm \[alg:allgraphs\] to obtain a circulation $C'^*$ of $G'=(V,E \cup \{e'\})$ such that, by Lemma \[lemma:circcost\], $c(C'^*) \le 6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G')}}$. Using this circulation, we apply the generalized version of Lemma \[lemma:costcirc\]. However, if $e' \notin E$ and it is used in the solution (i.e., it was added as a shortest path), we have to replace $e'$ by a shortest path between $s$ and $t$ in $G$. This is equivalent to using ${\ensuremath{dist(s,t)}}$ from $G$ instead of $G'$ in Lemma \[lemma:costcirc\]. Therefore, in the following ${\ensuremath{dist(s,t)}}$ always refers to the distance in $G$ and we obtain a solution to [graph-TSPP]{}of at most $$\begin{aligned}
&&\frac{4}{3}n + \frac{2}{3}(6(1-\sqrt{2})n + (4\sqrt{2} -3){\ensuremath{OPT_{LP}(G')}}) -
\frac{2}{3} + \frac{{\ensuremath{dist(s,t)}}}{3} \nonumber\\
&=&(16/3 - 4\sqrt{2})n + {\ensuremath{dist(s,t)}}/3 + (8 \sqrt{2}/3 - 2)({\ensuremath{OPT_{LP}(G')}}) - 2/3\end{aligned}$$ edges.
In the following, let $d={\ensuremath{dist(s,t)}}/n$ and $\zeta = ({\ensuremath{OPT_{LP}(G')}}-1)/n$. Then, using the lower bound ${\ensuremath{OPT_{LP}(G')}}-1$ on ${\ensuremath{OPT_{LP}(G,s,t)}}$, the approximation ratio achieved by our algorithm is at most $$\label{eqn:hppapprox}
\frac{16/3-4\sqrt{2}+d/3}{\zeta} + 8\sqrt{2}/3 - 2 + \epsilon_1,$$ where $\zeta \ge 1 - 1/n$ and $\epsilon_1 = (8\sqrt{2}/3-8/3)/{\ensuremath{OPT_{LP}(G')}}$. In the following calculations, we omit $\epsilon_1$, since it decreases with the input size. Similarly, we assume $\zeta \ge 1$. We will consider the deviation, however, in the final result.
Since (\[eqn:hppapprox\]) depends on $\zeta$, similar to the case of [graph-TSP]{}we employ a second algorithm to obtain an upper bound independent of $\zeta$.
Let $\mathcal{A}$ be the following simple approximation algorithm for [graph-TSPP]{}which can be considered folklore. First, $\mathcal{A}$ computes a spanning tree $T$ of cost $n-1$ in $G$. Then $\mathcal{A}$ doubles all edges but those on the unique path between $s$ and $t$ in $T$.
The output of $\mathcal{A}$ is clearly a valid solution to [graph-TSPP]{}and it computes a solution of at most $2\cdot(n-1)-{\ensuremath{dist(s,t)}}$ edges. Similar to (\[eqn:hppapprox\]), this results in an approximation ratio of at most $$\label{eqn:hppapproxtwo}
(2-d)/\zeta.$$
Note that for $\zeta=1$ and $d= \sqrt{2}-1$, disregarding $\varepsilon$, (\[eqn:hppapproxtwo\]) is the approximation ratio we are aiming for. Any increase of $\zeta$ or $d$ can only improve this ratio. Therefore we may restrict the analysis to values of $d$ in the range $[0,\sqrt{2}-1]$.
We will first analyze the approximation ratio depending on $d$ and determine afterwards the value of $d$ where the minimum of the two approximation ratios is maximized.
For any fixed $d$ within the considered range, (\[eqn:hppapprox\]) is monotonically increasing with respect to $\zeta$, whereas (\[eqn:hppapproxtwo\]) is monotonically decreasing. Since we are interested in the minimum of the ratios, in the worst case both ratios are equal. This happens when $$\label{eqn:equalhpp}
\zeta = \frac{12\sqrt{2}-4d-10}{8\sqrt{2}-6}.$$
We now replace $\zeta$ by (\[eqn:equalhpp\]) in (\[eqn:hppapproxtwo\]) to obtain the worst case approximation ratio depending on $d$ $$\frac{8\sqrt{2}-6-4d\sqrt{2}+3d}{6\sqrt{2}-2d-5}.$$ Since this ratio can be seen to be monotonically increasing with respect to $d$ within the considered range, the worst case appears when $d=\sqrt{2}-1$, and thus we obtain as upper bound on the approximation ratio $$\frac{8\sqrt{2}-6-4(\sqrt{2}-1)\sqrt{2}+3(\sqrt{2}-1)}{6\sqrt{2}-2(\sqrt{2}-1)-5}
= \frac{15\sqrt{2}-17}{4\sqrt{2}-3} = 3 - \sqrt{2}.$$
To conclude the proof, we still have to consider $\varepsilon_1$ and the case where $\zeta<1$. For any $\varepsilon>0$, we determine an $n_0$ based on $\varepsilon_1$ and $\zeta$ similar to the degree bounded case and solve [graph-TSPP]{}on graphs with fewer than $n_0$ vertices exactly. Altogether, we obtain an approximation ratio of at most $$3-\sqrt{2} + \varepsilon$$ (see also Figure \[fig:ratioshpp\]).
![The minimum of the approximation ratios (\[eqn:hppapprox\]) and (\[eqn:hppapproxtwo\]) depending on $d$ and $\zeta$.[]{data-label="fig:ratioshpp"}](hpp.mps){width="10cm"}
Conclusions
===========
We have introduced a framework of removable pairings to find Eulerian multigraphs. This framework proved to be useful to obtain an approximation algorithm for [graph-TSP]{}with an approximation ratio smaller than $1.461$ and to obtain a tight upper bound on the integrality gap of the Held-Karp relaxation for a restricted class of graphs that contains degree three bounded and claw-free graphs. In particular, we showed that in subcubic $2$-vertex-connected graphs we can always find a solution to [graph-TSP]{}of at most $4n/3 - 2/3$ edges, which settles a conjecture from [@BSSS11] affirmatively.
Our framework is not restricted to [graph-TSP]{}. With the same techniques and a more detailed analysis, our result translates to the traveling salesman path problem on graphic metrics with prespecified start and end vertex. In this way, one is guaranteed to obtain an approximation ratio smaller than $1.586$ and, for the degree three bounded case, the approximation ratio gets arbitrarily close to $1.5$.
We note that the framework of removable pairings is straightforward to generalize to general metrics, but the problem of finding a large enough removable pairing in such graphs in order to improve on Christofides’ algorithm remains open.
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Example of Circulation Network
==============================
![The circulation network $C(G,T)$ of a graph $G$ with depth-first-tree $T$. In-vertices and out-vertices of the circulation network is depicted in white and black, respectively.[]{data-label="fig:circreplace_appendix"}](circulationcost){width="12cm"}
Omitted Proofs
==============
Proof of Lemma \[lemma:2connTSP\] {#app:2connTSP}
---------------------------------
We prove the more general lemma from Section \[sec:tspp\] that also applies to the traveling salesman path problem.
[**\[lemma:2connTSP\]**]{} (Restated) *Let $G$ be a graph and let $\mathcal{A}$ be an algorithm that, given a $2$-vertex-connected subgraph $H$ of $G$ and $s,t \in V(H)$, returns a [graph-TSPP]{} solution to $(H,s,t)$ with cost at most $r \cdot
{\ensuremath{OPT_{LP}(H,s,t)}}$. Then there is an algorithm $\mathcal{A}'$ that returns a [graph-TSPP]{} solution to $(G,s,t)$ for any $s,t\in V(G)$ with cost at most $r \cdot {\ensuremath{OPT_{LP}(G,s,t)}}$. Furthermore, the running time of $\mathcal{A}'$ is a polynomial in the running time of $\mathcal{A}$.*
We define an $r$-approximation algorithm $\mathcal{A'}$ for $G$ as follows:
1. If $G$ is $2$-vertex connected then return the [graph-TSPP]{} solution obtained by running $\mathcal{A}$ on $(G,s,t)$.
2. Otherwise, let $v$ be a cut vertex whose removal results in components $C_1, C_2,\ldots,C_l$ with $l >1$. Recursively run $\mathcal{A'}$ on the $l$ sub-instances $(G_1,s_1, t_1), \dots, (G_l, s_l, t_l)$ and return the union of the obtained solutions, where $G_i$ denotes the subgraph of $G$ induced by $C_i \cup \{v\}$, $$s_i = \begin{cases}
s & \mbox{if }s\in C_i \\
v & \mbox{otherwise}
\end{cases}\qquad \mbox{and} \qquad t_i = \begin{cases}
t & \mbox{if }t\in C_i \\
v & \mbox{otherwise}
\end{cases}.$$
As a vertex is selected to be a cut vertex at most once, $\mathcal{A'}$ terminates in time bounded by a polynomial in the running time of $\mathcal{A}$. It remains to verify that it returns a [graph-TSPP]{} solution to $(G,s,t)$ with cost at most $r \cdot
{\ensuremath{OPT_{LP}(G,s,t)}}$. We do so by induction on the depth of the recursion. In the base case no recursive calls are made so the solution is that returned by $\mathcal{A}$ which by assumption is a [graph-TSPP]{} solution to $(G,s,t)$ with cost at most $r \cdot
{\ensuremath{OPT_{LP}(G,s,t)}}$.
Now consider the inductive step when a cut vertex $v$ of $G$ is selected whose removal results in components $C_1, C_2, \ldots, C_l$ with $l > 1$. Let $E_i$ be the multiset of edges of the obtained [graph-TSPP]{} solution to $(G_i, s_i, t_i)$. With this notation the edge set returned by $\mathcal{A}'$ is $\bigcup_{i=1}^\ell E_i$ and we need to prove that
- it is a feasible [graph-TSPP]{} solution to $(G, s,t)$, i.e, the edge set $\bigcup_{i=1}^\ell E_i \cup \{s,t\}$ forms a spanning Eulerian subgraph; and
- $\sum_{i=1}^\ell |E_i| \leq r \cdot {\ensuremath{OPT_{LP}(G,s,t)}}.$
We start by proving (a). By the induction hypothesis, the edge set $E_i \cup \{s_i, t_i\}$ forms a spanning Eulerian subgraph of $G_i$ and, consequently, $\bigcup_{i=1}^\ell \left(E_i \cup \{s_i,
t_i\}\right)$ forms a spanning Eulerian subgraph of $G$. That $\bigcup_{i=1}^\ell E_i \cup \{s, t\}$ is a spanning Eulerian subgraph of $G$ now follows from that the endpoints of $\{s_1,t_1\}, \{s_2,
t_2\}, \ldots, \{s_\ell, t_\ell\}$ can be partitioned so that one is $s$, one is $t$ and the remaining $2(\ell-1)$ endpoints are $v$ ( possibly not different from $s$ and $t$).
We proceed by proving (b). By the induction hypothesis, $ \sum_{i=1}^\ell |E_i| \leq r \cdot \sum_{i=1}^\ell {\ensuremath{OPT_{LP}(G_i, s_i,
t_i)}} $ and it is thus sufficient to prove $\sum_{i=1}^\ell {\ensuremath{OPT_{LP}(G_i,
s_i, t_i)}} \leq {\ensuremath{OPT_{LP}(G,s,t)}}$. To this end, Let $x$ be an optimal solution to ${\ensuremath{LP(G,s,t)}}$ and let $x^i$ denote its restriction to the subgraph $G_i$ with start vertex $s_i$ and end vertex $t_i$. By the definition of $G_i, s_i, t_i$ and the fact that $v$ is a cut vertex, it is easy to see that each constraint in ${\ensuremath{LP(G_i, s_i, t_i)}}$ has an identical constraint in ${\ensuremath{LP(G,s,t)}}$. Therefore, $x^i$ corresponds to a solution to [$LP(G_i, s_i, t_i)$]{} and hence ${\ensuremath{OPT_{LP}(G,s,t)}} \geq
\sum_{i=1}^\ell {\ensuremath{OPT_{LP}(G_i,s_i,t_i)}}, $ which completes the inductive step and the proof of the lemma.
Maximum Profit to Size Ratio {#sec:maxsizratio}
----------------------------
We verify that $\max_{0\leq T\leq 1} \frac{1-T}{2-T} T$ is obtained when $T= 2-\sqrt{2}$. Let $f(T) = \frac{1-T}{2-T} T = \frac{T}{2-T} -
\frac{T^2}{2-T}$ and consider its first derivative $$\frac{d}{dT}f(T) = \frac{1}{2-T} + \frac{T}{(2-T)^2} - \left(\frac{2T}{2-T} + \frac{T^2}{(2-T)^2}\right) = \frac{1-2T}{2-T} + \frac{T-T^2}{(2-T)^2}.$$ From this it follows that $\frac{d}{dT}f(T) = 0$ when $$(1-2T)(2-T) + T-T^2 = 0 \Leftrightarrow T^2 - 4T + 2 = 0 \Leftrightarrow T = 2 \pm \sqrt{2}.$$ It is now easy to verify that the unique maximum of $f(T)$ for $0\leq
T\leq 1$ is obtained when $T= 2-\sqrt{2}$.
[^1]: This research was supported by ERC Advanced investigator grant 226203.
|
---
abstract: |
We introduce a complete many-valued semantics for two normal lattice-based modal logics. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal analysis of the interaction between (competing) scientific theories.\
[**Keywords:**]{} Non distributive modal logic, Graph-based semantics, Competing theories.
author:
- |
[**Willem Conradie$^a$**]{} and [**Andrew Craig$^b$**]{} and [**Alessandra Palmigiano$^{b,c}$**]{} and [**Nachoem Wijnberg$^{d,e}$**]{}\
$^a$School of Mathematics, University of the Witwatersrand\
$^b$Department of Mathematics and Applied Mathematics, University of Johannesburg,\
$^c$Faculty of Technology, Policy and Management, Delft University of Technology\
$^d$ College of Business and Economics, University of Johannesburg\
$^e$ Amsterdam Business School, University of Amsterdam
bibliography:
- 'BIBeusflat2019.bib'
title: '**Modelling competing theories**'
---
Introduction
============
The contributions of this paper lie at the intersection of several strands of research. They are rooted in the generalized Sahlqvist theory for normal LE-logics [@CoPa:non-dist; @conradie2016constructive], i.e. those logics algebraically captured by varieties of normal lattice expansions (LEs) [@gehrke2001bounded]. Via canonical extensions and discrete duality, basic normal LE-logics of arbitrary signatures and a large class of their axiomatic extensions can be uniformly endowed with complete relational semantics of different kinds, of which those of interest to the present paper are relational structures based on [*formal contexts*]{} [@gehrke2006generalized; @galatos2013residuated; @conradie2016categories; @Tarkpaper; @greco2018algebraic] and [*reflexive graphs*]{} [@conradie2015relational; @graph-based-wollic]. In a mathematical setting in which the original discrete duality for perfect normal LEs has been relaxed to a discrete adjunction for complete normal LEs, these semantic structures have yielded uniform theoretical developments in the algebraic proof theory [@greco2018algebraic] and in the model theory [@conradie2018goldblatt] of LE-logics, and also insights on possible interpretations of LE-logics which have generated new opportunities for applications. In particular, via polarity-based semantics, in [@conradie2016categories], the basic non-distributive modal logic and some of its axiomatic extensions are interpreted as [*epistemic logics of categories and concepts*]{}, and in [@Tarkpaper], the corresponding ‘common knowledge’-type construction is used to give an epistemic-logical formalization of the notion of [*prototype*]{} of a category; in [@roughconcepts; @ICLA2019paper], polarity-based semantics for non-distributive modal logic is proposed as an encompassing framework for the integration of rough set theory [@pawlak] and formal concept analysis [@ganter2012formal], and in this context, the basic non-distributive modal logic is interpreted as the logic of [*rough concepts*]{}; via graph-based semantics, in [@graph-based-wollic], the same logic is interpreted as the logic of [*informational entropy*]{}, i.e. an inherent boundary to knowability due e.g. to perceptual, theoretical, evidential or linguistic limits. In the graphs $(Z, E)$ on which the relational structures are based, the relation $E$ is interpreted as the indiscernibility relation induced by informational entropy, much in the same style as Pawlak’s approximation spaces in rough set theory. However, the key difference is that, rather than generating modal operators which associate any subset of $Z$ with its definable $E$-approximations, $E$ generates a complete lattice (i.e. the lattice of $E^c$-concepts). In this approach, concepts are not definable approximations of predicates, but rather they represent ‘all there is to know’, i.e. the theoretical horizon to knowability, given the inherent boundary encoded into $E$ (in their turn, $E^c$-concepts are approximated by means of the additional relations of the graph-based relational structures from which the semantic modal operators arise). Interestingly, $E$ is required to be reflexive but in general neither transitive nor symmetric, which is in line with proposals in rough set theory [@wybraniec1989generalization; @yao1996-Sahlqvist; @vakarelov2005modal] that indiscernibility does not need to give rise to equivalence relations.
In this paper, we start exploring the many-valued version of the graph-based semantics of [@graph-based-wollic] for two axiomatic extensions of the basic normal non-distributive modal logic, and in particular their potential for modelling situations in which informational entropy derives from the theoretical frameworks under which empirical studies are conducted.
Preliminaries
=============
This section is based on [@graph-based-wollic Section 2.1] and [@roughconcepts Section 7.2].
Basic normal nondistributive modal logic {#sec:logics}
----------------------------------------
Let ${\mathsf{Prop}}$ be a (countable or finite) set of atomic propositions. The language $\mathcal{L}$ of the [*basic normal nondistributive modal logic*]{} is defined as follows: $$\varphi := \bot \mid \top \mid p \mid \varphi \wedge \varphi \mid \varphi \vee \varphi \mid \Box \varphi \mid \Diamond\varphi,$$ where $p\in {\mathsf{Prop}}$. The [*basic*]{}, or [*minimal normal*]{} $\mathcal{L}$-[*logic*]{} is a set $\mathbf{L}$ of sequents ${\varphi}\vdash\psi$ with ${\varphi},\psi\in\mathcal{L}$, containing the following axioms: and closed under the following inference rules: An [*$\mathcal{L}$-logic*]{} is any extension of $\mathbf{L}$ with $\mathcal{L}$-axioms ${\varphi}\vdash\psi$. Relevant to what follows are the axiomatic extensions of $\mathbf{L}$ generated by $\Box\bot\vdash \bot$ and $\top\vdash \Diamond\top$, and by $\Box{\varphi}\vdash {\varphi}$ and ${\varphi}\vdash\Diamond{\varphi}$. Let $\mathbf{L}_0$ (resp. $\mathbf{L}_1$) be the axiomatic extension obtained by adding $\Box\bot\vdash \bot$ (resp. $\Box p\vdash p$) to $\mathbf{L}$. Notice that $\mathbf{L}_1$ is an extension of $\mathbf{L}_0$.
Many-valued enriched formal contexts
------------------------------------
Throughout this paper, we let $\mathbf{A} = (D, 1, 0, \vee, \wedge, \otimes, \to)$ denote an arbitrary but fixed complete frame-distributive and dually frame-distributive, commutative and associative residuated lattice (understood as the algebra of truth-values) such that $1\to \alpha = \alpha$ for every $\alpha\in D$. For every set $W$, an $\mathbf{A}$-[*valued subset*]{} (or $\mathbf{A}$-[*subset*]{}) of $W$ is a map $u: W\to \mathbf{A}$. We let $\mathbf{A}^W$ denote the set of all $\mathbf{A}$-subsets. Clearly, $\mathbf{A}^W$ inherits the algebraic structure of $\mathbf{A}$ by defining the operations and the order pointwise. The $\mathbf{A}$-[*subsethood*]{} relation between elements of $\mathbf{A}^W$ is the map $S_W:\mathbf{A}^W\times \mathbf{A}^W\to \mathbf{A}$ defined as $S_W(f, g) :=\bigwedge_{z\in W }(f(z)\rightarrow g(z)) $. For every $\alpha\in \mathbf{A}$, let $\{\alpha/ w\}: W\to \mathbf{A}$ be defined by $v\mapsto \alpha$ if $v = w$ and $v\mapsto \bot^{\mathbf{A}}$ if $v\neq w$. Then, for every $f\in \mathbf{A}^W$, $$\label{eq:MV:join:generators}
f = \bigvee_{w\in W}\{f(w)/ w\}.$$ When $u, v: W\to \mathbf{A}$ and $u\leq v$ w.r.t. the pointwise order, we write $u\subseteq v$. An $\mathbf{A}$-[*valued relation*]{} (or $\mathbf{A}$-[*relation*]{}) is a map $R: U \times W \rightarrow \mathbf{A}$. Two-valued relations can be regarded as $\mathbf{A}$-relations. In particular for any set $Z$, we let $\Delta_Z: Z\times Z\to \mathbf{A}$ be defined by $\Delta_Z(z, z') = \top$ if $z = z'$ and $\Delta_Z(z, z') = \bot$ if $z\neq z'$. An $\mathbf{A}$-relation $R: Z\times Z\to \mathbf{A}$ is [*reflexive*]{} if $\Delta_Z \subseteq R$. Any $\mathbf{A}$-valued relation $R: U \times W \rightarrow \mathbf{A}$ induces maps $R^{(0)}[-] : \mathbf{A}^W \rightarrow \mathbf{A}^U$ and $R^{(1)}[-] : \mathbf{A}^U \rightarrow \mathbf{A}^W$ defined as follows: for every $f: U \to \mathbf{A}$ and every $u: W \to \mathbf{A}$,
---------------- ---------------------------------------------------------
$R^{(1)}[f]:$ $ W\to \mathbf{A}$
$ x\mapsto \bigwedge_{a\in U}(f(a)\rightarrow R(a, x))$
$R^{(0)}[u]: $ $U\to \mathbf{A} $
$a\mapsto \bigwedge_{x\in W}(u(x)\rightarrow R(a, x))$
---------------- ---------------------------------------------------------
A [*formal*]{} $\mathbf{A}$-[*context*]{}[^1] or $\mathbf{A}$-[*polarity*]{} (cf. [@belohlavek]) is a structure $\mathbb{P} = (A, X, I)$ such that $A$ and $X$ are sets and $I: A\times X\to \mathbf{A}$. Any formal $\mathbf{A}$-context induces maps $(\cdot)^{\uparrow}: \mathbf{A}^A\to \mathbf{A}^X$ and $(\cdot)^{\downarrow}: \mathbf{A}^X\to \mathbf{A}^A$ given by $(\cdot)^{\uparrow} = I^{(1)}[\cdot]$ and $(\cdot)^{\downarrow} = I^{(0)}[\cdot]$. These maps are such that, for every $f\in \mathbf{A}^A$ and every $u\in \mathbf{A}^X$, $$S_A(f, u^{\downarrow}) = S_X(u, f^{\uparrow}),$$ that is, the pair of maps $(\cdot)^{\uparrow}$ and $(\cdot)^{\downarrow}$ form an $\mathbf{A}$-[*Galois connection*]{}. In [@belohlavek Lemma 5], it is shown that every $\mathbf{A}$-Galois connection arises from some formal $\mathbf{A}$-context. A [*formal*]{} $\mathbf{A}$-[*concept*]{} of $\mathbb{P}$ is a pair $(f, u)\in \mathbf{A}^A\times \mathbf{A}^X$ such that $f^{\uparrow} = u$ and $u^{\downarrow} = f$. It follows immediately from this definition that if $(f, u)$ is a formal $\mathbf{A}$-concept, then $f^{\uparrow \downarrow} = f$ and $u^{\downarrow\uparrow} = u$, that is, $f$ and $u$ are [*stable*]{}. The set of formal $\mathbf{A}$-concepts can be partially ordered as follows: $$(f, u)\leq (g, v)\quad \mbox{ iff }\quad f\subseteq g \quad \mbox{ iff }\quad v\subseteq u.$$ Ordered in this way, the set of the formal $\mathbf{A}$-concepts of $\mathbb{P}$ is a complete lattice, which we denote $\mathbb{P}^+$.
An [*enriched formal $\mathbf{A}$-context*]{} (cf. [@roughconcepts Section 7.2]) is a structure $\mathbb{F} = (\mathbb{P}, R_\Box, R_\Diamond)$ such that $\mathbb{P} = (A, X, I)$ is a formal $\mathbf{A}$-context and $R_\Box: A\times X\to \mathbf{A}$ and $R_\Diamond: X\times A\to \mathbf{A}$ are $I$-[*compatible*]{}, i.e. $R_{\Box}^{(0)}[\{\alpha / x\}]$, $R_{\Box}^{(1)}[\{\alpha / a\}]$, $R_{\Diamond}^{(0)}[\{\alpha / a\}]$ and $R_{\Diamond}^{(1)}[\{\alpha / x\}]$ are stable for every $\alpha \in \mathbf{A}$, $a \in A$ and $x \in X$. The [*complex algebra*]{} of an enriched formal $\mathbf{A}$-context $\mathbb{F} = (\mathbb{P}, R_\Box, R_\Diamond)$ is the algebra $\mathbb{F}^{+} = (\mathbb{P}^{+}, [R_{\Box}], \langle R_{\Diamond} \rangle )$ where $[R_{\Box}], \langle R_{\Diamond} \rangle : \mathbb{P}^{+} \to \mathbb{P}^{+}$ are defined by the following assignments: for every $c = ({[\![{c}]\!]}, {(\![{c}]\!)}) \in \mathbb{P}^{+}$,
------------------------------------ ------ --------------------------------------------------------------------------------------------
$[R_{\Box}]c$ $ =$ $ (R_{\Box}^{(0)}[{(\![{c}]\!)}], (R_{\Box}^{(0)}[{(\![{c}]\!)}])^{\uparrow})$
$ \langle R_{\Diamond} \rangle c $ $ =$ $ ((R_{\Diamond}^{(0)}[{[\![{c}]\!]}])^{\downarrow}, R_{\Diamond}^{(0)}[{[\![{c}]\!]}])$.
------------------------------------ ------ --------------------------------------------------------------------------------------------
\[prop:fplus\] (cf. [@roughconcepts Lemma 15]) If $\mathbb{F} = (\mathbb{X}, R_{\Box}, R_{\Diamond})$ is an enriched formal $\mathbf{A}$-context, $\mathbb{F}^+ = (\mathbb{X}^+, [R_{\Box}], \langle R_{\Diamond}\rangle)$ is a complete normal lattice expansion such that $[R_\Box]$ is completely meet-preserving and $\langle R_\Diamond\rangle$ is completely join-preserving.
Many-valued graph-based frames {#sec:frames}
==============================
A reflexive $\mathbf{A}$-[*graph*]{} is a structure ${\mathbb{X}}= (Z, E)$ such that $Z$ is a nonempty set, and $E: Z\times Z\to \mathbf{A}$ is reflexive. From now on, we will assume that all $\mathbf{A}$-graphs we consider are reflexive even when we drop the adjective.
\[def:A-lifting of a graph\] For any reflexive $\mathbf{A}$-graph ${\mathbb{X}}= (Z, E)$, the formal $\mathbf{A}$-context associated with ${\mathbb{X}}$ is $$\mathbb{P_X}: = (Z_A, Z_X, I_E),$$ where $Z_A := \mathbf{A}\times Z$ and $Z_X := Z$, and $I_E: Z_A\times Z_X\to \mathbf{A}$ is defined by $I_E((\alpha, z), z') = E(z, z')\rightarrow \alpha$. We let ${\mathbb{X}}^+: = \mathbb{P_X}^+$.
Any $R: Z\times Z\to \mathbf{A}$ admits the following [*liftings*]{}:
-------- --------------------------------------------------------
$I_R:$ $ Z_A\times Z_X \to \mathbf{A}$
$ ((\alpha, z), z')\mapsto R(z, z')\rightarrow \alpha$
$J_R:$ $ Z_X\times Z_A \to \mathbf{A}$
$ (z, (\alpha, z'))\mapsto R(z, z')\rightarrow \alpha$
-------- --------------------------------------------------------
Recall that for all $f: \mathbf{A}\times Z\to \mathbf{A}$, and $u: Z\to \mathbf{A}$, the maps[^2] $f^{\uparrow}: Z\to \mathbf{A}$ and $u^{\downarrow}: \mathbf{A}\times Z\to \mathbf{A}$ are respectively defined by the assignments $$z\mapsto \bigwedge_{(\alpha, z')\in Z_A}[ f(\alpha, z')\to (E(z', z)\to\alpha)]$$ $$(\alpha, z)\mapsto \bigwedge_{z'\in Z_X}[ u(z')\to (E(z, z')\to\alpha)].$$
\[def:graph:based:frame:and:model\] A [*graph-based $\mathbf{A}$-frame*]{} is a structure $\mathbb{G} = (\mathbb{X}, R_{\Diamond}, R_{\Box})$ where $\mathbb{X} = (Z,E)$ is a reflexive $\mathbf{A}$-graph, and $R_{\Diamond}$ and $R_{\Box}$ are binary $\mathbf{A}$-relations on $Z$ such that the structure $\mathbb{F_G}: = (\mathbb{P_X}, I_{R_\Box}, J_{R_\Diamond})$ is an enriched formal $\mathbf{A}$-context. That is, $R_{\Diamond}$ and $R_{\Box}$ satisfy the following $E$-[*compatibility*]{} conditions: for any $z\in Z$ and $\alpha, \beta\in\mathbf{A}$, $$\begin{aligned}
(R_\Box^{[0]}[\{\beta / z \}])^{[10]} &\subseteq R_\Box^{[0]}[\{\beta / z \}] \\
(R_\Box^{[1]}[\{\beta / (\alpha, z) \}])^{[01]} &\subseteq R_\Box^{[1]}[\{\beta / (\alpha, z) \}]\\
(R_\Diamond^{[1]}[\{\beta / z\}])^{[10]} &\subseteq R_\Diamond^{[1]}[\{\beta / z \}] \\
(R_\Diamond^{[0]}[\{\beta / (\alpha, z) \}])^{[01]} &\subseteq R_\Diamond^{[0]}[\{\beta / (\alpha, z) \}].\end{aligned}$$ where for all $f: \mathbf{A}\times Z\to \mathbf{A}$ and $u: Z\to \mathbf{A}$,
------------------------- --------------------------------------------------------------------------
$u^{[0]} = E^{[0]}[u]:$ $\mathbf{A}\times Z\to \mathbf{A}$
$(\alpha, z)\mapsto I_E^{(0)}[u](\alpha, z) = u^{\downarrow}(\alpha, z)$
$f^{[1]} = E^{[1]}[f]:$ $Z\to \mathbf{A}$
$z\mapsto I_E^{(1)}[f](z) = f^{\uparrow}(z)$
------------------------- --------------------------------------------------------------------------
---------------------- -------------------------------------------------------
$R_{\Box}^{[0]}[u]:$ $\mathbf{A}\times Z\to \mathbf{A}$
$(\alpha, z)\mapsto I_{R_{\Box}}^{(0)}[u](\alpha, z)$
$R_{\Box}^{[1]}[f]:$ $ Z\to \mathbf{A}$
$z\mapsto I_{R_\Box}^{(1)}[f](z)$
---------------------- -------------------------------------------------------
-------------------------- -----------------------------------------------------------
$R_{\Diamond}^{[0]}[f]:$ $Z\to \mathbf{A}$
$z\mapsto J_{R_\Diamond}^{(0)}[f]( z)$
$R_{\Diamond}^{[1]}[u]:$ $\mathbf{A}\times Z\to \mathbf{A}$
$(\alpha, z)\mapsto J_{R_\Diamond}^{(1)}[u](\alpha, z)$.
-------------------------- -----------------------------------------------------------
Hence, for any $z\in Z$ and $\alpha\in \mathbf{A}$,
-------------------------------------------------------------------------------------------------------------------
$E^{[0]}[u](\alpha, z): = \bigwedge_{z'\in Z_X}[u(z')\to (E(z, z') \to \alpha)]$
$E^{[1]}[f](z): = \bigwedge_{(\alpha, z')\in Z_A}[f(\alpha, z')\to (E(z', z) \to \alpha)]$.
$R_{\Box}^{[0]}[u](\alpha, z): = \bigwedge_{z'\in Z_X}[u( z')\to (R_{\Box}(z, z') \to \alpha)]$
$R_{\Box}^{[1]}[f](z): = \bigwedge_{(\alpha, z')\in Z_A}[f(\alpha, z')\to (R_{\Box}(z', z) \to \alpha)]$
$R_{\Diamond}^{[0]}[f](z): = \bigwedge_{(\alpha, z')\in Z_A}[f(\alpha, z')\to (R_{\Diamond}(z, z') \to \alpha)]$
$R_{\Diamond}^{[1]}[u](\alpha, z): = \bigwedge_{z'\in Z_X}[u(z')\to (R_{\Diamond}(z', z) \to \alpha)]$.
-------------------------------------------------------------------------------------------------------------------
The [*complex algebra*]{} of a graph-based $\mathbf{A}$-frame $\mathbb{G}= (\mathbb{X}, R_{\Diamond}, R_{\Box})$ is the algebra $\mathbb{G}^+ = (\mathbb{X}^+, [R_\Box], \langle R_\Diamond\rangle),$ where $\mathbb{X}^+: = \mathbb{P_X}^+$, and $[R_\Box]$ and $\langle R_\Diamond\rangle$ are unary operations on $\mathbb{X}^+$ defined as follows: for every $c = ({[\![{c}]\!]}, {(\![{c}]\!)}) \in \mathbb{X}^+$,
-------------------------------- ------ -------------------------------------------------------------------------------------
$[R_\Box]c$ $ =$ $ (R_{\Box}^{[0]}[{(\![{c}]\!)}], (R_{\Box}^{[0]}[{(\![{c}]\!)}])^{[1]})$
$ \langle R_\Diamond\rangle c$ $ =$ $ ((R_{\Diamond}^{[0]}[{[\![{c}]\!]}])^{[0]}, R_{\Diamond}^{[0]}[{[\![{c}]\!]}])$.
-------------------------------- ------ -------------------------------------------------------------------------------------
By definition, it immediately follows that
If $\mathbb{G}$ is a graph-based $\mathbf{A}$-frame, $\mathbb{G}^+ = \mathbb{F_G}^+$.
Hence, by the lemma above and Lemma \[prop:fplus\],
If $\mathbb{G} = (\mathbb{X}, R_{\Box}, R_{\Diamond})$ is a graph-based $\mathbf{A}$-frame, $\mathbb{G}^+ = (\mathbb{X}^+, [R_{\Box}], \langle R_{\Diamond}\rangle)$ is a complete normal lattice expansion such that $[R_\Box]$ is completely meet-preserving and $\langle R_\Diamond\rangle$ is completely join-preserving.
The following lemma is an immediate consequence of [@roughconcepts Lemma 14] applied to $\mathbb{F_G}$.
\[equivalence of I-compatible-mv\] For every graph-based $\mathbf{A}$-graph $\mathbb{G} = (\mathbb{X}, R_{\Box}, R_{\Diamond})$,
1. the following are equivalent:
1. $(R_{\Box}^{[0]}[\{\alpha/z\}])^{[10]}\subseteq R_{\Box}^{[0]}[\{\alpha/z\}]$ for every $z\in Z$ and $\alpha\in\mathbf{A}$;
2. $(R_{\Box}^{[0]}[u])^{[10]}\subseteq R_{\Box}^{[0]}[u]$ for every $u: Z_X\to\mathbf{A}$;
3. $R_{\Box}^{[1]}[f^{[10]}]\subseteq R_{\Box}^{[1]}[f]$ for every $f:Z_A\to\mathbf{A}$.
2. the following are equivalent:
1. $(R_{\Box}^{[1]}[\{\alpha/(\beta, z)\}])^{[01]}\subseteq R_{\Box}^{[1]}[\{\alpha/(\beta, z)\}]$ for every $z\in Z$ and $\alpha, \beta\in\mathbf{A}$;
2. $(R_{\Box}^{[1]}[f])^{[01]}\subseteq R_{\Box}^{[1]}[f]$ for every $f: Z_A\to\mathbf{A}$;
3. $R_{\Box}^{[0]}[u^{[01]}]\subseteq R_{\Box}^{[0]}[u]$ for every $u:Z_X\to\mathbf{A}$.
3. the following are equivalent:
1. $(R_{\Diamond}^{[0]}[\{\alpha/(\beta, z)\}])^{[01]}\subseteq R_{\Diamond}^{[0]}[\{\alpha/(\beta, z)\}]$ for every $z\in Z$ and $\alpha, \beta\in\mathbf{A}$;
2. $(R_{\Diamond}^{[0]}[f])^{[01]}\subseteq R_{\Diamond}^{[0]}[f]$ for every $f: Z_A\to\mathbf{A}$;
3. $R_{\Diamond}^{[1]}[u^{[01]}]\subseteq R_{\Diamond}^{[1]}[u]$ for every $u:Z_X\to\mathbf{A}$.
4. the following are equivalent :
1. $(R_{\Diamond}^{[1]}[\{\alpha/z\}])^{[10]}\subseteq R_{\Diamond}^{[1]}[\{\alpha/z\}]$ for every $z\in Z$ and $\alpha\in\mathbf{A}$;
2. $(R_{\Diamond}^{[1]}[u])^{[10]}\subseteq R_{\Diamond}^{[1]}[u]$ for every $u: Z_X\to\mathbf{A}$;
3. $R_{\Diamond}^{[0]}[f^{[10]}]\subseteq R_{\Diamond}^{[0]}[f]$ for every $f:Z_A\to\mathbf{A}$.
Many-valued graph-based models
==============================
Let $\mathcal{L}$ be the language of Section \[sec:logics\].
A [*graph-based*]{} $\mathbf{A}$-[*model*]{} of $\mathcal{L}$ is a tuple $\mathbb{M} = (\mathbb{G}, V)$ such that $\mathbb{G} = (\mathbb{X}, R_\Box, R_\Diamond)$ is a graph-based $\mathbf{A}$-frame and $V: \mathcal{L}\to \mathbb{G}^+$ is a homomorphism. For every ${\varphi}\in \mathcal{L}$, let $V({\varphi}): = ({[\![{{\varphi}}]\!]}, {(\![{{\varphi}}]\!)})$, where ${[\![{{\varphi}}]\!]}: Z_A\to \mathbf{A}$ and ${(\![{{\varphi}}]\!)}: Z_X\to \mathbf{A}$ are s.t. ${[\![{{\varphi}}]\!]}^{[1]} = {(\![{{\varphi}}]\!)}$ and ${(\![{{\varphi}}]\!)}^{[0]} = {[\![{{\varphi}}]\!]}$. Hence:
--------------------------- --- ------------------------------------------------------------------------------------------------------
$V(p)$ = $({[\![{p}]\!]}, {(\![{p}]\!)})$
$V(\top)$ = $(1^{\mathbf{A}^{Z_A}}, (1^{\mathbf{A}^{Z_A}})^{[1]})$
$V(\bot)$ = $((1^{\mathbf{A}^{Z_X}})^{[0]}, 1^{\mathbf{A}^{Z_X}})$
$V({\varphi}\wedge \psi)$ = $({[\![{{\varphi}}]\!]}\wedge{[\![{\psi}]\!]}, ({[\![{{\varphi}}]\!]}\wedge{[\![{\psi}]\!]})^{[1]})$
$V({\varphi}\vee \psi)$ = $(({(\![{{\varphi}}]\!)}\wedge{(\![{\psi}]\!)})^{[0]}, {(\![{{\varphi}}]\!)}\wedge{(\![{\psi}]\!)})$
$V(\Box{\varphi})$ = $(R^{[0]}_\Box[{(\![{{\varphi}}]\!)}], (R^{[0]}_\Box[{(\![{{\varphi}}]\!)}])^{[1]})$
$V(\Diamond{\varphi})$ = $((R^{[0]}_\Diamond[{[\![{{\varphi}}]\!]}])^{[0]}, R^{[0]}_\Diamond[{[\![{{\varphi}}]\!]}])$.
--------------------------- --- ------------------------------------------------------------------------------------------------------
Valuations induce $\alpha$-[*support relations*]{} between value-state pairs and formulas for each $\alpha\in \mathbf{A}$ (in symbols: $\mathbb{M}, (\beta, z)\Vdash^\alpha {\varphi}$), and $\alpha$-[*refutation relations*]{} between states of models and formulas for each $\alpha\in \mathbf{A}$ (in symbols: $\mathbb{M}, z\succ^\alpha {\varphi}$) such that for every ${\varphi}\in \mathcal{L}$, all $z\in Z$and all $\beta\in \mathbf{A}$,
------------------------------------------------- ----- ------------------------------------------------
$\mathbb{M}, (\beta, z)\Vdash^\alpha {\varphi}$ iff $ \alpha\leq {[\![{{\varphi}}]\!]}(\beta, z),$
$\mathbb{M},z \succ^\alpha {\varphi}$ iff $ \alpha\leq {(\![{{\varphi}}]\!)}( z).$
------------------------------------------------- ----- ------------------------------------------------
This can be equivalently expressed as follows:
\[def:Sequent:True:In:Model\] A sequent ${\varphi}\vdash \psi$ is *true in a model* $\mathbb{M} = (\mathbb{G}, V)$ (notation: $\mathbb{M} \models {\varphi}\vdash \psi$) if ${[\![{{\varphi}}]\!]}\subseteq {[\![{\psi}]\!]}$, or equivalently, if ${(\![{\psi}]\!)}\subseteq {(\![{{\varphi}}]\!)}$. A sequent ${\varphi}\vdash \psi$ is *valid on a graph-based frame* $\mathbb{G}$ (notation: $\mathbb{G} \models {\varphi}\vdash \psi$) if ${\varphi}\vdash \psi$ is true in every model $\mathbb{M} = (\mathbb{G}, V)$ based on $\mathbb{G}$.
\[remark:Monotone:Vals\] It is not difficult to see that for all stable valuations, if $p\in {\mathsf{Prop}}$ and $\beta, \beta'\in \mathbf{A}$ such that $\beta\leq \beta'$, then ${[\![{p}]\!]}(\beta, z) \leq {[\![{p}]\!]}(\beta', z)$ for every $z\in Z$, and one can readily verify that this condition extends compositionally to every ${\varphi}\in \mathcal{L}$.
Before moving on to the case study, let us expand on how to understand informally the notion of $\alpha$-support at value-state pairs. To this end, it is perhaps useful to start analysing $\mathbb{M}, (\beta, z)\Vdash^\alpha \bot$. By definition, this is the case iff $ \alpha\leq\beta$. Hence, the role of $\beta$ in the pair $(\beta, z)$ is to indicate the maximum extent $\alpha$ to which the ‘state’ $(\beta, z)$ is allowed to $\alpha$-support a false statement. Equivalently, $(\beta, z)$ does not $\alpha$-support the falsehood for any $\alpha\not\leq \beta$. Hence, when $\mathbf{A}$ is linearly ordered, $\beta$ indicates the ‘threshold’ beyond which (i.e. overcoming which by going up) $\alpha$-support becomes meaningful at the given ‘state’ $(\beta, z)$. These observations open the way to the possibility of imposing extra conditions on the extension functions ${[\![{{\varphi}}]\!]}: \mathbf{A}\times Z\to \mathbf{A}$, depending on the given situation to be modelled. These extra conditions are not required by the general semantic environment, but are accommodated by it. For instance, if considering this threshold $\beta$ is not relevant to the given case at hand, then one can choose to restrict oneself to valuations such that, if $p\in {\mathsf{Prop}}$, then ${[\![{p}]\!]}(\beta, z) = {[\![{p}]\!]}(\beta', z)$ for every $z\in Z$ and every $\beta, \beta'\in \mathbf{A}$. One can readily verify that this condition extends compositionally to every ${\varphi}\in \mathcal{L}$.
Case study: competing theories {#sec:Case:Study}
==============================
In the previous sections, we have illustrated how many-valued semantics for modal logic [@fitting1991many; @bou2011minimum] can be generalized from Kripke frames to graph-based structures. This generalization is the parametric version (where $\mathbf{A}$ is the parameter) of the graph-based semantics of [@graph-based-wollic], and the main motivation for introducing it is that it allows for a rich description of certain essentials (in the present case, of the role of theories in the practice of empirical sciences), while still using basic intuitions from the crisp setting. For instance, in the many-valued setting, the basic intuition still holds that the generalization from classical Kripke frame semantics to graph-based semantics consists in thinking of the graph-relation $E$ as encoding an inherent boundary to knowability (referred to as [*informational entropy*]{}) which disappears in the classical setting (in which $E$ coincides with the identity relation). Informational entropy can be due to many factors (e.g. technological, theoretical, linguistic, perceptual, cognitive), and in [@graph-based-wollic], examples are discussed in which the nature of these limits is perceptual and linguistic. In the present section, we discuss how the [*theoretical frameworks*]{} adopted by empirical scientists can be a source of informational entropy.
For the purpose of this analysis, we consider graph-based structures $(Z, E, \{R_{X_i}\mid X_i\subseteq \mathsf{Var}, 0\leq i\leq n\})$ in which $\mathbb{X} = (Z, E)$ represents a network of databases, and $\mathsf{Var}$ is a set of variables which includes the variables structuring the information contained in the databases of $Z$. In this context, $\mathcal{L}$-formulas can be thought of as hypotheses which will be assigned truth values (more specifically, truth-degrees) at value-state pairs of models based on these frames. We will refer to any such pair $(\beta, z)$ as a [*situation*]{}, the $\beta$ component of which is understood as the maximum degree of flexibility in operationalizing variables in that given situation. This truth value assignment of a formula (hypothesis) at a value-state pair (situation) is then intended to represent the significance of the correlation posited by the hypothesis, when tested in the given database according to the degree of flexibility allowed at that situation, with higher truth values indicating higher levels of significance.[^3] This interpretation is coherent with the property mentioned in Remark \[remark:Monotone:Vals\]: indeed, the higher the flexibility in operationalizing variables, the more leeway to obtain a higher $\alpha$-support of hypotheses at situations.
Moreover, in the context of the graph-based structures above, an empirical theory is characterized by (and here identified with) a certain subset $X$ of variables which are [*relevant*]{} to the given theory; also, in what follows, for all databases $z_j\in Z$, we let $X_j$ denote the set of variables structuring the data contained in $z_j$.[^4] Hence, the $\mathbf{A}$-relation $E$ encodes to what extent database $z_2$ is similar to $z_1$ (e.g. by letting $E(z_1, z_2)$ record the percentage of variables of $z_1$ that also occur in $z_2$), while the relations $R_X$ encode to what extent one database is similar to another, relative to $X$ (e.g. by letting $R_X(z_1, z_2)$ record the percentage of variables of $X_1\cap X$ that also occur in $X_2$). Below, we give a more concrete illustration of this environment by means of an example about [*dietary theories*]{}.
The first theory, known since antiquity, is that body-fat loss of individuals depends on what they ate and how much exercise they did. We refer to it as the [*ancient theory*]{} ($A$), on the basis of which Aretaeus the Cappadocian might have created a database $z_A$ recording how many [*kochliaria*]{} of olive oil and of honey, how many [*minas*]{} of bread, of olives, of lamb, and how many [*kyathoi*]{} of wine a group of athletes and a group of rhetoric students ate each day, and how many [*stadia*]{} they walked or ran each day, and how many [*minas*]{} each individual weighed each day.
The second one, the [*modern theory*]{} ($M$), was developed in Victorian times by Wilbur Olin Atwater.[^5] In line with the Taylorist view on labour efficiency, the modern theory explains body-fat loss in terms of a negative balance between the daily caloric intake of individuals provided by food and their daily caloric expenditure, due e.g. to maintaining body temperature or to exercise. The modern theory improves over the ancient in that it provides a common ground of commensurability, which was absent in the ancient theory, between the variables relative to food intake and those relative to exercise, by reducing all of them to their energetic import, measured in calories. Hence, an imaginary database $z_M$ built by Atwater on the basis of this theory would record how many calories individuals got from food and how many calories they spent per day, and their mass in kilograms measured each day.
The third theory, referred to as the [*hormonal response theory*]{} ($H$),[^6] postulates that body-fat loss is governed by a hormone, insulin, which is released in response to the intake of certain macronutrients: namely, it is maximally released in response to intake of carbohydrates, less so but still significantly released in response to protein intake, while fat intake does not trigger any significant insulin response.[^7] This theory posits that as long as insulin values are high, the body cannot access its own fat and use it as energy source, no matter how severe the caloric restriction. An imaginary database $z_H$ built by Banting and Best (who famously discovered insulin and its function) on the basis of this theory would record how many calories [*from carbohydrates*]{}, how many calories [*from proteins*]{}, how many calories [*from fat*]{} individuals got from food, how many calories they spent per day, and their mass in kilograms measured each day.
This scenario can be modelled as the graph-based $\mathbf{A}$-frame $(Z, E, R_A, R_M, R_H)$, where $\mathbf{A}$ is the 11-element [Ł]{}ukasiewicz chain, $Z: = \{z_A, z_M, z_H\}$, and $E: Z\times Z\to \mathbf{A}$ is as indicated in the following diagram:
(ancient) at (-2,0) [$z_A$]{}; (modern) at (0,0) [$z_M$]{}; (hormonal) at (2,0) [$z_H$]{}; (modern) – (ancient) node \[midway,below\][1]{}; (modern) – (hormonal) node \[midway,below\][1]{}; (ancient) to node \[midway,above\][$0.2$]{} (modern); (hormonal) to node \[midway,above\][$0.4$]{} (modern); (modern) to node \[midway,below\][$1$]{} (modern); (ancient) to node \[midway,above\][$1$]{} (ancient); (hormonal) to node \[midway,above\][$1$]{} (hormonal); (ancient) to node \[midway,above\][$0.6$]{} (hormonal); (hormonal) to node \[midway,below\][$1$]{} (ancient);
For any arrow in the diagram above, its value (i.e. the similarity degree of the target to the source) intuitively represents to which extent the target database provides information about variables relevant to the theory according to which the source database has been built. This similarity relation is reflexive by definition. Moreover, all arrows have non-zero values because the three databases constructed on the basis of the three different theories have a minimal common ground, namely they all include the weight of individuals (which, as we will see, is the dependent variable in the hypotheses tested on them). We assign value 1 to all the arrows which have $z_M$ as source or $z_A$ as target, since the information relevant to the modern theory can be fully retrieved from all databases, and the information relevant to the modern and hormonal response theories can be fully retrieved from the variables in $z_A$. The arrow from $z_A$ to $z_M$ is assigned the lowest non-zero value, since from the information about the caloric intake and expenditure contained in $z_M$ one cannot retrieve the actual types of food the individuals ingested or the exercise they did.
For $X\in \{A, M, H\}$, and for any $z, z'\in Z$, the value of the $R_X$-arrow from $z$ to $z'$ represents the similarity degree of $z'$ to $z$ [*relative to*]{} $X$. A concrete way to picture this is the following: assume that a scientist adopting theory $X$ is asked to which extent s/he would swap database $z$ for database $z'$. If the scientist in question is Aretaeus, and he is asked e.g. to give up $z_H$ for $z_M$, he would not be very happy, for although $z_H$ is not particularly good for his purposes and requires a substantial guesswork from him, he would be even worse off with $z_M$, and he would suffer the same loss of information captured by the value $E(z_H, z_M)$. This justifies letting $R_A(z_H, z_M): = E(z_H, z_M) = 0.4$. However, Aretaeus would certainly be willing to swap $z_M$ for $z_H$, since whatever little he can do with $z_M$ can be certainly done with $z_H$, and in fact possibly more. Hence we let again $R_A(z_M, z_H): = E(z_M, z_H) = 1$, and so on. Hence, $R_A: = E$. If the scientist in question is Atwater, and he was asked to give up $z_A$ for $z_H$, he would be fine with it, because both databases provide all the information relevant to the theoretical framework he has adopted. In fact, he would be fine with swapping any database for any other database: that is, the relation $R_M: Z\times Z\to \mathbf{A}$ maps every tuple of databases to $1$. An analogous reasoning justifies the following definition for the relation $R_H: Z\times Z\to \mathbf{A}$:
(ancient) at (-2,0) [$z_A$]{}; (modern) at (0,0) [$z_M$]{}; (hormonal) at (2,0) [$z_H$]{}; (modern) – (ancient) node \[midway,below\][1]{}; (modern) – (hormonal) node \[midway,below\][1]{}; (ancient) to node \[midway,above\][$0.3$]{} (modern); (hormonal) to node \[midway,above\][$0.5$]{} (modern); (modern) to node \[midway,below\][$1$]{} (modern); (ancient) to node \[midway,above\][$1$]{} (ancient); (hormonal) to node \[midway,above\][$1$]{} (hormonal); (ancient) to node \[midway,above\][$0.9$]{} (hormonal); (hormonal) to node \[midway,below\][$1$]{} (ancient);
where $R_H$ and $E$ only differ in the value of the arrow from $z_A$ to $z_H$. The relations above are $E$-compatible (cf. Definition \[def:graph:based:frame:and:model\]), and $E$-reflexive (i.e. $E\subseteq R$ for any $R\in \{R_A, R_M, R_H\}$, see [@graph-based-wollic] Definition 6) hence $(Z, E, R_A, R_M, R_H)$ is a graph-based $\mathbf{A}$-frame for a multi-modal language with modalities $\Box_A, \Box_M, \Box_H$ and $\Diamond_A, \Diamond_M, \Diamond_H$, in which the axioms $\Box_i {\varphi}\vdash {\varphi}$ and $ {\varphi}\vdash \Diamond_i {\varphi}$ are valid for every $i\in \{A, M, H\}$ (cf. Proposition \[prop:correspondence\]).[^8]
Let the formula ${\varphi}$ be the hypothesis stating that individuals who restrict their daily caloric intake to less than 20 calories per kilogram of body mass will lose weight over time. This hypothesis is phrased in terms of the variables relevant to the modern theory, and hence it can be tested on [*all databases*]{} in $Z$. Let us assume that the results of the tests of ${\varphi}$ do not vary from one situation to another situation with the same degree of flexibility $\beta$, and it turns out that, though 80% of the individuals restricting their daily caloric intake to less than 20 calories per kilogram of body mass indeed lost a bit of weight, generally not too much, 10% of individuals remained at the same weight, and 10% even gained weight. Let us assume that in the statistical model this results in moderate effect size, but a p-value of 0.1, which is considered to yield too low a level of significance to reject the null-hypothesis (that caloric restriction has no effect). So we propose, for the sake of this example, to assign ${[\![{{\varphi}}]\!]}: Z_A \to \mathbf{A}$ according to the following table[^9]:
$\beta$ $z_A$ $z_M$ $z_H$
--------- ------- ------- -------
$0.0$ $0.5$ $0.5$ $0.5$
$0.1$ $0.6$ $0.6$ $0.6$
$0.2$ $0.7$ $0.7$ $0.7$
$0.3$ $0.8$ $0.8$ $0.8$
$0.4$ $0.9$ $0.9$ $0.9$
$0.5$ $1.0$ $1.0$ $1.0$
$0.6$ $1.0$ $1.0$ $1.0$
$0.7$ $1.0$ $1.0$ $1.0$
$0.8$ $1.0$ $1.0$ $1.0$
$0.9$ $1.0$ $1.0$ $1.0$
$1.0$ $1.0$ $1.0$ $1.0$
Let the formula $\psi$ be the hypothesis stating that individuals who restrict their daily caloric intake to less than 20 calories per kilogram of body mass and who let at least 80% of their caloric intake come from fat will lose more weight than individuals on the same daily caloric regime but getting less than 80% of their calories from fat. This hypothesis is phrased in terms of the variables relevant to the hormonal response theory, and hence it can certainly be tested on $z_A$ and $z_H$. We wish to make a case that, modulo some guesswork that of course will make the results less reliable, the hypothesis $\psi$ [*can*]{} be tested on $z_M$ as well. For instance, if the database $z_M$ built by Atwater was based on a group of individuals living in Connecticut in the years 1890-1895, and for some fortuitous circumstances an independent database exists about their eating habits (e.g. the list of customers of the local grocer’s and their weekly orders, and by chance these customers also include the people in $z_M$), then it would be possible to make some estimates about which individuals in the sample of $z_M$ let at least 80% of their daily caloric intake come from fat. This is of course an easy way out in our fictitious example, but it reflects a very common situation in the practice of empirical research, that databases do not perfectly match the hypotheses that scientists wish to test on them, and that some guesswork is needed to a greater or lesser extent. Notice that the imperfect match between the observations in databases and variables in hypotheses is independent from the (maximum) degree of flexibility in operationalising variables (formally encoded in the value $\beta$ of the pairs which we refer to as ‘situations’). Specifically, the degree of flexibility in operationalising any variables is an [*a priori*]{} parameter that we fix for each ‘situation’, independently of the hypotheses tested in the given situation. In contrast, the discussion in the paragraph above is relative to the test of a specific hypothesis on a database, and hence depends inherently on the given hypothesis. Furthermore, once such a suitable translation is found, its suitability will not depend on how the target variable is operationalised in each situation, but will depend only on the match between the theory according to which the database has been built and the theory to which the hypothesis pertains.
Let us imagine that $\psi$ is confirmed for 95% of the individuals in the samples of all databases. Let us assume that in the statistical model this results in a high effect size for coefficient of the (dummy) variable recording whether the high-fat diet was followed or not and a p-value of 0.01, which corresponds to a level of significance generally considered to be high enough to reject the null-hypothesis (that the type of macronutrients from which [*restricted*]{} caloric intake proceeds has no effect on weight loss). In short, the results in respect to this hypothesis seem very strong and credible. So we propose, for $\beta = 0$, to assign a truth-value of 0.8 to $\psi$ at $z_A$ and $z_H$, and a truth-value of 0.4 to $\psi$ at $z_M$, a strong discount due to the guesswork needed to accommodate the testing of $\psi$ on $z_M$.[^10] The following table gives the complete specification of ${[\![{\psi}]\!]}$:
$\beta$ $z_A$ $z_M$ $z_H$
--------- ------- ------- -------
$0.0$ $0.8$ $0.4$ $0.8$
$0.1$ $0.9$ $0.5$ $0.9$
$0.2$ $1.0$ $0.6$ $1.0$
$0.3$ $1.0$ $0.7$ $1.0$
$0.4$ $1.0$ $0.8$ $1.0$
$0.5$ $1.0$ $0.9$ $1.0$
$0.6$ $1.0$ $1.0$ $1.0$
$0.7$ $1.0$ $1.0$ $1.0$
$0.8$ $1.0$ $1.0$ $1.0$
$0.9$ $1.0$ $1.0$ $1.0$
$1.0$ $1.0$ $1.0$ $1.0$
We are now in a position to compute the extensions of $\Box_M{\varphi}$, $\Box_H{\varphi}$, $\Box_M\psi$ and $\Box_H\psi$.[^11] Intuitively, $\Box_X\chi$ can be understood as what becomes of hypothesis $\chi$ when ‘seen through the lenses’ of theory $X$.[^12]
It can be verified that:
$$\begin{aligned}
&{[\![{\Box_M \psi}]\!]}\\
= &\left[\begin{tabular}{r|lll}
$\beta$ &$z_A$ &$z_M$ &$z_H$\\
\hline
$0.0$ &$0.4$ &$0.4$ &$0.4$\\
$0.1$ &$0.5$ &$0.5$ &$0.5$\\
$0.2$ &$0.6$ &$0.6$ &$0.6$\\
$0.3$ &$0.7$ &$0.7$ &$0.7$\\
$0.4$ &$0.8$ &$0.8$ &$0.8$\\
$0.5$ &$0.9$ &$0.9$ &$0.9$\\
$0.6$ &$1.0$ &$1.0$ &$1.0$\\
$0.7$ &$1.0$ &$1.0$ &$1.0$\\
$0.8$ &$1.0$ &$1.0$ &$1.0$\\
$0.9$ &$1.0$ &$1.0$ &$1.0$\\
$1.0$ &$1.0$ &$1.0$ &$1.0$
\end{tabular}\right]
\leq &\left[\begin{tabular}{r|lll}
$\beta$ &$z_A$ &$z_M$ &$z_H$\\
\hline
$0.0$ &$0.5$ &$0.5$ &$0.5$\\
$0.1$ &$0.6$ &$0.6$ &$0.6$\\
$0.2$ &$0.7$ &$0.7$ &$0.7$\\
$0.3$ &$0.8$ &$0.8$ &$0.8$\\
$0.4$ &$0.9$ &$0.9$ &$0.9$\\
$0.5$ &$1.0$ &$1.0$ &$1.0$\\
$0.6$ &$1.0$ &$1.0$ &$1.0$\\
$0.7$ &$1.0$ &$1.0$ &$1.0$\\
$0.8$ &$1.0$ &$1.0$ &$1.0$\\
$0.9$ &$1.0$ &$1.0$ &$1.0$\\
$1.0$ &$1.0$ &$1.0$ &$1.0$
\end{tabular}\right]\\
&& = {[\![{{\varphi}}]\!]}\end{aligned}$$
and
$$\begin{aligned}
&{[\![{\Box_M \psi}]\!]}\\
= &\left[\begin{tabular}{r|lll}
$\beta$ &$z_A$ &$z_M$ &$z_H$\\
\hline
$0.0$ &$0.4$ &$0.4$ &$0.4$\\
$0.1$ &$0.5$ &$0.5$ &$0.5$\\
$0.2$ &$0.6$ &$0.6$ &$0.6$\\
$0.3$ &$0.7$ &$0.7$ &$0.7$\\
$0.4$ &$0.8$ &$0.8$ &$0.8$\\
$0.5$ &$0.9$ &$0.9$ &$0.9$\\
$0.6$ &$1.0$ &$1.0$ &$1.0$\\
$0.7$ &$1.0$ &$1.0$ &$1.0$\\
$0.8$ &$1.0$ &$1.0$ &$1.0$\\
$0.9$ &$1.0$ &$1.0$ &$1.0$\\
$1.0$ &$1.0$ &$1.0$ &$1.0$
\end{tabular}\right]
\leq &\left[\begin{tabular}{r|lll}
$\beta$ &$z_A$ &$z_M$ &$z_H$\\
\hline
$0.0$ &$0.8$ &$0.4$ &$0.8$\\
$0.1$ &$0.9$ &$0.5$ &$0.9$\\
$0.2$ &$1.0$ &$0.6$ &$1.0$\\
$0.3$ &$1.0$ &$0.7$ &$1.0$\\
$0.4$ &$1.0$ &$0.8$ &$1.0$\\
$0.5$ &$1.0$ &$0.9$ &$1.0$\\
$0.6$ &$1.0$ &$1.0$ &$1.0$\\
$0.7$ &$1.0$ &$1.0$ &$1.0$\\
$0.8$ &$1.0$ &$1.0$ &$1.0$\\
$0.9$ &$1.0$ &$1.0$ &$1.0$\\
$1.0$ &$1.0$ &$1.0$ &$1.0$
\end{tabular}\right]\\
&&= {[\![{\psi}]\!]}\end{aligned}$$
It can also be checked that ${[\![{\Box_H {\varphi}}]\!]} = {[\![{{\varphi}}]\!]}$, ${[\![{\Box_H \psi}]\!]} = {[\![{\psi}]\!]}$ and ${[\![{\Box_M {\varphi}}]\!]} = {[\![{{\varphi}}]\!]}$.
These identities and inequalities can be interpreted as follows: each theory leaves unchanged the hypotheses formulated in terms of its own variables, or proper subsets thereof; however, if a hypothesis formulated according to a more expressive theory is ‘seen through the lenses’ of a less expressive theory (this is the case of $\Box_M \psi$), it is expected to score worse. Finally, ${\varphi}$ and $\psi$ are prima facie incomparable. But is it really so?
Epilogue {#Sec:epilogue}
========
Although very stylised and simplified, the scenario above illuminates a number of interesting notions and their interrelations. First, we have identified each theory with the set of its relevant variables. This move naturally provides a connection with a strand of research we have been recently developing, based on the idea that lattice-based modal logics can be interpreted as the logics of categories or formal concepts [@conradie2016categories; @Tarkpaper; @roughconcepts]. This connection can be articulated in general terms by modelling [*theories as categories*]{}, extensionally captured by sets of hypotheses, and intensionally captured by their relevant variables.
Having identified theories with sets of variables has allowed us to associate states of the models (understood as databases) with theories, thereby giving a very simple and concrete representation of the otherwise abstract idea that ‘observations are theory-laden’, and that this [*theory-ladennes*]{} lays at the core of the informational entropy that this paper sets out to studying. In the toy example of the previous section, states (databases) bijectively correspond to theories, but this does not need to be the case in general.
Related to this, we have captured a local and a global way in which [*similarity*]{} ensues from theory-driven informational entropy. Specifically, the relation $E$ captures the [*local*]{} perspective, in which a given database $z'$ is similar to a database $z$ to the extent to which $z'$ is amenable to test hypotheses formulated using the variables of the theory associated with $z$, that is, to the extent to which $z'$ is suitable to answer questions pertaining to ‘the theory of $z$’, while the relations $R_X$ capture the [*global*]{} perspective; i.e., $R_X$ encodes information on how similar any one database is to another in respect to their relative performances in testing hypotheses formulated using variables of $X$.
These formal tools can be used to illuminate a very common situation in the practice of empirical research, namely that databases do [*not*]{} perfectly match the hypotheses that scientists wish to test on them, and that a key underlying aspect in empirical research concerns precisely how to address this imperfect match. In this paper, we have laid the groundwork for addressing this issue with bespoke logical tools, by means of the modal operators interpreted using the relations $R_X$, which, as discussed above, translate hypotheses from the ‘language’ (variables) of one theory to the ‘language’ (variables) of another, and what is lost in translation depends on the relationship between the two theories.
Finally, we can try and discuss whether the formalization above throws light on the following two questions: what does it mean for [*theories*]{} to [*compete*]{}? And how do we assess whether one theory has outcompeted the other?
We propose the following view: theories can compete, most obviously when hypotheses that belong to different theories (as ${\varphi}$ and $\psi$ in our example belong to $M$ and $H$ respectively) predict the same dependent variable (weight loss in our example). The most direct way in which two theories (e.g. $M$ and $H$) can compete is when their respective hypotheses (${\varphi}$ and $\psi$) are tested on each member of a set of databases. Each of these databases will be more or less suitable to test a given hypothesis. In particular, any hypothesis is expected to get its best scores[^13] if tested either on databases which are constructed in accordance with the theory in the variables of which the hypothesis is formulated, or on databases that are [*maximally similar*]{} to those. We refer to all these databases as being ‘home-ground’ to that given hypothesis. For instance, in the case study of the previous section, every database is ‘home-ground’ to ${\varphi}$, while only $z_A$ and $z_H$ are ‘home-ground’ to $\psi$. When a hypothesis is tested on a database that is not its ‘home-ground’, it will typically find no or less adequate values for its variables. The solution, as in our example of testing $\psi$ on the database $z_M$, is to look for proxies that represent to some extent the missing variable (or recover the values of the missing variables by some motivated guesswork). These proxies are often second-best or worse, making the results of the test less credible, even if they lead to accepting the hypotheses. This results in assigning the hypothesis a lower truth value at the database where proxies/guesswork were needed to test the hypothesis. However, precisely due to the disadvantage of not being on ‘home-ground’, if a hypothesis pertaining to theory $X$ is tested on a database $z$ which is not ‘home-ground’ to it and gets more than half as good results as the competing hypothesis, pertaining to theory $Y$, to which $z$ is ‘home-ground’, this should be considered an impressive victory of theory $X$ and its hypothesis, just as is the effect of the rule that goals scored by soccer teams (in e.g. the Champion’s League) in away matches count double. Applying this view to the case of ${\varphi}$ and $\psi$, we can hence argue that $\psi$ outcompetes ${\varphi}$, since, as discussed above, every database is ‘home-ground’ to ${\varphi}$, while only $z_A$ and $z_H$ are ‘home-ground’ to $\psi$, and moreover, $\psi$ scores systematically better than ${\varphi}$ on each database that is ‘home-ground’ to both and, even when $\beta = 0$, the lower score of $\psi$ on $z_M$ (0.4) is very close to the score of ${\varphi}$ on $z_M$ (0.5).
Conclusions
===========
In this paper, we have introduced a complete many-valued semantic environment for (multi) modal languages based on the logic of general (i.e. not necessarily distributive) lattices, and, by means of a toy example, we have illustrated its potential as a tool for the formal analysis of situations arising in the theory and practice of empirical science. As this is only a preliminary exploration, many questions arise, both technical and conceptual, of which here we list a few.
#### A range of protocols for comparing theories.
As discussed in Sections \[sec:Case:Study\] and \[Sec:epilogue\] hypotheses that look prima facie incomparable can become comparable through the lenses of the modal operators. In this paper we do not insist on a specific protocol to establish a winner among competing theories. However, we wish to highlight that the framework introduced can accommodate a wide range of possible protocols, including those involving common-knowledge-type constructions.
#### More expressive languages.
We conjecture that the proof of completeness of the logic of Section \[sec:logics\] given in Appendix \[sec:completeness\] can be extended modularly to more expressive languages that display essentially “many valued" features in analogy with those considered in [@bou2011minimum]. This is current work in progress.
#### Sahlqvist theory for many-valued non-distributive logics.
A natural direction of research is to develop the generalized Sahlqvist theory for the logics of graph-based $\mathbf{A}$-frames, by extending the results of [@CMR] on Sahlqvist theory for many-valued logics on a distributive base.
#### Towards an analysis theory dynamics.
We have shown that using the many-valued environment allows us to discuss competition between theories in an intuitively appealing and formally sound manner. This lays the basis for a host of further developments to model the dynamics of theories and databases, to better understand what distance between theories means, as well as studying the hierarchical relations between theories, in analogy with the hierarchical structure of categories.
#### Socio-political theories and scientific theories.
The present semantic environment naturally lends itself not only to the analysis of competition of scientific theories but also to the analysis of a wide spectrum of phenomena in which theories, broadly construed, play a key role. For instance, building on the present work, in [@socio-political], a formal environment is introduced in which the similarities can be analysed between the competition among political theories (both in their institutional incarnations as political parties, and in their social incarnations as social blocks or groups) and the competition between scientific theories.
The authors would like to thank Apostolos Tzimoulis for insightful discussions and for his substantial contributions to the proof of the completeness theorem.
Correspondence results {#sec:correspondence}
======================
In what follows, for every graph-based $\mathcal{L}$-frame $\mathbb{G}$, we let $R_{\blacksquare}: Z\times Z\to\mathbf{A}$ be defined by the assignment $(z, z')\mapsto R_{\Diamond}(z', z)$.
\[prop:correspondence\] The following are equivalent for every graph-based $\mathcal{L}$-frame $\mathbb{G}$:
1. $\mathbb{G}\models \Box\bot\vdash \bot$ iff for any $(\beta, z)\in Z_A$, $$\bigwedge_{z'\in Z_X}[R_{\Box}(z, z') \to \beta] \leq \bigwedge_{z'\in Z_X}[E(z, z') \to \beta].$$
2. $\mathbb{G}\models \top\vdash \Diamond\top$ iff for any $z\in Z_X$, $$\bigwedge_{(\alpha, z')\in Z_A}[R_{\Diamond}(z, z') \to \alpha]\leq \bigwedge_{(\alpha, z')\in Z_A}[E(z', z) \to \alpha].$$
3. $\mathbb{G}\models \Box p\vdash p\quad$ iff $\quad E\subseteq R_{\Box}$.
4. $\mathbb{G}\models p\vdash \Diamond p\quad $ iff $\quad E\subseteq R_{\blacksquare}$.
1.
----- ------------------------------------------------------------------------------------------------------------------------
$\Box\bot\leq \bot$
iff $R^{[0]}_{\Box}[{(\![{\bot}]\!)}]\leq {[\![{\bot}]\!]}$
iff $R^{[0]}_{\Box}[1^{\mathbf{A}^{Z_X}}]\leq (1^{\mathbf{A}^{Z_X}})^{[0]}$
iff $R^{[0]}_{\Box}[1^{\mathbf{A}^{Z_X}}](\beta, z)\leq (1^{\mathbf{A}^{Z_X}})^{[0]}(\beta, z)$ $\forall(\beta, z)\in Z_A$
iff $\bigwedge_{z'\in Z_X}[1(z')\to (R_{\Box}(z, z') \to \beta)]$
$\leq \bigwedge_{ z'\in Z_X}[1(z')\to (E(z, z') \to \beta)]$ $\forall(\beta, z)\in Z_A$
iff $\bigwedge_{z'\in Z_X}[R_{\Box}(z, z') \to \beta]$
$\leq \bigwedge_{z'\in Z_X}[E(z, z') \to \beta]$ for any $(\beta, z)\in Z_A$.
----- ------------------------------------------------------------------------------------------------------------------------
2.
----- -------------------------------------------------------------------------------------------------------
$\top\leq \Diamond\top$
iff $R^{[0]}_{\Diamond}[{[\![{\top}]\!]}]\leq {(\![{\top}]\!)}$
iff $R^{[0]}_{\Diamond}[1^{\mathbf{A}^{Z_A}}]\leq (1^{\mathbf{A}^{Z_A}})^{[1]}$
iff $R^{[0]}_{\Diamond}[1^{\mathbf{A}^{Z_A}}](z)\leq (1^{\mathbf{A}^{Z_A}})^{[0]}( z)$ for any $z\in Z_X$
iff $\bigwedge_{(\alpha, z')\in Z_A}[1(\alpha, z')\to (R_{\Diamond}(z, z') \to \alpha)]$
$\leq \bigwedge_{(\alpha, z')'\in Z_A}[1(\alpha,z')\to (E(z', z) \to \alpha)]$ $\forall z\in Z_X$
iff $\bigwedge_{(\alpha, z')\in Z_A}[R_{\Diamond}(z, z') \to \alpha]$
$\leq \bigwedge_{(\alpha, z')\in Z_A}[E(z', z) \to \alpha]$ for any $z\in Z_X$.
----- -------------------------------------------------------------------------------------------------------
3.
----- ------------------------------------------------------------------------------------------- --------------------------------------------
$\forall p [\Box p \leq p]$
iff $\forall {\mathbf{m}}[\Box {\mathbf{m}}\leq {\mathbf{m}}]$ (ALBA [@CoPa:non-dist])
iff $\forall \alpha \forall z [R^{[0]}_{\Box}[\{\alpha/z\}^{[01]}] \leq \{\alpha/z\}^{[0]}]$ (${\mathbf{m}}:= \{\alpha/z\}$)
iff $\forall \alpha \forall z [R^{[0]}_{\Box}[\{\alpha/z\}] \leq \{\alpha/z\}^{[0]}]$ (Lemma \[equivalence of I-compatible-mv\])
iff $\forall \alpha, \beta \forall z, w[\alpha \to (R_{\Box}(w, z) \to\beta) $
$\leq \alpha \to (E(w, z) \to\beta)$ ($\ast$)
iff $E\subseteq R_{\Box}$. ($\ast\ast$)
----- ------------------------------------------------------------------------------------------- --------------------------------------------
To justify the equivalence to ($\ast$) we note that $R^{[0]}_{\Box}[\{\alpha/z\}](\beta, w) = \bigwedge_{z' \in Z_X}[\{\alpha/z\}(z') \to (R_{\Box}(w, z')\to\beta)] = \alpha \to (R_{\Box}(w, z) \to\beta)$, and moreover $\{\alpha/z\}^{[0]}(\beta, w) = \bigwedge_{z' \in Z_X}(\{\alpha/z\}(z') \to (E(w, z')\to\beta)) = \alpha \to (E(w, z)\to\beta)$. For the equivalence to ($\ast\ast$), note that instantiating $\alpha: = 1$ and $\beta: = R_{\Box} (w, z)$ in ($\ast$) yields $1 \leq E(w, z) \to R_{\Box} (w, z)$ which, by residuation, is equivalent to ($\ast\ast$). The converse direction is immediate by the monotonicity of $\to$ in the second coordinate and its antitonicity in the first coordinate.
4.
----- ----------------------------------------------------------------------------------------------------------------------- --------------
$p\leq \Diamond p$
iff $\forall {\mathbf{j}}[{\mathbf{j}}\leq \Diamond {\mathbf{j}}]$
iff $\forall \alpha,\beta \forall z [R^{[0]}_{\Diamond}[\{\alpha/(\beta, z)\}^{[01]}] \leq \{\alpha/(\beta, z)\}^{[1]}]$
iff $\forall \alpha, \beta \forall z [R^{[0]}_{\Diamond}[\{\alpha/(\beta, z)\}] \leq \{\alpha/(\beta, z)\}^{[1]}]$
iff $\forall \alpha, \beta \forall z, w[\alpha \to (R_{\Diamond}(w, z)\to\beta) $
$\leq \alpha \to (E(z, w) \to\beta)$ ($\ast$)
iff $E\subseteq R_{\blacksquare}$. ($\ast\ast$)
----- ----------------------------------------------------------------------------------------------------------------------- --------------
As to the equivalence to ($\ast$), note that $R^{[0]}_{\Diamond}[\{\alpha/(\beta, z)\}] (w) = \bigwedge_{(\gamma, z') \in Z_A}[\{\alpha/(\beta, z)\}(\gamma, z') \to (R_{\Diamond}(w, z')\to\gamma)] = \alpha \to (R_{\Diamond}(w, z) \to\beta)$, and that $\{\alpha/(\beta, z)\}^{[1]}(w) = \bigwedge_{(\gamma, z') \in Z_A}(\{\alpha/(\beta, z)\}(\gamma, z') \to (E(z',w)\to\gamma)) = \alpha \to (E(z, w)\to\beta)$. For the equivalence to ($\ast\ast$) note that instantiating $\alpha := 1$ and $\beta: = R_{\Diamond} (w, z) = R_{\blacksquare}(z, w)$ in ($\ast$) yields $1 \leq E(z, w) \to R_{\blacksquare}(z, w)$ which, by residuation, is equivalent to ($\ast\ast$). The converse direction is immediate by the monotonicity of $\to$ in the second coordinate and its antitonicity in the first coordinate.
Clearly, $E$-reflexivity (i.e. condition $E\subseteq R_{\Box}$ in Proposition \[prop:correspondence\].3) implies the inequality in Proposition \[prop:correspondence\].1; however, this inequality is also verified under weaker but practically relevant assumptions. For instance, if $\mathbf{A}$ is a finite chain, the inequality in Proposition \[prop:correspondence\].1 is equivalent to $\mathrm{min}\{R_{\Box}(z, z') \to \beta\mid z'\in Z_X\}\leq \mathrm{min}\{E(z, z') \to \beta\mid z'\in Z_X\} = E(z, z)\to\beta = 1\to\beta = \beta$. Hence, this condition is equivalent to the condition that for every $\beta\in \mathbf{A}$ and $z\in Z$, some $z'\in Z$ exists such that $R_{\Box}(z, z') \to \beta\leq \beta$. This condition is satisfied if for every $z\in Z$ some $z'\in Z$ exists such that $R_{\Box}(z, z') = 1$. Similar considerations apply to the remaining items of the proposition above.
Completeness {#sec:completeness}
============
This section is an adaptation and expansion of the completeness result of [@socio-political Appendix B], of which Apostolos Tzimoulis and Claudette Robinson are prime contributors. We will use the validity of $\Box \bot\vdash \bot$ in the proof of the $\Box$ case in Lemma \[lemma:truthlemma\]. As discussed in Section \[sec:Case:Study\], this axiom is valid in the model of our case study.
For the sake of uniformity with previous settings (cf. e.g. [@roughconcepts Section 7.2]) in this section, we work with graph-based frames $\mathbb{G} = (\mathbb{X}, R_{\Box}, R_{\Diamond})$ the associated complex algebras of which are order-dual to the one in Definition \[def:graph:based:frame:and:model\]. That is, for the sake of this section, we define the enriched formal context $\mathbb{P}_{\mathbb{G}}: = (Z_A, Z_X, I_{E} , I_{R_\Box}, J_{R_\Diamond})$ by setting $Z_A: = Z$, $Z_X: = \mathbf{A}\times Z$ and $I_{E}: Z_A\times Z_X\to \mathbf{A}$ and $I_{R_\Box}: Z_A\times Z_X\to \mathbf{A}$ and $J_{R_\Diamond}: Z_X\times Z_A\to \mathbf{A}$ be defined by the assignments $(z, (\alpha, z'))\mapsto E(z, z')\to \alpha$, $(z, (\alpha, z'))\mapsto R_{\Box}(z, z')\to \alpha$ and $((\alpha, z), z')\mapsto R_{\Diamond}(z, z')\to \alpha$, respectively.
For any lattice $\mathbb{L}$, an $\mathbf{A}$-[*filter*]{} is an $\mathbf{A}$-subset of $\mathbb{L}$, i.e. a map $f:\mathbb{L}\to \mathbf{A}$, which is both $\wedge$- and $\top$-preserving, i.e. $f(\top) = 1$ and $f(a\wedge b) = f(a)\wedge f(b)$ for any $a, b\in\mathbb{L}$. Intuitively, the $\wedge$-preservation encodes a many-valued version of closure under $\wedge$ of filters. An $\mathbf{A}$-filter is [*proper*]{} if it is also $\bot$-preserving, i.e. $f(\bot)= 0$. Dually, an $\mathbf{A}$-[*ideal*]{} is a map $i:\mathbb{L}\to \mathbf{A}$ which is both $\vee$- and $\bot$-reversing, i.e. $i(\bot) = \top$ and $i(a\vee b) = i(a)\wedge i(b)$ for any $a, b\in\mathbb{L}$, and is [*proper*]{} if in addition $i(\top) = 0$. The [*complement*]{} of a (proper) $\mathbf{A}$-ideal is a map $u: \mathbb{L}\to \mathbf{A}$ which is both $\vee$- and $\bot$-preserving, i.e. $u(\bot) = 0$ and $u(a\vee b) = u(a)\vee u(b)$ for any $a, b\in\mathbb{L}$ (and in addition $u(\top) = 1$). Intuitively, $u(a)$ encodes the extent to which $a$ does not belong to the ideal of which $u$ is the many-valued complement. We let $\mathsf{F}_{\mathbf{A}}(\mathbb{L})$, $\mathsf{I}_{\mathbf{A}}(\mathbb{L})$ and $\mathsf{C}_{\mathbf{A}}(\mathbb{L})$ respectively denote the set of proper $\mathbf{A}$-filters, proper $\mathbf{A}$-ideals, and the complements of proper $\mathbf{A}$-ideals of $\mathbb{L}$. For any $\mathcal{L}$-algebra $(\mathbb{L}, \Box, \Diamond)$, and any $\mathbf{A}$-subset $k:\mathbb{L}\to \mathbf{A}$, let $k^{-\Diamond}: \mathbb{L}\to \mathbf{A}$ be defined as $k^{-\Diamond}(a)=\bigvee\{k(b)\mid \Diamond b\leq a\}$ and let $k^{-\Box}: \mathbb{L}\to \mathbf{A}$ be defined as $k^{-\Box}(a)=\bigwedge \{k(b)\mid a \leq \Box b \}$. By definition one can see that $k(a)\leq k^{-\Diamond}(\Diamond a)$ and $k^{-\Box}(\Box a) \leq k(a)$ for every $a\in \mathbb{L}$. Let $\mathbf{Fm}$ (resp. $\mathbf{Fm}_0$, $\mathbf{Fm}_1$) be the Lindenbaum-Tarski algebra of the basic $\mathcal{L}$-logic $\mathbf{L}$ (resp. $\mathbf{L}_0$, $\mathbf{L}_1$). Moreover, in what follows we write $\mathbf{Fm}_{\ast}$ for the Lindenbaum-Tarski algebra of an arbitrary (not necessarily proper) extension $\mathbf{L}_{\ast}$ of $\mathbf{L}$. In the remainder of this section, we abuse notation and identify formulas with their equivalence class in $\mathbf{Fm}_{\ast}$.
\[lemma:f minus diam is filter\]
1. If $f:\mathbb{L}\to \mathbf{A}$ is an $\mathbf{A}$-filter, then so is $f^{-\Diamond}$.
2. If $f:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is a proper $\mathbf{A}$-filter, then so is $f^{-\Diamond}$.
3. If $u:\mathbb{L}\to \mathbf{A}$ is the complement of an $\mathbf{A}$-ideal, then so is $u^{-\Box}$.
4. If $u:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is the complement of a proper $\mathbf{A}$-ideal, then so is $u^{-\Box}$.
5. If ${\varphi}, \psi\in \mathbf{Fm}_{\ast}$, then ${\varphi}\vee\psi = \top$ implies that ${\varphi}= \top$ or $\psi = \top$.
6. If ${\varphi}, \psi\in \mathbf{Fm}_{\ast}$, then ${\varphi}\not\vdash \bot$ and $\psi \not\vdash \bot$ imply that ${\varphi}\wedge\psi \not\vdash \bot$.
1\. For all $a, b\in \mathbb{L}$,
----------------------- --- -------------------------------------------
$f^{-\Diamond}(\top)$ = $\bigvee\{f(b)\mid \Diamond b\leq \top\}$
= $\bigvee\{f(b)\mid b\in \mathbb{L}\}$
= $f(\top)$
= $1$
----------------------- --- -------------------------------------------
--- --------------------------------------------------------------------------------------------- ------------
$f^{-\Diamond}(a)\wedge f^{-\Diamond}( b)$
= $ \bigvee\{f(c_1)\mid \Diamond c_1\leq a\} \wedge \bigvee\{f(c_2)\mid \Diamond c_2\leq b\}$
= $ \bigvee\{f(c_1)\wedge f(c_2)\mid \Diamond c_1\leq a\mbox{ and }\Diamond c_2\leq b\}$ ($\star$)
= $ \bigvee\{f(c_1\wedge c_2)\mid \Diamond c_1\leq a\mbox{ and }\Diamond c_2\leq b\}$ ($\sharp$)
= $ \bigvee\{f(c)\mid \Diamond c\leq a\mbox{ and }\Diamond c\leq b\}$ ($\ast$)
= $\bigvee\{f(c)\mid \Diamond c\leq a\wedge b\}$
= $f^{-\Diamond}(a\wedge b)$,
--- --------------------------------------------------------------------------------------------- ------------
the equivalence marked with ($\star$) being due to frame distributivity, the one marked with ($\sharp$) to the fact that $f$ is and $\mathbf{A}$-filter, and the one marked with ($\ast$) to the fact that $\Diamond(c_1\wedge c_2)\leq \Diamond c_1\wedge \Diamond c_2$.
2\. In general, $f^{-\Diamond}$ need not be a *proper* filter, even if $f$ is. However, let us show that this is the case when $f$ is a proper filter of $\mathbf{Fm}$. Indeed, in this algebra, $f^{-\Diamond}(\bot) = \bigvee\{f([{\varphi}])\mid [\Diamond {\varphi}] \leq [\bot]\} = \bigvee\{f([{\varphi}])\mid \Diamond {\varphi}\vdash \bot \} = \bigvee\{f([{\varphi}])\mid {\varphi}\vdash \bot \} = f([\bot]) = 0$. The crucial inequality is the third to last, which holds since $\Diamond {\varphi}\vdash \bot$ iff ${\varphi}\vdash \bot$. The right to left implication can be easily derived in $\mathbf{L}$. For the sake of the left to right implication we appeal to the completeness of $\mathbf{L}$ with respect to the class of all normal lattice expansions of the appropriate signature [@CoPa:non-dist] and reason contrapositively. Suppose ${\varphi}\not\vdash \bot$. Then, by this completeness theorem, there is a lattice expansion $\mathbb{C}$ and assignment $v$ on $\mathbb{C}$ such that $v({\varphi}) \neq 0$. Now consider the algebra $\mathbb{C}'$ obtained from $\mathbb{C}$ by adding a new least element $0'$ and extending the $\Diamond$-operation by declaring $\Diamond 0' = 0'$. We keep the assignment $v$ unchanged. It is easy to check that $\mathbb{C}'$ is a normal lattice expansion, and that $v(\Diamond {\varphi}) \geq 0 > 0'$ and hence $\Diamond {\varphi}\not \vdash \bot$.
Items 3 and 4 are proven by arguments which are dual to the ones above.
5\. As to proving item 5, we reason contrapositively. Suppose $\top \not\vdash {\varphi}$ and $\top \not\vdash \psi$. By the completeness theorem to which we have appealed in the proof of item 2, there are lattice expansions $\mathbb{C}_1$ and $\mathbb{C}_2$ and corresponding assignments $v_i$ on $\mathbb{C}_i$ such that $v_1({\varphi}) \neq \top^{\mathbb{C}_1}$ and $v_2(\psi) \neq \top^{\mathbb{C}_2}$. Consider the algebra $\mathbb{C}'$ obtained by adding a new top element $\top'$ to $\mathbb{C}_1\times \mathbb{C}_2$, defining the operation $\Diamond' = \Diamond^{\mathbb{C}'}$ by the same assignment of $\Diamond^{\mathbb{C}_1\times \mathbb{C}_2}$ on $\mathbb{C}_1\times \mathbb{C}_2$ and mapping $\top'$ to $(\Diamond \top)^{\mathbb{C}_1\times \mathbb{C}_2}$, and the operation $\Box' = \Box^{\mathbb{C}'}$ by the same assignment of $\Box^{\mathbb{C}_1\times \mathbb{C}_2}$ on $\mathbb{C}_1\times \mathbb{C}_2$, and mapping $\top'$ to itself. The normality (i.e. finite meet-preservation) of $\Box'$ and the monotonicity of $\Diamond'$ follow immediately by construction. The normality (i.e. finite join-preservation) of $\Diamond'$ is verified by cases: if $a\lor b\neq \top'$, then it immediately follows from the normality of $\Diamond^{\mathbb{C}_1\times\mathbb{C}_2}$. If $a\lor b=\top'$, then by construction, either $a=\top'$ or $b=\top'$ (i.e. $\top'$ is join-irreducible), and hence, the join-preservation of $\Diamond'$ is a consequence of its monotonicity. Consider the valuation $v': \mathsf{Prop}\to \mathbb{C}'$ defined by the assignment $p\mapsto e (v_1(p), v_2(p))$, where $e: \mathbb{C}_1\times \mathbb{C}_1\to \mathbb{C}'$ is the natural embedding. Let us show, for all $\chi \in \mathcal{L}$, that if $(v_1(\chi), v_2(\chi)) \neq \top^{\mathbb{C}_1 \times \mathbb{C}_2}$, then $v'(\chi) \neq \top'$. We proceed by induction on $\chi$. The cases for atomic propositions and conjunction are immediate. The case for disjunction uses the join-irreducibility of $\top'$. When $\chi := \Diamond \theta$, then $v'(\Diamond \theta) = \Diamond' v'(\theta)\neq \top'$, since, by construction, $\top'$ is not in the range of $\Diamond'$.
If $\chi: = \Box \theta$, then $v'(\chi) = v'(\Box \theta) = \Box' v'(\theta)$. Then the assumption that $(v_1(\chi), v_2(\chi)) \neq \top^{\mathbb{C}_1 \times \mathbb{C}_2}$ implies that $v'(\theta) \neq \top'$. Indeed, if $v'(\theta) = \top'$, then, by induction hypothesis, $(v_1(\theta), v_2(\theta)) = (\top^{\mathbb{C}_1}, \top^{\mathbb{C}_2})$ and hence $(v_1(\Box \theta), v_2(\Box \theta)) = \top^{\mathbb{C}_1 \times \mathbb{C}_2}$. Therefore, from $v'(\theta) \neq \top'$, it follows from the definition of $\Box'$ that $v'(\Box \theta) = \Box' v'(\theta) \neq \top'$, which concludes the proof of the claim.
Clearly, $v_1({\varphi}) \neq \top^{\mathbb{C}_1}$ and $v_2(\psi) \neq \top^{\mathbb{C}_2}$ imply that $(v_1({\varphi}), v_2({\varphi})) \neq \top^{\mathbb{C}_1 \times \mathbb{C}_2}$ and $(v_1(\psi), v_2(\psi)) \neq \top^{\mathbb{C}_1 \times \mathbb{C}_2}$. So, by the above claim, $v'({\varphi}) \neq \top'$ and $v'(\psi) \neq \top'$, and hence, since $\top'$ is join-irreducible, $v'({\varphi}\vee \psi) \neq \top'$.
The proof of item 6 is dual to the one above.
\[eq:premagicnew2\] For any $f\in \mathsf{F}_{\mathbf{A}}(\mathbb{L})$ and any $u\in \mathsf{C}_{\mathbf{A}}(\mathbb{L})$,
1. $\bigwedge_{b\in \mathbb{L}} (f^{-\Diamond}(b)\to u(b)) = \bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))$;
2. $\bigwedge_{b\in \mathbb{L}} (f(b) \to u^{-\Box}(b)) = \bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a))$.
For (1) we use the fact that $f(a)\leq f^{-\Diamond}(\Diamond a)$ implies that $f^{-\Diamond}(\Diamond a)\to u(\Diamond a)\leq f(a)\to u(\Diamond a)$ for every $a\in \mathbb{L}$, which is enough to show that $\bigwedge_{b\in \mathbb{L}} (f^{-\Diamond}(b)\to u(b)) \leq \bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a)).$ Conversely, to show that $$\bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))\leq \bigwedge_{b\in \mathbb{L}} (f^{-\Diamond}(b)\to u(b)),$$ it is enough to show that, for every $b\in \mathbb{L}$, $$\bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))\leq f^{-\Diamond}(b)\to u(b),$$ i.e. by definition of $f^{-\Diamond}(b)$ and the fact that $\to$ is completely join-reversing in its first coordinate, $$\bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))\leq \bigwedge_{\Diamond c\leq b} (f(c)\to u(b)).$$ Hence, let $c\in \mathbb{L}$ such that $\Diamond c\leq b$, and let us show that $$\bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))\leq f(c)\to u(b).$$ Since $u$ is $\vee$-preserving, hence order-preserving, $\Diamond c\leq b$ implies $u(\Diamond c)\leq u(b)$, hence $$\bigwedge_{a\in \mathbb{L}} (f(a)\to u(\Diamond a))\leq f(c)\to u(\Diamond c) \leq f(c)\to u(b),$$ as required. For (2), we use $u^{-\Box}(\Box a) \leq u(a)$ and the fact that $\to$ is order-preserving in the second coordinate to show the inequality $\bigwedge_{b\in \mathbb{L}} (f(b) \to u^{-\Box}(b)) \leq \bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a))$. To show $$\bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a)) \leq \bigwedge_{b\in \mathbb{L}} (f(b) \to u^{-\Box}(b))$$ we can show that for any $b \in \mathbb{L}$ $$\bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a)) \leq f(b) \to u^{-\Box}(b).$$ After applying the definition of $u^{-\Box}(b)$ and the fact that $\to$ is completely meet-preserving in its second coordinate, the above inequality is equivalent to $$\bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a)) \leq \bigwedge_{b \leq \Box c} ( f(b) \to u(c)).$$ Let $c \in \mathbb{L}$ with $b \leq \Box c$. Since $f$ is order-preserving we get $$\bigwedge_{a\in \mathbb{L}} (f(\Box a)\to u(a)) \leq f(\Box c) \to u(c) \leq f(b) \to u(c).$$
\[def:canonical frame\] The [*canonical*]{} graph-based $\mathbf{A}$-[*frame*]{} associated with any $\mathbf{Fm}_{\ast}$ is the structure $\mathbb{G}_{\mathbf{Fm}_{\ast}} = (Z, E, R_{\Diamond}, R_\Box)$ defined as follows:[^14] [ $$Z:=\{(f,u)\in \mathsf{F}_{\mathbf{A}}(\mathbf{Fm}_{\ast})\times \mathsf{C}_{\mathbf{A}}(\mathbf{Fm}_{\ast})\mid \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f({\varphi})\to u({\varphi}))=1\}.$$ ]{}
For any $z\in Z$ as above, we let $f_z$ and $u_z$ denote the first and the second coordinate of $z$, respectively. Then $E: Z\times Z\to \mathbf{A}$, $R_\Diamond :Z\times Z\to \mathbf{A}$ and $R_{\Box}: Z\times Z \to \mathbf{A}$ are defined as follows: $$E(z, z'): = \bigwedge_{{\varphi}\in \mathbf{Fm_{\ast}}}(f_z({\varphi})\to u_{z'}({\varphi}));$$ [ $$R_\Diamond(z, z') := \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_{z'}^{-\Diamond}({\varphi})\to u_z({\varphi})) = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_{z'}({\varphi})\to u_z(\Diamond{\varphi}));$$ ]{} [$$R_{\Box}(z,z'):= \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_z({\varphi}) \to u_{z'}^{-\Box}({\varphi})) = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}
(f_z(\Box {\varphi})\to u_{z'}({\varphi})).$$ ]{} We will write $\mathbb{G} = (Z, E, R_{\Diamond}, R_\Box)$ for $\mathbb{G}_{\mathbf{Fm}_{\ast}} = (Z, E, R_{\Diamond}, R_\Box)$ whenever ${\mathbf{Fm}_{\ast}}$ is clear from the context.
\[lemma:compatibility\] The structure $\mathbb{G}_{\mathbf{Fm}_{\ast}}$ of Definition \[def:canonical frame\] is a graph-based $\mathbf{A}$-frame, in the sense specified at the beginning of the present section.
We need to show that $R_{\Diamond}$ is $E$-compatible, i.e., $$\begin{aligned}
(R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}])^{[10]} &\subseteq R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}] \\
(R_\Diamond^{[0]}[\{\beta / z \}])^{[01]} &\subseteq R_\Diamond^{[0]}[\{\beta / z \}],\end{aligned}$$ and that $R_{\Box}$ is $E$-compatible, i.e., $$\begin{aligned}
(R_\Box^{[0]}[\{\beta / (\alpha, z) \}])^{[10]} &\subseteq R_\Box^{[0]}[\{\beta / (\alpha, z) \}] \\
(R_\Box^{[1]}[\{\beta / z \}])^{[01]} &\subseteq R_\Box^{[1]}[\{\beta / z \}].\end{aligned}$$ Considering the second inclusion for $R_{\Diamond}$, by definition, for any $(\alpha, w)\in Z_X$,
--- --------------------------------------------------------------------------------------------------
$R_{\Diamond}^{[0]}[\{\beta / z \}](\alpha, w)$
= $\bigwedge_{z'\in Z_A}[\{\beta / z \}(z')\to (R_{\Diamond}(w, z') \to \alpha)]$
= $\beta\to (R_{\Diamond}(w, z) \to \alpha)$
$(R_\Diamond^{[0]}[\{\beta / z \}])^{[01]} (\alpha, w)$
= $\bigwedge_{z'\in Z_A}[(R_\Diamond^{[0]}[\{\beta / z \}])^{[0]}(z')\to (E(z', w)\to \alpha) ],$
--- --------------------------------------------------------------------------------------------------
and hence it is enough to find some $z'\in Z$ such that $$(R_\Diamond^{[0]}[\{\beta / z \}])^{[0]}(z')\to (E(z', w)\to \alpha)\leq \beta\to (R_{\Diamond}(w, z) \to \alpha),$$ i.e.
-- ------------------------------------------------------------------------------------------------------------ ----------
$\left(\bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Diamond(v, z)\to\gamma)]\to (E(z', v)\to\gamma)\right)$
$\to (E(z', w)\to \alpha) \leq \beta\to (R_{\Diamond}(w, z) \to \alpha)$ ($\ast$)
-- ------------------------------------------------------------------------------------------------------------ ----------
Let $z'\in Z$ such that $u_{z'}: {\mathbf{Fm}_{\ast}}\to\mathbf{A}$ maps $\bot$ to $0$ and every other ${\varphi}\in {\mathbf{Fm}_{\ast}}$ to $1$, and $f_{z'}: = f_z^{-\Diamond}$ (cf. Lemma \[lemma:f minus diam is filter\].2). Then
------------ --- ---------------------------------------------------------------------------------------
$E(z', w)$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}} f_z^{-\Diamond}({\varphi})\to u_{w}({\varphi})$
= $R_{\Diamond}(w, z)$,
------------ --- ---------------------------------------------------------------------------------------
and likewise $E(z', v) = R_{\Diamond}(v, z)$. Therefore, for this choice of $z'$, inequality ($\ast$) can be rewritten as follows:
-- ----------------------------------------------------------------------------------------------------------------------
$\left(\bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Diamond(v, z)\to\gamma)]\to (R_{\Diamond}(v, z)\to\gamma)\right)$
$\to (R_{\Diamond}(w, z)\to \alpha) \leq \beta\to (R_{\Diamond}(w, z) \to \alpha)$
-- ----------------------------------------------------------------------------------------------------------------------
The inequality above is true if $$\beta \leq \bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Diamond(v, z)\to\gamma)]\to (R_{\Diamond}(v, z)\to\gamma),$$ i.e. if for every $(\gamma, v)\in Z_X$, $$\beta \leq [\beta\to (R_\Diamond(v, z)\to\gamma)]\to (R_{\Diamond}(v, z)\to\gamma),$$ which is an instance of a tautology in residuated lattices.
Let us show that $(R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}])^{[10]} \subseteq R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}]$. By definition, for every $w\in Z_A$,
--- ----------------------------------------------------------------------------------------------------------------------------
$R_\Diamond^{[1]}[\{\beta / (\alpha, z)\}] (w)$
= $\bigwedge_{(\gamma, z')\in Z_X}[\{\beta / (\alpha, z)\}(\gamma, z')\to (R_{\Diamond}(z', w)\to \gamma)]$
= $\beta\to (R_{\Diamond}(z, w)\to \alpha)$
$(R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}])^{[10]}(w)$
= $\bigwedge_{(\gamma, z')\in Z_X}[(R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}])^{[1]}(\gamma, z')\to (E(w, z')\to \gamma)]$.
--- ----------------------------------------------------------------------------------------------------------------------------
Hence it is enough to find some $(\gamma, z')\in Z_X$ such that $$\begin{gathered}
(R_\Diamond^{[1]}[\{\beta / (\alpha, z) \}])^{[1]}(\gamma, z')\to (E(w, z')\to \gamma) \\
\leq \beta\to ( R_{\Diamond}(z, w)\to \alpha),\end{gathered}$$ i.e.
-- ------------------------------------------------------------------------------------------------------ ----------
$\left( \bigwedge_{v\in Z} (\beta\to (R_{\Diamond}(z, v)\to \alpha))\to (E(v, z')\to \gamma)\right)$
$\to (E(w, z')\to \gamma)\leq \beta\to (R_{\Diamond}(z, w)\to \alpha)$ ($\ast$)
-- ------------------------------------------------------------------------------------------------------ ----------
Let $(\gamma, z') := (\alpha, z')$ such that $f_{z'}: \mathbf{Fm}_{\ast}\to\mathbf{A}$ maps $\top$ to $1$ and every other ${\varphi}\in \mathbf{Fm}_{\ast}$ to $0$, and $u_{z'}: \mathbf{Fm}_{\ast} \to\mathbf{A}$ is defined by the assignment $$u_{z'}({\varphi}) = \left\{\begin{array}{ll}
1 & \text{if } \top\vdash {\varphi}\\
u_z(\Diamond{\varphi}) & \text{otherwise. } \\
\end{array}\right.$$ by definition, $u_{z'}$ maps $\top$ to 1 and $\bot$ to 0; moreover, using Lemma \[lemma:f minus diam is filter\].5, it can be readily verified that $u_{z'}$ is $\vee$-preserving. Then
------------ --- ------------------------------------------------------------------------------------------------
$E(v, z')$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_v({\varphi})\to u_{z'}({\varphi}))$
= $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_v({\varphi})\to u_{z}(\Diamond {\varphi}))$
= $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_v^{-\Diamond}({\varphi})\to u_{z}({\varphi}))$
= $R_{\Diamond}(z, v)$,
------------ --- ------------------------------------------------------------------------------------------------
and likewise $E(w, z') = R_{\Diamond}(z, w)$. Therefore, for this choice of $z'$, inequality ($\ast$) can be rewritten as follows:
-- -------------------------------------------------------------------------------------------------------------- ----------
$\left( \bigwedge_{v\in Z} (\beta\to R_{\Diamond}(z, v)\to \alpha)\to (R_{\Diamond}(z, v)\to \alpha)\right)$
$\to (R_{\Diamond}(z, w)\to \alpha)\leq \beta\to (R_{\Diamond}(z, w)\to \alpha)$ ($\ast$)
-- -------------------------------------------------------------------------------------------------------------- ----------
which is shown to be true by the same argument as the one concluding the verification of the previous inclusion.
Let us show that $(R_\Box^{[0]}[\{\beta / (\alpha, z) \}])^{[10]} \subseteq R_\Box^{[0]}[\{\beta / (\alpha, z) \}] $. For any $w \in Z_A$,
--- -------------------------------------------------------------------------------------------------------------------
$R_{\Box}^{[0]}[\{\beta / (\alpha,z) \}](w)$
= $\bigwedge_{(\gamma,z')\in Z_X}[\{\beta / (\alpha,z) \}(\gamma,z')\to (R_{\Box}(w, z') \to \gamma)]$
= $\beta\to (R_{\Box}(w, z) \to \alpha)$,
$(R_\Box^{[0]}[\{\beta /(\alpha,z) \}])^{[10]} (w)$
= $\bigwedge_{(\gamma,z')\in Z_X}[(R_\Box^{[0]}[\{\beta /(\alpha,z)\}])^{[1]}(\gamma,z')\to (E(w,z')\to \gamma) ].$
--- -------------------------------------------------------------------------------------------------------------------
Hence, it is enough to find some $(\gamma, z')\in Z_X$ such that
----------- ---------------------------------------------------------------------------------
$(R_\Box^{[0]}[\{\beta /(\alpha,z)\}])^{[1]}(\gamma,z')\to (E(w,z')\to \gamma)$
$ \leq $ $\beta\to (R_{\Box}(w, z) \to \alpha)$,
----------- ---------------------------------------------------------------------------------
i.e.
-- -------------------------------------------------------------------------------------------------- ----------
$\left( \bigwedge_{v\in Z} (\beta\to (R_{\Box}(v, z)\to \alpha))\to (E(v, z')\to \gamma)\right)$
$\to (E(w,z')\to \gamma) \leq \beta\to (R_{\Box}(w, z) \to \alpha)$ ($\ast$)
-- -------------------------------------------------------------------------------------------------- ----------
Let $(\gamma, z') := (\alpha, z')$ such that $f_{z'}: \mathbf{Fm}_{\ast}\to\mathbf{A}$ maps $\top$ to $1$ and every other ${\varphi}\in \mathbf{Fm}_{\ast}$ to $0$, and $u_{z'}: = u^{-\Box}_{z}$ (cf. Lemma \[lemma:f minus diam is filter\].4). Then
------------ --- -------------------------------------------------------------------------------------
$E(v, z')$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}} [f_v({\varphi})\to u_{z}^{-\Box}({\varphi})]$
= $R_{\Box}(v, z)$,
------------ --- -------------------------------------------------------------------------------------
and likewise $E(w, z') = R_{\Box}(w, z)$. Therefore, for this choice of $z'$, inequality ($\ast$) can be rewritten as follows:
-- -------------------------------------------------------------------------------------------------------- --
$\left( \bigwedge_{v\in Z} (\beta\to (R_{\Box}(v, z)\to \alpha))\to (R_{\Box}(v, z)\to \alpha)\right)$
$\to (R_{\Box}(w, z) \to \alpha) \leq \beta\to (R_{\Box}(w, z) \to \alpha)$.
-- -------------------------------------------------------------------------------------------------------- --
The inequality above is true if $$\beta \leq \bigwedge_{v\in Z} (\beta\to (R_{\Box}(v, z)\to \alpha))\to (R_{\Box}(v, z)\to \alpha),$$ i.e. if for every $v\in Z_A$, $$\beta \leq (\beta\to (R_{\Box}(v, z)\to \alpha))\to (R_{\Box}(v, z)\to \alpha),$$ which is an instance of a tautology in residuated lattices.
For the last inclusion, for any $(\alpha, w)\in Z_X$,
--- ----------------------------------------------------------------------------------------------
$R_{\Box}^{[1]}[\{\beta / z \}](\alpha,w)$
= $\bigwedge_{z'\in Z_A}[\{\beta / z \}(z')\to (R_{\Box}(z',w) \to \alpha)]$
= $\beta\to (R_{\Box}(z, w) \to \alpha)$,
$(R_\Box^{[1]}[\{\beta / z \}])^{[01]} (\alpha, w)$
= $\bigwedge_{z'\in Z_A}[(R_\Box^{[1]}[\{\beta / z \}])^{[0]}(z')\to (E(z', w)\to \alpha) ]$,
--- ----------------------------------------------------------------------------------------------
and hence it is enough to find some $z'\in Z_A$ such that $$(R_\Box^{[1]}[\{\beta / z \}])^{[0]}(z')\to (E(z', w)\to \alpha) \leq \beta\to (R_{\Box}(z, w) \to \alpha),$$ i.e.
-- -------------------------------------------------------------------------------------------------------- ----------
$\left(\bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Box(z, v)\to\gamma)]\to (E(z', v)\to\gamma)\right)$
$\to (E(z', w)\to \alpha) \leq \beta\to (R_{\Box}(z, w) \to \alpha)$ ($\ast$)
-- -------------------------------------------------------------------------------------------------------- ----------
Let $z'\in Z_A$ such that $u_{z'}: \mathbf{Fm}_{\ast}\to\mathbf{A}$ maps $\bot$ to $0$ and every other ${\varphi}\in \mathbf{Fm}_{\ast}$ to $1$, and $f_{z'}: \mathbf{Fm}_{\ast}\to\mathbf{A}$ is defined by the assignment $$f_{z'}({\varphi}) = \left\{\begin{array}{ll}
0 & \text{if } {\varphi}\vdash \bot\\
f_z(\Box{\varphi}) & \text{otherwise. } \\
\end{array}\right.$$ by definition, $f_{z'}$ maps $\top$ to 1 and $\bot$ to 0; moreover, using Lemma \[lemma:f minus diam is filter\].6, it can be readily verified that $f_{z'}$ is $\wedge$-preserving. Then
------------ --- --------------------------------------------------------------------------------------------
$E(z', w)$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_{z'}({\varphi})\to u_{w}({\varphi}))$
= $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_z(\Box{\varphi})\to u_{w}({\varphi}))$
= $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} (f_z({\varphi})\to u_{w}^{-\Box}({\varphi}))$
= $R_{\Box}(z, w)$,
------------ --- --------------------------------------------------------------------------------------------
and likewise $E(z', v) = R_{\Box}(z, v)$. Therefore, for this choice of $z'$, inequality ($\ast$) can be rewritten as follows:
-- ------------------------------------------------------------------------------------------------------------ --
$\left(\bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Box(z, v)\to\gamma)]\to (R_\Box(z, v)\to\gamma)\right)$
$\to (R_{\Box}(z, w) \to \alpha) \leq \beta\to (R_{\Box}(z, w) \to \alpha)$.
-- ------------------------------------------------------------------------------------------------------------ --
The inequality above is true if $$\beta \leq \bigwedge_{(\gamma, v)\in Z_X}[\beta\to (R_\Box(z, v)\to\gamma)]\to (R_\Box(z, v)\to\gamma),$$ i.e. if for every $(\gamma, v)\in Z_X$, $$\beta \leq [\beta\to (R_\Box(z, v)\to\gamma)]\to (R_\Box(z, v)\to\gamma),$$ which is an instance of a tautology in residuated lattices.
\[def:canonical model\] The [*canonical graph-based*]{} $\mathbf{A}$-[*model*]{} associated with $\mathbf{Fm}_{\ast}$ is the structure $\mathbb{M}_{\mathbf{Fm}_{\ast}} =(\mathbb{G}_{\mathbf{Fm}_{\ast}}, V)$ such that $\mathbb{G}_{\mathbf{Fm}_{\ast}}$ is the canonical graph-based $\mathbf{A}$-frame of Definition \[def:canonical frame\], and if $p \in \mathsf{Prop}$, then $V(p) = ({[\![{p}]\!]}, {(\![{p}]\!)})$ with ${[\![{p}]\!]}: Z_A\to \mathbf{A}$ and ${(\![{p}]\!)}: Z_X\to \mathbf{A}$ defined by $z\mapsto f_z(p)$ and $(\alpha, z)\mapsto u_z(p)\to \alpha$, respectively.[^15]
The structure $\mathbb{G}_{\mathbf{Fm}_{\ast}}$ of Definition \[def:canonical model\] is a graph-based $\mathbf{A}$-model.
It is enough to show that ${[\![{p}]\!]}^{[1]}={(\![{p}]\!)}$ and ${[\![{p}]\!]}={(\![{p}]\!)}^{[0]}$ for any $p \in \mathsf{Prop}$. To show that ${(\![{p}]\!)} (\alpha, z)\leq {[\![{p}]\!]}^{[1]}(\alpha, z)$ for any $(\alpha, z)\in Z_X$, by definition, we need to show that $$u_z(p)\to\alpha\leq \bigwedge_{z'\in Z_A}({[\![{p}]\!]}(z')\to (E(z', z)\to\alpha)),$$ i.e. that for every $z'\in Z_A$, $$u_z(p)\to\alpha\leq {[\![{p}]\!]}(z')\to (E(z', z)\to\alpha).$$ By definition, the inequality above is equivalent to $$u_z(p)\to\alpha\leq f_{z'}(p)\to (\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} [f_{z'}({\varphi})\to u_z({\varphi})]\to\alpha).$$ Since $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}} [f_{z'}({\varphi})\to u_z({\varphi})]\leq f_{z'}(p)\to u_z(p)$ and $\to$ is order-reversing in its first coordinate, it is enough to show that $$u_z(p)\to\alpha\leq f_{z'}(p)\to [(f_{z'}(p)\to u_z(p))\to\alpha].$$ By residuation the inequality above is equivalent to $$u_z(p)\to\alpha\leq [f_{z'}(p)\otimes (f_{z'}(p)\to u_z(p))]\to\alpha,$$ which is equivalent to $$[f_{z'}(p)\otimes (f_{z'}(p)\to u_z(p))]\otimes [u_z(p)\to\alpha]\leq \alpha,$$ which is the instance of a tautology in residuated lattices. Conversely, to show that $ {[\![{p}]\!]}^{[1]}(\alpha, z)\leq {(\![{p}]\!)} (\alpha, z)$, i.e. $$\bigwedge_{z'\in Z_A}({[\![{p}]\!]}(z')\to (E(z', z)\to\alpha))\leq u_z(p)\to\alpha,$$ it is enough to show that $$\label{eqq}
{[\![{p}]\!]}(z')\to (E(z', z)\to\alpha))\leq u_z(p)\to\alpha$$ for some $z'\in Z$. Let $z': = (f_p, u)$ such that $u:\mathbf{Fm}_{\ast}\to \mathbf{A}$ maps $\bot$ to $0$ and every other element of $\mathbf{Fm}_{\ast}$ to $1$, and $f_p:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$f_p({\varphi}) = \left\{\begin{array}{ll}
1 & \text{if } p\vdash {\varphi}\\
0 & \text{otherwise.} \\
\end{array}\right.$$ Hence, $E(z', z) = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_p({\varphi})\to u_z({\varphi}))=\bigwedge_{p\vdash{\varphi}}u_z({\varphi}) = u_z(p)$, the last identity holding since $u_z$ is order-preserving. Therefore, ${[\![{p}]\!]}(z')\to (E(z', z)\to\alpha)) = f_p(p)\to (u_z(p)\to \alpha) = 1\to (u_z(p)\to \alpha) = u_z(p)\to \alpha$, which shows .
By adjunction, the inequality ${(\![{p}]\!)} \leq {[\![{p}]\!]}^{[1]}$ proven above implies that ${[\![{p}]\!]}\leq{(\![{p}]\!)}^{[0]}$. Hence, to show that ${[\![{p}]\!]}={(\![{p}]\!)}^{[0]}$, it is enough to show ${(\![{p}]\!)}^{[0]}(z)\leq{[\![{p}]\!]}(z)$ for every $z\in Z$, i.e. $$\bigwedge_{(\alpha, z')\in Z_X} {(\![{p}]\!)}(\alpha, z')\to (E(z, z')\to \alpha)\leq f_z(p),$$ and to show the inequality above, it is enough to show that $$\label{eqqq} {(\![{p}]\!)}(\alpha, z')\to (E(z, z')\to \alpha)\leq f_z(p)$$ for some $(\alpha, z')\in Z_X$. Let $\alpha: =f_z(p)$ and $z': = (f_{z'}, u_{p})$ be such that $u_{z'} = u_{p}: \mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the following assignment: $$u_{p}({\varphi}) = \left\{\begin{array}{ll}
0 & \text{if } {\varphi}\vdash \bot\\
f_z(p) & \text{if } {\varphi}\vdash p \mbox{ and } {\varphi}\not\vdash \bot\\
1 & \text{if } {\varphi}\not\vdash p.
\end{array}\right.$$ By construction, $u_{z'}$ is $\vee$-, $\bot$- and $\top$-preserving. Moreover, ${(\![{p}]\!)}(\alpha, z') = u_{z'}(p)\to\alpha = f_z(p)\to f_z(p) = 1$, and $E(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_z({\varphi})\to u_{z'}({\varphi})) = \bigwedge_{{\varphi}\vdash p}(f_z({\varphi})\to f_z(p)) = 1$. Hence, the left-hand side of can be equivalently rewritten as $1\to (1\to f_z(p)) = f_z(p)$, which shows and concludes the proof.
Recall that $\mathbf{Fm}_0$ is the Lindenbaum-Tarski algebra of the logic $\mathbf{L}_0$ which is the axiomatic extension of $\mathbf{L}$ with the axiom $\Box \bot \vdash \bot$. We write $\mathbf{L}_{0\ast}$ to denote an arbitrary (possible non-proper) extension of $\mathbf{L}_{0}$ and $\mathbf{Fm}_{0\ast}$ for the corresponding Lindenbaum-Tarski algebra. The axiom $\Box \bot \vdash \bot$ is required only in the inductive step for $\Box$-formulas in the following lemma. To emphasise this, we work mostly in the context of $\mathbf{Fm}_{\ast}$ and switch to $\mathbf{Fm}_{0\ast}$ only when required.
\[lemma:truthlemma\] For every ${\varphi}\in \mathbf{Fm}_{0\ast}$, the maps ${[\![{{\varphi}}]\!]}: Z_A\to \mathbf{A}$ and ${(\![{{\varphi}}]\!)}: Z_X\to \mathbf{A}$ coincide with those defined by the assignments $z\mapsto f_z({\varphi})$ and $(\alpha, z)\mapsto u_z({\varphi})\to \alpha$, respectively.
We proceed by induction on ${\varphi}$. If ${\varphi}: = p\in \mathsf{Prop}$, the statement follows immediately from Definition \[def:canonical model\].
If ${\varphi}: =\top$, then ${[\![{\top}]\!]}(z) = 1 = f_z(\top)$ since $\mathbf{A}$-filters are $\top$-preserving. Moreover,
------------------------------- ----- ------------------------------------------------------------------------
${(\![{\top}]\!)}(\alpha, z)$ $=$ ${[\![{\top}]\!]}^{[1]}(\alpha, z)$
$=$ $ \bigwedge_{z'\in Z_A} [{[\![{\top}]\!]}(z')\to (E(z', z)\to\alpha)]$
$=$ $ \bigwedge_{z'\in Z_A} [f_{z'}(\top)\to (E(z', z)\to\alpha)]$
$=$ $ \bigwedge_{z'\in Z_A} [E(z', z)\to\alpha]$.
------------------------------- ----- ------------------------------------------------------------------------
So, to show that $u_z(\top)\to \alpha\leq {(\![{\top}]\!)}(\alpha, z)$, we need to show that for every $z'\in Z$, $$u_z(\top)\to \alpha\leq E(z', z)\to\alpha,$$ and for this, it is enough to show that $$\bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_{z'}(\psi)\to u_z(\psi)]\leq u_z(\top),$$ which is true, since by definition, $u_z(\top) = 1$. To show that ${(\![{\top}]\!)}(\alpha, z)\leq u_z(\top)\to \alpha$, i.e. that $$\bigwedge_{z'\in Z_A} [E(z', z)\to\alpha]\leq u_z(\top)\to \alpha,$$ it is enough to find some $z'\in Z$ such that $E(z', z)\to\alpha\leq u_z(\top)\to \alpha$. Let $z': = (f_\top, u)$ such that $u:\mathbf{Fm}_{\ast}\to \mathbf{A}$ maps $\top$ to $1$ and every other element of $\mathbf{Fm}_{\ast}$ to $0$, and $f_\top:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$f_\top({\varphi}) = \left\{\begin{array}{ll}
1 & \text{if } \top\vdash {\varphi}\\
0 & \text{otherwise. } \\
\end{array}\right.$$By definition, $E(z', z) = \bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_{z'}(\psi)\to u_z(\psi)] = \bigwedge_{\top\vdash \psi}[1\to u_z(\psi)]= \bigwedge_{\top\vdash \psi}u_z(\psi)
\geq u_z(\top)$, the last inequality being due to the fact that $u_z$ is order-preserving. Hence, $E(z', z)\to\alpha\leq u_z(\top)\to \alpha$, as required.
If ${\varphi}: =\bot$, then ${(\![{\bot}]\!)}(\alpha, z) = 1 = u_z(\bot)\to \alpha$ since complements of $\mathbf{A}$-ideals are $\bot$-preserving. Let us show that ${[\![{\bot}]\!]}(z) = f_z(\bot)$. The inequality $f_z(\bot)\leq {[\![{\bot}]\!]}(z) $ follows immediately from the fact that $f_z$ is proper and hence $f_z(\bot) = 0$. To show that ${[\![{\bot}]\!]}(z) \leq f_z(\bot)$, by definition ${[\![{\bot}]\!]}(z)={(\![{\bot}]\!)}^{[0]}(z) = \bigwedge_{(\alpha, z')\in Z_X} [(u_{z'}(\bot)\to\alpha)\to (E(z, z')\to\alpha)]$, hence, it is enough to find some $(\alpha, z')\in Z_X$ such that $$\label{eqqqqqq}(u_{z'}(\bot)\to\alpha)\to (E(z, z')\to\alpha) \leq f_z(\bot).$$ Let $\alpha: = f_z(\bot)$ and let $z': = (f_{\top}, u_{\bot})$ such that $f_{\top}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}: = \top$, and $u_{\bot}:\mathbf{Fm}_{\ast} \to \mathbf{A}$ is defined by the assignment $$u_{\bot}(\psi) = \left\{\begin{array}{ll}
0 & \text{if } \psi\vdash \bot\\
1 & \text{if }\psi\not\vdash \bot.
\end{array}\right.$$ By definition and since $f_z$ is order-preserving and $\bot$-preserving, $E(z, z') = \bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_z(\psi)\to u_{\bot}(\psi)] = 1$. Hence, can be rewritten as follows: $$(f_z(\bot)\to f_z(\bot))\to f_z(\bot)\leq f_z(\bot),$$ which is true since $f_z(\bot)\to f_z(\bot) =1$ and $1\to f_z(\bot) = f_z(\bot)$.
If ${\varphi}: = {\varphi}_1\wedge{\varphi}_2$, then ${[\![{{\varphi}_1\wedge{\varphi}_2}]\!]}(z) = ({[\![{{\varphi}_1}]\!]} \wedge {[\![{{\varphi}_2}]\!]})(z) = {[\![{{\varphi}_1}]\!]}(z) \wedge {[\![{{\varphi}_2}]\!]}(z) = f_z({\varphi}_1) \wedge f_z({\varphi}_2) = f_z({\varphi}_1 \wedge {\varphi}_2)$. Let us show that ${(\![{{\varphi}_1 \wedge {\varphi}_2}]\!)}(\alpha, z) = u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha$. By definition,
--- -----------------------------------------------------------------------------------------------
${(\![{{\varphi}_1 \wedge {\varphi}_2}]\!)}(\alpha, z)$
= ${[\![{{\varphi}_1 \wedge {\varphi}_2}]\!]}^{[1]}(\alpha, z)$
= $\bigwedge_{z'\in Z}[{[\![{{\varphi}_1 \wedge {\varphi}_2}]\!]}(z')\to (E(z', z)\to \alpha)]$
= $\bigwedge_{z'\in Z}[f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to (E(z', z)\to \alpha)]$.
--- -----------------------------------------------------------------------------------------------
Hence, to show that $u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha\leq {(\![{{\varphi}_1 \wedge {\varphi}_2}]\!)}(\alpha, z)$, we need to show that for every $z'\in Z$, $$u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha\leq f_{z'}({\varphi}_1\wedge {\varphi}_2)\to (E(z', z)\to \alpha).$$ Since by definition $E(z', z) = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}[f_{z'}({\varphi})\to u_z({\varphi})]\leq f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to u_z({\varphi}_1 \wedge {\varphi}_2)$ and $\to$ is order-reversing in the first coordinate and order-preserving in the second one, it is enough to show that for every $z'\in Z$,
-------- ------------------------------------------------------------------------------------------------------------------------------------------
$u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha$
$\leq$ $ f_{z'}({\varphi}_1\wedge {\varphi}_2)\to ((f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to u_z({\varphi}_1 \wedge {\varphi}_2))\to \alpha)$.
-------- ------------------------------------------------------------------------------------------------------------------------------------------
By residuation, the above inequality is equivalent to
-------- ---------------------------------------------------------------------------------------------------------------------------------------------
$u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha$
$\leq$ $[f_{z'}({\varphi}_1\wedge {\varphi}_2)\otimes (f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to u_z({\varphi}_1 \wedge {\varphi}_2))]\to \alpha$.
-------- ---------------------------------------------------------------------------------------------------------------------------------------------
The above inequality is true if $$f_{z'}({\varphi}_1\wedge {\varphi}_2)\otimes (f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to u_z({\varphi}_1 \wedge {\varphi}_2))\leq u_z({\varphi}_1 \wedge {\varphi}_2),$$ which is an instance of a tautology in residuated lattices.
To show that ${(\![{{\varphi}_1 \wedge {\varphi}_2}]\!)}(\alpha, z)\leq u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha$, it is enough to find some $z'\in Z$ such that $$f_{z'}({\varphi}_1 \wedge {\varphi}_2)\to (E(z', z)\to \alpha)\leq u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha.$$ Let $z': = (f_{{\varphi}_1 \wedge {\varphi}_2}, u_\bot)$ such that $u_\bot:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}: = \bot$, and $f_{{\varphi}_1 \wedge {\varphi}_2}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$f_{{\varphi}_1 \wedge {\varphi}_2}(\psi) = \left\{\begin{array}{ll}
1 & \text{if } {\varphi}_1 \wedge {\varphi}_2\vdash \psi\\
0 & \text{otherwise. } \\
\end{array}\right.$$ For $z': = z$, since $f_{z'}({\varphi}_1 \wedge {\varphi}_2) = 1$ and $1\to (E(z', z)\to \alpha) = E(z', z)\to \alpha$, the inequality above becomes $$E(z', z)\to \alpha\leq u_z({\varphi}_1 \wedge {\varphi}_2)\to \alpha,$$ to verify which, it is enough to show that $u_z({\varphi}_1 \wedge {\varphi}_2)\leq E(z', z)$. Indeed, by definition, $E(z', z) = \bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_{z'}(\psi)\to u_z(\psi)] = \bigwedge_{{\varphi}_1 \wedge {\varphi}_2\vdash \psi}[1\to u_z(\psi)]= \bigwedge_{{\varphi}_1 \wedge {\varphi}_2\vdash \psi}u_z(\psi)\geq u_z({\varphi}_1 \wedge {\varphi}_2)$, the last inequality being due to the fact that $u_z$ is order-preserving.
If ${\varphi}: = {\varphi}_1\vee{\varphi}_2$, then ${(\![{{\varphi}_1\vee{\varphi}_2}]\!)}(\alpha, z) = ({(\![{{\varphi}_1}]\!)} \wedge {(\![{{\varphi}_2}]\!)})(\alpha, z) = {(\![{{\varphi}_1}]\!)}(\alpha, z) \wedge {(\![{{\varphi}_2}]\!)}(\alpha, z) = (u_z({\varphi}_1)\to\alpha) \wedge (u_z({\varphi}_2)\to \alpha) = (u_z({\varphi}_1)\vee u_z({\varphi}_2))\to \alpha)= u_z({\varphi}_1 \vee {\varphi}_2)\to \alpha$. Let us show that ${[\![{{\varphi}_1 \vee {\varphi}_2}]\!]}(z) = f_z({\varphi}_1 \vee {\varphi}_2)$. By definition,
--- -----------------------------------------------------------------------------------------------------------------
${[\![{{\varphi}_1 \vee {\varphi}_2}]\!]}(z)$
= ${(\![{{\varphi}_1 \vee {\varphi}_2}]\!)}^{[0]}(z)$
= $\bigwedge_{(\alpha, z')\in Z_X}[{(\![{{\varphi}_1 \vee {\varphi}_2}]\!)}(\alpha, z')\to (E(z, z')\to \alpha)]$
= $\bigwedge_{(\alpha, z')\in Z_X}[(u_{z'}({\varphi}_1 \vee {\varphi}_2)\to\alpha)\to (E(z, z')\to \alpha)]$.
--- -----------------------------------------------------------------------------------------------------------------
Hence, to show that $f_z({\varphi}_1 \vee {\varphi}_2)\leq {[\![{{\varphi}_1 \vee {\varphi}_2}]\!]}(z)$, we need to show that for every $(\alpha, z')\in Z_X$, $$f_z({\varphi}_1 \vee {\varphi}_2)\leq (u_{z'}({\varphi}_1 \vee {\varphi}_2)\to\alpha)\to (E(z, z')\to \alpha).$$ Since by definition $E(z, z') = \bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_{z}(\psi)\to u_{z'}(\psi)]\leq f_{z}({\varphi}_1 \vee {\varphi}_2)\to u_{z'}({\varphi}_1 \vee {\varphi}_2)$ and $\to$ is order-reversing in the first coordinate and order-preserving in the second one, it is enough to show that for every $(\alpha, z')\in Z_X$, $$\begin{gathered}
f_z({\varphi}_1 \vee {\varphi}_2) \leq \\
(u_{z'}({\varphi}_1 \vee {\varphi}_2)\to\alpha)\to ((f_{z}({\varphi}_1 \vee {\varphi}_2)\to u_{z'}({\varphi}_1 \vee {\varphi}_2))\to \alpha).\end{gathered}$$
By residuation, associativity and commutativity of $\otimes$, the inequality above is equivalent to
-- ---------------------------------------------------------------------------------------------------------------------------
$ f_z({\varphi}_1 \vee {\varphi}_2)\otimes (f_{z}({\varphi}_1 \vee {\varphi}_2)\to u_{z'}({\varphi}_1 \vee {\varphi}_2))$
$ \otimes (u_{z'}({\varphi}_1 \vee {\varphi}_2)\to\alpha) \leq \alpha$,
-- ---------------------------------------------------------------------------------------------------------------------------
which is a tautology in residuated lattices.
To show that ${[\![{{\varphi}_1 \vee {\varphi}_2}]\!]}(z)\leq f_z({\varphi}_1 \vee {\varphi}_2)$, it is enough to find some $(\alpha, z')\in Z_X$ such that $$\label{eqqqqq}
(u_{z'}({\varphi}_1 \vee {\varphi}_2)\to\alpha)\to (E(z', z)\to \alpha)\leq f_z({\varphi}_1 \vee {\varphi}_2).$$ Let $\alpha: = f_z({\varphi}_1 \vee {\varphi}_2)$ and let $z': = (f_{\top}, u_{{\varphi}_1 \vee {\varphi}_2})$ such that $f_{\top}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}:= \top$, and $u_{{\varphi}_1 \vee {\varphi}_2}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$u_{{\varphi}_1 \vee {\varphi}_2}(\psi) = \left\{\begin{array}{ll}
0 & \text{if } \psi\vdash \bot\\
f_z({\varphi}_1 \vee {\varphi}_2) & \text{if } \psi\not\vdash \bot \mbox{ and } \psi\vdash {\varphi}_1 \vee {\varphi}_2 \\
1 & \text{if }\psi\not\vdash {\varphi}_1 \vee {\varphi}_2.
\end{array}\right.$$ By definition and since $f_z$ is order-preserving and proper, $E(z, z') = \bigwedge_{\psi\in \mathbf{Fm}_{\ast}}[f_z(\psi)\to u_{{\varphi}_1 \vee {\varphi}_2}(\psi)] = \bigwedge_{\bot\not\dashv \psi\vdash{\varphi}_1 \vee {\varphi}_2} [f_z(\psi)\to f_z({\varphi}_1 \vee {\varphi}_2) ] = 1$. Hence, can be rewritten as follows: $$(f_z({\varphi}_1 \vee {\varphi}_2)\to f_z({\varphi}_1 \vee {\varphi}_2))\to f_z({\varphi}_1 \vee {\varphi}_2)\leq f_z({\varphi}_1 \vee {\varphi}_2),$$ which is true since $f_z({\varphi}_1 \vee {\varphi}_2)\to f_z({\varphi}_1 \vee {\varphi}_2) =1$ and $1\to f_z({\varphi}_1 \vee {\varphi}_2) = f_z({\varphi}_1 \vee {\varphi}_2)$.
If ${\varphi}: = \Diamond\psi$, let us show that ${(\![{\Diamond\psi}]\!)} (\alpha, z) = u_z(\Diamond \psi)\to \alpha$. By definition,
---------------------------------------- --- ------------------------------------------------------------------------------------
${(\![{\Diamond\psi}]\!)} (\alpha, z)$ = $R^{[0]}_\Diamond[{[\![{\psi}]\!]}](\alpha, z)$
= $\bigwedge_{z'\in Z_A}[{[\![{\psi}]\!]}(z')\to (R_{\Diamond}(z, z') \to \alpha)]$
= $\bigwedge_{z'\in Z_A}[f_{z'}(\psi)\to (R_{\Diamond}(z, z') \to \alpha)]$,
---------------------------------------- --- ------------------------------------------------------------------------------------
Hence, to show that $u_z(\Diamond\psi)\to \alpha\leq {(\![{\Diamond\psi}]\!)}(\alpha, z)$, we need to show that for every $z'\in Z$, $$u_z(\Diamond\psi)\to \alpha\leq f_{z'}(\psi)\to (R_{\Diamond}(z, z') \to \alpha).$$ By definition we have $R_\Diamond(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_{z'}({\varphi}) \to u_{z}(\Diamond{\varphi}))
\leq f_{z'}(\psi)\to u_{z}(\Diamond \psi)$, and since $\to$ is order-reversing in the first coordinate and order-preserving in the second one, it is enough to show that for every $z'\in Z$, $$u_z(\Diamond\psi)\to \alpha\leq f_{z'}(\psi)\to ((f_{z'}(\psi)\to u_{z}(\Diamond \psi)) \to \alpha).$$ By residuation, associativity and commutativity of $\otimes$, the inequality above is equivalent to $$[f_{z'}(\psi)\otimes (f_{z'}(\psi)\to u_{z}(\Diamond \psi))] \otimes (u_z(\Diamond\psi)\to \alpha) \leq \alpha$$ which is a tautology in residuated lattices.
To show that ${(\![{\Diamond\psi}]\!)}(\alpha, z)\leq u_z(\Diamond\psi)\to \alpha$, it is enough to find some $z'\in Z$ such that $$\label{eqqqqqqq}
f_{z'}(\psi)\to (R_{\Diamond}(z, z') \to \alpha)\leq u_z(\Diamond\psi)\to \alpha.$$ Let $z': = (f_{\psi}, u_\bot)$ such that $u_\bot:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}: = \bot$, and $f_{\psi}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$f_{\psi}({\varphi}) = \left\{\begin{array}{ll}
1 & \text{if } \psi\vdash {\varphi}\\
0 & \text{otherwise. } \\
\end{array}\right.$$ By definition and Lemma \[eq:premagicnew2\],
--------------------- -------- ----------------------------------------------------------------------------------------------------
$R_\Diamond(z, z')$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_{z'}^{-\Diamond}({\varphi})\to u_{z}({\varphi})) $
= $ \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}(f_{z'}({\varphi})\to u_{z}(\Diamond {\varphi}))$
= $ \bigwedge_{\psi\vdash {\varphi}} u_{z}(\Diamond {\varphi})$
$\geq$ $ u_{z}(\Diamond \psi)$,
--------------------- -------- ----------------------------------------------------------------------------------------------------
the last inequality being due to the fact that $u_z$ and $\Diamond$ are order-preserving. Since $\to$ is order reversing in the first coordinate and order-preserving in the second one, to show it is enough to show that $$f_{z'}(\psi)\to (u_{z}(\Diamond \psi) \to \alpha)\leq u_z(\Diamond\psi)\to \alpha.$$ This immediately follows from the fact that, by construction, $f_{z'}(\psi) = 1$.
Let us show that ${[\![{\Diamond\psi}]\!]}(z) = f_z(\Diamond\psi)$. By definition,
--- ------------------------------------------------------------------------------------------------
${[\![{\Diamond\psi}]\!]} (z)$
= ${(\![{\Diamond\psi}]\!)}^{[0]}(z)$
= $\bigwedge_{(\alpha, z')\in Z_X}[{(\![{\Diamond\psi}]\!)}(\alpha, z')\to (E(z, z')\to\alpha)]$
= $\bigwedge_{(\alpha, z')\in Z_X}[(u_{z'}(\Diamond\psi)\to \alpha)\to (E(z, z')\to\alpha)].$
--- ------------------------------------------------------------------------------------------------
Hence, to show that $f_z(\Diamond\psi)\leq {[\![{\Diamond\psi}]\!]}(z)$, we need to show that for every $(\alpha, z')\in Z_X$, $$f_z(\Diamond\psi)\leq (u_{z'}(\Diamond\psi)\to \alpha)\to (E(z, z')\to\alpha).$$ Since by definition $E(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}[f_{z}({\varphi})\to u_{z'}({\varphi})]\leq f_{z}(\Diamond\psi)\to u_{z'}(\Diamond\psi)$ and $\to$ is order-reversing in the first coordinate and order-preserving in the second one, it is enough to show that for every $(\alpha, z')\in Z_X$, $$f_z(\Diamond\psi)\leq (u_{z'}(\Diamond\psi)\to \alpha)\to ((f_{z}(\Diamond\psi)\to u_{z'}(\Diamond\psi))\to\alpha).$$ By residuation, associativity and commutativity of $\otimes$, the inequality above is equivalent to $$[f_z(\Diamond\psi)\otimes (f_{z}(\Diamond\psi)\to u_{z'}(\Diamond\psi))] \otimes (u_{z'}(\Diamond\psi)\to \alpha) \leq \alpha$$ which is a tautology in residuated lattices.
To show that ${[\![{\Diamond\psi}]\!]}(z)\leq f_z(\Diamond\psi)$, it is enough to find some $(\alpha, z')\in Z_X$ such that $$\label{eqqqqqqqq}(u_{z'}(\Diamond\psi)\to \alpha)\to (E(z, z')\to\alpha)\leq f_z(\Diamond\psi).$$ Let $\alpha: = f_z(\Diamond\psi)$ and let $z': = (f_{\top}, u_{\Diamond\psi})$ such that $f_{\top}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}: = \top$, and $u_{\Diamond\psi}:\mathbf{Fm}_{\ast}\to \mathbf{A}$ is defined by the assignment $$u_{\Diamond\psi}({\varphi}) = \left\{\begin{array}{ll}
0 & \text{if } {\varphi}\vdash \bot\\
f_z(\Diamond\psi) & \text{if } {\varphi}\not\vdash \bot \mbox{ and } {\varphi}\vdash \Diamond\psi \\
1 & \text{if }{\varphi}\not\vdash \Diamond\psi.
\end{array}\right.$$ By definition and since $f_z$ is order-preserving and proper, $E(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm}_{\ast}}[f_z({\varphi})\to u_{\Diamond\psi}({\varphi})] = \bigwedge_{{\varphi}\vdash\Diamond\psi} [f_z({\varphi})\to f_z(\Diamond\psi) ] = 1$. Hence, can be rewritten as follows: $$(f_z(\Diamond\psi)\to f_z(\Diamond\psi))\to f_z(\Diamond\psi)\leq f_z(\Diamond\psi),$$ which is true since $f_z(\Diamond\psi)\to f_z(\Diamond\psi) =1$ and $1\to f_z(\Diamond\psi) = f_z(\Diamond\psi)$.
If ${\varphi}:=\Box \psi$ we will show that ${[\![{\Box \psi}]\!]}(z)=f_z(\Box \psi)$. By definition:
---------------------------- --- ------------------------------------------------------------------------------------------------
${[\![{\Box\psi}]\!]} (z)$ = $R^{[0]}_\Box[{(\![{\psi}]\!)}](z)$
= $\bigwedge_{(\alpha,z') \in Z_X} [{(\![{\psi}]\!)}(\alpha,z') \to (R_{\Box}(z,z')\to \alpha)]$
= $\bigwedge_{(\alpha,z') \in Z_X} [(u_{z'}(\psi) \to \alpha) \to (R_{\Box}(z,z')\to \alpha)]$
---------------------------- --- ------------------------------------------------------------------------------------------------
Hence to show that $f_z(\Box \psi) \leq {[\![{\Box \psi}]\!]}(z)$ we must show that for every $(\alpha,z') \in Z_X$ we have $$f_z(\Box \psi) \leq (u_{z'}(\psi) \to \alpha) \to (R_{\Box}(z,z')\to \alpha).$$
We have $R_{\Box}(z,z')=\bigwedge_{{\varphi}\in \mathbf{Fm}_{0\ast}} (f_z(\Box {\varphi}) \to u_{z'}({\varphi}))$ so $R_{\Box}(z,z')\leq f_z(\Box \psi) \to u_{z'}(\psi)$. Since $\to$ is order-reversing in the first coordinate and order-preserving in the second coordinate, it will be enough to show that $$f_z(\Box \psi) \leq (u_{z'}(\psi) \to \alpha) \to ((f_z(\Box \psi) \to u_{z'}(\psi))\to \alpha).$$ Using residuation, and associativity and commutativty of $\otimes$ we can see that this is an instance of a tautology in residuated lattices.
To show ${[\![{\Box \psi}]\!]}(z)\leq f_z(\Box \psi)$ we must find $(\alpha, z') \in Z_X$ such that
$$\label{eq:Box1}
(u_{z'}(\psi) \to \alpha) \to (R_{\Box}(z,z')\to \alpha) \leq f_z(\Box \psi).$$
Let $\alpha: = f_z(\Box \psi)$ and let $z': = (f_{\top}, u_{\psi})$ such that $f_{\top}:\mathbf{Fm}_{0\ast}\to \mathbf{A}$ is defined as indicated above in the base case for ${\varphi}:= \top$, and $u_{\psi}:\mathbf{Fm}_{0\ast}\to \mathbf{A}$ is defined by the assignment $$u_{\psi}(\chi) = \left\{\begin{array}{ll}
0 & \text{if } \chi\vdash \bot\\
f_z(\Box\psi) & \text{if } \chi\not\vdash \bot \mbox{ and } \chi\vdash \psi \\
1 & \text{if }\chi\not\vdash \psi.
\end{array}\right.$$ By definition, and since $f_{z}$ is order-preserving and $\Box\bot\leq \bot$ is valid,[^16] $R_{\Box}(z, z') = \bigwedge_{\chi\in\mathbf{Fm}_{0\ast}}(f_{z}(\Box\chi)\to u_{z'}(\chi)) = 1$. Hence, can be rewritten as follows: $$(f_{z}(\Box\psi)\to f_z(\Box\psi)) \to (1\to f_z(\Box\psi)) \leq f_z(\Box \psi),$$ which is a tautology.Next, we want to show that ${(\![{\Box \psi}]\!)}(\alpha, z)=u_z(\Box\psi) \to \alpha$. By definition
----------------------------------- --- -------------------------------------------------------------------------------
${(\![{\Box\psi}]\!)} (\alpha,z)$ = ${[\![{\Box \psi}]\!]}^{[1]}(\alpha,z)$
= $\bigwedge_{z' \in Z_A} [{[\![{\Box \psi}]\!]}(z') \to (E(z',z) \to \alpha)]$
= $\bigwedge_{z' \in Z_A} [f_{z'}(\Box \psi) \to (E(z',z) \to \alpha)]$
----------------------------------- --- -------------------------------------------------------------------------------
To show that ${(\![{\Box \psi}]\!)}(\alpha, z) \leq u_z(\Box\psi) \to \alpha$ we just need to find $z' \in Z_A$ such that $f_{z'}(\Box \psi) \to (E(z',z) \to \alpha) \leq u_z(\Box \psi) \to \alpha$. Define $f_{\Box \psi} \colon \mathbf{Fm}_{0\ast} \to \mathbf{A}$ by $$f_{\Box \psi}(\chi) = \begin{cases} 1 & \text{ if } \Box \psi \vdash \chi \\ 0 & \text{otherwise.}\end{cases}$$ Now let $z'=(f_{\Box \psi}, u_{\bot})$ where $u_{\bot}$ is as defined in the base case. Clearly $f_{z'}(\Box \psi)=1$ and so $f_{z'}(\Box \psi) \to (E(z',z) \to \alpha) = E(z',z) \to \alpha$. Now
----------- -------- ---------------------------------------------------------------------------------------
$E(z',z)$ = $\bigwedge_{{\varphi}\in \mathbf{Fm}_{0\ast}} (f_{z'}({\varphi}) \to u_z({\varphi}))$
= $\bigwedge_{\Box \psi \vdash \chi} (f_{z'}(\chi) \to u_z(\chi))$
= $\bigwedge_{\Box \psi \vdash \chi} (1\to u_z(\chi)) $
= $\bigwedge_{\Box \psi \vdash \chi} u_z(\chi) $
$\geq$ $u_z(\Box \psi)$.
----------- -------- ---------------------------------------------------------------------------------------
The last inequality follows from the fact that $u_z$ is order-preserving. Since $\to$ is order-reversing in the first coordinate we have $$f_{z'}(\Box \psi) \to (E(z',z) \to \alpha) = E(z',z) \to \alpha \leq u_z(\Box \psi) \to \alpha.$$
To show that $ u_z(\Box\psi) \to \alpha\leq {(\![{\Box \psi}]\!)}(\alpha, z)$, we must show that for all $z' \in Z_A$ we have $u_z(\Box\psi) \to \alpha\leq f_{z'}(\Box \psi) \to (E(z',z) \to \alpha)$. By definition we have $E(z',z)=\bigwedge_{{\varphi}\in \mathbf{Fm}_{0\ast}} (f_{z'}({\varphi}) \to u_{z}({\varphi})) \leq f_{z'}(\Box \psi)\to u_{z}(\Box \psi)$. Therefore the desired inequality will follow if we can show $$u_z(\Box\psi) \to \alpha\leq f_{z'}(\Box \psi) \to ((f_{z'}(\Box \psi)\to u_{z}(\Box \psi)) \to \alpha).$$ By residuation, the above inequality is equivalent to $$u_z(\Box\psi) \to \alpha\leq [f_{z'}(\Box \psi) \otimes (f_{z'}(\Box \psi)\to u_{z}(\Box \psi))] \to \alpha.$$ This last inequality is true if $$f_{z'}(\Box \psi) \otimes (f_{z'}(\Box \psi)\to u_{z}(\Box \psi)) \leq u_z(\Box\psi),$$ which is an instance of a tautology in residuated lattices.
As a consequence of the truth lemma we get:
1. The axiom $\Box\bot\vdash \bot$ is valid in $\mathbb{G}_{\mathbf{Fm}_{0\ast}}$.
2. The axiom $\Box p \vdash p$ is valid in $\mathbb{G}_{\mathbf{Fm}_{1\ast}}$.
1\. Clearly, since the axiom above does not contain atomic propositions, its validity on $\mathbb{G}_{\mathbf{Fm}_{0\ast}}$ coincides with its satisfaction on $\mathbb{M}_{\mathbf{Fm}_{0\ast}}$. Hence, $\mathbb{M}_{\mathbf{Fm}_{0\ast}}\models \Box\bot\vdash \bot$ iff ${(\![{\bot}]\!)}\subseteq {(\![{\Box\bot}]\!)}$ iff ${(\![{\bot}]\!)}(\alpha, z)\leq {(\![{\Box\bot}]\!)}(\alpha, z)$ for every $(\alpha, z)\in Z_X$, iff (by Lemma \[lemma:truthlemma\]) $u_z(\bot)\to \alpha\leq u_z(\Box \bot)\to \alpha$, iff $u_z(\Box\bot)\leq u_z(\bot)$, which is true, since $u_z: \mathbf{Fm}_{0\ast}\to \mathbf{A}$ is order-preserving.
2\. By Proposition \[prop:correspondence\].3 it is enough to show that $E \subseteq R_{Box}$ in $\mathbb{G}_{\mathbf{Fm}_{1\ast}}$, i.e. that $E(z,z') \leq R_{\Box}(z,z')$ for all $z,z' \in Z$.[^17] Recall that $$E(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm_{\ast}}}(f_z({\varphi})\to u_{z'}({\varphi}))$$ and $$R_{\Box}(z, z') = \bigwedge_{{\varphi}\in \mathbf{Fm_{\ast}}}(f_z({\varphi})\to u^{-\Box}_{z'}({\varphi})).$$ Hence it is enough to show that $u_{z'}({\varphi}) \leq u^{-\Box}_{z'}({\varphi}) := \bigwedge \{u_{z'}(\psi) \mid {\varphi}\vdash \Box \psi \}$, i.e. that if ${\varphi}\vdash \Box \psi$, then $u_{z'}({\varphi}) \leq u_{z'}(\psi)$. Indeed, ${\varphi}\vdash \Box \psi$ and $\Box \psi \vdash \psi$ imply ${\varphi}\vdash \psi$ and hence $u_{z'}({\varphi}) \leq u_{z'}(\psi)$ follows from the monotonicity of $u_{z'}$.
\[thm:completeness\] For $i \in \{0,1\}$, the normal $\mathcal{L}$-logic $\mathbf{L}_i$ is sound and complete w.r.t. its corresponding class of graph-based $\mathbf{A}$-frames.
Consider an $\mathcal{L}$-sequent ${\varphi}\vdash \psi$ that is not derivable in $\mathbf{L}_i$. In order to show that $\mathbb{M}_{\mathbf{Fm}_{i}} \not\models {\varphi}\vdash \psi$ (Definition \[def:Sequent:True:In:Model\]), we need to show that ${[\![{{\varphi}}]\!]}(z)\nleq{[\![{\psi}]\!]}(z) = {(\![{\psi}]\!)}^{[0]}(z)$ for some $z\in Z$. Consider the proper filter $f_{{\varphi}}$ and complement of proper ideal $u_{\psi}$ given by $$f_{{\varphi}}(\chi) = \left\{\begin{array}{ll}
1 &\text{if } {\varphi}\vdash \chi\\
0 &\text{if } {\varphi}\not \vdash \chi
\end{array} \right.$$ and $$u_{\psi}(\chi) = \left\{\begin{array}{ll}
0 &\text{if } \chi \vdash \psi\\
1 &\text{if } \chi \not \vdash \psi
\end{array} \right.$$ Then $\bigwedge_{\chi\in \mathbf{Fm}_i}(f_{{\varphi}}(\chi)\to u_{\psi}(\chi)) = 1$, for else there would have to be a $\chi \in \mathbf{Fm}_i$ such that $f_{{\varphi}}(\chi) = 1$ and $u_{\psi}(\chi) = 0$, which would mean that ${\varphi}\vdash \chi$ and $\chi \vdash \psi$ and hence that ${\varphi}\vdash \psi$, in contradiction with the assumption that ${\varphi}\vdash \psi$ is not derivable. It follows that $z: = (f_{{\varphi}}, u_{\psi})$ is a state in the canonical model $\mathbb{M}_{\mathbf{Fm}_i}$. By the Truth Lemma, ${[\![{{\varphi}}]\!]}(z) = f_{{\varphi}}({\varphi}) = 1$, and moreover
-------- --------------------------------------------------------------------------------------
${(\![{\psi}]\!)}^{[0]}(z)$
= $\bigwedge_{(\alpha, z')\in Z_X}{(\![{\psi}]\!)}(\alpha, z')\to (E(z, z')\to\alpha)$
$\leq$ ${(\![{\psi}]\!)}(0, z)\to (E(z, z)\to 0)$
= $(u_{\psi}(\psi) \to 0)\to (E(z, z)\to 0)$
= $(0 \to 0)\to (1\to 0)$
= 0,
-------- --------------------------------------------------------------------------------------
which proves the claim.
[^1]: In the crisp setting, a [*formal context*]{} [@ganter2012formal], or [*polarity*]{}, is a structure $\mathbb{P} = (A, X, I)$ such that $A$ and $X$ are sets, and $I\subseteq A\times X$ is a binary relation. Every such $\mathbb{P}$ induces maps $(\cdot)^\uparrow: \mathcal{P}(A)\to \mathcal{P}(X)$ and $(\cdot)^\downarrow: \mathcal{P}(X)\to \mathcal{P}(A)$, respectively defined by the assignments $B^\uparrow: = I^{(1)}[B]$ and $Y^\downarrow: = I^{(0)}[Y]$. A [*formal concept*]{} of $\mathbb{P}$ is a pair $c = ({[\![{c}]\!]}, {(\![{c}]\!)})$ such that ${[\![{c}]\!]}\subseteq A$, ${(\![{c}]\!)}\subseteq X$, and ${[\![{c}]\!]}^{\uparrow} = {(\![{c}]\!)}$ and ${(\![{c}]\!)}^{\downarrow} = {[\![{c}]\!]}$. The set $L(\mathbb{P})$ of the formal concepts of $\mathbb{P}$ can be partially ordered as follows: for any $c, d\in L(\mathbb{P})$, $$c\leq d\quad \mbox{ iff }\quad {[\![{c}]\!]}\subseteq {[\![{d}]\!]} \quad \mbox{ iff }\quad {(\![{d}]\!)}\subseteq {(\![{c}]\!)}.$$ With this order, $L(\mathbb{P})$ is a complete lattice, the [*concept lattice*]{} $\mathbb{P}^+$ of $\mathbb{P}$. Any complete lattice $\mathbb{L}$ is isomorphic to the concept lattice $\mathbb{P}^+$ of some polarity $\mathbb{P}$.
[^2]: \[footnote: abbreviations\] Applying this notation to a graph ${\mathbb{X}}= (Z, E)$, we will abbreviate $E^{[0]}[u]$ and $E^{[1]}[f]$ as $u^{[0]}$ and $f^{[1]}$, respectively, for each $u, f$ as above, and write $u^{[01]}$ and $f^{[10]}$ for $(u^{[0]})^{[1]}$ and $(f^{[1]})^{[0]}$, respectively. Then $u^{[0]} = I_{E}^{(0)}[u] = u^{\downarrow}$ and $f^{[1]} = I_{E}^{(1)}[f] = f^{\uparrow}$, where the maps $(\cdot)^\downarrow$ and $(\cdot)^\uparrow$ are those associated with the polarity $\mathbb{P_X}$.
[^3]: However, in this paper we do not intend to set up a systematic correlation between significance levels and truth values. The values chosen in the example below are only supposed to be intuitively plausible.
[^4]: It is interesting to notice that this basic environment naturally captures the idea that evidence is laden with theory: that is, we can think of each database $z_j$ as being constructed on the basis of the theory corresponding to set of variables $X_j$ associated with $z_j$.
[^5]: Source: `https://www.sciencehistory.org/ distillations/magazine/counting-calories`
[^6]: Source: `https://idmprogram.com/ can-make-thininsulin-hormonal-obesity-v/`
[^7]: Proviso: for the purpose of keeping this example simple, we are oversimplifying the hormonal response theory.
[^8]: Notice that, although the setting of [@graph-based-wollic] is crisp, the correspondence results in [@graph-based-wollic] Proposition 4 remain verbatim the same when passing to the many-valued setting. This is a phenomenon already observed in the correspondence theory for many-valued logics [@CMR].
[^9]: It can be checked that this valuation is stable.
[^10]: For higher values of $\beta$, these values increase accordingly. It can be checked that the valuation as specified in the table, is stable.
[^11]: Since $R_A: = E$, the modal operators $\Box_A$ and $\Diamond_A$ coincide with the identity on $\mathbb{X}^+$.
[^12]: These modal operators can be used to reason about “comparative studies" which span across all databases and establish the degree of similarity between each databases and the focal one.
[^13]: We do not mean ‘best scores’ in absolute terms, but in relative terms: that is, a bad hypothesis will score low on every database, but it will still get its higher scores on databases that are maximally compatible with the theory in the language of which the hypothesis is formulated.
[^14]: Recall that for any set $W$, the $\mathbf{A}$-[*subsethood*]{} relation between elements of $\mathbf{A}$-subsets of $W$ is the map $S_W:\mathbf{A}^W\times \mathbf{A}^W\to \mathbf{A}$ defined as $S_W(f, g) :=\bigwedge_{w\in W}(f(w)\rightarrow g(w)) $. If $S_W(f, g) =1$ we also write $f\subseteq g$.
[^15]: We write $\mathbb{M}$ for $\mathbb{M}_{\mathbf{Fm}_{\ast}}$ when $\mathbf{Fm}_{\ast}$ is clear from the context.
[^16]: This is the only place in the completeness proof where we are using an extra assumption.
[^17]: Recall the setting of the present section is different from that in Appendix \[sec:correspondence\] (cf. the discussion at the beginning of the present section). However, it is not difficult to verify that Proposition \[prop:correspondence\].3 holds verbatim also in the present setting, by suitably adapting the chains of equivalences used in the proof.
|
---
abstract: 'Succinct data structures give space-efficient representations of large amounts of data without sacrificing performance. They rely on cleverly designed data representations and algorithms. We present here the formalization in Coq/SSReflect of two different tree-based succinct representations and their accompanying algorithms. One is the Level-Order Unary Degree Sequence, which encodes the structure of a tree in breadth-first order as a sequence of bits, where access operations can be defined in terms of Rank and Select, which work in constant time for static bit sequences. The other represents dynamic bit sequences as binary balanced trees, where Rank and Select present a low logarithmic overhead compared to their static versions, and with efficient insertion and deletion. The two can be stacked to provide a dynamic representation of dictionaries for instance. While both representations are well-known, we believe this to be their first formalization and a needed step towards provably-safe implementations of big data.'
author:
- Reynald Affeldt
- Jacques Garrigue
- Xuanrui Qi
- Kazunari Tanaka
bibliography:
- 'bib.bib'
title: Proving tree algorithms for succinct data structures
---
Introduction
============
Succinct data structures [@navarro2016] represent combinatorial objects (such as bit vectors or trees) in a way that is space-efficient (using a number of bits close to the information theoretic lower bound) and time-efficient (i.e., not slower than classical algorithms). This topic is attracting all the more attention as we are now collecting and processing large amounts of data in various domains such as genomes or text mining. As a matter of fact, succinct data structures are now used in software products of data-centric companies such as Google [@kudo2011efficient].
The more complicated a data structure is, the harder it is to process it. A moment of thought is enough to understand that constant-time access to bit representations of trees requires ingenuity. Succinct data structures therefore make for intricate algorithms and their importance in practice make them perfect targets for formal verification [@tanaka2016icfem].
In this paper, we tackle the formal verification of tree algorithms for succinct data structures. We first start by formalizing basic operations such as counting () and searching () bits in arrays. This is an important step because the theory of these basic operations sustains most succinct data structures. Next, we formally define and verify a bit representation of trees called Level-Order Unary Degree Sequence (hereafter LOUDS). It is for example used in the Mozc Japanese input method [@kudo2011efficient]. The challenge there is that this representation is based on a level-order (i.e., breadth-first) traversal of the tree, which is difficult to describe in a structural way. Nonetheless, like most succinct data structures, this bit representation only deals with static data. Last, we further explore the advanced topic of dynamic bit vectors. The implementation of the latter requires to combine static bit vectors from succinct data structures with classical balanced trees. We show in particular how this can be formalized using a flavor of red-black trees where the data is in the leaves (rather than in the internal nodes, as in most functional implementations).
In both cases, our code can be seen as a verified functional specification of the algorithms involved. We were careful to use the right abstractions in definitions so that this specification could be easily translated to efficient code using arrays. For LOUDS we only rely on the and functions; we have already provided an efficient implementation for [@tanaka2016icfem]. For dynamic bit vectors, while the code we present here is functional, it closely matches the algorithms given in [@navarro2016]. We did prove all the essential correctness properties, by showing the equivalence of each operation with its functional counterpart (functions on inductive trees for LOUDS, and on sequences of bits for dynamic bit vectors).
Independently of this verified functional specification, we identify two technical contributions, that arised while doing this formalization. One is the notion of level-order traversal up to a path in a tree, which solves the challenge of performing path-induction on a level-order traversal. Another is our experience report with using small-scale reflection to prove algorithms on inductive data, which we hope could provide insights to other researchers.
The rest of this paper is organised as follows. The next section introduces and . Section \[sec:louds\] describes our formalization of LOUDS, including the notion of level-order traversal up to a path. Section \[sec:dynamic\_vectors\] uses trees to represent bit vectors, defining not only and , but also insertion and deletion. Section \[sec:proof\_techniques\] reports on our experience. Section \[sec:related\_work\] compares with the litterature, and Section \[sec:conclusion\] concludes.
Two functions to build them all {#sec:rank_and_select}
===============================
The and functions are the most basic blocks to form operations on succinct data structures: counts bits while searches for their position. The rest of this paper (in particular Sect. \[sec:louds\_functions\] and Sect. \[sec:dynamic\_vectors\]) explains how they are used in practice to perform operations on trees. In this section, we just briefly explain their formalization and theory.
Counting bits with {#sec:counting_rank}
-------------------
The function counts the number of elements (most often bits) in the prefix (i.e., up to some index ) of an array . It can be conveniently formalized using standard list functions:
``` {.ssr}
Definition rank b i s := count_mem b (take i s).
```
Figure \[fig:rankselect\] provides several examples of queries. The mathematically-inclined reader can alternatively[^1] think of as the cardinal of the number of indices of bits in a tuple :
``` {.ssr}
Definition Rank (i : nat) (B : n.-tuple T) :=
#|[set k : [1,n] | (k <= i) && (tacc B k == b)]|.
```
In this definition, denotes[^2] sequences of of length ; is the type of integers between $1$ and $n$; and accesses the tuple counting the indices from $1$.
(a0) at (0,0) [[*bitstring*]{}]{}; (a1) at (40pt,0) [1001]{}; (a2)\[right of=a1,xshift=-7pt\] [0100]{}; (a3)\[right of=a2,xshift=-7pt\] [1110]{}; (a4)\[right of=a3,xshift=-7pt\] [0100]{}; (a5)\[right of=a4\] [1101]{}; (a6)\[right of=a5,xshift=-7pt\] [0000]{}; (a7)\[right of=a6,xshift=-7pt\] [1111]{}; (a8)\[right of=a7,xshift=-7pt\] [0100]{}; (a9)\[right of=a8\] [1001]{}; (a10)\[right of=a9,xshift=-7pt\] [1001]{}; (a11)\[right of=a10,xshift=-7pt\] [0100]{}; (a12)\[right of=a11,xshift=-7pt\] [0100]{}; (a13)\[right of=a12\] [0101]{}; (a14)\[right of=a13,xshift=-7pt\] [0101]{}; (a15)\[right of=a14,xshift=-12pt\] [10]{}; in [a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15]{} [ ($(\from.west)+(+0.1,0)$) – +(0,-0.6); at ($(\from.west)+(+0.2,-0.5)$) ; ]{} ; ($(a1.north west)+(0,5pt)$) – ($(a15.north east)+(0,5pt)$) ; (lbl) at ($(a7.north)+(10pt,10pt)$) [length $n = 58$]{} ;
($(a1.south west)+(0,-17pt)$)–($(a1.south east)+(0,-17pt)$) ; at ($(a1.south)+(0,-25pt)$) ; at ($(a1.south)+(0,-35pt)$) ;
($(a1.south west)+(0,-15pt)$)–($(a9.south east)+(0,-15pt)$) ; at ($(a6.south)+(0,-24pt)$) ; at ($(a6.south)+(0,-34pt)$) ;
($(a1.south west)+(0,-8pt)$)–($(a15.south east)+(0,-8pt)$) ; at ($(a12.south)+(0,-23pt)$) ; at ($(a12.south)+(0,-33pt)$) ;
Finding bits with {#sec:finding_select}
------------------
Intuitively, compared with , performs the converse operation: it returns the index of the -th occurrence of , i.e., the [*minimum*]{} index whose rank is . It is conveniently specified using the construct of the library [@ssrman]:
``` {.ssr}
Variables (T : eqType) (b : T) (n : nat).
Lemma select_spec (i : nat) (B : n.-tuple T) :
exists k, ((k <= n) && (Rank b k B == i)) || (k == n.+1) && (count_mem b B < i).
Definition Select i (B : n.-tuple T) := ex_minn (select_spec i B).
```
With this definition, returns the index of the sought bit [*plus one*]{} (counting indices from $0$); ing the $0^{\rm th}$ bit always returns $0$; when no adequate bit is found, returns the size of the array plus one. The need for the $0$ case explains why it makes sense to return indices starting from $1$. Figure \[fig:rankselect\] provides several examples to illustrate the function.
The theory of and {#sec:theory_rank_select}
------------------
The and functions are used in a variety of applications whose formal verification naturally calls for a shared library of lemmas. Our first work is to identify and isolate this theory. Its lemmas are not all difficult to prove. For instance, the fact that cancels directly follows from the definitions:
``` {.ssr}
Lemma SelectK n (s : n.-tuple T) (j : nat) :
j <= count_mem b s -> Rank b (Select b j s) s = j.
```
However, as often with formalization, it requires a bit of work and try-and-error to find out the right definitions and the right lemmas to put in the theory of and . For example, how appealing the definition of above may be, proving its equivalence with a functional version such as
``` {.ssr}
Fixpoint select i (s : seq T) : nat :=
if i is i.+1 then
if s is a :: s' then (if a == b then select i s' else select i.+1 s').+1
else 1
else 0.
```
turns out to add much comfort to the development of related lemmas.
As a consequence, the resulting theory of and sometimes looks technical and we therefore refer the reader to the source code [@compact] to better appreciate its current status. Here, we just provide for the sake of completeness the definition of two derived functions that are used later in this paper.
### The and functions {#sec:succ}
In a bitstring, the function computes the position of the next $0$-bit or $1$-bit. It will find its use when dealing with LOUDS operations in Sect. \[sec:children\]. More precisely, given a bitstring , returns the index of the next following index . This operation is achieved by a combination of and . First, a call to counts the number of ’s up to index ; let be this number. Second, a call to searches for the $\coqin{(N+1)}^{\rm th}$ [@navarro2016 p. 89]:
``` {.ssr}
Definition succ (b : T) (s : seq T) y := select b (rank b y.-1 s).+1 s.
```
In particular, there is no in the set $\{ \coqin{s}_i \,|\, \coqin{y} \leq i < \coqin{succ b s y} \}$:
``` {.ssr}
Lemma succP b n (s : n.-tuple T) (y : [1, n]) :
b \notin \bigcup_(i : [1,n] | y <= i < succ b s y) [set tacc s i].
```
Conversely, the function computes the position of the previous bit and will find its use in Sect. \[sec:parent\]. It is similar to , so that we only provide its definition for reference:
``` {.ssr}
Definition pred (b : T) (s : seq T) y := select b (rank b y s) s.
```
LOUDS formalization {#sec:louds}
===================
Operationally, a LOUDS encoding consists in turning a tree into an array of bits via a level-order traversal. Figure \[fig:louds\] provides a concrete example. The resulting array is the ordered concatenation of the bit representation of each node. Each node is represented by a list of bits that contains as many $1$-bits as there are children and that is terminated by a $0$-bit.
The significance of the LOUDS encoding is that it preserves the branching structure of the tree without pointers, making for a compact representation in memory. Moreover, read-only operations can be implemented using and , which can be implemented in constant-time.
We explain how we formalize the LOUDS encoding in Sect. \[sec:louds\_formalization\] and how we formally verify the correctness of operations on trees built out of and in Sect. \[sec:louds\_functions\].
[0.1]{}
child [ node [level 1]{} [ child [ node [level 2]{} [ child [ node [level 3]{} ]{} ]{} ]{} ]{} ]{} ;
[0.44]{}
child [ node\[xshift=-2mm\][2]{} child [ node\[xshift=-2mm\][5]{} ]{} child [ node\[xshift=2mm\][6]{} ]{} ]{} child [ node [3]{} ]{} child [ node [4]{} child [ node [7]{} ]{} child [ node [8]{} child [node [10]{} ]{} ]{} child [ node [9]{} ]{} ]{} ;
\
[level-ordered enumeration of nodes:\
$\begin{array}{|l|l|l|l|}
\hline
\mathrm{level}\;0 & \mathrm{level}\;1 & \mathrm{level}\;2 & \mathrm{level}\;3 \\
\hline
1 & 2,3,4 & 5,6,7,8,9 & 10 \\
\hline
\end{array}$]{}
[0.44]{}
child [ node [1110]{} child [ node\[xshift=-2mm\][110]{} child [ node\[xshift=-2mm\][0]{} ]{} child [ node\[xshift=2mm\][0]{} ]{} ]{} child [ node [0]{} ]{} child [ node [1110]{} child [ node [0]{} ]{} child [ node [10]{} child [node [0]{} ]{} ]{} child [ node [0]{} ]{} ]{} ]{} ;
\
[Bitstring encoding:\
$\begin{array}{|l|l|l|l|l|}
\hline
& \mathrm{level}\;0 & \mathrm{level}\;1 & \mathrm{level}\;2 & \mathrm{level}\;3 \\
\hline
10 & 1110 & 11001110 & 000100 & 0 \\
\hline
\end{array}$]{}
LOUDS encoding formalized in {#sec:louds_formalization}
-----------------------------
We define arbitrarily-branching trees by an inductive type:
``` {.ssr}
Variable A : Type.
Inductive tree := Node : A -> seq tree -> tree.
Definition forest := seq tree.
Definition children_of_node : tree -> forest := ...
Definition children_of_forest : forest -> forest := flatten \o map children_of_node.
```
where is the function composition operator (i.e., $\circ$), and where the type is the type of labels. We also introduce the abbreviation for a list of trees, and functions to obtain children. With this definition of trees, a leaf is a node with an empty list of children. For example, the tree of Fig. \[fig:louds\] becomes in :
``` {.ssr}
Definition t : tree nat := Node 1
[:: Node 2 [:: Node 5 [::]; Node 6 [::]];
Node 3 [::];
Node 4 [:: Node 7 [::];
Node 8 [:: Node 10 [::]];
Node 9 [::]]].
```
### Height-recursive level-order traversal {#sec:lo_traversal}
The intuitive definition of level-order traversal iterates on a forest, returning first the toplevel nodes of the forest, then their children (applying ), etc. We parameterize the definition with an arbitrary function for generality.
``` {.ssr}
Variables (A B : Type) (f : tree A -> B).
Fixpoint lo_traversal' n (s : forest A) :=
if n is n'.+1 then map f s ++ lo_traversal' n' (children_of_forest s) else [::].
Definition lo_traversal t := lo_traversal' (height t) [:: t].
```
The parameter is filled here with the maximum height of the forest, meaning that we iterate just the right number of times for the forest to become empty.
Yet, this definition is not fully satisfactory. One reason is that it is not structural: we are not recursing on a tree, but iterating on a forest, using its height as recursion index. Another one is that, as we will see in Sect. \[sec:louds\_functions\], the name [*level-order*]{} is misleading. For many proofs, we are not interested in complete traversal of the tree, level by level, but rather by partial traversal along a path in the tree, where the forest we consider actually overlaps levels.
### A structural level-order traversal {#sec:structural_lo_traversal}
At first it may seem that the non-structurality is inherent to level-order traversal. There is no clear way to build the sequence corresponding to the traversal of a tree from those of its children. However, Gibbons and Jones [@Gibbons91; @Jones93] showed that this can be achieved by splitting the output into a list of levels. One can combine two such structured traversals by [*zipping*]{} them, i.e., concatenating corresponding levels, and recover the usual traversal by flattening the list. Since concatenation of lists forms a monoid, zipping of traversals also forms a monoid.
``` {.ssr}
Variable (A : Type) (e : A) (M : Monoid.law e).
Fixpoint mzip (l r : seq A) : seq A := match l, r with
| (l1::ls), (r1::rs) => (M l1 r1) :: mzip ls rs
| nil, s | s, nil => s
end.
Lemma mzipA : associative mzip.
Lemma mzip1s s : mzip [::] s = s.
Lemma mzips1 s : mzip s [::] = s.
Canonical mzip_monoid := Monoid.Law mzipA mzip1s mzips1.
```
Here , from the module of , denotes an operator together with its neutral element (here ) and the required monoidal equations, which are also satisfied by .
We now define our traversal by instantiating to the concatenation monoid. The resulting is a structure of type that can be used as an operator of type enjoying the properties of a monoid.
``` {.ssr}
Variables (A : eqType) (B : Type) (f : tree A -> B).
Definition mzip_cat := mzip_monoid (cat_monoid B).
Fixpoint level_traversal t :=
[:: f t] :: foldr (mzip_cat \o level_traversal) nil (children_of_node t).
Lemma level_traversalE t :
level_traversal t =
[:: f t] :: \big[mzip_cat/nil]_(i <- children_of_node t) level_traversal i.
Definition lo_traversal_st t := flatten (level_traversal t).
Theorem lo_traversal_stE t : lo_traversal_st t = lo_traversal f t.
```
To let recognize the structural recursion, we have to use the recursor in the definition of . Yet, the intended equation is the one expressed by , i.e., first output the image of the node, and then combine the traversals of the children. Then can be proved equal to the previously defined . Deforestation can furthermore improve the efficiency of [^3].
### LOUDS encoding {#sec:louds_encoding}
Finally, the LOUDS encoding is obtained by instantiating with an appropriate function (called the of a node), and flattening once more:
``` {.ssr}
Definition node_description s := rcons (nseq (size s) true) false.
Definition children_description t := node_description (children_of_node t).
Definition LOUDS t := flatten (lo_traversal_st children_description t).
```
Here, adds to the end of the sequence , while creates a sequence consisting of copies of . Note that we chose here not to add the usual “[10]{}” prefix [@navarro2016 p. 212] shown in Fig. \[fig:louds\], as it appeared to just complicate definitions. It can be easily recovered by adding an extra root node, as “[10]{}” is the representation of a node with 1 child.
For example, we can recover the encoding displayed in Fig. \[fig:louds\] with this definition of :
``` {.ssr}
Lemma LOUDS_t : LOUDS (Node 0 [:: t]) =
[:: true; false; true; true; true; false;
true; true; false; false; true; true; true; false;
false; false; false; true; false; false; false].
```
We can also prove some properties of this representation, such as its size:
``` {.ssr}
Lemma size_LOUDS t : size (LOUDS t) = 2 * number_of_nodes t - 1.
```
This is an easy induction, remarking that is a morphism between and .
LOUDS functions using and {#sec:louds_functions}
--------------------------
In this section, we formalize LOUDS functions and prove their correctness. These functions are essentially built out of and . Their correctness statements establish a correspondence between operations on trees defined inductively and operations on their LOUDS encoding. We start by explaining how we represent positions in trees and then comment on the formal verification of LOUDS operations using representative examples.
### Positions in trees {#sec:positioning}
For a tree defined inductively, we represent the position of a node as usual: using a , i.e., a list that records the branches taken from the root to reach the node. For example, the position of the node $8$ in Fig. \[fig:a\_sample\_tree\] is . Not all positions are valid; we sort out the valid ones by means of the predicate (definition omitted for brevity).
(0,0) – (1.8,-1.8) – (0.6,-1.8) – (0.6,-2.2) – (-2.2,-2.2) – cycle ; (0.6,-1.8) – (1.8,-1.8) – (2.2,-2.2) – (0.6,-2.2) – cycle; (0.6,-2.2) – (-2.2,-2.2) – (-2.6,-2.6) – (0.6,-2.6) – cycle; (0,0) – (-3,-3) – (3,-3) – cycle ; (0,0) – (0,-0.6) – (0.6,-1.2) – (0.6,-1.8);
(2.5,0) rectangle (3,-0.2) node\[anchor=west,yshift=4pt\] [ traversed tree]{} ; (2.5,-0.5) rectangle (3,-0.7) node\[anchor=west,yshift=3pt\][ fringe]{} ;
In contrast, the position of nodes in the LOUDS encoding is not immediate. We define it as the length of the generated LOUDS up to the corresponding path. To do that, we first need to define a notion of level-order traversal up to a path, which collects all the nodes preceding the one referred by that path (which need not be valid):
``` {.ssr}
Definition split {T} n (s : seq T) := (take n s, drop n s).
Variables (A : eqType) (B : Type) (f : tree A -> B).
Fixpoint lo_traversal_lt (s : forest A) (p : seq nat) : seq B := match p, s with
| nil, _ | _, nil => nil
| n :: p', t :: s' =>
let (fs, ls) := split n (children_of_node t) in
map f (s ++ fs) ++ lo_traversal_lt (ls ++ children_of_forest (s' ++ fs)) p'
end.
```
This new traversal appears to be the key to clean proofs of LOUDS properties. In a previous attempt using the height-recursive level-order traversal of Sect. \[sec:lo\_traversal\], proofs were unwieldy (one needed to manually set up inductions) and lemmas did not arise naturally. We expect this new traversal to have applications to other uses of level-order traversal.
This definition may seem scary, but it closely corresponds to the imperative version of level-order traversal, which relies on a queue: to get the next node, take it from the front of the queue, and add its children to the back of the queue. We define our traversal so that the node we have reached is the one at the front of the queue . To move to its $n^{\rm th}$ child (indices starting from 0), we first output all the nodes in the queue, and its children up to the previous one, and proceed with a new queue containing the remaining children (starting from the $n^{\rm th}$) and the children of the other nodes we have just output. Figure \[fig:generating\] shows how the traversal progresses. The point is that as soon as the queue spans all the fringe of the traversed tree, it is able to generate the remainder of the traversal. We can verify that indeed qualifies as a level-order traversal by proving that its output converges to the full level-order traversal when the length of reaches the height of the tree:
``` {.ssr}
Theorem lo_traversal_ltE (t : tree A) (p : seq nat) :
size p >= height t -> lo_traversal_lt [:: t] p = lo_traversal_st f t.
```
We also introduce a function that computes the fringe of the traversal up to , i.e., the forest generating the remainder of the traversal.
``` {.ssr}
Fixpoint lo_fringe (s : forest A) (p : seq nat) : forest A := ...
Lemma lo_traversal_lt_cat s p1 p2 :
lo_traversal_lt s (p1 ++ p2) =
lo_traversal_lt s p1 ++ lo_traversal_lt (lo_fringe s p1) p2.
```
We omit the definition but the lemma states exactly this property. It decomposes the traversal generated by a path, allowing induction from either end of the list representing the position.
Using the path-indexed traversal function, we can directly obtain the index of a node in the level-order traversal of a tree:
``` {.ssr}
Definition lo_index (s : forest A) (p : seq nat) := size (lo_traversal_lt id s p).
```
The expression counts the number of nodes in the traversal of before the position . Similarly, we give an alternative definition of the LOUDS encoding, and use it to map a position in the tree to a position in its encoding (i.e., the index of the first bit of the representation of a node):
``` {.ssr}
Definition LOUDS_lt s p := flatten (lo_traversal_lt children_description s p).
Definition LOUDS_position s p := size (LOUDS_lt s p).
```
Here the position in the whole tree is obtained as , but we can also compute relative positions by using where is a generating forest whose front node is the one we start from. Note that both and return indices starting from 0.
For example, in Fig. \[fig:louds\], the position of the node $8$ is in the inductively defined tree and in the LOUDS encoding:
``` {.ssr}
Definition p8 := [:: 2; 1].
Eval compute in LOUDS_position [:: Node 0 [:: t]] (0 :: p8). (* 17 *)
```
Finally, here is one of the essential lemmas for proofs on LOUDS, which relates and using :
``` {.ssr}
Lemma LOUDS_position_select s p p' : valid_position (head dummy s) p ->
LOUDS_position s p = select false (lo_index s p) (LOUDS_lt s (p ++ p')).
```
Namely if the index of is $n$, then its position in the LOUDS encoding is the index of its $n^{\rm th}$ 0-bit (recall that counts indices starting from 1). Here allows us to complete to a path of sufficient length, so that converges to .
### Number of children using {#sec:children}
As a first example, let use formalize the LOUDS function that counts the number of children of a node. For a tree defined inductively, this operation can be achieved by first walking down the path to the node and then looking at the list of its children.
``` {.ssr}
Fixpoint subtree (t : tree) (p : seq nat) :=
if p is n :: p' then subtree (nth t (children_of_node t) n) p' else t.
Definition children t p := size (children_of_node (subtree t p)).
```
To count the number of children of a node using a LOUDS encoding, one first has to notice that each node is terminated by a $0$-bit. Given such a $0$-bit (or equivalently the corresponding node), one can find the number of children by computing the distance with the next $0$-bit [@navarro2016 p. 214]. Finding this bit is the purpose of the function of Sect. \[sec:succ\]:
``` {.ssr}
Definition LOUDS_children (B : bitseq) (v : nat) : nat := succ false B v.+1 - v.+1.
```
The offset comes from the fact computes on indices starting from 1.
is correct because, when applied to the of a position , it produces the same result as the function :
``` {.ssr}
Theorem LOUDS_childrenE (t : tree A) (p p' : seq nat) :
let B := LOUDS_lt [:: t] (p ++ 0 :: p') in
valid_position t p -> LOUDS_children B (LOUDS_position [:: t] p) = children t p.
```
### Parent and child node using and {#sec:parent}
A path in a tree defined inductively gives direct ancestry information. In particular, removing the last index denotes the parent, and adding an extra index denotes the corresponding child. It takes more ingenuity to find parent and child using a LOUDS representation and functions from Sect. \[sec:rank\_and\_select\] alone. The idea is to count the number of nodes and branches up to the position in question [@navarro2016 p. 215]. More precisely, given a LOUDS position , let be the number of nodes up to ( computes this number). Then, looks for the -th down-branch, which is the branch leading to the node of position . Last, this branch belongs to a node whose position can be recovered using the function (from Sect. \[sec:succ\]). Reciprocally, one computes the $i^{\rm th}$ child by using and . This leads to the following definitions:
``` {.ssr}
Definition LOUDS_parent (B : bitseq) (v : nat) : nat :=
let j := select true (rank false v B) B in pred false B j.
Definition LOUDS_child (B : bitseq) (v i : nat) : nat :=
select false (rank true (v + i) B).+1 B.
```
One can check the correctness of and as follows. Consider a node reached by the path . Its parent is the node reached by the path , and conversely it is the $i^{\rm th}$ child of this node. We can formally prove that the LOUDS position of (respectively ) and the position computed by (respectively ) coincide:
``` {.ssr}
Variables (t : tree A) (p p' : seq nat) (i : nat).
Hypothesis HV : valid_position t (rcons p i).
Let B := LOUDS_lt [:: t] (rcons p i ++ p').
Theorem LOUDS_parentE :
LOUDS_parent B (LOUDS_position [:: t] (rcons p i)) = LOUDS_position [:: t] p.
Theorem LOUDS_childE :
LOUDS_child B (LOUDS_position [:: t] p) i = LOUDS_position [:: t] (rcons p i).
```
The approach that we explained so far shows how to carry out the formal verification of the LOUDS operations that are listed in [@navarro2016 Table 8.1]. However, how useful they may be for many big-data applications, these operations assume static compact data structures. The next section explains how to extend our approach to deal with dynamic structures.
Dynamic bit vectors {#sec:dynamic_vectors}
===================
In some applications bit vectors need to support dynamic operations—not just static queries. We formalize such , and implement and verify “dynamic operations” on them: inserting a bit into a bit vector, and deleting a bit from one.
In Sect. \[sec:representing\_dynamic\_vectors\], we explain the data structure that allows for an efficient implementation of dynamic operations. In Sect. \[sec:basic\_queries\], we formalize the and queries. Sections \[sec:insert\] and \[sec:delete\] are dedicated to the formalization of the more difficult insertion and deletion.
Representing dynamic bit vectors {#sec:representing_dynamic_vectors}
--------------------------------
The choice of representation for dynamic bit vectors is motivated by complexity considerations. Insertion into a linear array has time complexity $O(n)$, but we can improve this by using a balanced binary search tree to represent the bit array, which enables us to handle insertions in at most $O(w)$ time, with a trade-off of $O(n/w)$ bits of extra space, where $w$ is a parameter controlling the width of each tree node and should no more than the size of a native machine word in bits[^4] [@navarro2016]: i.e., for a typical 64-bit machine, we would set $w$ to 32 or 64.
On a side note, balanced binary trees are certainly not the most compact data structure that could be used here. In fact, various data structures with better complexity have been designed [@navsad14; @ramanetal01], however those structures are complicated and are unlikely to offer practical improvements over the structure presented here [@navarro2016]. As a result, we choose to work only with balanced binary trees, which are much easier to reason about.
\[every node/.style=[draw,level distance=8mm]{},level 1/.style=[sibling distance=40mm,level distance=9mm]{},level 2/.style=[sibling distance=18mm]{}\]
child [ node\[fill=black, circle, label=180:[$\substack{\text{num} = 8 \\ \text{ones} = 2}$]{}\] child [ node[10000010]{} ]{} child [ node[00000100]{} ]{} ]{} child [ node\[fill=black, circle, label=0:[$\substack{\text{num} = 16 \\ \text{ones} = 5}$]{}\] child [ node\[fill=red, circle, label=[\[xshift=-1cm\]70:[$\substack{\text{num} = 8 \\ \text{ones} = 2}$]{}]{}\] child [ node[00001010]{} ]{} child [ node[00001011]{} ]{} ]{} child [ node[10000001]{} ]{} ]{};
In our formalization of the dynamic bit vector’s algorithms, we use a red-black tree as our balanced tree structure. Each node holds a color and meta-data about the bit vector, and each leaf holds a (i.e., list-based) bit array. Following Navarro [@navarro2016], we store two natural numbers in each node: the size and the rank of the left subtree (recorded as “num” and “ones” in Fig. \[fig:dynamicbitvector\]).
``` {.ssr}
Inductive color := Red | Black.
Inductive btree (D A : Type) : Type :=
| Bnode of color & btree D A & D & btree D A
| Bleaf of A.
Definition dtree := btree (nat * nat) (seq bool).
```
Our first step is to formalize the structural invariant of our tree representation of bit vectors, which is required to prove the correctness of queries and updates on it. It states that the numbers encoded in each node are the left child’s size and rank, and that leaves contain a number of bits between and .
``` {.ssr}
Variables low high : nat. (* instantiated as w^2 / 2 and w^2 * 2 *)
Fixpoint wf_dtree (B : dtree) := match B with
| Bnode _ l (num, ones) r => [&& num == size (dflatten l),
ones == count_mem true (dflatten l),
wf_dtree l & wf_dtree r]
| Bleaf arr => low <= size arr < high
end.
```
Here, the function defines the semantics of our tree representation of a bit vector () by converting it to a flat representation of that vector:
``` {.ssr}
Fixpoint dflatten (B : dtree) := match B with
| Bnode _ l _ r => dflatten l ++ dflatten r
| Bleaf s => s
end.
```
Verifying basic queries {#sec:basic_queries}
-----------------------
The basic query operations can be easily defined via traversal of the tree. We implement the queries , $\select_{1}$, and $\select_{0}$ as the functions , , and . For example, is implemented as follows, using the (static) function from Sect. \[sec:counting\_rank\]:
``` {.ssr}
Fixpoint drank (B : dtree) (i : nat) := match B with
| Bnode _ l (num, ones) r =>
if i < num then drank l i else ones + drank r (i - num)
| Bleaf s => rank true i s
end.
```
We prove that our function indeed computes the query using a custom induction principle , corresponding to the predicate :
``` {.ssr}
Lemma drankE (B : dtree) i : wf_dtree B -> drank B i = rank true i (dflatten B).
Proof. move=> wf; move: B wf i. apply: dtree_ind. (* ... *) Qed.
```
Note that our implementation is only correct on well-formed trees.
The formalization and verification of the queries proceed along the same lines.
Implementing and verifying insertion {#sec:insert}
------------------------------------
Insertion is significantly harder to implement than static queries. We need to maintain the invariant on the size of the leaves, which means that we have to split a leaf if it becomes too big, and in that case we may need to rebalance the tree, to maintain the red-black invariant, updating the meta-data on the way.
We translate the algorithm given by Navarro [@navarro2016] directly into . Here, is the maximum number of bits a leaf can contain before it needs to be split up:
``` {.ssr}
Definition dins_leaf s b i :=
let s' := insert1 s b i in (* insert element b in sequence s at position i *)
if size s + 1 == high then
let n := size s' %/ 2 in let sl := take n s' in let sr := drop n s' in
Bnode Red (Bleaf _ sl) (n, count_mem true sl) (Bleaf _ sr)
else Bleaf _ s'.
Fixpoint dins (B : dtree) b i : dtree := match B with
| Bleaf s => dins_leaf s b i
| Bnode c l d r =>
if i < d.1 then balanceL c (dins l b i) r (d.1.+1, d.2 + b)
else balanceR c l (dins r b (i - d.1)) d
end.
Definition dinsert (B : btree D A) b i : btree D A :=
match dins B b i with
| Bleaf s => Bleaf _ s
| Bnode _ l d r => Bnode Black l d r
end.
```
recurses on the tree, searching for the leaf where the insertion must be done, calling then , which inserts a bit in the leaf, eventually splitting it if required. On its way back, calls balancing functions and to maintain the red-black invariant. We omit the code of the balancing functions (see [@compact]). Like the standard version, they fix imbalances possibly occurring on the left and on the right, respectively, but they must also adjust the meta-data in the nodes. is a simple wrapper over that completes the insertion by painting the root black. The real definitions are more abstract [@compact]; we chose to instantiate them in this paper for readability.
Verifying requires verifying three different properties: must preserve the data, maintain the structural invariants of the tree, and return a balanced red-black tree. Properties and are related, in that the latter is required by the former.
``` {.ssr}
Notation wf_dtree_l := (wf_dtree low high).
Definition wf_dtree' t := if t is Bleaf s then size s < high else wf_dtree_l t.
Lemma wf_dtree_dtree' t : wf_dtree_l t -> wf_dtree' t.
Lemma wf_dtree'_dtree t : wf_dtree' t -> wf_dtree 0 high t.
Lemma dinsertE (B : dtree) b i :
wf_dtree' B -> dflatten (dinsert B b i) = insert1 (dflatten B) b i.
Lemma dinsert_wf (B : dtree) b i : wf_dtree' B -> wf_dtree' (dinsert B b i).
```
A subtle point here is that we may start from a tree formed of a single small leaf, i.e., a leaf smaller than . To handle this situation we introduce , which does not enforce the lower bound on this single leaf. This new predicate is entailed by the original invariant (it removes one check), but interestingly it also entails it if we set the lower bound to 0. Since the queries of Sect. \[sec:basic\_queries\] were proved with abstract lower and upper bounds, their proofs are readily usable through this weakening. However, we need to use when we prove properties of , as it modifies the tree.
Proving and involves no theoretical difficulty. We explain in Sect. \[sec:proof\_techniques\] some techniques to write short proofs: about 100 lines in total for both properties, including lemmas for and , which involve large case analyses.
Property about never breaking the red-black tree invariant is notoriously more challenging. More importantly, we want to eliminate cases where the “height balance” at a node is broken. It is easy to model the property that no red node has a red child; the “height balance” property is modeled using the black-depth. We can thus model the red-black tree invariant with a recursive function that takes as arguments the “color context” (the color of the parent’s node) and the black-depth of the node :
``` {.ssr}
Fixpoint is_redblack (B : dtree) (ctxt : color) (bh : nat) := match B with
| Bleaf _ => bh == 0
| Bnode c l _ r => match c, ctxt with
| Red, Red => false
| Red, Black => is_redblack l Red bh && is_redblack r Red bh
| Black, _ => (bh > 0) && is_redblack l Black bh.-1
&& is_redblack r Black bh.-1
end end.
```
To show that preserves the red-black tree property, we define and prove a number of weaker structural lemmas that are basically equivalent to stating that a tree returned by is structurally valid if the root is painted black. We do not describe the proof in detail because the technique is well-known [@okasaki98] and has been formalized in multiple sources (see Sect. \[sec:related\_work\]). Using these weaker lemmas, we can prove the following structural validity lemma:
``` {.ssr}
Lemma dinsert_is_redblack (B : dtree) b i n :
is_redblack B Red n -> exists n', is_redblack (dinsert B b i) Red n'.
```
Deletion: searching for invariants {#sec:delete}
----------------------------------
\[every node/.style=[draw,sibling distance=12mm,level distance=10mm]{}\]
;
(LHS4) – (RHS4) node\[midway, above, draw=none\] [delete 2${}^\mathrm{nd}$ bit]{}; (LHS5) – (RHS5) node\[midway, above, draw=none\] [delete 2${}^\mathrm{nd}$ bit]{};
Deletion in dynamic bit vectors is difficult for two reasons. One is that, in order to maintain the upper and lower bounds on the size of leaves, which is required to attain simultaneously space and time efficiency, deleting a bit in a leaf may require some rearrangement of the surrounding nodes. Figure \[fig:deletion\] shows the result of deleting a bit in a leaf of the tree, when this leaf has already the smallest allowed size. This can be resolved by borrowing a bit from a sibling (left case), or merging two siblings (right case), but depending on the configurations of nodes, this may require to first rotate the tree.
The other is that deletion in a functional red-black tree is a complex operation [@kahrs01], and that finding how to adapt the invariants of the litterature to our specific case proved to be non-trivial. Therefore, we took a twofold approach. First, we searched for invariants in a concrete tree structure with invariants encoded using dependent types. Then, we removed dependent types and implemented and proved its correctness (more details in Sect. \[sec:proof\_techniques\]).
Contrary to insertion, knowing the color of the modified child is not sufficient to rebalance its parent correctly after deletion, and recompute its meta-data. We need to propagate two more pieces of information: whether the black-height decreased ( below), and the meta-data corresponding to the deleted bit (). We encapsulate these in a “tree state”:
``` {.ssr}
Record deleted_dtree: Type := MkD { d_tree :> dtree; d_down: bool; d_del: nat*nat }.
```
Note that is automatically coerced to .
Now, we can define in the natural way, but we need to take care about balance operations and invariants on the size of leaves. Specifically, the balance operations must be reimplemented as and , which need to satisfy the following invariants, i.e., the resulting “balanced” tree is *deleted-red-black* (i.e., a red-black tree, either with the same black height, or with a black root and decreased black height), given that the unproblematic subtree is red-black, while the unbalanced one is deleted-red-black.
``` {.ssr}
Definition balanceL' (c:color)(l:deleted_dtree)(d:nat*nat)(r:dtree):deleted_dtree :=
Definition balanceR' (c:color)(l:dtree)(d:nat*nat)(r:deleted_dtree):deleted_dtree :=
Definition is_deleted_redblack tr (c : color) (bh : nat) :=
if d_down tr then is_redblack tr Red bh.-1 else is_redblack tr c bh.
Lemma balanceL'_Black_deleted_is_redblack l r n c :
0 < n -> is_deleted_redblack l Black n.-1 -> is_redblack r Black n.-1 ->
is_deleted_redblack (balanceL' Black l r) c n.
Lemma balanceL'_Red_deleted_is_redblack l r n :
is_deleted_redblack l Red n -> is_redblack r Red n ->
is_deleted_redblack (balanceL' Red l r) Black n.
(* similar statements with respect to balanceR' *)
```
Regarding leaves, we need special processing in the base cases of , as illustrated in Fig. \[fig:deletion\]. might have to “borrow” a bit from a sibling of a target leaf or combine target siblings (possibly after a rotation), to preserve the size invariants. Afterwards, will recursively rebalance the whole .
Thus we implement (as ), and prove its correctness as follows:
``` {.ssr}
Fixpoint ddel (B : dtree) (i : nat) : deleted_dtree := ...
Lemma ddeleteE B i : wf_dtree' B -> dflatten (ddel B i) = delete (dflatten B) i.
Lemma ddelete_wf (B : dtree) n i :
is_redblack B Black n -> i < dsize B -> wf_dtree' B -> wf_dtree' (ddel B i).
Lemma ddelete_is_redblack B i n :
is_redblack B Red n -> exists n', is_redblack (ddel B i) Red n'.
```
These statements are variants of the properties , and of Sect. \[sec:insert\]. The proofs are complicated by the huge number of cases, handled using the proof techniques discussed in the next section.
Using small-scale reflection with inductive data {#sec:proof_techniques}
================================================
The small-scale reflection approach is known to be beneficial for mathematical proofs [@mathcompbook]. However, while tactics are now widely used in the community, it is not always clear how to write proofs of programs using inductive data structures in an idiomatic style, in particular in presence of deep case analysis.
In the first part of the paper, concerning level-order traversal, the question is not so acute, as the induction principle we need for LOUDS is not structural on the shape of trees, but rather on paths, represented as lists, which are already well supported by the library. Thus the question was the more traditional one of which definitions to use, so that we can obtain natural lemmas. This proved to be a time consuming process, which led to gradually build a library of lemmas, resulting in proofs that match the intuition, using almost only case analysis and rewriting.
However, the second part, about dynamic bit vectors, uses heavily structural induction on binary trees, and required developing some proof techniques to streamline the proofs.
A basic idea of small-scale reflection is to use recursive Boolean predicates (i.e., recursive computable functions) rather than inductive propositions. We have already presented two examples: and . Properly designed, they allow one to prune case analysis by reducing to on impossible cases. On the other hand, they do not decompose naturally in inductive proofs, which led us first to apply a standard technique: define a specialized induction principle for trees satisfying ( in Sect. \[sec:basic\_queries\]). Using it, the correctness of static queries and non-structural modification operations (i.e., setting and clearing of bits) were easy to prove, as the case analysis was trivial.
Properties of , , and their auxiliary functions are trickier to prove, as they require complex case analyses and delicate re-balancing of branches. Nevertheless, we essentially applied the same principle of solving goals through direct case analysis. With this approach, the correctness lemmas (which state that our operations are semantically correct) were largely automated, consistent with prior research [@nipkow16]. The structural lemmas were harder to prove, mainly due to the sheer number of cases involved and the complexity of invariants. Our proofs proceed by first applying case analysis to the tree up to the required depth, and then decomposing all assumptions to repeatedly rewrite the goal using them until it is solved. This proof pattern is captured by the following tactic:
``` {.ssr}
Ltac decompose_rewrite :=
let H := fresh "H" in case/andP || (move=>H; rewrite ?H ?(eqP H)).
```
It is reminiscent of the tactic, a generic tactic for intuitionistic logic which breaks both hypotheses and goals into pieces; here we rather rely on rewriting inside Boolean conjunctions to solve goals piecewise. For , this approach instantly finishes most of our proofs, especially those about red-black tree invariants; the few cases that require manual treatment being usually handled in one single . This is true for most auxiliary functions of too, with one caveat: where has us generate a dozen cases, requires hundreds. To cope with this, we had first to decompose the case analysis in steps, solving most cases on the way, which means losing some simplicity to speed up proof search. The proof is still mostly automatic: apply , and throw in relevant lemmas. When possible, it appears that using instead of speeds up by a factor of 2 or more, which matters when the lemma takes more than 1 minute to prove. We have only 3 such time-consuming case analyses, one for each invariant. Among the 12 lemmas involved in proving the invariants, only the inductive proof of well-formedness for seems to show the limit of this approach, as it required specific handling for each case of the function definition.
Contents ( in [dynamicredblack.v]{}) Lines of code Lines of proof
-------------------------------------- --------------- ----------------
Definitions (, ) 19 18
Queries () 38 58
Insertion (, ) 65 208
Set/clear a bit () 25 120
Deletion (, ) 98 215
: Implementation of dynamic bit vectors (see Table \[tab:implementation\_overview\] for the whole implementation)[]{data-label="tab:implementation_rb_trees"}
File in [@compact] Section in this paper
----------------------- ----------------------------------------------------------------------------------------
[rankselect.v]{} Sections \[sec:counting\_rank\], \[sec:finding\_select\], \[sec:theory\_rank\_select\]
[predsucc.v]{} Sect. \[sec:succ\]
[tree\_traversal.v]{} Sections \[sec:lo\_traversal\], \[sec:structural\_lo\_traversal\]
[louds.v]{} Sections \[sec:louds\_encoding\], \[sec:louds\_functions\]
[dynamicredblack.v]{} Sect. \[sec:dynamic\_vectors\]
: Formalization overview [@compact] (see Table \[tab:implementation\_rb\_trees\] for the details about dynamic bit vectors)[]{data-label="tab:implementation_overview"}
For comparison, Table \[tab:implementation\_rb\_trees\] provides the size of code and proof required for each of our proof script. This does not include lemmas about the list-based reference implementation. Note that we count all Boolean predicates used to model properties as proofs.
The proofs of and , which we did not describe here, may seem relatively verbose. We prove the same properties (a,b,c) as in Sect. \[sec:insert\], but the number of lines hides a disparity between proofs of (a) correctness and (c) red-blackness, which are almost immediate, as the structure of the tree is unchanged, and (b) invariants of the meta-data, for which switching a bit requires to propagate the difference back to the root, with extra local invariants.
Last, we mention our experience with alternative approaches. In parallel with our developement using small-scale reflection, we attempted to formalize dynamic bit vectors using dependent types, where all invariants are encoded in the type of the data itself. While this guarantees that we never forget an invariant, difficulties with the [@sozeauthesis] environment led us to write some functions using tactics [@compact]. As written in Sect. \[sec:delete\], this direct connection between code and proof actually helped us discover some tricky invariants. However, the resulting code does not lend itself to further analysis, hence our choice here to stick to a more conventional separation between code and proof. We did eventually succeed in re-implementing the dependently-typed version using the environment, but at the price of very verbose definitions [@compact].
Table \[tab:implementation\_overview\] gives an at-a-glance overview of our entire Coq development, with a list of files and their corresponding sections in this paper.
Related work {#sec:related_work}
============
has been used to formalize a constant-time, $o(n)$-space function that was furthermore extracted to efficient OCaml code [@tanaka2016icfem] and C code [@tanaka2017jip]. This work focuses on the query for static bit arrays while our work extends the toolset for succinct data structures with more queries (, , etc.) and dynamic structures.
The functions and of Sect. \[sec:structural\_lo\_traversal\] match functions given in squiggle notation in related work by Jones and Gibbons [@Jones93]. In this work, the function of Sect. \[sec:structural\_lo\_traversal\] also appears and is called “long zip with plussle”. To the best of our knowledge, the function is original to our work.
Larchey-Wendling and Matthes recently studied the certification and extraction of breadth-first traversals [@Larchey2019mpc]. They too define , but then prove it equivalent to a queue based algorithm, which they extract to efficient OCaml code. Their goal is orthogonal to ours, as for succinct data structures what matters is not the efficiency of the traversal, but the correctness of the parent/child navigation functions, which by definition require a constant number of queries.
One may use any kind of balanced binary tree to represent dynamic bit vectors [@navarro2016]. There are many purely-functional balanced binary search trees, such as AVL trees [@avl] and weight-balanced trees [@adams93], but purely functional red-black trees [@kahrs01; @okasaki98] are most widely studied and preferred by us. As a matter of fact, they have already been formalized in [@appel11; @chlipala13; @filliatre2004esop], Agda [@oster11], and Isabelle [@nipkow16].
We had to re-implement red-black trees due to the difference of stored contents. Above formalizations are intended to represent sets, and maintain the ordering invariant. Our trees represent vectors, and maintain both that the contents (as concatenation of the leaves) are unchanged, and that meta-data in inner nodes is correct (see Sect. \[sec:representing\_dynamic\_vectors\]). Still, we found many hints in related work. For example, in Sect. \[sec:insert\] about insertion, the balancing functions use Okasaki’s well-known purely functional balance algorithm [@okasaki98], and we formulate our invariants and propositions similarly to above formalizations.
There are now many proofs of programs that use , but we could not find much discussion trying to synthesize the new techniques put at work. Sergey et al. used for teaching [@sergey_pnp; @sergey_nanevski_introducing], observing benefits for clarity and maintainability, but also giving examples of custom tactics needed to prove programs. Gonthier et al. [@gonthier_ziliani_nanevski_dreyer_2013] have shown how, in some cases, one can avoid relying on ad hoc tactics through an advanced technique involving overloading of lemmas. The techniques we describe in Sect. \[sec:proof\_techniques\], while more rudimentary, are simple and efficient, yet we have not seen them described elsewhere.
Conclusion {#sec:conclusion}
==========
We reported on an effort to formalize succinct data structures. We started with a foundational theory of the and functions for counting and searching bits in immutable arrays. Using this theory, we formalized a standard compact representation of trees (LOUDS) and proved the correctness of its basic operations. Last, we formalized dynamic bit vectors: an advanced topic in succinct data structures.
Our work is a first step towards the construction of a formal theory of succinct data structures. We already overcame several technical difficulties while dealing with LOUDS trees: it took much care to find suitable recursive traversals and to sort out the off-by-one conditions when specifying basic operations. Similarly, the formalization of dynamic vectors could not be reduced to the matter of extending conservatively an existing formalization of balanced trees: we needed to re-implement them to accommodate specific invariants.
As for future work, we plan to enable code extraction for the functions we have been verifying, and prove their complexity, so as to complete previous work [@tanaka2016icfem] and ultimately achieve a formally verified implementation of succinct data structures. We have already shown that the LOUDS representation of a tree with $n$ nodes uses just $2n$ bits of data. For the LOUDS operations, constant time complexity is a direct consequence of their being implemented using a constant number of and operations. For dynamic bit vectors, we will first need to properly define a framework for space and time complexity.
[^1]: This is actually the definition that appears in Wikipedia at the time of this writing.
[^2]: The notation is a idiom for a suffix operator. Similarly we use and for successor and predecessor.
[^3]: See in [@compact].
[^4]: The complexity bounds referred to in this section are dependent on the model of computation used. Here, we assume that we are working with a sequential RAM machine, where we have $O(w) = O(\log n)$ as we can only address at most $2^w$ bits of memory.
|
---
abstract: 'We use fluctuating hydrodynamics to evaluate the enhancement of thermally excited fluctuations in laminar fluid flow using plane Couette flow as a representative example. In a previous publication \[J. Stat. Phys. [**144**]{} (2011) 774\] we derived the energy amplification arising from thermally excited wall-normal fluctuations by solving a fluctuating Orr-Sommerfeld equation. In the present paper we derive the energy amplification arising from wall-normal vorticity fluctuation by solving a fluctuating Squire equation. The thermally excited wall-normal vorticity fluctuations turn out to yield the dominant contribution to the energy amplification. In addition, we show that thermally excited streaks, even in the absence of any externally imposed perturbations, are present in laminar fluid flow.'
author:
- 'J. M. Ortiz de Zárate'
- 'J. V. Sengers'
title: '[Hydrodynamic Fluctuations in Laminar Fluid Flow. II. Fluctuating Squire Equation]{}'
---
Introduction
============
The presence of gradients, such as temperature gradients, concentration gradients, or velocity gradients, always causes non-equilibrium enhancements of thermal fluctuations that are spatially long ranged, even when the system is far away from any hydrodynamic instability [@DorfmanKirkpatrickSengers; @BOOK]. The present paper is part of a detailed study of the nature of thermally excited fluctuations in laminar fluid flow using plane Couette flow as a representative example. It has been verified that fluctuating hydrodynamics, originally developed for thermal fluctuations in equilibrium states [@LandauLifshitz; @FoxUhlenbeck1], can be extended to deal with thermal fluctuations in non-equilibrium states [@BOOK]. In fluctuating hydrodynamics the usual deterministic hydrodynamic equations are supplemented with random dissipative fluxes of thermal (natural) origin. In the case of laminar fluid flows one has to consider a random stress tensor to account for intrinsic thermal noise that always will be present. This noise will be amplified by the presence of a velocity gradient. Energy amplification induced by the flow has attracted the attention of many investigators. In many of the studies reported in the literature the noise is not of thermal origin and does not obey a fluctuation-dissipation relation [@FarrellIoannou; @FarrellIoannou2; @BamiehDahleh; @JovanovicBamieh; @EckhardtPandit]. Fluctuating hydrodynamics provides a systematic method for assessing the nature of spontaneous fluctuations in laminar flow induced by intrinsic noise. The application of fluctuating hydrodynamics to shear flows has been initiated by some previous investigators, but without considering confinement effects [@TremblayEtAl; @LD85; @LDD89; @LD02; @WS03]. However, the long-ranged nature of the fluctuations is highly anisotropic and for certain directions of the wave vector the fluctuations encompass the entire fluid system, so that boundary effects will affect these fluctuations.
Previously we have derived the appropriate fluctuating hydrodynamics equations for laminar fluid flow. Specifically, we have shown how the thermally excited wall-normal velocity fluctuations can be described by a stochastic Orr-Sommerfeld equation [@miCouette] and the thermally excited vorticity fluctuations by a stochastic Squire equation [@miCouette2]. Accounting for realistic boundary conditions we obtained solutions of the stochastic Sommerfeld and Squire equations based on semi-quantitative Galerkin approximations[@miCouette; @miCouette2; @miJNNFM]. We now have derived more exact solutions of these stochastic equations in terms of an expansion of the eigenfunctions (hydrodynamic modes) of the hydrodynamic operator. The more exact solution of the stochastic Orr-Sommerfeld equation for the wall-normal velocity fluctuations has been presented in a previous article in this series, to be referred to as paper I [@miORR]. We found that the flow-induced enhancement of the wall-normal velocity fluctuations and the resulting energy amplification increases with the Reynolds number Re approximately as Re$^2$. The actual enhancement of the velocity fluctuations strongly depends on the wave number. For large wave numbers (in the bulk of the fluid), this enhancement varies as the fourth power of the inverse wave number, independent of any boundary conditions. For small wave numbers the enhancement vanishes as the square of the wave number due to the presence of boundaries. Our previous approximate solution based on a Galerkin approximation [@miCouette] did reproduce the correct dependence of the non-equilibrium enhancement of the fluctuations on the wave number but underestimated the magnitude of the enhancement at intermediate wave numbers [@miORR].
The present paper is concerned with an analysis of the solution of the stochastic Squire equation for the wall-normal vorticity fluctuations. We shall proceed as follows. In Sect. \[S02\] we recall the expressions for the stochastic Orr-Sommerfeld and Squire equations in terms of suitable dimensionless variables. The Squire equation for the wall-normal vorticity fluctuations includes a coupling with the Orr-Sommerfeld equation for the wall-normal velocity fluctuations. Hence, the solution of the stochastic Squire equation for the wall-normal vorticity fluctuations to be obtained in the present paper will depend on the solution of the stochastic Orr-Sommerfeld equation for the wall-normal velocity fluctuations obtained in our preceding paper [@miORR]. In Sect. \[S02B\] we describe the procedure for solving the stochastic Squire equation. For this purpose we expand the solution in terms of the eigenfunctions of the linear hydrodynamic operator associated with the Squire equation. In Sect. \[S3\] we derive the corresponding hydrodynamic modes and decay rates. In Sect. \[S04\] we then obtain the expressions for both the equilibrium and nonequilibrium contributions to the intensity of the vorticity fluctuations, while in Sect. \[S05\] we deduce the nonequilibrium energy amplification arising from these vorticity fluctuations. The nonequilibrium energy enhancement turns out to be proportional to the square of the Reynolds number. Specifically, we evaluate the nonequilibrium energy amplification associated with the vorticity fluctuations for fluctuations with wave vector in the spanwise direction which appears to be the most interesting case. We also find good agreement between the exact solution, obtained in this paper, and the semi-quantitative solution previously obtained in a Galerkin approximation [@miCouette2]. We conclude with some general comments in Sect. \[S07\].
Fluctuating Hydrodynamics of Shear Flow. The Stochastic Orr-Sommerfeld and Squire Equations\[S02\]
==================================================================================================
We consider a liquid with uniform temperature $T$ under incompressible laminar flow with uniform density $\rho$ between two horizontal plates separated by a distance $2L$. As in our previous publication [@miORR] we adopt a coordinate system with the $X$-axis in the streamwise direction, the $Y$-axis in the spanwise direction, and the $Z$-axis in the wall-normal direction [@DrazinReid]. Thus the mean flow velocity $\mathbf{v}_0=\{\dot{\gamma} z,0,0\}$ is in the $X$-direction with $\dot{\gamma}$ representing a constant shear rate in the $Z$-direction. The bounding upper plate, at the position $z=+L$, moves in the positive $X$-direction with velocity $\dot{\gamma}L$, while the lower plate, at the position $z=-L$, moves with the same velocity in the opposite direction. This flow configuration is commonly referred to as plane Couette flow. It is convenient to use a dimensionless position variable $\mathbf{r}$, measured in terms of the length $L$, a dimensionless time $t$ obtained by multiplying the actual time with the shear rate $\dot{\gamma}$, a dimensionless fluid velocity $\mathbf{v}$ in terms of the product $\dot{\gamma}L$, and a dimensionless stress tensor $\boldsymbol{\Pi}$ in terms of $\rho{L}^2\dot{\gamma}$.
We want to study velocity fluctuations around the stationary flow solution of the Navier-Stokes equation. Specifically, we are interested in fluctuations of thermal origin, *i.e.*, fluctuations resulting from the intrinsically stochastic nature of molecular motions. Such fluctuations are always present and are unavoidably linked to any dissipative processes that are present in the system, Newton’s law of viscosity in the present case. Fluctuating hydrodynamics provides a general and systematic framework for describing such thermal fluctuations, even when the system is in a non-equilibrium state [@DorfmanKirkpatrickSengers; @BOOK]. The idea is that the linear phenomenological laws representing dissipation in the system are to be supplemented with random dissipative fluxes (thermal noise), whose statistical properties are given by the fluctuation-dissipation theorem, see *e.g.* [@Kubo]. The goal is then to obtain the correlation functions of the fluctuating thermodynamic fields, velocity fluctuations $\delta\mathbf{v}$ in our case, in terms of the statistical properties of the thermal noise, the stochastic stress tensor $\delta\boldsymbol{\Pi}$ in our case. Implementing this procedure we have shown in previous publications that the fluctuations $\delta{v}_z$ of the wall-normal velocity component $v_z$ satisfy a stochastic Orr-Sommerfeld equation [@miCouette] $$\partial_{t}(\nabla^2\delta{v}_z)+
z~\partial _{x}(\nabla^2\delta{v}_z) - \frac{1}{\mathrm{Re}}
\nabla^{4}(\delta{v}_z) = -
\left\{\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\left[\boldsymbol{\nabla}\left(\delta\boldsymbol\Pi
\right)\right]\right\}_z,\label{Eq09A}$$ and the vorticity fluctuations $\delta{w}_z=\partial_x\delta{v}_y-\partial_y\delta{v}_x$ a stochastic Squire equation [@miCouette2] $$\partial_{t}(\delta{w}_z)+
z~\partial _{x}(\delta{w}_z) -
\partial_y\delta{v}_z - \frac{1}{\mathrm{Re}}
\nabla^{2}(\delta{w}_z) =
\left\{\boldsymbol{\nabla}\times\left[\boldsymbol{\nabla}\left(\delta\boldsymbol\Pi
\right)\right]\right\}_z. \label{Eq09B}$$ In these equations Re is the Reynolds number $$\text{Re}=\frac{\rho\dot{\gamma} L^2}{\eta}$$ with $\eta$ being the shear viscosity of the fluid. By combining Eqs. and with the incompressibility assumption, $\boldsymbol{\nabla}\cdot\delta\mathbf{v}$, one can obtain the three components of the fluctuating velocity field $\delta\mathbf{v}$.
The difference between the stochastic Orr-Sommerfeld equation and the stochastic Squire equation and their deterministic counterparts [@DrazinReid; @SchmidHenningson] is the presence of noise terms on the right-hand side (RHS). These additive noise terms appear as derivatives of the random stress $\delta\boldsymbol{\Pi}$; their correlation functions can be deduced from the fluctuation dissipation theorem which in this case reads [@BOOK] $$\langle \delta \Pi _{ij}(\mathbf{r},t)\cdot \delta \Pi _{kl}(\mathbf{r}%
^{\prime },t^{\prime })\rangle =2\tilde{S} \left(
\delta_{ik}\delta _{jl}+\delta _{il}\delta _{jk}\right) \delta (\mathbf{r}-\mathbf{%
r}^{\prime })~\delta (t-t^{\prime }).\label{FDT1}$$ In this equation $\tilde{S}$ is the dimensionless strength of the thermal noise [@miORR]: $$\label{Eq04B}
\tilde{S}=\frac{k_\text{B} T}{\dot{\gamma}^3 L^7}
\frac{\eta}{\rho_0^2} = \frac{k_\mathrm{B} T}{\rho_0 L^3}
\frac{1}{\dot{\gamma}^2 L^2} \frac{1}{\mathrm{Re}}.$$ where $k_\text{B}$ is Boltzmann’s constant. We note that the actual correlation function for the fluctuating stress tensor only depends on the properties of the fluid, namely temperature, density, and viscosity; the shear rate and the Reynolds number only appears in as a consequence of the manner in which the stress has been made dimensionless.
Equations and form a pair of coupled stochastic differential equations which have to be solved for the velocity and vorticity fluctuations subject to appropriate no-slip boundary conditions: $$\label{BC1}
\delta{v}_z(\mathbf{r},t)=\partial_z\delta{v}_z(\mathbf{r},t)=\delta{w}_z(\mathbf{r},t)=0,\hspace*{20pt}\text{at}~z=\pm{1}.$$
In our earlier work we have accounted for these boundary conditions by solving Eqs. and using a Galerkin method that allowed for analytical but approximate expressions for the correlation functions of the wall-normal velocity and vorticity fluctuations [@miCouette; @miCouette2]. We have summarized these approximate solutions of the two stochastic equations in a subsequently publication [@miJNNFM]. While the Galerkin-approximation technique enabled us to obtain relatively simple analytical results, the approximation is somewhat uncontrolled. Hence, we found it desirable to compare the approximate analytical solutions with exact numerical solutions that can be obtained through expansions in eigenfunctions of the hydrodynamic operators. We have implemented this project for the solution of the Orr-Sommerfeld stochastic equation , yielding the correlation function for the wall-normal velocity fluctuations in paper I in this series [@miORR]. In the present paper we analyze the solution of the stochastic Squire equation for the wall-normal vorticity fluctuations. There is an important difference between the Orr-Sommerfeld equation and the Squire equation . The solution of the Orr-Sommerfeld equation for the velocity fluctuations is independent of the solution of the Squire equation for the vorticity fluctuations. On the other hand, the Squire equation is coupled with the Orr-Sommerfeld equation though the presence of the term $\partial_y\delta{v}_z$ in Eq. . Physically it means that the wall-normal velocity fluctuations involve only a coupling between the same (viscous) hydrodynamic mode (at different wave numbers) to which we have referred as “self-coupling” [@miCouette]. However, the solution of the Squire equation for the wall-normal vorticity fluctuations is related to two mode-coupling mechanisms: a self-coupling between vorticity fluctuations and a cross-coupling between velocity and vorticity fluctuations. We shall recover our earlier observation [@miCouette2] that the self-coupling mechanism in the Squire equation determines the intensity of the vorticity fluctuations in equilibrium and that the cross-coupling mechanism in the Squire equation determines the nonequilibrium intensity enhancement of the wall-normal vorticity fluctuations.
To put our present work in the context of previous investigations of the effect of externally imposed stochastic forcing on shear flows we may mention the following. In their original papers, Farrell and Ionannou [@FarrellIoannou; @FarrellIoannou2] and Bamieh and Dahleh [@BamiehDahleh] introduced a stochastic forcing term directly into the RHS of the Orr-Sommerfeld and Squire equations and instead of the thermal forcing incorporated by us. More recently, Jovanovic and Bamieh [@JovanovicBamieh] introduced the forcing in the more basic Navier-Stokes equation, resulting in a more transparent interpretation of the random terms as actual forces. Moreover, Jovanovic and Bamieh [@JovanovicBamieh] allowed for some flexibility in the spatial spectrum of the noise, distinguishing between structured and unstructured noise. However, in spite of this flexibility, thermal noise was not included as a particular case considered by the authors [@JovanovicBamieh]. Indeed, from the flutuation-dissipation relation , it follows that the spatial spectrum of the resulting stochastic forcing in the RHS of Eqs. and is not contained in the expressions of Jovanovic and Bamieh [@JovanovicBamieh]. Hence, the results obtained here will differ from previous investigations.
In terms of the input-output nomenclature that, borrowed from the dynamics and control literature, has become popular lately [@JovanovicBamieh; @HwangCossu1]; we take as input the thermal noise and our output (through the linearized Navier-Stokes equations) is the wall-normal vorticity autocorrelation function.
Procedure for Solving the Stochastic Squire Equation\[S02B\]
============================================================
Just as the procedure used for solving the Sommerfeld-Orr equation [@miORR], to solve the Squire equation we apply a Fourier transform in time and in the horizontal $XY$-plane parallel to the walls: $$\left[\mathrm{i}\omega+\mathcal{H}\right]\cdot\delta{w}_z(\omega, \mathbf{q}_\parallel,z)=
\mathrm{i}q_y\delta{v}_z(\omega,\mathbf{q}_\parallel,z)+S_z(\omega,\mathbf{q}_\parallel,z),\label{Eq10}$$ where $\omega$ is the frequency of the fluctuations $\delta{w}_z$ and $\mathbf{q}_\parallel=\{q_x,q_y\}$ the corresponding wave vector in the plane parallel to the walls. In Eq. , $\mathcal{H}$ represents a linear hydrodynamic operator: $$\label{E49}
\mathcal{H}=\mathrm{i} zq_x - \dfrac{1}{\mathrm{Re}}\left(\partial_z^2-q_\parallel^2\right),$$ The RHS of Eq. contains two stochastic forcing terms. The first one, $\text{i}q_y\delta{v}_z$, accounts for a stochastic forcing originating from the wall-normal velocity fluctuations, which can be represented in terms of the exact solution of the Orr-Sommerfeld equation with thermal forcing derived in paper I [@miORR]. The second forcing term $S_z$ is given by the Fourier transform, in time and in the $XY$-plane, of the combination of derivatives of the random stress $\delta\boldsymbol{\Pi}$ in the RHS of Eq. : $$\label{E6}
S_z=\mathrm{i}\partial_z[q_x\delta\Pi_{zy}-q_y\delta\Pi_{zx}]-q_x^2\delta\Pi_{xy}+q_y^2\delta\Pi_{yx}+q_xq_y[\delta\Pi_{xx}-\delta\Pi_{yy}].$$ Since the hydrodynamic operator is linear we find the solution of Eq. by expanding in a set of right eigenfunctions, $R_N(\mathbf{q}_\parallel,z)$, or hydrodynamic modes as: $$\delta{\omega}_z(\omega,\mathbf{q}_\parallel,z)= \sum_{N=0}^\infty G_N(\omega,\mathbf{q}_\parallel)
~R_N(\mathbf{q}_\parallel,z),\label{EX33}$$ where the hydrodynamic modes are the solution of: $$\mathcal{H}\cdot{R}_N(\mathbf{q}_\parallel,z) =
\Gamma_N(\mathbf{q}_\parallel)~{R}_N(\mathbf{q}_\parallel,z)\label{E50}$$ with $\Gamma_N(\mathbf{q}_\parallel)$ being the corresponding eigenvalue or decay rate. In Eq. $R_N(\mathbf{q}_\parallel,z)$ must satisfy the boundary conditions $R_N(\mathbf{q}_\parallel,\pm1)=0$. In Eqs. - we anticipated the fact, to be discussed in more detail in Sect. \[S3\], that the right eigenvalue problem of the Squire operator, Eq. , has indeed an infinite numerable set of solutions. Next, to evaluate the coefficients $G_N(\omega,\mathbf{q}_\parallel)$ of the series expansion we use the property that the complex conjugates ${R}_N^*(\mathbf{q}_\parallel,z)$ are the left eigenfunctions of the Squire operator , with corresponding eigenvalues $\Gamma_N^*(\mathbf{q}_\parallel)$. Indeed, by using the boundary conditions it can be readily shown that the adjoint of the operator $\mathcal{H}$ is simply its complex conjugate and the left eigenfunction is just the complex conjugate of the right eigenfunction. As a consequence, the biorthogonality [@CourantHilbert] condition reads: $$\label{E60}
\int_{-1}^{1}
{R}_M(\mathbf{q}_\parallel,z)~{R}_N(\mathbf{q}_\parallel,z)
~ dz= B_N(\mathbf{q}_\parallel)~\delta_{NM}.$$ Equation determines the normalization of the eigenfunctions. Next, we evaluate the coefficients $G_N(\omega,\mathbf{q}_\parallel)$ in the expression for $\delta{\omega}_z(\omega,\mathbf{q}_\parallel,z)$ by substituting Eq. into Eq. . The resulting expression is then projected onto ${R}_M(\mathbf{q}_\parallel,z)$ and using the biortogonality condition , one readily obtains: $$\label{E10}
G_N^{\mathrm{R}}(\omega,\mathbf{q}_\parallel)=\frac{F_N(\omega,\mathbf{q}_\parallel)}
{B_N(\omega,\mathbf{q}_\parallel)[\mathrm{i}\omega +
\Gamma_N(\mathbf{q}_\parallel)]},$$ where $F_N(\omega,\mathbf{q}_\parallel)$ are the projections of the RHS of Eq. onto the ${R}_N(\mathbf{q}_\parallel,z)$ functions, namely: $$\label{E11}
F_N(\omega,\mathbf{q}_\parallel) = \int_{-1}^{1}
R_N(\mathbf{q}_\parallel,z)\left[\mathrm{i}q_y\delta{v}_z(\omega,\mathbf{q}_\parallel,z)+S_z(\omega,\mathbf{q}_\parallel,z) \right]~ dz.$$ For the evaluation of the vorticity fluctuations, we need the correlation functions $\langle{F}_N^*(\omega,\mathbf{q}_\parallel) \cdot{F}_M(\omega^\prime,\mathbf{q}_\parallel^\prime)\rangle$. These, in turn, can be deduced from the correlations:
\[E12\] $$\begin{aligned}
\langle\delta{v}_z^*(\omega,\mathbf{q}_\parallel,z) \cdot\delta{v}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle &= (2\pi)^3~\delta(\omega-\omega^\prime)~\delta(\mathbf{q}_\parallel-\mathbf{q}_\parallel^\prime)~C_{zz}(\omega,\mathbf{q}_\parallel,z,z^\prime),\label{E12A}\\
\langle{S}_z^*(\omega,\mathbf{q}_\parallel,z) \cdot{S}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle &=(2\pi)^3~\delta(\omega-\omega^\prime)~\delta(\mathbf{q}_\parallel-\mathbf{q}_\parallel^\prime)\notag\\&\hspace*{60pt}\times2\tilde{S}~q_\parallel^2(q_\parallel^2+\partial_z\partial_{z^\prime})\delta(z-z^\prime), \label{E12B}\\
\langle{S}_z^*(\omega,\mathbf{q}_\parallel,z) \cdot\delta{v}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle &=
\langle\delta{v}_z^*(\omega,\mathbf{q}_\parallel,z) \cdot{S}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle=0.\label{E12C}\end{aligned}$$
Equation represents the solution of the stochastic Orr-Sommerfeld equation obtained in paper I [@miORR] (see Eq. (24) in [@miORR]), where explicit expressions for the function $C_{zz}(\omega,\mathbf{q}_\parallel,z,z^\prime)$ in terms of the hydrodynamic modes and decay rates of the Orr-Sommerfeld operator have been presented. We shall not repeat those expression here, although they shall be used in some of the following calculations. Equation is a consequence of the fact that the random noise terms in the stochastic Orr-Sommerfeld and Squire equations are uncorrelated, see Eq. (18) in Ref. [@miCouette2]. Finally, Eq. , which is new in this paper, is readily obtained from the definition of ${S}_z(\omega,\mathbf{q}_\parallel,z)$ and the fluctuation-dissipation theorem for the random stress tensor. The expression for the prefactor $\tilde{S}$ is given by Eq. .
With the help of Eqs. , the correlations $\langle{F}_N^*(\omega,\mathbf{q}_\parallel)\cdot{F}_M(\omega^\prime,\mathbf{q}_\parallel^\prime)\rangle$ between the coefficients defined in Eq. , can be expressed as: $$\label{E13}
\langle{F}_N^*(\omega,\mathbf{q}_\parallel) \cdot{F}_M(\omega^\prime,\mathbf{q}_\parallel^\prime)\rangle= \left[\Xi^\text{(E)}_{NM}(\mathbf{q}_\parallel) + \Xi^\text{(NE)}_{NM}(\omega,\mathbf{q}_\parallel)\right]
(2\pi)^3~\delta(\omega-\omega^\prime)~\delta(\mathbf{q}_\parallel-\mathbf{q}_\parallel^\prime),$$ with mode-coupling coefficients
\[E14\] $$\begin{aligned}
\Xi^\text{(E)}_{NM}(\mathbf{q}_\parallel)&=2\tilde{S}~q_\parallel^2 \int_{-1}^{1}dz~R_N^*(q_\parallel,z) \left[q_\parallel^2-\partial_z^2\right] R_M(q_\parallel,z),\label{E14A}\\
\Xi^\text{(NE)}_{NM}(\omega,\mathbf{q}_\parallel)&=q_y^2\iint_{-1}^{1}\hspace*{-6pt}dz~dz^\prime~R_N^*(q_\parallel,z) ~C_{zz}(\omega,\mathbf{q}_\parallel,z,z^\prime)~R_M(q_\parallel,z^\prime).\label{E14B}\end{aligned}$$
The first expression, , is obtained from Eqs. and if one performs integrations by parts to move the derivatives in from the delta function to the hydrodynamic modes, and uses the boundary conditions $R_N(\mathbf{q}_\parallel,\pm1)=0$. In Eqs. and we have introduced superscripts (E) and (NE) to distinguish between the two contributions, anticipating the fact (to be discussed at length later) that the first set of mode-coupling coefficients will contribute only to the equilibrium equal-time vorticity fluctuations, while the second set of mode-coupling coefficients (due to the coupling of wall-normal velocity and vorticity fluctuations) contains the nonequilibrium amplification (enhancement) of those fluctuations.
Equation for $\Xi^\text{(E)}_{NM}(\mathbf{q}_\parallel)$ can be further transformed by integrating by parts while making use of the eigenvalue problem , to obtain an expression more useful for future use: $$\label{E15}
\Xi^\text{(E)}_{NM}(\mathbf{q}_\parallel)=\tilde{S}\text{Re}~q_\parallel^2\left[\Gamma_N^*(q_\parallel)+ \Gamma_M(q_\parallel)\right] \int_{-1}^{1}d\xi~R_N^*(q_\parallel,\xi)~R_M(q_\parallel,\xi),$$ where we renamed the integration variable as $\xi$.
Hydrodynamic Modes and Decay Rates\[S3\]
========================================
The hydrodynamic modes $R_N(\mathbf{q}_\parallel,z)$ of the operator $\mathcal{H}$ are obtained by solving Eq. with the appropriate boundary conditions, *i.e.*, $R_N(\mathbf{q}_\parallel,\pm1)=0$. In view of the definition of $\mathcal{H}$, the general solution of Eq. can be expressed as a linear combination of Airy functions. The boundary condition at $z=1$ can be easily accommodated by a convenient selection of the coefficients. Then the solution of Eq. satisfying the boundary condition at $z=1$ can be written in the form [@Romanov]: $$\begin{gathered}
\label{E72}
R_N(z) = \mathrm{Bi}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i})\right]~\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i}z)
\right] \\-
\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i})\right]~\mathrm{Bi}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i}z)
\right].\end{gathered}$$ In principle, the parameter $a_N$ in can be any complex number. The decay rate of the hydrodynamic mode is expressed in terms of this parameter $a_N$ as: $$\Gamma_N(\mathbf{q}_\parallel) = q_x~a_N(\mathbf{q}_\parallel)
+ \frac{q_\parallel^2}{\mathrm{Re}}.\label{Eq51}$$ For the hydrodynamic mode to satisfy the boundary condition at $z=-1$ we need to impose the condition: $$\begin{gathered}
\label{E74}
0=\mathrm{Bi}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i})\right]
~\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N+\mathrm{i})\right]
\\-\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i})\right]
~\mathrm{Bi}\left[(q_x\mathrm{Re})^{1/3}(a_N+\mathrm{i})\right].\end{gathered}$$ Because of the oscillatory character of the Airy functions, Eq. has an infinite numerable set of complex roots $a_N$. This fact has been anticipated in Eq. , and the index $N$ has been used throughout to distinguish among the various modes. In principle, the decay rates $a_N$ depend on the parallel wave vector $\mathbf{q}_\parallel=\{q_x,q_y\}$ and on the Reynolds number Re. However, as a consequence of the structure of Eq. , the decay rates only depend on the magnitude $q_\parallel$ of the wave vector $\mathbf{q}_\parallel$ in the plane parallel to the plates and on an effective Reynolds number $$\label{EN21}
\overline{\mathrm{Re}}=\mathrm{Re}\cos\varphi,$$ where $\varphi$ is the azimuthal angle of the wave vector $\mathbf{q}_\parallel$, measured with respect to the stream-wise $X$-direction. Hence, $q_x\text{Re}=q_\parallel\overline{\text{Re}}$. This simplification is commonly referred to as Squire symmetry [@DrazinReid]. As was discussed in detail in the preceding paper [@miORR], the eigenvalues and the eigenfunctions of the Orr-Sommerfeld hydrodynamic operator obey the same Squire symmetry. However, in contrast to the decay rates of the Orr-Sommerfeld hydrodynamic operator, the decay rates $a_N$ of the Squire hydrodynamic operator have an additional symmetry property, namely $$\label{EN22}
a_N(q_\parallel,\overline{\mathrm{Re}})=a_N(q_\parallel/\lambda,\overline{\mathrm{Re}}\lambda)$$ for any real parameter value $\lambda$. As a consequence, it is sufficient to determine the solutions of Eq. for a single value of the effective Reynolds number, such as $\overline{\mathrm{Re}}=1$.
In Fig. \[F1\] we show, as a function of the wavenumber $q_\parallel$, the decay rates of the Squire hydrodynamic operator $\mathcal{H}$ for $\overline{\mathrm{Re}}=1$. Use of the explicit exact expression for the eigenfunctions (and, hence, Eq. ) enables us for a much simpler computation of the data displayed in Fig. \[F1\] when compared to a direct numerical integration of Eq. performed by Gustavsson and Hultgren [@GustavssonHultgren]. As expected, the results are the same: see, *e.g.*, Fig. 1 of Ref. [@GustavssonHultgren]. For small $q_\parallel$ the decay rates are real numbers and a simple perturbative calculation allows to obtain the first terms in a series expansion in powers of $q_\parallel$, namely: $$\left.a_N(q_\parallel)\right|_{\overline{\text{Re}}=1} = \frac{N^2\pi^2}{4~q_\parallel} + \mathcal{O}(q_\parallel),\hspace*{50pt}\text{for}~N=1,2,3\dots$$ For larger $q_\parallel$ and depending on the order $N$, the decay rates merge in pairs of complex conjugate numbers. For even larger $q_\parallel\to\infty$ the real part of $a_N$ decay to zero as $q_\parallel^{-1/3}$, while the imaginary parts of each pair approach $\pm\mathrm{i}$. As explained above, this general landscape of decay rates is maintained for any Reynolds number because of the scaling relation . In particular, none of the decay rates becomes zero for any value of the wave vector or the Reynolds number. The same is true for the Orr-Sommerfeld equation and, as is well known, there is no linear hydrodynamic instability in plane Couette flow.
One difference with the landscape of eigenvalues of the Orr-Sommerfeld operator is that there is no transient merging of eigenvalues in a limited “window" of wave numbers, nor crossing of eigenvalues of different order. Hence, the curious behavior of the decay rates associated with the Orr-Sommerfeld hydrodynamic operator, as illustrated in Fig. 2 of paper I [@miORR] or in Fig. 2 of Gustavsson and Hultgren [@GustavssonHultgren], is absent in the case of the decay rates associated with the Squire hydrodynamic operator. The general nature of the eigenvalue map for the Squire operator is the same as depicted in Fig. \[F1\] here, independent of the Reynolds number.
In view of the structure of the eigenfunctions and of the boundary conditions $R_N(q_\parallel,\pm1)=0$, the normalization constants can be evaluated exactly by using both the known integrals of products of the Airy functions [@Albright] and the Wronskian of Airy functions. This procedure yields: $$\label{E19B}
B_N(q_\parallel)=\int_{-1}^{1}dz~[R_N(q_\parallel,z)]^2=\frac{-\mathrm{i}}{\pi^2(q_x\mathrm{Re})^{1/3}} \left\{ 1-\frac{\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N-\mathrm{i})\right]} {\mathrm{Ai}\left[(q_x\mathrm{Re})^{1/3}(a_N+\mathrm{i})\right]}\right\},$$ and a similar expression in terms of the Bi functions, see Eq. . Again, as a consequence of the Squire symmetry, the normalization constant $B_N(q_\parallel)$ only depends on the product $q_x\mathrm{Re}=q_\parallel\overline{\text{Re}}$.
Nonequilibrium Vorticity Fluctuations\[S04\]
============================================
Starting from the general theory presented in Sect. \[S02B\] and using the expressions for the hydrodynamic modes and decay rates derived in Sect. \[S3\] we can now evaluate the autocorrelation function $\langle\delta{w}^*_z(\omega,\mathbf{q}_\parallel,z)\cdot
\delta{w}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle$ of the wall-normal vorticity fluctuations. By combining Eqs. , , and , one readily obtains: $$\label{E19}
\langle\delta{w}^*_z(\omega,\mathbf{q}_\parallel,z)\cdot
\delta{w}_z(\omega^\prime,\mathbf{q}_\parallel^\prime,z^\prime)\rangle = W_{zz}(\omega,q_\parallel,z,z^\prime) ~(2\pi)^3~\delta(\omega-\omega^\prime)~\delta(\mathbf{q}_\parallel-\mathbf{q}_\parallel^\prime)$$ with $$\label{E20}
W_{zz}(\omega,q_\parallel,z,z^\prime)=\sum_{N,M=0}^\infty \frac{\Xi^\text{(E)}_{NM} + \Xi^\text{(NE)}_{NM}(\omega)}{B_N^*~B_M[-\mathrm{i}\omega+\Gamma_N^*][\mathrm{i}\omega+\Gamma_M]}~R_N^*(z)~R_M(z^\prime).$$ To simplify the notation we have suppressed on the RHS of Eq. the explicit dependence of the quantities on the wave vector $\mathbf{q}_\parallel$. In principle, Eq. enables us to investigate not only the intensity of the vorticity fluctuations, but also the time-dependent correlation function characterizing the dynamics these fluctuations. However, just as in the case of the velocity fluctuations derived in paper I [@miORR], we consider here only the intensity of the nonequilibrium vorticity fluctuations, which is given by the equal-time correlation function $\langle\delta\omega^*_z(\mathbf{q}_\parallel,z,t)\cdot\delta{w}_z(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle$. This equal-time correlation function is obtained by applying a double inverse Fourier transform, in frequencies $\omega$ and $\omega^\prime$, to Eq. so that $$\label{E21}
\langle\delta\omega^*_z(\mathbf{q}_\parallel,z,t)\cdot\delta{w}_z(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle = W_{zz}(\mathbf{q}_\parallel,z,z^\prime)~(2\pi)^2~\delta(\mathbf{q}_\parallel-\mathbf{q}_\parallel^\prime),$$ with $$\label{E22}
W_{zz}(\mathbf{q}_\parallel,z,z^\prime)=\frac{1}{2\pi} \int_{-\infty}^{\infty}d\omega~W_{zz}(\omega,\mathbf{q}_\parallel,z,z^\prime).$$ Substituting Eq. into Eq. , and performing the integration over the frequency $\omega$ of the fluctuations, one readily obtains an expression for the amplitude of the equal-time correlation function. In view of the structure of Eq. , we conclude that the resulting expression will contain two additive contributions, namely an equilibrium contribution (E) and a nonequilibrium contribution (NE): $$\label{E29B}
W_{zz}(\mathbf{q}_\parallel,z,z^\prime)= W_{zz}^\text{(E)}(\mathbf{q}_\parallel,z,z^\prime) + W_{zz}^\text{(NE)}(\mathbf{q}_\parallel,z,z^\prime).$$
Equilibrium contribution to the intensity of the fluctuations
-------------------------------------------------------------
Since the mode-coupling coefficients $\Xi^\text{(E)}_{NM}$ do not depend on the frequency $\omega$, the integration in Eq. can be readily performed for the equilibrium contribution. Combining the result with the expression for the “equilibrium" mode-coupling coefficients we arrive at: $$\label{E23}
\begin{split}
W_{zz}^\text{(E)}(\mathbf{q}_\parallel,z,z^\prime)&=\tilde{S}\text{Re}~q_\parallel^2 \sum_{N,M=0}^\infty \int_{-1}^{1}d\xi\frac{R_N^*(\xi)~R_M(\xi)}{B_N^*~B_M}~R_N^*(z)~R_M(z^\prime),\\
&=\tilde{S}\text{Re}~q_\parallel^2 \sum_{N=0}^\infty \int_{-1}^{1}d\xi\frac{R_N^*(\xi)}{B_N^*}~R_N^*(z)~\delta(\xi-z^\prime),\\
&=\tilde{S}\text{Re}~q_\parallel^2 ~\delta(z-z^\prime),
\end{split}$$ where we have made use of the expansion of the delta function in terms of the hydrodynamic modes $R_N(\mathbf{q}_\parallel,z)$, $$\label{E24}
\delta(\xi-z)=\sum_{N=0}^\infty \frac{1}{B_N(q_\parallel)}~R_N(q_\parallel,\xi)~R_N(q_\parallel,z),$$ which is obtained by using the orthogonality condition . It is obvious that $\delta(\xi-z)$, as a function of $z$, satisfies the relevant boundary conditions. Expression is valid for any value of the wave number $q_\parallel$ or the effective Reynolds number $\overline{\text{Re}}$. The important result is that the expression for $W_{zz}^\text{(E)}(\mathbf{q}_\parallel,z,z^\prime)$ indeed reproduces the intensity of the equal-time autocorrelation function of the wall-normal vorticity fluctuations for a fluid in equilibrium. Note that the prefactor $\tilde{S}\text{Re}$ appears in Eq. as a consequence of the adoption of dimensionless variables; when one reverts to physical variables the resulting prefactor is indeed independent of the shear rate.
Equation justifies the use of superscripts (E) and (NE) in Eq. . In addition, it shows that the nonequilibrium contribution to the intensity of vorticity fluctuations arises only from the coupling with the wall-normal velocity fluctuations in the stochastic Squire equation , and not from the self-coupling also present in the stochastic Squire equation . This result was already found previously [@miCouette2] on the basis of a Galerkin approximation. We now see that this is exact, and not just a consequence of the simplicity of the approximation used previously [@miCouette2]. However, the self-coupling in the Squire contribution does contribute to the time-dependent nonequilibrium correlation function which is not considered here.
The separation of the effects of thermal noise into an equilibrium and a nonequilibrium contribution is equivalent to what was found in the study of stochastic forcing, for instance, Eq. (17) of Ref. [@BamiehDahleh] or Eq. (4.3) of Jovanovic and Bamieh [@JovanovicBamieh]. The terms proportional to the cube of the Reynolds number in those expressions [@BamiehDahleh; @JovanovicBamieh] vanish when there is no flow in the system, but physically, the terms proportional to Re exists even when there is no flow. Indeed, as is the case in our Eq. , one power of $\text{Re}$ in Refs. [@BamiehDahleh; @JovanovicBamieh] is due to dimensionality reasons. A forcing (stochastic or not) must have units of force and the dimensionless time is in unit of the (inverse) shear rate. An important feature of our calculation is that we recover the well-known expression for the vorticity fluctuations in equilibrium [@BoonYip; @HansenMcDonald].
Nonequilibrium contribution to the intensity of the fluctuations
----------------------------------------------------------------
Upon substitution of the (NE) part of Eq. into Eq. , one obtains the nonequilibrium contribution $W_{zz}^\text{(NE)}(\mathbf{q}_\parallel,z,z^\prime)$ to the vorticity fluctuations. By making further use of Eq. for the mode-coupling coefficients, we arrive at the explicit expression: $$\begin{gathered}
\label{E26}
W_{zz}^\text{(NE)}(z,z^\prime)=\frac{q_y^2}{2\pi} \sum_{N,M=0}^\infty\hspace*{-6pt} \frac{R_N^*(z)~R_M(z^\prime)}{B_N^*~B_M}\iint_{-1}^1\hspace*{-6pt}d\xi d\xi^\prime R_N^*(\xi)~R_M(\xi^\prime)\\ \times \int_{-\infty}^{\infty} \frac{C_{zz}(\omega,\xi,\xi^\prime)~d\omega} {[-\mathrm{i}\omega+\Gamma_N^*][\mathrm{i}\omega+\Gamma_M]}.\end{gathered}$$ In principle one needs to substitute the autocorrelation function $C_{zz}(\omega,\xi,\xi^\prime)$ of the wall-normal velocity fluctuations from Paper I [@miORR] into Eq. , and perform the various integrations and summations to obtain the function of interest. This procedure yields rather complicated expressions and only marginal analytical progress can be made, at the expense of very large and cumbersome expressions. Therefore, unlike our previously solution obtained on the basis of a Galerkin approximation [@miCouette2], only a numerical computation of $W_{zz}^\text{(NE)}(z,z^\prime)$ is generally possible, and even this numerical procedure turns out to be rather long and difficult.
There is one important issue that should be mentioned. Because of the presence of the term $q_y^2$ as a prefactor in Eq. , it turns out that the intensity of the nonequilibrium vorticity fluctuations has a maximum in the spanwise direction ($q_x=0$), while it is zero in the streamwise direction. This is opposite to the wave-vector dependence of the wall-normal velocity fluctuations discussed in Paper I [@miORR]. In that case the intensity of the fluctuations has a maximum in the streamwise direction ($q_y=0$) and is zero in the spanwise direction. Moreover, for the same Reynolds number the intensity of the vorticity fluctuations is substantially larger than the intensity of the wall-normal velocity fluctuations. Hence, an important conclusion that can be derived from Eq. is that the most important effect of the flow on thermal fluctuations is the enhancement of wall-normal vorticity fluctuations with wave vector in the spanwise direction; or, equivalently, fluctuations that are constant in the streamwise direction. The same conclusion was obtained from our previous approximate Galerkin solution [@miCouette; @miCouette2], see in particular Fig. 6 in Ref. [@miJNNFM]. A similar conclusion is obtained both from direct numerical simulations of the full Navier-Stokes equations or from analytical studies of transient growth (amplification) of perturbations, see, for instance refs. [@GaymeEtAl1; @HwangCossu1]
Not only are the vorticity fluctuations in the spanwise direction the most dominant and interesting. In addition, further analytical progress is also possible for this case. This is the reason why previous investigators, who considered externally imposed forcing, have focused on vorticity response in the spanwise direction [@FarrellIoannou; @BamiehDahleh; @EckhardtPandit]. Indeed, when $q_x =0$ the (Fourier transformed) Orr-Sommerfeld and Squire equations simplify notably. The eigenfunctions of the Squire operator can be simply expressed in terms of trigonometric functions, while the corresponding eigenvalues can be obtained analytically. The eigenfunctions of the Orr-Sommerfeld operator in this limit can be written as combinations of trigonometric and hyperbolic functions, while the eigenvalues can be obtained numerically by solving relatively simple algebraic equations, as first discussed by Dolph and Lewis [@DolphLewis]. One important property is that in both cases the eigenfunctions possess a well-defined parity, and can be naturally classified into odd eigenfunctions and even eigenfunctions. We note that this is not true in the general case $q_x\neq0$. Hence, in the remainder of this paper we shall focus on the vorticity fluctuations in the spanwise direction, and their effect on the nonequilibrium energy amplification induced by the fluid flow.
Nonequilibrium Energy Amplification\[S05\]
==========================================
We recall that in the derivation of the stochastic Orr-Sommerfeld and Squire equations we have used the incompressibility assumption $\boldsymbol{\nabla}\cdot\delta\mathbf{v}=0$ [@miCouette; @miCouette2]. Hence, only two components of the velocity fluctuate independently. Most investigators on the subject have been interested in the so-called kinetic-energy amplification, that can be obtained from the sum of the equal-time autocorrelation functions $\langle\delta{v}^*_i(\mathbf{q}_\parallel,z,t)\cdot\delta{v}_i(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle$. And indeed, as elucidated in more detail in our previous analysis on the basis of a Galerkin approximation [@miCouette2], the spatial spectrum of the kinetic-energy amplification is proportional to the vertical average $$\begin{aligned}
\frac{1}{2}&\int_{-1}^{1}\int_{-1}^{1}\hspace*{-6pt}dz~dz^\prime~\sum_i
\langle\delta{v}^*_i(\mathbf{q}_\parallel,z,t)\cdot
\delta{v}_i(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle\label{Ex42}
\\
&=\frac{1}{2}\int_{-1}^{1}\int_{-1}^{1}\hspace*{-6pt}dz~dz^\prime \left\{\frac{1}{q_\parallel^2}
\langle\delta{w}^*_z(\mathbf{q}_\parallel,z,t)\cdot
\delta{w}_z(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle
+\langle\delta{v}^*_z(\mathbf{q}_\parallel,z,t)\cdot\delta{v}_z(\mathbf{q}_\parallel^\prime,z^\prime,t)\rangle\right\},\notag\end{aligned}$$ where on the RHS of this equation we only need to consider the wall-normal velocity and vorticity fluctuations because of the divergence-free condition $\boldsymbol{\nabla}\cdot\delta\mathbf{v}=0$. In addition, the fact that there is no cross-correlation between $\delta{w}_z$ and $\delta{v}_z$ has also been employed.
The contribution in Eq. arising from the wall-normal velocity fluctuations has been studied extensively in Paper I [@miORR]. We focus here on the contribution from the vorticity fluctuations which, in view of Eqs. -, will be proportional to the quantity $$\label{E28}
W_{zz}(\mathbf{q}_\parallel) = \frac{1}{2} \int_{-1}^{1}\int_{-1}^{1}\hspace*{-6pt}dz~dz^\prime~W_{zz}(\mathbf{q}_\parallel,z,z^\prime),$$ which is a function of the horizontal wave vector $\mathbf{q}_\parallel$ of the fluctuations that we shall investigate in this section.
First of all, one notes that because of the structure of the mode-coupling coefficients, the spectrum $W_{zz}(\mathbf{q}_\parallel)$ can be expressed as the sum of an equilibrium and a nonequilibrium contribution which we prefer to write in the form $$\label{E29}
W_{zz}(\mathbf{q}_\parallel)=W_{zz}^\text{(E)}({q}_\parallel)\left[1+\Delta{W_{zz}^\text{(NE)}}(\mathbf{q}_\parallel)\right],$$ where $\Delta{W_{zz}^\text{(NE)}}(\mathbf{q}_\parallel)$ represents the nonequilibrium energy enhancement. The equilibrium contribution $W_{zz}^\text{(E)}({q}_\parallel)$ in Eq. is obtained by substituting Eq. into Eq. $$\label{E36}
W_{zz}^\text{(E)}({q}_\parallel)=\tilde{S}\text{Re}~q_\parallel^2,$$ which is the same as in the absence of any flow.
The nonequilibrium enhancement $\Delta{W_{z}^\text{NE}}(\mathbf{q}_\parallel)$ in Eq. , resulting from the wall-normal vorticity fluctuations, can be obtained by substituting Eq. into Eq. . As already mentioned before, in general, this procedure yields a complicated expression that can only be evaluated numerically.
Enhancement of streamwise-constant fluctuations (*i.e.*, with wave vector in the spanwise direction)
----------------------------------------------------------------------------------------------------
A particular simple case is that of fluctuations constant in the streamwise direction with $\mathbf{q}$ in the spanwise direction, *i.e.*, for which $q_x=0$ and $q_y=q_\parallel=q$. As mentioned above, under this condition the working equations simplify greatly, and a more compact analytical expression can be obtained for the nonequilibrium enhancement. Because of its obvious simplicity, this particular case has been analyzed in detail by some previous investigators [@FarrellIoannou; @BamiehDahleh; @JovanovicBamieh] but for externally imposed stochastic forcing, not thermal noise. Hence, their results differ from the ones obtained here. For this reason, we present our explicit results for the enhancement of vorticity fluctuations induced by thermal noise with the (horizontal) wave vector in the spanwise direction, namely: $$\begin{gathered}
\left.\Delta{W_{zz}^\text{NE}}(q)\right|_{q_x=0} =\frac{\text{Re}^2}{8q^3}\left[\frac{9-\tanh^2q}{2q}-\frac{\tanh{q}}{\cosh^2q}-\frac{9\tanh{q}}{2q^2}\right]\\ + \text{Re}^2 \sum_{N=0}^\infty\frac{(2a_N^2+3q^2)\tanh\sqrt{a_N^2+2q^2}+\dfrac{q^2\sqrt{a_N^2+2q^2}}{\cosh^2\sqrt{a_N^2+2q^2}}}{(a_N^2+q^2)^2(a_N^2+2q^2)^2\left[\dfrac{q\tanh{q}}{\sqrt{a_N^2+2q^2}}- \tanh\sqrt{a_N^2+2q^2} \right]},\label{E37}\end{gathered}$$ with $$a_N=\frac{\pi}{2}(2N+1),$$ so that $(a_N^2+q^2)/\text{Re}$ are the eigenvalues of the Squire operator in the spanwise direction with corresponding eigenfunctions of even parity. As already mentioned, in this particular case (spanwise direction) the hydrodynamic modes (eigenfunctions) have a well-defined parity. Because of the $z$-integrations in Eq. for the mode-coupling coefficients, modes with different parity do not couple. In addition, because of the integrations in Eq. , only the even eigenfunctions or modes do finally contribute to the nonequilibrium enhancement. To obtain Eq. we have closely followed a procedure used by Bamieh and Dahleh [@BamiehDahleh]; in particular we used the auxiliary function $g(z)$ introduced in their *Lemma 4*. However, we obtain a different result because the thermal noise, considered here, has a special spatial spectrum given by the fluctuation-dissipation theorem , that is different from the spectrum of the externally imposed stochastic forcing considered by Bamieh and Dahleh [@BamiehDahleh]. One power of the Reynolds number appears in the energy amplification because of dimensionality reasons. Physically, the flow-induced amplification of thermal noise is proportional to the square of the Reynolds number (shear rate), not to the cube of the Reynolds number as stated by Bamieh and Dahleh [@BamiehDahleh] or Jovanovic and Bamieh [@JovanovicBamieh].
In Fig. \[F2\] we show a plot of the nonequilibrium enhancement $\Delta{W_{zz}^\text{NE}}(q)$ of the wall-normal vorticity fluctuations with wave vector in the spanwise direction ($q_x=0$, or streamwise constant) as given by Eq. . The data in Fig. \[F2\] are for $\text{Re}=500$, but we note that the Reynolds number appears in Eq. only as as a prefactor, so that the ratio $\Delta{W_{zz}^\text{NE}}(q)/\text{Re}^2$ does not depend on the Reynolds number. We conclude from Fig. \[F2\] that the spanwise energy amplification can be very well visualized as a simple crossover between the two asymptotic behaviors, at large and small $q$, that can be easily obtained from Eq. , namely $$\begin{aligned}
\left.\Delta{W_{zz}^\text{NE}}(q)\right|_{q_x=0}\xrightarrow{q\to{0}}&~\text{Re}^2q^2\left[\frac{34}{315}-\frac{256}{\pi^7} \sum_{N=0}^\infty \frac{2+\cosh{(2N+1)\pi}}{(2N+1)^7 \sinh{(2N+1)\pi}} \right]\notag\\&~\simeq 8.14\times10^{-3}~\text{Re}^2q^2,\label{E39}\\
\left.\Delta{W_{zz}^\text{NE}}(q)\right|_{q_x=0}\xrightarrow{q\to{\infty}}&~\frac{\text{Re}^2}{2q^4}.\label{E40}\end{aligned}$$ We recover in Eq. the $q^{-4}$ behavior that is typical of nonequilibrium fluctuations [@BOOK] at large wave numbers corresponding to wavelengths smaller than the spacing between the walls. For such wavelengths the fluctuations are not affected by the boundary conditions and we recover, as an asymptotic limit for large $q$, results obtained by previous investigators for nonequilibrium fluctuations in fluids under shear [@TremblayEtAl; @LD85; @LDD89; @LD02; @WS03]. On the other hand, the vanishing of the intensity of fluctuations with small $q$, as implied by Eq. , is to be expected in the sense that the walls (boundary conditions) effectively suppress fluctuations of very long wavelength, comparable with the separation distance between walls. From Fig. \[F2\] we note that the flow at $\text{Re}=500$ causes an enhancement of the thermal energy about 1000 times larger that the thermal energy that would be expected from a local equilibrium assumption. Such a profound enhancement of the fluctuations is a general phenomenon in fluids in nonequilibrium states [@BOOK].
Figure \[F2\] shows that the main effect of the flow on the thermal fluctuations is to select and maximally amplify the wall-normal vorticity fluctuations with a particular value of the wave vector $\mathbf{q}_\text{m}$. As already discussed, the wave vector maximally enhanced is in the spanwise direction, and from Eq. we find numerically its magnitude to be $q_\text{m}\simeq1.4103$, which is the location of the maximum in Fig. \[F2\]. Therefore, in real space, the thermal noise amplified by the flow will manifest itself mainly as a set of vortices of size $\simeq(2\pi/1.41)L\simeq4.5L$ distributed in the spanwise direction and that are constant (extremely elongated) in the streamwise direction[^1]. The intensity of these fluctuating vortices will be proportional to $k_\text{B}T$ and to the square of the Reynolds number, see Eq. . But we note that close to the center of the layer, where the base flow velocity is zero, the velocity fluctuations might be of the same order as the mean (base) velocity. These fluctuating vortices deform the base flow, developing a series of streaks, *i.e.*, narrow regions where the streamwise velocity is larger or smaller than the average, as indicated in Fig. \[F2B\] where we have tried to illustrate schematically this physical situation. These streaks will be typically separated by a distance of about 4.5 times the half gap between the walls, as also shown in Fig. \[F2B\]. This optimal spanwise wave number is independent of the Reynolds number.
At this point it is interesting to note that several authors have identified the appearance of a set of fluctuating streaks in sheared flows as a precursor of the instability [@Waleffe1; @HofEtAl]. More quantitatively, extensive numerical simulations of the unstabilization of plane Couette flow [@KomminahoEtAl; @DuguetEtAl; @GaymeEtAl1; @HwangCossu1] have revealed large-scale coherent streaks with typical spanwise wavelength of $\approx3L-4L$. For instance, most recently Gayme et al. [@GaymeEtAl1] performed an extensive analysis of direct numerical simulation data by Tsukahara et al. [@TsukaharaEtAl], obtaining an optimal spanwise wavelength of 1.8 times the distance between the plates, or 3.6 times the half distance, to be compared with our result of $4.5L$. Of course, the scope of our present work is restricted by the use of linear equations, so that its relevance to shear flow instability has to be considered, at most, as tentative. It is well-known and widely accepted that a complete understanding of unstabilization of shear flows requires a fully nonlinear theory [@Waleffe1; @HofEtAl]. In any case, the identification of the modes that are maximally enhanced in a linear theory may provide useful insights for developing simplified nonlinear theories, like the single mode model recently discussed by Gayme et al. [@GaymeEtAl1]. Furthermore, it is intriguing to know that a linear theory predicts that thermal (natural) noise develops into “streaks", of intensity $\propto k_\text{T}\text{Re}^2$, separated by a distance about $4.5L$, in agreement with numerical simulations of the nonlinear problem [@HwangCossu1].
To conclude this section, we mention that the algebraic $\propto{q}^{-4/3}$ dependence found for the enhancement of bulk fluctuations vanishes in the spanwise direction [@miCouette2]. That is the reason why the “shoulder" at intermediate $q$ shown in the bottom panel of Fig. 2 of Ref. [@miCouette2] does not appear in our current results. Bulk nonequilibrium fluctuations in sheared fluids have been first investigated by Tremblay et al. [@TremblayEtAl], who described the $q^{-4}$ behavior. Subsequently, Dufty and Lutsko [@LD85; @LD02] included the algebraic wave-number dependence that, in general, appears at shorter wavelengths. Although these earlier papers [@TremblayEtAl; @LD85; @LD02] refer specifically to wall-normal velocity fluctuations, we have found elsewhere [@miCouette2] that these features are also present for vorticity fluctuations in bulk.
Comparison with Galerkin approximation
--------------------------------------
In our previous publications [@miCouette2; @miJNNFM] the same problem considered here was investigated on the basis of a simple Galerkin approximation. In these previous publications we did not present explicit analytical expressions for the intensity of the vorticity fluctuations (on the basis of a Galerkin approximation), but only displayed the results graphically. However, as in the case of the exact solution, also in case of the Galerkin approximation the solution simplifies greatly for vorticity fluctuations in the spanwise direction. Specifically, for the fluctuations in the spanwise direction the expression in terms of the Galerkin approximation in Refs. [@miCouette2; @miJNNFM] reduces to $$\label{E41}
W_{zz}(q) = \tilde{S}\frac{5}{6} \text{Re}q^2\left[1+\frac{27\text{Re}^2q^2}{7(2q^2+5)(4q^4+23q^2+78)} \right].$$ Equation contains an equilibrium contribution that is about 17% percent lower than the exact result, Eq. . This is a shortcoming of the Galerkin-approximation scheme, as discussed elsewhere [@miCouette2]. But our purpose here is to compare the nonequilibrium enhancement, given by the second term inside the square brackets in Eq. , with the exact result given by Eq. . For this purpose we show in Fig \[F3\] the exact $\Delta{W_{z}^\text{NE}}(q)$ in the spanwise direction from Eq. as a solid curve, together with the Galerkin approximation of this quantity as a dashed curve. The agreement is rather good qualitatively. Quantitatively, the Galerkin approximation for large $q$ underpredicts the exact asymptotic limit by 4% (indistinguishable on the scale of Fig. \[F3\]), whereas in the small $q$ limit Galerkin approximation overpredicts the exact asymptotic limit by about 60%. The prediction, on the basis of the Galerkin approximation , of the wave number of maximum enhancement $q_\text{m}$ is quite good, differing by about 1% from the true value.
Although we have here compared explicitly the exact solution with the Galerkin approximation only in the spanwise direction, from the good results obtained we may infer that the Galerkin approximation developed in Ref. [@miCouette2] will also yield a good representation of the nonequilibrium enhancement of the wall-normal vorticity fluctuations in any direction. This expectation has been confirmed by some preliminary calculations for arbitrary $\mathbf{q}_\parallel$ on the basis of Eq. .
Concluding remarks\[S07\]
=========================
We have shown that even in the absence of any external perturbations, already the thermally excited fluctuations, which are always present, cause a substantial energy amplification in laminar fluid flow with the main contribution to the energy amplification arising from wall-normal vorticity fluctuations as a result of a coupling of these wall-normal vorticity fluctuations with the wall-normal velocity fluctuations. On the other hand, in computational fluid dynamics, to destabilize shear flows and to investigate the transition to turbulence, some externally imposed random initial conditions are customarily introduced [@DuguetEtAl]. We have demonstrated that such externally imposed initial conditions are not needed physically. Indeed, thermal noise causes already the base flow to spontaneously develop streaks, which are currently expected to be the first step in the transition to turbulence, even in a linear approximation. We have evaluated the typical spanwise distance between these thermally excited streaks, finding excellent agreement with what is observed in simulations. Of course, the problem of the nonlinear evolution of these streaks needs to be further investigated. For this purpose direct numerical simulations of fluctuating hydrodynamics may be of interest.
In the last few decades there has been considerable interest in the fluid dynamics community in these and similar problems, like optimal disturbances, transient growth of perturbations, and energy amplification. At the same time, and somewhat independently, in the statistical physics community there has been an interest in the nonequilibrium enhancement of thermal fluctuations. The purpose of this paper, as well of the previous paper [@miORR], has been to make a connection between the approaches used in statistical physics and fluid dynamics.
The authors acknowledge stimulating discussions with Andreas Acrivos, Bruno Eckhardt and Mihailo Jovanovic. The research was supported by the Spanish Ministry of Science and Innovation through research grant FIS2008-03801.
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[^1]: This is by approximating the complicated spatial spectrum of the thermal velocity-fluctuations, as shown for instance in Fig. 6 of Ref. [@miJNNFM], by just two delta functions located at the symmetric maxima (of the fluctuating wall-normal vorticity) at $q_x=0$, $q_y=\pm1.4103$.
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abstract: 'The competition in the pinning of a directed polymer by a columnar pin and a background of random point impurities is investigated systematically using the renormalization group method. With the aid of the mapping to the noisy-Burgers’ equation and the use of the mode-coupling method, the directed polymer is shown to be marginally localized to an arbitrary weak columnar pin in $1+1$ dimensions. This weak localization effect is attributed to the existence of large scale, nearly degenerate optimal paths of the randomly pinned directed polymer. The critical behavior of the depinning transition above $1+1$ dimensions is obtained via an $\epsilon$-expansion.'
address:
- |
School of Natural Sciences\
Institute for Advanced Study\
Princeton, NJ 08540, U.S.A.
- |
Institut für Theoretische Physik\
Universität zu Köln\
D-50937 Köln, Germany
author:
- 'Terence Hwa [@addr]'
- Thomas Nattermann
title: Disorder Induced Depinning Transition
---
= 10000
Introduction
============
The statistical mechanics of an elastic manifold embedded in a medium of random point defects has been the subject of many studies in the past decade [@nat0; @hh; @kz]. Such systems are encountered in a variety of contexts, ranging from the fluctuations of domain walls in random magnets [@nat0; @hh; @dw; @manifold], to the pinning of magnetic flux lines in dirty superconductors [@creep; @nat1; @bmy]. Over the years, many theoretical methods have been developed to understand these systems [@cayley; @migdal; @bethe; @p; @mp; @fh; @hhf; @iv; @hf]. In particular, through a mapping to stochastic hydrodynamics [@hhf; @iv; @hf], we now know many properties of one-dimensional manifolds (directed polymers) in random media. There have also been numerous numerical simulations; some recent studies can be found in Ref. [@mezard; @kim; @hhk].
Recently, strong flux pinning effects [@civale; @koncz; @budhani] exhibited by samples of high temperature superconductor with [*extended*]{} defects such as columnar faults and twin planes lead naturally to the investigation of competition between extended and point defects [@hnv; @balents]. It has been argued in the case of many interacting flux lines that pinning by extended defects are [*weakened*]{} by point defects [@hnv]. There have also been a number of studies on the competing effects between extended and point defects on a [*single*]{} flux line or a directed polymer [@mk; @zhh; @tl; @bk1; @kolo1; @kolo2; @bk2; @kolo3]. An early study of this type was done by Kardar [@mk], who investigated numerically the pinning of a directed polymer to a [*line defect*]{} in the presence of a background of point defects in 1+1 dimensions. The polymer was found to depin from the line defect, if the pinning potential of the line defect is smaller than a certain threshold value. Critical behavior associated with the depinning transition was later investigated in more detail by Zapatocky and Halpin-Healy [@zhh]. The results of Refs. [@mk] and [@zhh] have been challenged by Tang and Lyuksyutov [@tl], who argued against the existence of a depinning transition in two dimensions, based on large scale simulations and approximate renormalization-group analysis on a hierarchical lattice. Tang and Lyuksyutov proposed instead that a directed polymer is [*always*]{} localized, although weakly, to the line defect in 1+1 dimensions, and that the depinning transition only exists above 1+1 dimensions. This conclusion is supported by a recent study of Balents and Kardar [@bk2], who performed numerical simulations, and also developed a functional renormalization group analysis by describing the directed polymer as a generalized $D$-dimensional manifold and studying the depinning transition using a [*double*]{} expansion (in the dimensionality of the manifold $D=4-\delta$ and the dimensionality of the embedding space $d = d_c(D)+\epsilon$.) On the other hand, analysis of the directed polymer itself by Kolomeisky and Straley have led to different conclusions when different renormalization group Ansatz were used [@kolo1; @kolo2; @kolo3].
In this paper, we give a [*systematic*]{} analysis of the competition in pinning between point and line defects for a directed polymer. By exploiting known knowledge of the randomly pinned directed polymer in 1+1 dimensions in the absence of any extended defects [@hf], we construct a renormalization-group analysis [*directly*]{} in 1+1 dimensions, the critical dimension for this problem. Our results prove the conclusion of Ref. [@tl], that the polymer is always pinned at and below 1+1 dimensions, and that the depinning transition only occurs above two dimensions. The existence of weak localization at the critical dimension is understood in terms of the anomolous large scale excitations of the directed polymer, known as the “droplet excitations" [@hf]. Critical scaling behaviors at the depinning transition are then obtained using a $1+1+\epsilon$ dimensional expansion.
The paper is organized as follows: We first introduce the known properties of the randomly pinned directed polymer in Section II. We consider the effect of pinning by a line defect using phenomenological scaling arguments, which establishes 1+1 dimensions to be the critical dimension. In Section III, we attempt to solve the problem using the replica method at the critical dimension. Despite the existence of an exact solution via the Bethe Ansatz [@bethe; @p] when the extended defect is absent, we failed to develop a systematic and controlled way of incorporating a weak extended defect. We next use an uncontrolled Hartree approximation, which contains the right physical ingredients, but overestimates the effect of point disorder. A [*systematic*]{} analysis taking advantage of known mapping of the directed polymer to the noisy-Burgers’ equation [@hf] is presented in Section IV. Through explicit calculations, facilitated by the use of the mode coupling approximation in $1+1$ dimensions, we construct the renormalization-group recursion relation and derive various scaling behaviors near the critical dimension. We show that the results can be understood naturally in terms of the anomolous droplet excitations [@hf]. In Section V, we generalize our method to describe the directed polymer pinned by a variety of extended and point defects, including columnar pin of extended range and trajectory, and point defects with long range correlations. Some useful relations about the randomly pinned directed polymer, in particular, the mode coupling approximation, is summarized in Appendix A, and a number of detail calculations are relegated to Appendix B.
Phenomenological scaling analysis
=================================
We describe the statistical mechanics of a randomly pinned directed polymer of length $t$ by the Hamiltonian $${\cal H}_0 = \int^t_0 dz\left\{\frac{\kappa}{2}\left(\frac{d{\mbox{\boldmath $\xi$}}}
{dz}\right)^2 + \eta[{\mbox{\boldmath $\xi$}}(z),z]\right\}, \label{H0}$$ where ${\mbox{\boldmath $\xi$}}(z)\in \Re^{d_\perp}$ denotes the transverse displacement of the polymer in $d = d_\perp + 1$ embedding dimensions (with $d_\perp$ being the co-dimension), $\kappa$ is the polymer line tension, and $\eta({{\bf x}},z)$ describes a background medium of uncorrelated point defects. The random potential is assumed to be Gaussian distributed with mean zero and the variance $$\overline{\eta({{\bf x}},z)\eta({{\bf x}}',z')}
= 2\Delta\delta^{d_\perp}({{\bf x}}- {{\bf x}}')\delta(z-z'), \label{eta}$$ where the overbar denotes disorder average.
The randomly pinned directed polymer has been the topic of detailed investigations during the last decade [@cayley]–[@hhk]. The emerging qualitative picture for a long polymer ($t\to\infty$) is the following: At low temperatures, the static properties of the polymer are dominated by the random potential and are controlled by one or a few optimal path(s) which minimize the total free energy [@fh; @hf]. To take advantage of fluctuations in the random potential $\eta$, the optimal path executes large transverse wandering \[solid line of Fig. 1(a)\]. If we fix one end of the polymer, then the root-mean-square displacement of the other end of an optimal path is $$X(t) = B t^\zeta, \label{Xrms}$$ where $3/4 > \zeta >1/2$ is an universal exponent which depends on $d$. The result $\zeta =2/3$ in $d=1+1$ has been obtained exactly by a number of methods [@bethe; @hhf]. Numerical calculations give $\zeta \approx 5/8$ in $d=2+1$. Displacement of the fixed end of the polymer by a distance $r \ll X(t)$ typically result in a rearrangement of the optimal path within a segment of length $\tau \sim (r/B)^{1/\zeta}$ from the fixed end, as shown in Fig. 1(a). For sufficiently large displacement, i.e., for $ r \gg B t^{\zeta}$, the optimal path is completely changed \[dotted line of Fig. 1(a)\], with a completely different free energy \[Fig. 1(b)\]. The typical free energy difference $\Delta F_0$ between such independent paths is $$\Delta F_0 \approx At^{\theta}, \label{Frms}$$ with an identity $$\theta=2\zeta-1 \label{id}$$ relating the two exponents. $\Delta F_0$ also sets the scale of the free energy difference between the optimal path and a typical path.
Thermal fluctuations do not change this picture qualitatively. They merely wipe out the fine structure of the random potential, leading to temperature dependent amplitudes. For instance, $B \propto (\Delta/\kappa T)^{1/3}$ and $A \propto \kappa B^2$ in $d=1+1$. In $d \le 2+1$ dimensions, this picture is in fact correct for all temperatures. In $d > 2+1$, the polymer undergoes a continuous phase transition and becomes dominated by thermal fluctuations rather than disorder at sufficiently high temperatures. We shall however be focused on the more interesting low temperature phase throughout this paper.
Let us consider the influence of an additional pinning potential, $U_p({{\bf x}})$, in the form of a line defect with a short range $a$, located at the origin. We model the defect by the Hamiltonian $${\cal H}_1 = \int^t_0 dz \ U_p[{\mbox{\boldmath $\xi$}}(z)], \label{H1}$$ with $U_p({{\bf x}}) = - U$ for $|{{\bf x}}| < a$, and $U_p({{\bf x}}) = 0$ for $|{{\bf x}}| > a$. Clearly, such an attractive potential favors configurations of the polymer close to the origin, i.e., it attempts to [*localize*]{} the polymer. On the other hand, the polymer will try to wander [*away*]{} from the defect in order to take advantage of fluctuations in the background random potential $\eta({{\bf x}},z)$. Depending on the outcome of this competition, the pinning potential $U_p$ may have a number of possible effects: It may always or never localize the polymer for any finite pinning strengths $U$, or it may localize the polymer only above a certain critical pinning strength $U_c$, thus giving rise to a depinning transition.
To gain an understanding of the competing effects, we first consider the pure problem with $\eta({{\bf x}},z) = 0$. Let $$W^{(0)}({{\bf x}},t) = \int^{({{\bf x}},t)}_{(0,0)} {\cal D}[{\mbox{\boldmath $\xi$}}]
e^{-({\cal H}_0^{(0)} + {\cal H}_1)/T}$$ be the Boltzmann weight of propagating the polymer from $(0,0)$ to $({{\bf x}},t)$, with ${\cal H}_0^{(0)} = {\cal H}_0[\eta=0]$ and the superscript $(0)$ indicating the absence of the random potential, then $W^{(0)}$ satisfies the diffusion equation $$T \frac{\partial}{\partial t} W^{(0)}({{\bf x}},t)
= \frac{T^2}{2\kappa}\nabla^2 W^{(0)} - U_p({{\bf x}})W^{(0)}({{\bf x}},t).
\label{Seqn.0}$$ Our problem is equivalent to that of a $d_\perp$-dimensional (imaginary-time) quantum mechanical particle with a pinning well $U_p$ at the origin. In the thermodynamic limit $t \to \infty$, the statistical mechanics of the polymer is given by the ground state of the quantum problem whose solution is well known: There is always a bound state for arbitrarily weak pinning potential in $d_\perp \le 2$, where the polymer is always bounded to the line defect. For $d_\perp > 2$, a critical pinning strength of the order $U^{(0)}_0 \equiv T^2/(2\kappa a^2)$ is necessary to have a bound state. A depinning transition occurs at $U = U^{(0)}_c \sim U^{(0)}_0$. Fig. 2 summarizes the phase diagram of the pure system in various dimensions.
It is useful to understand these results by simple physical considerations: Assume first that the polymer is completely localized within the pinning well. Then the energy gained (compared to a free polymer) is $U$ per length, and the entropy cost (or the kinetic energy cost of localizing a quantum particle) is of the order $U_0^{(0)}$ per length. Thus if $U \gg U_0^{(0)}$, the total free energy of the polymer is $ F^{(0)} = -U + U_0^{(0)} < 0$, and the polymer is localized. However, if $U \lesssim U_0^{(0)}$, then $F^{(0)} >0$ and the polymer will not be completely localized to the well since the free energy exceeds that of a free polymer in this case. An alternative scenario when $U \lesssim U_0^{(0)}$ is to have the polymer completely delocalized, ignoring the existence of the weak pinning potential. This is self-consistent for $d_\perp>2$, since for a polymer with one end fixed at the origin, the accumulated probability of a long free polymer returning to the origin is $t^{(2-d_\perp)/2} \to 0$. Thus we see that for $d_\perp >2$, the polymer is essentially free for $U \ll U_0^{(0)}$, and is completely localized if $U \gg
U_0^{(0)}$. The simplest phase diagram is then to have a depinning transition separating the pinned and free phases, occuring at $U = U_c^{(0)} \sim U_0^{(0)}$. On the other hand, for $d_\perp \le 2$, the probability of a long free polymer returning to the origin is of order $1$. Thus the completely delocalized phase is not self consistent here, and the effect of a weak pinning potential can never be ignored in the limit $t \to \infty$.
The above can be stated more quantitatively by comparing the average pinning energy $\langle {\cal H}_1 \rangle^{(0)}$ with the available thermal energy $T$. A useful parameter to focus on is the dimensionless ratio $g^{(0)} = \langle {\cal H}_1 \rangle^{(0)} / T$. The polymer is free if $g^{(0)} \ll 1$ and is completely localized if $g^{(0)} \gg 1$ in the limit $t \to \infty$. If we just perform the thermal average $\langle \ldots \rangle^{(0)}$ by the Wiener measure of a [*free*]{} polymer, $e^{-{\cal H}_0^{(0)}/T}$, then we obtain $$g^{(0)}_0 = \frac{1}{\epsilon^{(0)}}\left[\left(\frac{t}{a_\|^{(0)}}\right)
^{\epsilon^{(0)}} - 1 \right]
\frac{U}{U_0^{(0)}} \label{g00}$$ where $\epsilon^{(0)} = (2- d_\perp)/2$ and $a_\|^{(0)} = (\kappa/T) a^2$ is the short distance cutoff along the length of the polymer. Thus the free polymer phase is obtained for $d_\perp > 2$ (i.e., $\epsilon^{(0)} < 0 $) if $ U < U_c^{(0)} \propto
|\epsilon^{(0)}| U_0^{(0)}$. For $U > U_c^{(0)}$, the polymer is free only for $$t < l_\parallel^{(0)} = a_\|^{(0)} \left[\frac{U}{U_c^{(0)}}-1\right]
^{-\frac{1}{|\epsilon^{(0)}|}}.$$ $l_\|^{(0)}$ is a crossover length above which the polymer becomes localized to the pinning potential. From it, we can define a localization length $$l_\perp^{(0)} =\left[\frac{T}{\kappa} l_\|^{(0)}\right]^{1/2}
= a \left[\frac{U}{U_c^{(0)}}-1\right]^{-\frac{1}{2|\epsilon^{(0)}|}},
\qquad d_\perp > 2\label{L0.1}$$ which characterize the typical transverse excursion of the polymer. On the other hand, for $d_\perp < 2$ where $\epsilon^{(0)} > 0$, $g^{(0)}_0 \gg 1$ for arbitrary small $U$ in the limit of long polymer $t \to \infty$. Defining the crossover and localization lengths in the same way, we have $$l_\perp^{(0)} = a [U/U_c^{(0)}]^{-\frac{1}{2|\epsilon^{(0)}|}},
\qquad \epsilon^{(0)} > 0 \label{L0.2}$$ with the polymer delocalized only in the limit $U \to 0$. Qualitatively similar behavior is obtained in $d_\perp = 2$, with $$l_\perp^{(0)} = a e^{U_c^{(0)}/U}. \label{L0.3}$$ Eqs. (\[L0.1\]) – (\[L0.3\]) are valid as long as $l_\perp^{(0)} \gg
a$. Throughout this paper, we shall only be interested in this critical regime, where the scaling behavior of localization length is insensitive to the detail shape of $U_p$.
Let us now return to the problem of a [*randomly pinned*]{} directed polymer, i.e., with $\eta({{\bf x}},z) \ne 0$, and estimate the competition between the pinning and the random potential that occurs in this case. Just as the localization of a polymer costs entropy in the pure problem, here, localization prevents the polymer from seeking out favorable regions of the random potential $\eta$ far from the origin, and therefore leads to a [*loss* ]{} in the random component of the energy that could otherwise be gained even as $T\to 0$. If a polymer is localized within a distance $l_\perp$ about the origin, then it is consisted of a number of uncorrelated segments of length $l_\| \approx (l_\perp/B)^{1/\zeta}$, each of which having a free energy of order $Al_\|^\theta$ higher than that of the delocalized, optimal path (see Fig. 3). Thus the free energy cost of localization is of the order $ A (l_\perp/B)^{(\theta-1)/\zeta}$ per length. This plays the role of the entropy cost in the localization of a free polymer. For a very strong pinning potential, the polymer is again always localized completely within the pinning well because the energy per length gained, $U$, always exceeds the random energy cost of localization, which is of the order $U_0 = A (a/B)^{(\theta-1)/\zeta}$ per length.
The effect of a weak pinning potential can be estimated perturbatively from $\langle {\cal H}_1 \rangle$ as just described for the pure problem, except that it now depends on the realization of the random potential $\eta$. The natural quantity to examine is the average energy gained by the polymer in the presence of the pinning potential, $$\delta F = \overline{ \langle {\cal H}_1 \rangle },$$ given that one end is fixed at the origin (as shown in Fig. 3). As in the case of the pure problem, this is just the average of the accumulated return probability of the polymer to the origin. Since the rms displacement is $B t^\zeta$ in the absence of the pinning potential $U_p$, then $\delta F_0 \propto U (a/Bt^\zeta)^{d_\perp}t$ to leading order in $U$. To determine the effect of the pinning potential, it is necessary to compare $\delta F_0$ to $\Delta F_0 \approx A t^\theta$, the intrinsic variations in the free energy \[see Fig. 1(b)\]. The dimensionless ratio that characterizes the strength of pinning is now $$g_0 = \frac{\delta F_0}{\Delta F_0}
\propto \frac{U}{U_0} \left(\frac{t}{a_\|}\right)^\epsilon,$$ to leading order in $U/U_0$, with $a_\| = (a/B)^{1/ \zeta}$ being the short distance cutoff along the length of the polymer, and $$\epsilon(d) = 1 - d_\perp\zeta - \theta = 2 - (d+1) \zeta(d). \label{eps}$$ The critical dimension is $d=1+1$ since $\zeta(2) = 2/3$ exactly [@bethe; @hhf]. If $g_0 \ll 1$, which is the case when $\epsilon < 0$, then the weak pinning potential is irrelevant in the thermodynamic limit. If $\epsilon > 0$, then $g_0 \gg 1$ and even a small $U$ will completely change the energy landscape of the randomly pinned directed polymer. For example, the free energy of the pinned state in Fig. 3 now becomes much lower than that of the optimal path of the unperturbed system. The apparent difference between the problem with point disorder and the pure problem is the behavior of $g_0$ at the critical dimension where $\epsilon = 0$: For the pure problem, $g^{(0)}_0$ diverges logarithmically according to Eq. (\[g00\]), indicating the marginal [*relevance*]{} of the pinning potential. But with point disorder, $$g_0 \propto U/U_0 \propto U a / (A B) \label{g0}$$ remains finite. This naively suggests the [*irrelevance*]{} of a small pinning potential at the critical dimension. If this is true, then there will have to be a depinning transition at the critical dimension $d=1+1$, since a strong enough pinning potential still produces a localized state. The simplest phase diagram in this case \[Fig. 4(a)\] will look quite different from that of the pure problem (Fig. 2). Fig. 4(a) is actually consistent with early numerical results of Kardar [@mk] and Zapatocky and Halpin-Healy [@zhh], who find a depinning transition in $d=1+1$. However, through a systematic renormalization-group analysis presented in Sec. IV, we will find that $U$ itself is renormalized and diverges logarithmically, while all other parameters such as $A$ and $B$ only suffer finite renormalization. These results lead to a logarithmically diverging $g$ and hence the marginal [*relevance*]{} of pinning at the critical dimension. Thus the phase diagram for the randomly pinned directed polymer is like that of the pure problem (Fig. 2), but with a shifted critical dimension and $\epsilon$, as shown in Fig. 4(b). This result is supported by the recent large scale numerical studies of Refs. [@tl; @bk2]. Renormalization group analysis can then be used to obtain the critical behavior at the depinning transition, in particular, the divergence of the correlation length, $$l_\perp = a\left(\frac{U}{U_c} - 1\right)^{\nu_\perp},$$ where $\nu_\perp$ is the liberation exponent.
Analysis in Replica space
=========================
A convenient way to treat one dimensional objects like the directed polymer is the transfer matrix approach described in Sec. II. The [*full*]{} Boltzmann weight (or the restricted partition function) $W({{\bf x}},t)$ of a polymer propagating from $({\bf 0},0)$ to $({{\bf x}},t)$ in a random medium $\eta({{\bf x}},t)$ is described by an equation analogous to Eq. (\[Seqn.0\]), except with $U_p({{\bf x}})$ replaced by $U_p({{\bf x}})+\eta({{\bf x}},t)$. All physical properties follow from disorder averages of the free energy $\overline{F({{\bf x}},t)}=-T \overline{\ln W({{\bf x}},t)}$ and derivatives thereof. One approach to performing this average is the replica method, which exploits the identity $\ln W({{\bf x}},t)=\lim_{n\to 0}\frac{1}{n}(W^n({{\bf x}},t)-1)$. Up to an exchange of the $n\to 0$ and the thermodynamic limit (see below), $\overline{F({{\bf x}},t)}$ is given by the $n^{th}$ moment of $W$, $\overline{W^n({{\bf x}},t)}\equiv {\cal W}({{\bf x}},...{{\bf x}};t)$, where $${\cal W}({{\bf x}}_1,..., {{\bf x}}_n;t)=\prod^n_{\alpha=1}
\int^{({{\bf x}}_{\alpha},t)}
_{({\bf 0},0)}
{\cal D}[{\mbox{\boldmath $\xi$}}_{\alpha}]e^{-{{\cal H}_n}/T}, \label{Wn}$$ $\alpha \in \{1, \ldots, n\}$ is the replica index, and ${\cal H}_n$ denotes the replica Hamiltonian $${\cal H}_n = \sum ^n_{\alpha=1} \int^t_0 dz
\left\{\frac{\kappa}{2}\left(\frac{d{\mbox{\boldmath $\xi$}}_\alpha}{dz}\right)^2
+ U_p[{\mbox{\boldmath $\xi$}}_{\alpha}(z)]-\sum_{\beta\neq \alpha}\frac{\Delta}{T}
\delta^{d_\perp} [{\mbox{\boldmath $\xi$}}_{\alpha}(z)-{\mbox{\boldmath $\xi$}}_{\beta}(z)] \right\}. \label{Hn}$$ For analytic simplicity, we approximate the columnar pinning potential by a $\delta$-function, i.e., $U_p({{\bf x}}) = - u \delta^{d_\perp}({{\bf x}})$ with $u \approx U a^{d_\perp}$, from here on. Applying the transfer matrix approach directly to Eq. (\[Wn\]) leads to the Schrödinger-like equation $$T\frac{\partial}{\partial t}{\cal W} ({{\bf x}}_1,...,{{\bf x}}_n;t)
=-\hat{\cal H }(n)\ {\cal W}({{\bf x}}_1,...,{{\bf x}}_n;t), \label{Seqn}$$ where $$\hat{\cal H}(n)=\sum^n_{\alpha=1}
\left\{-\frac{T^2}{2\kappa}{\mbox{\boldmath $\nabla$}}^2_\alpha
-u\delta^{d_\perp}({{\bf x}}_{\alpha})-\sum_{\beta\not=\alpha}
\frac{\Delta}{T}\delta^{d_\perp}({{\bf x}}_ {\alpha}-{{\bf x}}_{\beta})\right\},
\label{hartree.H}$$
As usual, Eq. (\[Seqn\]) can be solved with the Ansatz $${\cal W}({{\bf x}}_1,..., {{\bf x}}_n ;t)=\sum ^{\infty}_{j=0} a_j e^{-E_j(n) t/T}
\Phi_j({{\bf x}}_1,...,{{\bf x}}_n)$$ and $\hat{\cal H}(n) \Phi _j = E_j(n) \Phi _j$. In $d_\perp = 1$ and for $U_p = 0$, the ground state wave function is easily found by using the Bethe-Ansatz [@bethe], $\Phi_0(x_1,...,x_n) \propto \exp\{ - \frac{\Delta \kappa}{T^3}
\sum_{\alpha < \beta}|x_\alpha - x_\beta| \}$. A similar solution follows for the ground state of a randomly pinned directed polymer confined to the semi-infinite plane $x \ge 0$ and subject to an attractive potential $U_p$ at $x=0$ [@bethe]. Upon changing the strength of the attractive potential, one obtains a depinning transition for the half-plane problem when the ground state energy becomes larger than that for $U_p =0$. However, searches for a similar solution to the Hamiltonian (\[hartree.H\]) have not been successful. A naive application of the perturbation theory for small $U_p$ starting from the wave function $\Phi_0(x_1,...,x_n)$ fails likewise, because the low energy [*excited*]{} states of $\hat{\cal H}(n)$ are not uniquely defined in the limit the number of particles $n \to 0$ [@p]. This ambiguity, arising from the exchange of the $n\to 0$ and the thermodynamic limits, is a well-known “trouble spot” for the replica method, and often calls for elaborate schemes with broken replica symmetry [@mp]. For the problem at hand, the ground state itself does not involve replica symmetry breaking [@bethe; @p]. However, it is demonstrated in Ref. [@p] that states with broken replica symmetry could have energies arbitrary close to the ground state energy $E_0(n)$. These will dominate in any perturbative calculations.
We shall circumvent the problem of replica symmetry breaking by introducing a completely different method in Sec. IV. For now, we consider another limit, $\Delta \to 0$, where the system almost decouples into $n$ one-particle problems. Here an attractive potential $U_p$ always gives a bounded ground state in $d_\perp = 1$. In term of the reduced length $r = x/x_0$ where $x_0=T^2/(2u\kappa)$, the reduced Hamiltonian $\hat{h}(n) = \hat{\cal H}(n)/\frac{T^2}{2\kappa x_0^2}$ becomes $$\hat{h}(n) = - \sum_\alpha \left\{ \frac{\partial^2}{\partial r_\alpha^2} + 2
\delta(r_\alpha) + \frac{2\Delta}{uT}\sum_{\beta\neq \alpha}
\delta(r_\alpha - r_\beta) \right\}. \label{h.hat}$$ We now attempt to treat the case of [*weak*]{} inter-replica interaction $\Delta \ll uT$ by using the Hartree approximation [@no]. Making the Ansatz for the wave function $\Phi(r_1, r_2, \ldots, r_n) = \phi_1(r_1)\phi_2(r_2)...\phi_n(r _n)$ and minimizing the reduced energy ${\cal E}(n) =
\int dr_1 \ldots dr_n\Phi\hat{h}(n) \Phi$ under the conditions $\int dr \phi^2_{\alpha}(r) = 1$ which we impose by Lagrangian multipliers ${\varepsilon}_{\alpha}$, we get a set of Hartree equations [@no]. These simplify to a single equation if we impose symmetry between the replicas $\phi_{\alpha}(r)=\phi(r)$ and ${\varepsilon}_\alpha = {\varepsilon}$ for all $\alpha$’s, $$\left[ \frac{\partial^2}{\partial r^2} +2 \delta(r)+4(n-1)g^{-1}_0
{\phi}^2(r)-{\varepsilon}\right] \phi (r) = 0,\label{hartree.eqn}$$ where $$g_0^{-1}= \Delta/(uT). \label{g0.2}$$ This equation can be solved easily with the boundary condition $\phi(r) = \phi'(r) = 0 $ for $|r| \to \infty$, giving a localized wave function $$\phi(r) = e^{-r \sqrt{{\varepsilon}}}\left[\frac{\sqrt{{\varepsilon}}+\sqrt{{\varepsilon}-2(n-1)g_0^{-1}\phi^2(r)}}{\sqrt{1+{\varepsilon}}}\right] \label{hartree.1}$$ with a negative ground state energy $${\cal E}_0(n)=-\frac{n}{3}(1+\sqrt{{\varepsilon}}+{\varepsilon})
= -n\left[1+(n-1)g_0^{-1}+\frac{1}{3}(n-1)^2g^{-2}_0\right]
\label{hartree.2}$$ where ${\varepsilon}=[1 + (n-1)g_0^{-1}]^2$, and the localization length is $$\l_{\perp} =x_0/\sqrt{{\varepsilon}}
\propto \left( 1+ (n-1)g_0^{-1}\right)^{-1}. \label{hartree.3}$$ Note that the Hartree results Eqs. (\[hartree.1\]) – (\[hartree.3\]) are in fact the [*exact*]{} solutions of the Hamiltonian (\[hartree.H\]) in the limit $g_0^{-1} \to 0$, where the very strong pinning potential always localizes the directed polymer within $l_\perp$ of the origin. However, we see that in the limit $n\to 0$, the random potential characterized by $g_0^{-1}$ tends to [*increase*]{} the localization length. If we extrapolate the Hartree result Eq. (\[hartree.3\]) to finite value of $g_0^{-1}$, then we find an instability, i.e., $l_\perp \to \infty$ as $g_0^{-1} \to 1^{-}$. An obvious interpretation of this instability is the occurrence of a depinning transition.
On the other hand, an analogous Hartree calculation for the problem of an attractive potential in the half plane yields again the solution (\[hartree.1\]), except with ${\varepsilon}= (1+2(n-1)g_0^{-1})^2$. This result suggests a delocalization transition at $g_{c}^{-1} = 1/2$ with $l_\perp \propto (g_0 - g_c)^{-1}$, or $\nu _\perp =1$ for the half-plane problem. However, the exact Bethe-Ansatz calculation [@bethe] gave $g_{c}^{-1} =1$ and $\nu _\perp =2$. Thus we see that while the Hartree solution gives the qualitative effect of a background random potential — a tendency to [*depin*]{} the directed polymer from the origin — it [*overestimates*]{} the influence of randomness and cannot be used reliably to characterize the depinning transition quantitatively. Indeed, we will show in the following section that the directed polymer is always pinned to the origin for [*all*]{} finite values of $g_0$ in $d_\perp = 1$ dimensions.
\[ Note: Although our Hartree equation (\[hartree.eqn\]) agrees with that considered previously by Zhang [@zhang] if we set $u=0$, the physics of our solution is completely different: In the limit $n \to 0$ which we consider, the disorder leads to a [*repulsive*]{} interaction $g_0^{-1}\delta(x_\alpha - x_\beta)$ between the replicas, which makes a change of the roughness exponent $\zeta$ from $1/2$ for free polymers to the larger value $2/3$ possible. It is the pinning potential $U_p$ that confines the solution close to the origin. In contrast, Zhang considered the [*large*]{}-$n$ limit where the interaction between the replicas is [*attractive*]{}, leading to the unphysical solution of an exponentially decreasing wave function even if $u=0$. \]
The Renormalization-Group Analysis
==================================
As we have seen in the Sections II and III, both naive scaling arguments and the uncontrolled Hartree solution of the replicated system suggest that the polymer undergoes a depinning transition in the critical dimension $d=1+1$, consistent with the phase diagram sketched in Fig. 4(a). However, we have not been able to construct a [*controlled*]{} perturbative study for weak pinning potential $U_p$ within the replica formalism, despite the knowledge of the exact ground state. The technical problem encountered is the occurrence of replica symmetry breaking for the low energy excited states. In this section, we shall formulate a perturbative renormalization group study of the depinning transition in $1+1$ dimensions without ever introducing the notion of replica. We shall take advantage of the mapping of the directed polymer to the hydrodynamics of the noisy-Burgers’ equation [@hhf; @iv; @hf]. Many details of the mapping have been discussed in Ref. [@hf] and some are summarized in Appendix A. Here, we will outline our approach and quote the results. A number of detailed calculations are given in Appendix B.
Formalism
---------
It is useful to introduce first a formal language to characterize the properties of the directed polymer in the absence of the pinning potential $U_p$. For convenience, we shall fix one end ${\mbox{\boldmath $\xi$}}(t)$ of the polymer at some arbitrary point ${{\bf x}}$. This is implemented by introducing the one-point restricted partition function, $$Z_0({{\bf x}},t) = \int {\cal D}[{\mbox{\boldmath $\xi$}}] \ \delta^{d_\perp}({\mbox{\boldmath $\xi$}}(t)-{{\bf x}})
e^{-{\cal H}_0/T}, \label{Z0}$$ where the subscript $0$ is used to indicate $U_p = 0$. All thermal averages taken with respect to $Z_0({{\bf x}},t)$ will be denoted by $\langle \ldots \rangle_{{{\bf x}},t}$. The disorder averaged probability of finding a segment ${\mbox{\boldmath $\xi$}}(t_0)$ of the polymer at position ${{\bf y}}$ given that the other end is fixed at $({{\bf x}},t)$ is $$G^{(1,1)}_t({{\bf x}}-{{\bf y}},t-t_0) \equiv \overline{\langle
\delta^{d_\perp}({\mbox{\boldmath $\xi$}}(t_0)-{{\bf y}}) \rangle}_{{{\bf x}},t}, \label{G}$$ where we use the subscript $t$ to emphasize the dependence of $G^{(1,1)}$ on the total polymer length $t$. The functional form of $G^{(1,1)}$ has been investigated in details elsewhere [@hf] and is summarized in Appendix A. Here, we just mention the qualitative behavior, $G_t^{(1,1)}({{\bf r}},\tau) = (B \tau^\zeta)^{-d_\perp}$ for $r \ll B \tau^\zeta$ and vanishes rapidly as $r \gg B\tau^\zeta$.
It is also useful to consider the free energy of the one-point restricted polymer, $$F_0({{\bf x}},t) = - T \log Z_0({{\bf x}},t),$$ which satisfies the noisy-Burgers’ equation [@fns; @kpz; @medina] $$\partial_t F_0({{\bf x}},t) = \frac{T}{2\kappa}{\mbox{\boldmath $\nabla$}}^2 F_0 - \frac{1}{2\kappa} ({\mbox{\boldmath $\nabla$}}F_0)^2 + \eta({{\bf x}},t). \label{kpz.eqn}$$ The free energy correlation function is $$C_t({{\bf x}}-{{\bf y}}) = \overline{\left[F_0({{\bf x}},t)-F_0({{\bf y}},t)\right]^2}, \label{C}$$ with $$\begin{aligned}
&C_t({{\bf r}}) = 2 A^2 t^{2\theta} \qquad &{\rm for} \qquad r \gg Bt^\zeta,\label{ct}\\
&C_t({{\bf r}}) \propto A^2(r/B)^{2\theta/\zeta} \qquad
&{\rm for} \qquad r \ll Bt^\zeta.\label{cr}\end{aligned}$$ The sample-to-sample free energy variation introduced in Eq. (\[Frms\]) is just $$\Delta F_0 = \sqrt{C_t(\infty)/2} = At^\theta.$$ Again, the scaling form for $C_t$ is summarized in Appendix A. As we shall see, the functions $G$, $C$, together with a number of other distribution functions, will allow us to compute the effect of an additional pinning potential perturbatively.
In terms of $Z_0$, the full partition function $Z({{\bf x}},t)$ of the polymer in the presence of the pinning potential $U_p({\mbox{\boldmath $\xi$}}(z)) = - u \delta^{d_\perp}({\mbox{\boldmath $\xi$}}(z))$ is just $$Z({{\bf x}},t) = Z_0({{\bf x}},t) \langle e^{-{\cal H}_1/T} \rangle_{{{\bf x}},t}, \label{Z}$$ where ${\cal H}_1 = \int_0^t dz \ U_p$ as in Eq. (\[H1\]). For convenience, we again use the continuum delta function supplemented by a short distance cutoff $a$ (with $u=Ua^{d_\perp}$) to model the pin.
The Renormalization Group Analysis
----------------------------------
We now investigate the effect of ${\cal H}_1$ on the “bare" system $Z_0$ perturbatively. Clearly the analysis is complicated by the fact that the “bare" system itself is glassy and thus highly nontrivial. However, to understand whether the phase diagram belongs to that of Fig. 4(a) or Fig. 4(b), we only need to compute the marginal relevancy of the pinning potential in $1+1$ dimensions. Fortunately, the randomly pinned directed polymer in $d=1+1$ is one of the very few glassy systems for which a great deal is known. In particular we note that there are two pieces of exact information: (i) The disorder averaged thermal displacement is the same as that of the pure system, i.e., $$\overline{ \langle [\xi(t) - \xi(0)]^2 \rangle_{x,t}^c }
= \frac{T}{\kappa} t,\label{gi}$$ due to a statistical tilt symmetry [@hf; @schulz] preserved by the point disorder. (ii) A fluctuation-dissipation theorem for the noisy-Burgers’ equation [@fns; @medina] which states that $$\overline{\partial_x F_0(x,t)\partial_y F_0(y,t)} = \frac{\Delta \kappa}
{T} \delta (x-y) \label{fdt}$$ in the limit of large $t$ (see Appendix A). Eq. (\[fdt\]) implies that $C_t(r) = (\Delta\kappa/T) |r|$. Comparison with Eq. (\[cr\]) yields $2\theta/\zeta= 1$, and $A^2/B \propto \Delta\kappa/T$. Along with the exponent identity Eq. (\[id\]), we have $\theta = 1/3$, $\zeta=2/3$, and dimensional analysis gives $B\propto (\Delta/\kappa T)^{1/3}$. We see that our “bare” problem, the fixed point of the randomly pinned directed polymer, is a fixed plane spanned by the axis $\kappa$ and $\Delta\kappa/T$, or alternatively by the parameters $A\propto \kappa B^2$ and $B$. Hence all disorder averaged functions depend [*only*]{} on these two parameters in the limit of large $t$. For example, the bare coupling constant of the problem is defined as $g_0 = \delta F_0/\Delta F_0$ where $\Delta F_0 = A t^{1/3}$ and $$\begin{aligned}
&\delta F_0 & = \overline{\langle{\cal H}_1\rangle_{0,t}} = -u \int_0^t dz
\overline{\langle \delta[\xi(z)] \rangle_{0,t}} \nonumber \\
& & = -u \int_0^t dz \ G_t^{(1,1)}(0,t-z). \label{dF}\end{aligned}$$ Scaling form for $G^{(1,1)}$ (see Appendix A) yields $\delta F_0 \propto -(u/B)t^{1/3}$, hence $g_0 \propto u/(AB)
\propto uT/\Delta$ as in Eqs. (\[g0\]) and (\[g0.2\]). In subsequent calculations, we shall use $g_0 \equiv
u/(\kappa B^3)$. Following standard renormalization group treatment, we now consider the renormalization of the parameters $\kappa$, $B$, and $u$ due to the perturbation ${\cal H}_1$. \[Note that the effect of the pinning potential will obviously depend on the position of the fixed end $\xi(t)$. If $\xi(t)$ is sufficiently far away from the origin, the polymer will never feel the effect of $U_p$. In our calculations, we shall fix $\xi(t)$ at the origin (as shown in Fig. 3) to evaluate the maximal effect of the pinning potential.\]
We begin with the renormalization of the stiffness $\kappa$, since the presence of a pinning potential breaks the statistical tilt symmetry and hence the exact relation Eq. (\[gi\]). Adding a term $- h \int_0^t dz (d\xi/dz)$ to ${\cal H}$ and using the transformation $\xi (z)
\rightarrow \xi (z) + h (t-z)/ \kappa $ and the property $\eta(\xi (z) + h(t-z) / \kappa,z) = \eta (\xi(z),z)$ in the statistical sense, we obtain the renormalized free energy (with one end fixed at the origin) $$\overline{F[0,t;h]} = \overline {F_0(0,t)}-\frac{h^2}{2\kappa}t
- u \int^t_0 dz \overline{\langle\delta[\xi(z)+h(t-z)/\kappa]\rangle}_{0,t}$$ to the lowest order in $u$. Note that $\overline{F_0}$ does not depend on $h$ due to the statistical symmetry. Remembering that $\overline{ \langle\langle [\xi(t) - \xi(0)]^2
\rangle\rangle^c_{x,t} }
= - T \partial_h^2 \overline{F}|_{h=0}$ where $\langle\langle\ldots\rangle\rangle$ denotes thermal average using the full Hamiltonian ${\cal H}$, and defining the renormalized stiffness constant from $$\overline{ \langle\langle [\xi(t) - \xi(0)]^2 \rangle\rangle^c_{0,t} }
\equiv \frac{T}{\widetilde{\kappa}} t,$$ we have $$\begin{aligned}
&\widetilde{\kappa}^{-1} &= \kappa ^{-1}- u\frac{\partial^2}
{\partial h^2} \int_0^t \frac{dz}{t}
G_t^{(1,1)}(-h(t-z)/\kappa,t-z)|_{h=0} \label{rg1.b} \\
& &= \kappa^{-1} ( 1 - C_\kappa g_0 ). \label{rg1.a}\end{aligned}$$ In Appendix B, we find $C_\kappa$ to be a finite constant given by Eq. (\[Ckappa\]). Thus, there is only a [*finite*]{} renormalization of $\kappa$ which could be absorbed in its redefinition.
Next, we consider the renormalization of the amplitude of the mean square displacement of the polymer $$\widetilde{X}^2 \equiv \int dr \ r^2 G_t^{(1,1)}(r,t).$$ Expansion to the lowest order in $u$ yields $$\begin{aligned}
&\widetilde{X}^2 \equiv (\widetilde{B} t^\zeta)^2
&= \int^\infty_{-\infty} dy \ y^2 \left[ G^{(1,1)}_t(y,t) + \frac{1}{T}
\overline{\langle\delta[\xi(0)-y]{\cal H}_1\rangle^c_{x,t}}\right] \nonumber \\
& &=(Bt^\zeta)^2 + u \int^\infty_{-\infty} dy y^2 \int_0^t dz
G_t^{(2,1)}(0,t-z;y,t) \label{rg3.a}\end{aligned}$$ where $$G_t^{(2,1)}(x-y_1,t-z_1;x-y_2,t-z_2)
= -\frac{1}{T} \ \overline{\langle\delta(\xi(z_1)-y_1)
\delta(\xi(z_2)-y_2)\rangle^c_{x,t}} \label{G21}$$ is a nonlinear response function of the noisy-Burgers’ equation. In Appendix A, we obtain an expression for $G^{(2,1)}$ in term of $G^{(1,1)}$ using the mode coupling approximation, the validity of which will be discussed shortly. Using the relations (\[response.A\]) and (\[G21.A\]), we obtain from Appendix B $$\widetilde{B} = B(1-C_B g_0), \label{rg3.b}$$ where $C_B$ is another finite constant given in Eq. (\[CB\]). As for $\kappa$, the small correction to $B$ can be absorbed in its definition.
To check the above result in a different way, we calculated also corrections to $\Delta F_0$, or the relation (\[fdt\]), due to the perturbation ${\cal H}_1$. Expanding the total free energy to the lowest order in $u$ gives $$\overline{\partial_x F(x,t) \partial_y F(y,t)}
=\frac{\Delta\kappa}{T} \delta(x-y)
-u\frac{\partial ^2}{\partial x \partial y} \int_0^t dz
G_t^{(1,2)}(x-y,0;\frac{x+y}{2},t) \label{rg2.a}$$ where $$G_t^{(1,2)}(x-y,0;\frac{x+y}{2},t)
= \overline{F_0(x,t)\langle\delta[\xi(z)]\rangle}_{y,t}
+\overline{F_0(y,t)\langle\delta[\xi(z)]\rangle}_{x,t} \label{G12}$$ is another three-point response function of the noisy-Burgers equation. Again, we use the mode-coupling approximation to relate $G^{(1,2)}$ to the elementary functions $G^{(1,1)}$ and $C$ in a simple way. Using the relations (\[response.A\]), (\[fdt.A\]), and (\[G12.A\]) , we obtain $$\overline{\partial_x F(x,t) \partial_y F(y,t)} = \frac{\Delta\kappa}{T}
\left[\delta(x-y) - g_0 f(x,y)\right], \label{rg2.b}$$ with $f(x,y) \sim |x-y|^{-1}$ if $x + y =0$, and a faster decay for $x+y \ne 0$ [@note.1]. Thus, the perturbation ${\cal H}_1$ changes the form of the correlation function. However, it does not lead to a [*singular*]{} renormalization of $\Delta F_0$, which is obtained by integrating Eq. (\[rg2.b\]) over [*both*]{} the $x$ and $y$ coordinates.
Finally, we consider the renormalization of $u$ itself. This follows from the expansion of the disorder averaged free energy up to second order in $u$: $$\begin{aligned}
&\delta F(x,t) &\equiv\overline{F(x,t)}-\overline{F_0(x,t)} \nonumber\\
& &=\overline{\langle{\cal H}_1\rangle}_{x,t}
- \frac{1}{2T}\overline{\langle{\cal H}^2_1\rangle^c_{x,t}}.\end{aligned}$$ It will be convenient to integrate over the fixed point $x$. Using the normalization of $G^{(1,1)}$, $\int_{-\infty}^\infty
dx G^{(1,1)}_t(x,\tau) = 1$, we have $$\int dx \delta F(x,t)
= - u t + \frac{u^2}{2}\int_0^t dz_1 dz_2 \int_{-\infty}^\infty dx
G_t^{(2,1)}(x,t-z_1;x,t-z_2). \label{rg4.a}$$
Using Eqs. (\[response.A\]) and (\[G21.A\]), we get the following correction to $u$ $$\widetilde{u} = u [ 1 + C_u g_0 \log (t/a_\|) ] \label{rg4.b}$$ where $C_u$ is a finite positive constant given by Eq. (\[Cu\]), and $a_\| \approx (a/B)^{3/2}$ is the short distance cutoff along the length of the directed polymer (see Appendix B). The logarithmic divergence in Eq. (\[rg4.b\]) indicates the breakdown of the small-$u$ perturbation for $g_0 \log (t/a_\| ) \gg 1$. Thus the pinning potential is [*strongly relevant*]{} beyond the localization length $$l_{\|}= a_\| e^{1/g_0}, \label{lperp}$$ and the directed polymer becomes localized to the line defect.
Since the only quantity that suffers nontrivial renormalization is the pinning strength $u$, it is straightforward to form a renormalization-group recursion relation which allows one to obtain information about the polymer in dimensions $d > d_c$ using the calculations performed at the critical dimensionality $d_c = 1+1$. From Eq. (\[rg4.b\]) and with the rescaling $ t'= b t$, we obtain the recursion relation $$b\frac{dg}{db} = \epsilon(d) g + C_u g^2 \label{rr}$$ where $\epsilon(d) = 2 - \zeta(d) (d+1)$ as given in Eq. (\[eps\]). Thus the coupling constant $g(b)$, which is a dimensionless measure of the pinning strength at scale $b$, flows to large values from any non-zero initial value, leading to the pinning of the directed polymer to the line defect at large scales if $\epsilon \ge 0$, i.e., if $d \le 1+1$. But for $\epsilon < 0$ or $d > 1+1$, the pinning potential is only effective if its strength exceeds a critical value, $g_c = \epsilon/C_u$. Thus the phase diagram of the directed polymer is in fact given by Fig. 4(b), like that of the pure problem (Fig. 2), rather than Fig. 4(a) as suggested by naive scaling arguments and the uncontrolled Hartree calculation.
The Depinning Transition
------------------------
We now investigate the critical behavior of the directed polymer close to the depinning transition. Right at the depinning point $g_c$, the wandering exponent $\zeta_c$ and the energy exponent $\theta_c$ are simply $$\zeta_c = \zeta(d) \qquad {\rm and} \qquad \theta_c =
\theta(d) \label{expc}$$ to $O(\epsilon)$ since none of the parameters $\kappa$, $A$ and $B$ pick up divergent renormalization. The divergence of the correlation lengths, as one approaches the depinning transition from the pinned side, can be read off from Eq. (\[rr\]). We obtain $$l_\| =a_\| \left(\frac{g_0}{g_c}-1\right)^{-\nu_\|} \qquad {\rm and} \qquad
l_\perp =a \left(\frac{g_0}{g_c}-1\right)^{-\nu_\perp}$$ with the liberation exponents $$\nu_{\|} = 1 / |\epsilon|, \qquad {\rm and} \qquad \nu_\perp = \zeta_c \nu_\| =
\zeta(d)/|\epsilon|, \label{nu}$$ again valid to $O(\epsilon)$. In $d=2+1$ where $\zeta \approx
5/8$ and $\epsilon = 1/2$, Eq. (\[nu\]) gives $\nu_\perp \approx 1.25$ which is comparable to the results of numerical simulations: $\nu_\perp =1.3\pm 0.6$ by Balents and Kardar [@bk1], and $\nu_\perp =1.8\pm 0.6$ by Tang and Lyuksyutov [@tl]. It will be interesting to see whether Eqs. (\[expc\]) and (\[nu\]) might be valid to all orders in $\epsilon$ (up to some upper critical dimensions), as they do in the case of the pure problem without point disorders [@kolo2]. This can be directly probed numerically in $d=1+1$ by using correlated point disorder or by modifying the form of $U_p$ (see Sec. V).
The results Eqs. (\[expc\]) and (\[nu\]) have been conjectured earlier by Kolomeisky and Staley [@kolo1; @kolo2], who combined the renormalization-group flow equations for the pure depinning problem with $\eta = 0$ and the one for only point disorder with $U_p=0$ in an [*ad hoc*]{} way. In particular, Kolomeisky and Straley neglected to consider the renormalization of the stiffness $\kappa$, $B$, and the amplitude of the disorder potential $\Delta$ due to the pinning potential $U_p$. It is the lack of any divergent renormalization of these quantities, as obtained through explicit calculations in this study, that ensures the validity of Eq. (\[expc\]), at least to $O(\epsilon)$. The lack of such divergent renormalization follows from the large transverse wandering of the polymer at the depinning transition, thus diminishing the effect of the pinning potential. (Similar behaviors have been found for the renormalization of the surface tension of a wetting layer at the wetting transition [@wetting].)
Balents and Kardar [@bk2] obtained similar conclusions, Eqs. (\[expc\]) and (\[nu\]), by generalizing the directed polymer to a $D$-dimensional manifold, and then computing the analogy of the coefficients $C_\kappa$, $C_B$ and $C_u$ in a functional renormalization group (FRG) analysis to first order in $\delta = 4- D$. Since the $O(\epsilon)$ results (\[expc\]) and (\[nu\]) do not depend on the numerical values of the $C$’s, the FRG approach is effective despite the large expansion parameter ($\delta = 3$) for the directed polymer.
Finally, we comment on the validity of the mode coupling approximation used in the evaluation of the expressions leading to Eqs. (\[rg1.b\]), (\[rg3.b\]), (\[rg2.b\]) and (\[rg4.b\]). Due to the combination of a statistical tilt symmetry and a fluctuation dissipation theorem in $d=1+1$, the mode coupling approach gives the correct [*scaling*]{} behavior for the functions $G^{(m,n)}$ (see Refs. [@mode; @mode2] and Appendix A). This is all that is needed here since, as explained above, the results (\[expc\]) and (\[nu\]) are independent of the numerical values of the coefficients $C$’s. On the other hand, the mode coupling method is known to give very good quantitative results even for the scaling functions [@hhk; @mode; @mode2]. Thus the coefficient $C$’s computed in this way should be quantitatively accurate.
\[ Note added: Very recently, Kolomeisky and Staley [@kolo3] modified their renormalization-group analysis and obtained a power-law rather than exponential divergence of the localization length at the critical dimension $1+1$. We disagree with their conclusions and point out what we believe to be the key difference between Ref. [@kolo3] and the present work: The analysis of Ref. [@kolo3] is a one-loop calculation with respect to the [*pure*]{} directed polymer. The effect of point disorder is taken into account by an [*Ansatz*]{} which makes appropriate rescaling of the coefficients. On the other hand, the analysis presented in this section [*starts*]{} from the low-temperature fixed point of the randomly-pinned directed polymer, and takes into account of the extended pinning potential in a [*systematic*]{} treatment. The two approaches should eventually lead to the same scaling behavior (see Sec. IV.D). However, the approach of Ref. [@kolo3] requires a careful, consistent formulation. In particular, the choice of parameters \[Eq. (2.4)\] used in Ref. [@kolo3] implies that not all “temperatures” renormalize in the same way (since Galilean invariance must be respected). This introduces some ambiguities in the scaling ansatz used there. \]
Physical Interpretations
------------------------
As shown by the renormalization group analysis, the logarithmic divergence of the effective pinning strength $\widetilde{u}$ at the critical dimension is the single most important element leading to the phase diagram Fig. 4(b) and the exponents (\[expc\]) and (\[nu\]). Discussions of the preceding paragraphs also illustrates the robustness of these results — the same results are obtained from a variety of methods, some with drastic unjustified approximations, as long as the [*scaling*]{} behaviors are properly included. We shall now provide a physical picture of the weak localization using a phenomenological scaling theory, and argue that the logarithmic divergence of $\widetilde{u}$ should indeed be expected at the critical dimension.
In Sec. II, we established the energy gained by the directed polymer (due to the presence of a weak pinning potential) to be $\delta F_0 \propto u \int_0^t d\tau (B\tau)^{-d_\perp\zeta}\sim t^{1/3}$ in $1+1$ dimensions. This corresponds to describing the optimal path of the directed polymer as a generalized random walk (with the wandering exponent $\zeta$ rather than $1/2$), and then equating the energy gained to the accumulated return probability of the random walk. However, such a treatment of the optimal path is oversimplified. As discussed in detail in Ref. [@hf], while the optimal path in a typical sample (or a typical region of a very large sample) is unique, there is a nonzero probability that there exists a different path whose free energy is within $O(1)$ of that of the optimal path. The probability of finding two such paths a distance $\Delta$ apart is $p(\Delta) \sim \Delta^{-3/2}$ in $1+1$ dimension [@hf], and the “droplet” formed (i.e., the difference between the two paths) typically have a length $\tau \sim \Delta^{1/\zeta}$, which is of $O(\Delta^{3/2})$ in $1+1$ dimension. Thus, if one of the optimal path encounters the origin, then the probability of having a nearly degenerate optimal path also encountering the origin is $p(\tau^{2/3}) \sim
\tau^{-1}$ in $1+1$ dimensions. In this way, we see that the [*accumulated*]{} effect of the statistically droplets existing at different scales leads to a logarithmic divergence, i.e., $\int^t d\tau p(\tau^{2/3}) \sim
\log t$. This makes the columnar pin marginally relevant at the critical dimension, and results in a phase diagram of the type depicted in Fig. 4(b) rather than that in Fig. 4(a). Note that the important ingredients of the above argument are only the probability of droplet formation and the shape of the droplets. It therefore suggests the marginal relevance of an extended defect at the critical dimension to be a general consequence of the droplet scaling theory.
Technically, the droplets manifest themselves in the expression describing the renormalization of the pinning strength in Eq. (\[rg4.a\]), a diagrammatic representation of which is shown in Fig. 6. The similarity of the droplet configuration in Fig. 5 and the loop diagram in Fig. 6 is striking. As explained in detail in Ref. [@hf], the distribution of droplets is given by the function $G^{(2,1)}$. However, the particular distribution discussed in Ref. [@hf] has to do with droplets with a given width $\Delta$ at the same vertical coordinate, whereas the distribution needed here is the one with a given length $\tau$ at the same transverse coordinate. Arguments leading to the logarithmic divergence in the preceding paragraphs assumes $\tau \sim \Delta^{1/\zeta}$. This is validated by the more detailed calculation given in Appendix B.
In some ways, the depinning transition here is similar to the de Almeida-Thouless [@AT] line in a spin glass. The columnar pin which breaks the statistical translational symmetry of the directed polymer plays a role similar to the external magnetic field which breaks the statistical up-down symmetry of the Ising spin glass. It should be interesting to study the depinning transition in the replica formalism. The encounter of replica symmetry breaking alluded to in Sec. III is no longer mysterious now — it is simply how the droplet excitations are manifested within the replica formalism [@mp; @hf].
Related Depinning Problems
==========================
The results of Sec. IV can be easily generalized to long range correlated point disorders, other forms of pinning potentials, and higher dimensional elastic manifolds. Here we shall describe a few interesting cases to illustrate the method.
Following Nattermann [@nat2] and Medina [*et al*]{} [@medina], we consider a random potential with long-range correlations of the form $$\overline{\eta ({{\bf x}},z) \eta ({{\bf x}}', z')} = 2 \Delta \delta (z - z')
{|{{\bf x}}-{{\bf x}}'| }^{2\rho - d +1}. \label{corr}$$ We shall be focus on the case $d=1+1$, where the model (\[H0\]) describes the domain wall of the low temperature phase of a random Ising model in two dimensions. The correlator (\[corr\]) extrapolates smoothly between the case of random bond ($\rho=0$) and random field ($\rho=1$) [@medina; @nat2]. It is known that the roughness exponent $\zeta (\rho)$ stays at its value of $2/3$ for short range correlated disorder as long as $\rho \le 1/4$. For $ 1 \ge \rho \ge 1/4$ it takes on the Flory value of $\zeta(\rho) = 3/{(5-2\rho)}$. From Eq. (\[rr\]) and $\epsilon
= 2-\zeta(d+1)$, we see that $\epsilon = 0$ for $\rho \le 1/4$, and $\epsilon = (1-4\rho)/(5-2\rho) < 0$ for $\rho > 1/4$. Thus for $\rho >
1/4$, a small pinning potential is irrelevant. As one varies $u$, there will be a depinning transition with exponents $ \nu_\| = 1/|\epsilon| = (5-2\rho)/(4\rho-1)$ and $\nu _{\perp} = \nu _{\|}\zeta_c = 3/{(4 \rho - 1)}$ to $O(\epsilon)$. If the exponent $\nu_\|=1/|\epsilon|$ is indeed exact to all orders as in the pure case, then the above expression will be valid for all $1\ge \rho > 1/4$. In particular we should obtain a depinning transition with the exponents $\nu _{\perp} = \nu _{\|} = 1$ for $\rho=1$. A numerical simulation of the depinning transition of the domain wall of the 2d random field Ising model should therefore be an efficient way of probing the “exactness" of the $O(\epsilon)$ result.
We can generalize the methods described in this paper to study the pinning of a directed polymer by other forms of the pinning potential $U_p({{\bf r}})$. If we write ${\cal H}_1 = \int d^{d_\perp}{{\bf r}}U_p({{\bf r}})
\int_0^t dz \delta^{d_\perp}[{{\bf r}}-{\mbox{\boldmath $\xi$}}(z)]$, then the lowest order correction to the free energy is $$\delta F_0(t) = \overline{\langle{\cal H}_1 \rangle}
= \int_0^t dz \int d^{d_\perp}{{\bf r}}\ U_p({{\bf r}})\ G^{(1,1)}_t({{\bf r}},t-z), \label{df2}$$ where $G^{(1,1)}$ is the one-point distribution function defined in Eq. (\[G\]), with the scaling properties described in Appendix A. Suppose $U_p({{\bf r}})$ has a long tail, i.e., $U_p = u/r^s$, then $\delta F_0(t) \sim t^{1-s\zeta}$. Comparing this to $\Delta F_0 \sim
t^{\theta}$ and recalling the exponent identity Eq. (\[id\]), we find the critical dimension of $U_p$ to be $$2 = \zeta(d_c) (2+s).$$ We expect a depinning transition at finite $u$ above the critical dimension $d_c$, with the liberation exponent $\nu_\| =
1/|2-\zeta(d)(2+s)|$.
We may also consider a pinning potential $U_p({{\bf r}}) = -u \delta^{d_\perp}
[{{\bf r}}- {\bf R}(z)]$, where the trajectory of the pin ${\bf R}(z)$ is itself an arbitrary (but fixed) function of $z$. For example, we may have $R(z) \sim z^{\zeta_R}$, describing the quenched defect trajectory of a superconductor subject to random collision by heavy ions. In this case, Eq. (\[df2\]) becomes $$\delta F_0(t) = - u \int_0^t dz \ G_t({\bf R}(z),t-z).$$ If $R(z) < |t-z|^\zeta$ or $\zeta_R < \zeta$, i.e., if the transverse fluctuation of the defect trajectory is smaller than that of the randomly pinned directed polymer, then the $z$-dependence of the pinning potential is irrelevant and we have $\delta F_0(t) \sim u t^{1- d_\perp \zeta}$ as before, and the defect acts like a straight columnar pin. However, if $R(z) > |t-z|^\zeta$, as for example is the case for a misoriented columnar pin where $R(z) \propto |t-z|$ [@splay], then $G^{(1,1)}$ is sharply cutoff and $\delta F_0(t)$ becomes finite for large $t$. In this case, a weak pinning potential is irrelevant compared to the random energy gain $\Delta F_0 \sim t^\theta$. This type of analysis can be extended to study a large variety of pinning potentials, including the case where ${\bf R}(z)$ itself is the trajectory of another directed polymer [@hwa].
Another interesting ramification of our considerations pertains to $D$-dimensional oriented manifolds in $d$ embedding dimensions, i.e., $\vec{z} \in \Re^D$ and ${\mbox{\boldmath $\xi$}}(\vec{z}) \in \Re^{d-D}$. $D=1$ describes a directed polymer or a flux line, and $D=2$ describes, for example, the domain wall of a three dimensional ferromagnet. The free energy scales in this case with an exponent $\theta = 2\zeta+ D-2$, which is the $D$-dimensional analog of the exponent identity (\[id\]). Following Ref. [@bk1], we also generalize the extended defect to be $n$-dimensional, i.e., $U_p({{\bf r}}) = -u\delta^{d-n}({{\bf r}})$, and ${\cal H}_1 = \int d^D \vec{z} \
U_p[{\mbox{\boldmath $\xi$}}(\vec{z})]$. The columnar pinning potential discussed corresponds to $n=1$, and planar defects such as grain boundaries correspond to $n=2$. Straightforward generalization of the scaling arguments of Sec. II and Eq. (\[dF\]) gives $\delta F_0(t) = \overline{\langle{\cal H}_1 \rangle} \sim - u t^{D
- \zeta(d-n)}$, where $t$ is now the linear size of the manifold. Comparing this to $\Delta F_0 \sim t^\theta$, we find the condition for the relevance of $U_p$ to be given by $$2 > \zeta (d)(2 + d - n). \label{dc.2}$$ An important application here is the pinning of an interface to a planar defect, say the localization of a domain wall to a grain boundary in a random ferromagnet. In this case, $D=n = 2$, $d=3$, and Eq. (\[dc.2\]) reads $\zeta (3) < 2/3$. For the interface of a random bond Ising magnet, this is always fulfilled and the interface is always pinned by the planar defect. For random field systems however, $\zeta(3) = 2/3$ exactly in 3 dimensions, and our criterion (\[dc.2\]) is inconclusive. Straightforward generalization of the scaling argument given in Sec. IV.D shows that the interface is again weakly pinned by an arbitrary weak pinning potential.
We are grateful to many helpful discussions with L. Balents, D. S. Fisher, T. Halpin-Healy, M. Kardar, E. B. Kolomeisky, J. Krug, D. R. Nelson, and L.-H. Tang. TH is supported by US Department of Energy Contract No. DE-FG02-90ER40542. TN acknowledges the financial support of the Volkswagen Foundation and the hospitality of Harvard University where a part of this work was done.
Useful Relations for the directed polymer
=========================================
In order to make the paper more self-contained, we summarize here some useful results on the randomly pinned directed polymer obtained in earlier publications [@hf; @medina; @mode]. The restricted partition function of a directed polymer of length $t$ with one end ${\mbox{\boldmath $\xi$}}(t)$ fixed at ${{\bf x}}$ is $$Z_0({{\bf x}},t) = \int {\cal D}[{\mbox{\boldmath $\xi$}}]\delta^{d_\perp}[{{\bf x}}-{\mbox{\boldmath $\xi$}}(t)]
e^{-{\cal H}_0/T}.$$ The free energy $F_0({{\bf x}},t) = -T \log Z_0({{\bf x}},t)$ fulfills the noisy-Burgers’ equation [@fns; @kpz; @medina] $$\partial_t F_0({{\bf x}},t) = \frac {T}{2\kappa}{{\mbox{\boldmath $\nabla$}}}^2 F_0
- \frac{1}{2\kappa}({\mbox{\boldmath $\nabla$}}F_0)^2 + \eta({{\bf x}},t). \label{kpz.A}$$
In Sec. IV, we introduced various correlation and distribution functions to evaluate the effect of the pinning potential ${\cal H}_1$. To begin with, the one-point distribution function is $$\overline{\langle\delta[{\mbox{\boldmath $\xi$}}(z) - {{\bf y}}]\rangle}_{{{\bf x}},t}
= G_t^{(1,1)} ( {{\bf x}}- {{\bf y}}, t-z ).$$ Again, $\langle\ldots\rangle_{{{\bf x}},t}$ denotes thermal average using the partition function $Z_0({{\bf x}},t)$ and the subscript $t$ is used in $G^{(1,1)}$ to denote the explicit $t$-dependence of the distribution function. By simple scaling and normalization requirements, we have $$G^{(1,1)}_t({{\bf r}},\tau) = (B\tau^\zeta)^{-d_\perp}
{\widetilde{g} }_{t/\tau}(r/B\tau^\zeta),\label{response.A}$$ where ${\widetilde{g} }_{t/\tau}(0)$ is finite, ${\widetilde{g} }_{t/\tau}(s)$ decreases sharply for $s \gg 1$, and $\int d^{d_\perp} s \ {\widetilde{g} }_{t/\tau}(s) = 1$ for all $t/\tau$. The rms displacement $X^2_t(\tau)$ is given by the second moment, $$\begin{aligned}
&X^2_t(\tau) &\equiv \int d^{d_\perp}{{\bf r}}\ {{\bf r}}^2 G_t^{(1,1)}({{\bf r}},\tau)
\nonumber\\
& &= B^2 \tau^{2\zeta} \int d^{d_\perp}s\ s^2 {\widetilde{g} }_{t/\tau}(s),\end{aligned}$$ which is a weak function of $t/\tau$. It is found numerically [@kim; @hhk] that $X^2_t(\tau)/\tau^{2\zeta}$ is the same order of magnitude for $t = \tau$ and $t \gg \tau$. We fix $B$ by the rms displacement of the free end, $X^2_t(t) = B^2
t^{2\zeta}$ (see Eq. (\[Xrms\])). This is accomplished by choosing $$\int d^{d_\perp}s \ s^2 {\widetilde{g} }_1(s) = 1. \label{width.A}$$
Other than $G^{(1,1)}$, we are also interested in the higher-order distribution functions, $$\begin{aligned}
&G_t^{(2,1)}({{\bf x}}-{{\bf y}}_1,t-z_1;&{{\bf x}}-{{\bf y}}_2,t-z_2) = -\frac{1}{T}
\overline{\langle\delta[{\mbox{\boldmath $\xi$}}(z_1) - {{\bf y}}_1] \delta[{\mbox{\boldmath $\xi$}}(z_2) - {{\bf y}}_2 ]
\rangle_{{{\bf x}},t}^{(c)}}\ ,\\
&G_{[t_1,t_2]}^{(1,2)}({{\bf x}}_1-{{\bf x}}_2,t_1-t_2;&\frac{{{\bf x}}_1+{{\bf x}}_2}{2}-{{\bf y}},
\frac{t_1+t_2}{2}-z) \nonumber \\
& &= \left[\overline{F_0(x_1,t_1)\langle\delta[\xi(z)-{{\bf y}}]
\rangle_{{{\bf x}}_2,t_2}}
+ \overline{F_0(x_2,t_2)\langle\delta[\xi(z)-{{\bf y}}]
\rangle_{{{\bf x}}_1,t_1} } \right] ,\end{aligned}$$ as well as the free energy correlation function [@corr.note] $$C_{[t_1,t_2]}({{\bf x}}_1-{{\bf x}}_2,t_1-t_2)
= \overline{ [F_0({{\bf x}}_1,t_1) - F_0({{\bf x}}_2,t_2)]^2 },$$ where the subscript $[t_1,t_2]$ denotes the larger of $t_1$ and $t_2$. In analogy to $G^{(1,1)}$, the correlation function has a similar scaling form, $$C_t({{\bf r}},\tau) = A^2 t^{2\theta} {\widetilde{c} }_{t/\tau}(r/B\tau^\zeta).$$ The scaling function ${\widetilde{c} }$ gives the following scaling properties for $C$, $$C_t({{\bf r}},\tau) \left\{
\begin{array}{ll}
= 2 A^2 t^{2\theta} & B\tau^\zeta \ll B t^\zeta \ll r, \\
\propto 2 A^2 (r/B)^{2\theta/\zeta} & B\tau^\zeta \ll r \ll B t^\zeta, \\
\propto 2 A^2 \tau^{2\theta} & r \ll B\tau^\zeta \ll B t^\zeta.
\end{array}
\right. \label{corr.A}$$ For convenience, we also write the correlation function as $$C_{[t_1,t_2]}({{\bf x}}_1-{{\bf x}}_2,t_1-t_2) = G^{(0,2)}_{t_1}(0,0) +
G^{(0,2)}_{t_2}(0,0) - 2 G^{(0,2)}_{[t_1,t_2]}({{\bf x}}_1-{{\bf x}}_2,t_1-t_2) \label{G02}$$ where $$G^{(0,2)}_{[t_1,t_2]}({{\bf x}}_1-{{\bf x}}_2,t_1-t_2)
= \overline{F_0({{\bf x}}_1,t_1)F_0({{\bf x}}_2,t_2)}.$$ The scaling properties of $G^{(0,2)}_t({{\bf r}},\tau)$ is easily obtained from Eq. (\[corr.A\]). For example, from Eq. (\[G02\]) and $G^{(0,2)}_t({{\bf r}}\to \infty,\tau) \to 0$, we have $G^{(0,2)}_t(0,0) = \frac{1}{2}C_t(\infty) = A^2 t^{2\theta}$. Below, we shall provide the approximate forms for all of the $G^{(m,n)}$’s.
It was shown in Ref. [@hf] that $G^{(m,n)}$ can be obtained simply by adding a source term ${\widetilde{J} }({{\bf x}},t)$ to right hand side of the equation of motion (\[kpz.A\]) and then taking appropriate derivatives, i.e., $$G^{(m,n)} = \overline{\frac{\delta}{\delta{\widetilde{J} }({{\bf y}}_1,z_1)}\ldots
\frac{\delta}{\delta{\widetilde{J} }({{\bf y}}_m,z_m)} [F_0({{\bf x}}_1,t_1) \ldots F_0({{\bf x}}_n,t_n)]}.$$ In the context of the stochastic hydrodynamics of the noisy-Burgers’ equation, the above is nothing but the generalized [*response function*]{}. In Ref. [@hf], it was shown that the nonlinear response function $G^{(2,1)}$ gives the statistics of the rare but singular “droplet excitations”, which are connected to replica symmetry breaking in the replica formalism [@p; @mezard]. Here we encounter them again in perturbation theory, as we already did when using the replica formalism in Sec. III.
However, unlike Sec. III where we failed to develop a perturbative expansion due to the lack of knowledge of replica-symmetry broken excited states, here we can construct the forms of the nonlinear response functions $G^{(m,n)}$ rather straightforwardly by exploiting a Fluctuation Dissipation Theorem (FDT), which the equation of motion (\[kpz.A\]) satisfies in $1+1$ dimensions [@medina; @mode]. For example, the FDT gives $$\frac{\partial}{\partial x} \frac{\partial}{\partial y}
G^{(0,2)}_t(x-y,\tau)
= \frac{\Delta\kappa}{T} G^{(1,1)}_t(x-y,\tau), \qquad \tau > 0. \label{fdt.A}$$ Taking the limit $\tau \to 0$ in Eq. (\[fdt.A\]) and using the definition of $G^{(1,1)}$, we immediately obtain $$\overline{\partial_x F_0(x,t) \partial_y F_0(y,t)} = \frac{\Delta\kappa}{T} \delta(x-y).$$ Similarly, by integrating Eq. (\[fdt.A\]) and using the scaling forms for $G^{(1,1)}$, we can recover the scaling properties of $C$ given in Eq. (\[corr.A\]).
The combination of the FDT and a Galilean invariance (corresponding to the statistical rotational symmetry) in $1+1$ dimensions allows one to use a mode-coupling scheme [@mode; @mode2; @bks] to obtain the forms of the functions $G^{(m,n)}$. In particular, $G^{(1,1)}$ and $G^{(0,2)}$ are given, to a very good approximation, by the following set of self-consistent integral equations, $$\begin{aligned}
&G_t^{(1,1)}(x-y,t-z) &= {\widehat{G} }^{(1,1)}(x-y,t-z) \nonumber\\
& &+ \frac{1}{\kappa^2} \int^\infty_{-\infty} dx' dy'
\int^t_0 dt' \int_0^{t'} dz'{\widehat{G} }^{(1,1)}(x-x',t-t')
\frac{\partial}{\partial x'}G_{t'}^{(1,1)}(x'-y',t'-z') \nonumber \\
& &\quad\frac{\partial}{\partial x'}\frac{\partial}{\partial y'}
G_{t'}^{(0,2)}(x'-y',t'-z')
\frac{\partial}{\partial y'} G_{z'}^{(1,1)}(y'-y,z'-z), \label{mc.1}\\
&G_{[t_1,t_2]}^{(0,2)}(x_1-x_2,t_1-t_2) &=
{\widehat{G} }^{(0,2)}_{[t_1,t_2]}(x_1-x_2,t_1-t_2) \nonumber \\
& &+ \frac{1}{2\kappa^2} \int_{-\infty}^\infty dx_1' dx_2' \int^{t_1}_0 dt_1'
\int_0^{t_2} dt_2' G^{(1,1)}_{t_1}(x_1-x_1',t_1-t_1') \nonumber \\
& &\quad G^{(1,1)}_{t_2}(x_2-x_2',t_2-t_2')
\left[\frac{\partial}{\partial x_1'}\frac{\partial}{\partial x_2'}
G_{[t_1',t_2']}^{(0,2)}(x_1'-x_2',t_1'-t_2') \right]^2, \label{mc.2}\end{aligned}$$ where $${\widehat{G} }^{(1,1)}(r,\tau) = \sqrt{\frac{T}{2\pi\kappa\tau}}
\exp\left[-\frac{T}{\kappa}\frac{r^2}{\tau}\right], \qquad \tau > 0$$ is the “bare” response function, and $${\widehat{G} }^{(0,2)}_{[t_1,t_2]}(x_1-x_2,t_1-t_2)
= 2\Delta \int_{-\infty}^{\infty} dx' \int_0^{t_2}dt'
{\widehat{G} }^{(1,1)}(x_1-x',t_1-t'){\widehat{G} }^{(1,1)}(x_2-x',t_2-t')$$ is the “bare" correlation function. The mode-coupling equations (\[mc.1\]) and (\[mc.2\]) can be solved by using the scaling forms (\[response.A\]) and (\[corr.A\]) for $G^{(1,1)}$ and $G^{(0,2)}$. The scaling functions ${\widetilde{g} }$ and ${\widetilde{c} }$ obtained in this way are in very good agreement with those from numerical simulations [@kim; @hhk; @mode]. It is found that ${\widetilde{g} }_\sigma(s)$ is approximately a Gaussian with a weak $\sigma$ dependence, and the width of the “Gaussian" is fixed by the condition (\[width.A\]) to be $1$.
The functions $G^{(1,1)}$ and $G^{(0,2)}$ can now be used to construct higher order response functions via the mode-coupling scheme. For instance, $$\begin{aligned}
&G_t^{(2,1)}(x-y_1,t-z_1;&x-y_2,t-z_2) \nonumber \\
& &=-\frac{1}{\kappa}\int_{-\infty}^\infty dx' \int_0^t dt'
G_t^{(1,1)}(x-x',t-t') \nonumber \\
& & \quad \frac{\partial}{\partial x'}G_{t'}^{(1,1)}(x'-y_1,t'-z_1)
\frac{\partial}{\partial x'}G_{t'}^{(1,1)}(x'-y_2,t'-z_2) \label{G21.A}\\
&G_{[t_1,t_2]}^{(1,2)}(x_1-x_2,t_1-t_2;
&\frac{x_1+x_2}{2}-y,\frac{t_1+t_2}{2}-z) \nonumber \\
& &= -\frac{1}{\kappa} \int_{-\infty}^\infty dx_1' \int_0^{t_1} dt_1'
G_t^{(1,1)}(x_1-x_1',t_1-t_1') \nonumber \\
& & \quad \quad\frac{\partial}{\partial x_1'}G_{t_1'}^{(1,1)}(x_1'-y,t_1'-z)
\frac{\partial}{\partial x_1'}G_{[t_1',t_2]}^{(0,2)}(x_2-x_1',t_2-t_1')
\nonumber \\
& & \quad + {\rm permutation of} [ (x_1,t_1) \leftrightarrow (x_2,t_2) ]. \label{G12.A}\end{aligned}$$ The validity of the above mode coupling approximation for $G^{(2,1)}$ was discussed in detail in Ref. [@hf]. Eqs. (\[G21.A\]) and (\[G12.A\]) should capture the key scaling properties but may not be quantitatively accurate. We shall nevertheless use the above expressions to evaluate the integrals obtained in the RG analysis of Sec. IV. As our main concern is the existence of logarithmic divergence in the renormalization of various parameters, rather than the numerical value of any particular integrals, the use of the mode coupling approximation should be adequate.
Results of Calculations
=======================
In this appendix, we compute the perturbative effect of ${\cal H}_1$ on the parameters $\kappa$, $B$, and $u$, by using the expression for $G^{(m,n)}$’s obtained from the mode coupling approximation (see Appendix A). We are particularly interested in the $t$-dependence of the renormalized parameters $\tilde\kappa(t)$, $\tilde B(t)$ and $\tilde u(t)$ in the limit $t\to \infty$.
We start with the renormalization of the stiffness $\kappa$. From Eq. (\[rg1.a\]) and the scaling form (\[response.A\]) for $G^{(1,1)}$, we have $$\begin{aligned}
&\widetilde{\kappa}^{-1} & = \kappa^{-1} - u \int_0^t \frac{dz}{t}
\frac{(t-z)^2}{\kappa^2[B(t-z)^\zeta]^3} \frac{\partial^2}{\partial s^2}
{\widetilde{g} }_{t/(t-z)}(s)|_{s=0} \nonumber \\
& & = \frac{1}{\kappa} -\frac{u}{\kappa^2 B^3} \int_0^1 d\sigma
{\widetilde{g} }''_{1/\sigma}(0), \label{tkappa}\end{aligned}$$ where ‘primes’ indicate derivatives of ${\widetilde{g} }$ and $\zeta = 2/3$ in $d=1+1$. Thus we obtain Eq. (\[rg1.b\]) with $g_0 = u/(\kappa B^3)$ and $$C_\kappa =\int_0^1 d\sigma {\widetilde{g} }''_{1/\sigma}(0) \label{Ckappa}$$ which is finite.
Next we consider the renormalization of the transverse wandering coefficient $B$. From Eq. (\[rg3.a\]), we have $$\widetilde{B}^2 t^{2\zeta} = B^2 t^{2\zeta} + I_B,$$ with $$\begin{aligned}
&I_B &= u \int_{-\infty}^{\infty} dy \ y^2 \int_0^t dz \
G_t^{(2,1)}(0,t-z;y,t) \nonumber \\
& &= -\frac{u}{\kappa} \int_0^t dz \int_z^t dt' \int_{-\infty}^\infty dx'
\int_{-\infty}^\infty dy \ y^2 G_t^{(1,1)}(x',t-t') \nonumber \\
& & \quad \frac{\partial}{\partial x'} G_{t'}^{(1,1)}(x'-y,t')
\frac{\partial}{\partial x'} G_{t'}^{(1,1)}(x',t'-z), \label{IB}\end{aligned}$$ where we used the mode-coupling approximation Eq. (\[G21.A\]) for $G^{(2,1)}$. Using the scaling form (\[response.A\]) for $G^{(1,1)}$, and noting that ${\widetilde{g} }(s)$ is symmetric in $s$, we find $$I_B = - (B^2 t^{2\zeta}) 2 g_0 C_B$$ where $$C_B = \int_0^1 \frac{d\tau}{\tau^\zeta} \int_0^1 d\sigma \int_{-\infty}
^\infty ds (-s) {\widetilde{g} }'_{(1-\tau)/\sigma}(s) \
{\widetilde{g} }_{1/\tau}( s \sigma^\zeta/\tau^\zeta). \label{CB}$$ Again $C_B$ is finite since ${\widetilde{g} }$ are normalized and sharply cutoff for large argument. We thus obtain the result Eq. (\[rg3.b\]).
Finally, we consider the renormalization of $u$. The expression given by Eq. (\[rg4.a\]) can be described diagrammatically as in Fig. 6. If we use the mode-coupling approximation Eq. (\[G21.A\]) for $G^{(2,1)}$, and note the normalization condition $\int dx G_t^{(1,1)}(x,\tau) = 1$, we find $$\begin{aligned}
& \delta F(t) &= - u t+ \frac{u^2}{2} \int_0^t dz_1 dz_2
\int_{-\infty}^\infty dx G_t^{(2,1)}(x,t-z_1;x,t-z_2)\nonumber \\
& &= - u t - \frac{u^2}{2\kappa } \int_0^t dt' \int_0^{t'} dz_1 \int_0^{t'} dz_2
\int_{-\infty}^\infty dx' \frac{\partial}{\partial x'}
G_t^{(1,1)}(x',t'-z_1) \frac{\partial}{\partial x'}
G_t^{(1,1)}(x',t'-z_2).\end{aligned}$$ Using the scaling form (\[response.A\]) for $G^{(1,1)}$ again, we obtain $$\delta F(t) = - u t - \frac{u^2}{2\kappa B^3} \int_0^t dt' I_u(t'),
\label{dFt}$$ where $$I_u(\tau) = \int_0^\tau \frac{dt_1}{t_1^\zeta} \frac{dt_2}{t_2^{2\zeta}}
\int_{-\infty}^\infty ds \ {\widetilde{g} }'_{\tau/t_1}(s)
\ {\widetilde{g} }'_{\tau/t_2}(s t_1^\zeta/t_2^\zeta).
\label{Iu}$$ We shall see that $I_u$ is actually divergent. To regularize the integral, we insert a ultra-violet cutoff scale $a_\| \propto (a/B)^{1/\zeta}$, since in our model, the columnar pin $U_p$ is really a potential well of finite size $a$; it is only approximated by a delta function at scales much larger than $a$. Eq. (\[Iu\]) then becomes $$I_u(\tau) = \int_{a_\|}^\tau\frac{dt_1}{t_1} F(t_1/\tau,a_\|/\tau),
\label{Iu.2}$$ where $$F(\hat{t},\hat{a}) = \int_{\hat{a}/\hat{t}}^{1/\hat{t}} d\sigma
\sigma^{-2\zeta} \int_{-\infty}^{\infty} ds \ {\widetilde{g} }'_{1/\hat{t}}(s)
\ {\widetilde{g} }'_{1/(\sigma\hat{t})}(s/\sigma^\zeta).$$ Clearly the divergent part of $I_u(\tau)$ comes from the limit $\tau/a_\|\to 0$ in Eq. (\[Iu.2\]). So to leading order, we have $I_u(\tau) = \log(\tau/a_\|) C_u$, where $$\begin{aligned}
&C_u &= \lim_{\hat{a}\to 0} F(\hat{a},\hat{a}) \nonumber \\
& &= \lim_{\hat{a}\to 0} \int_1^{1/\hat{a}} \frac{d\sigma}{\sigma^{2\zeta}}
\int_{-\infty}^\infty ds \ {\widetilde{g} }'_{1/\hat{a}}(s)
\ {\widetilde{g} }'_{1/(\sigma\hat{a})}(s/\sigma^\zeta) \nonumber \\
& &\approx \int_1^{\infty} \frac{d\sigma}{\sigma^{2\zeta}}
\int_{-\infty}^\infty ds \ {\widetilde{g} }'_{\infty}(s) \ {\widetilde{g} }'_{\infty}(s/\sigma^\zeta),
\label{Cu}\end{aligned}$$ which is positive definite since $2\zeta = 4/3 > 1$ and ${\widetilde{g} }$ is well behaved. The effective pinning potential $\tilde u$ can now be defined as $$\tilde u \equiv -\frac{\partial}{\partial t} \delta F(t)
= u [ 1 + g_0 C_u \log(t/a_\|) ],$$ which is quoted in Eq. (\[rg4.b\]).
On leave from: Department of Physics, SUNY Stony Brook, Stony Brook, NY 11794.
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To simplify notation, we define our energy scale such that the mean free energy is zero, i.e., $\overline{F_0({{\bf x}},t)} = 0$.
|
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abstract: 'The effects of the initial state interactions on the $K^--p$ radiative capture branching ratios are examined and found to be quite sizable. A general coupled-channel formalism for both strong and electromagnetic channels using a particle basis is presented, and applied to all the low energy $K^--p$ data with the exception of the [*1s*]{} atomic level shift. Satisfactory fits are obtained using vertex coupling constants for the electromagnetic channels that are close to their expected SU(3) values.'
address:
- 'California State Polytechnic University, Pomona CA 91768'
- |
Service de Physique Nucléaire, CEA/DSM/DAPNIA,\
Centre d’Études de Saclay, F-91191 Gif–sur–Yvette, France
author:
- 'Peter B. Siegel'
- Bijan Saghai
title: ' Initial State Interactions for $K^-$-Proton Radiative Capture'
---
=1.7 in
PACS numbers: 13.75.Jz, 24.10.Eq, 25.40.Lw, 25.80.Nv
**Introduction**
================
The $K^-$-proton interaction is a strong multichannel process \[1\], with the $\Lambda (1405)$ resonance just below the $K^--p$ threshold at 1432 MeV. At low energies, the $K^-$ can elastically scatter off the proton, charge exchange to $\overline{K^0}-n$, or scatter to ($\Sigma^+ \pi^-$), ($\Sigma^0 \pi^0$), ($\Sigma^- \pi^+$) or ($\Lambda \pi^0$) final states. In addition, the electromagnetic radiative capture processes $K^- p \rightarrow
\Lambda \gamma (\Sigma^0 \gamma)$ are also possible \[2\]. Besides, the amplitudes of these latter reactions can be related to those of the associated strangeness photoproduction, i.e. $\gamma p \rightarrow
K^+ \Lambda (K^+ \Sigma^0)$, by the crossing symmetry. Much effort has been done experimentally and theoretically to understand this system. In particular, experiments to measure the branching ratios of the radiative capture reactions $K^-p \rightarrow \Lambda \gamma$ and $K^-p \rightarrow \Sigma^0 \gamma$ were recently performed \[3\] to help clarify the details of the reaction mechanism, with a special interest in the nature of the $\Lambda (1405)$ resonance \[4\]. However, on one hand the result for the $\Lambda \gamma$ channel is unexpectably smaller than both the previous measured value \[5\] and those obtained through phenomenological models \[2\], and on the other hand the measured branching ratio for the $\Sigma^0 \gamma$ final states comes out significantly higher than the one for the other channel, producing yet another mystery to this already complicated problem.
The most recent measurements of the threshold branching ratios with stopped kaons, done at Brookhaven \[3\] are:
$$R_{\Lambda \gamma} = {{\Gamma(K^-p \rightarrow \Lambda \gamma)} \over
{\Gamma(K^-p \rightarrow all)}} = .86 \pm .07 \pm .09 \times
10^{-3}~,$$ and
$$R_{\Sigma \gamma} = {{\Gamma(K^-p \rightarrow \Sigma^0 \gamma)} \over
{\Gamma(K^-p \rightarrow all)}} = 1.44 \pm .20 \pm .11 \times
10^{-3}.$$
Existing calculations \[2\] overestimate $R_{\Lambda \gamma}$ by a factor of 3 or 4 (except a few recent phenomenological analysis \[6,7\] of the kaon photoproduction processes). The pioneer calculations considered the $\Lambda (1405)$ in two different ways: as an [*s*]{}-channel resonance \[8\] or as a quasi-bound ($K-N,\Sigma \pi$) state \[9-11\]. In the quark-model approaches, this hyperon is considered as a pure $q^3$ state \[12,13\], a quasi-bound $\overline{K}$N state \[14,15\] or still as a hybrid ($q^3$+$q^4$$\overline{q}$...) state \[16-18\]. A number of potential model fits to the scattering data incorporate the $\Lambda (1405)$ as a quasi-bound ($K-N,\Sigma \pi$) resonance \[19-21\]. It would be interesting to see if the radiative capture channels can also be understood within a single model. Specially, since the radiative capture data were taken to help distinguish between these two possibilities.
The diversified nature of the low energy data challenges theoretical models. Even if the analysis is restricted to the hadronic sector, difficulties arise when trying to understand the $K^-p$ [*1s*]{} atomic level shift. The sign of the $K^-p$ scattering length, extracted from this experiment, is opposite to that determined from K-matrix and potential model fits to the other hadronic data. This conflict poses interesting questions which are discussed in Ref.\[19,20,22\]. Only one potential, Ref. \[20\], has been published which is compatible with all the low energy hadronic data. We will examine the initial state interactions for the radiative capture branching ratio for this potential.
Calculations which focus on the radiative capture branching ratios usually do not include the initial state interactions. Only one group \[15\], which uses the cloudy bag quark model, has included these in the electromagnetic branching ratio calculation. Since the interactions are strongly coupled among the various channels, any meaningful comparison with the data needs to include channel couplings. We find here that the effect of the initial state interactions is far from being negligible. One limitation with this latter calculation is that no comparison is made with the strong branching ratio data. The other threshold branching ratios are \[23,24\]:
$$\gamma = {{\Gamma(K^-p \rightarrow \pi^+ \Sigma^-)} \over
{\Gamma(K^-p \rightarrow \pi^- \Sigma^+)}} = 2.36 \pm .04 ,$$
$$R_c = {{\Gamma(K^-p \rightarrow charged ~particles)} \over
{\Gamma(K^-p \rightarrow all)}} = .664 \pm .011 ,$$
and
$$R_n = {{\Gamma(K^-p \rightarrow \pi^0 \Lambda)} \over
{\Gamma(K^-p \rightarrow ~all~ neutral~ states)}} = .189 \pm .015 .$$
They put tight constraints on the threshold amplitudes and potential coupling strengths \[17,20,21\]. In fact, the five branching ratios are amongst the most precise data in the strangeness sector. However, at present there is no comprehensive analysis which includes both the hadronic and electromagnetic branching ratios of the $K^-p$ system.
The aim of this paper is to examine all the low energy data within a single model and determine if it can be understood using known coupling strengths and minimal SU(3) symmetry breaking for relevant vertices in the electromagnetic channels. In doing so, we focus on how the two radiative capture branching ratios are affected by the initial state interactions among the different channels. In order to unravel the essential physics from the many channel system, in section 2 we will first set up a general procedure to separate the strong (or initial state) interactions from the electromagnetic ones. As shown in section 2, the initial state hadronic interactions can be described with 6 complex numbers. Thus we need a model of the interaction between the strong channels to produce these 6 numbers. In section 3 we examine two phenomenological potential models, each of which fit the low-energy scattering data, the resonance at 1405 MeV, and the hadronic branching ratios at threshold. In one potential, present work, the relative potential strengths between the various channels are guided by SU(3) symmetry. For this potential, the parameters are adjusted to fit all the low energy $K^-p$ data with the exception of the [*1s*]{} atomic level shift value of the $K^-p$ scattering length. The other potential is from Ref.\[20\], in which the scattering length is compatible with the atomic level shift data.
To clarify our discussion, we wish to underline here the nature of the fitting parameters in the potential guided by SU(3) symmetry. For the hadronic channels, the relative potential strengths are given by a value determined from SU(3) symmetry times a “breaking factor”, which is equal to 1, if the the relative channel couplings are SU(3) symmetric. This SU(3) structure is motivated by chiral symmetry \[14\]. For a good fit to the low energy data we need to vary the relative strengths somewhat, allowing the breaking factor to deviate from 1. The final values of this factor have no obvious physical significance. The potential enables one to estimate the effects of the initial state interactions from a potential which gives a good fit to the low-energy data. In the electromagnetic channels, the radiative capture amplitudes are derived from first order “Born” photoproduction processes, which involve the meson-baryon-baryon coupling constants, $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$. These coupling constants are related by SU(3) symmetry to the well known $\pi NN$ coupling constant. For the fit, these coupling constants are allowed to vary up to $\pm
50 \%$ from their SU(3) values. Here the final values of these 4 parameters will have physical significance and can be compared to values derived from other analyses. Thus we will examine if the radiative capture branching ratio data can be understood using vertex coupling constants for the electromagnetic channels that are close to their expected SU(3) values when initial state interactions are included from a hadronic interaction which fits the low energy hadronic data.
**General Formalism**
=====================
Consider a coupled-channel system consisting of [*n*]{} hadronic channels and one electromagnetic channel. We assume that the interaction between channels can be represented for each partial wave $l$ by real potentials of the form $V_{ij}^l(\sqrt{s},k_i,k_j)$ where $\sqrt{s}$ is the total energy and $k_i$ is the momentum of channel [*i*]{} in the center-of-mass frame. We will use the notation where the indices [*i,j*]{} are integers for the strong channels and is $\gamma $ for the electromagnetic channels. We will also assume that the transition matrix element for each partial wave from channel [*i*]{} to channel [*j*]{} can be derived from a coupled-channel Lippmann-Schwinger equation:
$$T_{ij}(\sqrt{s},k_i,k_j) = V_{ij}(\sqrt{s},k_i,k_j) +
\sum\limits_{m} \int V_{im}(\sqrt{s},k_i,q) G_m(\sqrt{s},q)
T_{mj}(\sqrt{s},q,k_j) q^2 dq, \eqno(1)$$
where $G_i(\sqrt{s},q)$ is the propagator for channel [*i*]{}. We suppress the index $l$, since for our problem only the $l=0$ partial wave is of interest. The electromagnetic coupling is weak, and to a very good approximation we can neglect the back coupling of the photon channels. Thus the T-matrix for radiative capture can be written as:
$$T_{i \gamma}(\sqrt{s},k_i,k_{\gamma}) =
V_{i \gamma}(\sqrt{s},k_i,k_{\gamma}) +
\sum\limits_{m\ne \gamma} \int V_{im}(\sqrt{s},k_i,q) G_m(\sqrt{s},q)
T_{m \gamma}(\sqrt{s},q,k_{\gamma}) q^2 dq . \eqno(2)$$
Note that in the above equation there is no integration over the photon’s momentum. There is only an integration over the hadronic momentum $k_i$ in the $V_{i \gamma}$ potential. This means that only half off-shell information is needed for the hadron-photon potential. Since $\sqrt{s}$ and $k_\gamma$ are fixed in the integral, we can write $V_{i \gamma}(\sqrt{s},q,k_{\gamma})$ as:
$$V_{i \gamma}(\sqrt{s},q,k_\gamma) = {{V_{i \gamma}(\sqrt{s},q,k_\gamma)}
\over {V_{i \gamma}(\sqrt{s},k_i,k_\gamma)}}
V_{i \gamma}(\sqrt{s},k_i,k_\gamma),$$
or
$$V_{i \gamma}(\sqrt{s},q,k_\gamma) = v_{i \gamma}(q)
V_{i \gamma}(\sqrt{s},k_i,k_\gamma).
\eqno(3)$$
Substituting this form for $V_{i \gamma}$ into equation (2) we obtain for the T-matrix:
$$T_{i \gamma}(\sqrt(s),k_i,k_\gamma) = \sum \limits_{m \ne \gamma}
M_{im}(\sqrt{s}) V_{m \gamma}(\sqrt{s},k_m,k_\gamma),$$
with the matrix $M_{im}$ defined as
$$\begin{aligned}
M_{im} &\equiv& \delta_{im} + \int V_{i,m}(\sqrt{s},k_i,q)
G_m(\sqrt{s},q) v_{m \gamma}(q) q^2 dq \\
& & + \sum_{n \ne \gamma} \int \int V_{in}(\sqrt{s},k_i,q')
G_n(\sqrt{s},q') V_{nm}(\sqrt{s},q',q) G_m(\sqrt{s},q)
v_{m \gamma}(q) q'^2 dq' q^2 dq
+ \cdots .\end{aligned}$$
The state “m” is the last hadronic state before the photon is produced. Since all the on-shell momenta are determined from $\sqrt{s}$ we have
$$T_{i \gamma}(\sqrt{s}) = \sum \limits_{m \ne \gamma} M_{im}(\sqrt{s})
V_{m \gamma}(\sqrt{s}).
\eqno(4)$$
This form for the transition matrix to the photon channels is very convenient, since it separates out the strong part from the electromagnetic part of the interaction. The matrix [*M*]{} is determined entirely from the hadronic interactions and vertices. In the absence of channel-coupling M is the unit matrix. Any deviation from unity is related to the initial state interactions. Note that [*no assumptions were made on the form of the propagator or the potentials connecting the hadronic channels*]{}. They need not be separable.
Labeling the $K^--p$ channel as number 5, and defining $A_m(\sqrt{s})$ as $M_{5m}(\sqrt{s})$ we can write the scattering amplitude to the photon channels as:
$$F_{K^-p \rightarrow \Lambda \gamma (\Sigma^0 \gamma)} =
\sum A_m(\sqrt{s})
f_{m \rightarrow \Lambda \gamma (\Sigma^0 \gamma)}. \eqno(5)$$
where the $f_m$’s are the amplitudes to go from the hadronic channel [*m*]{} to the appropriate photon channel. These amplitudes are derivable from diagrams representing the photoproduction process. The quantities $A_m$ are unitless complex numbers, and contain all the information about the initial state interactions for radiative capture. Generally the sum over m is restricted to states which have charged hadrons. For the $K^--p$ process the problem is greatly simplified, since there are only 3 channels which have charged hadrons: $\pi^+ \Sigma^-$, $\pi^- \Sigma^+$ and $K^-p$. To a very good approximation (see section 3), the $A_m$’s are the same for both the $\Lambda \gamma$ and the $\Sigma^0 \gamma$ channels. Thus, three complex numbers, determined from the hadronic interactions, describe all the initial state interactions for decays to both $\Lambda \gamma$ and $\Sigma^0 \gamma$ final states.
The result of Eq. (5) is essentially Watson’s Theorem \[25\] using a particle basis. Watson’s Theorem, which also relates information about the strong interaction to that of the electromagnetic process, uses an isospin basis. The photoproduction amplitude is shown to have a phase equal to the hadronic phase shift for a given isospin. Eq. (5) reduces to this result if there is only one strong channel. In this case, $A$ is proportional to $e^{i \delta}$ where $\delta$ is the phase-shift of the strong channel. For pion-nucleon photoproduction it is useful to use an isospin basis since the T-matrix is diagonal and both the photoproduction amplitude and the hadronic phase shift can be determined from experiment. It is especially useful if one isospin dominates (i.e. the $P_{33}$). However, the T-matrix (or potential) for $K-N$, $\Sigma \pi$, $\Lambda \pi$ system is not diagonal in an isospin basis. Watson’s Theorem would apply to the eigenphases of the coupled $K-N$, $\Sigma \pi$ system for I=0, and the coupled $K-N$, $\Sigma \pi$, $\Lambda \pi$ system for I=1. Since these phases are not easily determined from experiment the results of Watson’s Theorem are not as useful in this case. Also, in the next section we point out that isospin breaking effects are very important at low $K^--p$ energies. Thus, in analyzing threshold branching ratios, a particle basis is necessary. Another advantage of using Eq. (5) is that the interference of the “Born Amplitudes” $f$ due to the initial state interactions of the hadrons is made transparent.
**Results and Discussion**
==========================
The potentials for the strong channels
--------------------------------------
The two parts in determining the photoproduction rates in Eq. (4) are the $A_m$, which are determined from the strong part of the interaction, and the channel amplitudes $f_{m \rightarrow \Lambda \gamma
(\Sigma \gamma)}$. For notation, we will label the channels 1-8 as $\pi^+ \Sigma^-$, $\pi^0 \Sigma^0$, $\pi^- \Sigma^+$, $\pi^0 \Lambda$, $K^- p$, $\overline{K^0} n$, $\Lambda \gamma$, and $\Sigma^0 \gamma$ respectively. We begin by discussing the determination of the $A_m$. These were obtained by using a separable potential and fitting to the available low energy data on the strong channels. We took $v_{i \gamma}$ in Eq. 3 to be equal to $v_i$ in Eq. 6 below. Two different separable potentials were used: one guided by $SU(3)$ symmetry for the relative channel couplings which fits all the low energy data except the [*1s*]{} atomic-level shift, and one from Ref. \[20\] which fits all the low energy data including the sign of the scattering length from the [*1s*]{} atomic-level shift. Values for the $A_m$ at the $K^-p$ threshold for each fit are listed in Table I.
[[**TABLE I.**]{} The $A_i$ values from Eq. (5) for two different strong potentials. The potential with approximate SU(3) symmetry fits all low energy hadronic data except the [*1s*]{} $K^-p$ atomic level shift. The potential of Ref. \[20\] fits the atomic level shift as well. ]{}
------- ---------------------------- --------------------------
$A_i$ Potential with Approximate Potential of
$SU(3)$ Symmetry Tanaka and Suzuki \[22\]
$A_1$ (1.20, 0.52) (1.49, -0.28)
$A_2$ (-1.02, -.14) (-1.10, 0.52)
$A_3$ (0.83, -.23) (0.71, -0.75)
$A_4$ (-0.16, -0.34) (-0.30, -0.39)
$A_5$ (-0.15, 1.06) (2.01, 2.55)
$A_6$ (1.18, -0.41) (2.08, -1.12)
------- ---------------------------- --------------------------
Following Ref. \[21\] the separable potentials for the strong channels are taken to be of the form:
$$V^I_{ij}(k,k') = {{g^2} \over {4 \pi}} C^I_{ij} b^I_{ij}
v_i(k) v_j(k'), \eqno(6)$$
where the $C^I_{ij}$ are determined from $SU(3)$ symmetry. The $b^I_{ij}$ are “breaking parameters” which are allowed to vary slightly from unity. The $v_i(k)$ are form factors, taken for this analysis to be equal to $\alpha_i^2 / (\alpha_i^2 + k^2)$, and [*g*]{} is an overall strength constant. These potentials are used in a coupled-channel Lippmann-Schwinger equation with a non-relativistic propagator to solve for the cross-sections to the various channels. The data used in the fit are from Refs. \[26-30\]. The resonance at an energy of 1405 MeV was also fitted. As discussed in Ref. \[21\] it was not possible to fit all the low-energy data using potentials that had $b^I_{ij}=1$ for all i and j. To get an acceptable fit without including the radiative capture data, it is necessary to vary the $b^I_{ij}$ by at least $\pm 15 \%$ from unity. To get a very good fit to all the data and determine the range of the $A_m$, we let the $b^I_{ij}$ vary from $0.5$ to $1.5$. In Table II we list the values we used for the $I=0$ and $I=1$ potentials for our “best fit”. This “best fit” also included the radiative capture data, and is discussed in
[[**TABLE II.**]{} The “best fit” values of $C^I_{ij}(b^I_{ij})$ for the potential of Eq. (6). ]{}
----------------------------------------------------------------------------------------------------------------------
$C^{I=0}_{ij}$ $\Sigma \pi$ $KN$
---------------------------- -------------------------------- -------------------------------- -----------------------
$\Sigma \pi$ $-2$ (0.50) $-{{\sqrt{6}} \over 4}$ (1.29)
$KN$ $-{{\sqrt{6}} \over 4}$ (1.29) $-{3 \over 2}$ (1.43)
$C^{I=1}_{ij}$ $\Sigma \pi$ $\Lambda \pi$ $KN$
$\Sigma \pi$ $-1$ (0.50) 0 $-{1 \over 2}$ (1.37)
$\Lambda \pi$ 0 0 $ {\sqrt{6} \over 4}$
$KN$ $-{1 \over 2}$ (1.37) $ {\sqrt{6} \over 4}$ $-{1 \over 2}$ (0.50)
$\alpha_{\Sigma \i} = 974$ $\alpha_{\Lambda \pi} = 886$ $\alpha_{KN} = $g^2 = 1.19 fm^2$
445 $
----------------------------------------------------------------------------------------------------------------------
the next section. The elements are listed as a product of the $C^{I}_{ij}$ values from SU(3) times $b^I_{i}$ which was allowed to vary from $0.5$ to $1.5$. Also listed are the values for $\alpha_i$ in MeV/c and the overall strength $g^2$ from Eq. (6). We note that for this fit the $\Lambda (1405)$ is produced as a $K-N (\Sigma \pi)$ bound state resonance \[21\] (see Fig. 4).
The $A_i$ are a measure of how much the initial state interactions enhance the single scattering amplitude. Not all the $A_i$ are needed in the $K^-p$ radiative decay calculation, since only channels which have charged particles contribute. Thus only $A_1$, $A_3$, and $A_5$ enter the calculation. Also due to isospin symmetry in the $\Sigma \pi$ sector $A_1$, $A_2$, and $A_3$ must satisfy the relation: $A_1 + A_3 = -2A_2$. In the absence of initial state interactions, $A_1=A_3=0$ and $A_5=1$. As can be seen in Table I, the magnitude of $A_1$, $A_3$, and $A_5$ are between 0.8 and 1.3. Since $F_{K^-p \rightarrow \Lambda \gamma} = A_1 f_{\Sigma^- \pi^+
\rightarrow \Lambda \gamma} + A_3 f_{\Sigma^+ \pi^-
\rightarrow \Lambda \gamma} + A_5 f_{K^-p \rightarrow \Lambda \gamma}$, cancellations amongst the various amplitudes can make the radiative capture probability very sensitive to the initial state interactions. Unfortunately, the $A_i$ cannot be directly determined experimentally, and will have some model dependencies. We tried to estimate the model dependency for the potential of Eq. 6 by allowing the $b^I_{ij}$ to vary different amounts between 0.5 and 1.5 and see how much the $A_i$ changed. For acceptable fits to the data, excluding the atomic [*1s*]{} level shift, the $A_i$ varied only $\pm 20 \%$ in magnitude.
An important aspect of the problem is to include the appropriate isospin breaking effects due to the mass differences of the particles. This was done as described in Ref. \[21\] by using the correct relativistic momenta and reduced energies in the propagator. The effects are very important in calculating the threshold branching ratios, since the masses of $\overline{K^0}-n$ are 7 MeV greater than the masses of $K^-p$. As shown in Ref.\[31\], the Coulomb potential can be neglected when calculating the branching ratios. In Fig. 1 we plot the branching ratios as a function kaon laboratory momentum $P_{Lab}$. The three different curves for each ratio correspond to different types of SU(3) breaking to be discussed later. Note that the energy dependence is particularly strong for branching ratios $\gamma$, $R_n$, $R_{\Lambda \gamma}$ and $R_{\Sigma \gamma}$.
For $\gamma$, which is the ratio of $\Sigma^- \pi^+$ production to $\Sigma^+ \pi^-$ the energy dependence is easy to understand. For a model as the one presented here which does not include the $\Lambda (1405)$ as an [*s*]{}-channel resonance, the reaction $K^-p \rightarrow \pi^+ \Sigma^-$ cannot occur in a single step. This is a double charge exchange reaction, and needs to undergo two single charge exchange steps with the middle one being neutral. From the total cross-section data (See Fig. 2), the most important neutral channel in low-energy $K^-p$ scattering comes out to be $\overline{K^0} n$. This causes the ratio $\gamma$ to have a strong energy dependence near the $\overline{K^0} n$ threshold. Since the $\overline{K^0}
n$ channel is also important in $\Lambda \pi$ production, the ratio $R_n$ also varies rapidly with energy near threshold. Thus for an accurate comparison with the data, one needs to use a particle basis in calculating the photoproduction branching ratios at threshold. We note that if an [*s*]{}-channel resonance was the dominating process in the $\Sigma \pi$ reaction, then the ratio $\gamma$ would not have as rapid an energy dependence near the $K^-n$ threshold. It is also interesting that $\Gamma (K^-p \rightarrow \Lambda \gamma)$ is substantially less than $\Gamma (K^-p \rightarrow \Sigma^0 \gamma)$ at energies below the $\overline{K^0} n$ threshold and greater at energies above. Experimental data of these branching ratios near the $K^-p$ threshold would help clarify the nature of the $\Lambda (1405)$.
The electromagnetic channels
----------------------------
We now turn our attention to the most important part of the calculation, the amplitudes for the $\Lambda \gamma$ and $\Sigma^0 \gamma$ channels. As discussed previously, only the amplitudes for the three charged channels will contribute to the radiative capture amplitude in Eq. (5): $K^-p \rightarrow \Lambda \gamma (\Sigma^0
\gamma)$, $\Sigma^{\pm} \pi^{\mp} \rightarrow \Lambda \gamma (\Sigma^0
\gamma)$. Here we will use the amplitudes obtained from the diagrams shown in Fig. 3. These diagrams are the leading order contributions to photoproduction \[32\]. We include the most important amplitudes which are the “extended Born terms”, including the $\Lambda$ and the $\Sigma^0$ exchange terms, and the vector meson exchange terms ($K^*$, $\rho$). The expressions for these terms and their relative importance are given in Appendix I.
There are four coupling constants which enter in the photoproduction amplitudes: $g_{KN \Lambda}$, $g_{KN \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \Lambda}$. Since there are only two branching ratios to fit, we need to limit the method of our search. We investigated three cases: [*a*]{}) assume exact SU(3) symmetry for the coupling constants with $g_{\pi NN} = 13.4$ and vary the F-D mixing ratio $\alpha$ to best fit the data, [*b)*]{} assume $\alpha = 0.644$, $g_{\pi NN} = 13.4$ and vary the coupling constants slightly from their SU(3) values for a best fit, and [*c*]{}) use the $A_m$ from the potential of Ref. \[20\] and SU(3) symmetry for the coupling constants to see if a fit of the radiative decay branching ratios is possible. For the search, we weighted each data point equally, and thus the two radiative capture branching ratios did not have a great affect on the hadronic parameters.
In the first case, we assumed exact SU(3) symmetry with $g_{\pi NN} = 13.4$ and varied the F-D mixing ratio $\alpha$ for a best fit to the data. We found an acceptable fit with a $\chi^2$ per data point of 2.47 for $\alpha = 1.0$. The branching ratios for this fit are $ \gamma$ = 2.25, $ R_c$ = 0.66 and $ R_n$ = 0.17 for the strong channels, and $ R_{\Lambda \gamma}$ =$1.22 \times 10^{-3}$ and $ R_{\Sigma \gamma}$ =$1.47 \times 10^{-3}$ for the electromagnetic channels. Although this is not the accepted value for $\alpha$, it is remarkable to get a fit with only one adjustable variable.
In the next case, we fix $\alpha$ to be 0.644. The search is done using MINUIT code \[33\] on 13 parameters: the three ranges for the strong channels, the six breaking factors for the strong channels $b^I_{ij}$, $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$. We allowed the $b^I_{ij}$ to vary $\pm 50 \%$, $\pm 40 \%$, and $\pm 30 \%$ from unity while the 4 coupling constants $g_{Kp \Lambda}$, $g_{Kp \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \lambda}$ varied by $\pm 50 \%$, $\pm 40 \%$, and $\pm 30 \%$ from their SU(3) values respectively. The range parameters $\alpha$ were allowed to vary from 200 to 1000 MeV/c. The results for the branching ratios and the reduced $\chi^2$ are listed in Table III. The first column lists the percentage that the parameters, except $g_{Kp \Lambda}$, were allowed to vary from their SU(3) values (or in the case of the $b^I_{ij}$’s from unity). The second column lists the percentage that $g_{Kp \Lambda}$ was allowed to vary from its SU(3) value of -13.2. We also tried to find a satisfactory fit in which $g_{Kp \Lambda}$ was as close to -13.2 as possible. A fit was found in which $g_{Kp \Lambda}$ was varied only $\pm 20 \%$, while the other parameters were allowed to vary $\pm 50 \%$. The first row of Table III shows these results. We call this our “best fit” since the most well determined coupling constants, $g_{KN \Lambda}$ and $g_{KN \Sigma}$ are close to their SU(3) values, with $g_{KN \Sigma}$ only $50 \%$ high. Our best fit values for the coupling constants are $g_{KN \Lambda}=-10.6$, $g_{KN \Sigma}=5.8$, $g_{\pi \Sigma \Sigma}=-7.2$, and $g_{\pi \Sigma \Lambda}=-5.0$. Notice that our values for the two first coupling constants are in agreement with those obtained from strangeness photoproduction \[7,34\] and hadronic sector \[35,36\] analyses. The electromagnetic branching ratios change drastically if the initial state interactions are excluded from the calculation. We obtain $R_{\Lambda \gamma}=.56 \times 10^{-3}$ and $R_{\Sigma \gamma} = .12 \times 10^{-3}$ without the initial state interactions. The two branching ratios are hence decreased by roughly a factor of 2 and more than one order of magnitude, respectively, by switching off the initial state interactions.
Graphs of the different fits for the five branching ratios and total cross sections as a function of kaon laboratory momentum are shown in Fig. 1 and 2, respectively. In Fig. 4 we plot the $\Sigma \pi$ spectrum normalized to the data of Hemmingway \[37\]. As in Ref.\[14\], we plot $k^\pi_{c.m.} |T_{\Sigma \pi \rightarrow \Sigma \pi}|^2$, where $T_{\Sigma \pi \rightarrow \Sigma \pi}$ is the T-matrix in the I=0 sector for $\Sigma \pi \rightarrow \Sigma \pi$ scattering. In each figure, the solid line corresponds to the “best fit” parameters, the dotted line to $\pm 40 \%$ SU(3) breaking for all the parameters, and the dashed line to $\pm 30 \%$ breaking for all the parameters. In each case the $\Lambda (1405)$ is produced as a bound state resonance as in Ref \[21\].
The $K^-p$ scattering length obtained from our best fit is (-.63 + .76 i) fm. This compares closely with the value from Ref. \[32\] of (-.66 + .64 i) fm. These values, however, have the opposite sign for the real part from that extracted from the [*1s*]{} $K^-p$ atomic level shift data \[38\]. Since the atomic level shift data is still puzzling \[39\], we did not try to fit it in our search. This discrepancy has been discussed in detail in Ref. \[20\] with some interesting results. Hence, we used the $A_m$ obtained from the potential of Ref. \[20\] which fit all the hadronic low energy data and has the same sign for the scattering length as the atomic level shift data. We were able to reproduce their results using their non-relativistic potentials. From Table I we see that $A_1$ and $A_3$ do not differ too much from those obtained with the “SU(3) guided” potential. However, $A_5$ is much different in magnitude and its real part has the opposite sign. Perhaps this is because the atomic [*1s*]{} shift and hence the scattering length has the opposite sign. For the potential of \[20\] the resulting radiative capture branching ratios using coupling constants from SU(3) symmetry are $R_{\Lambda \gamma} =
17.5 \times 10^{-3}$ and $R_{\Sigma \gamma} = 3.29 \times 10^{-3}$, which are far from the experimental values. For satisfactory agreement with the radiative capture branching ratios, the coupling constants would have to deviate from their SU(3) values by an unreasonable amount. The reason for the bad agreement is that $A_5$ is very large and its real part is positive. In order to obtain a small value for $\Lambda \gamma$ production, the amplitudes have to cancel in Eq. (5). Since the relative signs of the $f_{m \rightarrow \Lambda \gamma (\Sigma^0 \gamma)}$ are fixed by SU(3) symmetry, the $A_m(\sqrt{s})$ have to have appropriate relative phases to cause this cancellation. The $A_m$ from the potential guided by SU(3) symmetry have this feature.
[[**TABLE III.**]{} Branching ratios and $\chi^2$ per data point for different amounts of SU(3) breaking. Column 2 lists the variation in the coupling constant $g_{Kp \Lambda}$. Column 1 lists the variation in the other parameters. ]{}
All except $g_{Kp \Lambda}$ $g_{Kp \Lambda}$ $\chi^2/N$ $\gamma$ $R_c$ $R_n$ $R_{\Lambda \gamma} \times 10^{3}$ $R_{\Sigma \gamma} \times 10^{3}$
----------------------------- ------------------ ------------ ---------------- ----------------- ----------------- ------------------------------------ -----------------------------------
$\pm 50 \%$ $\pm 20 \%$ 1.76 2.31 .661 .164 1.09 1.55
$\pm 50 \%$ $\pm 50 \%$ 1.21 2.35 .659 .194 0.89 1.46
$\pm 40 \%$ $\pm 40 \%$ 1.54 2.32 .659 .179 1.04 1.53
$\pm 30 \%$ $\pm 30 \%$ 2.94 2.20 .652 .174 1.31 1.65
Experiment 2.36 $\pm$ .04 .664 $\pm$ .011 .189 $\pm$ .015 .86 $\pm$.07 1.44 $\pm$ .20
**Conclusions**
===============
We have done a comprehensive analysis of all the low energy data, except the [*1s*]{} atomic level shift, on the $K^-p$ system. To facilitate the analysis, we derived an expression for the radiative capture cross section which separates out the strong interaction from the electromagnetic ones. The initial state interactions can be described by six complex amplitudes, $A_m$, with only three of them relevant to the radiative capture process. For the strong part of the interaction we choose a separable potential whose relative potential strengths were guided by SU(3) symmetry. This potential is phenomenological and serves to produce appropriate $A_m$ from the low energy scattering and resonance data. The radiative capture amplitudes are derived from first order “Born” photoproduction processes, and are determined from meson-baryon-baryon coupling constants, whose values are related by SU(3) symmetry to the well known $\pi NN$ coupling constant.
We found a number of good fits in which the coupling constants were close to their expected SU(3) values. For these fits, the relative coupling strengths in the strong channels were guided by $SU(3)$ symmetry. In all of the fits, the $\Lambda (1405)$ is produced as a bound $K-N (\Sigma \pi)$ resonance, and the initial state interactions were very important for the radiative capture branching ratios. The ratio $R_{\Lambda \gamma}$ varies roughly by a factor of 2, and the ratio $R_{\Sigma \gamma}$ by more than a factor of 10 due to the initial state interactions.
Results presented in this paper, reproduce well enough the existing strong and electromagnetic data from threshold up to $P_{K}^{\rm
lab}
\approx$ 200 MeV/c. Our predictions, specially for the branching ratios, show clearly the need for more experimental investigations ; one of the main motivations being to clarify the nature of the $\Lambda (1405)$ resonance. Such measurements are planned at DA$\Phi$NE \[40\] using the tagged low energy kaon beam and may also be achieved at Brookhaven and KEK.
**Acknowledgements**
====================
We would like to thank J.C. David, C. Fayard, G.H. Lamot, Andreas Steiner and Wolfram Weise for many helpful discussions and suggestions regarding this work. We are grateful to the Institute for Nuclear Theory (Seattle), for an stimulating and pleasant stay, where the idea of this collaboration emerged. One of us (PS) would like to thank the Centre d’Etudes de Saclay for the hospitality extended to him.
\
In this Appendix we will summarize the contributions to the photoproduction amplitudes shown in Fig. 3. Here we will write the expressions for the $K^- p \rightarrow \Lambda \gamma$ amplitude. The amplitudes for the $K^- p \rightarrow \Sigma^0
\gamma$, $\Sigma^{\pm} \pi^{\mp} \rightarrow \Lambda \gamma$, and $\Sigma^{\pm} \pi^{\mp} \rightarrow \Sigma^0 \gamma$ processes will be the same with appropriate masses and coupling constants.
To lowest order the amplitude for the $K^-p \rightarrow \Lambda
\gamma$ reaction f is the sum of three amplitudes:
$$f_{K^-p \rightarrow \Lambda \gamma} = F_{Born} + F_\Sigma + F_{K^*},$$
which correspond to the the Born, the $\Sigma^0$, and the $K^*$ diagrams shown in Fig 3.
The Born amplitude is derived in Ref. \[32\] and is given by:
$$F_{Born} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}}
{{g_{Kp \Lambda} e} \over {2 m_p}}
(1 + {{k_{\gamma}} \over {E_{\Lambda}+m_{\Lambda}}}
(1 + \kappa_p + \kappa_{\Lambda})),$$
at the $K^-p$ threshold. The “$\Sigma$” term is also derived in Ref. \[32\] and is given by:
$$F_{\Sigma} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}}
{{g_{Kp \Sigma} e} \over {2 m_p}}
\kappa_{\Sigma \Lambda} {{\sqrt{s}-m_\Lambda} \over
{\sqrt{s} + m_\Sigma}}.$$
In a similar manner, the $K^*$ exchange term can be evaluated. In this case, there is a vector and a tensor piece. We obtain for the amplitude:
$$F_{K^*} = - \sqrt{{E_{\Lambda} + m_{\Lambda}} \over {2 m_{\Lambda}}}
[ {{g^V_{K^*p \Lambda} e} \over {2 m_p}}
{{\kappa_{K^*K} k^2_\gamma (\sqrt{s}-m_p)} \over
{(t - m^2_{K^*})(E_\Lambda + m_\Lambda)}}
+ {{g^T_{K^*p \Lambda} e} \over {2 m_p}}
{{\kappa_{K^*K} k^2_\gamma (\sqrt{s}-m_p)} \over
{(t - m^2_{K^*})2 m_p}}
({{m_\Lambda + m_p} \over {m_\Lambda + E_\Lambda}})].$$
In the absence of initial state interactions, the differential cross section is given by:
$${{d \sigma} \over {d \Omega}} = {{(E_\Lambda + m_\Lambda)
(E_p + m_p)} \over {64 \pi^2 s}}
{{P_\gamma} \over {P_K}} |f_{K^-p \rightarrow
\Lambda \gamma}|^2.$$
The first part of $F_{Born}$ has the largest magnitude. The other pieces are reduced by kinematical factors, with $F_{K^*}$ giving the smallest contribution. The $K^*$ exchange makes up about $2 \%$ of the amplitude. Thus, the uncertainty in the vector and tensor coupling constants are not so important, and we fixed them to be the SU(3) values. Also, since the calculation is not particularly sensitive to the values of the electromagnetic couplings, we fixed them at their excepted values. Values for the constants which were held fixed during the search are listed in Table IV.
[[**TABLE IV.**]{} Values of the coupling constants which were held constant. ]{}
--------------------------------- ------------------------------------
$\kappa_p = 1.793$ $g^V_{K^*p \Lambda} = -4.5 $
$\kappa_\Lambda = -.613$ $g^T_{k^*p \Lambda} = -16.6 $
$\kappa_{\Sigma \Lambda} = 1.6$ $g^V_{K^*p \Sigma} = -2.6 $
$\kappa_{K^*K} = 1.58$ $g^T_{K^*p \Sigma} = 3.2 $
$\kappa_{\rho \pi} = 1.41$ $g^V_{\rho \Sigma \Lambda} = 0$
$\kappa_{\Sigma^-} = -2.157$ $g^T_{\rho \Sigma \Lambda} = 11.1$
$\kappa_{\Sigma^+} = 1.42$ $g^V_{\rho \Sigma \Sigma} = -5.2$
$\kappa_{\Sigma^0} = .619$ $g^T_{\rho \Sigma \Sigma} = -12.8$
--------------------------------- ------------------------------------
The search was done on the more important coupling constants, $g_{K p \Lambda}$, $g_{K p \Sigma}$, $g_{\pi \Sigma \Sigma}$, and $g_{\pi \Sigma \Lambda}$. At the $K^-p$ threshold, using the coupling constants of Table III, the radiative capture amplitudes are:
$$f_{K^-p \rightarrow \Lambda \gamma} = {e \over {2 m_p}}
(-g_{Kp \Lambda}(1.28) - g_{Kp \Sigma}(.2) -.75),$$
$$f_{K^-p \rightarrow \Sigma^0 \gamma} = {e \over {2 m_p}}
(-g_{Kp \Sigma}(1.32) - g_{Kp \Lambda}(.15) + .02),$$
$$f_{\pi^+ \Sigma^- \rightarrow \Lambda \gamma} = {e \over {2 m_\Sigma}}
(-g_{\pi \Sigma \Lambda}(.835) + g_{\pi \Sigma \Sigma}(.19) + .21),$$
$$f_{\pi^+ \Sigma^- \rightarrow \Sigma^0 \gamma} = {e \over {2 m_\Sigma}}
(-g_{\pi \Sigma \Sigma}(.95) + g_{\pi \Sigma \Lambda}(.153) + .19),$$
$$f_{\pi^- \Sigma^+ \rightarrow \Lambda \gamma} = {e \over {2 m_\Sigma}}
(g_{\pi \Sigma \Lambda}(1.16) - g_{\pi \Sigma \Sigma}(.19) -.21),$$
$$f_{\pi^- \Sigma^+ \rightarrow \Sigma^0 \gamma} = {e \over {2 m_\Sigma}}
(-g_{\pi \Sigma \Sigma}(1.28) + g_{\pi \Sigma \Lambda}(.153) + .19),$$
where the three terms in parenthesis correspond to the three amplitudes described above. The above equations show the relative importance of the different contributions to radiative capture at the $K^-p$ threshold. The best fit values for these coupling constants are summarized in Table V.
[[**TABLE V.**]{} “Best fit” values of the coupling constants for the electromagnetic amplitudes. ]{}
-------------------------------- ---------------------------------
$g_{Kp \Lambda} = -10.6$ $g_{Kp \Sigma} = 5.8$
$g_{\pi \Sigma \Sigma} = -7.2$ $g_{\pi \Sigma \Lambda} = -5.0$
-------------------------------- ---------------------------------
\
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**Figure Captions**
Figure 1. The five branching ratios, defined in the text, are plotted as a function of $K^-$ laboratory momentum: [*a*]{}) branching ratio $\gamma$, [*b*]{}) $R_c$, [*c*]{}) $R_n$, [*d*]{}) $R_{\Lambda \gamma}$, and [*e*]{}) $R_{\Sigma
\gamma}$. The three curves correspond to different amounts of SU(3) breaking as listed in Table III. The solid curve corresponds to our “best fit” parameters: the vertex couplings of Table V and the strong couplings of Table II. This corresponds to the first line in Table III. The dotted curve is for $\pm 40 \%$ variation in all the parameters, the third line in Table III. The dashed curve is for $\pm 30 \%$ variation in all the parameters, the last line in Table III. The data points at threshold \[3,19,20\] with error bars are also shown. The $\overline{K^0}-n$ threshold is at 89.4 MeV/c.
Figure 2. Cross sections are compared with the experimental data for the six strong channels and the two electromagnetic channels: [*a*]{}) $K^-p$ elastic scattering, [*b*]{}) $K^-p \rightarrow \overline{K^0} n$, [*c*]{}) $K^-p \rightarrow \pi^0 \Lambda$, [*d*]{}) $K^-p \rightarrow \pi^+ \Sigma^-$, [*e*]{}) $K^-p \rightarrow \pi^0 \Sigma^0$, [*f*]{}) $K^-p \rightarrow \pi^- \Sigma^+$, g) $K^-p \rightarrow \Lambda
\gamma$, and h) $K^-p \rightarrow \Sigma^0 \gamma$. The three curves for each cross section correspond to different amounts of SU(3) breaking as in Fig. 1.
Figure 3. The main diagrams which contribute to the radiative capture amplitude for $K^-p \rightarrow \Lambda \gamma$: [*a), b), c)*]{} and [*f)*]{} are the “Born” terms, [*d)*]{} the “$\Sigma$” or cross term, and [*e)*]{} the vector meson $K^*$ exchange term. The $K^-p \rightarrow \Sigma^0 \gamma$ reaction and the $\Sigma \pi$ reactions $\pi^{\pm} \Sigma^{\mp} \rightarrow \Lambda \gamma (\Sigma^0
\gamma)$ will have similar terms.
Figure 4. The $\Sigma \pi$ mass spectrum normalized to the data of Ref. \[37\] is plotted as a function of the $\Sigma \pi$ center of mass energy. The three curves correspond to different amounts of SU(3) breaking as in Fig. 1.
|
---
abstract: 'Muon tomographic visualization techniques try to reconstruct a 3D image as close as possible to the real localization of the objects being probed. Statistical algorithms under test for the reconstruction of muon tomographic images in the Muon Portal Project are here discussed. Autocorrelation analysis and clustering algorithms have been employed within the context of methods based on the Point Of Closest Approach (POCA) reconstruction tool. An iterative method based on the log-likelihood approach was also implemented. Relative merits of all such methods are discussed, with reference to full <span style="font-variant:small-caps;">Geant</span>4 simulations of different scenarios, incorporating medium and high-Z objects inside a container.'
address:
- 'INAF - Osservatorio Astrofisico di Catania, Italy'
- 'Dip. di Fisica e Astronomia, Università di Catania, Italy'
- 'INFN Section of Catania, Italy'
author:
- 'S. Riggi'
- 'V. Antonuccio-Delogu'
- 'M. Bandieramonte'
- 'U. Becciani'
- 'A. Costa'
- 'P. La Rocca'
- 'P. Massimino'
- 'C. Petta'
- 'C. Pistagna'
- 'F. Riggi'
- 'E. Sciacca'
- 'F. Vitello'
title: 'Muon tomography imaging algorithms for nuclear threat detection inside large volume containers with the *Muon Portal* detector'
---
Muon tomography ,imaging algorithms ,clustering methods ,autocorrelation analysis ,maximum likelihood ,EM algorithm
25.30.Mr ,87.57.Q- ,87.57.nf
Introduction {#IntroductionSection}
============
Muon tomography is a technique employing the scattering of secondary cosmic muons inside a material, to reconstruct a 3D image as close as possible to the true localization of the objects inside the volume to be inspected. Due to the dependence of the scattering angle on the atomic number $Z$ of the material, this technique is particularly promising to search for the presence of high-$Z$ materials inside large volumes - such as containers - even in presence of additional, low- and medium-$Z$, objects.\
To reach a good precision in the reconstruction of the tomographic image a good tracking muon detector is required, able to reconstruct on a event-by-event basis the track of the muon before and after traversing the volume, even in presence of multiple hits generated by the structure itself (mechanical structure, empty container, …) or by the surrounding materials (roof, buildings, soil, …). Spatial and angular resolutions are then mandatory in this respect, to provide a good description of the incoming and outgoing muon tracks. Such performances may be achieved with several detection technologies, some of which have inherent good spatial resolution (for instance multiwire gas detectors) whereas others with worse intrinsic resolution (such as segmented scintillator strips) may reach good results depending on the number of detection planes and relative distance between them. On the other side, imaging algorithms of good quality and able to produce results in a small CPU time are an essential tool for the reconstruction of tomographic images. Relatively short computing times, few minutes at most, are infact required at the inspection site, in order to follow the real container flux without significantly interfering with the usual harbour activities. A large effort is at present pursued by the different groups working in the field to set up appropriate software tools for such task, which involve not only a proper reconstruction, in nearly real-time, of the image, but also good rendering techniques to provide the user with an easy-to-interpret tomographic image.\
In the framework of a new Project aiming at the construction of a real scale prototype of scanning muon detector for containers, an important part of our efforts has been concentrated on testing both traditional and new numerical techniques devoted to this task.\
In the present paper we report a comparison of several methods which, starting from raw data, produce a tomographic image of the objects contained in a large volume. Data for such comparison were provided by detailed <span style="font-variant:small-caps;">Geant4</span> simulations incorporating the knowledge of the mechanical structure of the detector and of the container under inspection and the physical interactions of the muons (with realistic energy and angular distributions) with all materials. In addition to the simplest method, based on the reconstruction of the Point-Of-Closest-Approach, also methods based on autocorrelation analysis, clustering and log-likelihood algorithms have been tested under different scenarios.\
Section \[DetectorSection\] reports a brief description of the detector prototype under construction, while Section \[AlgorithmSection\] discusses the main ingredients of all such methods. Section \[ResultsSection\] reports the application of these algorithms to different sets of simulated data. Finally, Section \[SummarySection\] is devoted to a discussion of the main aspects which could be improved in the future, in order to arrive to a real-time tomographic analysis for such detector.
Overview of the *Muon Portal* project {#DetectorSection}
=====================================
The Muon Portal project [@RiggiECRS; @LoPresti; @MuonPortalWebsite] is a prototype of a dedicated particle detector for the inspection of harbour containers through the technique of muon tomography. The experimental setup is based on four XY detector planes, each providing the X and Y position measurements, two placed below and two above the volume to be inspected. The size of each plane is optimized to fit that of a real TEU (Twenty-foot Equivalent Units) container, namely 5.9 m $\times$ 2.4 m $\times$ 2.4 m.\
To favour the detector assembly and its maintenance, each plane is divided into 6 modules of size 1 m $\times$ 3 m arranged to cover the above specified detector area with minimal dead surfaces. Each module is hosted inside a dedicated casing providing the mechanical support for the detector planes. The mechanical structure is designed to minimize the amount of material traversed by the cosmic ray muons. A dummy mechanical structure is also being designed to be inserted between the intermediate detector planes to emulate a real container volume.\
Each module is segmented into 100 strips of extruded plastic scintillators (1 $\times$ 1 $\times$ 300 cm$^3$) with two embedded wavelength-shifting (WLS) fibres to collect the light produced inside the scintillator bars. Each fiber is coupled at one end to Silicon Photomultipliers (SiPMs), designed ad-hoc for the project to maximize the light yield with reasonable cost requirements.\
More details concerning the detector geometry, electronic readout and channel reduction mechanism can be found in [@RiggiECRS; @LoPresti].\
The expected detector acceptance $\mathcal{A}$ to a flux of cosmic ray muons has been evaluated from detailed detector simulations, yielding $\mathcal{A}$=10 m$^{2}$sr, corresponding to a number of expected events of $\sim$2$\times$10$^{5}$ for a scanning time of $\Delta$t=5 minutes and a standard cosmic ray flux $\phi_{\mu}$= 1 m$^{-2}$s$^{-1}$ integrated over the solid angle.\
The angular accuracy for muon track reconstruction has been found of the order of 0.25$^{\circ}$ from toy simulations in which only the impact of the position resolution is considered. It increases at $\sim$0.5$^{\circ}$ in detailed <span style="font-variant:small-caps;">Geant4</span> simulations including also multiple scattering effects.\
The precision on the determination of the scattering data, scattering angle and lateral displacement, which are relevant for tomography imaging studies, needs to be estimated too. The analysis yields a scattering angle uncertainty of $\sim$0.7$^{\circ}$ and a lateral displacement uncertainty of the order of 2 cm. This imposes a limit on the minimum size of the threat objects that can be identified with reasonable accuracy inside the container volume and within reasonable scanning times, typically not smaller than 5 cm.
Statistical methods for muon tomography imaging {#AlgorithmSection}
===============================================
In this section we briefly report details on the statistical algorithms adopted for tomographic image reconstruction. They have been developed in C++ using links to <span style="font-variant:small-caps;">Geant4</span> [@GEANT4] and <span style="font-variant:small-caps;">Root</span> [@ROOT] frameworks for detector geometry building and navigation and mathematical routines.
<span style="font-variant:small-caps;">POCA</span>-based methods
----------------------------------------------------------------
The simplest algorithm in the field is the *Point Of Closest Approach* (<span style="font-variant:small-caps;">Poca</span>) which makes the simplified assumption that the muon scattering occurs in a single-point. It therefore searches for the geometrical point of closest approach $\mathbf{P}_{poca}$=$\frac{1}{2}(\mathbf{P}_{in}+\mathbf{P}_{out})$ between the incoming $\mathbf{u}_{in}$ and outcoming $\mathbf{u}_{out}$ reconstructed track directions with respect to the inspected volume (see sketch in Figure \[POCASketchFig\]): $$\begin{aligned}
\mathbf{P}_{in,out}&=& \mathbf{P_{_{0}}}_{in,out}+t_{in,out}\mathbf{u}_{in,out}\\
t_{in}&=& (b e-c d)/\Delta\\
t_{out}&=& (a e-b d)/\Delta\end{aligned}$$ where $\mathbf{P_{_{0}}}_{in,out}$ are two points on the incoming and outgoing tracks, $a=\mathbf{u}_{in}\cdot\mathbf{u}_{in}$, $b=\mathbf{u}_{in}\cdot\mathbf{u}_{out}$, $c=\mathbf{u}_{out}\cdot\mathbf{u}_{out}$, $d=\mathbf{u}_{in}\cdot\mathbf{w}$, $e=\mathbf{u}_{out}\cdot\mathbf{w}$, $\Delta=ac-b^{2}$, $w=\mathbf{P_{_{0}}}_{in}-\mathbf{P_{_{0}}}_{out}$.
Such method is of easy implementation and provides fast results, useful as first-order approximation to the problem or as a starting approximation for more detailed algorithms. However, it neglects the multiple scatterings throught the volume material and therefore has the drawback of providing poor-resolution images, it is quite sensitive to the presence of shield materials located above or below the potential threat and cannot localize very well materials at the volume borders. This motivates the implementation of the log-likelihood algorithm discussed below, which is based on more realistic physical and statistical assumptions and allows to face the problems encountered with the <span style="font-variant:small-caps;">Poca</span> algorithm. It is also desirable to have additional “grid-free” statistical methods for tomography analysis, which do not require to assume a predefined grid dividing the inspected volume into three-dimensional voxels. We explored in next paragraphs two alternative methods, based on the POCA observable: the two-points autocorrelation analysis and density-based clustering algorithms.
### Autocorrelation analysis {#AutocorrelationSection}
The two-point autocorrelation function, hereafter denoted *2pt-ACF* for brevity, is one of the main statistics generally used to describe the distribution of galaxies and to search for localized excess of data observations at certain scales in a volume with respect to a homogeneous random distribution. It is therefore well suited also for the problem of tomography imaging where we need to search for a density excess of POCA observations inside the container with respect to a normal situation, for instance an empty container.\
Following Peebles [@Peebles] the 2pt-ACF $\xi(r)$ defines the probability $dP$ to find simultaneously two objects at a distance $r$ from each other within two volume elements $dV_{1}$ and $dV_{2}$ in a data sample with event density $n$: $$dP = n^{2}[1 + \xi(r)]dV_{1}dV_{2}$$ A positive correlation ($\xi>$0) at distance $r$ indicate clustering at such scale, anticorrelation ($\xi<$0) indicate that the objects tend to avoid each other, while $\xi\sim$0 is relative to an homogeneous distribution without significative clusters.\
For practical purposes the *2pt-ACF* can be computed from a sample of objects counting the pairs of observations at different separations $r$. Four estimators are generally used in literature: Peebles-Hauser $\hat{\xi}_{PH}$ [@PeeblesHauser], Davis-Peebles $\hat{\xi}_{DP}$ [@DavisPeebles], Hamilton $\hat{\xi}_{H}$ [@Hamilton] and Landy-Szalay $\hat{\xi}_{LS}$ [@LandySzalay]. They require to calculate the number of pairs $DD(r)$, $RR(r)$, $DR(r)$ at distance $r$ respectively present in the data set (data-data), in a random data set (random-random) and in the data-random set (data-random): $$\begin{aligned}
\hat{\xi}_{PH}(r)&=& \frac{N_{RR}}{N_{DD}}\frac{DD(r)}{RR(r)}-1\\\hat{\xi}_{DP}(r)&=& \frac{N_{DR}}{N_{DD}}\frac{DD(r)}{DR(r)}-1\\\hat{\xi}_{H}(r)&=& \frac{N_{DR}^{2}}{N_{DD}N_{RR}}\frac{DD(r)RR(r)}{[DR(r)]^{2}}-1\\\hat{\xi}_{LS}(r)&=& 1+\frac{N_{RR}}{N_{DD}}\frac{DD(r)}{RR(r)}-2\frac{N_{RR}}{N_{DR}}\frac{DR(r)}{RR(r)}$$ with $N_{D}$, $N_{R}$ total number of observations present in the data and random data sets and where $N_{DD}= N_{D}(N_{D}-1)/2$, $N_{RR}= N_{R}(N_{R}-1)/2$ and $N_{DR}= N_{D}N_{R}$ are the total number of corresponding pairs in the data-data, random-random, data-random catalogues. Such estimators take into account the edge effect due to the fact that it is not always possible to fit in complete spheres of radius $r$ at every position within a survey volume, for example at the container borders.\
The above estimators define spatial correlation only. To seach for both spatial and angular correlations we introduced a weight $w_{ij}=\theta_{i}^{\alpha}+\theta_{j}^{\alpha}$ ($\alpha$=2) for each observation pair $ij$ with scattering angles ($\theta_{i}$, $\theta_{j}$) and computed a weighted correlation estimator $\xi_{w}(r)$.
### Clustering algorithms {#ClusteringSection}
High-Z materials are imaged with the POCA method as regions of higher densities of unspecified shape with respect to the background. It is therefore a natural choice to employ density-based clustering methods in the tomography reconstruction. They connect data points within a certain distance threshold $\epsilon$, satisfying a density criterion defined as the minimum number of objects $N_{min}$ within $\epsilon$. A cluster of arbitrary shape, in contrast to other clustering methods, is in this way defined by all density-connected objects plus all objects within the distance range.\
The most popular clustering algorithms is <span style="font-variant:small-caps;">dbscan</span> [@DBSCAN]. It requires $\epsilon$ and $N_{min}$ as input parameters and it is based on the following steps:
1. Choose an arbitrary unvisited data point as starting point;
2. Find the neighborhood of this point, e.g. all points within the radius $\epsilon$. If more than $N_{min}$ neighborhoods are found around this point then a cluster is started and the point marked as visited, otherwise the point is labelled as noise;
3. If a point is found to be a part of the cluster then its $\epsilon$ neighborhood is also the part of the cluster and the above procedure from step 2 is repeated for all $\epsilon$ neighborhood points. This is repeated until all points in the cluster are determined.
4. A new unvisited point is retrieved and processed, leading to the discovery of a further cluster or noise.
5. This process continues until all points are marked as visited.
Several clustering algorithms were tested, including <span style="font-variant:small-caps;">dbscan</span>. The friends-of-friends algorithm [@fof; @fof1], hereafter denoted as <span style="font-variant:small-caps;">fof</span>, has been found to provide the best results. The <span style="font-variant:small-caps;">fof</span> is a percolation algorithm normally used to identify dark matter halos from $N$-body simulations. It defines uniquely groups that contain all the particles separated by a distance smaller than a given linking length $l_{link}$. Once the linking length is defined, the algorithm identifies all pairs of particles which have a mutual distance smaller than the linking one. These pairs are designated friends, and clusters are defined as sets of particles that are connected by one or more of the friendly relations, so that they are friends of friends. The linking length is related to a parameter $l$ that represents the mean interparticle separation in simulations (related to the mean number density $\langle n\rangle$ as $l={\langle n\rangle}^{-1/3}$). Another parameter in FOF algorithm is the minimum number of particles $N_{min}$, in a cluster. The aim is to reject spurious clusters, that is groups of friends who do not form persistent objects in the simulation. Choosing $N_{min}$ sufficiently large allows to eliminate spurious clusters. In fact it is much more likely that a spurious cluster (noise) involves a small number of points and not viceversa.
\
Maximum likelihood method
-------------------------
A better statistical treatment of the scattering processes can be done using a log-likelihood approach. It assumes the volume to be imaged divided into $N_{voxels}$ three-dimensional voxels or pixels of size $N_{x}\times N_{y}\times N_{z}$. Following the well known Rossi’s formula, describing the variance of the scattering angle of a particle of momentum $p_{0}$ traversing a material of radiation length $X_{0}$, a scattering density $\lambda$ is defined for each voxel and given by: $$\label{ScatteringDensityDefinition}
\lambda(X_{0})= \left(\frac{15\;\mbox{MeV}}{p_{0}}\right)^{2}\frac{1}{X_{0}}$$ The determination of $\lambda_{j}$ ($j$=1,…$N_{voxels}$) can be done by fitting the scattering data $\mathbf{x}_{i}$= ($\Delta\theta_{x,y}$, $\Delta_{x,y}$) for each $i$-th muon event for both x and y coordinates. Such joint distribution for a given scattering layer is with good approximation[^1] modelled with a bivariate gaussian with covariance matrix $\mathbf{\Sigma}_{i}$ given by: $$\Sigma_{i}= E_{i}+p_{r,i}^{2}\sum_{j=1}^{N_{voxels}}W_{ij}\lambda_{j}$$ where $E_{i}$ is the measurement error matrix, $W_{ij}$ is the scattering covariance matrix through the $j$ voxel and $p_{r,i}= p/p_{0}$ is the ratio between the muon momentum $p$ and the reference momentum $p_{0}$.\
The log-likelihood $\mathcal{L}$ of a data sample of $N$ muon events is therefore given by: $$P(\mathbf{x}|\lambda)= \prod_{i=1}^{N}\frac{1}{2\pi|\Sigma_{i}|^{1/2}}\exp\left(-\frac{1}{2}\mathbf{x}_{i}^{T}\mathbf{\Sigma}_{i}\mathbf{x}_{i}\right)$$ $$\mathcal{L}(\mathbf{x}|\lambda)= \frac{1}{2}\sum_{i}^{N}(\log|\Sigma_{i}^{-1}|-y_{i}^{T}\Sigma_{i}^{-1}y_{i})$$ The scattering densities $\lambda_{j}$ are estimated by maximizing the above log-likelihood. Traditional algorithms, such as those based on Newton-Raphson optimization, are limited by the large number of parameters to be determined, i.e. $\sim$5$\times$10$^{4}$ for voxels of size 10 cm, and by considerable computation and storage required to compute the Hessian matrix. Schultz et al [@Schultz] provided a closed form solution to the problem in the EM formulation, leading to the following iterative estimation for $\lambda_{j}$: $$\begin{aligned}
\label{EMIterationFormula}
\lambda_{j}^{(k+1)}&= \frac{1}{M_{j}}\sum_{i}S_{ij}^{(k)}\\
S_{ij}&= 2\lambda_{j}^{(k)}+p_{r,i}^{2}(\lambda_{j}^{(k)})^{2}(y_{i}^{T}\Sigma_{i}^{-1}W_{ij}\Sigma_{i}^{-1}y_{i}-\mbox{Tr}(\Sigma_{i}^{-1}W_{ij}))\end{aligned}$$ where $M_{j}$ is the number of events traversing voxel $j$. A formula to compute $W_{ij}$ is also available in [@Schultz]. In the following we will therefore denote this method as *EM-ML* for brevity.\
The algorithm requires the following stages:
- Init
1. Reconstruct the scattering data ($\Delta\theta_{x,y}$, $\Delta_{x,y}$)$_{i}$ for each event;
2. Compute the weight matrices $W_{ij}$ for each event $i$ on the basis of the muon path length through the $j$ voxel. The latter can be estimated assuming a straight line connecting entrance and exit points from the inspected volume, eventually passing from the POCA point, if this is available or trustable. Figure \[EMLLSketchFig\] shows a sketch of the algorithm raytracing principle. The path length calculation is achieved with a standalone <span style="font-variant:small-caps;">Geant4</span> navigator allowing a fast navigation through the container voxelized geometry;
- Imaging
1. Assume an initial estimate $\lambda_{j}^{0}$ for $\lambda_{j}$;
2. Iterate formula (\[EMIterationFormula\]) until convergence or early stopping;
It is well known that with iterative algorithms the image reconstruction can be deteriorated as the iteration proceeds. An early stopping criterion is therefore often used. Here we decided to stop the iterative procedure when the average relative $\lambda_{j}$ variation drops below a prespecified threshold $\varepsilon$, namely: $$\frac{1}{N_{voxels}}\sum_{j}^{N_{voxels}} \frac{\lambda_{j}^{(k+1)}-\lambda_{j}^{(k)}}{\lambda_{j}^{(k)}}<\varepsilon\;\;\;\;$$ where we assumed $\varepsilon$=1% and we required the criterion to be fulfilled during a given number of consecutive iterations (i.e. 5). Typically 20-30 iterations are needed to match the above criterion for the considered tomographic scenarios.
\
\
Application to simulated data {#ResultsSection}
=============================
To validate the imaging methods described in the previous section in realistic conditions we developed a detailed <span style="font-variant:small-caps;">Geant4</span> simulation of the detector. More details on the simulation procedure as well as on the data reconstruction and quality selection are reported in the next sections (\[EventSimulationSection\], \[EventReconstructionSection\]). Typical results obtained over different tomographic scenarios are presented in section \[ResultsSection\].
\
\
End-to-end detector simulation {#EventSimulationSection}
------------------------------
The developed simulation incorporates all relevant detector elements (scintillators, WLS fibers, …) and also the relevant mechanical structures responsible for the possible muon scatterings along its path. These includes the support rack, the container volume (essentially roof and floor) eventually with potential threat objects and a ground layer placed below the detector structure.\
Cosmic ray muons are injected in the detector with realistic energy and angular distributions, as derived from <span style="font-variant:small-caps;">Corsika</span> [@CORSIKA] simulations for proton-induced showers, generated for the Catania location (sea level, 37$^{\circ}$30’4“68 N, 15$^{\circ}$4’27”12 E, ($B_{x}$,$B_{z}$)= (27.16,-35.4)) with a E$^{-2.6}$ energy spectrum in the range 10$^{9}$-10$^{15}$ eV and isotropic angular distribution. The energy distributions of the secondaries is approximately log-normal peaked at $\sim$3 GeV. The angular distributions are $\propto\sin\theta\cos^{2}\theta$, peaked at $\sim$30$^{\circ}$. The particle distributions obtained with <span style="font-variant:small-caps;">Corsika</span> have been parametrized for fast generation.\
Two kind of simulations are available. Full simulations include the explicit trasport of photons inside the scintillator bars and WLS fibres and are typically used only for detector design studies. Fast simulations, in which optical processes are switched off, are instead used for event reconstruction studies as well as to provide tomographic scenarios to test the imaging algorithms.\
In Figure \[DetectorFig\] we show a sample simulated $\mu$ event of energy 1 GeV together with hits produced in each detector plane. The color scale in the plots represents the energy deposited in each hit. Hits with red borders are effectively due to the muon while others are spurious.
Event reconstruction and selection {#EventReconstructionSection}
----------------------------------
The event reconstruction procedure is done according to the following stages:
- *Hit selection*: Strips are considered to be triggered if the particle energy deposit $dE/dX$ (or the number of produced photoelectrons) is above a pre-specified threshold. A trigger threshold of 1 MeV is assumed for strip triggering. Furthermore we select strips triggering within a given time interval $\Delta t$. Events triggering at least four planes, hereafter denoted as 4-fold events, are selected and hits from each X-Y planes are collected to form a list of candidate track points. Each of these points is smeared with a Gaussian of width equal to the detector position resolution $\sigma$=1 cm/$\sqrt{12}\sim$2.9 mm.
- *Cluster finding*: The list of candidate track points in each plane is scanned to find cluster candidates, defined by adjacent hit strips. The obtained clusters can be eventually re-splitted in single hits afterwards if the cluster multiplicity is above a given threshold. Finally single hits are replaced with the cluster barycenter and passed to the track finding stage.
- *Track finding*: Valid track candidates (at least one hit in each plane, $\theta_{rel}<60^{\circ}$) are collected and then reconstructed using a Kalman-Filter approach [@KalmanFilter]. In case of multiple track candidates, the track selection is done according to minimum $\chi^{2}$ criterion.
For tomography studies a practical choice is to select events with only one cluster per plane. Multi-cluster events with larger multiplicity will be anyway recorded for cosmic ray physics studies.\
To reject spurious events and therefore to reduce the chance of getting false positive in the imaging phase, we applied in the reconstruction algorithms a further quality selection to the available data:
- POCA, Clustering, 2pt-ACF: Events with scattering angles $\theta$ larger than 2 degrees are selected to reduce the noise due to other scatterers;
- EM-ML: Only events crossing the entire container from the top plane to the bottom plane are considered. To limit the chances of uncorrect raytracing, leading to misidentifications and fakes, the POCA information is used to define the muon path inside the container only for events with a “trustable” POCA reconstruction, e.g. those preliminarly selected in the clustering analysis stage. After the selection chain, only a small percentage ($\sim$5%) of the total events was found to be rejected.
\
Imaging results {#ImagingResultsSection}
---------------
We validated the implemented algorithms using <span style="font-variant:small-caps;">Geant4</span> simulations with different tomographic scenarios, shown in Figure \[TomographyScenariosFig\]:
- *Scenario A*: Four threat boxes ($W$, $U$, $Pb$, $Sn$) of size 10 cm$\times$10 cm$\times$10 cm inserted at the center of a empty container. The container load relative to the scene is $\sim$100 kg.
- *Scenario B*: A “<span style="font-variant:small-caps;">muon</span>” shape built with voxels of size 10 cm$\times$10 cm$\times$10 cm inserted at the center of an empty container. Each letter is made of different materials: <span style="font-variant:small-caps;">m</span>= Uranium, <span style="font-variant:small-caps;">u</span>= Iron, <span style="font-variant:small-caps;">o</span>= Lead, <span style="font-variant:small-caps;">n</span>= Aluminium. The container load relative to the scene is $\sim$480 kg.
- *Scenario C*: Same of scenario B. A denser environment is assumed inside the container volume, filled with layers of washing machine-like elements. These are made by an aluminium casing with an iron engine inside with relative support bars and a concrete block. The container load relative to the scene is $\sim$3500 kg.
A number of muon events of 5$\times$10$^{5}$, corresponding to $\sim$10 minutes scanning time, has been simulated for each scenario using a realistic energy spectrum with range 0.1-100 GeV.\
In Figure \[POCAFig\] we report the results relative to the POCA method for the three scenarios under test. On the left panels we report the tomographic XY section of the container for a fixed $z$ level equal to $z$= 5cm (volume center). The right panels show a 3D volume rendering of the entire container. The volume has been divided into cells of volume 10 cm$\times$10 cm$\times$10 cm and the color scale represents the POCA signal for each bin $i$, namely $\sum_{j=1}^{N_{i}}\theta_{j}^{2}$ with $N_{i}$ number of POCA events in bin $i$. As can be seen, all three scenarios are successfully identified. Due to the intrinsic resolution of the POCA method[^2] a persistent halo is present and consequentely the imaged objects are slightly increased in size with respect to the real dimensions, particularly along the vertical z-axis. A considerable noise, related to the engine elements, is present in the dense environment scenario. In such case we needed to adopt a more stringent quality cut in the scattering angle ($\theta>$6$^{\circ}$), to achieve the identification of the threat objects.\
We report in Figure \[ACFFig\] the results relative to the autocorrelation analysis, computed using a random data sample of 5$\times10^{6}$ simulated events in an empty container volume, e.g. ten times larger than the data sample under investigation. Upper plots refer to the standard 2pt-ACF estimators, reported with different color lines, while in the bottom panels we report the weighted correlation function. As can be seen, in all cases we obtain a significant excess with respect to the background at distance scales around 5 cm. The excess can be largely enhanced by using the scattering angle information (weighted 2pt-ACF) together with the spatial information.\
In Figure \[ClusteringFig\] we report the results obtained with the clustering method for the three scenarios. The correlation scale found with ACF analysis provides a useful estimation of the linking length parameter $\epsilon$ to be used in the clustering reconstruction. We assumed $\epsilon$=5 cm and a minimum point threshold of $N_{pts}$= 30. The color scale for the $i$-th cluster indicates the cluster weight $w_{i}$, defined as $w_{i}$= $\sum_{j=1}^{N_{i}}\theta_{j}^{2}/N_{j}$ with $N_{i}$ number of POCA points in cluster $i$. As one can see, a good accuracy is achieved in the identification of the three different scenarios, with a larger presence of noisy clusters in the reconstructed scenario C.\
Finally in Figure \[EMLLFig\] we report the results obtained with the EM-ML method. As can be observed the target objects are reconstructed with a considerably better resolution. The halo responsible for the object deformation observed with the POCA reconstruction is almost absent. As already discussed for the other algorithms, the last scenario required a different strategy with respect to the scenarios A and B. In particular we observed that in presence of noise the convergence criterion adopted for the other reconstructions ($\epsilon$=1%) is not sufficient to achieve a tomography image with quality comparable to scenario B. A smaller tolerance parameter was therefore assumed ($\epsilon=$0.05%) to achieve better results (see Figs. \[EMLLFig5\], \[EMLLFig6\]), at the cost of significantly increasing the number of required iterations and the computing times. This scenario demonstrates that a finer tuning of the likelihood algorithm is currently desirable over different, more realistic, noisy scenarios with the aim of determining a unique configuration for real time analysis.\
To assess the reconstructed image quality we made use of the structural similarity index *SSIM* introduced in [@SSIMPaper] for two-dimensional images, extended to our three-dimensional images. It allows luminosity, contrast and structure comparisons on a local basis between the reconstructed tomographic image and a reference image, built by considering the known scattering densities in each scenario as defined in formula \[ScatteringDensityDefinition\]. After proper normalization of both maps to the same range, we considered a comparison window around each image pixel $j$ and calculated over it the pixel means ($\mu_{j,rec}$, $\mu_{j,ref}$), standard deviations ($\sigma_{j,rec}$, $\sigma_{j,ref}$) and covariance $\sigma_{j,rec-ref}$ for the reconstructed and reference images respectively. The *SSIM* index for pixel $j$ is then defined as:\
$$\begin{aligned}
\mbox{\emph{SSIM}}_{j} = \frac{(2\mu_{j,rec}\mu_{j,ref}+C_{1})(2\sigma_{j,rec-ref}+C_{2})}{(\mu_{j,rec}^{2}+\mu_{j,ref}^{2}+C_{1})
(\sigma_{j,rec}^{2}+\sigma_{j,ref}^{2}+C_{2})}\end{aligned}$$ where $C_{1}$ and $C_{2}$ are small constants introduced to avoid numerical instabilities when $(\mu_{j,rec}^{2}+\mu_{j,ref}^{2})$ or $(\sigma_{j,rec}^{2}+\sigma_{j,ref}^{2})$ are very close to zero. An index close to one indicates strong agreement with the reference image, while, on the contrary, a nearly null index is symptomatic of a bad reconstruction. It is possible to define also a mean similarity index *MSSIM*, obtained by averaging the previous index over all pixels or over a given region of interest.\
In Table \[SSIMIndexTable\] we report the mean similarity index computed for the three tomographic scenarios under study and for the POCA, clustering and EM-ML methods. We considered a 3$\times$3$\times$3 pixel window around each pivot pixel and we calculated the mean similarity index over a spatial region surrounding the threats. As the visual analysis already suggested, the computed indexes are close to unity, indicating an overall accurate reconstruction. The computed index effectively reflects the better reconstruction performances of the EM-ML method with respect to the other implemented methods.
-------------- -------- -------------- ---------
**Scenario**
*POCA* *Clustering* *EM-ML*
A 0.94 0.94 0.99
B 0.89 0.89 0.98
C 0.83 0.83 0.89
-------------- -------- -------------- ---------
: Similarity index SSIM obtained with the POCA, clustering and EM-ML methods for the three tomographic scenarios under analysis.[]{data-label="SSIMIndexTable"}
Towards a real-time tomographic analysis {#SummarySection}
========================================
Concerning the software tools, two working lines are currently under progress within the project in view of the complete operation of the Muon Portal, one aiming to develop a graphical user interface for the tomography and visualization tasks, and the other focusing on the optimization of the designed algorithms for real-time application.\
To be compatible with the real container traffic at the harbours, the tomography analysis must be performed in reasonable small times, few minutes at most. The POCA algorithm, with its simplicity, guarantees the smallest computation times, perfectly matching the port requirements. No optimizations are therefore needed in such case. This is not the case for the other designed algorithms.\
The EM-ML algorithm typically requires $\sim$30 minutes on a Xeon QuadCore E5620 2.40Ghz processor for a typical scanning run of $\sim$ 10 minutes and 20-30 iterations, and therefore cannot match the requirements of a real time image processing, at least in its serial implementation. However, both the init and imaging step of the algorithm are embarrassingly parallelizable as being based on independent event loops. We are therefore planning to implement also a parallel version of the algorithm using the <span style="font-variant:small-caps;">MPI</span> library [@MPI]. The achieved speed-up with respect to a serial implementation would be remarkable. The parallel implementation allows a real time application of the method even with a modest number of computing machines.\
The computation of the 2pt ACF is a very time-consuming task, proportional to $N^{2}$ ($N$ size of the data sample). Optimized serial implementations, based on building a kd-tree [@KdTree] with the data, allow to drop the algorithm complexity at the level of $N\log(N)$. Significative speed-up can then be obtained afterwards with ad hoc optimization and parallelization techniques [@ACFParallelized] or by making use of GPUs [@ACFGPU]. At the present status a brute implementation is available. To maintain the computation time at reasonable level, the pair calculation is limited to adjacent three-dimensional voxels with size matching the maximum desired correlation scale and to observations with scattering angles larger than a predefined threshold. An optimized version of the algorithm is however currently being designed.\
The density-based clustering algorithm suffers from the same problematic discussed for the ACF computation, as it requires the computation of the nearest neighbour of each point in the volume (O(N$^{2}$)). The algorithm has been optimized by using kd trees and further optimization strategies to achieve a O($N\log(N)$) complexity which makes the algorithm reliable for real time analysis.\
In conclusion we are pursuing a large efforts in combining different reconstruction and visualization tools for a reliable and fast image processing of a muon tomography. While standard algorithms have already been implemented and their use in a real time processing of a tomographic image may be achieved even by a single standard processor, the use of alternative, more accurate, algorithms requires additional work, possible on their parallelization, to provide a comparatively fast tool. Work along this line has already started and the results will be reported in a future paper.
[99]{} S. Riggi et al (The Muon Portal Collaboration), J. Phys. Conf. Ser. 409 (2013) 012046. D.Lo Presti et al (The Muon Portal Collaboration), IEEE Nuclear Science Symposium and Medical Imaging Conference Record (NSS/MIC) N1-2 (2012). *http://muoni.oact.inaf.it:8080/* S. Agostinelli et al, Nucl. Instr. and Methods A 506 (2003) 250; K. Amako et al, IEEE Transactions on Nuclear Science 53 (2006) 270. R. Brun and F. Rademakers, Proceedings of the AIHENP’96 Workshop, Lausanne (1996), Nucl. Instr. and Meth. A 389 (1997) 81. See also *http://root.cern.ch/* P.J.E. Peebles, *The Large-Scale Structure of the Universe*, Princeton University Press, Princeton, New Jersey, 1980. P.J.E. Peebles and M.J. Hauser, ApJSS 28 (1974) 19. M. Davis and P.J.E. Peebles, ApJ 267 (1983) 465. A.J.S. Hamilton, ApJ 417 (1993) 19. S.D. Landy and A.S. Szalay ApJ 412 (1993) L64. M. Ester et al, Proceedings of 2nd International Conference on Knowledge Discovery and Data Mining (KDD-96), Portland (1996). D.W. Pfitzner, J.K. Salmon and T. Sterling, Data Mining and Knowledge Discovery 1 (1997) 419. S. More et al, ApJ Supplement Series 195 (2011) 4. L. Schultz et al, IEEE Transactions on Image Processing 16 (2007) 1985. D. Heck et al, Report FZKA-6019 (1998). R. Frühwirth, Nucl. Instr. Meth. A 262 (1987) 444; R. Mankel, Rep. Prog. Phys. 67 (2004), 553. Z. Wang et al, IEEE Transactions on Image Processing 13 (2004) 600. *http://www.mpich.org/*. J.L. Bentley, Communications of the ACM 18 (1975) 509-517. J. Dolence and R.J. Brunner, Proceedings of the 9th LCI International Conference on High-Performance Clustered Computing, Urbana Illinois (2008); W.B. March et al, Proceedings of the 18th ACM SIGKDD Conference on Knowledge discovery and data mining, Beijing (2012); A. Moore at al, Proceedings of MPA/MPE/ESO Conference “Mining the Sky”, Garching (2000); R. Ponce at al, Proceedings of the ADASS XXI Conference, Paris (2011), arXiv:1204.6630.
[^1]: Actually long tails are present in the distribution and $\sim$2% of the data cannot be well described by the single gaussian assumption. A gaussian mixture is often used to reproduce the tails.
[^2]: We performed dedicated simulations of cosmic ray muons traversing a single uranium layer of thickness 10 cm and reconstructed the POCA information for each event. About 20% of the events have the POCA information reconstructed outside the expected threat volume, falling in particular in the first surrounding voxels.
|
---
abstract: 'Opinion mining from customer reviews has become pervasive in recent years. Sentences in reviews, however, are usually classified independently, even though they form part of a review’s argumentative structure. Intuitively, sentences in a review build and elaborate upon each other; knowledge of the review structure and sentential context should thus inform the classification of each sentence. We demonstrate this hypothesis for the task of aspect-based sentiment analysis by modeling the interdependencies of sentences in a review with a hierarchical bidirectional LSTM. We show that the hierarchical model outperforms two non-hierarchical baselines, obtains results competitive with the state-of-the-art, and outperforms the state-of-the-art on five multilingual, multi-domain datasets without any hand-engineered features or external resources.'
author:
- Sebastian Ruder
- Parsa Ghaffari
- 'John G. Breslin'
bibliography:
- 'hierarchical\_absa.bib'
title: 'A Hierarchical Model of Reviews for Aspect-based Sentiment Analysis'
---
Introduction
============
Sentiment analysis [@Pang2008] is used to gauge public opinion towards products, to analyze customer satisfaction, and to detect trends. With the proliferation of customer reviews, more fine-grained aspect-based sentiment analysis (ABSA) has gained in popularity, as it allows aspects of a product or service to be examined in more detail.
Reviews – just with any coherent text – have an underlying structure. A visualization of the discourse structure according to Rhetorical Structure Theory (RST) [@Mann1988] for the example review in Figure \[fig:rst\_structure\] reveals that sentences and clauses are connected via different rhetorical relations, such as *Elaboration* and *Background*.
Intuitively, knowledge about the relations and the sentiment of surrounding sentences should inform the sentiment of the current sentence. If a reviewer of a restaurant has shown a positive sentiment towards the quality of the food, it is likely that his opinion will not change drastically over the course of the review. Additionally, overwhelmingly positive or negative sentences in the review help to disambiguate sentences whose sentiment is equivocal.
Neural network-based architectures that have recently become popular for sentiment analysis and ABSA, such as convolutional neural networks [@Severyn2015a], LSTMs [@Vo2015], and recursive neural networks [@Nguyen2015a], however, are only able to consider intra-sentence relations such as *Background* in Figure \[fig:rst\_structure\] and fail to capture inter-sentence relations, e.g. *Elaboration* that rely on discourse structure and provide valuable clues for sentiment prediction.
We introduce a hierarchical bidirectional long short-term memory (H-LSTM) that is able to leverage both intra- and inter-sentence relations. The sole dependence on sentences and their structure within a review renders our model fully language-independent. We show that the hierarchical model outperforms strong sentence-level baselines for aspect-based sentiment analysis, while achieving results competitive with the state-of-the-art and outperforming it on several datasets without relying on any hand-engineered features or sentiment lexica.
Related Work
============
**Aspect-based sentiment analysis.** Past approaches use classifiers with expensive hand-crafted features based on n-grams, parts-of-speech, negation words, and sentiment lexica [@Pontiki2014a; @Pontiki2015]. The model by Zhang and Lan is the only approach we are aware of that considers more than one sentence. However, it is less expressive than ours, as it only extracts features from the preceding and subsequent sentence without any notion of structure. Neural network-based approaches include an LSTM that determines sentiment towards a target word based on its position [@Tang2015] as well as a recursive neural network that requires parse trees [@Nguyen2015a]. In contrast, our model requires no feature engineering, no positional information, and no parser outputs, which are often unavailable for low-resource languages. We are also the first – to our knowledge – to frame sentiment analysis as a sequence tagging task.
**Hierarchical models.** Hierarchical models have been used predominantly for representation learning and generation of paragraphs and documents: Li et al. use a hierarchical LSTM-based autoencoder to reconstruct reviews and paragraphs of Wikipedia articles. Serban et al. use a hierarchical recurrent encoder-decoder with latent variables for dialogue generation. Denil et al. use a hierarchical ConvNet to extract salient sentences from reviews, while Kotzias et al. use the same architecture to learn sentence-level labels from review-level labels using a novel cost function. The model of Lee and Dernoncourt is perhaps the most similar to ours. While they also use a sentence-level LSTM, their class-level feed-forward neural network is only able to consider a limited number of preceding texts, while our review-level bidirectional LSTM is (theoretically) able to consider an unlimited number of preceding *and* successive sentences.
Model
=====
{width="0.75\linewidth"}
In the following, we will introduce the different components of our hierarchical bidirectional LSTM architecture displayed in Figure \[fig:hierarchical\_lstm\].
Sentence and Aspect Representation
----------------------------------
Each review consists of sentences, which are padded to length $l$ by inserting padding tokens. Each review in turn is padded to length $h$ by inserting sentences containing only padding tokens. We represent each sentence as a concatentation of its word embeddings $x_{1:l}$ where $x_t \in \mathbb{R}^k$ is the $k$-dimensional vector of the $t$-th word in the sentence.
Every sentence is associated with an aspect. Aspects consist of an entity and an attribute, e.g. `FOOD#QUALITY`. Similarly to the entity representation of Socher et al. , we represent every aspect $a$ as the average of its entity and attribute embeddings $\frac{1}{2} (x_e + x_a) $ where $x_e, x_a \in \mathbb{R}^m$ are the $m$-dimensional entity and attribute embeddings respectively[^1].
LSTM
----
We use a Long Short-Term Memory (LSTM) [@Hochreiter1997], which adds input, output, and forget gates to a recurrent cell, which allow it to model long-range dependencies that are essential for capturing sentiment.
For the $t$-th word in a sentence, the LSTM takes as input the word embedding $x_t$, the previous output $h_{t-1}$ and cell state $c_{t-1}$ and computes the next output $h_t$ and cell state $c_t$. Both $h$ and $c$ are initialized with zeros.
Bidirectional LSTM
------------------
Both on the review and on the sentence level, sentiment is dependent not only on preceding but also successive words and sentences. A Bidirectional LSTM (Bi-LSTM) [@Graves2013a] allows us to look ahead by employing a forward LSTM, which processes the sequence in chronological order, and a backward LSTM, which processes the sequence in reverse order. The output $h_t$ at a given time step is then the concatenation of the corresponding states of the forward and backward LSTM.
Hierarchical Bidirectional LSTM
-------------------------------
Stacking a Bi-LSTM on the review level on top of sentence-level Bi-LSTMs yields the hierarchical bidirectional LSTM (H-LSTM) in Figure \[fig:hierarchical\_lstm\].
The sentence-level forward and backward LSTMs receive the sentence starting with the first and last word embedding $x_{1}$ and $x_l$ respectively. The final output $h_l$ of both LSTMs is then concatenated with the aspect vector $a$[^2] and fed as input into the review-level forward and backward LSTMs. The outputs of both LSTMs are concatenated and fed into a final softmax layer, which outputs a probability distribution over sentiments[^3] for each sentence.
Experiments
===========
Datasets
--------
For our experiments, we consider datasets in five domains (restaurants, hotels, laptops, phones, cameras) and eight languages (English, Spanish, French, Russian, Dutch, Turkish, Arabic, Chinese) from the recent SemEval-2016 Aspect-based Sentiment Analysis task [@SemEval2016:task5], using the provided train/test splits. In total, there are 11 domain-language datasets containing 300-400 reviews with 1250-6000 sentences[^4]. Each sentence is annotated with none, one, or multiple domain-specific aspects and a sentiment value for each aspect.
**Language** **Domain** `Best` `XRCE` `IIT-TUDA` `CNN` `LSTM` `H-LSTM` `HP-LSTM`
-------------- ------------- ---------- ---------- ------------ ------- -------- ---------- -----------
English Restaurants **88.1** **88.1** 86.7 82.1 81.4 83.0 85.3
Spanish Restaurants **83.6** - **83.6** 79.6 75.7 79.5 81.8
French Restaurants **78.8** **78.8** 72.2 73.2 69.8 73.6 75.4
Russian Restaurants 77.9 - 73.6 75.1 73.9 **78.1** 77.4
Dutch Restaurants 77.8 - 77.0 75.0 73.6 82.2 **84.8**
Turkish Restaurants **84.3** - **84.3** 74.2 73.6 76.7 79.2
Arabic Hotels 82.7 - 81.7 82.7 80.5 82.8 **82.9**
English Laptops **82.8** - **82.8** 78.4 76.0 77.4 80.1
Dutch Phones 83.3 - 82.6 83.3 81.8 81.3 **83.6**
Chinese Cameras **80.5** - - 78.2 77.6 78.6 78.8
Chinese Phones 73.3 - - 72.4 70.3 **74.1** 73.3
Training Details
----------------
Our LSTMs have one layer and an output size of 200 dimensions. We use 300-dimensional word embeddings. We use pre-trained GloVe [@Pennington2014] embeddings for English, while we train embeddings on `frWaC`[^5] for French and on the Leipzig Corpora Collection[^6] for all other languages.[^7] Entity and attribute embeddings of aspects have 15 dimensions and are initialized randomly. We use dropout of 0.5 after the embedding layer and after LSTM cells, a gradient clipping norm of 5, and no $l_2$ regularization.
We unroll the aspects of every sentence in the review, e.g. a sentence with two aspects occurs twice in succession, once with each aspect. We remove sentences with no aspect[^8] and ignore predictions for all sentences that have been added as padding to a review so as not to force our model to learn meaningless predictions, as is commonly done in sequence-to-sequence learning [@Sutskever2014]. We segment Chinese data before tokenization.
We train our model to minimize the cross-entropy loss, using stochastic gradient descent, the Adam update rule [@Kingma2015], mini-batches of size 10, and early stopping with a patience of 10.
Comparison models
-----------------
We compare our model using random (`H-LSTM`) and pre-trained word embeddings (`HP-LSTM`) against the best model of the SemEval-2016 Aspect-based Sentiment Analysis task [@SemEval2016:task5] for each domain-language pair (`Best`) as well as against the two best single models of the competition: `IIT-TUDA` [@Kumar2016], which uses large sentiment lexicons for every language, and `XRCE` [@Brun2016], which uses a parser augmented with hand-crafted, domain-specific rules. In order to ascertain that the hierarchical nature of our model is the deciding factor, we additionally compare against the sentence-level convolutional neural network of Ruder et al. (`CNN`) and against a sentence-level Bi-LSTM (`LSTM`), which is identical to the first layer of our model.[^9]
Results and Discussion
======================
We present our results in Table \[tab:results\]. Our hierarchical model achieves results superior to the sentence-level CNN and the sentence-level Bi-LSTM baselines for almost all domain-language pairs by taking the structure of the review into account. We highlight examples where this improves predictions in Table \[tab:predictions\].
In addition, our model shows results competitive with the best single models of the competition, while requiring no expensive hand-crafted features or external resources, thereby demonstrating its language and domain independence. Overall, our model compares favorably to the state-of-the-art, particularly for low-resource languages, where few hand-engineered features are available. It outperforms the state-of-the-art on four and five datasets using randomly initialized and pre-trained embeddings respectively.
**Id** **Sentence** **LSTM** **H-LSTM**
-------- ------------------------ ------------ ------------
1.1 No Comparison *negative* *positive*
It has great sushi and
even better service.
Green Tea creme
brulee is a must!
Don’t leave the
restaurant without it.
: Example sentences where knowledge of other sentences in the review (not necessarily neighbors) helps to disambiguate the sentiment of the sentence in question. For the aspect in 1.1, the sentence-level LSTM predicts *negative*, while the context of the service and food quality in 1.2 allows the H-LSTM to predict *positive*. Similarly, for the aspect in 2.2, knowledge of the quality of the green tea crème brulée helps the H-LSTM to predict the correct sentiment.[]{data-label="tab:predictions"}
Pre-trained embeddings
----------------------
In line with past research [@Collobert2011a], we observe significant gains when initializing our word vectors with pre-trained embeddings across almost all languages. Pre-trained embeddings improve our model’s performance for all languages except Russian, Arabic, and Chinese and help it achieve state-of-the-art in the Dutch phones domain. We release our pre-trained multilingual embeddings so that they may facilitate future research in multilingual sentiment analysis and text classification[^10].
Leveraging additional information
---------------------------------
As annotation is expensive in many real-world applications, learning from only few examples is important. Our model was designed with this goal in mind and is able to extract additional information inherent in the training data. By leveraging the structure of the review, our model is able to inform and improve its sentiment predictions as evidenced in Table \[tab:predictions\].
The large performance differential to the state-of-the-art for the Turkish dataset where only 1104 sentences are available for training and the performance gaps for high-resource languages such as English, Spanish, and French, however, indicate the limits of an approach such as ours that only uses data available at training time.
While using pre-trained word embeddings is an effective way to mitigate this deficit, for high-resource languages, solely leveraging unsupervised language information is not enough to perform on-par with approaches that make use of large external resources [@Kumar2016] and meticulously hand-crafted features [@Brun2016].
Sentiment lexicons are a popular way to inject additional information into models for sentiment analysis. We experimented with using sentiment lexicons by Kumar et al. but were not able to significantly improve upon our results with pre-trained embeddings[^11]. In light of the diversity of domains in the context of aspect-based sentiment analysis and many other applications, domain-specific lexicons [@Hamilton2016] are often preferred. Finding better ways to incorporate such domain-specific resources into models as well as methods to inject other forms of domain information, e.g. by constraining them with rules [@Hu2016] is thus an important research avenue, which we leave for future work.
Conclusion
==========
In this paper, we have presented a hierarchical model of reviews for aspect-based sentiment analysis. We demonstrate that by allowing the model to take into account the structure of the review and the sentential context for its predictions, it is able to outperform models that only rely on sentence information and achieves performance competitive with models that leverage large external resources and hand-engineered features. Our model achieves state-of-the-art results on 5 out of 11 datasets for aspect-based sentiment analysis.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the anonymous reviewers, Nicolas Pécheux, and Hugo Larochelle for their constructive feedback. This publication has emanated from research conducted with the financial support of the Irish Research Council (IRC) under Grant Number EBPPG/2014/30 and with Aylien Ltd. as Enterprise Partner as well as from research supported by a research grant from Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289.
[^1]: Averaging embeddings produced slightly better results than using a separate embedding for every aspect.
[^2]: We experimented with other interactions, e.g. rescaling the word embeddings by their aspect similarity, an attention-like mechanism, as well as summing and multiplication, but found that simple concatenation produced the best results.
[^3]: The sentiment classes are *positive*, *negative*, and *neutral*.
[^4]: Exact dataset statistics can be seen in [@SemEval2016:task5].
[^5]: <http://wacky.sslmit.unibo.it/doku.php?id=corpora>
[^6]: <http://corpora2.informatik.uni-leipzig.de/download.html>
[^7]: Using 64-dimensional Polyglot embeddings [@Al-Rfou2013] yielded generally worse performance.
[^8]: Labeling them with a `NONE` aspect and predicting *neutral* slightly decreased performance.
[^9]: To ensure that the additional parameters do not account for the difference, we increase the number of layers and dimensions of `LSTM`, which does not impact the results.
[^10]: <https://s3.amazonaws.com/aylien-main/data/multilingual-embeddings/index.html>
[^11]: We tried bucketing and embedding of sentiment scores as well as filtering and pooling as in [@Vo2015]
|
---
abstract: 'The discovery of a hyper metal-poor star with total metallicity of $\le 10^{-5}$ Z$_\odot$, has motivated new investigations of how such objects can form from primordial gas polluted by a single supernova. In this paper we present a shock-cloud model which simulates a supernova remnant interacting with a cloud in a metal-free environment at redshift $z=10$. Pre-supernova conditions are considered, which include a multiphase neutral medium and H<span style="font-variant:small-caps;">ii</span> region. A small dense clump ($n=100$ cm$^{-3}$), located 40 pc from a 40 M$_\odot$ metal-free star, embedded in a $n=10$ cm$^{-3}$ ambient cloud. The evolution of the supernova remnant and its subsequent interaction with the dense clump is examined. We include a comprehensive treatment of the non-equilibrium [hydrogen and helium]{} chemistry and associated radiative cooling that is occurring at all stages of the shock-cloud model, covering the temperature range $10-10^9$ K. [Deuterium chemistry and its associated cooling are not included because the UV radiation field produced by the relic H<span style="font-variant:small-caps;">ii</span> region and supernova remnant is expected to suppress deuterium chemistry and cooling.]{} We find a $10^{3}\times$ density enhancement of the clump (maximum density $\approx 78000$ cm$^{-3}$) within this metal-free model. This is consistent with Galactic shock-cloud models considering solar metallicity gas with equilibrium cooling functions. Despite this strong compression, the cloud does not become gravitationally unstable. We find that the small cloud modelled here is destroyed for shock velocities $\gtrsim 50\,$kms$^{-1}$, and not significantly affected by shocks with velocity $\lesssim 30\,$kms$^{-1}$. Rather specific conditions are required to make such a cloud collapse, and substantial further compression would be required to reduce the local Jeans mass to subsolar values.'
author:
- |
H. Dhanoa$^{1}$[^1], J. Mackey$^{2}$, and J. Yates$^{1}$\
$^{1}$ Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT\
$^{2}$ Argelander-Institut für Astronomie, Auf dem Hügel 71, Bonn D-53121, Germany
bibliography:
- 'mf\_letter.bib'
title: 'Pressure-driven fragmentation of multi-phase clouds at high redshift'
---
\[firstpage\]
High redshift - stars: formation - supernovae: general - molecules
Introduction
============
The first galaxies are thought to have formed around redshift $z\ge 10$ when the universe was less than 500 Myrs old. These nascent environments are considered to be the key sites where the transition from Population III to Population II stars took place. A possible fossil from this era is SDSS J102915+172927, which is a low-mass ($M<0.8$ $M_{\odot}$) star with a total metallicity of Z$\,<10^{-5}$Z$_{\odot}$ [@Caffau11]. As a result of such low metallicity, it is deduced that the star formed from primordial gas which was polluted by a single supernova. This star has challenged the theory that a critical metallicity is needed to form sub-solar-mass Population II star [@Klessen12]. A better understanding of star formation and its feedback effects at high redshifts is extremely important in relation to the formation of such objects.
While it is important to study star formation at very low metallicity [@Nagakura09; @Chiaki13], one cannot evaluate the effects of tiny metal abundances without also studying primordial gas. The metal-free problem is the limiting case, and is therefore very useful as a baseline study for comparison to later calculations for gas that is polluted by trace amounts of metals. It is also very interesting in its own right, because we still do not know if stars of mass $<1$ M$_\odot$ can form at zero metallicity (see e.g.the interesting proposal presented by @Stacy14).
Here we examine the shock-cloud interaction model developed by @MacBroHer03, in which shock compression and subsequent cooling can decrease the Jeans mass in primordial gas, thereby forming lower-mass stars than would form without the shock collision. Our work is the first to investigate this problem with detailed multi-dimensional simulations for metal-free gas. Radiative cooling is the critical factor in promoting hydrodynamic and gravitational instabilities. Therefore in this paper we focus on the non-equilibrium cooling that dominates this system. This can only be captured correctly by including non-equilibrium chemistry (linked to thermal models) for the full temperature range associated with a supernova shock model. [We have focused on hydrogen and helium chemistry because we expect that the environment surrounding a progenitor Population III star is dominated by H$_2$ cooling. Both the relic H<span style="font-variant:small-caps;">ii</span> region and the supernova remnant are sources of diffuse UV radiation that suppresses HD cooling, so we have not included deuterium chemistry in this work. @WolHai11 showed that HD cooling is strongly suppressed by UV radiation fields that are up to five orders of magnitude weaker than what is required to suppress H$_2$ cooling. ]{}
@Kitayama05 and @Vasiliev08 highlighted the important link between the radial distribution of primordial gas prior to the supernova explosion and the subsequent evolution of the supernova remnant, and therefore the formation of extremely metal-poor stars. Consequently, we include both the H<span style="font-variant:small-caps;">ii</span> region and neutral medium, to obtain a realistic supernova shell evolution. Once the supernova shock begins to travel within neutral matter, it interacts with a multi-phase medium [@Reach05], which cannot be characterised by a single density. @Greif08 have found that turbulence driven by cold accretion onto a protogalaxy produces a primordial interstellar medium with a large range of densities and temperatures. The pressure-driven compression and fragmentation of dense clumps found in this neutral matter could be a possible site for low-mass star formation.
At present most supernova shock models for the early universe only include non-equilibrium cooling for temperatures below $10^4$ K and focus on the fragmentation of the supernova shell itself. @Machida05 were the first to investigate primordial low-mass star formation at high redshift via this method. The authors included non-equilibrium cooling from H$_2$ and HD molecules, coupled to a semi-analytic dynamic model. They found that shell fragmentation was possible for explosion energies $\ge 10^{51}$ erg and ambient density $n>3$ cm$^{-3}$. The contraction of the fragments was studied, and the Jeans mass was reduced to $\sim$1 M$_\odot$. @Nagakura09 extended this model to include metal-line cooling for low-metallicity gas coupled to a 1D hydrodynamic code. They use linear perturbation analysis of the expanding shell to constrain the criteria for fragmentation and found that there is little dependency on metallicity in the range 10$^{-4}-10^{-2}\, Z_{\odot}$. Compared to @Machida05, they found that fragmentation only occurred in higher ambient uniform densities ($n\ge 100$ cm$^{-3}$ for a $10^{51}$ erg explosion and $n\ge 10$ cm$^{-3}$ for a $10^{52}$ erg explosion), and eventually form fragments of mass $10^2-10^3$ M$_\odot$.
@Chiaki13 developed a 1D supernova model that considers a gas with metallicity $10^{-5}\,\rm{Z}_\odot$. The authors include metal-free non-equilibrium chemistry for temperatures below $10^4$ K, with separate calculated rates for metal-line cooling. However, above $10^4$ K the authors utilise the collisional ionization equilibrium cooling function by @Sutherland93. The authors find that the supernova shell becomes gravitationally unstable for a wide range of explosion energies ($10^{51}-3\times 10^{52}$ erg) and ambient uniform densities ($n\ge 10$ cm$^{-3}$). The thermal evolution of a shell fragment was followed using a one-zone model (a point calculation) which includes low-metallicity chemistry and dust cooling. They expect the fragment to evolve into a high density core ($10^{13}$ cm$^{-3}$), which will eventually form multiple clumps of mass $0.01 -0.1$ M$_\odot$.
Using a one-zone model, @MacBroHer03 modelled an equilibrium primordial gas cloud that is shocked by a supernova. The shocked cloud is heated to a higher entropy state and it is assumed to cool isobarically back to its original equilibrium temperature, but now at a much higher density than before. In this way the Jeans mass of the gas could be reduced by a large factor, allowing much lower-mass stars to form. This argument also applies to smooth ISM distributions, as discussed above [@Machida05; @Nagakura09; @Chiaki13], as long as isobaric conditions hold in the decelerating shell.
The one-zone model of @MacBroHer03 also crucially depends on the isobaric assumption to increase the gas density in the cooling cloud. In reality, however, pressure is a decreasing function of time in a supernova remnant, because the explosion is (by definition) vastly over-pressurised compared to its surroundings. As long as the expansion timescale of the supernova $t_\mathrm{exp}=R_\mathrm{sh}/\dot{R_\mathrm{sh}}$ (where $R_\mathrm{sh}$ is the shock radius and $\dot{R_\mathrm{sh}}$ its velocity) is short compared to the local timescale for gravitational effects (i.e the free-fall time $t_\mathrm{ff}=1/\sqrt{G\rho}$, where $\rho$ is the gas density and $G$ the gravitational constant) then the time-dependence of the external pressure is an important part of the solution. The passage of a strong shock through a dense cloud can also have catastrophic consequences for the cloud [@KleMcKCol94] through turbulent hydrodynamic instabilities. Both of these considerations are best addressed with multi-dimensional hydrodynamic simulations and cannot be captured in one-zone models.
@Melioli06 investigated star formation triggered in the Galactic environment, via the interaction of a supernova shell and molecular cloud. The authors produce constraints on cloud collapse (and therefore possible star formation) in the ‘supernova remnant radius vs. cloud density’ parameter space. This was achieved by an analytic study comparing the gravitational free-fall time and destruction time scale of the cloud (which depends on a number of parameters including radiative cooling). By running a suite of 3D hydrodynamic simulations, they were able to confirm that these numerical models were consistent with their analytic constraints. The authors recognise that using an approximate polytropic pressure equation to represent radiative cooling maybe an over simplification and more realistic cooling functions are required.
@Johansson13 have concentrated on the compression of smaller clouds (radius $\sim$ 1 pc) found in the local interstellar medium as a method of triggered star formation. Their MHD simulations (without self-gravity) concentrate on the radiative interaction between the shock and the cloud. The cooling function utilised is a piecewise power-law given by @Sanchez02 and @Slyz05, and assumes collisional ionization equilibrium. They find that the cloud fragments into small dense cool clumps and do not become Jeans unstable. Importantly they find that initial density enhancements within the cloud can increase by a factor of $10^3 -10^5$, which eventually relaxes to a final density enhancement of $10^2-10^3$. This is consistent with results by @Vaidya13, who have a similar model which includes self gravity. They find that gravity does not contribute to the large increase in density but plays an important role by preventing the re-expansion of the high density region.
These studies have highlighted that radiative cooling is a crucial process in the interaction between shocks and clouds. In this paper we simulate a supernova exploding in a metal-free environment and include the non-equilibrium radiative cooling that occurs at all stages of its evolution and subsequent collision with a multiphase neutral cloud. The diffusion of the metals is neglected and the system is approximated by primordial chemistry. Hence we present a model which includes the non-equilibrium metal-free chemistry and its associated cooling for the evolution of a supernova shell and its subsequent interaction of a small dense clump embedded in a neutral cloud at redshift $z=10$. In section §\[sec:model\] we outline how the initial conditions are generated by the pre-supernova model, and introduce the chemo-dynamic modelling of the supernova remnant. The results describing the generation of the pre-supernova model, the 1D Supernova model and the 2D interaction of the clump and shock, are presented in section §\[sec:results\]. Finally, in sections §\[sec:dicusssion\] and §\[sec:conc\] we discuss our findings and give a summary of the conclusions.
Methods and initial conditions {#sec:model}
==============================
We have modelled the interaction of a supernova shell with a dense clump in three stages:
1. the pre-supernova phase, where the dynamical effects of photoionization heating from the star are modelled;
2. the post-supernova phase, where the supernova blast wave expands into the relic H<span style="font-variant:small-caps;">ii</span> region left by the star; and
3. the shock-cloud interaction, where the expanding supernova shell compresses a dense cloud.
The first two stages are simulated in one dimension with spherical symmetry, whereas the third stage is simulated in two dimensions with rotational symmetry using a. This is because compression and fragmentation of the clump cannot be captured within 1D models. However, it is possible to achieve a good representation of the evolution of the supernova remnant in 1D models, assuming that the shell has not interacted with any dense clumps [@Jun96].
For the 1D simulations we use reflective boundary conditions at the origin (imposed by the symmetry of the problem), and a zero gradient outflow condition at the large radius boundary. For the 2D simulations with cylindrical coordinates $(R,z)$ we use a reflective boundary at $R=0$ (again imposed by symmetry) and zero gradient at $R=R_\mathrm{max}$, an inflow boundary at $z=z_\mathrm{min}$, and zero gradient at $z=z_\mathrm{max}$. The inflow boundary condition is justified because the post-shock flow variables change slowly for $\approx 5-7$ pc behind the blast wave (see Fig. 2).
[As argued in the Introduction, we do not expect HD cooling to be important because the supernova shell and dense clump are exposed to UV radiation from the nearby relic H<span style="font-variant:small-caps;">ii</span> region and expanding supernova remnant. HD cooling is much more readily suppressed by UV radiation than H$_2$ cooling [@WolHai11], so we focus here only on the hydrogen and helium chemistry and cooling ]{}
Pre-supernova phase
-------------------
We use the radiation-magnetohydrodynamics code <span style="font-variant:small-caps;">pion</span> [@Mackey10; @MacLim11] for the simulations presented here, first in 1D with spherical symmetry and later in 2D with rotational (axi-)symmetry. <span style="font-variant:small-caps;">pion</span> uses an explicit, finite-volume, integration scheme that is accurate to second order in time and space [@Fal91]. Here only the Euler equations of hydrodynamics are solved, together with the ionization rate equation of hydrogen and associated non-equilibrium heating and cooling processes. The microphysical processes of ionization, recombination, heating and cooling are coupled to hydrodynamics using Algorithm 3 in @Mac12.
We consider a metal-free star exploding in a small galaxy at redshift $z=10$, sweeping up the ambient medium to form an expanding shell. The simplified initial condition consists of a uniform neutral interstellar medium with hydrogen number density $n=10\,\mathrm{cm}^{-3}$. Into this we place a dense cloud with (uniform) number density $n=100\,\mathrm{cm}^{-3}$, radius $r_\mathrm{c}=1.3$pc, and located at $r=40$pc from the star (which is at the origin). The gas is comprised of atomic hydrogen and helium (number density ratio of 1.00:0.08) and is cooled via atomic processes. We assume the star has formed in a sufficiently large galaxy that gravitational potential gradients can be neglected in the hydrodynamical evolution of the system. This is the simplest possible model for feedback from the massive star to a nearby cloud.
For the star’s properties we take the 40 M$_\odot$ metal-free model from @Schaerer02 with no mass loss. This has a lifetime of 3.86Myr, an effective temperature $T_\mathrm{eff}=10^{4.9}$K, and a time-averaged H-ionising photon luminosity $Q_0=2.47\times10^{49}$s$^{-1}$. For simplicity we distribute these photons according to a blackbody spectrum with the star’s $T_\mathrm{eff}$. We ignore any post main sequence evolutionary effects because this comprises a small fraction of the star’s life, and because the evolution is very uncertain. This model in @Schaerer02 also remains relatively blue for its full lifetime, thus supporting our approximation of excluding a red supergiant phase.
Supernova Remnant phase
-----------------------
A supernova remnant is dominated by non-equilibrium cooling, therefore we developed a microphysics module which links the non-equilibrium chemistry and its associated cooling. This was accomplished by solving the following set of equations: $$\begin{aligned}
\frac{\partial{E}}{\partial{t}}&=&-\Lambda(\Sigma x_m,\rho,\mbox{T}) + \Gamma(\Sigma x_n,\rho,\mbox{T})\\
\frac{\partial x_i}{\partial t}&=& C_i \left( x_j,\rho, \rm{T}\right) -D_i\left( x_j,\rho,\rm{T}\right) x_i\label{chemical_ODE_eqn}\end{aligned}$$ where $E$ is the internal energy density (in ergcm$^{-3}$), $\Lambda$ is the cooling function of the gas (in ergcm$^{-3}$s$^{-1}$), $\Gamma$ is the heating function of the gas (in ergcm$^{-3}$s$^{-1}$), $x_i$ is the fractional abundance of a chemical species, $i$, for a total number of chemical species $N_\mathrm{s}$, T is the temperature of the gas (K), $\rho$ is the total mass density of the gas (g cm$^{-3}$), C is the formation rate of the species and D is the destruction rate of the species. We use a chemical network of 11 species (H, He, H$_2$, H$^+$, H$_2^+$, H$_3^+$, HeH$^+$, He$^+$, He$^{++}$, H$^-$ and e$^-$) and 42 reactions. The chemical rates cover the temperature range $10-10^{9}$ K, which are described in appendix \[appen:chem\]. The atomic species and electron fraction are treated numerically as conservation equations.
The supernova is modelled by injecting thermal energy, not kinetic (i.e. we ignore the free-expansion phase). Therefore at very early times the newly shocked gas has an artificially high temperature (T$>10^9$ K), and at these temperatures we utilise the value of the reaction rates at $10^9$ K. To avoid artificial overcooling at early times, we only switch on the cooling when the gas adiabatically cooled down to $10^8$ K. The thermal model includes atomic cooling [@Fukugita94; @Hummer94], Bremmstrahlung cooling [@Hummer94; @Shapiro87], inverse Compton scattering [@Peebles71] and molecular line cooling from H$_2$, H$_2^+$ and H$_3^+$ [@Glover08; @Hollenbach79; @Glover09]. The heating processes included in the model are CMB heating (assumed equal to $\Lambda(\rm{T}_{CMB})$) and cosmic ray heating [@Glover07]. We set the cosmic ray ionization rate at $\zeta=10^{-18}\,\rm{s}^{-1}$ assuming the supernova remnant to be their source. The chemical model, together with tests of the chemistry and dynamics, are presented in the appendices. Both the 1D pre-supernova and post-supernova models consist of 5120 grid points to cover a 50 pc range, and are run until it the SN shell reaches 4 pc from the clump centre. The output of this phase ii model (both chemical and dynamic properties) is then mapped onto a 2D grid which covers an area of $9.60\times3.20$ pc ($480\times160$ grid zones, 0.02 pc per zone) to study the shock-cloud interaction (phase iii).
Results {#sec:results}
=======
Parameters
--------------------------- ----------------
Shell thickness 0.08 pc
Maximum shell density 1976 cm$^{-3}$
Minimum shell temperature 920 K
Shell velocity 39 km s$^{-1}$
Clump radius 1.3 pc
Maximum clump density 104 cm$^{-3}$
Minimum clump temperature 872 K
: Initial conditions of 2D model[]{data-label="IC_table"}
Pre-supernova phase
-------------------
The radial profile of the initial conditions and the pre-supernova ISM are plotted in Fig. (\[fig\_10cc\_HII\]). The gas density inside the photoionised H<span style="font-variant:small-caps;">ii</span> region ($r<33$pc) has decreased compared to the initial conditions (to close to $n=1\,\mathrm{cm^{-3}}$) because photoheating has driven its expansion. In this phase we only include atomic cooling, we assume that the H$_2$ within the gas has been destroyed as a result of Lyman-Werner radiation from the star. The shocked neutral ISM has only weak atomic coolants and so has not formed a shell, and remains very close to the initial ISM density. The cloud (or in 1D a shell) has been pushed outwards by the H<span style="font-variant:small-caps;">ii</span> region expansion, and is moving out at $v\approx2$kms$^{-1}$ (Fig. \[fig\_10cc\_HII\]b). The wave reflected back inwards is driving the negative velocity seen between $16<r<30$pc, and this is a transient feature imposed by the assumed spherical symmetry (which forces waves to reflect back and forth between the origin and any strong discontinuities). It has little effect on the overall solution except to marginally increase the density in this radius range. The temperature profile of the H<span style="font-variant:small-caps;">ii</span> region is typical of that produced by hot stars in metal-free gas [@IliCiaAlvEA06].
![ Plots of gas number density (a), velocity (b), temperature (c), and H$^{+}$ fraction (d) as a function of distance from the star. The dashed lines show the initial conditions and the solid lines the conditions at the pre-supernova stage. []{data-label="fig_10cc_HII"}](fig_1a "fig:"){width="40.00000%"} ![ Plots of gas number density (a), velocity (b), temperature (c), and H$^{+}$ fraction (d) as a function of distance from the star. The dashed lines show the initial conditions and the solid lines the conditions at the pre-supernova stage. []{data-label="fig_10cc_HII"}](fig_1b "fig:"){width="40.00000%"} ![ Plots of gas number density (a), velocity (b), temperature (c), and H$^{+}$ fraction (d) as a function of distance from the star. The dashed lines show the initial conditions and the solid lines the conditions at the pre-supernova stage. []{data-label="fig_10cc_HII"}](fig_1c "fig:"){width="40.00000%"} ![ Plots of gas number density (a), velocity (b), temperature (c), and H$^{+}$ fraction (d) as a function of distance from the star. The dashed lines show the initial conditions and the solid lines the conditions at the pre-supernova stage. []{data-label="fig_10cc_HII"}](fig_1d "fig:"){width="40.00000%"}
Supernova Remnant phase
-----------------------
![ Gas number density (a), expansion velocity (b), temperature (c), and species fractions (d) as a function of distance from the star for the 1D post-supernova evolution, at $t=0.2012$Myr after the supernova explosion. Note that panel (d) has a different $x$-axis to the other panels, zoomed in to show only the chemistry of the supernova shell and the overdense cloud (smaller and larger radii show little variation). The supernova shell is at $r\approx41.7$pc, and the overdense cloud at $r\approx44.6-46.6$pc. []{data-label="fig_10cc_1DSN"}](fig_2a "fig:"){width="40.00000%"} ![ Gas number density (a), expansion velocity (b), temperature (c), and species fractions (d) as a function of distance from the star for the 1D post-supernova evolution, at $t=0.2012$Myr after the supernova explosion. Note that panel (d) has a different $x$-axis to the other panels, zoomed in to show only the chemistry of the supernova shell and the overdense cloud (smaller and larger radii show little variation). The supernova shell is at $r\approx41.7$pc, and the overdense cloud at $r\approx44.6-46.6$pc. []{data-label="fig_10cc_1DSN"}](fig_2b "fig:"){width="40.00000%"} ![ Gas number density (a), expansion velocity (b), temperature (c), and species fractions (d) as a function of distance from the star for the 1D post-supernova evolution, at $t=0.2012$Myr after the supernova explosion. Note that panel (d) has a different $x$-axis to the other panels, zoomed in to show only the chemistry of the supernova shell and the overdense cloud (smaller and larger radii show little variation). The supernova shell is at $r\approx41.7$pc, and the overdense cloud at $r\approx44.6-46.6$pc. []{data-label="fig_10cc_1DSN"}](fig_2c "fig:"){width="40.00000%"} ![ Gas number density (a), expansion velocity (b), temperature (c), and species fractions (d) as a function of distance from the star for the 1D post-supernova evolution, at $t=0.2012$Myr after the supernova explosion. Note that panel (d) has a different $x$-axis to the other panels, zoomed in to show only the chemistry of the supernova shell and the overdense cloud (smaller and larger radii show little variation). The supernova shell is at $r\approx41.7$pc, and the overdense cloud at $r\approx44.6-46.6$pc. []{data-label="fig_10cc_1DSN"}](fig_2d "fig:"){width="40.00000%"}
![ The upper plot displays the maximum density of within the clump as the shock passes through, along with the temperature of the maximum density point and associated Jeans mass. The lower plot displays the mass within the clump as a function of different densities. []{data-label="fig_clump"}](jeans_mass "fig:"){width="49.00000%"} ![ The upper plot displays the maximum density of within the clump as the shock passes through, along with the temperature of the maximum density point and associated Jeans mass. The lower plot displays the mass within the clump as a function of different densities. []{data-label="fig_clump"}](clump_variables "fig:"){width="49.00000%"}
The output from the pre-supernova model is utilised as the initial conditions of the 1D supernova model. The clump has been moved to 45 pc due to the weak shock driven by dynamical expansion of the H<span style="font-variant:small-caps;">ii</span> region (Figure \[fig\_10cc\_HII\]a). When mapping the chemical species, we assume the percentage of ionised hydrogen and helium (He$^+$) are equal, and the initial molecular fractions are set to zero. A $10^{52}$ erg explosion is initiated and a shell starts to form at around $27$ pc. After 0.2012Myr the supernova shock is well into the radiative phase, so a thin shell has formed that is about 200$\times$ denser than the pre-shock gas. This agrees well with the isothermal shock jump conditions, where the overdensity is equal to the Mach number ($\mathcal{M}$) squared. In the shell, the isothermal sound speed is $a\approx2.5$kms$^{-1}$, so $\mathcal{M}^2 \approx (39/2.5)^2\approx240$. This is also similar to the maximum overdensity obtained from the test calculation in Appendix \[app:SNtest\]. In the interior of the supernova remnant the usual Sedov-Taylor solution remains imprinted on the fluid quantities: the density and velocity tend to zero at the origin, and the temperature increases to maintain the constant interior pressure. The molecular fractions are all negligible in the hot interior, and have a maximum in the shocked shell because here the density is highest but there is also still a non-negligible electron fraction from heating in the shell’s forward shock. The maximum H$_2$ fraction in the shell is $x(\mathrm{H}_2)\approx0.002$, in agreement with previous work [@Machida05].
The 1D supernova model is terminated when the shell reaches 41.9 pc (before it collides with the clump) and the output of this simulation (Figure \[fig\_10cc\_1DSN\]) is mapped onto a 2D axisymmetric grid. The initial conditions for the 2D model are outlined in Table \[IC\_table\]. The supernova shell is already travelling within the neutral ambient medium and is proceeding towards a dense spherical clump ($\sim19$ M$_\odot$) at a velocity of 39 km s$^{-1}$. The clump centre is 46 pc from the progenitor star. Figure \[fig\_10cc\_SN\] displays the evolution of the clump as the supernova shell collides and compresses it. The upper half plane of the plots display the log of the number density (log$_{10}$ $n_{\rm{H}}$/cm$^{-3}$) and corresponding lower half plane plots log of gas temperature reflected about the axis of symmetry. The black contour shows where the H$_2$ fraction equals $10^{-3}$.
After 0.31 Myr the shock has passed through half of the clump (upper plot in Figure \[fig\_10cc\_SN\]), we can see from Figure \[fig\_clump\] the maximum density of clump is $\sim6000$ cm$^{-3}$ with an associated temperature of $\sim1000$ K. The supernova shell has passed through the clump completely by 0.41 Myrs (middle plot in Figure \[fig\_10cc\_SN\]), and due to the decline in pressure the maximum density has decreased to $\sim 5200$ cm$^{-3}$. The shock has caused an increase in free electrons, which catalyse the formation of H$_2$. Hence the temperature of the high density gas has decreased to $\sim 400$ K. As the supernova shell passes through and around the clump, the region of strong shear at the clump’s edge undergoes adiabatic expansion and cools to close to the CMB temperature (middle panel of Fig. 3). This is not radiative cooling; the minimum temperature of the densest gas (with the strongest cooling) is $\sim400$ K. The clump reaches its maximum density of $\approx78000$ cm$^{-3}$ around 0.47 Myrs after the initial supernova explosion (bottom panel of Fig. 3). Again we see the densest gas is not the coldest gas, with a temperature of $\sim300$ K. The high density gas ($10^4$ cm$^{-3} \lesssim n \lesssim 10^5$ cm$^{-3}$) does not cool below $\sim 148$ K at any time. The re-expanding outer layers of the cloud are significantly colder with $T\approx60$ K, because of adiabatic expansion. The turbulence that can been seen in the passing shock is due to the thin-shell instability. During the shock-cloud interaction, the clump mass has increased from 19 M$_\odot$ to 40 M$_\odot$. We do not expect this clump to be gravitationally unstable as the minimum Jeans mass is 1000 M$_\odot$ (Figure \[fig\_clump\]).
After the passage of the shock the dense cloud is embedded in the high pressure, hot, low density interior of the supernova remnant. Our simulations do not have the spatial resolution to resolve the boundary layer between these two phases (we also do not include thermal conduction or model the external irradiation of the cloud), so the details of the boundary layer are probably not very reliable. The dominant physical process, however, is the simple pressure confinement of the cloud, and this is well-captured by our calculation. By the time the cloud is accelerated off the simulation domain it is entering an equilibrium phase of a pressure-confined cloud, similar to the cometary phase for irradiated clouds [@BerMcK90].
Discussion {#sec:dicusssion}
==========
----------- ------------------ --------------------- ---------------------------------------------------------- --------------- -------------- ---------------- ---------------- ------------------
Model No. Supernova Energy Ambient cloud H<span style="font-variant:small-caps;">ii</span> region Clump density Temperature Clump distance Shock velocity Clump fate
($10^{51}$ erg) density (cm$^{-3}$) included (cm$^{-3}$) of clump (K) (pc) (km s$^{-1}$)
M01 10 10 Yes 100 872 46 39 compressed clump
M02 2.0 10 Yes 100 872 46 - shell stalled
M03 1.0 10 Yes 100 872 46 - shell stalled
M04 0.6 10 Yes 100 872 46 - shell stalled
M05 10 1 No 100 200 50 200 destroyed
M06 2.0 1 No 100 200 50 46 small fragments
M07 1.0 1 No 100 200 50 26 destroyed
M08 0.6 1 No 100 200 50 16 destroyed
M09 1.0 1 No 100 200 40 49 small fragments
----------- ------------------ --------------------- ---------------------------------------------------------- --------------- -------------- ---------------- ---------------- ------------------
We have made a first investigation of the importance of non-equilibrium cooling processes occurring at all temperatures in primordial cloud-shock interactions (i.e. a SN shell interacting with primordial gas at redshift $z = 10$). This is an interesting case to study in its own right, for predicting the minimum mass that a metal-free star could potentially have. It is also the limiting case of considering shock-cloud interactions at extremely low metallicity, and so is useful for establishing a control simulation, against which models with non-zero metallicity can later be compared (Dhanoa et al., in prep). The progenitor gas cloud for the hyper metal-poor star SDSS J102915+172927 [@Caffau11] (with a total metallicity $Z\lesssim 10^{-5}$ Z$_\odot$) may have formed in a similar environment that was metal-free, but which became slightly polluted with supernova ejecta.
We include non-equilibrium chemistry to capture the radiative cooling occurring during the interaction of a shock and a small cloud, to establish if it is possible to form low-mass stars via this method. Considering a primordial chemistry for this process may be a simplification; because metals from the supernova ejecta would interact and mix within the shell once the discontinuity between the shell and the ejecta is disrupted by the impact of the clump [@Tenorio96]. However, the metallicity of the shell is expected to be near zero [@Salvaterra04] and according to @CenR the shock velocity ensures that the clump remains mostly unaffected by metals. If this is true then modelling the shock and cloud as metal-free is a good approximation.
We calculate the minimum Jeans mass of the of the compressed clump with only H$_2$ cooling (i.e. the minimum possible), and therefore represent a limiting case for shock-cloud interactions for both low-metallicity models and primordial models which include deuterium cooling. We find that the fractional abundance of H$_2$ in the high density region exceeds $3\times10^{-3}$, hence deuterium cooling may become important in this interaction [@Nakamura02]. On the other hand, @WolHai11 have shown that HD cooling is supressed by UV radiation fields that are five orders of magnitude weaker than what is required to supress H$_2$ cooling. Thus we expect that models with no HD cooling are applicable to a wider range of environments than models with HD cooling, once stars have begun forming in the vicinity.
We assume that the progenitor star is formed in a dark matter halo that is large enough so that edge effects do not need to be taken into account for a radius of $r \leq 50$ pc. @Vasiliev08 highlighted an important link between the radial distribution of primordial gas prior to the supernova explosion and the subsequent evolution of the supernova remnant; the state of the supernova shell directly influences the formation of extremely metal-poor stars. This distribution is heavily dependent on the size of the H<span style="font-variant:small-caps;">ii</span> region prior to the star’s explosion. Studies which reproduce the abundance patterns in extremely metal-poor stars by modelling the evolution and explosion of metal-free stars stars [@Nomoto06; @Joggerst09; @Joggerst10], suggest that metal-poor stars are formed by metal-free stars within a mass range of $15- 40$ M$_\odot$. The explosion mechanism for metal-free stars is uncertain, especially above 30 M$_\odot$, and so the star can have a range of explosion energies from $0.6 - 10 \times 10^{51}$ erg, which are associated with core collapse supernovae and hypernovae.
A clump initially at distance $r \ge 40$ pc from the star can safely be assume to be neutral, because Figure \[fig\_10cc\_HII\] shows that the clump does not interact with any ionising radiation. Clouds found closer to the progenitor star may evaporate, or at a minimum, have a different thermal state to a neutral cloud. Radiation between $11.18-13.6$ eV photodissociates H$_2$ molecules and so has a knock-on heating effect on the gas. This dissociation radiation propagates further than ionising radiation, and without any dust present we expect that clump is completely atomic in the pre-supernova stage. In the 2D model dissociative photons from the hot gas is assumed to be negligible [@Vasiliev08], but the possible effects of UV radiation on the clump should be investigated in more detail in future work.
After exploring a number of explosion energies (see models M01$-$M04 in Table \[model-table\]), we found that only the shock formed from a hypernova explosion ($10^{52}$ erg) reached and compressed the clump. When extending our study by exploring other ambient cloud densities (models M05$-$M09 in Table \[model-table\]), it emerges that the shock velocity determines the fate of the neutral clump. If the shock is too fast the clump is destroyed. When the supernova shock is too slow, the clump is only slightly compressed but inevitably destroyed. This is because the initial shock causes a secondary shock to travel through the rest of the clump, finally the gas disperses and flows downstream with the supernova shock. We therefore find that a small range of shock velocities ($30-50$ km s$^{-1}$) which can cause the clump to compress or fragment. Here the cooling time is equal to or less than the collapse/compression time and the velocity of the shock causes at least half of the clump to be compressed. Shock velocities above 40 km s$^{-1}$ cause the clump to fragment into smaller clumps, while below this velocity we find the clump is compressed.
The clump is near a supernova remnant so it will be exposed to cosmic rays, but the cosmic ray spectrum and intensity is unknown because of uncertainties in the expected interstellar magnetic field and the explosion mechanism for metal-free stars. We have assumed that the spectrum with be close to the observed spectrum in the Galactic environment, in keeping with @Stacy07. In this model we include a background cosmic ray ionization rate of $10^{-18}$ s$^{-1}$, as this rate was found to produce an overall cooling effect. We have not explored X-rays in this work, which would be produced by the supernova remnant. This would increase the H$_2$ abundance of gas ahead of the shell by increasing the free electron content [@Ferrara98; @Haiman97] and should be subject to further investigation. The effects of a range of cosmic ray ionization rates ($10^{-18}-10^{-15}$ s$^{-1}$) and their associated heating on the shock-clump interactions will also be explored future work.
The shocked clump of model M01 implodes because of the passage of the supernova shock (Figure\[fig\_10cc\_SN\]). This is the same behaviour seen in 3D simulations of clouds interacting with clumps [@Melioli06; @Leao09; @Johansson13], and earlier 2D work [e.g. @KleMcKCol94]. We find that in our simulation the clump gains a maximum density of $\sim 78000$ cm$^{-3}$, which is a density enhancement of $10^{2.89}$ but does not become Jeans unstable. @Vaidya13 show that self-gravity has no effect on the clump at this point of the shock interaction, where the implosion is pressure-driven and the clump reaches its maximum density. This gives us confidence that the implosion phase is correctly captured by our simulation. @Johansson13 investigate the compression of a $n=17$cm$^{-3}$ cloud (with radius 1.5 pc) and find higher densities enhancements of $10^3 - 10^5$. They also conclude that the clump will not become Jeans unstable. It is worth noting that their work considers solar metallicity gas with an equilibrium cooling function. Hence this may change when the model is refined to include non-equilibrium cooling.
Dust is assumed to be the major coolant in low-metallicity environments [@Klessen12; @Schneider12]. How quickly it can form in a primordial supernova ejecta and the extent of mixing that would occur during this cloud-shock interaction are still open questions. It is believed that dust is quickly destroyed in the reverse shocks formed when the supernova shell begins to travel within the multiphase neutral medium [@Cherchneff10; @Silvia10]. Without much dust in the environment, we cannot expect metal-line cooling to drastically lower the Jean mass, especially at metallicities $\le10^{-5}\,\rm{Z}_\odot$. In light of this, much further work is required to investigate the effects of cosmic rays and external radiation fields (especially X-ray and UV) on this process, because there may be important positive feedback effects [@Ricotti02; @Oshea05] that have not been considered so far.
Conclusion {#sec:conc}
==========
We have presented a metal-free shock-cloud model, which simulates a supernova remnant interacting with a cloud at redshift $z=10$. We model a dense clump ($n=100$ cm$^{-3},\, r =1.3$ pc) embedded in a 10 cm$^{-3}$ ambient cloud, which is 40 pc from the progenitor star. We consider realistic pre-supernova conditions by including the effects of stellar radiation from a 40 M$_\odot$ metal-free star on the multi-phase neutral medium. At the end of the star’s main-sequence lifetime, a hypernova ($10^{52}$ erg) is initiated and the evolution of the supernova shell and its subsequent interaction with the dense clump is studied. Radiative cooling is a crucial process in the shock-cloud interaction, allowing the formation of dense cold gas that may be susceptible to gravitational collapse. During this process we have comprehensively modelled the radiative (non-equilibirum) cooling taking place.
We followed the evolution of the supernova remnant and its interaction with the surrounding ionised and neutral medium. When the radiative shell interacts with the metal-free clump, it reaches a maximum of density $\sim78000$ cm$^{-3}$. This is a $10^{2.89}$ density enhancement and is consistent with Galactic shock-cloud models considering solar metallicity gas with equilibrium cooling functions. The clump undergoes a reduction in Jeans mass from $10^5$ M$_\odot$ to $10^3$ M$_\odot$, but does not become gravitationally unstable. Further work is required to ascertain the effect of cosmic rays, X-rays and UV radiation on the clump during the supernova phase.
In this work, we found an optimal range of shock velocities ($30- 50$ km s$^{-1}$) which compress small metal-free clouds. Below this range the cloud is slightly perturbed by the supernova shock and is not subject to any appreciable density enhancement. Above this range the clumps are destroyed, therefore the results by @MacBroHer03 are overoptimistic, as they assume the cloud survives a 200 km s$^{-1}$ interaction.
In this initial study we have only considered a single clump with fixed radius and density, varying the supernova energy and the density of the medium that the clump is embedded in. We have shown that the Jeans mass is indeed reduced significantly by the shock-cloud interaction, but not sufficiently to form stars with $<1$ M$_\odot$. In order to draw more general conclusions about the possibility of forming such low mass stars from metal free gas, we plan to follow up this work by considering a range of clump sizes and central densities.
When investigating model M01, we have achieved an appreciable Jean mass reduction of a small dense clump and a density enhancement comparable to Galactic studies, by including non-equilibrium metal-free radiative cooling. Further refinement of this model by including low-metallicity chemistry plus positive feedback effects from cosmic rays, X-rays and UV radiation, may cause a further reduction in Jeans mass. Galactic models should be extended to include non-equilibrium cooling, as this work has shown that it is the dominant process in shock-cloud interactions.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors would like to thank S.C.O. Glover and J.M.C. Rawlings for their helpful discussions. This work used the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST/K000373/1 and STFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the National E-Infrastructure. JM acknowledges funding during this project by a fellowship from the Alexander von Humboldt Foundation and from the Deutsche Forschungsgemeinschaft priority program 1573, “Physics of the Interstellar Medium”.
Chemistry Network {#appen:chem}
=================
The full chemical network is displayed in Table \[chem\_network\]. All the molecular reaction rates (R07 -R42) have been adapted for the temperature range ($10-10^{9}$ K) have been divided into two categories: i) Formation rates (listed in Table \[Formation\_molcules\]) and ii) Destruction rates (listed in Table \[Extended\_molcules\]).
Most of the UMIST 06 rates are valid until 41,000K. If a formation rate is valid up to a lower temperature, the value at the maximum temperature range is kept constant for temperatures above until 41,000K. Above 41,000 K all formation rates are cut-offand the reaction rates take on the the following forms: $$\begin{aligned}
K_1&=&k \times \rm{exp}\left(1.0 -\frac{T}{41000.0}\right)\\
K_2&=&k \times \rm{exp}\left(10 \times\left(1.0 -\frac{T}{41000.0}\right)\right)\end{aligned}$$ where $k$ is the value of the rate at 41000K. The details of how each formation reaction is treated, can be found in Table \[Formation\_molcules\].
The destruction rates are extrapolated above their valid temperature range. Above this temperature, if there is a maximum value after which the rate decreases (T$_{ex}$), this maximum value is kept constant for all higher temperatures (T $>$ T$_{ex}$). All the destruction rates, with the corresponding maximum extrapolation temperatures and temperatures ranges are displayed in Table \[Extended\_molcules\].
[l l r]{} Reaction No. &Reaction & References for rate coefficients\
R01 & H[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ H + $\gamma$ & H\
R02 & He[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ He + $\gamma$ & VF\
R03 & He[$^{+}$]{}[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ He[$^{+}$]{}+ $\gamma$ & VF\
R04 & H + e[$^{-}$]{}$\rightarrow$ H[$^{+}$]{}+ e[$^{-}$]{}+ e[$^{-}$]{}& V\
R05 & He + e[$^{-}$]{}$\rightarrow$ He[$^{+}$]{}+ e[$^{-}$]{}+ e[$^{-}$]{}& V\
R06 & He[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ He[$^{+}$]{}[$^{+}$]{}+ e[$^{-}$]{}+ e[$^{-}$]{}& V\
R07 & H$_2$ + H $\rightarrow$ H + H + H & GA08\
R08 & H[$^{-}$]{}+ H $\rightarrow$ H + H + e[$^{-}$]{}& GA08\
R09 & H[$^{-}$]{}+ He $\rightarrow$ He + H + e[$^{-}$]{}& GA08\
R10 & H$_2$ + H$_2$ $\rightarrow$ H$_2$ + H + H & UM06\
R11 & H$^-$ + e$^-$ $\rightarrow$ H + e$^-$ + e$^-$ & JR\
R12 & H$_2$ + He$^{+}$ $\rightarrow$ He + H$^+$ + H & UMO6\
R13 & H$_2$ + e$^-$ $\rightarrow$ H + e$^-$ + H & UM06\
R14 & [${\rm{H}_2}^+$]{}+ e$^-$ $\rightarrow$ H$^+$ + e$^-$ + H & R14\*\
R15 & HeH[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ He$^+$ + e$^-$ + H & R14\*\
R16 & H$^+$ + H $\rightarrow$ [${\rm{H}_2}^+$]{}+ $\gamma$ & UM06, GA08\
R17 & [$\rm{H}^{+}$]{}+ He $\rightarrow$ HeH[$^{+}$]{}+ $\gamma$ & UM06\
R18 & H + e$^-$ $\rightarrow$ H$^-$ + $\gamma$ & UM06, GA08\
R19 & HeH[$^{+}$]{}+ e[$^{-}$]{}$\rightarrow$ He + H & UM06\
R20 & [${\rm{H}_2}^+$]{}+ e$^-$ $\rightarrow$ H + H & UM06\
R21 & ${{\rm{H}_3}^+}$ + e$^-$ $\rightarrow$ H + H + H & UM06\
R22 & ${{\rm{H}_3}^+}$ + e$^-$ $\rightarrow$ H$_2$ + H & UM06\
R23 & H[$^{-}$]{}+ [${\rm{H}_2}^+$]{}$\rightarrow$ H + H + H & GA08\
R24 & H + He$^{+}$ $\rightarrow$ He + H$^+$ & UM06,hd\
R25 & H$_2$ + He$^{+}$ $\rightarrow$ He + [${\rm{H}_2}^+$]{}& UM06\
R26 & [$\rm{H}^{+}$]{}+ H$^{-}$ $\rightarrow$ H + H & UM06\
R27 & H$^-$ + [${\rm{H}_2}^+$]{}$\rightarrow$ H$_2$ + H & UM06\
R28 & H$^-$ + He$^+$ $\rightarrow$ He + H & UM06\
R29 & H + [${\rm{H}_2}^+$]{}$\rightarrow$ H$_2$ + [$\rm{H}^{+}$]{}& UM06\
R30 & [${\rm{H}_2}^+$]{}+ H$_2$ $\rightarrow$ ${{\rm{H}_3}^+}$ + H & UM06\
R31 & H$^-$ + ${{\rm{H}_3}^+}$ $\rightarrow$ H$_2$ + H$_2$ & UM06\
R32 & H + HeH$^+$ $\rightarrow$ He + [${\rm{H}_2}^+$]{}& UM06\
R33 & H$_2$ + HeH[$^{+}$]{}$\rightarrow$ He + ${{\rm{H}_3}^+}$ & UM06\
R34 & [${\rm{H}_2}^+$]{}+ He $\rightarrow$ HeH[$^{+}$]{}+ H & UM06\
R35 & H[$^{-}$]{}+ H[$^{+}$]{}$\rightarrow$ [${\rm{H}_2}^+$]{}+ e[$^{-}$]{}& SK87\
R36 & H$^-$ + H $\rightarrow$ H$_2$ + e$^-$ & UM06\
R37 & H + CR $\rightarrow$ [$\rm{H}^{+}$]{}+ e$^-$ & UM06\
R38 & He + CR $\rightarrow$ He[$^{+}$]{}+ e$^-$ &UM06\
R39 & H$_2$ + CR $\rightarrow$ [$\rm{H}^{+}$]{}+ H + e$^-$ &UM06\
R40 & H$_2$ + CR $\rightarrow$ H + H &UM06\
R41 & H$_2$ + CR $\rightarrow$ [$\rm{H}^{+}$]{}+ H$^-$ &UM06\
R42 & H$_2$ + CR $\rightarrow$ [${\rm{H}_2}^+$]{}+ e$^-$ &UM06\
\
\[chem\_network\]
---------- ------------------- --------------- --------------- --------------
Reaction Valid Temperature Below Minimum Above Maximum Cut off Type
Number Range (K) Temperature Temperature T $> $41000K
R16 S:10 – 32000 - C CT2
R17 16 – 100 C E CT
R18 S:10 – 41000 - - CT2
R29 10 – 41000 - - CT
R30 10 – 41000 - - CT
R33 10 – 41000 - - CT
R34 10 – 41000 - - CT
R35 10 – 41000 - - CT
R36 S:10 – 41000 - - CT
---------- ------------------- --------------- --------------- --------------
: Molecular reactions that are cut-off at 41000 K: E= rate extrapolated; C= max/min value kept constant and extended; - = Not Applicable; S= switching between different reaction rates within temperature range; CT2= $k\exp{(10.0\times(1.0-\rm{T}/41000))}$ and CT= $k\exp{(1.0-\rm{T}/41000})$ are exponential cut-off for T> 41000K and $k$ is the value of the reaction rate at 41000 K []{data-label="Formation_molcules"}
[|c||c|c|c|c|]{} Reaction & Valid Temperature & Below & Above & Maximum Extrapolation\
Number & Range of Rate (K) & Range& Range& Temperature T$_{ex}$ (K)\
R07 & 1833 – 41000 & E & E & $10^9$\
R08 & 10 – 10000 & - & C & -\
R09 & 10 – 10000 & - & C & -\
R10 & 2803 – 41000 & E & E & $10^7$\
R11 & 10 – 41000& - & E & $10^5$\
R12 & 100 – 300 & E & E & $10^8$\
R13 & 3400 –41000 & E & E & $10^8$\
R14 & 3400 –41000 & E & E & $10^8$\
R15 & 3400 –41000 & E & E & $10^8$\
R19 & 10 – 300 & - & E & $10^9$\
R20 & 10 – 300 & - & E & $10^9$\
R21 & 10 – 1000 & - & E & $10^9$\
R22 & 10 – 1000 & - & E & $10^9$\
R23 & 10 – 10000 & - & C & -\
R24 & S:10 – 41000 & - & C & -\
R25 & 10 – 300 & - & E & $10^9$\
R26 & 10 – 300 & - & E & $10^4$\
R27 & 10 – 300 & - & E & $10^9$\
R28 & 10 – 300 & - & E & $10^9$\
R31 & 10 – 300 & - & E & $10^9$\
R32 & 10 – 41000 & - & E & $10^9$\
R37& 10 – 41000 &- & C & -\
R38& 10 – 41000 &- & C & -\
R39& 10 – 41000 &- & C & -\
R40& 10 – 41000 &- & C & -\
R41& 10 – 41000 &- & C & -\
R42& 10 – 41000 &- & C & -\
Cooling test {#app:cooltest}
============
Fig. \[chem\_test\] displays a comparison of the primordial chemistry network presented in this work (DMY) and that of @Glover08 [GA] which includes 32 reactions that contain hydrogen and helium species only. [The GA deuterium reactions are not included]{}. Notably GA have included three-body reactions and density dependent reactions for: $$\begin{aligned}
&\rm{H}_2 + H_2 \rightarrow H + H +H_2\\
&\rm{H}_2 + He \rightarrow H + H +He.\end{aligned}$$ These reactions have been neglected in our network. However, @Glover08 do not include H$_3^+$ and HeH$^+$.
In this test we adopt a one-zone constant density model, where both chemistry networks are linked to the [same set of cooling functions, i.e. the H$_2$ and H$_2^+$ cooling functions provided by @Glover08 and @Hollenbach79 plus the atomic cooling functions given by @Fukugita94, @Hummer94, @Shapiro87 and @Peebles71]{}. The initial temperature of the gas is $10^4$ K and three densities are investigated: $n=1\,{\rm{cm}^{-3}}$, $n=100\,{\rm{cm}^{-3}}$ and $n=10^4\,{\rm{cm}^{-3}}$. The gas is allowed to chemically evolve and cool over $5\times10^7$ years.
For the low density test (i.e. $n = 1\, {\rm{cm}^{-3}}$) both microphysics modules reach the same temperature of 244 K. In the test for $n = 10^4\, {\rm{cm}^{-3}}$ the temperatures are very close; our module cools down to 6115 K and the GA module cools to 6090 K. At this density the H$_2$ cooling is within the LTE regime. The temperatures are very close, the difference is due to the rates we have included and not because the three body reactions were excluded. Three body reactions are dominant for densities $n \ge10^5\, {\rm{cm}^{-3}}$, and can be neglected as we do not expect the densities in the module to reach this value. Finally for $n = 100\, {\rm{cm}^{-3}}$ we obtain 69 K whilst the GA module obtains 90 K. When we include the reactions that are missing from our module, we still obtain 69 K. This highlights that the differences in temperature are due to the differences between the rates used in UMIST 06 database and the GA module.
1D supernova shell expansion {#app:SNtest}
============================
![ Supernova shell expansion as a function of time for an adiabatic calculation, a calculation with atomic line cooling only, and a calculation with atomic and molecular cooling switched on. The expansion radius is compared to the analytic Sedov-Taylor solution in the upper plot. The lower plot shows the maximum gas number density in the shell as a function of time for the same three models. []{data-label="fig_SNexp"}](shell_position_overall "fig:"){width="49.00000%"} ![ Supernova shell expansion as a function of time for an adiabatic calculation, a calculation with atomic line cooling only, and a calculation with atomic and molecular cooling switched on. The expansion radius is compared to the analytic Sedov-Taylor solution in the upper plot. The lower plot shows the maximum gas number density in the shell as a function of time for the same three models. []{data-label="fig_SNexp"}](density_overall "fig:"){width="49.00000%"}
Fig. \[fig\_SNexp\] shows the results of a 1D test, in which the expansion of a blastwave is followed using different chemistry/cooling assumptions: adiabatic with no chemistry, including chemistry but only atomic coolants, and including chemisty with atomic and molecular coolants. The radius of the SN forward shock (upper panel) and maximum density in the shell (lower panel) are plotted as a function of time since explosion. We used uniform radial grid with 5120 grid zones between $r=0$ and $r=130$pc, and input $10^{51}$ergs of thermal energy in the 8 grid zones closest to the origin. The ISM is a constant density medium with $\rho=2.44\times10^{-24}\,$gcm$^{-3}$ at a redshift of 20. The initial ISM temperature is $T=10^4$K (corresponding to a pressure of $p\approx1.5\times10^{-12}$dynecm$^{-2}$). Without any cooling this can be compared to the Sedov-Taylor solution, and when cooling and chemistry are included we compare to the results of @Machida05.
The adiabatic calculation matches the Sedov-Taylor solution until about 0.8Myr, after which the shock runs ahead of this solution. The explanation for this is that the shock weakens as it slows down at late times, and the ISM ambient pressure is no longer negligible. This breaks the scale-free nature of the analytic solution, and the result is that the shock radius advances faster than predicted at late times [cf. @RagCanRodEA12].
At about 0.05Myr the simulations with cooling start to decelerate and deviate from the adiabatic solution. The expansion rate changes from the Sedov-Taylor value $R_\textrm{sh}\propto t^{2/5}$ to the momentum-conserving value $R_\textrm{sh}\propto t^{1/4}$. Atomic cooling is initially much stronger than molecular cooling, so both of these runs match each other until the molecular cooling begins to affect the shell and the ISM at $t\approx0.2$Myr. At later times the shell density in the cooling model decreases steadily because it can no longer cool, and the weak forward shock keeps adding lower entropy gas to the shell. The molecular cooling model has a higher density shell once molecular cooling becomes important at $t\approx0.2$Myr, because it can cool to much lower temperatures. This has the further effect that the shell remains at a high density for much longer.
The molecular cooling calculation shows that we get compression factors of $>100\times$ in the shell at $t\geq0.2\,$Myr. This model disagrees strongly with @Machida05 (see their fig. 4), who found only weak density increase in the supernova shell for times up to $10^7$ years. The density in their analytic model was set by the imposed pressure-confining boundary conditions on the shell, so we suspect that one of the boundary conditions was incorrect.
\[lastpage\]
[^1]: E-mail: hd@star.ucl.ac.uk (HD); jmackey@astro.uni-bonn.de (JM)
|
---
abstract: 'We give explicit formulas for the coproduct and the antipode in the Connes-Moscovici Hopf algebra $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$. To do so, we first restrict ourselves to a sub-Hopf algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ containing the nontrivial elements, namely those for which the coproduct and the antipode are nontrivial. There are two ways to obtain explicit formulas. On one hand, the algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is isomorphic to the Faà di Bruno Hopf algebra of coordinates on the group of identity-tangent diffeomorphism and computations become easy using substitution automorphisms rather than diffeomorphisms. On the other hand, the algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is isomorphic to a sub-Hopf algebra of the classical shuffle Hopf algebra which appears naturally in resummation theory, in the framework of formal and analytic conjugacy of vector fields. Using the very simple structure of the shuffle Hopf algebra, we derive once again explicit formulas for the coproduct and the antipode in $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$.'
author:
- Frédéric Menous
bibliography:
- 'main.bib'
title: 'Formulas for the Connes-Moscovici Hopf algebra'
---
Introduction.
=============
The Connes-Moscovici Hopf algebra $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$ was introduced in [[@cm]]{} in the context of noncommutative geometry. Because of its relation with the Lie algebra of formal vector fields, it was also proved in [[@cm]]{} that its subalgebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is isomorphic to the Faà di Bruno Hopf algebra of coordinates of identity-tangent diffeomorphisms (see [[@cm]]{},[[@fig]]{}). In the past years, it appeared that this Hopf algebra was strongly related to the Hopf algebras of trees (see [[@ck0]]{}) or graphs (see [[@ck1]]{},[[@ck2]]{}) underlying perturbative renormalization in quantum field theory.
Our aim is to give explicit formulas for the coproduct and the antipode in $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$, since only recursive formulas seem to be known.
We remind in section \[cm\] the definition of the Connes-Moscovici Hopf algebra, as well as its properties and links with the Faà di Bruno Hopf algebra and identity-tangent diffeomorphisms (for details, see [[@cm]]{},[[@fig]]{}). The formulas are given in section \[for\]. We present a proof based on the isomorphism between identity-tangent diffeomorphisms and substitution automorphisms which are easier to handle in the computations. These manipulations on substitution automorphisms are very common in J. Ecalle’s work on the formal classification of differential equations, vector fields, diffeomorphism... (see [[@et1]]{},[[@et2]]{},[[@et3]]{},[[@esn]]{}). In fact, the first proof for these formulas was based on mould calculus and shuffle Hopf algebras, which we shortly describe in section \[moulds\]. Sections \[mor\] and \[ini\] give the outlines of the initial proof based on a Hopf morphism from $\mathcal{H}^1 \subset \mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$ in a shuffle Hopf algebra.
Connes-Moscovici and Faà di Bruno Hopf algebras.\[cm\]
======================================================
The Connes-Moscovici Hopf algebra
---------------------------------
The Connes-Moscovici Hopf algebra $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$ defined in [[@cm]]{} is the enveloping algebra of the Lie algebra which is the linear span of $Y$, $X$, $\delta_n$, $n \geq 1$ with the relations, $$[ X, Y ] = X, [ Y, \delta_n ] = n \delta_n, [ \delta_n, \delta_m ] = 0, [ X,
\delta_n ] = \delta_{n + 1} \label{crocm}$$ for all $m, n \geq 1$. The coproduct $\Delta$ in $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$ is defined by $$\Delta ( Y ) = Y \otimes 1 + 1 \otimes Y, \Delta ( X ) = X \otimes 1 + 1
\otimes X + \delta_1 \otimes Y, \Delta ( \delta_1 ) = \delta_1 \otimes 1 + 1
\otimes \delta_1$$ where $\Delta ( \delta_n )$ is defined recursively, using equation \[crocm\] and the identity $$\forall h_1, h_2 \in \mathcal{H}_{{\ensuremath{\operatorname{CM}}}}, \quad \Delta ( h_1 h_2 ) =
\Delta ( h_1 ) \Delta ( h_2 )$$ The coproduct of $X$ and $Y$ is given, whereas the coproduct of $\delta_n$ is nontrivial. Nonetheless, the algebra generated by $\{ \delta_n, \hspace{1em} n
\geq 1 \}$ is a graded sub-Hopf algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}} \subset
\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}$ where the graduation is defined by $${\ensuremath{\operatorname{gr}}} ( \delta_{n_1} \ldots \delta_{n_s} ) = n_1 + \ldots + n_s$$ As mentioned in [[@cm]]{}, the Hopf algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is strongly linked to Faà di Bruno Hopf algebra.
The Faà di Bruno Hopf algebra
-----------------------------
Let us consider the group of formal identity tangent diffeomorphisms : $$G_2 = \{ f ( x ) = x + \sum_{n \geq 1} f_n x^{n + 1} \in \mathbbm{R}[ [ x ]
] \}$$ with, by convention, the product $\mu : G_2 \times G_2 \rightarrow G_2$ : $$\mu ( f, g ) = g \circ f$$ For $n \geq 0$, the functionals on $G_2$ defined by $$a_n ( f ) = \frac{1}{( n + 1 ) !_{}} ( \partial_x^{n + 1} f ) ( 0 ) = f_n
\hspace{1em} a_n : G_2 \rightarrow \mathbbm{R}$$ are called de Faà di Bruno coordinates on the group $G_2$ and $a_0 = 1$ being the unit, they generates a graded unital commutative algebra $$\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}} =\mathbbm{R}[ a_1, \ldots, a_n, \ldots ]
\hspace{1em} ( {\ensuremath{\operatorname{gr}}} ( a_n ) = n )$$ Moreover, the action of these functionals on a product in $G_2$ defines a coproduct on $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ that turns to be a graded connected Hopf algebra (see [[@fig]]{} for details). For $n \geq 0$, the coproduct is defined by $$a_n \circ \mu = m \circ \Delta ( a_n ) \label{copfdb}$$ where $m$ is the usual multiplication in $\mathbbm{R}$, and the antipode reads $$S \circ a_n = a_n \circ {\ensuremath{\operatorname{rec}}}$$ where ${\ensuremath{\operatorname{rec}}} ( \varphi ) = \varphi^{- 1}$ is the composition inverse of $\varphi$.
For example if $f ( x ) = x + \sum_{n \geq 1} f_n x^{n + 1}$ and $g ( x ) = x
+ \sum_{n \geq 1} g_n x^{n + 1}$ then if $h = \mu ( f, g ) = g \circ f$ and $h
( x ) = x + \sum_{n \geq 1} h_n x^{n + 1}$, $$\begin{array}{ccccccc}
a_0 ( h ) & = & 1 = a_0 ( f ) a_0 ( g ) & \rightarrow & \Delta a_0 & = &
a_0 \otimes a_0\\
a_1 ( h ) & = & f_1 + h_1 & \rightarrow & \Delta a_1 & = & a_1 \otimes
a_0 + a_0 \otimes a_1\\
a_2 ( h ) & = & f_2 + f_1 g_1 + g_2 & \rightarrow & \Delta a_2 & = & a_2
\otimes a_0 + a_1 \otimes a_1 + a_0 \otimes a_2
\end{array}$$ As proved in [[@cm]]{} and [[@fig]]{}, there exists a Hopf isomorphism between $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ and $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$.
Connes-Moscovici coordinates
----------------------------
Following [[@cm]]{}, one can define new functionals on $G_2 $ by $\gamma_0 =
a_0$=1 (unit) and for $n \geq 1$, $$\gamma_n ( f ) = ( \partial^n_x \log ( f' ) ) ( 0 )$$ These functionals, which may be called the Connes-Moscovici coordinates on $G_2$, freely generates the Faà di Bruno Hopf algebra : $$\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}} =\mathbbm{R}[ a_1, \ldots, a_n, \ldots ]
=\mathbbm{R}[ \gamma_1, \ldots, \gamma_n, \ldots ] \hspace{1em} {\ensuremath{\operatorname{gr}}} (
a_n ) = {\ensuremath{\operatorname{gr}}} ( \gamma_n ) = n$$ and their coproduct is given by the formula \[copfdb\]. Now, see [[@cm]]{}, [[@ck0]]{} :
The map $\Theta$ defined by $\Theta ( \delta_n ) = \gamma_n$ is a graded Hopf isomorphism between $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ and $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$
This means that the coproduct and the antipode in $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ can be rather computed in $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$. Unfortunately, if the coproduct and the antipode is well-known for the functionals $a_n$, using the Faà di Bruno formulas for the composition and the inverse of diffeomorphisms in $G_2$, it seems that formulas for the $\gamma_n$ cannot be easily derived. In order to do so, we will either work with substitution automorphism which are easier to handle than diffeomorphisms (see section \[for\], or identify $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ as a sub-Hopf algebra of a shuffle Hopf algebra and use mould calculus (see sections \[moulds\], \[mor\], \[ini\]).
Formulas in $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$.\[for\]
======================================================================
Notations
---------
In the sequel we note $$\mathcal{N}= \{ {\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in (\mathbbm{N}^{\ast}
)^s, \hspace{1em} s \geq 1 \}$$ For ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}$, $$\| {\ensuremath{\boldsymbol{n}}} \| = n_1 + \ldots + n_s, \hspace{1em} l ( {\ensuremath{\boldsymbol{n}}} ) =
s$$ and if $n \geq 1$, $$\mathcal{N}_n = \{ {\ensuremath{\boldsymbol{n}}} \in \mathcal{N} \hspace{1em} ; \hspace{1em}
\| {\ensuremath{\boldsymbol{n}}} \| = n \}$$ For a tuple ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}$, we note ${\ensuremath{\boldsymbol{n}}}! = n_1 ! \ldots n_s !$. More over, ${\ensuremath{\operatorname{Split}}} ({\ensuremath{\boldsymbol{n}}})$ is the subset of $\bigcup_{t \geq 1} \mathcal{N}^t$ such that $({\ensuremath{\boldsymbol{n}}}^1,
\ldots,{\ensuremath{\boldsymbol{n}}}^t ) \in {\ensuremath{\operatorname{Split}}} ({\ensuremath{\boldsymbol{n}}})$ if and only if the concatenation of $({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )$ is equal to ${\ensuremath{\boldsymbol{n}}}$ : $${\ensuremath{\operatorname{Split}}} ({\ensuremath{\boldsymbol{n}}}) = \{ ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )
\in \mathcal{N}^t, \hspace{1em} {\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t
={\ensuremath{\boldsymbol{n}}} \}$$
In summation formulas, we will use the fact that $$\bigcup_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} {\ensuremath{\operatorname{Split}}} ({\ensuremath{\boldsymbol{n}}}) =
\bigcup_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n}
\mathcal{N}_{n_1} \times \ldots \times \mathcal{N}_{n_s}$$ so that if $f$ is a function on $\mathcal{N}$ and $g$ is a function on $\bigcup_{t \geq 1} \mathcal{N}^t$, for $n \geq 1$, $$\begin{array}{l}
\displaystyle \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^1 \in \mathcal{N}_{n_1}\\
{ \vdots}\\
{\ensuremath{\boldsymbol{m}}}^s \in \mathcal{N}_{n_s}
\end{array}$}}} f ({\ensuremath{\boldsymbol{n}}}) g ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s
) =\\
\hspace{10em} \displaystyle \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n}
\sum_{{\ensuremath{\boldsymbol{m}}}^1
\ldots {\ensuremath{\boldsymbol{m}}}^s ={\ensuremath{\boldsymbol{n}}}} f ( \| {\ensuremath{\boldsymbol{m}}}^1 \|, \ldots, \|
{\ensuremath{\boldsymbol{m}}}^s \| ) g ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s )
\end{array}$$ where $\displaystyle \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^s
={\ensuremath{\boldsymbol{n}}}}$is the sum over ${\ensuremath{\operatorname{Split}}} ({\ensuremath{\boldsymbol{n}}})$.
Finally, for $({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t ) \in \mathcal{N}^t$ ($t
\geq 1$), $$A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t ) = \frac{1}{l ({\ensuremath{\boldsymbol{n}}}^1 ) !
\ldots l ({\ensuremath{\boldsymbol{n}}}^t ) !} \prod_{i = 1}^t \frac{1}{\| {\ensuremath{\boldsymbol{n}}}^i \|
+ 1}$$ and, for $k \geq 1$, $$B_k ( {\ensuremath{\boldsymbol{n}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{n}}}^t ) = C^{l ( {\ensuremath{\boldsymbol{n}}}^t
)}_k \prod_{i = 1}^{t - 1} C_{\| {\ensuremath{\boldsymbol{n}}}^{i + 1} \| + \ldots + \|
{\ensuremath{\boldsymbol{n}}}^t \| + k}^{l ( {\ensuremath{\boldsymbol{n}}}^i )}$$
Main formulas
-------------
We will now prove the following formulas :
\[th2\]For $n \geq 1$, $$\begin{array}{rcl}
\Delta ( \delta_n )& =& \delta_n \otimes 1 + 1 \otimes \delta_n \\
& &\displaystyle+ \sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_{s + 1} ) \in \mathcal{N}_n\\
s \geq 1
\end{array}$}}} \frac{n!}{n_1 ! \ldots n_{s + 1} !} \alpha^{n_1, \ldots,
n_s}_{n_{s + 1}} \delta_{n_1} \ldots \delta_{n_s} \otimes \delta_{n_{s
+ 1}}\end{array}$$ and, for ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}$ ($l
({\ensuremath{\boldsymbol{n}}}) = s$) and $m \geq 1$, $$\alpha^{{\ensuremath{\boldsymbol{n}}}}_m = \sum_{t = 1}^{l ({\ensuremath{\boldsymbol{n}}})} C^{t_{}}_m
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}
\end{array}$}}} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )$$ where, for ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}$, $l
({\ensuremath{\boldsymbol{n}}}) = 1$, $\| {\ensuremath{\boldsymbol{n}}} \| = n_1 + \ldots + n_s$ and with the convention $C^t_m = \displaystyle \frac{m!}{t! ( m - t ) !} = 0$ if $t > m$.
For the antipode $S$ :
\[th3\]For $n \geq 1$, $$S ( \delta_n ) = \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}\\
n_1 + \ldots + n_s = n
\end{array}$}}} \frac{n!}{n_1 ! \ldots n_s !} \beta^{n_1, \ldots, n_s}
\delta_{n_1} \ldots \delta_{n_s}$$ with $\beta^{n_1} = - 1$ and, if ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_{s + 1} )
\in \mathcal{N}$ ($s \geq 1$), $$\beta^{n_1, \ldots, n_s, n_{s + 1}} = \sum_{t = 1}^s
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}
\end{array}$}}} U^{\| {\ensuremath{\boldsymbol{n}}}^1 \|, \ldots, \| {\ensuremath{\boldsymbol{n}}}^t \|}_{n_{s
+ 1}} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )$$ where, if ${\ensuremath{\boldsymbol{m}}}= ( m_1, \ldots, m_t ) \in \mathcal{N}/ \{ \emptyset
\}$ and $k \geq 1$, $$U^{{\ensuremath{\boldsymbol{m}}}}_k = \sum_{i = 1}^{l ({\ensuremath{\boldsymbol{m}}})} ( - 1 )^{i - 1}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^i ={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} B_k ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^i )$$
We will now give the more recent proof of this formulas. These formulas were first conjectured and then proved using a Hopf morphism between $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ and a shuffle Hopf algebra noted ${\ensuremath{\operatorname{sh}}}
(\mathbbm{N}^{\ast} )$. We will come back later on this morphism and the afferent proofs. Let us first look at the correspondence between FdB coordinates and the CM coordinates on $G_2$.
Coordinates on $G_2$
--------------------
Let $\varphi ( x ) = x +\displaystyle \sum_{n \geq 1} \varphi_n x^{n +
1}$. We have for $n
\geq 1$ : $$a_n ( \varphi ) = \varphi_n, \hspace{1em} \gamma_n ( \varphi ) = (
\partial^n_x \log ( \varphi' ) ) ( 0 ) = f_n$$ If $f ( x ) = \displaystyle \sum_{n \geq 1} \frac{f_n}{n!} x^n$, then $$f ( x ) = \log ( \varphi' ( x ) ) \hspace{1em} \varphi ( x ) = \int_0^x e^{f
( t )} d t \label{corr}$$ For any sequence $( u_n )_{n \geq 1}$, we note $$\forall {\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}, \hspace{1em}
u_{{\ensuremath{\boldsymbol{n}}}} = u_{n_1} \ldots u_{n_s}$$ Using equation \[corr\], we get easily that $$\begin{array}{ccc}
f ( x ) & = &\displaystyle \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}}
\frac{( - 1 )^{l ( {\ensuremath{\boldsymbol{n}}} )}}{l ( {\ensuremath{\boldsymbol{n}}} )} ( n_1 + 1 ) \ldots
( n_s + 1 ) \varphi_{{\ensuremath{\boldsymbol{n}}}} x^{\| {\ensuremath{\boldsymbol{n}}} \|}\\
\varphi ( x ) & = & \displaystyle x + \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in
\mathcal{N}} \frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) ! {\ensuremath{\boldsymbol{n}}} !}
\frac{f_{{\ensuremath{\boldsymbol{n}}}}}{\| {\ensuremath{\boldsymbol{n}}} \| + 1} x^{\| {\ensuremath{\boldsymbol{n}}} \| + 1}
\end{array} \label{e19}$$ and these formulas establish the correspondence between FdB and CM coordinates on $G_2$. In order to prove theorems \[th2\] and \[th3\], we need to understand how these coordinates read on $\varphi^{- 1}$ and $\mu ( \varphi,
\psi ) = \psi \circ \varphi$ ($\varphi, \psi \in G_2$). To do so, we will rather work with substitution automorphisms than with diffeomorphism.
Taylor expansions and substitution automorphisms\[aut\]
-------------------------------------------------------
Let $\tilde{G}_2$ be the set of linear maps from $\mathbbm{R}[ [ x ] ]$ to $\mathbbm{R}[ [ x ] ]$ such that
- For $F \in \tilde{G}_2$, the image $F ( x )$ by $F$ of the series $x$ is in $G_2$.
- For any two series $A$ and $B$ in $\mathbbm{R}[ [ x ] ]$, we have $$F ( A.B ) = F ( A ) .F ( B ) \label{prod}$$
The elements of $\tilde{G}_2$ are called substitution automorphisms and
$\tilde{G}_2$ is a group for the composition and the map : $$\begin{array}{ccccc}
\tau & : & \text{$\tilde{G}_2$} & \rightarrow & G_2\\
& & F & \mapsto & \varphi ( x ) = F ( x )
\end{array}$$ defines an isomorphism between the groups $\tilde{G}_2$ and $G_2$. Moreover, for $A \in \mathbbm{R}[ [ x ] ]$, $$F ( A ) = A \circ \tau ( F )$$
If $F \in \widetilde{G_{}}_2$, then, thanks to equation \[prod\], for $k
\geq 0$, $$\text{$F ( x^k ) = \left( F ( x ) \right)^k = ( \tau ( F ) ( x ) )^k = (
\varphi ( x ) )^k$}$$ thus, for $A ( x ) = \sum_{k \geq 0} A_k x^k \in \mathbbm{R} [ [ x ] ]$, $$\begin{array}{ccc}
F ( A ) ( x ) & = & F \left( \sum_{k \geq 0} A_k x^k \right)\\
& = & \sum_{k \geq 0} A_k F ( x^k )\\
& = & \sum_{k \geq 0} A_k ( \varphi ( x ) )^k\\
& = & A \circ ( \tau ( F ) ) ( x )
\end{array}$$ This proves that $\tau$ is injective and for any $\varphi \in G_2$ the map $$\begin{array}{ccccc}
F & : & \mathbbm{R}[ [ x ] ] & \rightarrow & \mathbbm{R}[ [ x ] ]\\
& & A & \mapsto & A \circ \varphi
\end{array}$$ is a substitution automorphism of $\tilde{G}_2$ such that $\tau ( F ) =
\varphi$. The map $\tau$ is a bijection. Now, for $F$ and $G$ in $\tilde{G}_2$, $$\tau ( F \circ G ) ( x ) = F ( G ( x ) ) = \tau ( G ) \circ \tau ( F ) ( x
) = \mu ( \tau ( F ), \tau ( G ) ) ( x )$$ and if $H = \tau^{- 1} ( ( \tau ( F ) )^{- 1} )$ then $F \circ H = H \circ F
= {\ensuremath{\operatorname{Id}}}$. This ends the proof.
$^{}$
Using Taylor expansion, we also get formulas for $\tau^{- 1} ( \varphi )$, $\varphi \in G_2$,
Let $\varphi ( x ) = {\displaystyle}x + \sum_{n \geq 1} \varphi_n x^{n + 1} \in G_2$ and $F
= \tau^{- 1} ( \varphi )$, then $$F = {\ensuremath{\operatorname{Id}}} + \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}}
\frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}} x^{\| {\ensuremath{\boldsymbol{n}}}
\| + l ( {\ensuremath{\boldsymbol{n}}} )} \partial_x^{l ( {\ensuremath{\boldsymbol{n}}} )}$$
This also means that $F$ can be decomposed in homogeneous components : $$F = {\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} F_n \hspace{1em}, \hspace{1em} F_n =
\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n} \frac{1}{l (
{\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}} x^{\| {\ensuremath{\boldsymbol{n}}} \| + l (
{\ensuremath{\boldsymbol{n}}} )} \partial_x^{l ( {\ensuremath{\boldsymbol{n}}} )}$$ such that $$\forall n \geq 1, \hspace{1em} \forall k \geq 1, \hspace{1em} \exists c \in
\mathbbm{R}, \hspace{1em} F_n ( x^k ) = c x^{n + k}$$
If $\varphi ( x ) = x + {\displaystyle}\sum_{n \geq 1} \varphi_n x^{n + 1} = x +
\bar{\varphi} ( x ) \in G_2$, then, if $F = \tau^{- 1} ( \varphi )$, then for $A \in \mathbbm{R} [ [ x ] ]$, $$\begin{array}{ccc}
F ( A ) ( x ) & = & A ( x + \bar{\varphi} ( x ) )\\
& = & {\displaystyle}A ( x ) + \sum_{s \geq 1} \frac{\left( \bar{\varphi} ( x ) )^s
\right.}{s!} A^{( s )} ( x )\\
& = & {\displaystyle}A ( x ) + \sum_{s \geq 1} \sum_{n_1 \geq 1, \ldots, n_s \geq 1}
\frac{1}{s!} \varphi_{n_1} \ldots \varphi_{n_s} x^{n_1 + \ldots + n_s +
s} A^{( s )} ( x )\\
& = & {\displaystyle}\left( {\ensuremath{\operatorname{Id}}} + \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in
\mathcal{N}} \frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}}
x^{\| {\ensuremath{\boldsymbol{n}}} \| + l ( {\ensuremath{\boldsymbol{n}}} )} \partial_x^{l (
{\ensuremath{\boldsymbol{n}}} )} \right) \left( A ( x ) \right)
\end{array}$$
The automorphism $F$ can be seen as a differential operator acting on $\mathbbm{R}[ [ x ] ]$ and from now on we note multiplicatively the action of such operators : $$F. \varphi = F ( \varphi )$$
As this will be of some use later, let us give the following formula : If ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_{}$ and $k \geq 1$, $$\begin{array}{ccc}
F_{{\ensuremath{\boldsymbol{n}}}} .x^k & = & F_{n_1} \ldots F_{n_s} .x^k\\
& = & {\displaystyle}\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i\leq s\\
\end{array}$}}} \left( \frac{\varphi_{{\ensuremath{\boldsymbol{m}}}^1}x^{n_1 + l (
{\ensuremath{\boldsymbol{m}}}^1 )}}{ l ( {\ensuremath{\boldsymbol{m}}}^1
) !} \partial_x^{l (
{\ensuremath{\boldsymbol{m}}}^1 )} \right) \ldots \left(
\frac{\varphi_{{\ensuremath{\boldsymbol{m}}}^s}x^{n_s + l (
{\ensuremath{\boldsymbol{m}}}^s )} }{l
( {\ensuremath{\boldsymbol{m}}}^s ) !}
\partial_x^{l ( {\ensuremath{\boldsymbol{m}}}^s )} \right) .x^k\\
& = & {\displaystyle}\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i \leq s\\
\end{array}$}}} B_k ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s )
\varphi_{{\ensuremath{\boldsymbol{m}}}^1} \ldots \varphi_{{\ensuremath{\boldsymbol{m}}}^s} x^{\|
{\ensuremath{\boldsymbol{n}}} \| + k}
\end{array}$$ where $$B_k ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s ) = C^{l ( {\ensuremath{\boldsymbol{m}}}^s
)}_k \prod_{i = 1}^{s - 1} C_{\| {\ensuremath{\boldsymbol{m}}}^{i + 1} \| + \ldots + \|
{\ensuremath{\boldsymbol{m}}}^s \| + k}^{l ( {\ensuremath{\boldsymbol{m}}}^i )}$$ With these results one can already derive formulas for the ${\ensuremath{\operatorname{FdB}}}$ coordinates on $G_2$.
Formulas in $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$
-------------------------------------------------------------
We recover the usual formulas :
\[prop7\]We have for $n \geq 1$, $$\Delta ( a_n ) = a_n \otimes 1 + 1 \otimes a_n + \sum_{k = 1}^{n - 1}
\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_k} C^{l (
{\ensuremath{\boldsymbol{n}}} )}_{n - k + 1} a_{{\ensuremath{\boldsymbol{n}}}} \otimes a_{n - k}$$ and $$S ( a_n ) = \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} \left(
\sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^s = {\ensuremath{\boldsymbol{n}}}} ( - 1 )^s B_1 (
{\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s ) \right) a_{{\ensuremath{\boldsymbol{n}}}}$$
Let $\varphi ( x ) = x + {\displaystyle}\sum_{n \geq 1} \varphi_n x^{n + 1}$ and $\psi ( x
) = x + {\displaystyle}\sum_{n \geq 1} \psi_n x^{n + 1}$ two elements of $G_2$ and $\eta =
\mu ( \varphi, \psi ) = \psi \circ \varphi$ with $$\eta ( x ) = x + \sum_{n \geq 1} \eta_n x^{n + 1}$$ If $F$, $G$ and $H$ are the substitution automorphisms corresponding to $\varphi$, $\psi$ and $\eta$, then $H = F \circ G$ : $$\begin{array}{cccc}
H & = & {\displaystyle}{\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} H_n & \\
& = & {\displaystyle}\left( {\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} F_n \right) \left( {\ensuremath{\operatorname{Id}}} +
\sum_{n \geq 1} G_n \right) & \\
& = &{\displaystyle}{\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} \sum_{k = 0}^n F_k G_{n - k} & ( F_0
= G_0 = {\ensuremath{\operatorname{Id}}} )
\end{array}$$ But for $l \geq 1$, $G_l ( x ) = \psi_l x^{l + 1}$ and then, for $k \geq 1$, $$\begin{array}{ccc}
F_k G_l .x & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in
\mathcal{N}_k} \frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}}
x^{\| {\ensuremath{\boldsymbol{n}}} \| + l ( {\ensuremath{\boldsymbol{n}}} )} \partial_x^{l ( {\ensuremath{\boldsymbol{n}}}
)} ( \psi_l x^{l + 1} )\\
& = &{\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_k}
\psi_l \frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}} \frac{( l +
1 ) !}{( l + 1 - l ( {\ensuremath{\boldsymbol{n}}} ) ) !} x^{\| {\ensuremath{\boldsymbol{n}}} \| + l +
1}\\
& = &{\displaystyle}\left( \sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in
\mathcal{N}_k} C^{l ( {\ensuremath{\boldsymbol{n}}} )}_{l + 1} \varphi_{{\ensuremath{\boldsymbol{n}}}}
\psi_l \right) x^{k + l + 1}
\end{array}$$ and then, for $n \geq 1$, $$\eta_n = \varphi_n + \psi_n + \sum_{k = 1}^{n - 1} \sum_{{\ensuremath{\boldsymbol{n}}}= (
n_1, \ldots, n_s ) \in \mathcal{N}_k} C^{l ( {\ensuremath{\boldsymbol{n}}} )}_{l + 1}
\varphi_{{\ensuremath{\boldsymbol{n}}}} \psi_{n - k}$$ If now $\tilde{\varphi} = \varphi^{- 1}$ and $\tilde{F} = \tau^{- 1} (
\tilde{\varphi} )$, then, as $\tilde{F} F = {\ensuremath{\operatorname{Id}}}$ we get $$\tilde{F} = {\ensuremath{\operatorname{Id}}} + \sum_{s \geq 1} ( - 1 )^s F_{n_1} \ldots F_{n_s} =
{\ensuremath{\operatorname{Id}}} + \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}} ( - 1 )^{l ( {\ensuremath{\boldsymbol{n}}}
)} F_{{\ensuremath{\boldsymbol{n}}}}$$ but for ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_{}$, $$F_{{\ensuremath{\boldsymbol{n}}}} ( x ) = \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i\leq s\\
\end{array}$}}} B_1 ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s )
\varphi_{{\ensuremath{\boldsymbol{m}}}^1} \ldots \varphi_{{\ensuremath{\boldsymbol{m}}}^s} x^{\|
{\ensuremath{\boldsymbol{n}}} \| + 1}$$ Now $$\tilde{\varphi}_n = \sum_{\text{${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in
\mathcal{N}_n$}} ( - 1 )^s \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i\leq s\\
\end{array}$}}} B_1 ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s )
\varphi_{{\ensuremath{\boldsymbol{m}}}^1} \ldots \varphi_{{\ensuremath{\boldsymbol{m}}}^s} \label{e38}$$ and this gives the attempted result.
Using the same ideas, we will finally prove theorems \[th2\] and \[th3\]
Proof of Theorems \[th2\] and \[th3\]
-------------------------------------
As before, let $\varphi ( x ) = x + {\displaystyle}\sum_{n \geq 1} \varphi_n x^{n + 1}$ and $\psi ( x ) = x + {\displaystyle}\sum_{n \geq 1} \psi_n x^{n + 1}$ two elements of $G_2$ and $\eta = \mu ( \varphi, \psi ) = \psi \circ \varphi$ with $$\eta ( x ) = x + \sum_{n \geq 1} \eta_n x^{n + 1}$$ If $$\begin{array}{cccccc}
f ( x ) & = & \log ( \varphi' ( x ) ) & = & {\displaystyle}\sum_{n \geq 1}
\frac{f_n}{n!}
x^n & \hspace{1em} ( f_n = \gamma_n ( \varphi ) )\\
g ( x ) & = & \log ( \psi' ( x ) ) & = & {\displaystyle}\sum_{n \geq 1} \frac{g_n}{n!}
x^n & \hspace{1em} ( g_n = \gamma_n ( \psi ) )\\
h ( x ) & = & \log ( \eta' ( x ) ) & = & {\displaystyle}\sum_{n \geq 1} \frac{h_n}{n!}
x^n & \hspace{1em} ( h_n = \gamma_n ( \eta ) )
\end{array}$$ then $$\begin{array}{rcl}
h ( x ) & = & \log ( ( \psi \circ \varphi )' ( x ) )\\
& = & \log ( \varphi' ( x ) . \psi' ( \varphi ( x ) )\\
& = & \log ( \varphi' ( x ) ) + ( \log \psi' ) \circ \varphi ( x )\\
& = & f ( x ) + F ( g ) ( x )
\end{array}$$ where $F$ is the substitution automorphism associated to $\varphi$. We remind that $F = {\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} F_n$. Because of equation \[e19\], $$\begin{array}{ccc}
F_n & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n}
\frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \varphi_{{\ensuremath{\boldsymbol{n}}}} x^{\| {\ensuremath{\boldsymbol{n}}}
\| + l ( {\ensuremath{\boldsymbol{n}}} )} \partial_x^{l ( {\ensuremath{\boldsymbol{n}}} )}\\
& = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n}
\frac{1}{l ( {\ensuremath{\boldsymbol{n}}} ) !} \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i\leq s\\
\end{array}$}}} \frac{A ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s )
f_{{\ensuremath{\boldsymbol{m}}}^1} \ldots f_{{\ensuremath{\boldsymbol{m}}}^s}}{ {\ensuremath{\boldsymbol{m}}}^1 ! \ldots
{\ensuremath{\boldsymbol{m}}}^s !} x^{\| {\ensuremath{\boldsymbol{n}}} \| + l ( {\ensuremath{\boldsymbol{n}}} )}
\partial_x^{l ( {\ensuremath{\boldsymbol{n}}} )}\\
& = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n}
\frac{f_{{\ensuremath{\boldsymbol{n}}}}}{{\ensuremath{\boldsymbol{n}}} !} \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots
{\ensuremath{\boldsymbol{m}}}^s = {\ensuremath{\boldsymbol{n}}}} A ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s
) \frac{1}{s!} x^{n + s} \partial_x^s
\end{array} \label{e42}$$ But for $k \geq 1$, $$\begin{array}{ccc}
F_n \left( \frac{g_k}{k!} x^k \right) & = &
\end{array} \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} \frac{f_{{\ensuremath{\boldsymbol{n}}}}
g_k}{{\ensuremath{\boldsymbol{n}}} !k!} \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^s =
{\ensuremath{\boldsymbol{n}}}} A ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s ) C_k^s x^{n + k}$$ and we obtain immediately the formula for the coproduct.
Let now $\tilde{\varphi} = \varphi^{- 1}$ and $$\tilde{f} ( x ) =
\log ( \tilde{\varphi}' ( x ) ) = \sum_{n \geq 1}
\frac{\tilde{f}_n}{n!} x^n \quad ( \tilde{f}_n = \gamma_n (
\tilde{\varphi} ) )$$ Since $\tilde{\varphi} \circ \varphi ( x ) = x$, $$0 = \log ( ( \tilde{\varphi} \circ \varphi )' ( x ) ) = f ( x ) + F.
\tilde{f} ( x )$$ thus $$\tilde{f} ( x ) = - \tilde{F} .f ( x ) = - f ( x ) - \sum_{{\ensuremath{\boldsymbol{n}}} \in
\mathcal{N}} ( - 1 )^{l ( {\ensuremath{\boldsymbol{n}}} )} F_{{\ensuremath{\boldsymbol{n}}}} ( f ) ( x )$$ But, once again, $$\begin{array}{lll}
f_{{\ensuremath{\boldsymbol{n}}}, k} ( x ) & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} ( -
1 )^{l ( {\ensuremath{\boldsymbol{n}}} )} F_{{\ensuremath{\boldsymbol{n}}}} ( \frac{f_k}{k!} x^k )\\
& = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} ( - 1 )^{l ( {\ensuremath{\boldsymbol{n}}} )}
\frac{f_k}{k!} \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}\\
1\leq i\leq s\\
\end{array}$}}} B_k ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{} {\ensuremath{\boldsymbol{m}}}^s )
\varphi_{{\ensuremath{\boldsymbol{m}}}^1} \ldots \varphi_{{\ensuremath{\boldsymbol{m}}}^s} x^{\|
{\ensuremath{\boldsymbol{n}}} \| + k}\\
& = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots
{\ensuremath{\boldsymbol{m}}}^s = {\ensuremath{\boldsymbol{n}}}} ( - 1 )^s B_k ( {\ensuremath{\boldsymbol{m}}}^1, \ldots,^{}
{\ensuremath{\boldsymbol{m}}}^s ) \varphi_{{\ensuremath{\boldsymbol{n}}}} \frac{f_k}{k!} x^{\| {\ensuremath{\boldsymbol{n}}}
\| + k}\\
& = & - {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} U_k ( {\ensuremath{\boldsymbol{n}}} )
\varphi_{{\ensuremath{\boldsymbol{n}}}} \frac{f_k}{k!} x^{\| {\ensuremath{\boldsymbol{n}}} \| + k}
\end{array}$$ Now, replacing $\varphi_{{\ensuremath{\boldsymbol{n}}}}$ as in equation \[e42\], $$\begin{array}{lll}
f_{{\ensuremath{\boldsymbol{n}}}, k} ( x ) & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} ( -
1 )^{l ( {\ensuremath{\boldsymbol{n}}} )} F_{{\ensuremath{\boldsymbol{n}}}} ( \frac{f_k}{k!} x^k )\\
& = & - {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} U_k ( {\ensuremath{\boldsymbol{n}}} )
\varphi_{{\ensuremath{\boldsymbol{n}}}} \frac{f_k}{k!} x^{\| {\ensuremath{\boldsymbol{n}}} \| + k}\\
& = & - {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} \frac{f_{{\ensuremath{\boldsymbol{n}}}}
f_k}{{\ensuremath{\boldsymbol{n}}} !k!} \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^s =
{\ensuremath{\boldsymbol{n}}}} A ( {\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s ) U_k ( \|
{\ensuremath{\boldsymbol{m}}}^1 \|, \ldots, || {\ensuremath{\boldsymbol{m}}}^s \| ) x^{n + k}
\end{array}$$ Now, for $l \geq 1$, $$\tilde{f}_l = - f_l + \sum_{n = 1}^{l - 1} \sum_{{\ensuremath{\boldsymbol{n}}} \in
\mathcal{N}_n} \frac{ l!f_{{\ensuremath{\boldsymbol{n}}}} f_{l - n}}{{\ensuremath{\boldsymbol{n}}} ! ( l - n )
!} \sum_{{\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^s = {\ensuremath{\boldsymbol{n}}}} A (
{\ensuremath{\boldsymbol{m}}}^1, \ldots, {\ensuremath{\boldsymbol{m}}}^s ) U_{l - n} ( \| {\ensuremath{\boldsymbol{m}}}^1
\|, \ldots, || {\ensuremath{\boldsymbol{m}}}^s \| )$$ and this gives immediately the attempted formula.
This ends the proofs for our formulas but, as we said before, the first proofs were derived from mould calculus and we will give the main ideas in the next sections.
Mould calculus and the shuffle Hopf algebra ${\ensuremath{\operatorname{sh}}}
(\mathbbm{N}^{\ast} )$.\[moulds\]
=============================================================================
An example of mould calculus
----------------------------
### Formal Conjugacy of equations
Mould calculus, as defined by J. Ecalle (see [[@et2]]{},[[@et3]]{},[[@esn]]{}), appears in the study of formal or analytic conjugacy of differential equations, vector fields, diffeomorphisms. In order to introduce it, we give here a very simple but useful example.
Let $u \in G_2$ and the associated equation $$( E_u ) \hspace{2em} \partial_t x = u ( x ) = x + \sum_{n \geq 1} u_n x^{n
+ 1}$$ For $u$ and $v$ in $G_2$ the equations $( E_u )$ and $( E_v )$ are formally conjugated if there exists an element $\varphi$ of $G_2$ such that, if $x$ is a solution of $( E_u )$ then $y = \varphi ( x )$ is a solution of $( E_v )$. This defines an equivalence relation on the set of such equations and one can easily check that there is only one class : For any equation $( E_u )$, there exist a unique $\varphi_{}$ of $G_2$ such that, if $x$ is a solution of $( E_u
)$ then $y = \varphi_{} ( x )$ is a solution of $$( E_0 ) \hspace{2em} \partial_t y = y$$ The equation for $\varphi_{}$ reads $$u ( x ) \varphi'_{} ( x ) = \varphi_{} ( x )$$ and, if $$\varphi ( x ) = x + \sum_{n \geq 1} \varphi_n x^{n + 1}$$ then $$\begin{array}{ccc}
u_1 + 2 \varphi_1 & = & \varphi_1\\
u_2 + 2 \varphi_1 u_1 + 3 \varphi_2 & = & \varphi_2\\
& \vdots & \\
u_n + \sum_{k = 1}^{n - 1} ( k + 1 ) u_{n - k} \varphi_k + ( n + 1 )
\varphi_n & = & \varphi_n
\end{array}$$ Recursively, one can determine the values $a_n ( \varphi ) = \varphi_n$ and thus the diffeomorphism $\varphi$. This does not give a direct formula for the coefficients of $\varphi$. Among other properties that may be useful for more sophisticated equations, we will see that the mould calculus will give explicit formulas.
Mould calculus, for this example, is based on two remarks which are detailed in the next two sections.
### Diffeomorphisms an substitution automorphisms
As we have seen in section \[aut\], to any diffeomorphism $\varphi \in G_2$ one can associate a substitution automorphism $F \in \tilde{G}_2$ $$F = {\ensuremath{\operatorname{Id}}} + \sum_{n \geq 1} F_n$$ Moreover, the action of such an operator on a product of formal power series induces a coproduct $$\Delta F_{} = F_{} \otimes F_{} \hspace{1em} ( F_{} ( f g ) = ( F_{} f ) (
F_{} g ) )$$ which also reads $$\forall n \geq 1, \hspace{1em} \Delta F_n = F_n \otimes {\ensuremath{\operatorname{Id}}} + \sum_{k
= 1}^{n - 1} F_k \otimes F_{n - k} + {\ensuremath{\operatorname{Id}}} \otimes F_n$$
### Symmetral moulds and shuffle Hopf algebra
Now, for $u \in G_2$, the equation $( E_u )$ reads $$\partial_t x = \left( \mathbbm{B}_0 + \sum_{n \geq 1} u_n \mathbbm{B}_n
\right) .x =\mathbbm{B}.x \hspace{1em} \text{with} \hspace{1em}
\mathbbm{B}_n = x^{n + 1} \partial_x$$ Instead of computing the conjugating map $\varphi$ we could look for its associated substitution automorphism $F_{}$ in the following shape : $$F = {\ensuremath{\operatorname{Id}}} + \sum_{s \geq 1} \sum_{n_1 \geq 1, \ldots, n_s \geq 1}
M^{n_1, \ldots, n_s} \mathbbm{B}_{n_1} \ldots \mathbbm{B}_{n_s}$$ As we will see later, in order to get a substitution automorphism, is is sufficient to impose that for any sequences ${\ensuremath{\boldsymbol{k}}}= ( k_1, \ldots, k_s
)$ and ${\ensuremath{\boldsymbol{l}}}= ( l_1, \ldots, l_t )$, $$M^{{\ensuremath{\boldsymbol{k}}}} M^{{\ensuremath{\boldsymbol{l}}}} = \sum_{{\ensuremath{\boldsymbol{m}}}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{l}}} }_{{\ensuremath{\boldsymbol{m}}}} M^{{\ensuremath{\boldsymbol{m}}}}$$ where ${\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{l}}} }_{{\ensuremath{\boldsymbol{m}}}}$ is the number of shuffling of the sequences ${\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{l}}}$ that gives the sequence ${\ensuremath{\boldsymbol{m}}}$. The set of such coefficients is called a symmetral mould. Moreover the conjugacy equation reads $$\mathbbm{B}F_{} .x = F_{} \mathbbm{B}_0 .x$$ Now we can solve the equation $\mathbbm{B}F_{} = F_{} \mathbbm{B}_0$ by noticing that, for $( n_1, \ldots, n_s ) \in (\mathbbm{N}^{\ast} )^s$, $$\left[ \mathbbm{B}_0,\mathbbm{B}_{n_1} \ldots \mathbbm{B}_{n_s} \right] = (
n_1 + \ldots + n_s )\mathbbm{B}_{n_1} \ldots \mathbbm{B}_{n_s}$$ and using this commutation relations, one can check that for $s = 1$ and a sequence $( n_1$) we get $$u_{n_1} + n_1 M^{n_1} = 0$$ and for $s \geq 2$ and a sequence $( n_1, \ldots, n_s ) \in
(\mathbbm{N}^{\ast} )^s$, $$u_{n_1} M^{n_2, \ldots, n_s} + ( n_1 + \ldots + n_s ) M^{n_1, \ldots, n_s} =
0$$ This defines a symmetral mould, for $s \geq 1$ and $( n_1, \ldots, n_s ) \in
(\mathbbm{N}^{\ast} )^s$, $$M^{n_1, \ldots, n_s} = \frac{( - 1 )^s u_{n_1} \ldots u_{n_s}}{( n_1 +
\ldots + n_s ) ( n_2 + \ldots + n_s ) \ldots ( n_{s - 1} + n_s ) n_s}$$ thus we get explicit formulas for $F$ and $\varphi ( x ) = F_{} .x$ : For $n
\geq 1$, $$\varphi_n x^{n + 1} = \sum_{s = 1}^n \sum_{{\text{\scriptsize $\begin{array}{c}
n_1 + \ldots + n_s = n\\
n_i \geq 1
\end{array}$}}} M^{n_1, \ldots, n_s} \mathbbm{B}_{n_1} \ldots
\mathbbm{B}_{n_s} .x$$ and $$\varphi_n = \sum_{s = 1}^n \sum_{{\text{\scriptsize $\begin{array}{c}
n_1 + \ldots + n_s = n\\
n_i \geq 1
\end{array}$}}} ( n_s + 1 ) ( n_{s - 1} + n_s + 1 ) \ldots ( n_2 + \ldots +
n_s + 1 ) M^{n_1, \ldots, n_s}$$
We just gave the outlines of the method here. The important idea is that we only used the commutation of $\mathbbm{B}_0$ with the over derivations $\mathbbm{B}_n$ ($n \geq 1$), which means that we worked as these derivations were free of other relations. This can be interpreted in the following algebraic way.
The free group and its Hopf algebra of coordinates
--------------------------------------------------
### Lie algebra and substitution automorphisms
Let $\mathcal{A}^1$ the Lie algebra of formal vector fields generated by the derivations
$$\forall n \geq 1, \quad \mathbbm{B}_n = x^{n + 1} \partial_x$$
Its enveloping algebra $\mathcal{U}(\mathcal{A}^1 )$ is a graded Hopf algebra and, see [[@cm]]{}, the Hopf algebra $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is the dual of $\mathcal{U}(\mathcal{A}^1 )$. Note that this dual is well-defined as the graded components of $\mathcal{U}(\mathcal{A}^1 )$ are vector spaces of finite dimension. If $G (\mathcal{A}^1 ) \subset
\mathcal{U}(\mathcal{A}^1 )$ is the group of the group-like elements of $\mathcal{U}(\mathcal{A}^1 )$, this is exactly the group of substitution automorphism describe above and it is isomorphic to the group $G_2$ $$\forall F \in G (\mathcal{A}^1 ), \forall f \in \mathbbm{R}[ [ x ] ] \quad
F.f = f \circ \varphi^{}, \quad \varphi \in G_2$$ In other terms, $G (\mathcal{A}^1 ) = \widetilde{G_{}}_2$.
### The free group and its Hopf algebra of coordinates
Our previous mould calculus suggests to introduce, by analogy with $\mathcal{A}^1$, the graded free Lie algebra $A^1$ generated by a set of primitive elements $X_n$, $n \geq 1$, $$\Delta ( X_n ) = X_n \otimes 1 + 1 \otimes X_n$$ The enveloping algebra $\mathcal{U}( A^1 )$ is a Hopf algebra which is also called the concatenation Hopf algebra in combinatorics (see [[@reut]]{}). If the unity is $X_{\emptyset} = 1$ ($\emptyset$ is the empty sequence), then an element ${\ensuremath{\boldsymbol{U}}}$ of $\mathcal{U}( A^1 )$ can be written $$\begin{array}{lll}
{\ensuremath{\boldsymbol{U}}} & = & {\displaystyle}U^{\emptyset} X_{\emptyset} + \sum_{s \geq 1}
\sum_{n_1, \ldots, n_s \geq 1} U^{n_1, \ldots, n_s} \mathcal{} X_{n_1}
\ldots X_{n_s}\\
& = &{\displaystyle}U^{\emptyset} X_{\emptyset} + \sum_{s \geq 1} \sum_{n_1, \ldots,
n_s \geq 1} U^{n_1, \ldots, n_s} \mathcal{} X_{n_1, \ldots, n_s}\\
& = &{\displaystyle}\sum U^{\bullet} X_{\bullet}
\end{array}$$ where the collection of coefficients $U^{\bullet}$ is called a *mould.* The structure of the enveloping algebra $\mathcal{U}( A^1 )$ can be described as follows : the product is given by $$\forall {\ensuremath{\boldsymbol{m}}},{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}, \quad X_{{\ensuremath{\boldsymbol{m}}}}
X_{{\ensuremath{\boldsymbol{n}}}} = X_{{\ensuremath{\boldsymbol{m}}} {\ensuremath{\boldsymbol{n}}}} \quad
\text{(concatenation),}$$ the coproduct is $$\Delta ( X_{{\ensuremath{\boldsymbol{n}}}} ) = \sum_{{\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2} {\ensuremath{\operatorname{sh}}}
\left( \begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2\\
{\ensuremath{\boldsymbol{n}}}
\end{array} \right) X_{{\ensuremath{\boldsymbol{n}}}^1} \otimes X_{{\ensuremath{\boldsymbol{n}}}^2}$$ where ${\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2}_{{\ensuremath{\boldsymbol{n}}}}$ is the number of shuffling of the sequences ${\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2$ that gives ${\ensuremath{\boldsymbol{n}}}$. Finally, the antipode $S$ is defined by $$S ( X_{n_1, \ldots, n_s} ) = ( - 1 )^s X_{n_s, \ldots, n_1}$$
Once again one can define the group $G ( A^1 )$ and if ${\ensuremath{\boldsymbol{F}}} \in G (
A^1 )$ then $${\ensuremath{\boldsymbol{F}}}= \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N} \cup \{ \emptyset \}}
F^{{\ensuremath{\boldsymbol{n}}}} X_{{\ensuremath{\boldsymbol{n}}}}$$ where the mould $F^{\bullet}$ is *symmetral* : $F^{\emptyset} = 1$ and $$\forall {\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2, \quad F^{{\ensuremath{\boldsymbol{n}}}^1}
F^{{\ensuremath{\boldsymbol{n}}}^2} = \sum_{{\ensuremath{\boldsymbol{n}}}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2}_{{\ensuremath{\boldsymbol{n}}}} F^{{\ensuremath{\boldsymbol{n}}}}$$ Moreover, if ${\ensuremath{\boldsymbol{G}}}$ is the group inverse of ${\ensuremath{\boldsymbol{F}}}$, then its associated mould is given by the formulas $$G^{n_1, \ldots, n_s} = ( - 1 )^s F^{n_s, \ldots, n_1}$$ Thanks to the graduation on $\mathcal{U}( A^1 )$, its dual $H^1$ is a Hopf algebra, the Hopf algebra of coordinates on $G ( A^1 )$ and, if the dual basis of $\{ X_{{\ensuremath{\boldsymbol{n}}}}, \hspace{1em} {\ensuremath{\boldsymbol{n}}} \in \mathcal{N} \}$ is $\{
Z^{{\ensuremath{\boldsymbol{n}}}}, \hspace{1em} {\ensuremath{\boldsymbol{n}}} \in \mathcal{N} \}$ then the product in $H^1$ is defined by : $$\forall {\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2, \quad Z^{{\ensuremath{\boldsymbol{n}}}^1}
Z^{{\ensuremath{\boldsymbol{n}}}^2} = \sum_{{\ensuremath{\boldsymbol{n}}}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1,{\ensuremath{\boldsymbol{n}}}^2}_{{\ensuremath{\boldsymbol{n}}} }
Z^{{\ensuremath{\boldsymbol{n}}}}$$ The coproduct is : $$\Delta ( Z^{{\ensuremath{\boldsymbol{n}}}} ) = Z^{{\ensuremath{\boldsymbol{n}}}} \otimes 1 + 1 \otimes
Z^{{\ensuremath{\boldsymbol{n}}}} + \sum_{{\ensuremath{\boldsymbol{n}}}^1 {\ensuremath{\boldsymbol{n}}}^2 ={\ensuremath{\boldsymbol{n}}}}
Z^{{\ensuremath{\boldsymbol{n}}}^1} \otimes Z^{{\ensuremath{\boldsymbol{n}}}^2}$$ where ${\ensuremath{\boldsymbol{n}}}^1 {\ensuremath{\boldsymbol{n}}}^2$ is the concatenation of the two nonempty sequences ${\ensuremath{\boldsymbol{n}}}^1$ and ${\ensuremath{\boldsymbol{n}}}^2$ and $Z^{\emptyset} = 1$ is the unity. Finally, the antipode is given by $$S ( Z^{n_1, \ldots, n_s} ) = ( - 1 )^s Z^{n_s, \ldots, n_1}$$ The structure of $H^1$ (coproduct, antipode, ...) is fully explicit. This will be of great use since our previous mould calculus suggests that there exists a surjective morphism from $A^1$ on $\mathcal{A}^1$ that induces an injective morphism from $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ into $H^1$. In other words, $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ can be identified to a sub-Hopf algebra of $H^1$ and, as everything is explicit in $H^1$, one can derive formulas for the coproduct and the antipode in $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}^1$.
Morphisms.\[mor\]
=================
The application defined by $\rho ( X_n ) =\mathbbm{B}_n = x^{n + 1}
\partial_x$ obviously determines a morphism from $A^1$ (resp. $\mathcal{U}(
A^1 )$, resp. $G ( A^1 )$) on $\mathcal{A}^1$ (resp. $\mathcal{U}(\mathcal{A}^1 )$, resp. $G (\mathcal{A}^1 ) \simeq G_2$) and it is surjective : If $\varphi \in G_2$ and $F = \tau^{- 1} ( \varphi ) \in G
(\mathcal{A}^1 ) = \tilde{G}_2$, then, if $$b ( x ) = x + \sum_{n \geq 1} b_n x^{n + 1} = \frac{\varphi ( x )}{\varphi'
( x )}$$ then $\varphi$ is the unique diffeomorphism of $G_2$ that conjugates $( E_b )$ to $( E_0 )$ thus $$F = {\ensuremath{\operatorname{Id}}} + \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}} M^{{\ensuremath{\boldsymbol{n}}}}
\mathbbm{B}_{{\ensuremath{\boldsymbol{n}}}} = \rho \left( X_{\emptyset} + \sum_{{\ensuremath{\boldsymbol{n}}}
\in \mathcal{N}} M^{{\ensuremath{\boldsymbol{n}}}} X_{{\ensuremath{\boldsymbol{n}}}} \right)$$ By duality, it induces a morphism $\rho^{\ast}$ from $\mathcal{H}^1$ to $H^1$ by $$\forall \gamma \in \mathcal{H}^1, \quad \rho^{\ast} ( \gamma ) = \gamma
\circ \rho$$ and, since $\rho$ is surjective, $\rho^{\ast}$ is injective : $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}^1$ is isomorphic to the sub-Hopf algebra $\rho^{\ast}
(\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}} ) \subset H^1$. Using this injective morphism, we define $$\forall n \geq 1, \quad \Gamma_n = \rho^{\ast} ( \gamma_n )$$ and $\rho^{\ast} (\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}^1 )$ is then the Hopf algebra generated by the $\Gamma_n$. In order to get formulas in $\mathcal{H}_{{\ensuremath{\operatorname{CM}}}}^1$, we will use the algebra $\rho^{\ast}
(\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}} )$ and express the $\Gamma_n$ in terms of the $Z^{{\ensuremath{\boldsymbol{n}}}}$ :
\[th8\]For $n \geq 1$, $$\begin{array}{ccc}
\Gamma_n & = & {\displaystyle}n! \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n
\end{array}$}}} \sum_{t = 1}^s \frac{( - 1 )^{t - 1}}{t}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}
\end{array}$}}} Z^{{\ensuremath{\boldsymbol{n}}}^1} \ldots Z^{{\ensuremath{\boldsymbol{n}}}^t}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = &{\displaystyle}n! \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n
\end{array}$}}} Q^{{\ensuremath{\boldsymbol{n}}}} Z^{{\ensuremath{\boldsymbol{n}}}}
\end{array}$$ where $S^{n_1, \ldots, n_s} = \prod_{i = 1}^s ( n_i + n_{i + 1} + \ldots +
n_s + 1 ) = \prod_{i = 1}^s ( \hat{n}_i + 1 )$ and $Q^{n_1, \ldots, n_s} = (
n_s + 1 ) \prod_{i = 2}^s \hat{n}_i$ with $Q^{n_1} = ( n_1 + 1 )$.
Let ${\ensuremath{\boldsymbol{F}}}= X_{\emptyset} + \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}}
F^{{\ensuremath{\boldsymbol{n}}}} X_{{\ensuremath{\boldsymbol{n}}}} \in G ( A^1 )$. If $F = \rho ({\ensuremath{\boldsymbol{F}}})
\in G (\mathcal{A}^1 )$, then $$\Gamma_n ({\ensuremath{\boldsymbol{F}}}) = \gamma_n ( F ) = \gamma_n ( \varphi ) = (
\partial^n_x \log ( \varphi' ) ( x ) )_{x = 0}$$ where $\varphi \in G_2$ is defined by : $$\varphi^{} ( x ) = \rho ({\ensuremath{\boldsymbol{F}}}) .x = F.x = x +
\sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}
\end{array}$}}} F^{n_1, \ldots, n_s} \mathbbm{B}_{n_1} \ldots
\mathbbm{B}_{n_s} .x$$ Then $$\varphi' ( x ) = 1 + \sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}
\end{array}$}}} F^{n_1, \ldots, n_s} S^{n_1, \ldots, n_s} x^{n_1 + \ldots +
n_s}$$ Using the logarithm and derivation, one easily gets the formula $$\Gamma_n ({\ensuremath{\boldsymbol{F}}}) = n! \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n
\end{array}$}}} \sum_{t = 1}^s \frac{( - 1 )^{t - 1}}{t}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}
\end{array}$}}} F^{{\ensuremath{\boldsymbol{n}}}^1} \ldots F^{{\ensuremath{\boldsymbol{n}}}^t}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}$$
We prove the second part of the formula in section \[ini\], using the fact that $F^{\bullet}$ is symmetral. As $$\Gamma_n ({\ensuremath{\boldsymbol{F}}}) = n! \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n
\end{array}$}}} F^{{\ensuremath{\boldsymbol{n}}}} Q^{{\ensuremath{\boldsymbol{n}}}}$$ and $Z^{{\ensuremath{\boldsymbol{n}}}} .{\ensuremath{\boldsymbol{F}}}= F^{{\ensuremath{\boldsymbol{n}}}}$, theorem \[th8\] will be proved.
As $\rho^{\ast} ( \delta_n ) = \Gamma_n \in H^1$, and, since the coproduct and the antipode are explicit in $H^1$, we can once again obtain the formulas given in theorems \[th2\] and \[th3\].
Initial Proofs. {#ini}
===============
Proof of theorem \[th8\]
------------------------
We already proved that, for $n \geq 1$, $$\Gamma_n = n! \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}_n
\end{array}$}}} \sum_{t = 1}^s \frac{( - 1 )^{t - 1}}{t}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}
\end{array}$}}} Z^{{\ensuremath{\boldsymbol{n}}}^1} \ldots Z^{{\ensuremath{\boldsymbol{n}}}^t}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}$$ Extending the notion of shuffling, for $t \geq 1$, if ${\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^t,{\ensuremath{\boldsymbol{m}}}$ are $t + 1$ sequences, then ${\ensuremath{\operatorname{sh}}}^{
{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^t}_{{\ensuremath{\boldsymbol{m}}}}$ is the number of ways to obtain the sequence ${\ensuremath{\boldsymbol{m}}}$ by shuffling the sequences ${\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^t$. Then, $$\begin{array}{ccc}
{\displaystyle}\frac{1}{n!} \Gamma_n & = &{\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in
\mathcal{N}_n} \sum_{t
= 1}^{l ({\ensuremath{\boldsymbol{n}}})} \frac{( - 1 )^{t - 1}}{t}
\sum_{
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}}
Z^{{\ensuremath{\boldsymbol{n}}}^1} \ldots Z^{{\ensuremath{\boldsymbol{n}}}^t}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = &{\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_n} \sum_{t = 1}^{l
({\ensuremath{\boldsymbol{n}}})} \frac{( - 1 )^{t - 1}}{t} \sum_{
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}} \left(
\sum_{{\ensuremath{\boldsymbol{m}}}} {\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1,
\ldots,{\ensuremath{\boldsymbol{n}}}^t}_{{\ensuremath{\boldsymbol{m}}} }
Z^{{\ensuremath{\boldsymbol{m}}}} \right) S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots
S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = &{\displaystyle}\sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n} \left( Z^{{\ensuremath{\boldsymbol{m}}}}
\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}})} \frac{( - 1 )^{t - 1}}{t}
\sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t \in \mathcal{N}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{m}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t} \right.
\end{array}$$ Note that in these equations, we had $\| {\ensuremath{\boldsymbol{m}}} \| = \| {\ensuremath{\boldsymbol{n}}}
\|$ and $l ({\ensuremath{\boldsymbol{m}}}) = l ({\ensuremath{\boldsymbol{n}}})$. For a given sequence ${\ensuremath{\boldsymbol{m}}} \in \mathcal{N}$, let $$Q^{{\ensuremath{\boldsymbol{m}}}} = \sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}})} \frac{( - 1 )^{t - 1}}{t}
\sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t \in \mathcal{N}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{m}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}$$ it remains to prove that, if ${\ensuremath{\boldsymbol{m}}}= ( m_1, \ldots, m_s )$ then $Q^{m_1, \ldots, m_s} = ( m_s + 1 ) \prod_{i = 2}^s \hat{m}_i$ with $Q^{m_1} =
( m_1 + 1 )$. We prove this formula by induction on $l ({\ensuremath{\boldsymbol{m}}})$.
If $l ({\ensuremath{\boldsymbol{m}}}) = 1$, then ${\ensuremath{\boldsymbol{m}}}= ( m_1 )$ and $$Q^{m_1} = \frac{( - 1 )^0}{1} \sum_{{\ensuremath{\boldsymbol{n}}}^1 \in \mathcal{N}}
{\ensuremath{\operatorname{sh}}}^{ {\ensuremath{\boldsymbol{n}}}^1}_{
{\ensuremath{\boldsymbol{m}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} = S^{m_1} = m_1 + 1$$ If $l ({\ensuremath{\boldsymbol{m}}}) = s \geq 2$, then let ${\ensuremath{\boldsymbol{m}}}= ( m_1, \ldots, m_s
)$ and ${\ensuremath{\boldsymbol{p}}}= ( m_2, \ldots, m_s )$. For any sequence ${\ensuremath{\boldsymbol{n}}}= (
n_1, \ldots, n_k )$, we note $m_1 {\ensuremath{\boldsymbol{n}}}= ( m_1, n_1, \ldots, n_k )$. If a shuffling of $t \geq 1$ sequences ${\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t$ gives ${\ensuremath{\boldsymbol{m}}}$ then
- Either there exists $1 \leq i \leq t$ such that ${\ensuremath{\boldsymbol{n}}}^i = (
m_1 )$ (but then $t \geq 2$), and, omitting ${\ensuremath{\boldsymbol{n}}}^i= (
m_1 )$, the corresponding shuffling of the $t - 1$ remaining sequences gives ${\ensuremath{\boldsymbol{p}}}$.
- Either there exists $1 \leq i \leq t$ such that ${\ensuremath{\boldsymbol{n}}}^i = m_1
\tilde{{\ensuremath{\boldsymbol{n}}}}^i$ ($\tilde{{\ensuremath{\boldsymbol{n}}}}^i \not= \emptyset$) (necessarily, $t < l ({\ensuremath{\boldsymbol{m}}})$) and, replacing ${\ensuremath{\boldsymbol{n}}}^i$ by $\tilde{{\ensuremath{\boldsymbol{n}}}}^i$, the corresponding shuffling of the $t$ sequences gives ${\ensuremath{\boldsymbol{p}}}$.
This means that : $$\begin{array}{rcl}
Q^{{\ensuremath{\boldsymbol{m}}}} & = & {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}})} \frac{( - 1 )^{t -
1}}{t} \sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{m}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = & {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}}) - 1} \frac{( - 1 )^{t - 1}}{t}
\sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
\sum_{i = 1}^t S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{m_1
{\ensuremath{\boldsymbol{n}}}^i} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& & +{\displaystyle}\sum_{t = 2}^{l ({\ensuremath{\boldsymbol{m}}})} \frac{( - 1 )^{t - 1}}{t}
\sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^{t - 1}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^{t - 1} }_{{\ensuremath{\boldsymbol{p}}}}
\sum_{i = 0}^{t - 1} S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots
S^{{\ensuremath{\boldsymbol{n}}}^i} S^{m_1} S^{{\ensuremath{\boldsymbol{n}}}^{i + 1}} \ldots
S^{{\ensuremath{\boldsymbol{n}}}^{t - 1}}
\end{array}$$ but as $S^{m_1} = m_1 + 1$ and $S^{m_1 {\ensuremath{\boldsymbol{n}}}^i} = ( m_1 + \|
{\ensuremath{\boldsymbol{n}}}^i \| + 1 ) S^{{\ensuremath{\boldsymbol{n}}}^i}$, $$\begin{array}{rcl}
Q^{{\ensuremath{\boldsymbol{m}}}} & = & {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}}) - 1} \frac{( - 1
)^{t - 1}}{t} \sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
\sum_{i = 1}^t ( m_1 + \| {\ensuremath{\boldsymbol{n}}}^i \| + 1 )
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^i} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& & + {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}}) - 1} \frac{( - 1 )^t}{t + 1}
\sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
\sum_{i = 0}^t ( m_1 + 1 ) S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots
S^{{\ensuremath{\boldsymbol{n}}}^i} S^{{\ensuremath{\boldsymbol{n}}}^{i + 1}} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = & {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}}) - 1} \frac{( - 1 )^{t - 1}}{t} ( t (
m_1 + 1 ) + \| {\ensuremath{\boldsymbol{p}}} \| ) \sum_{{\ensuremath{\boldsymbol{n}}}^1,
\ldots,{\ensuremath{\boldsymbol{n}}}^t} {\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1,
\ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& & + {\displaystyle}\sum_{t = 1}^{l ({\ensuremath{\boldsymbol{m}}}) - 1} \frac{( - 1 )^t}{t + 1} ( t +
1 ) ( m_1 + 1 ) \sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = & {\displaystyle}\| {\ensuremath{\boldsymbol{p}}} \| \sum_{t = 1}^{l ({\ensuremath{\boldsymbol{p}}})} \frac{( - 1 )^{t
- 1}}{t} \sum_{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t }_{
{\ensuremath{\boldsymbol{p}}}}
S^{{\ensuremath{\boldsymbol{n}}}^1} \ldots S^{{\ensuremath{\boldsymbol{n}}}^t}\\
& = & \| {\ensuremath{\boldsymbol{p}}} \| Q^{{\ensuremath{\boldsymbol{p}}}}
\end{array}$$ And it obviously gives the right formula for $Q^{{\ensuremath{\boldsymbol{m}}}}$.
Proof of theorem \[th2\]
------------------------
Using the above formula we have $$\begin{array}{rcl}
\Delta \Gamma_n & = & {\displaystyle}n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n}
Q^{{\ensuremath{\boldsymbol{m}}}}_{} ( \Delta Z^{{\ensuremath{\boldsymbol{m}}}} )\\
& = & {\displaystyle}n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n} Q^{{\ensuremath{\boldsymbol{m}}}}_{} \left(
Z^{{\ensuremath{\boldsymbol{m}}}} \otimes 1 + 1 \otimes Z^{{\ensuremath{\boldsymbol{m}}}} +
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} Z^{{\ensuremath{\boldsymbol{p}}}} \otimes Z^{{\ensuremath{\boldsymbol{q}}}} \right)\\
& = & {\displaystyle}\left( n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n} Q^{{\ensuremath{\boldsymbol{m}}}}_{}
Z^{{\ensuremath{\boldsymbol{m}}}} \right) \otimes 1 + 1 \otimes \left( n!
\sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n} Q^{{\ensuremath{\boldsymbol{m}}}}_{} Z^{{\ensuremath{\boldsymbol{m}}}}
\right)\\
& & {\displaystyle}+ n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} Q^{{\ensuremath{\boldsymbol{m}}}}_{} Z^{{\ensuremath{\boldsymbol{p}}}} \otimes
Z^{{\ensuremath{\boldsymbol{q}}}}\\
& = & {\displaystyle}\Gamma_n \otimes 1 + 1 \otimes \Gamma_n + n! \sum_{{\ensuremath{\boldsymbol{m}}} \in
\mathcal{N}_n} \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} Q^{{\ensuremath{\boldsymbol{m}}}}_{} Z^{{\ensuremath{\boldsymbol{p}}}} \otimes
Z^{{\ensuremath{\boldsymbol{q}}}}\\
& = & {\displaystyle}\Gamma_n \otimes 1 + 1 \otimes \Gamma_n + \tilde{\Delta} \Gamma_n
\end{array}$$ Now if ${\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}= ( m_1, \ldots, m_s )$ with ${\ensuremath{\boldsymbol{p}}},{\ensuremath{\boldsymbol{q}}} \in \mathcal{N}$ ($s \geq 2$), then $$\frac{Q^{{\ensuremath{\boldsymbol{m}}}}}{Q^{{\ensuremath{\boldsymbol{q}}}}} = \frac{( m_s + 1 ) \prod_{i =
2}^s \hat{m}_i}{( m_s + 1 ) \prod^s_{i = l ({\ensuremath{\boldsymbol{p}}}) + 2} \hat{m}_i} =
\prod_{i = 2}^{l ({\ensuremath{\boldsymbol{p}}}) + 1} \hat{m}_i = \prod_{i = 2}^{l
({\ensuremath{\boldsymbol{p}}}) + 1} ( \hat{p}_i + \| {\ensuremath{\boldsymbol{q}}} \| ) =
R^{{\ensuremath{\boldsymbol{p}}}}_{\| {\ensuremath{\boldsymbol{q}}} \|}$$ with the convention that if $i = l ({\ensuremath{\boldsymbol{p}}}) + 1$, then $\hat{p}_i = 0$. As this coefficient only depends on ${\ensuremath{\boldsymbol{p}}}$ and $\| {\ensuremath{\boldsymbol{q}}} \|$, $$\begin{array}{rcl}
\tilde{\Delta} \Gamma_n & = &{\displaystyle}n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} Q^{{\ensuremath{\boldsymbol{m}}}}_{} Z^{{\ensuremath{\boldsymbol{p}}}} \otimes
Z^{{\ensuremath{\boldsymbol{q}}}}\\
& = &{\displaystyle}n! \sum_{{\ensuremath{\boldsymbol{m}}} \in \mathcal{N}_n}
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{p}}} {\ensuremath{\boldsymbol{q}}}={\ensuremath{\boldsymbol{m}}}
\end{array}$}}} ( R^{{\ensuremath{\boldsymbol{p}}}}_{\| {\ensuremath{\boldsymbol{q}}} \|} Z^{{\ensuremath{\boldsymbol{p}}}} )
\otimes Q^{{\ensuremath{\boldsymbol{q}}}} Z^{{\ensuremath{\boldsymbol{q}}}}\\
& = &{\displaystyle}n! \sum_{k = 1}^{n - 1} \left( \sum_{{\ensuremath{\boldsymbol{p}}} \in
\mathcal{N}_{n - k}} R^{{\ensuremath{\boldsymbol{p}}}}_{\| {\ensuremath{\boldsymbol{q}}} \|}
Z^{{\ensuremath{\boldsymbol{p}}}} \right) \otimes \left( \sum_{{\ensuremath{\boldsymbol{q}}} \in
\mathcal{N}_k} Q^{{\ensuremath{\boldsymbol{q}}}} Z^{{\ensuremath{\boldsymbol{q}}}} \right)\\
& = &{\displaystyle}\sum_{k = 1}^{n - 1} \left( \frac{n!}{k!} \sum_{{\ensuremath{\boldsymbol{p}}} \in
\mathcal{N}_k} R^{{\ensuremath{\boldsymbol{p}}}}_{\| {\ensuremath{\boldsymbol{q}}} \|} Z^{{\ensuremath{\boldsymbol{p}}}}
\right) \otimes \Gamma_k\\
& = &{\displaystyle}\sum_{k = 1}^{n - 1} P^n_k \otimes \Gamma_k
\end{array}$$ and it remains to prove that, for $n \geq 1$ and $1 \leq k \leq n - 1$, $$P^n_k = \sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}\\
n_1 + \ldots + n_s = n - k, s \geq 1
\end{array}$}}} \frac{n!}{n_1 ! \ldots n_s !k!} \alpha^{n_1, \ldots, n_s}_k
\Gamma_{n_1} \ldots \Gamma_{n_s}$$ with $$\alpha^{{\ensuremath{\boldsymbol{n}}}}_k = \sum_{t = 1}^{l ({\ensuremath{\boldsymbol{n}}})} C^{t_{}}_k
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t ={\ensuremath{\boldsymbol{n}}}\\
{\ensuremath{\boldsymbol{n}}}^i \not= \emptyset
\end{array}$}}} \frac{1}{l ({\ensuremath{\boldsymbol{n}}}^1 ) ! \ldots l ({\ensuremath{\boldsymbol{n}}}^t ) !}
\prod_{i = 1}^t \frac{1}{\| {\ensuremath{\boldsymbol{n}}}^i \| + 1}$$ This formula was first conjectured on the first values of $n$. Now let $$\begin{array}{rcl}
\tilde{P}^n_k & = & {\displaystyle}\sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}_{n - k}
\end{array}$}}} \frac{n!}{n_1 ! \ldots n_s !k!} \alpha^{n_1, \ldots, n_s}_k
\Gamma_{n_1} \ldots \Gamma_{n_s}\\
& = & {\displaystyle}\frac{n!}{k!} \sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}_{n - k}
\end{array}$}}} \alpha^{n_1, \ldots, n_s}_k \sum_{{\ensuremath{\boldsymbol{m}}}^i \in
\mathcal{N}_{n_i}} Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s}
Z^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Z^{{\ensuremath{\boldsymbol{m}}}^s}\\
& = & {\displaystyle}\frac{n!}{k!} \sum_{{\text{\scriptsize $\begin{array}{c}
( n_1, \ldots, n_s ) \in \mathcal{N}_{n - k}
\end{array}$}}} \alpha^{n_1, \ldots, n_s}_k \sum_{{\ensuremath{\boldsymbol{p}}}}
\sum_{{\ensuremath{\boldsymbol{m}}}^i \in \mathcal{N}_{n_i}}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{p}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s}
Z^{{\ensuremath{\boldsymbol{p}}}}\\
& = & {\displaystyle}\frac{n!}{k!} \sum_{{\ensuremath{\boldsymbol{p}}} \in \mathcal{N}_{n - k}}
Z^{{\ensuremath{\boldsymbol{p}}}} \sum_{s \geq 1} \sum_{{\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^s} \alpha^{\| {\ensuremath{\boldsymbol{m}}}^1 \|, \ldots, \|
{\ensuremath{\boldsymbol{m}}}^s \|}_k {\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{p}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s}
\end{array}$$ It remains to prove that for a given ${\ensuremath{\boldsymbol{p}}} \in \mathcal{N}_{n - k}$, we have $$\tilde{R}^{{\ensuremath{\boldsymbol{p}}}}_k = \sum^{l ({\ensuremath{\boldsymbol{p}}})}_{s = 1}
\sum_{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s} \alpha^{\| {\ensuremath{\boldsymbol{m}}}^1 \|,
\ldots, \| {\ensuremath{\boldsymbol{m}}}^s \|}_k {\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{p}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s} =
R^{{\ensuremath{\boldsymbol{p}}}}_k = \prod_{i = 2}^{l ({\ensuremath{\boldsymbol{p}}}) + 1} ( \hat{p}_i + k )$$ As in the previous proof, if $l ({\ensuremath{\boldsymbol{p}}}) = 1$ then $R^{p_1}_k = k$ and $$\tilde{R}^{p_1}_k = \alpha^{p_1}_k Q^{p_1} = C^1_k \frac{1}{l ({\ensuremath{\boldsymbol{p}}})
!} \frac{1}{p_1 + 1} ( p_1 + 1 ) = k$$ and if $l ({\ensuremath{\boldsymbol{p}}}) \geq 2$, as ${\ensuremath{\boldsymbol{p}}}= p_1 {\ensuremath{\boldsymbol{q}}}$, $$\begin{array}{rcl}
\tilde{R}^{{\ensuremath{\boldsymbol{p}}}}_k & = & \tilde{R}^{p_1 {\ensuremath{\boldsymbol{q}}}}_k\\
& = & {\displaystyle}\sum^{l ({\ensuremath{\boldsymbol{p}}}) - 1}_{s = 1} \sum_{{\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^s \atop 1\leq i\leq s} \alpha^{\| {\ensuremath{\boldsymbol{m}}}^1 \|,
\ldots, \| {\ensuremath{\boldsymbol{m}}}^i \| + p_1, \ldots, \| {\ensuremath{\boldsymbol{m}}}^s \|}_k
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{q}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{p_1 {\ensuremath{\boldsymbol{m}}}^i} \ldots
Q^{{\ensuremath{\boldsymbol{m}}}^s}\\
& & + {\displaystyle}\sum^{l ({\ensuremath{\boldsymbol{p}}}) - 1}_{s = 1}
\hspace{-2mm}\sum_{{\ensuremath{\boldsymbol{m}}}^1,
\ldots,{\ensuremath{\boldsymbol{m}}}^s\atop 0\leq i\leq s} \hspace{-2mm}
\alpha^{\| {\ensuremath{\boldsymbol{m}}}^1 \|,
\ldots, \| {\ensuremath{\boldsymbol{m}}}^i \|, p_1, \| {\ensuremath{\boldsymbol{m}}}^{i + 1} \|, \ldots, \|
{\ensuremath{\boldsymbol{m}}}^s \|}_k {\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s}_{
{\ensuremath{\boldsymbol{q}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s} Q^{p_1}
\end{array}$$ Since $Q^{p_1} = ( p_1 + 1 )$ and $Q^{p_1 {\ensuremath{\boldsymbol{m}}}^i} = \| {\ensuremath{\boldsymbol{m}}}^i
\| Q^{{\ensuremath{\boldsymbol{m}}}^i}$, we get $$\tilde{R}^{{\ensuremath{\boldsymbol{p}}}}_k = \sum^{l ({\ensuremath{\boldsymbol{p}}}) - 1}_{s = 1}
\sum_{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{q}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s} V^{\|
{\ensuremath{\boldsymbol{m}}}^1 \|, \ldots, \| {\ensuremath{\boldsymbol{m}}}^s ||}_{k, p_1}$$ where $$V^{n_1, \ldots, n_s}_{k, p_1} = \sum_{i = 1}^s n_i\alpha^{n_1, \ldots, n_i +
p_1 \ldots, n_s}_k + ( p_1 + 1 ) \sum_{i = 0}^s \alpha^{n_1, \ldots,
n_i, p_1, n_{i + 1} \ldots, n_s}_k$$ but $$\begin{array}{rcl}
V^{n_1, \ldots, n_s}_{k, p_1} & = & {\displaystyle}\sum_{i = 1}^s n_i \sum_{t = 1}^s
C^{t_{}}_k \sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = ( n_1, \ldots, n_i + p_1 \ldots,
n_s )
\end{array}$}}} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )\\
& & {\displaystyle}+ ( p_1 + 1 ) \sum_{i = 0}^s \sum_{t = 1}^{s + 1} C^{t_{}}_k
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = ( n_1, \ldots, n_i, p_1, \ldots,
n_s )
\end{array}$}}} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )
\end{array}$$ In the first term, we get a sequence ${\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = (
n_1, \ldots, n_i + p_1 \ldots, n_s )$ starting with a decomposition ${\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^t = ( n_1, \ldots, n_s )$ and adding $p_1$ to one element of one of the sequences ${\ensuremath{\boldsymbol{m}}}^i$. In the second term, ${\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = ( n_1, \ldots, n_i, p_1, \ldots,
n_s )$, then either $p_1$ is one of the sequences ${\ensuremath{\boldsymbol{n}}}^1
\ldots {\ensuremath{\boldsymbol{n}}}^t$, and, once it is omitted, we get a decomposition ${\ensuremath{\boldsymbol{m}}}^1 \ldots {\ensuremath{\boldsymbol{m}}}^{t - 1} = ( n_1,
\ldots, n_s )$, either we start with a decomposition ${\ensuremath{\boldsymbol{m}}}^1 \ldots
{\ensuremath{\boldsymbol{m}}}^t = ( n_1, \ldots, n_s )$ and $p_1$ is inserted in one of the sequences ${\ensuremath{\boldsymbol{m}}}^i$ : If ${\ensuremath{\boldsymbol{n}}}=( n_1, \ldots, n_s )$, then, $$\begin{array}{rcl}
V^{n_1, \ldots, n_s}_{k, p_1} & = &{\displaystyle}\sum_{t \geq 1}^{} C^{t_{}}_k
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = {\ensuremath{\boldsymbol{n}}}
\end{array}$}}} \sum_{i = 1}^t \frac{\| {\ensuremath{\boldsymbol{n}}}^i \| ( \|
{\ensuremath{\boldsymbol{n}}}^i \| + 1 )}{\| {\ensuremath{\boldsymbol{n}}}^i \| + p_1 + 1} A
({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )\\
& &{\displaystyle}+ ( p_1 + 1 ) \sum_{t \geq 1}^{} C^{t + 1_{}}_k
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = {\ensuremath{\boldsymbol{n}}}
\end{array}$}}} \frac{t + 1}{p_1 + 1} A ({\ensuremath{\boldsymbol{n}}}^1,
\ldots,{\ensuremath{\boldsymbol{n}}}^t )\\
& &{\displaystyle}+ ( p_1 + 1 ) \sum_{t \geq 1}^{} C^{t_{}}_k
\sum_{{\text{\scriptsize $\begin{array}{c}
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = {\ensuremath{\boldsymbol{n}}}
\end{array}$}}} \sum_{i = 1}^t \frac{\| {\ensuremath{\boldsymbol{n}}}^i \| + 1}{\|
{\ensuremath{\boldsymbol{n}}}^i \| + p_1 + 1} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t )
\end{array}$$ But $$( p_1 + 1 ) C^{t + 1_{}}_k \frac{t + 1}{p_1 + 1} = C^t_k ( k - t )$$ and $$\frac{\| {\ensuremath{\boldsymbol{n}}}^i \| ( \| {\ensuremath{\boldsymbol{n}}}^i \| + 1 )}{\| {\ensuremath{\boldsymbol{n}}}^i
\| + p_1 + 1} + ( p_1 + 1 ) \frac{\| {\ensuremath{\boldsymbol{n}}}^i \| + 1}{\|
{\ensuremath{\boldsymbol{n}}}^i \| + p_1 + 1} = \| {\ensuremath{\boldsymbol{n}}}^i \| + 1$$ thus $$\begin{array}{rcl}
V^{n_1, \ldots, n_s}_{k, p_1} & = & {\displaystyle}\sum_{t \geq 1}^{} C^{t_{}}_k
\sum_{
{\ensuremath{\boldsymbol{n}}}^1 \ldots {\ensuremath{\boldsymbol{n}}}^t = {\ensuremath{\boldsymbol{n}}}
} A ({\ensuremath{\boldsymbol{n}}}^1, \ldots,{\ensuremath{\boldsymbol{n}}}^t ) \left( ( k - t )
+ \sum_{i = 1}^t ( \| {\ensuremath{\boldsymbol{n}}}^i \| + 1 ) \right) \\
& = & {\displaystyle}( n_1 + \ldots + n_s + k ) \alpha^{n_1, \ldots, n_s}_k
\end{array}$$ Now by induction we get, if ${\ensuremath{\boldsymbol{p}}}= p_1 {\ensuremath{\boldsymbol{q}}}$, $$\begin{array}{rcl}
\tilde{R}^{{\ensuremath{\boldsymbol{p}}}}_k & = & {\displaystyle}\sum^{l ({\ensuremath{\boldsymbol{p}}}) - 1}_{s = 1}
\sum_{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s}
{\ensuremath{\operatorname{sh}}}^{{\ensuremath{\boldsymbol{m}}}^1, \ldots,{\ensuremath{\boldsymbol{m}}}^s }_{
{\ensuremath{\boldsymbol{q}}}}
Q^{{\ensuremath{\boldsymbol{m}}}^1} \ldots Q^{{\ensuremath{\boldsymbol{m}}}^s} ( \|
{\ensuremath{\boldsymbol{q}}} \| + k ) \alpha^{\| {\ensuremath{\boldsymbol{m}}}^1 \|, \ldots, \|
{\ensuremath{\boldsymbol{m}}}^s \|}_k\\
& = &{\displaystyle}( \| {\ensuremath{\boldsymbol{q}}} \| + k ) \tilde{R}^{{\ensuremath{\boldsymbol{q}}}}_k\\
& = &{\displaystyle}( \| {\ensuremath{\boldsymbol{q}}} \| + k ) R^{{\ensuremath{\boldsymbol{q}}}}_k\\
& = &{\displaystyle}( \| {\ensuremath{\boldsymbol{q}}} \| + k ) \prod_{i = 2}^{l ({\ensuremath{\boldsymbol{q}}}) + 1} (
\hat{q}_i + k )\\
& = &{\displaystyle}\prod_{i = 1}^{l ({\ensuremath{\boldsymbol{q}}}) + 1} ( \hat{q}_i + k )\\
& = &{\displaystyle}\prod_{i = 2}^{l ({\ensuremath{\boldsymbol{p}}}) + 1} ( \hat{q}_i + k )\\
& = &{\displaystyle}R^{{\ensuremath{\boldsymbol{p}}}}_k
\end{array}$$
We live the second proof of theorem \[th3\] to the reader : the ideas are the same, noticing that $$\begin{array}{rcl}
S ( \Gamma_n ) & = & {\displaystyle}\sum_{( n_1, \ldots, n_s ) \in \mathcal{N}_n}
Q^{n_1, \ldots, n_s} S ( Z^{n_1, \ldots, n_s} )\\
& = & {\displaystyle}\sum_{( n_1, \ldots, n_s ) \in \mathcal{N}_n} ( - 1 )^s Q^{n_1,
\ldots, n_s} Z^{n_s, \ldots, n_1}
\end{array}$$
Tables and conclusion.
======================
Some computations give the following tables.
The coproduct
-------------
The table gives the value of $\frac{n!}{n_1 ! \ldots n_{s + 1} !} \alpha^{n_1,
\ldots, n_s}_{n_{s + 1}}$ for a given sequence $( n_1, \ldots, n_{s + 1} )$ :
-------------------------------------------------------------------------------------------------
$( 1, 1 ) = 1$
------------------------ ----------------------- ------------------------- ----------------------
$( 1, 2 ) = 3$ $( 2, 1 ) = 1$ $( 1, 1, 1 ) = 1$
$( 1, 3 ) = 6$ $( 2, 2 ) = 4$ $( 3, 1 ) = 1$ $( 1, 1, 2 ) = 7$
$( 1,2, 1 ) = 3 / 2$ $( 2, 1, 1 ) = 3 / 2$ $( 1, 1, 1, 1 ) = 1$
$( 1, 4 ) = 10$ $( 2, 3 ) = 10$ $( 3, 2 ) = 5$ $( 4, 1 ) = 1$
$( 1,1, 3 ) = 25$ $( 1, 3, 1 ) = 2$ $( 3, 1, 1 ) = 2$ $( 1, 2,
2 ) = 25 / 2$
$( 2, 1, 2 ) = 25 / 2$ $( 2, 2, 1 ) = 3$ $( 1, 1, 1, 2 ) = 15$ $( 1, 1, 2, 1 ) = 2$
$( 1, 2, 1, 1 ) = 2$ $( 2, $( 1, 1, 1, 1, 1 ) = 1$
1, 1, 1 ) = 2$
-------------------------------------------------------------------------------------------------
This gives $$\begin{array}{ccc}
\tilde{\Delta} \Gamma_1 & = & 0\\
\tilde{\Delta} \Gamma_2 & = & \Gamma_1
\otimes \Gamma_1 \\
\tilde{\Delta} \Gamma_3 & = & ( \Gamma_2 + \Gamma_1^2 )
\otimes \Gamma_1 + 3 \Gamma_1 \otimes \Gamma_2 \\
\tilde{\Delta} \Gamma_4 & = & ( \Gamma_3 + 3 \Gamma_1^{}
\Gamma_2 + \Gamma_1^3 ) \otimes \Gamma_1 + ( 4 \Gamma_2 + 7 \Gamma_1^2 )
\otimes \Gamma_2 + 6 \Gamma_1 \otimes \Gamma_3 \\
\tilde{\Delta} \Gamma_5 & = & ( \Gamma_4 + 4
\Gamma_1
\Gamma_3 + 3 \Gamma_2^2 + 6 \Gamma^2_1 \Gamma_2 + \Gamma_1^4 ) \otimes
\Gamma_1\\
& & + ( 5 \Gamma_3 + 25 \Gamma_1^{} \Gamma_2 + 15 \Gamma_1^3 ) \otimes
\Gamma_2 + ( 10 \Gamma_2 + 25 \Gamma_1^2 ) \otimes \Gamma_3 + 10 \Gamma_1
\otimes \Gamma_4
\end{array}$$
The antipode
------------
The table gives the value of $\frac{( n_1 + \ldots + n_s ) !}{n_1 ! \ldots n_s
!} \beta^{n_1, \ldots, n_s}_{}$ for a given sequence $( n_1, \ldots, n_s )$ :
---------------------------------------------------------------------------------------------------------------
$( 1 ) = - 1$
---------------------------- -------------------------- --------------------------- ---------------------------
$( 2 ) = - 1$ $( 1, 1 ) = 1$
$( 3 ) = - 1$ $( 1, 2 ) = 3$ $( 2, 1 ) = 1$ $( 1, 1, 1 ) = - 2$
$( 4 ) = - 1$ $( 1, 3 ) = 6$ $( 2, 2 ) = 4$ $( 3, 1 ) = 1$
$( 1, 1, 2 ) = - 11$ $( 1, 2, 1 ) = - 9 / 2$ $( 2, 1, 1 ) = - 5 / 2$ $( 1, 1, 1, 1 ) = 6$
$( 5 ) = - 1$ $( 1, 4 ) = 10$ $( 2, 3 ) = 10$ $( 3, 2 ) = 5$
$( 4, 1) = 1$ $( 1, 1, 3 ) = - 35$ $( 1, 3, 1 ) = - 8$ $( 3, 1,
1 ) = - 3$
$( 1, 2, 2 ) = - 55 / 2$ $( 2, 1, 2 ) = - 35 / 2$ $( 2, 2, 1 ) = - 7$
$( 1, 1, 1, 2 ) = 50$ $( 1, 1, 2, 1 ) = 22$ $( 1, 2, 1, 1 ) = 29 / 2$ $( 2, 1, 1, 1 ) = 19 / 2$
$( 1, 1, 1, 1, 1 ) = - 24$
---------------------------------------------------------------------------------------------------------------
This gives : $$\begin{array}{lll}
S ( \Gamma_1 ) & = & - \Gamma_1\\
S ( \Gamma_2 ) & = & - \Gamma_2 + \Gamma_1^2\\
S ( \Gamma_3 ) & = & - \Gamma_3 + 4 \Gamma_1 \Gamma_2 - 2 \Gamma_1^3\\
S ( \Gamma_4 ) & = & - \Gamma_4 + 7 \Gamma_1 \Gamma_3 + 4 \Gamma_2^2 - 18
\Gamma_1^2 \Gamma_2 + 6 \Gamma_1^4\\
S ( \Gamma_5 ) & = & - \Gamma_5 + 11 \Gamma_1 \Gamma_4 + 15 \Gamma_2
\Gamma_3 - 46 \Gamma_1^2 \Gamma_3 - 52 \Gamma_1 \Gamma_2^2 + 96
\Gamma_1^3 \Gamma_2 - 24 \Gamma_1^5
\end{array}$$
This is the attempted result but the formulas in proposition \[prop7\], theorem \[th2\] and \[th3\] are not unique because $\mathcal{H}^1_{{\ensuremath{\operatorname{CM}}}}$ is commutative and, in the computations, it is much more ”simple” to consider that the algebra generated by the $\delta_n$ is somehow noncommutative. This situation calls for furthers investigations, since the coefficients appearing in proposition \[prop7\] for the Faà di Bruno coordinates seem to arise in the study of a noncommutative version of diffeomorphisms (see [[@fra]]{}).
|
Yu.A. BAUROV[^1]
*Central Research Institute of Machine Building,*
141070, Korolyov, Moscow Region, Russia
Yu.G. SOBOLEV[^2], V.F. KUSHNIRUK and E.A. KUZNETSOV
*Flerov Laboratory of Nuclear Reactions (FLNR),*
Joint Institute for Nuclear Research,
141980, Dubna, Moscow Region, Russia
A.A. KONRADOV[^3]
*Russian Academy of Sciences, Institute of Biochemical Physics,*
117977, Moscow, Russia
ABSTRACT
[The experimental data on continuous investigation of changes in $\beta$-decay count rate of ${}^{137}Cs$ and ${}^{60}Co$ from 9.12.98 till 30.04.99, are presented. The 27-day and 24-hour periods in these changes, inexplicable by traditional physics, have been found.]{} PACS numbers: 24.80+y, 23.90+w, 11.90+t In Refs. \[1-3\], periodic variations in $\beta$-decay rate of ${}^{60}Co$, ${}^{137}Cs$, and ${}^{90}Sr$, have been first discovered. An analysis of the 24-day period in $\beta$-decay of radioactive elements as well as of the daily rotation of the Earth in various seasons of the year has led to selection of some spatial direction characterized by the fact that near the points of the Earth’s surface where the latitude tangent line to a parallel passes through this direction, the decay count rate of radioactive elements changes. The main drawback of the experiments \[1-3\] was that their final results gave no possibility to clearly understand what was an effect of the “internal life” of the setup itself and what was due to the phenomenon of interest. In addition, the duration of these experiments was no more than three weeks, which did not allow to analyze long-period harmonics.
The aim of this paper is to find an answer to the above questions, using measurements of flux of $\gamma$-quanta in the process of $\beta$-decay of radioactive elements as in ref. [@1]. The experimental setup (Fig.1) consisted of three scintillation detectors, two of them being standard spectrometric scintillation detecting units [**BDEG2-23**]{} on the basis of [**NaI**]{}($Tl$)-scintillator (63mm in diameter, 63mm in height) and [**FEU-82**]{} photomultiplier ([**PM**]{}) with standard divider. One of these units was used to indicate the background radiation, and the second one was to present $\gamma$-radiation of ${}^{137}Cs$. The third detector was a [**BGO**]{}-scintillator (46mm in diameter, 60mm in height) and a [**FEU-143**]{} photomultiplier with standard divider. This detector was used to display the $\gamma$-radiation of a ${}^{60}Co$-source.
To diminish the influence of magnetic fields on the [**PM**]{}s, the detectors were placed into protecting screens made as cilynders from ten sheets of annealed permalloy 0.5 mm in thickness. The internal diameter of the cylinders was equal to 10 cm, and the height was 70 cm.
The detectors were placed in such a manner that the photocathodes of [**PM**]{}s were at a distance of one-half of height of the cylinder. The $\gamma$-sources were placed just on the end face surface of the scintillators through the center of the input window. All the detectors and the temperature-sensitive-element were positioned inside the metallic cube ($40\times40\times50$ cm$^3$) used as an additional magnetic shield. The thickness of the steel walls of the cube was equal to 3 mm. The detectors with $\gamma$-sources were surrounded by lead protection 5 cm in thickness. The system of registering information consisted of two subsystems. The first one was designed for accumulating information on the counting rate in ten-second intervals from scintillation detectors as well as on the temperature, power-source voltages (high voltage of [**PM**]{}s, voltage of [**CAMAC**]{} $= 6V, 24V$) and impulse noise of crate power supply. The second subsystem of information storage were made to record “marked” energetic distributions from scintillation detectors for the purpose of checking the stability of their amplitude distribution parameters (stability of discriminator thresholds, shape of amplitude distributions etc.). The set included three identical spectrometric registering sections (see Fig.1). Each section consisted of a preamplifier ([**PA**]{})-emitter follower matching the impedances, spectrometric amplifier ([**AFA**]{}) with active filters having shaping time constants $T_{int} = T_{dif} = 0.25 \mu s$ [@4], and system of fast discriminators ([**FD**]{}) of negative output signals of the amplifier and counters of gated pulses ([**SC**]{}). In addition, the positive output signal of the spectrometric amplifier [**AFA**]{} from each registering channel was fed to analog-to-digital converters ([**ADC**]{}) [@5] placed in a separate crate. To increase reliability of spectrometric section, all variable resistors in which, as almost 20-year operating experience has shown, sometimes the contact arm faults take place (when continuously adjust amplification in [**AFA**]{} and the threshold in [**FD**]{}), were replaced by the fixed resistors. The [**PM**]{}s of all sections had the general high-voltage power supply. In long-term experiments, the most important requirement upon the measuring system is the possibility of continuous control over its parameters as for detecting non-stable elements, units, and connections, so to refine possible correlations of measurable quantities with the environmental parameters. The experimental setup was powered from separate terminals of distributing board for diminishing the possible influence of additional parallel loads in the power network.
To monitor the temperature of the environment, a thermometric channel with high-sensitive temperature element and amplifier module was used. This element was made on the basis of assembly of semiconductor diodes with the summary thermoelectric coefficient about 10 mV/degree. The amplifier module gave stable bias current for the temperature-sensitive element and additionally amplified the signal up to the summary termoelectric coefficient of the measuring channel of 100 mV/degree. In the same module a transformer of voltage from high-voltage power supply of the scintillation units into low voltage for [**8ADC**]{} (see below) was arranged. The transformation ratio was about 3.3 V/kV.
In the measuring crate with the counters, amplifiers [**AFA**]{}, and the amplifier module of the thermometric channel, we have placed also a multichannel amplitude-to-digital converter [**8ADC**]{} for measuring high voltage ([**HV**]{}) of the scintillation detectors and monitoring the secondary power voltages $\pm 24V, \pm 6V$ of the crate [**CAMAC**]{} itself, as well as a special module to register the impulse noise of these secondary power sources. Any impulse input in the crate power line with an amplitude more than 10mV recorded “1” into the corresponding information bit of the word register of module data. The frequency spectrum of recorded impulse signals extended from tens Hz to several MHz. Thus we recorded impulse noise of the crate along with monitoring levels of constant high voltages of the power source of the scintillation units as well as low voltages of the crate power sources.
The start of measuring cycle and quantization of exposure time in the first recording subsystem were organized by a “Master-Trigger” [**MT1**]{}. It comprised a pulser with quartz stabilization of frequency of output pulses ([**QUARTZ**]{}) and a scaling circuit. Each cycle of measurements in the experiment started with generation of a ten-second exposure signal ([**GATE**]{}) by the unit [**MT1**]{}. This signal opened all counters of the setup ([**SC1-SC6**]{}). After the ten-second signal of exposure of the counters, [**MT1**]{} elaborated a signal “[**LAM**]{}” for the controller [**CC1**]{} of the measuring crate to organize a cycle of interrogation of the crate recorders and transmission of date to the storage [**PC1**]{}. The data file, transmitted to [**PC1**]{} in each interrogation cycle, included the following data words: - to 12.5cm[the number of readings in the counters [**SC1-SC6**]{} $8\times16$ bits]{},
- to 12.5cm[the codes of voltages of [**CAMAC**]{} sources $\pm 6V, \pm 24V$ $4\times15$ bits]{},
- the code of voltage of the high-voltage power source
to 13.5cm[of the scintillation units 15 bits,]{}
- to 12.5cm[the code of the recorder of impulse noise 4 bits.]{} The 15-digit codes with [**8ADC**]{} contained 12 bits of the voltage code and 3 bits of the channel number.
The characteristics of the sections: a) *Sensitivity (the exposure time 10 s with an accuracy of $10^{-6}$ s):*
---------------------- --- ----------------------------
“$\pm 6V$” = 5mV per channel;
“High-voltage power” = 750 mV per channel;
“$\pm 24V$” = 12.5mV per channel;
“Temperature” = $1^\circ$ for 40 channels.
---------------------- --- ----------------------------
b\) *Thresholds of the section ${}^{137}Cs$ [**NaI**]{}($Tl$) (calibration against $\gamma$-lines 662 keV, 1173 keV, 1332 keV):*
-------------------------------- --- ----------
the “low” threshold = 7 keV;
the threshold “under the peak” = 425 keV;
the threshold “on the peak” = 657 keV.
-------------------------------- --- ----------
c\) *Thresholds of the background section [**NaI**]{} ($Tl$) (calibration against $\gamma$-lines 662 keV, 1173 keV, 1332 keV):*
--------------------- --- ---------
the “low” threshold = 11 keV.
--------------------- --- ---------
d\) *Thresholds of the ${}^{60}Co$ [**BGO**]{}-section (calibration against $\gamma$-lines 662 keV, 1173 keV, 1332 keV):*
-------------------------------- --- ----------
the “low” threshold = 35 keV;
the threshold “under the peak” = 745 keV.
-------------------------------- --- ----------
The start of measurements in the second recording subsystem was organized by the “Master-Trigger” [**MT2**]{} from any signal of the discriminators [**FD1-FD6**]{} (chosen by the experimenter by way of switching from one channel to another in the module [**M**]{}). The unit [**MT2**]{} opened by its [**GATE**]{}-pulse the amplitude-code converters [**ADC1-ADC3**]{}, “spectrum mark” counter [**SC7**]{}, and triggered the cycle of recording information into the storage computer [**PC2**]{} after the time of amplitude-digital code transformation. The [**GATE**]{}-pulses from the [**MT1**]{} unit of the first recording subsystem were fed to the counter [**SC7**]{} input. Thus the counter gave information on numbers of ten-second exposure intervals of the first subsystem. This allowed to perform analysis (in “off-line” mode) of the amplitude distribution parameters of the chosen channel of recording in any combination of ten-second exposures.
The long-term dynamics of the radioactive decay of ${}^{137}Cs$ and ${}^{60}Co$ over the period from 9 December 1998 till 30 April 1999, was measured. The above described setup made it possible to perform precision measurements with monitoring parameters of the system at the different discrimination thresholds of decay energy. The spectra in the channels for ${}^{137}Cs$ are presented in Figs.2-4 with the corresponding thresholds. As an example, in Fig.5 the results of measurements over two-week time interval at the end of March, 1999, are shown, for 7 main variants of channels.
---------------------- ----------------------------------------------------------------------------
Variants of channels Measurements
1. [**BGO**]{}, the threshold of Fig.3-type;
2. [**BGO**]{}, the threshold of Fig.2-type;
3. [**NaI**]{}$^1$ with the threshold in Fig.3;
4. [**NaI**]{}$^1$ with the threshold in Fig.4;
6. [**NaI**]{}$^1$ with the threshold in Fig.2;
12. Internal temperature of the setup;
13. High voltage ([**HV**]{}) in channels [**NaI**]{}$^{1,2}$ and [**BGO**]{}.
---------------------- ----------------------------------------------------------------------------
In the present paper we shall analyze only the channel 6 corresponding to the minimum threshold of discrimination at which only low-energetic noise component was cut off, and the channel 12. From Fig.5 one can conclude that the channel with the low discrimination was the most stable though with a remarkable local dispersion: the data densely fill a relatively broad band.
The starting series have more than $1.2\times 10^6$ points in summary length over the whole time interval of observation. Each point corresponds to a ten-second interval of decay number accumulation. Hence, the summary duration of continuous measurements was $\sim 3347$ hours, or $\sim 140$ days.
When analyzing the periodical structure of the series we were interested in periods no shorter than several hours. In Fig.6 the results of normalized the starting series (i.e. reducing to the interval \[0,1\]) averaging over one-hour period, are given. With such hourly averaging, the “fast” component of dispersion disappeared, and the slow dynamic of the process was clearly seen. It is also clear from the Figure that the temperature inside the setup varied in antiphase with the count rate. This is well seen in the whole long series and, partially, in Fig.5. The cross-correlation function of these two series has a sharp minimum approaching -0.95 at the zero lag. This allowed us to take into account the temperature dependence of count rate measurements by way of simple addition of two normalized series. The Fourier-analysis (fast Fourier transformation - [**FFT**]{}) of the final temperature-compensated series have revealed two distinctly distinguishable periods. In Fig.7 a pronounced 27-day period is seen that may be caused, for example, by the influence of the Sun’s rotation around its axis (the synodic period of the Sun’s rotation relative to the Earth is equal to 27.28 days). In the hour-scale of the periods in Fig.8, a 24-hours period is well marked. It should be emphasized that this daily period is absent in the spectrum of the dynamic of the temperature itself (see Fig.9) and is found only in the dynamic of the radioactive decay, so that it can have an external cosmic reason, too.
Now let us consider the statistics of extremum values of the series of measurements. Evaluate more accurately the extent of nonuniformity of distribution of extremum values for the starting (10-second) series of observations in the low-threshold channel over the time of astronomical day. This procedure was described earlier [@3]. Here we give its brief presentation.
Under an extremum we mean here a value for which the modulus of difference with the average for the whole series is no lesser then two standard deviations. Ascribing to each extremum value that instant of day time at which this extremum was observed we shall have the resulting set of time instants in the interval from 0 till 24 hours when “jumps beyond two sigmas” were measured. The “null hypothesis” consists in that the extremum events occur with equal frequency in any time of day, i.e. the distribution of these instances is uniform over the day cycle. The hypothesis of uniformity of distribution can now be validated, for example, by the Kolmogoroff-Smirmoff’s test. In Figs.10 and 11 the results of computations are presented. The time of day laid as abscissa is expressed in degrees ($0-360^\circ$).
As a reference point, the time from beginning of observations is taken (the start on 9 Dec. 1998 at $23^h$ of astronomical time - the local time. The whole time of experiment was divided for this analysis into exact decades in days). The values of difference between the sample and uniform distribution functions for each moment of day time are plotted as ordinates (in degrees). With the dashed line the confidence levels of Kolmogoroff-Smirnoff’s criterion ($P<0.05$) are shown. An exit beyond these limits denotes a significant difference of the distribution from the uniform one, and the maximum point indicates the time of day (phase) when this nonuniformity was maximum. In Fig.10 the results for the maximum values, and in Fig.11 for minimum values are given.
The existence of reliable nonuniformity denotes presence of a daily period in the statistics of extremum values of the radioactive decay. A knowledge of phase (moment of maximum nonuniformity) as well as relation to the absolute time (from the beginning of the series) allows us to determine possible cosmic references connected with such a nonuniformity.
The analysis of the extremum jumps have shown that they overlie tangent lines to the Earth’s parallels making an angle of $\pm(35^\circ-45^\circ)$ with the direction having the right ascension coordinate $\alpha \approx 275^\circ$ that insignificantly ($\sim 5^\circ$) differs from the direction fixed in Refs.\[1-3,6\]. It should be noted also that, as background measurements in the channel 5 have shown (the flux of particles in this channel was no more then 50 particles per ten second), the oscillations of the background (due to the smallness of its flux as compared with that ($\sim 300$ per second) going beyond the scope of $2\sigma$) by no means could influence on the distribution of temporal coordinates of the extremum points.
[confr]{}
Yu.A.Baurov, V.L.Shutov, Prikladnaya Fizika, [**1**]{}, 1995, p.40 (in Russian).
Yu.A.Baurov, A.A.Konradov, V.F.Kushniruk, Yu.G.Sobolev. Scientific Report 1995-1996, “Heavy ion Physics”, [**E7-97-206**]{}, p.354-355, Dubna.
Yu.A.Baurov, A.A.Konradov, V.F.Kushniruk, Yu.G.Sobolev. Global anisotropy of space and experimental investigation of changes in $\beta$-decay count rate of radioactive elements, E.-print [**hep-ex/9809014**]{}, 16 Sep. 1998.
A.N.Kuznetsov, V.G. Subbotin, In the book: X Intern. Symp. on Nucl. Electronics, [**zfk-433**]{}, v.1, p.148-151, Dresden., 1981.
A.N.Kuznetsov, V.G. Subbotin, Preprint JINR, [**13-83-67**]{}, Dubna.
Yu.A.Baurov, Structure of physical space and new interaction in nature (theory and experiments), poster report on the International Workshop “Lorentz Group, CPT and Neutrinos” (Zacatecas, June 23-26, 1999, Mexico), E.-print [**hep-ph/9907239**]{}, 5 July 1999.
Fig.3.
Fig.4.
Fig.5.
Fig.6.
Fig.7.
Fig.8.
Fig.9.
Fig.10.
Fig.11.
[^1]: alex@theor.phys.msu.su
[^2]: ak@sky.chph.ras.ru
[^3]: sobolev@main1.jinr.dubna.su
|
---
abstract: 'We study the problem for the optimal charge distribution on the sites of a fixed Bravais lattice. In particular, we prove Born’s conjecture about the optimality of the rock-salt alternate distribution of charges on a cubic lattice (and more generally on a $d$-dimensional orthorhombic lattice). Furthermore, we study this problem on the two-dimensional triangular lattice and we prove the optimality of a two-component honeycomb distribution of charges. The results holds for a class of completely monotone interaction potentials which includes Coulomb type interactions. In a more general setting, we derive a connection between the optimal charge problem and a minimization problem for the translated lattice theta function.'
author:
- Laurent Bétermin
- Hans Knüpfer
bibliography:
- 'Bornconject.bib'
title: 'On Born’s conjecture about optimal distribution of charges for an infinite ionic crystal'
---
**AMS Classification:** Primary 49S99 Secondary 82B20\
**Keywords:** Calculus of variations; Lattice energy; Theta functions; Electrostatic energy; Ewald summation.\
Introduction and setting
========================
Introduction {#ss-intro}
------------
Ionic compounds are substances formed by charged ions, held together by electrostatic forces. The ions are typically aligned in regular crystalline structures, in an arrangement that minimizes the total interaction energy between the positive ions ([*cations*]{}) and negative ions ([*anions*]{}). A large class of such materials are salts, formed by a reaction of an acid and a base. The material properties of these ionic compounds such as their high melting point and their brittleness is determined by their specific lattice structure and the distribution of charges within the lattice. A variety of crystal lattice structures are observed as a function of the relative quantity and size of the ions. While prediction of the expected lattice structure have been made using contact number calculations [@Pauling], in general, the crystallization problem for ionic bounds has not been solved. Indeed, to investigate stability of ionic lattice structures, the total interaction energy between the particles needs to be calculated and compared for all possible lattice configurations and distributions of the ions on these lattices. In this paper, we consider the simpler question for optimizing the charge distribution on a given crystal lattice. In particular, we prove Born’s conjecture about the optimal charge distribution for charges located at the sites of a cubic or orthorhombic lattice.
The question asked by Born in [@Born-1921] (as recalled in [@Latticesums]) is the following:
“*How to arrange positive and negative charges on a simple cubic lattice of finite extent so that the electrostatic energy is minimal?’[^1]*
His conjecture is that the alternate distribution of charges, i.e. when $(-1)^{m+n+k}$ is the charge at the point $p=(m,n,k)\in \Z^3$, is the global minimizer of the electrostatic energy among all the distributions of charges with prescribed total charge. In [@Born-1921], Born proved this conjecture in dimension $d=1$ and he obtained the local minimality of the alternate structure in dimension $d=3$. In this paper, we prove Born’s conjecture in a general setting of $d$-dimensional lattices and for a large class of interaction energies. We further derive a connection of this problem to a minimization problem for the translated lattice theta function associated with the dual of the given lattice.
![Optimal charge distribution for the orthorhombic lattice for $d = 3$[]{data-label="fig-cubecharged"}](Salt.png){width="8cm"}
We consider an ensemble of charges $\phi_x$ located at the vertices $x \in X$ of a given $d$-dimensional Bravais lattice $X \subset \R^d$. For technical reasons, we also assume that the charges are $N$-periodic in each principal lattice direction for some $N \in \N$ and can be represented by the finite sublattice $K_N$ (for the precise definitions, we refer to the next section). The total interaction energy per lattice point can then be written as $$\begin{aligned}
\label{defenergyintro} \EE_{X,f}[\phi] \ = \frac{1}{2N^d} \sum_{y\in K_N} \sum_{x\in X\BS \{0\} } \phi_y
\phi_{x+y}f(x),\end{aligned}$$ for some radially symmetric interaction potential $f : \R^d \to [0,\infty)$. In the case, when $f$ is not absolutely summable on $X \BS \{ 0 \}$, the classic method of Ewald summation [@Ewald1] is used to give a meaning to the infinite sum in . The class of interaction potentials $f$ we consider in particular includes all potentials $f(x) = F(|x|^2)$ for some completely monotone function $F$. In particular, this includes all Riesz potentials of the form $f(x)=|x|^{-s}$ for some $s > 0$. We consider the minimization problem $$\begin{aligned}
\phi \mapsto \EE_{X,f}[\phi]\end{aligned}$$ for all periodic charge configurations satisfying a constraint for the total charge (see ) and for any given Bravais lattice $X\subset \R^d$.
In our first result, we show that this minimization problem is related to a minimization of the translated lattice theta function $$\begin{aligned}
\label{thet-1} z\mapsto \theta_{X^*+z}(\alp) = \sum_{p\in X^*} e^{-\pi \alp |p+z|^2}, \end{aligned}$$ associated to the dual lattice $X^*$ in terms of the variable $z\in \R^d$ for given $\alp > 0$ (see Theorem \[thm1\]). At the core of our proof lies an argument, originally due to Montgomery [@Mont] and generalized by one of the authors in [@BeterminPetrache]. Theorem \[thm1\] can be used to calculate the optimal charge distribution in specific Bravais lattices if the minimization problem for the translated lattice theta function can be solved for these lattices. We first consider the situation of a $d$-dimensional orthorhombic lattice. In this case, the minimizer $z$ of is the center of the primitive cell (for any $\alp$) [@BeterminPetrache]. For the triangular lattice case, the minimizers are the two barycenters of the primitive triangles forming the primitive rhombic cell [@Baernstein-1997]. In both case, the knowledge of these minimizers gives us the minimal configuration of charges, that are the alternate rock salt configuration in the orthorhombic case (Theorem \[thm-2\]), see Fig. \[fig-cubecharged\], and the honeycomb distribution for the triangular lattice (Theorem \[thm-3\]), see Fig. \[fig-triangcharged\]. In particular, Theorem \[thm-2\] gives an affirmative answer to Born’s conjecture, cited above. Let us note that, in the case of Riesz potentials which are not summable over the lattice, also the analytic continuation of Epstein’s zeta function has been used to describe the lattice energies in this case, see e.g. [@Emersleben]. Indeed, this approach yields the same energy as the Ewald method used in our approach.
We note that the translated lattice theta function appears in several mathematical models for physical systems with different kind of particles. For example, Ho and Mueller [@Mueller:2002aa] wrote the interaction between two Bose-Einstein condensates in terms of translated lattice theta functions, and the same is done by Trizac et al. [@Samaj12; @TrizacWigner16] in the context of Wigner bilayers. Mathematically, the problem of minimizing $z\mapsto \theta_{X+z}(\alp)$ was studied by Baernstein in [@Baernstein-1997] for $X=\Lambda_1$ a two-dimensional triangular lattice and by the first author in [@BeterminPetrache] for more general lattices.
Let us also recall related work for optimal lattice configurations for systems with the same kind of particles. The studies of lattice theta functions and Epstein zeta functions are originally due to Krazer and Prym [@KrazerPrym] and Epstein [@Epstein1]. Later, the problem of minimizing the Epstein zeta function among two-dimensional Bravais lattices with a fixed density was studied by Rankin [@Rankin], Ennola [@Eno2], Cassels [@Cassels] and Diananda [@Diananda] (see also the recent review [@Henn]). They proved the optimality of the triangular lattice (also called Abrikosov lattice in the context of superconductivity). Montgomery [@Mont] proved the optimality of the triangular lattice, among Bravais lattices with a fixed density, for the lattice theta function $X\mapsto \theta_X(\alp)$, see also [@NonnenVoros Appendix A.2.]. This result and its consequence [@CohnKumar] has been used to show that the triangular configuration is a ground state of total interaction energy for a class of interaction potentials. Let us cite the works of Sandier and Serfaty [@Sandier_Serfaty] on Coulomb gases and superconductivity and their consequences for the logarithmic energy on the $2$-sphere in [@Betermin:2014rr], but also the work of Aftalion, Blanc and Nier [@AftBN] on Bose-Einstein Condensates and that of Nonnenmacher-Voros [@NonnenVoros] on chaotic maps over a torus phase space. Furthermore, Montgomery’s result was used by the first author and Zhang in [@Betermin:2014fy; @BetTheta15] in order to prove the optimality of the triangular lattice at high density for more general interaction energies. In dimensions three, as recalled in [@Blanc:2015yu Section 2.5], the proof of the Sarnak-Strömbergsson conjecture [@SarStromb] about the optimality, for the lattice theta function, of the Body-Centered-Cubic (resp. Face-Centered-Cubic) lattice at high (resp. low) density , would be an important advance both in analytic number theory and in solid-state physics, see also [@SerfRoug15]. Some recent advances have been made by the first author in [@BeterminPetrache; @Beterminlocal3d], using recent results about Jacobi theta functions in [@Faulhuber:2016aa]. In higher dimensions, the local minimality of some lattices for the lattice theta function, the Epstein zeta function and other related lattice energies was studied by Coulangeon et al. in [@Coulangeon:kx; @Coulangeon:2010uq; @CoulLazzarini]. The above investigation are made under the assumption that the configuration can be expressed as a Bravais lattice. In a different approach without periodicity assumptions optimal lattice configurations have been studied e.g. in [@VN1; @VN2; @Rad2; @Rad3; @Crystal; @TheilFlatley; @ELi; @Stef1; @Stef2; @Luca:2016aa].
**Structure of the paper:** In the remaining parts of the first section, we introduce the mathematical formulation for the model, introduce some needed special functions and present useful identities for those. In Section \[sec-main\], we state our main results in Theorems \[thm1\]–\[thm-5\]. The proofs of these theorems are given in Section \[sec-proofs\].
**Notation:** We will write $(e_i)_{1\leq i\leq d}$ for the canonical basis of $\R^d$. For any $x,y \in \R^d$, we denote the Euclidean scalar product by $x\cdot y$. We also use the notation ${\ensuremath{\llbracket a, b \rrbracket}} := [a,b] \cap \Z$.
We recall that a Bravais lattice in $\R^d$ is a set points of the form $X=\bigoplus_{i=1}^d \Z u_i \subset \R^d$ for a given set of linearly independent vectors $u_i \in \R^d$ with $i \in {\ensuremath{\llbracket 1, d \rrbracket}}$. We call $A_X$ the generator matrix of the lattice $X$, i.e. $A_X \Z^d=X$. The associated quadratic form assigned with the Bravais lattice is given by $q_X(n) = \NNN{\sum_{i=1}^d n_i u_i}{}^2$ for $n \in \Z^d$. The dual lattice $X^*$ of the lattice $X$ is given by $$\begin{aligned}
X^*=\left\{ p \in \R^d \ : \ p\cdot x\in \Z \text{ for all $x\in X$}
\right\}.
\end{aligned}$$
The model {#ss-setting}
---------
We consider configurations of charges located on the sites of $d$-dimensional Bravais lattices $X \in \R^d$ for any $d \geq 1$. We will assume without loss of generality that all considered Bravais lattices have unit density, i.e. the unit cell $Q:= \sum_{i=1}^d [0,1) u_i$ of the lattice has unit volume. The general case can be recovered by rescaling the lattice.
We consider charged lattices, where a charge is assigned to every lattice point:
\[def-lattice\] Let $d \geq 1$.
1. A charged lattice $L = (X,\phi)$ is a Bravais lattice $X = \bigoplus_{i=1}^d \Z u_i$ together with a function $\phi : X \to \R$ such that for any $x\in X$, the point $x$ has charge $\phi_x = \phi(x)$.
2. We say that the charge distribution is $N$-periodic if it is periodic with period $N$ in any coordinate direction, i.e. $$\begin{aligned}
\phi(x+ N u_i) = \phi(x) &&\text{for any $x \in X$ and any $i\in {\ensuremath{\llbracket 1, d \rrbracket}}$}.
\end{aligned}$$ The charge distribution is periodic if it is periodic for some $N \in \N$. We define the finite sublattice $K_N\subset X$ by $$\begin{aligned}
\label{def-KN}
K_N \ := \ \Big\{x=\sum_{i=1}^d m_i u_i\in \Z^d \ : \ i \in {\ensuremath{\llbracket 1, d \rrbracket}} , m_i\in {\ensuremath{\llbracket 0, N-1 \rrbracket}} \Big\},\end{aligned}$$ and we call $K_N^*$ the corresponding sublattice in $X^*$.
3. The space of $N$-periodic functions on the lattice is denoted by $\Lam_N(X)$. It is equipped with inner product and norm by $$\begin{aligned}
\skpL{\phi}{\psi}{K_N} = \sum_{y \in K_N} \phi(y) \overline{\psi(y)},&& \NNN{\phi}{} = \sqrt{(\phi,\phi)_{K_N}}\ .\end{aligned}$$
Note that an $N$-periodic charge configuration is uniquely given by its values on the $N^d$ points on the finite sublattice $K_N$.
We will consider the following class of potentials:
\[def-FF\] For $d\geq 1$, we say that $f\in \FF$ if $f:\R^d\to [0,+\infty)$ and if, for any $x\in \R^d \BS \{ 0 \}$, we have $$\begin{aligned}
\label{FF-2} f(x)=\int_0^{\infty} e^{-|x|^2t} d\mu_f(t),
\end{aligned}$$ where $\mu_f$ is a non-negative Borel measure. If $f$ is absolutely summable over $X \BS \{ 0 \}$, we use the notation $f \in \ell^1(X \BS \{ 0 \})$.
Note that the assumption in Definition \[def-FF\] corresponds to the particular case of G-type potential defined in [@SaffLongRange Definition 1] where $\mu_f$ is non-negative.
\[rem-FF\] The class $\FF$ of admissible potentials is quite large. Indeed, by the Hausdorff-Bernstein-Widder theorem [@Bernstein-1929], is equivalent to $f(x) = F(|x|^2)$ for some completely monotone function $F:(0,\infty)\to \R$, i.e. for any $F$ which satisfies $$\begin{aligned}
(-1)^k F^{(k)}(r)\geq 0 && \forall r>0,\forall k \in \N.
\end{aligned}$$ In particular (also in view of Definition \[def-FF\](ii)), every potential of type $$\begin{aligned}
r^{-s}, \ e^{-\lam r^\alp}, \ \frac{e^{-\lam r}}r, \ \frac{e^{-\lam \sqrt{r}}}{\sqrt{r}} &&
\text{ for $s >0, $ $\lam>0$, $\alp\in (0,2]$},
\end{aligned}$$ is included in the class $\FF$. Further examples can be constructed by noting that if $f,g$ are completely monotone, then $\alp f+g$, $\alp > 0$ and $fg$ are completely monotone. On the other hand, the class of Lennard-Jones-type potentials of the form $V(r)= r^{-p} - b r^{-q}$ with $b > 0$ and $p>q$ are not included in $\FF$.
In two dimensions, the Coulomb potential $f(x)=-\log |x|$ does not belong to $\FF$. However, we believe that the results in this paper should hold for this potential as well. Indeed, this should follow by an approximation argument together with the formula $$\begin{aligned}
-\log |x|=\frac{1}{2} \lim_{\varepsilon\to 0^{+}}\Big( \int_\varepsilon^{\infty} \frac{e^{-t |x|^2}}{t}dt + \gamma +\log \varepsilon \Big),
\end{aligned}$$ where $\gamma \approx 0.577$ is the Euler-Mascheroni constant.
We assume that the interaction energy between two points of charges $\phi_x$, $\phi_y$ at positions $x,y \in X$ is given by $\phi_x \phi_y f(x-y)$ for some rotationally symmetric interaction potential $f \in \FF$. If $f$ is absolutely summable over $X \BS \{ 0 \}$, the total potential energy is obtained directly by summing the interaction energies for all points of the lattice. If $f$ is not summable, the method of Ewald summation [@Ewald1] (introduced by Ewald in his 1912 doctoral thesis) can be used to still define the total energy of the system assuming that the total net charge is zero. More precisely, as Born did in [@Born-1921] for the Coulomb potential, we use the classic Gaussian convergence factors method, as in [@deLeeuw:1980aa; @Ewaldpolytropic; @SaffLongRange]:
\[def-energy\] Let $N\geq 2$ and $L = (X,\phi)$ be a charged lattice with $N$-periodic charge distribution $\phi$ and let $f\in \FF$. If $f \nin \ell^1(X \BS \{ 0 \})$, we assume the charge neutrality condition, $$\begin{aligned}
\label{neutral} \sum_{y\in K_N} \phi_y = 0.
\end{aligned}$$ The energy per particle is given by $$\begin{aligned}
\label{ene} \EE_{X,f}[\phi] \ := \lim_{\eta \to 0} \Big(\frac{1}{2N^d} \sum_{y\in K_N} \sum_{x\in X\BS \{0\} } \phi_y \phi_{x+y}f(x) e^{-\eta
|x|^2} \Big). \end{aligned}$$
We note that there are a variety of different ways to define the lattice energy for non-integrable interaction potentials (see e.g. [@Latticesums]). The method of Ewald summation is commonly used to calculate the energy for different charged systems and has been optimized for computational speed, see e.g. [@deLeeuw:1980aa; @PerramLeeuw; @Ewaldpolytropic]. We also note that the Ewald summation can also be used to calculate an energy for the case of non-neutral charge configurations [@Ewaldpolytropic; @AlastueyJancovici]. If the interaction potential is given by a non-integrable Riesz potential, i.e. $f(x) = |x|^{-s}$ for $s \in (0,d]$, then another way to define the energy is by analytic extension of the Epstein zeta function, introduced in [@Epstein1]. We give the definition of the Epstein zeta function in slightly less generality as needed for our purposes.
Let $X\subset \R^d$, $d \geq 1$, be a Bravais lattice associated with the positive definite quadratic form $q_X$. Then Epstein’s zeta function is defined by $$\begin{aligned}
\textnormal{Z} \VECMOD{0;z}(q_X;s):=\sum_{n\in \Z^d\backslash \{0\}} \frac{e^{2i\pi n\cdot z}}{[q_X(n)]^{\frac s2}} && \text{for $z\in \R^d$, $s \in \{ \C : \Re s > d \}.$}
\end{aligned}$$
This function admits an analytic continuation beyond its domain of absolute convergence. For $s \in \C$ with $\Re s > 0$ and $z\not\in X^*$, the extension is given by $$\begin{aligned}
\label{AnalyticEpstein}
&\pi^{-\frac s2}\Gamma(\frac{s}{2})\textnormal{Z} \VECMOD{0;z} (q_X;s) \\
&\qquad = -\frac{2}{s}+\int_1^{\infty}\sum_{n\in \Z^d\backslash \{0\}} e^{2i\pi n\cdot z} e^{-\pi t q_X(n)} t^{\frac s2-1}dt
+\int_1^{\infty}\sum_{n\in \Z^d} e^{-\pi t q_{X^*}(n+z) } t^{\frac {d-s}2-1}dt, \NT\end{aligned}$$ where $q_{X^*}$ is the quadratic form associated with the dual lattice $X^*$, see [@Epstein1].
Discrete Fourier transform and convolution
------------------------------------------
The proofs are formulated in terms of the discrete Fourier transform (for an introduction, see e.g. [@DFT Chapter 6]) on the space of $N$-periodic functions $\Lam_N(X)$ on the lattice $X$. We first note that an orthonormal basis of $\Lam_N(X)$ is given by the functions , we define the functions by the functions $e^{(k)} \in \Lam_N(X)$ where $$\begin{aligned}
e^{(k)}(y)= \frac 1{N^{\frac d2}} e^{\frac{2\pi i}N y \cdot k}.\end{aligned}$$ The discrete Fourier transform is then defined as follows:
For any $\phi \in \Lam_N(X)$, its discrete Fourier transform $\widehat \phi \in \Lam_N(X^*)$ is given by $$\begin{aligned}
\widehat \phi(k) = \skpL{\phi}{e^{(k)}}{K_N} &&\text{for $k \in X^*$}.
\end{aligned}$$ where $e^{(k)}(y) := \frac 1{N^{\frac d2}} e^{\frac{2\pi i}N y \cdot k}$. For $\psi \in \Lam_N(X^*)$, the inverse Fourier transform is $$\begin{aligned}
{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\psi}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\psi}{\scalebox{-1}{\tmpbox}}}(x) = \skpL{\psi}{\overline{e^{(k)}}}{K_N^*} &&\text{for $x \in X$}.
\end{aligned}$$
Since the functions $e^{(k)}$ form an orthonormal basis, the Fourier transform is a bijective map $\Lam_N(X) \to \Lam_N(X)$ whose inverse is given by the inverse Fourier transform. Furthermore, Plancherel’s identity holds with constant $1$, i.e. $$\begin{aligned}
\skpL{\phi}{\phi}{K_N} = \skpL{\widehat \phi}{\widehat \phi}{K_N},\end{aligned}$$ since the discrete Fourier transform corresponds to the application of a unitary matrix. A simple calculation shows that $\widehat{\phi * \psi}(k) = \widehat \phi (k) \widehat \psi (k)$ for any $k\in X^*$, where the convolution is defined by $$\begin{aligned}
(\phi * \psi)(p) = \sum_{q \in K_N} \phi_q \psi_{p-q}.\end{aligned}$$
Theta functions and useful identities {#ss-theta}
-------------------------------------
Theta functions play an important role in different fields of mathematics. For the computation of lattice sums, the following Jacobi theta function is useful:
\[def-theta\] The third Jacobi theta function is defined by $$\begin{aligned}
\label{def-jacobitheta}
\vartheta_3(\xi; z) := \sum_{k\in \Z} e^{i\pi k^2 z + 2 i \pi k \xi}, \quad \Im (z)>0,\xi\in \C.\end{aligned}$$
We recall some useful identities for the Jacobi theta function restricted to the upper imaginary axis. This restriction has been considered by Montgomery in [@Mont] (he wrote $\theta(t,\beta) := \vartheta_3(\bet, it)$) in the context of lattice sums:
\[lem-theta\] Let $t > 0$ and let $\bet \in \R$. Then
1. \[theta-ii\] $\displaystyle \vartheta_3(\beta; it)=\prod_{r=1}^{\infty} \left( 1-e^{-2\pi rt}
\right)\left( 1+2e^{-(2r-1)\pi t}\cos(2\pi\beta)+e^{-2(2r-1)\pi t} \right).$
2. $\displaystyle \vartheta_3(\beta; it) > 0$.
3. The map $\beta\mapsto \vartheta_3(\beta; it)$ is $1$-periodic. Furthermore, we have $$\begin{aligned}
\label{bet-12} \vartheta_3\Big(\frac{1}{2}-\beta; it\Big)=\vartheta_3\Big(\frac{1}{2}+\beta; it\Big) && \text{ for any
$\beta\in [0,\tfrac
12]$.}
\end{aligned}$$
The product formula \[theta-ii\] is proved for example in [@SteinC Chapter 10, Theorem 1.3]. The positivity of $\vartheta_3$ follows by expressing the right hand side in the Jacobi product representation (i) of the theta function as $$\begin{aligned}
\vartheta_3(\beta; it)=\prod_{r=1}^{\infty}
\big( 1-e^{-2\pi rt}\big) \big(\sin^2 (2\pi \bet)) + \big[e^{-(2r-1)\pi t} + \cos (2 \pi \bet)\big]^2 \big) >0.
\end{aligned}$$ The periodicity is a direct consequence of formula . The statement finally follows from with the change of variables $q=-k$.
Finally, we introduce the translated lattice theta function (see e.g. [@ConSloanPacking Section 2.3]) which is a particular case of generalized lattice theta functions that have been studied by Krazer and Prym in [@KrazerPrym]:
\[def-thetaL\] Let $d\geq 1$ and let $X = \bigoplus_{i=1}^d \Z u_i \subset \R^d$ be a Bravais lattice. Then the theta function of the translated lattice $X+z$ or translated lattice theta function is defined by $$\begin{aligned}
\theta_{X+z}(\alp) \ := \ \sum_{x\in X} e^{-\pi \alp |x+z|^2} &&\text{for any $\alp>0$ $z\in \R^d$.}
\end{aligned}$$
The function $\theta_{X+z}(\alp)$ can be understood as follows: Consider a matrix of points at the lattice points of $X$, carrying a unit charge. Suppose that the charges induce an interaction potential $\exp(-\pi \alp |x|^2)$. Then $\theta_{X+z}(\alp)$ describes the (Gaussian) interaction energy between $z$ and $X$.
An interpretation of $\theta_{X+z}$ in terms of the heat flow is given by Baernstein in [@Baernstein-1997]. Let $P$ be the temperature at point $z$ and at time $t$, if at time $t=0$ a heat source of unit strength is placed at each point of $X$, i.e. $P$ is defined, for any $z\in \R^d$, any Bravais lattice $X\subset \R^d$ and any $t>0$ as the solution of $$\begin{aligned}
\left \{ \begin{array}{ll}
\partial_t P_X(z,t) = \Delta_z P_X \qqqquad &\text{for $(z,t) \in \R^d \times (0,\infty)$} \vspace{0.3ex} \\
P_X(z,0) = \sum_{p \in X} \delta_{p} \qqqquad &\text{for $z \in \R^d$},
\end{array}
\right.\end{aligned}$$ where $\delta_p$ is the Dirac measure at $p \in \R^d$. Then $$\begin{aligned}
\theta_{X+z}(\alp) = \frac{1}{\alpha^{\frac{d}{2}}} P_X\left(z,\frac 1{4\pi \alp}\right) &&\text{for $z \in \R^d$, $\alp > 0$.}\end{aligned}$$
We next recall some basic facts related to the translated lattice theta function, introduced in Definition \[def-thetaL\]. We first recall Jacobi’s Transformation Formula:
For any Bravais lattice $X\subset \R^d$ of density one, any $\alp>0$ and any $z\in \R^d$, we have $$\begin{aligned}
\label{prp-jacobi}
\theta_{X+z}(\alp) = \sum_{x\in X} e^{-\alp\pi |x+z|^2}=\frac{1}{\alp^{\frac d2}}\sum_{p\in X^*} e^{2\pi i p\cdot z}e^{-\frac{\pi |p|^2}\alp}.
\end{aligned}$$
A proof (based on Poisson’s summation formula) can be found e.g. in [@SaffLongRange Theorem A]. See also [@Bochnertheta] for a proof of a more general formula.
For one-dimensional lattices the translated lattice theta function can be expressed in terms of Jacobi theta function. Furthermore, we state other useful identities related to scaling and periodicity of the translated lattice theta function:
\[lem-thetaL\] Let $d \geq 1$ and let $X {\subset}\R^d$ be a Bravais lattice. Let $\alp > 0, \bet \in \R$, $z \in \R^d$ and let $\lam \neq 0$. Then
1. $\displaystyle \theta_{\Z+\beta}(\alp) = \frac{1}{\sqrt{\alp}} \vartheta_3
(\beta;i\alpha^{-1})$ for $X = \Z$.
2. $\displaystyle \theta_{\lam X + z}(\alp) = \theta_{X+\frac{z}{\lambda}}(\alp
\lambda^2)$,
3. the map $z \mapsto \theta_{X+z}(\alp)$ is periodic w.r.t. the unit cell $Q = \sum_{i=1}^d [0,1)u_i$.
In terms of $t := \frac 1\alp$, identity (i) is equivalent to $$\begin{aligned}
\label{i-eq} \vartheta_3 (\beta;it)=\frac{1}{\sqrt{t}}\sum_{n\in
\Z}e^{-\frac{\pi(n+\beta)^2}t}.
\end{aligned}$$ In turn is a direct application of Jacobi’s transformation formula (Proposition \[prp-jacobi\]) for $X=\Z$. The identity (ii) is easily obtained by $$\begin{aligned}
\theta_{\lam X + z}(\alp) =\sum_{x\in X} e^{-\pi \alp |\lam x + z|^2}=\sum_{x\in X} e^{-\pi \alp\lambda^2|x+\frac z\lambda|^2}=\theta_{X+\frac z {\lambda}}(\alp\lambda^2).
\end{aligned}$$ Finally, the statement (iii) follows from the periodicity of $X$.
Consequently, in view of (iii) for any fixed lattice $X$ and for given $\alp>0$, the problem of minimizing $z\mapsto \theta_{X+z}(\alp)$ can be restricted to $Q=\sum_{i=1}^d [0,1)u_i$.
\[lem-thetasymmetry\] Let $X=\bigoplus_{i=1}^d \Z u_i$ be a Bravais lattice and $T(z)=\sum_{i=1}^d u_i -z$ be the symmetry with respect to the center $c=\frac{1}{2}\sum_{i=1}^d u_i$ of the primitive cell $Q=\sum_{i=1}^d [0,1)u_i$. Then, for any $z\in \R^d$ and any $\alpha>0$, we have $$\begin{aligned}
\theta_{X+z}(\alpha)=\theta_{X+T(z)}(\alpha)
\end{aligned}$$
We have, for any $z\in \R^d$ and any $\alpha>0$, $$\begin{aligned}
\theta_{X+T(z)}(\alpha)=\sum_{x\in X} e^{-\pi \alpha |x+T(z)|^2}=\sum_{x\in X} e^{-\pi \alpha |x+\sum_{i=1}^d u_i-z|^2}=\sum_{x\in X} e^{-\pi \alpha |x-z|^2}=\theta_{X+z}(\alpha),
\end{aligned}$$ by the periodicity of the translated theta function (Lemma \[lem-thetaL\](iii)) and the symmetry $-X=X$.
Statement of main results {#sec-main}
=========================
We consider minimizer of the interaction energy among periodic charge configurations. As in the paper by Born [@Born-1921], we assume that the charges of the points of $K_N$ satisfy a constraint on the total charge per periodicity cell, i.e. $$\begin{aligned}
\label{totalcharge} \frac 1{N^d} \sum_{y\in K_N} \phi_y^2 \ = \ 1, && \phi(0) > 0.\end{aligned}$$ Our first result is a statement which connects the minimization of the energy for certain lattice energies with the minimization problem among vectors for the translated lattice theta function:
\[thm1\] Let $d\geq 1$, let $X=\bigoplus_{i=1}^d \Z u_i \subset \R^d$ be a Bravais lattice and let $f \in \FF$. Suppose that $$\begin{aligned}
\label{prop-minz0}
z_0 \ \in \ \operatorname{argmin}\Big\{ \theta_{X^*+z}(\alp) : z = \sum_{i=1}^d \lam_i u_i^* : \forall i\in {\ensuremath{\llbracket 1, d \rrbracket}}, \lam_i \in [0,1) \Big \} &&\forall \alp > 0,
\end{aligned}$$ i.e. $z_0$ is an absolute minimizer of the translated theta function associated to the dual lattice $X^*$ for all $\alp > 0$. Furthermore, suppose that $z_0\in \frac{1}{N}X^*$ for some $N\in \N$. Then the energy $\mathcal{E}_{X,f}$ is minimized among all periodic charge configurations $\phi$ which satisfy (and which satisfy if $f$ is nonsummable) by $$\begin{aligned}
\label{generalmin}
\phi^*(x) =c \cos \left(2\pi x\cdot z_0 \right) &&
\text{for
$x \in X$},
\end{aligned}$$ where the value of the constant $c$ is determined by .
Furthermore, if has at most two solutions, then the charge configuration is the unique minimizer of $\EE_{X,f}$, up to symmetries keeping $X$ invariant. Otherwise, if has more than two solutions then there are infinitely many charge configurations which are pairwise not related by symmetries keeping $X$ invariant.
The explicit form of the minimizers in implies in particular that the net charge for any minimizing configuration must be zero. We have the following corollary:
\[cor-neutral\] Suppose that $f \in \ell^1(X \BS \{ 0 \})$. Then for any minimizer $\phi^*$ of $\EE_{X,f}$, the total net charge of the configuration is zero, $$\begin{aligned}
\sum_{y\in K_N} \phi^*(y)=0.
\end{aligned}$$
We note that charge neutrality is assumed if $f$ is not absolutely summable. The corollary shows that charge neutrality also holds in the absolutely summable case.
Another interesting question is if there are lattices where there is no periodic optimal charge configuration. Here, Theorem \[thm1\] could be useful to prove such a non-existence result:
From the proof of Theorem \[thm1\], the following statement can be easily deduced: Let $d\geq 1$. Let $X=\bigoplus_{i=1}^d \Z u_i \subset \R^d$ be a Bravais lattice with dual lattice $X^*$. Suppose that $z_0$ is an absolute minimizer of the translated theta function for all $\alp > 0$, i.e. $z_0$ satisfies . Furthermore, suppose that $$\begin{aligned}
\operatorname{argmin}\Big\{ \theta_{X^*+z}(\alp) : z \in \R^d \Big\} \cap \left\{ \frac 1N X^* : N \in \N \right\} = \emptyset.
\end{aligned}$$ Then the absolute minimum of the energy $\mathcal{E}_{X,f}$ is not obtained within the class of periodic charge configurations.
Together with a result obtained in [@BeterminPetrache Cor. 3.17] about minimization of the translated lattice theta function for the orthorhombic lattice, we obtain the optimal charge distribution for orthorhombic (and in particular cubic) lattices.
\[thm-2\] Let $d\geq 1$, let $f\in \FF$ and let $X=\bigoplus_{i=1}^d \Z (a_i e_i) \subset \R^d$ be an orthorhombic lattice with $a_i > 0$ for $i \in {\ensuremath{\llbracket 1, d \rrbracket}}$. Then the minimizer $\phi^*_{\rm orth} : X \to \R$ of $\EE_{X,f}$ among all periodic charge distributions $\phi : X \to \R$ satisfying the normalization constraint , and the charge neutrality constraint if $f$ is nonsummable, is given by $$\begin{aligned}
\phi^*_{\rm orth}\Big(\sum_{i=1}^d m_i a_i e_i \Big) =(-1)^{\sum_{i=1}^d m_i} &&\text{for all $m_i \in \N$, $i \in {\ensuremath{\llbracket 1, d \rrbracket}}$}.
\end{aligned}$$ The minimizer is unique up to translations by a vector in $X$.
Theorem \[thm-2\] applies in particular for the case when $f$ is the Coulomb potential and $d=3$. Hence, Theorem \[thm-2\] proves and generalizes Born’s conjecture about optimal charge distributions in [@Born-1921] for cubic lattices and more general orthorhombic lattices. Moreover, we find again Born’s result [@Born-1921 Section 4] in dimension $d=1$.
From the proof of Theorem \[thm-2\], the following result for general lattices can be deduced:
\[cor-2\] Let $d\geq 1$, let $f\in \FF$ and let $X=\bigoplus_{i=1}^d \Z u_i \subset \R^d$ be a Bravais lattice. Furthermore, suppose that the equation only has a unique solution $z_0 \in \sum_{i=1}^d [0,1) u_i^*$. Then the unique, up to translations by a vector of $X$, periodic charge configuration $\phi$ which minimizes the energy $\EE_{X,f}$ is the alternating configuration $$\begin{aligned}
\phi^*_{\rm orth}\Big(\sum_{i=1}^d m_i u_i \Big) =(-1)^{\sum_{i=1}^d m_i} &&\text{for all $m_i \in \N$, $i \in {\ensuremath{\llbracket 1, d \rrbracket}}$}.
\end{aligned}$$
In the general case of arbitrary Bravais lattices, the minimizer of the translated lattice theta function $z\mapsto \theta_{X^*+z}(\alp)$ is not known. However, in the special case of the triangular lattice in two dimensions, this theta function is well understood and we obtain the optimal solution for the charge distribution problem. Recall that the triangular lattice of unit density in two dimensions is given by the set $$\begin{aligned}
\label{trilattice} \Lambda_1= \Z u_1\oplus \Z u_2, && \text{where
$u_1 = \sqrt{\tfrac{2}{\sqrt{3}}}
\VEC{1;0}$, \ \ $u_2 = \sqrt{\tfrac{2}{\sqrt{3}}}
\VEC{1/2;\sqrt{3}/2}$.}\end{aligned}$$ The situation here is slightly different than in the orthorhombic case:
\[thm-3\] Let $\Lam_1 \subset \R^2$ be the triangular lattice, defined in and let $f\in \FF$. Then the minimizer of $\EE_{\Lambda_1,f}$ among all periodic charge distributions $\phi : \Lam_1 \to \R $ which satisfy the normalization constraint , and the charge neutrality constraint if $f$ is nonsummable, is $$\begin{aligned}
\phi^*_{\rm tri}(m u_1+n u_2) = \sqrt{2} \cos\Big(\frac{2\pi}{3} (m +n ) \Big) &&\text{for $m,n \in \Z$.}
\end{aligned}$$ This minimizer is unique, up translations and rotations which keep $\Lam$ invariant.
We note that by definition the lattice points of the triangular lattice are $1$-periodic. The charges are, however, not periodic with the same periodicity. Indeed, the minimizer $\phi^*_{\rm tri}$ is $3$-periodic. The optimal charge distribution for the triangular lattice is sketched in Figure \[fig-triangcharged\]. One can think of the lattice as being realized by two different types of particles. The first (depicted in red) carries the charge $\sqrt{2}$, while the other type particle (depicted in blue) carries the charge $-\sqrt{2}/2$. Since there are twice as many particles of the second type than of the first type, the configuration is charge neutral.
![Optimal configuration $\phi^*_{\rm tri}$ in the case of triangular lattice[]{data-label="fig-triangcharged"}](Triangularcharged.png){width="9cm"}
This honeycomb structure with two kind of particles given by Figure \[fig-triangcharged\] appears in different simulations and experiments. In [@AssoudMessinaLowen], this structure arises as the minimizer of a binary mixture of particles interacting via the pair potential $V(r)=r^{-3}$ (up to a multiplicative constant) of parallel dipoles, at zero temperature, when the ratio of type particles is $(1/3,2/3)$ and at weak dipole-strength asymmetry. This result explains the experimental finding of [@Hay:2003aa Figure 5.(a)]. Furthermore, the triangular lattice with the same absolute values of charges (i.e. $\sqrt{2}$ and $\sqrt{2}/2$) as in Theorem \[thm-3\] is numerically identified in [@Xiao:1999aa; @Levashov:2003aa] as a minimizer of the Coulomb interaction energy when the particles, with positive charges, are fixed on a triangular lattice embedded in a negative background charge to ensure neutrality. However, contrary to our study, these works investigated the minimizer of the interaction energy when the charges are fixed and as the concentrations (or the dipole-strength asymmetry) of the species vary.
Let us note that the method of Ewald summation in our case yields the same formula for the energy (up to an $s$ dependent constant) as the one obtained by analytic extension of Epstein’s zeta function:
\[thm-5\] Let $d\geq 1$, $N\geq 2$, $X\subset \R^d$ be a Bravais lattice with generator matrix $A_X$, and $f_s(x)=|x|^{-s}$, $0< s\leq d$. For any $\phi$ satisfying and , for the $(\xi_k)_{k\in K_N^*}$ defined in , we have $\xi_k\geq 0$ for any $k$ and $$\begin{aligned}
\EE_{X,f_s}[\phi]=\frac{1}{2N^d}\sum_{k\in K_N^*} \xi_k \textnormal{Z} \VECMOD{0;A_X^{t}\frac{k}{N}} (q_X;s).\end{aligned}$$
Proofs {#sec-proofs}
======
Proof of Theorem \[thm1\] and Corollary \[cor-neutral\]
-------------------------------------------------------
We first give the proof of the Theorem in the case when the interaction potential is absolutely summable over $X \BS \{ 0 \}$. We then will extend the result to the case when the interaction potential is not absolutely summable.
#### The absolutely summable case.
In the first part of the proof (Lemma \[lem-s\]–\[lem-reverse\]), we follow the lines of the proof of Born in a slightly more general setting to write the energy in terms of the Fourier series [@Born-1921], see Lemma \[lem-xi\]. In the notation of Born, the corresponding plane waves $e^{\frac{2i \pi}{N} x\cdot k}$ are called “Grundpotentiale”. In the second step, we write the transformed expression of the energy in terms of the translated lattice theta functions $\theta_{X^*+\frac{k}{N}}(\alp)$.
We first note express the energy in terms of the autocorrelation function $s$:
\[lem-s\] Let $d\geq 1$ and let $X\subset \R^d$ be a Bravais lattice. Let $N\geq 2$ and let $\phi : X \to \R$ be an $N$-periodic charge distribution satisfying . Let $s : X \to \R$ be defined by $$\begin{aligned}
\label{def-s} s_x := \sum_{y\in K_N} \phi_y \phi_{y+x}.
\end{aligned}$$ Then $s \in \Lam_N(X)$, $s_{-x} = s_x$, $\displaystyle \sum_{x \in K_N} s_x = \Big(\sum_{x \in K_N} \phi_x\Big)^2$ and $s(0) = N^d$. The energy takes the form $$\begin{aligned}
\label{ene-s} \EE_{X,f}[\phi] \ &\upref{ene}= \frac 1{2 N^d} \sum_{x\in X\BS \{0\} } s_x f(x).
\end{aligned}$$
The formula follows directly from the definition of energy $$\begin{aligned}
\EE_{X,f}[\phi] \ \lupref{ene}= \frac{1}{2N^d} \sum_{y\in K_N} \sum_{x\in X\BS
\{0\} } \phi_y \phi_{y+x}f(x). \end{aligned}$$ by exchanging the sums. The periodicity of $\phi$ implies $s_{-x} = s_x$ for any $x\in X$. One can easily check that $s(0) = \NNN{\phi}{K_N}^2=N^d$. Furthermore, by periodicity of $\phi$, we obtain $$\begin{aligned}
\sum_{x\in K_N} s_x=\sum_{x\in K_N} \sum_{y\in K_N} \phi_y \phi_{y+x}=\sum_{y\in K_N} \phi_y \sum_{x\in K_N} \phi_{x+y}=\Big( \sum_{x\in K_N} \phi_x \Big)^2.
\end{aligned}$$
It is convenient to express the energy in terms of the discrete (inverse) Fourier transform $\xi$ of the autocorrelation function $s$. In the context of signal processing, $\xi$ has also been called power spectral density.
\[Energy expressed in dual variables\] \[lem-xi\] Let $d\geq 1$ and let $X\subset \R^d$ be a Bravais lattice. For $N\geq 2$, let $\phi \in \Lam_N(X)$ satisfy and let $s$ be defined as in . Let $\xi \in \Lam_N(X^*)$ be given by $$\begin{aligned}
\label{def-xi} \xi_k:=\frac{1}{N^{d}}\sum_{y\in K_N} s_y e^{\frac{2\pi i}{N}y\cdot
k} &&\text{for $k \in X^*$.}
\end{aligned}$$ Then $\xi_k \in \R$, $\xi_k \geq 0$, $\displaystyle \xi_0 =\frac{1}{N^d} \Big(\sum_{x \in K_N} \phi_x\Big)^2$ and $\xi_k=\xi_{-k}$ for all $k \in X^*$, and $$\begin{aligned}
\label{xi-totalcharge} \frac 1{N^{d}} \sum_{k \in K_N^*} \xi_k = 1.
\end{aligned}$$ Furthermore, the energy $E_{X,f}[\phi]$ can be equivalently written as $$\begin{aligned}
\label{ene-xi} \EE_{X,f}[\phi]=\frac{1}{2 N^{d}}\sum_{k\in K_N^*}\xi_k \sum_{x\in X\BS \{0\}} e^{\frac{2\pi i}N x \cdot k} f(x).
\end{aligned}$$
Note that $\xi = N^{-\frac d2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{s}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{s}{\scalebox{-1}{\tmpbox}}}$. Since $s_{-x} = s_x$ for any $x\in X$, we have $\xi \in \R$. The proof is based on Plancherel’s identity on $\Lam_N(X)$. We give the calculation in detail below, using for notational simplicity, the convention $f(0) := 0$. We have, using the symmetry $f(-q)=f(q)$, $$\begin{aligned}
\sum_{x \in X \BS \{ 0 \}} s_x f(x) &= \sum_{\ell \in X} \sum_{y \in K_N} s_y f(N\ell+y) =\sum_{\ell \in X} \sum_{y \in K_N} \Big( \sum_{k\in K_N^*} \xi_k e^{\frac{-2\pi i}N y \cdot k} \Big) f(N\ell+y) \\
&= \sum_{k \in K_N^*} \xi_k \sum_{x \in X \BS \{ 0 \}} e^{\frac{2\pi i}N x\cdot k} f(x).
\end{aligned}$$ Furthermore, $\xi = N^{-\frac d2}{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{s}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{s}{\scalebox{-1}{\tmpbox}}} =N^{-\frac d2} {\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\phi * (\phi \circ
P)}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\phi * (\phi \circ
P)}{\scalebox{-1}{\tmpbox}}} = N^{-\frac d2}|{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\phi}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\phi}{\scalebox{-1}{\tmpbox}}}|^2 \geq 0$, where $P(x) = -x$. The value of $\xi_0$ follows from $\sum_{x\in K_N} s_x=\left( \sum_{x\in K_N} \phi_x \right)^2$ (see Lemma \[lem-s\]). Finally, one can easily check that $\skpL{\xi}{1}{K_N^*} = s(0) = N^{d}$.
Note that due to the inversion symmetry of $s_x$ and $f(x)$, the sums on the right hand side of and are real numbers. By combining the sums over positive and negative indices, can e.g. be written as $$\begin{aligned}
\EE_{X,f}[\phi]=\frac{1}{2N^{d}}\sum_{k\in K_N^*}\xi_k \sum_{x\in X\BS \{0\}} \cos\Big( \frac{2\pi}N x \cdot k \Big) f(x).\end{aligned}$$
We also have the reverse transformation. Note that for given $\xi$, $\phi$ is not uniquely defined in general.
\[lem-reverse\] Let $\xi \in \Lam_N(X^*)$ satisfy $\xi \geq 0$, $\xi_{-k} = \xi_k$ and . Let $$\begin{aligned}
\label{phi-reverse} \phi_x=\frac{1}{N^{\frac d2}}\sum_{k\in K_N^*} \sqrt{\xi_k} \ \cos \Big(\frac{2 \pi}{N} x\cdot k \Big).
\end{aligned}$$ Then $\phi$ satisfies and the formulas and hold.
We define $s \in \Lam_N(X)$ by $s = N^{\frac d2} \widehat \xi$, i.e. $$\begin{aligned}
s_x = \sum_{k \in K_N^*} \xi_k e^{-\frac{2\pi i}N x \cdot k},
\end{aligned}$$ so that holds. A straightforward calculation now shows that the identities and are satisfied by the function $\phi$ defined in .
We turn to the proof of Theorem in the case when the interaction potential is summable:
\[prp-thm-11\] Suppose that the assumptions of Theorem \[thm1\] hold and suppose that $f \in \ell^1(X \BS \{ 0 \})$. Then the statement of Theorem \[thm1\] holds
Let $s$ and $\xi$ be defined as in Lemma \[lem-s\] and Lemma \[lem-xi\]. Then we have $$\begin{aligned}
\EE_{X,f}[\phi]=\frac{1}{2N^{d}}\sum_{k \in K_N^*}\xi_k \sum_{x\in X\BS \{0\}} e^{\frac{2 \pi i}N x\cdot k} f(x).
\end{aligned}$$ where $\xi \geq 0$ and $\xi$ satisfies . This suggests to consider $$\begin{aligned}
\label{ene-xi4}
E[k] \ :=\ \sum_{x\in X\BS \{0\}} e^{\frac{2 i\pi}N x\cdot k} f(x).
\end{aligned}$$ for $k \in X^*$. We note that $E[k]$ cannot be minimized for $k \in NX^*$. Indeed, in this case the energy decreases by just switching the sign of a single charge in the periodicity cell. In the following, we hence assume that $k\in X^*\backslash NX^*$.
In view of Definition \[def-FF\] and by Fubini’s Theorem, we get for any $k\in X^*\backslash NX^*$, $$\begin{aligned}
E[k] &= \sum_{x\in X\backslash \{0\}} \int_0^{\infty} e^{-|x|^2 t}e^{2\pi i x\cdot \frac{k}{N}}d\mu_f(t) = \int_0^{\infty} \Big( \sum_{x\in X}e^{-|x|^2 t}e^{2\pi i x\cdot \frac{k}{N}}-1\Big)d\mu_f(t).
\end{aligned}$$ By Jacobi’s transformation formula , this implies $$\begin{aligned}
E[k]&= \int_0^{\infty} \Big(\pi^{\frac d2} t^{-\frac{d}{2}}\sum_{p\in X^*} e^{-\frac{\pi^2}{t}|p+\frac{k}{N}|^2}- 1\Big) d\mu_f(t)\\
&=\int_0^{\infty} \Big(\pi^{\frac d2} t^{-\frac{d}{2}} \theta_{X^*+ \frac{k}{N}}(\tfrac \pi t) -1 \Big) d\mu_f(t).
\end{aligned}$$ Since $\mu_f$ is a non-negative measure, if $z_0$ is a minimizer on $X^*$ of $z\mapsto \theta_{X^*+z}(\alp)$ for all $\alp>0$, then $k_0=Nz_0$ minimizes $E$. By Lemma \[lem-reverse\], a minimizing configuration $\phi^*$ is then given, for any $x\in X$, by $$\begin{aligned}
\label{theform} \phi^*(x)= c \cos\big(2\pi x\cdot z_0 \big),
\end{aligned}$$ where $c$ is determined by the constraint . This concludes the proof of existence.
We turn to the proof of uniqueness. By assumption there are at most two minimizers $k^{(1)}$ and $k^{(2)}$ of $E[k]$. In view of Lemma \[lem-thetasymmetry\], these two minimizers are symmetry related, i.e. we have $\frac{k^{(2)}}{N} = \sum_{i=1}^d u_i^* - \frac{k^{(1)}}{N}$. Therefore, the minimizer of in the class of functions $\xi^*$ which satisfy $\xi^* \geq 0$, $\xi_{-k}^*=\xi_k^*$ for any $k\in X^*$ and is hence given by $\xi^* \in \Lam_N(X^*)$, defined by $\xi_{k_0}^*=\xi_{k_1}^*=\frac{N^d}{2}$, by periodicity and the fact that $\xi_{k_0}^*=\xi_{-k_0}^*$, and $\xi_{k}^*=0$ for $k \in K_N^* \BS \{ k_0 ,k_1\}$. It follows that the corresponding autocorrelation function $s^*$ is given, for any $x\in X$, by $$\label{uni-s} s^*_x= \frac{N^d}{2}\left(e^{\frac{2i \pi}{N}k_0\cdot x}+e^{\frac{2i \pi}{N}k_1\cdot x} \right)=\frac{N^d}{2}\left(e^{\frac{2i \pi}{N}k_0\cdot x}+e^{-\frac{2i \pi}{N}k_0\cdot x} \right)
$$ For any charge configuration $\phi^* \in \Lam_N(X)$, its (inverse) Fourier coefficients ${\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\phi^*_k}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\phi^*_k}{\scalebox{-1}{\tmpbox}}}$ satisfy the equation $|{\savestack{\tmpbox}{\stretchto{ \scaleto{ \scalerel*[\widthof{\ensuremath{\phi^*_k}}]{\kern-.6pt\bigwedge\kern-.6pt} {\rule[-\textheight/2]{1ex}{\textheight}} }{\textheight}}{0.5ex}}\stackon[1pt]{\phi^*_k}{\scalebox{-1}{\tmpbox}}}|^2 = \xi_k^*$ as a straightforward calculation shows. Any charge configuration with associated autocorrelation function $s$, given by therefore is of the form $$\label{uni-phi} \phi^*_x= \frac{\alp_1 N^d}{2} e^{\frac{2i
\pi}{N}k_0\cdot x}+ \frac{\alp_2 N^d}{2} e^{-\frac{2i \pi}{N}k_0\cdot x}$$ for coefficients $\alp_i \in \C$ with $|\alp_i| = 1$. We next use the fact that the charge configuration $\phi^*$ is real. Furthermore, by a shift of coordinates, we can assume that $\phi^*_0 > 0$ and that $\phi^*$ attains its maximum at $x = 0$. With these assumptions, one can show that $\alp_1 = 1$ and $\alp_2 = 1$. Therefore, in the case considered the charge distribution $\phi^*$ is uniquely determined by the autocorrelation function $s^*$ and is hence unique and given by .
Suppose that there are at least three solutions of . Then as above we can construct two symmetric functions $\xi^{(1)}, \xi^{(2)} \in \Lam_N(X^*)$ satisfying the properties of Lemma \[lem-xi\]. Then any convex combination of $\xi^{(\theta)} := \theta \xi^{(1)} + (1-\theta)\xi^{(2)}$ for $\theta \in [0,1]$ yields a minimizing charge configuration $\phi^{(\theta)}$ (by Lemma \[lem-reverse\]). We hence have constructed a one-parameter family of optimal charge configurations $\phi^{(\theta)}$ such that the corresponding autocorrelation functions $s_\theta$ are pairwise different.
Let $\phi^* \in \Lam_N(X)$ be a minimizer of $\EE_{X,f}$ and let $s^*$ and $\xi^*$ be defined by and . In view of the proof of Theorem \[thm1\], it then follows that $\xi^*$ is the convex combinations of functions $\xi$ with $\xi_0 = 0$. It follows that $\xi^*_0 = 0$. In view of Lemma \[lem-xi\], this shows that the configuration is charge neutral.
#### The nonsummable case
We turn to the case when the potential energy is not summable. In this case, the total energy of the lattice can be calculated using the Ewald summation method. The idea of Ewald summation with Gaussian convergent factor is to approximate the energy by replacing the interaction potential $f(x)$ by a family of screened interaction potentials $f(x) e^{-\eta |x|^2}$ (for some small parameter $\eta > 0$), and to split the screened interaction potential into a short-range part and a long-range part (for some cut-off parameter $\alp > 0$). We follow the strategy in [@Ewaldpolytropic] where the Ewald summation has been used to calculate the energy of the Riesz potentials $f_s(x)=|x|^{-s}$ in dimensions $d = 1, 2,3$ and for $s \geq 1$ in a general setting without assuming charge neutrality.
\[thm-riesz\] Let $d\geq 1$, let $X\subset \R^d$ be a Bravais lattice and $f\in \FF$. Then, for any $N\geq 2$, any $N$-periodic charge distribution $\phi$ satisfying and , and for any $\alp>0$, we have $$\begin{aligned}
\EE_{X,f}[\phi] &= \frac{1}{2N^d}\sum_{x \in X \BS \{ 0 \}} s_x f_{1}^{(\alp)}(x) +
\frac{1}{2N^d} \sum_{p\in X^* } \xi_{p}
f_{2}^{(\alp)}\left(\frac{{p}}{N}\right)
-\frac{\mu_f\left([0,\alpha^2] \right)}{2},
\end{aligned}$$ where $$\begin{aligned}
\label{def-fs12} f_1^{(\alpha)}(x)=\int_{\alpha^2}^{+\infty} e^{-t|x|^2}d\mu_f(t) , && f_{2}^{(\alp)}(x)=\pi^{\frac{d}{2}} \int_0^{\alpha^2} t^{-\frac d2} e^{-\frac{\pi^2}{t}|x|^2}d\mu_f(t).
\end{aligned}$$
We write the approximated interaction potential as $$\begin{aligned}
\label{alt-decompo}
f(x)e^{-\eta |x|^2}=\int_0^{\alpha^2} e^{-(t+\eta)|x|^2}d\mu_f(t)+\int_{\alpha^2}^{+\infty} e^{-(t+\eta)|x|^2}d\mu_f(t).\end{aligned}$$ The second integral in is absolutely integrable and the limit $\eta\to 0$ can be taken directly. We hence obtain $$\begin{aligned}
\EE_{X,f}[\phi] \ &= \lim_{\eta \to 0} \Big(\frac{1}{2N^d} \sum_{x\in X\BS \{0\} } s_x f(x) e^{-\eta |x|^2} \Big) = \frac{1}{2N^d} \sum_{x \in X \BS \{ 0 \}} s_x f_{1}^{(\alp)}(x) + I_2,
\end{aligned}$$ where $$\begin{aligned}
I_2 := \lim_{\eta \to 0} \Big( \frac 1{2 N^d } \sum_{x \in X \BS \{ 0 \}} s_x \int_0^{\alp^2} e^{-(t + \eta) |x|^2} \ d\mu_f(t)\Big).
\end{aligned}$$ For the term $I_2$, we transform into dual variables with help of Jacobi’s transformation formula . Taking into account the fact that $s_0 = N^d$, we have $$\begin{aligned}
\sum_{x \in X \BS \{ 0 \}} s_x e^{-(t + \eta) |x|^2} &= \sum_{\ell \in X}\sum_{y\in K_N} s_y e^{-(t + \eta) |y + N\ell|^2} - N^d\\
&= \frac {\pi^{\frac d2}}{(t+\eta)^{\frac d2} N^d} \sum_{y \in K_N} s_y \sum_{{p} \in X^*} e^{2\pi i {p} \cdot \frac yN} e^{- \frac{\pi^2}{(t+\eta)N^2}|{p}|^2} - N^d.
\end{aligned}$$ In view of the definition of $\xi$ in Lemma \[lem-xi\], we get $$\begin{aligned}
\sum_{x \in X \BS \{ 0 \}} s_x e^{-(t + \eta) |x|^2} &= \frac{\pi^{\frac d2}}{(t+\eta)^{\frac d2}} \sum_{{p} \in X^*} \xi_{p} e^{- \frac{\pi^2}{(t+\eta)N^2}|{p}|^2} -N^d.
\end{aligned}$$ Hence, since $\xi_0=0$ as a consequence of the charge neutrality of $\phi$, we arrive at $$\begin{aligned}
I_2 &=\frac{\pi^{\frac{d}{2}}}{2N^d}\lim_{\eta\to 0} \sum_{p\in X^*} \xi_p \int_0^{\alpha^2} (t+\eta)^{-\frac d2} e^{-\frac{\pi^2}{(t+\eta)N^2}|p|^2}d\mu_f(t)-\frac{1}{2}\int_0^{\alpha^2} d\mu_f(t)\\
&=\frac{\pi^{\frac{d}{2}}}{2N^d}\sum_{p\in X^*} \xi_p \int_0^{\alpha^2} t^{-\frac d2} e^{-\frac{\pi^2}{t}\left|\frac{p}{N}\right|^2}d\mu_f(t)-\frac{\mu_f\left( [0,\alpha^2] \right)}{2},
\end{aligned}$$ and the result is proved.
Following the same arguments as in the summable case, we get
\[Energy expressed in dual variables\] \[lem-xi-ns\] Let $d\geq 1$, $X\subset \R^d$ be a Bravais lattice and $f\in \FF$. Let $N\geq 2$ and let $\phi : X \to \R$ be a $N$-periodic charge distribution satisfying . Let $s$ and $\xi$ be defined as in and . Then $\xi$ is $N$-periodic and satisfies $\xi \in \R$, $\xi \geq 0$ and $\xi_{-k} = \xi_k$. The constraint takes the form $\skp{\xi}{1} = N^d$ and the energy $\EE_{X,f}[\phi]$ is expressed as $$\begin{aligned}
\EE_{X,f}[\phi]= &\frac{1}{2N^d} \sum_{k\in K_N^*} \xi_k \Big(\sum_{x \in X \BS \{ 0 \}} e^{\frac{2\pi i}{N} x \cdot k} f_{1}^{(\alp)}(x) + \sum_{q \in X^* } f_{2}^{(\alpha)}(q + \tfrac kN) \Big) -\frac{\mu_f\left( [0,\alpha^2] \right)}{2}.
\end{aligned}$$
The argument for both sums proceeds analogously as in the proof of Lemma \[lem-xi\], using the fact that $\xi_0=N^{-d}\sum_{p\in K_N} s_p=0$ because $\phi$ satisfies .
We are ready to give the proof of Theorem \[thm1\] in the non-summable case:
We have already shown that the statement of Theorem holds if $f$ is summable. We conclude the argument now for the non-summable case. By Lemma \[lem-xi-ns\], we have $$\begin{aligned}
\label{renorm-xis} \EE_{X,f}[\phi]= &\frac{1}{2N^d} \sum_{k\in K_N^*} \xi_k \Big(\sum_{x \in X \BS \{ 0 \}} e^{\frac{2\pi i}{N} x \cdot k} f_{1}^{(\alp)}(x) + \sum_{q \in X^* } f_{2}^{(\alp)}(q + \tfrac kN) \Big) -\frac{\mu_f\left( [0,\alpha^2] \right)}{2}.
\end{aligned}$$ Hence, it is enough to minimize $$\begin{aligned}
\label{FkforEpstein}
F[k] &:= \sum_{x \in X \BS \{ 0 \}} e^{\frac{2\pi i}{N} x \cdot k} f_{1}^{(\alp)}(x) + \sum_{q \in X^* } f_{2}^{(\alp)}(q + \tfrac kN) \\
&=\int_{\alpha^2}^{+\infty} \sum_{x\in X\backslash \{0\}} e^{\frac{2\pi i}{N} x \cdot k} e^{-t|x|^2}d\mu_f(t)++ \pi^{\frac{d}{2}}\int_0^{\alpha^2}\Big( \sum_{q \in X^*} e^{-\frac{\pi^2}{t}|q+\frac{k}{N}|^2} \Big)t^{-\frac{d}{2}}d\mu_f(t). \NT
\end{aligned}$$ By application of Jacobi’s transformation formula , this implies $$\begin{aligned}
F[k] &= \int_{\alp^2}^{\infty}\Big(\frac{\pi^{\frac{d}{2}}}{t^{\frac d2}} \sum_{p\in X^*} e^{-\frac{\pi^2}{ t}|p+\frac{k}{N}|^2} -1 \Big)d\mu_f(t) \\
&\qquad+ \pi^{\frac{d}{2}}\int_0^{\alpha^2}\Big( \sum_{q \in X^*} e^{-\frac{\pi^2}{t}|q+\frac{k}{N}|^2} \Big)t^{-\frac{d}{2}}d\mu_f(t),
\end{aligned}$$ and we can rewrite this expression in terms of theta functions: $$\begin{aligned}
F[k] &= \int_{\alp^2}^{\infty}\Big(\frac{\pi^{\frac{d}{2}}}{t^{\frac d2}} \theta_{X^*+\frac{k}{N}}\left( \frac{\pi}{t} \right) -1 \Big)d\mu_f(t) \\
&\qquad+ \pi^{\frac{d}{2}}\int_0^{\alpha^2} \theta_{X^*+\frac{k}{N}}\left( \frac{\pi}{t} \right)t^{-\frac{d}{2}} d\mu_f(t).\end{aligned}$$ We now conclude exactly as in the proof of the summable case.
Proof of Theorem \[thm-2\] and Corollary \[cor-2\]
--------------------------------------------------
In this section, we give the proof of Theorem \[thm-2\]. This proves Born’s conjecture for a general orthorhombic $d$-dimensional lattice distribution of charges and for any interacting potential $f\in \FF$.
Let us first recall the result the first author obtained in [@BeterminPetrache Cor. 3.17] about the global optimality, for $z\mapsto \theta_{X+z}(\alp)$, for any $\alp>0$, when $X$ is an orthorhombic lattice. The result in [@BeterminPetrache] is based on a result by Montgomery in [@Mont Lemma 1]. For the convenience of the reader, we present the argument adapted to the particular case of our setting:
\[minsquare\] Let $d\geq 1$ and $X=\bigoplus_{i=1}^d \Z (a_i e_i)$, where $a_i>0$ for any $1\leq i\leq d$. Then we have $$\begin{aligned}
\label{opticenter}
\theta_{X+z^*}(\alp)\leq \theta_{X+z}(\alp), && \text{for any $\alp>0$ and any $z\in Q$,}
\end{aligned}$$ where the unique minimizer $z^*$ is the center of gravity of $Q$, i.e. $$\begin{aligned}
z^*=\frac{1}{2}(a_1,...,a_d).
\end{aligned}$$
For any $z \in \R^d$ and in view of the bijection $\Z^d \to X$, $n \mapsto \sum_{i} a_i n_i$, we have $$\begin{aligned}
\theta_{X+z}(\alp)=\sum_{x\in X} e^{-\pi\alp |x+z|^2} =\sum_{n\in \Z^d} e^{-\pi \alp \sum_{i=1}^d (a_i n_i +z_i)^2} =\prod_{i=1}^d \theta_{a_i\Z +z_i}(\alp)
=\prod_{i=1}^d \theta_{\Z +\frac{z_i}{a_i}}(\alp a_i^2),
\end{aligned}$$ where we have used Lemma \[lem-thetaL\](ii). We hence have reduced the problem to the minimization of the translated theta function on the one-dimensional lattice $\Z$ and it is enough to show that $$\begin{aligned}
\label{oned-min} \theta_{ \Z + \frac{1}{2}}(\alp)\leq \theta_{\Z +\frac{z_i}{a_i}}(\alp)
\end{aligned}$$ for any $\alp > 0$. In order to show , we use the identity $$\begin{aligned}
\label{tran-clas} \frac{1}{\sqrt{t}}\theta_{\Z+\beta}(t^{-1}) = \vartheta_3(\beta;it)
\end{aligned}$$ of Lemma \[lem-thetaL\](i) which links the translated lattice theta function of the translated lattice to the Jacobi theta function. The Jacobi theta function can in turn be expressed in terms of the Jacobian product by $$\begin{aligned}
\label{theta-prod} \vartheta_3(\beta;it)=\prod_{r=1}^{\infty} \left( 1-e^{-2\pi rt} \right)\left(
1+2e^{-(2r-1)\pi t}\cos(2\pi\beta)+e^{-2(2r-1)\pi t} \right),
\end{aligned}$$ see Lemma \[lem-theta\]\[theta-ii\]. From the representation , it follows directly that, for any fixed $t>0$, the function $\beta \mapsto \vartheta_3(\beta;it)$ is decreasing on $[0,1/2]$ since each factor in is positive and decreasing. By Lemma \[lem-theta\](ii) and Lemma \[lem-theta\](iii), it hence follows that the theta function $\theta_{\Z+\beta}$ takes its minimum at $\beta\in [0,1/2]$. The estimate then follows in view of . This completes the proof.
Let $d\geq 1$, $N\geq 2$ and $f\in \FF$. If $X=\bigoplus_{i=1}^d \Z u_i$, then $X^*=\bigoplus_{i=1}^d \Z u_i^*$ where $u_i^*=a_i^{-1}e_i$. By Proposition \[minsquare\], the unique minimum of $z\mapsto \theta_{X^*+z}(\alp)$ is $$\begin{aligned}
z_0=\frac{1}{2}\sum_{i=1}^d u_i^*,
\end{aligned}$$ for all $\alp > 0$. Note that $z_0\in \frac{1}{N}X^*$ if only if $N \in
2\N$. Therefore, the unique minimizer of in the class of functions $\xi$ which satisfy $\xi \geq 0$, $\xi_{-k}=\xi_k$ for any $k\in X^*$ and is hence given by $\xi \in \Lam_N(X^*)$, defined by $\xi(k_0) = N^{d}$ and $\xi(k) = 0$ for $k \in K_N^* \BS \{ k_0 \}$. It follows that we get the autocorrelation function $s$ defined, for any $n\in \Z^d$ and any $x=\sum_{i=1}^d n_i u_i$, by $$s_x=N^d (-1)^{\sum_{i=1}^d n_i}.$$ Therefore, by Theorem \[thm1\], we can uniquely reconstruct the charge distribution $\phi$ which is $\phi^*(\sum_{i=1}^d n_i u_i)= (-1)^{\sum_{i=1}^d n_i}$.
By assumption, the set of points satisfying is given by a single point $z_0=\frac{k_0}{N}$ for some $k_0\in K_N^*$. In view of the proof of Theorem \[thm1\], we infer that $z_0$ is the unique minimizer of the translated lattice theta function in $\sum_{i=1}^d [0,1) u_i \BS \{ 0 \}$. In turn, by Lemma \[lem-thetasymmetry\], it then follows that $z_0=\frac{1}{2}\sum_{i=1}^d u_i^*$ is the center of the unit cell of $X^*$. The argument is concluded as in the proof of Theorem \[thm-2\].
Proof of Theorem \[thm-3\] {#sec-triangle}
--------------------------
The proof for Theorem \[thm-3\] follows by by using Theorem \[thm1\] together with a result by Baernstein in [@Baernstein-1997] about the minimizer for the translated theta function in the triangular lattice. We first note that the dual lattice of $\Lambda_1$ is the triangular lattice $\Lam_1^*$, defined by $$\begin{aligned}
\Lambda_1^*= \Z u_1^*\oplus \Z u_2^*, && \text{where
$u_1^* = \sqrt{\tfrac{2}{\sqrt{3}}}
\VEC{\sqrt{3}/2;-1/2}$, \ \ $u_2^* = \sqrt{\tfrac{2}{\sqrt{3}}}
\VEC{0;1}$,}\end{aligned}$$ i.e. $\Lambda_1^*$ and $\Lambda_1$ are the same lattice, up to rotation.
![Primitive cell $Q^*$ of $\Lambda_1^*$ formed by two primitive triangle with barycenters $z_0$ and $z_0'$.[]{data-label="Baernstein"}](Baernstein.png){width="8cm"}
For any $\alp>0$ $z\mapsto \theta_{\Lambda_1^*+z}(\alp)$ admits two minimizers in the set $Q^* :=[0,1)u_1^*+[0,1)u_2^*$, given by Baernstein’s result [@Baernstein-1997 Thm. 1]. These minimizers are the barycenters of the two primitive triangles forming $Q^*$ (see Fig. \[Baernstein\]), i.e. $$\begin{aligned}
\label{z0tri}
z_0 = \frac 13 (u_1^* + u_2^*) \quad \textnormal{and}\quad
z_0' = \frac 23 (u_1^* + u_2^*).\end{aligned}$$ We note that $z_0$ and $z_0'$ belong to $\frac{1}{N}\Lambda_1^*$ if and only if $N \in 3\N$. Consequently, by Theorem \[thm1\], the minimum among all the periodic configurations is achieved for any $N\in 3\N$, by the configurations defined, for any $(m,n)\in \Z^2$, by $$\begin{aligned}
\phi^*_{\rm tri}(mu_1+nu_2) = c \cos(2\pi (mu_1+nu_2)\cdot z_0)= c \cos\left(\frac{2\pi}{3}(m+n) \right).\end{aligned}$$ The value $c = \sqrt 2$ then follows from .
Proof of Theorem \[thm-5\]
--------------------------
We first show that $$\begin{aligned}
\label{fk-ep} F[k]= \textnormal{Z} \VECMOD{0;A_X^t \frac{k}{N}}(q_X;s) +\frac{2\pi^{\frac{s}{2}}}{s\Gamma(\frac{s}{2})},\end{aligned}$$ where $F[k]$ is given in . Using , we calculate $$\begin{aligned}
&\pi^{-\frac s2}\Gamma(\frac{s}{2})\textnormal{Z} \VECMOD{0;A_X^t \frac{k}{N}} (q_X;s) + \frac{2}{s} \\
&\qquad = \int_1^{\infty}\sum_{n\in \Z^d\backslash \{0\}} e^{2i\pi n\cdot A_X^t \frac{k}{N}} e^{-\pi t q_X(n)} t^{\frac s2-1}dt
+\int_1^{\infty}\sum_{n\in \Z^d} e^{-\pi t q_{X^*}(n+A_x^t \frac{k}{N}) } t^{\frac {d-s}2-1}dt, \NT \\
&\qquad = \int_1^{\infty}\sum_{x\in X \backslash \{0\}} e^{2i\pi x \cdot \frac{k}{N}} e^{-\pi t |x|^2} t^{\frac s2-1}dt
+\int_1^{\infty}\sum_{x \in X^*} e^{-\pi t |x+\frac{k}{N}|^2 } t^{\frac {d-s}2-1}dt, \NT\end{aligned}$$ Now, we know (see e.g. [@Laplacetransf Eq. (1.9)]) that if $f_s(x)=|x|^{-s}$, then $$\begin{aligned}
d\mu_{f_s}(t)=\frac{t^{\frac{s}{2}-1}}{\Gamma\left(\frac{s}{2} \right)} dt.\end{aligned}$$ Therefore, in view of for $\alpha=\sqrt{\pi}$, we hence get by a straightforward computation. Therefore, substituting $F[k]$ in , we finally obtain $$\begin{aligned}
\EE_{X,f_s}[\phi]&=\frac{1}{2N^d}\sum_{k\in K_N^*} \xi_k \textnormal{Z} \VECMOD{0;A_X^t \frac{k}{N}} (q_X;s)+\frac{1}{2N^d}\sum_{k\in K_N^*}\xi_k \frac{2\pi^{\frac{s}{2}}}{s\Gamma(\frac{s}{2})}-\frac{\pi^{s/2}}{s\Gamma(\frac{s}{2})}\\
&= \frac{1}{2N^d}\sum_{k\in K_N^*} \xi_k \textnormal{Z} \VECMOD{0;A_X^t \frac{k}{N}} (q_X;s)+\frac{\pi^{\frac{s}{2}}}{s\Gamma(\frac{s}{2})N^d}\left(\sum_{k\in K_N^*} \xi_k - N^d \right)\\
&= \frac{1}{2N^d}\sum_{k\in K_N^*} \xi_k \textnormal{Z} \VECMOD{0;A_X^t \frac{k}{N}} (q_X;s)\end{aligned}$$ by .
#### Acknowledgements.
LB is grateful for the support of MATCH. Both authors would like to thank Florian Nolte for interesting discussions.
[^1]: “Ein endliches Stück eines einfachen kubischen Raumgitters soll so mit gleich vielen positiven und negativen Ladungen von gleichem absoluten Betrage besetzt werden, dass die elektrostatische Energie des Systems möglichst klein wird."
|
---
author:
- Ignazio Scimemi
- Alexey Vladimirov
bibliography:
- 'TMD\_ref.bib'
title: 'Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum'
---
Introduction
============
The factorization theorem for differential cross-sections of a boson production (Drell-Yan process or DY in this paper) and semi-inclusive deep inelastic scattering (SIDIS) identifies clearly the sources of non-perturbative QCD effects as the transverse momentum dependent (TMD) distributions and, separately, their evolution kernel [@Collins:1989gx; @Bacchetta:2006tn; @Bacchetta:2008xw; @Becher:2010tm; @Collins:2011zzd; @GarciaEchevarria:2011rb; @Echevarria:2012js; @Echevarria:2014rua; @Chiu:2012ir; @Vladimirov:2017ksc; @Scimemi:2018xaf]. The extraction of these non-perturbative (NP) elements from data is then a major challenge for modern phenomenology [@Angeles-Martinez:2015sea].
In this article, we consider the unpolarized observables that have the simplest structure and are accessible in a relatively large number of experiments. It allows us to extract the quark unpolarized TMD distributions and the NP part of TMD evolution. There were plenty of extraction of these elements within various schemes [@Sun:2013hua; @Anselmino:2013lza; @Signori:2013mda; @DAlesio:2014mrz; @Aidala:2014hva; @Bacchetta:2017gcc; @Scimemi:2017etj; @Bertone:2019nxa; @Vladimirov:2019bfa]. The distinctive feature of this work is the simultaneous consideration of two kinds of reactions: DY and SIDIS. Previously, a global fit of both processes was performed only in ref. [@Bacchetta:2017gcc]. *We demonstrate that the global description of both processes is straightforward and does not meet any obstacle.* The description is based on the latest theory developments, such as next-to-next-to-leading order (NNLO) and N$^3$LO perturbative parts together with $\zeta$-prescription. In addition, we make a special emphasis on the following topics that previously did not attract serious attention: the universality and the theory uncertainties.
The theoretical work done in recent years for the development of the elements of TMD factorization has been noticeable. The significant efforts were committed in the perturbative calculations for TMD distributions at small-$b$ [@Gehrmann:2014yya; @Echevarria:2015byo; @Echevarria:2015usa; @Echevarria:2016scs; @Li:2016ctv; @Vladimirov:2016dll; @Luo:2019hmp]. Together with the N$^3$LO results for universal QCD anomalous dimensions [@Gehrmann:2010ue; @Baikov:2016tgj; @Moch:2017uml; @Moch:2018wjh; @Lee:2019zop], it leads to an extremely accurate perturbative input. The consistent composition of all elements is made employing the $\zeta$-prescription [@Scimemi:2018xaf; @Vladimirov:2019bfa]. The $\zeta$-prescription is essential for current and future TMD phenomenology because it grants a unified approach to observables irrespectively of the order of perturbative matching. So, the collinear matching procedure that is fundamental for resummation approaches (such as in refs. [@Landry:2002ix; @Qiu:2000hf; @Bozzi:2008bb; @Becher:2010tm; @Catani:2012qa; @Bizon:2018foh; @Bizon:2019zgf]) or $b^*$-like prescriptions (such as in refs.[@Collins:1981va; @Collins:2011zzd; @Aybat:2011zv; @Bacchetta:2017gcc]), is considered just as a part of the model for a TMD distribution in the $\zeta$-prescription. Therefore, *unpolarized TMD distributions (extracted in this work with NNLO matching) and the TMD evolution (extracted in this work with NNLO and N$^3$LO matching) are entirely universal and could be used for the description of other processes*, where the matching is not known at such a high order.
The factorization theorem declares a strong universality of NP elements. This statement refers to two important facts relevant for the present study: a) the evolution kernel is the same for all processes where the TMD factorization theorem is valid; b) the TMD parton distribution functions (TMDPDF) are the same in DY and SIDIS experiments. The test of universality needs an analysis of different types of experiments at the same time. Although the universality is a cornerstone of the approach, we have not found any dedicated tests in the literature. In order to check and proof universality properties of the TMD approach, we perform a three steps analysis:
- Firstly, we consider only the DY measurements, and analyze TMDPDF $f_1(x,b)$ and rapidity anomalous dimension (RAD), $\mathcal{D}(\mu,b)$. The DY data sets have a vast span in $x$ and $Q$, therefore, it is possible to extract $f_1$ (that dictates the $x$-dependence of the cross-section) and $\mathcal{D}$ (that dictates the $Q$-dependence of the cross-section) without a significant correlation between these functions. This analysis is conceptually similar to the previous work [@Bertone:2019nxa], albeit some improvements.
- Using the outcome of the previous step ($\mathcal{D}$ and $f_1$), we consider the SIDIS measurements and extract the TMDFF, $D_1$. Assuming the universality of TMD distributions, one should be able to describe the SIDIS cross-section with a single extra function $D_1$. It is a non-trivial statement since the SIDIS cross-section has 4-degrees of freedom, only two of which are affected by $D_1$. Additionally, the present SIDIS data are concentrated in a range of small-$Q$ that is unreachable for DY experiments.
- Finally, we perform a simultaneous fit of DY and SIDIS data. Given the excellent quality of the separate DY and SIDIS fits, this stage could provide only a fine-tune of non-perturbative parameters.
These three independent analyses provide a consistent and congruent picture of the TMD factorization and allow the extraction of three non-perturbative functions (unpolarized quark TMDPDF, TMDFF and quark evolution kernel). We find that our results are in full agreement with the depicted scenario, which gives a solid confirmation of the declared universality.
Apart from the described test of universality and extraction of TMD distributions in this work, we perform many additional tests and checks of the TMD approach: we test the phenomenological limits of the TMD factorization for SIDIS; we check the dependence of the TMD prediction on the collinear input; we perform an overall test of impact power suppressed contributions to the TMD factorization; we check the impact of experimental constraints on the final phase space configurations (like fiducial cross sections and lepton cuts at LHC, bin shapes in HERMES kinematics). Altogether the tests can form a comprehensive picture of TMD factorization and its accuracy. We have observed the impact of some input uncertainties, f.i. the ones from collinear PDF, to the prediction is unlucky large. Still, we restrict ourself to the indication of problematic regions, leaving it as an invitation for the further developments in the future.
Given the number of details needed for the presentation of this work, we split the discussions into almost independent parts. The first part, sec. \[sec:2\], contains the description of the TMD factorization theorem for unpolarized DY and SIDIS cases. In this section, we articulate all relevant formulas, including a lot of small corrections and details that were previously only mentioned. This part provides a comprehensive collection of theory results, which can be useful for comparison with other works and future tests, and it can be considered a theory review. The sec. \[sec:data\] is devoted to the review of the available SIDIS and DY data suitable for unpolarized TMD phenomenology. The sec. \[sec:fitprocedure\] presents the details of the comparison of the theory expression with the experimental data. It contains the definition of $\chi^2$-test, the interpretation of the experimental environment, and some details of the numerical implementation that is made by `artemide` package [@web]. The following sections \[sec:DY-fit\], \[sec:SIDIS-fit\] and \[sec:GLOBAL-fit\] describe the fit program outlined earlier, and they are devoted to DY(only), SIDIS(only), and DY and SIDIS(together) fits. Each of these sections contains several subsections describing impact tests of specific peculiarities. Finally, we collect the information on the resulting NP functions in sec. \[sec:final\].
Cross sections in TMD factorization {#sec:2}
===================================
In this section, we present in detail the cross sections of SIDIS and DY processes in the TMD factorization, skipping their derivation that can be found in refs. [@Collins:1989gx; @Bacchetta:2006tn; @Bacchetta:2008xw; @Becher:2010tm; @Collins:2011zzd; @GarciaEchevarria:2011rb; @Echevarria:2012js; @Echevarria:2014rua; @Chiu:2012ir; @Vladimirov:2017ksc; @Scimemi:2018xaf]. The main aim of the section is to collect all pieces of information on theoretical approximations and models that are used in the fit procedure.
SIDIS cross-section
-------------------
The (semi-inclusive) deep-inelastic scattering (SIDIS) is defined by the reaction $$\begin{aligned}
\ell(l)+H(P)\to \ell(l')+h(p_h)+X,\end{aligned}$$ where $l$ is a lepton, $H$ and $h$ are respectively the target and the fragmenting hadrons, and $X$ is the undetected final state. The vectors in brackets denote the momenta of each particle. The masses of the particles are $$\begin{aligned}
P^2=M^2,\qquad p_h^2=m^2,\qquad l^2=l'^2=m_l^2\simeq0.\end{aligned}$$ In the following, we neglect the lepton masses, but keep the effects of the hadron masses.
Approximating the interaction of a lepton and a hadron by a single photon exchange, one obtains the differential cross-section $$\begin{aligned}
\label{def:xSec-0}
d\sigma=\frac{2}{s-M^2} \frac{ \alpha^2_{\text{em}}}{(q^2)^2} L_{\mu\nu}W^{\mu\nu}\frac{d^3l'}{2E'}\frac{d^3p_h}{2E_{h}},\end{aligned}$$ with $q=l-l'$ being the momentum of the intermediate photon. Here, the scattering flux-factor, $((s-(m_l+M)^2)(s-(m_l-M)^2))^{-1/2}$ is evaluated at vanishing lepton mass; the factors $q^2$ come from the photon propagators $\Delta^{\mu\nu}=g^{\mu\nu}/(q^2+i0)$ and $\alpha_{\text{em}}=e^2/4\pi$ is QED coupling constant. The last factors in eq. (\[def:xSec-0\]) are the phase-space differentials for the detected hadron and lepton, with $E'$($E_h$) being their energies. The leptonic and hadronic tensors ($L^{\mu\nu}$ and $W^{\mu\nu}$) are $$\begin{aligned}
L_{\mu\nu}&=&e^{-2}\langle l'|J_{\mu}(0)|l\rangle \langle l|J^\dagger_{\nu}(0)|l'\rangle,{\nonumber}\\
W_{\mu\nu}&=&e^{-2}\int \frac{d^4x}{(2\pi)^4}e^{-i(x\cdot q)}\sum_X\langle P|J^\dagger_{\mu}(x)|p_h,X\rangle \langle p_h,X|J_{\nu}(0)|P\rangle,\end{aligned}$$ where $e$ is the lepton charge, and $J^\mu$ is the electro-magnetic current.
### Factorization for hadronic tensor
In this work we consider the transverse momentum dependence of the cross section which is factorizable in terms of transverse momentum dependent (TMD) distributions in the limit of $q_T\ll Q$, where $q_T$ is defined in eq. (\[def:qT<->pT\]) and $Q$ is the di-lepton invariant mass. We refer to the literature about the proof of factorization of the processes related to this work [@Collins:1989gx; @Becher:2010tm; @Collins:2011zzd; @GarciaEchevarria:2011rb; @Echevarria:2012js; @Echevarria:2014rua; @Chiu:2012ir; @Vladimirov:2017ksc]. In order to specify the properties of the TMD distributions and the factorized hadronic tensor, we start fixing the basic notation.
The formulation of the factorization theorem is done in the hadronic Breit frame (alternatively, we can call it “the factorization frame”), where the momenta of hadrons are almost light-like and back-to-back. The light-like direction to which the hadrons are aligned defines the decomposition of their momenta, as we detail for each type of experiment. For SIDIS we set $$\begin{aligned}
P^\mu&=&P^+ \bar n^\mu+\frac{M^2}{2P^+}n^\mu,\qquad p_h^\mu =p_h^- n^\mu+\frac{m^2}{2p^-_h}\bar n^\mu,\end{aligned}$$ with $n^2=\bar n^2=0,$ $ (n\bar n)=1$. Here, we have also introduced the common notation of a vector decomposition $$\begin{aligned}
v^\mu=v^+ \bar n^\mu+v^- n^\mu+v_T^\mu,\qquad v^+=(nv),\qquad v^-=(\bar nv),\qquad (nv_T)=(\bar nv_T)=0.\end{aligned}$$ The traverse component of a vector is extracted with the projector $$\begin{aligned}
\label{def:gT}
v_T^\mu=g_T^{\mu\nu}v_\nu,\qquad g_T^{\mu\nu}=g^{\mu\nu}-n^\mu \bar n^\nu-\bar n^\mu n^\nu.\end{aligned}$$ We also use the convention that the bold font denotes vectors that have only transverse components. So, they obey Euclidian scalar product: $$\begin{aligned}
{\bm{v}}_T^2=-v_T^2>0.\end{aligned}$$ For unpolarized hadrons, the factorized hadronic tensor in its complete form reads $$\begin{aligned}
\label{SIDIS:Wmunu}
W^{\mu\nu}&=&-2z_S \sum_{f}e_f^2 |C_V(Q^2,\mu^2)|^2\int \frac{d^2b}{(2\pi)^2}e^{-i(bq_T)}
\\{\nonumber}&&\times \Big[ g_T^{\mu\nu}
f_{1,f{\leftarrow}H}{\left(}x_{S},b;\mu,\zeta_1{\right)}D_{1,f\to h}{\left(}z_S,b;\mu,\zeta_2{\right)}\\{\nonumber}&& +(g_T^{\mu\nu}b^2-2b^\mu b^\nu)\frac{m\,M}{4} h_{1,f{\leftarrow}H}^\perp{\left(}x_S,b;\mu,\zeta_1{\right)}H_{1,f\to h}^\perp{\left(}z_S,b;\mu,\zeta_2{\right)}\Big]+
O{\left(}\frac{q_T^2}{Q^2}{\right)},\end{aligned}$$ where the index $f$ in the sum runs through all quark flavours (including anti-quarks), $e_f$ is a charge of a quark measured in units of $e$. The function $C_V$ is the matching coefficient for vector current to collinear/anti-collinear vector current and the factorization ($\mu$) and rapidity ($\zeta$) scales typical of the TMD factorization are shown explicitly. The variables $x_S$ and $z_S$ are the collinear fractions of parton momentum, $$\begin{aligned}
\label{def:x1z1}
x_S=-\frac{q^+}{P^+},\qquad z_S=\frac{p_h^-}{q^-},\end{aligned}$$ which are invariant under boosts along the direction of $n$, $\bar n$, but are not invariant for a generic Lorentz transformation.
The unpolarized TMDPDF and TMDFF from partons of flavor $f$ are defined as $$\begin{aligned}
\label{def:f1}
&&f_{1,f{\leftarrow}h}(x,b;\mu,\zeta)=
\\{\nonumber}&&\qquad\int \frac{d\lambda}{2\pi}e^{-ix\lambda p^+}
\sum_X \langle h(p)|\bar q(n \lambda+b) W^\dagger_n(n\lambda+b)\frac{\gamma^+}{2}|X\rangle\langle X|W_n(0)q(0)|h(p)\rangle,
\\\label{def:D1}
&&D_{1,f{\leftarrow}h}(z,b;\mu,\zeta)=
\\{\nonumber}&&\qquad\frac{1}{2zN_c}\int \frac{d\lambda}{2\pi}e^{i\lambda p^+/z}\sum_X\langle 0|\frac{\gamma^+}{2}W_n(n\lambda+b) q(n \lambda+b)|h(p),X\rangle\langle h(p),X|\bar q(0)W^\dagger_n(0)|0\rangle.\end{aligned}$$ Here, $W_v(x)$ are Wilson lines rooted at $x$ and pointing along vector $v$ to infinity. In the case of SIDIS, the Wilson lines in TMDPDF(TMDFF) points to future (past) infinity. The functions $h_1^\perp$ and $H_1^\perp$ are Boer-Mulders and Collins functions respectively and they are defined as $$\begin{aligned}
\label{def:h1}
&&iM \epsilon_T^{\alpha\beta}b_\beta h^\perp_{1,f{\leftarrow}h}(x,b;\mu,\zeta)=
\\{\nonumber}&&\qquad \int \frac{d\lambda}{2\pi}e^{-ix\lambda p^+}\sum_X\langle h(p)|\bar q(n \lambda+b)W^\dagger_n(n\lambda+b)\frac{i\sigma^{\alpha +}\gamma^5}{2}|X\rangle\langle X|W_n(0)q(0)|h(p)\rangle,
\\\label{def:H1}
&&iM \epsilon_T^{\alpha\beta}b_\beta H^\perp_{1,f{\leftarrow}h}(z,b;\mu,\zeta)=
\\{\nonumber}&& \frac{1}{2zN_c}\int \frac{d\lambda}{2\pi}e^{i\lambda p^+/z}\langle 0|\frac{i\sigma^{\alpha +}\gamma^5}{2}W_n(n\lambda+b)q(n \lambda+b)|h(p),X\rangle\langle h(p),X|\bar q(0)W^\dagger_n(0)|0\rangle,\end{aligned}$$ where $\epsilon_T^{\mu\nu}=\epsilon^{+-\mu\nu}$. In formulas (\[def:f1\]-\[def:H1\]) we have omit for brevity the obvious details of operator definitions, such as $T$($\bar T$)-ordering, color and spinor indices, rapidity and ultraviolet renormalization factors.
Boer-Mulders and Collins functions (\[def:h1\],\[def:H1\]) do not contribute to the angle averaged cross-section. However they can appear when cuts on the phase space distributions of final particles are introduced by the experimental setup. In this work we will not consider these effects, and leave their study for the future.
The TMD distributions depend on ${\bm{b}}^2$ only. Therefore, the angular dependence can be integrated explicitly with the result $$\begin{aligned}
\label{SIDIS:Wmunu-J}
W^{\mu\nu}&=&\frac{z_S}{\pi} \sum_{f}e_f^2 \Big[(-g_T^{\mu\nu})W^f_{f_1D_1}(Q,|q_T|,x_S,z_S)
\\{\nonumber}&&
+{\left(}g_T^{\mu\nu}-2\frac{q_T^\mu q_T^\nu}{q_T^2}{\right)}W^f_{h_1^\perp H_1^\perp}(Q,|q_T|,x_S,z_S)\Big]+
O{\left(}\frac{q_T^2}{Q^2}{\right)},\end{aligned}$$ where $$\begin{aligned}
\label{def:WfD}
W^f_{f_1D_1}(Q,q_T,x_S,z_S)&=&|C_V(Q^2,\mu^2)|^2
\\{\nonumber}&& \times\int_0^\infty db\,b J_0(bq_T)f_{1,f{\leftarrow}H}{\left(}x_S,b;\mu,\zeta_1{\right)}D_{1,f\to h}{\left(}z_S,b;\mu,\zeta_2{\right)},
\\\label{def:WhH}
W^f_{h^\perp_1H^\perp_1}(Q,q_T,x_S,z_S)&=&\frac{mM}{4}|C_V(Q^2,\mu^2)|^2
\\{\nonumber}&& \times \int_0^\infty db\,b^3 J_2(bq_T)h_{1,f{\leftarrow}H}^\perp{\left(}x_S,b;\mu,\zeta_1{\right)}H_{1,f\to h}^\perp{\left(}z_S,b;\mu,\zeta_2{\right)}.\end{aligned}$$ The functions $W^f$ are dimensionless and scale-independent functions. The experimental configurations are not usually provided in the factorization frame, and the correspondence between the measured quantities and the ones that appear in the factorization theorem is often non-trivial. It happens in fact, that a Lorentz transformation affects the power corrections to the cross section presented here. We detail this in the next sections.
### Kinematic variables
For the SIDIS cross-section one introduces the following scalar variables: $$\begin{aligned}
Q^2=-q^2,\qquad x=\frac{Q^2}{2(Pq)},\qquad y=\frac{(Pq)}{(Pl)},\qquad z=\frac{(Pp_h)}{(Pq)}.\end{aligned}$$ In the experimental environment one typically measures the transverse momentum defined as the one of the produced hadron with respect to the plane formed by vectors $q$ and $P$. The projector corresponding to these transverse components is given by the tensor $g_\perp^{\mu\nu}$ defined as $$\begin{aligned}
\label{def:gPerp-SIDIS}
g_\perp^{\mu\nu}&=&g^{\mu\nu}-\frac{1}{M^2Q^2+(Pq)^2}{\left[}Q^2 P^\mu P^\nu+(Pq)(P^\mu q^\nu+q^\mu P^\nu)-M^2 q^\mu q^\nu{\right]}\\{\nonumber}&=&g^{\mu\nu}-\frac{1}{Q^2(1+\gamma^2)}{\left[}2x^2 P^\mu P^\nu+2x(P^\mu q^\nu+q^\mu P^\nu)-\gamma^2 q^\mu q^\nu{\right]}.\end{aligned}$$ In what follows, we denote the transverse components of $v^\mu$ in the factorization frame as $v_T^\mu$, see eq. (\[def:gT\]), while transverse components projected by $g_\perp$ are $v_\perp^\mu$. To describe the target- and produced-mass corrections, it is convenient to use the following combinations $$\begin{aligned}
\label{def:mass-var}
\gamma=\frac{2Mx}{Q},\qquad \varsigma=\gamma\frac{m}{zQ},\qquad \varsigma^2_\perp=\gamma^2\frac{m^2+{\bm{p}}^2_{h\perp}}{z^2Q^2}.\end{aligned}$$ The definition of $\varsigma^2_\perp$ in eq. (\[def:mass-var\]) contains ${{\bm{p}}}^2_{h\perp}=p_{h \mu} p_{h \nu } g_\perp^{\mu\nu}$.
The measured transverse momentum $p_\perp$ is different from the one defined in the TMD factorization. The transverse momentum used in the factorization theorem $q_T$ is defined with respect to the hadron-hadron-plane. These corresponding transverse components are extracted by the tensor $g_T^{\mu\nu}$ in eq. (\[def:gT\]). In terms of hadron momenta the tensor $g_T$ reads $$\begin{aligned}
\label{def:gT-SIDIS}
g_T^{\mu\nu}&=&g^{\mu\nu}-\frac{1}{m^2M^2-(Pp_h)^2}{\left[}m^2 P^\mu P^\nu-(Pp_h)(P^\mu p_h^\nu+p_h^\mu P^\nu)+M^2 p_h^\mu p_h^\nu{\right]}\\{\nonumber}&=&g^{\mu\nu}+\frac{1}{Q^2(1-\varsigma^2)}{\left[}4\frac{x^2}{\gamma^2}\varsigma^2 P^\mu P^\nu-\frac{2x}{z}(P^\mu p_h^\nu+p_h^\mu P^\nu)+\frac{\gamma^2}{z^2} p_h^\mu p_h^\nu{\right]}.\end{aligned}$$ Using projectors (\[def:gT-SIDIS\]) and (\[def:gPerp-SIDIS\]), it is straightforward to derive the relation between $q_T^2=q_\mu q_\nu g_T^{\mu\nu}$ and $p_\perp^2=p_h^\mu p_h^\nu g_{\perp,\mu\nu}$: $$\begin{aligned}
\label{def:qT<->pT}
q_T^2=\frac{p_\perp^2}{z^2}\frac{1+\gamma^2}{1-\varsigma^2}.\end{aligned}$$
Using these definition we rewrite the elements of the SIDIS cross-section formula in the terms of observable variables. The differential volumes of the phase space are $$\begin{aligned}
\label{th:phase-elem-1}
\frac{d^3l'}{2E'}=\frac{y}{4x}dQ^2dx d\psi,\qquad \frac{d^3p_h}{2E_h}=\frac{1}{\sqrt{1-\varsigma^2_\perp}}\frac{dz d^2p_\perp}{2z}
=\frac{1}{\sqrt{1-\varsigma^2_\perp }}\frac{dz d{\bm{p}}^2_\perp d \varphi}{4z},\end{aligned}$$ where $\psi$ is the azimuthal angle of scattered lepton, and $\varphi$ is the azimuthal angle of the produced hadron. The variables $x_S$ and $z_S$ in eq. (\[def:x1z1\]) are $$\begin{aligned}
\label{def:SIDIS-x1}
x_S&=&-x\frac{2}{\gamma^2}{\left(}1-\sqrt{1+\gamma^2{\left(}1-\frac{{\bm{q}}_T^2}{Q^2}{\right)}}{\right)},
\\\label{def:SIDIS-z1}
z_S&=&-z\frac{1-\sqrt{1+\gamma^2{\left(}1-\frac{{\bm{q}}_T^2}{Q^2}{\right)}}}{\gamma^2}\frac{1+\sqrt{1-\varsigma^2}}{1-\frac{{\bm{q}}_T^2}{Q^2}}=z\frac{x_S}{x}\frac{1+\sqrt{1-\varsigma^2}}{2{\left(}1-\frac{{\bm{q}}_T^2}{Q^2}{\right)}},\end{aligned}$$ where we have used the variable ${\bm{q}}_T^2$ (\[def:qT<->pT\]) for simplicity.
The kinematic corrections presented above are usually small when $Q\gg M,m$. In this case the relation between observed and factorization variables simplifies $$\begin{aligned}
\label{eq:APPX}
{\bm{q}}_T^2\simeq \frac{{\bm{p}}_\perp^2}{z^2},\qquad x_S\simeq x{\left(}1-\frac{{\bm{q}}_T^2}{Q^2}{\right)},\qquad z_S\simeq z.\end{aligned}$$ Notice that the data of SIDIS at our disposal are taken at energies comparable with hadron masses and thus target mass correction could be significant. The contributions dependent of hadron masses could in principle be classified as power corrections. However we consider more appropriate to distinguish these corrections from others of different origin. Thus we will not use the approximate formulas in eq. (\[eq:APPX\]). The phenomenological test of this assumption is given in sec. \[sec:SIDIS-power-corr\].
### Leptonic tensor
The leptonic tensor for unpolarized SIDIS is $$\begin{aligned}
L_{\mu\nu}=2(l_\mu l'_\nu+l'_\mu l_\nu- (ll') g_{\mu\nu} ).\end{aligned}$$ In order to express the convolution of the leptonic tensor with a hadronic tensor we define the azimuthal angle of a produced hadron as [@Bacchetta:2006tn]: $$\begin{aligned}
\cos\phi=\frac{-l_\mu p_{h\nu}g_\perp^{\mu\nu}}{\sqrt{-l_\alpha l_\beta g_\perp^{\alpha\beta} }\sqrt{-p_{h\alpha'} p_{h\beta'} g_\perp^{\alpha'\beta'} }}\end{aligned}$$ and we define $$\varepsilon=\frac{1-y-\frac{\gamma^2y^2}{4}}{1-y+\frac{y^2}{2}+\frac{\gamma^2 y^2}{4}}.$$ As the result we obtain $$\begin{aligned}
\label{th:SIDIS-L1}
(-g_T^{\mu\nu}) L_{\mu\nu}&=&\frac{2Q^2}{1-\varepsilon}\Big[1+\frac{{\bm{p}}_\perp^2}{Q^2 z^2}\frac{\varepsilon-\frac{\gamma^2}{2}}{1-\varsigma^2}
\\{\nonumber}&&
-\cos\phi \frac{\sqrt{2\varepsilon(1+\varepsilon){\bm{p}}_\perp^2}}{zQ}\frac{\sqrt{1-\varsigma_\perp^2}}{1-\varsigma^2}
-\cos(2\phi)\frac{\varepsilon{\bm{p}}_\perp^2 \gamma^2 }{2 z^2 Q^2(1-\varsigma^2)}
\Big].\end{aligned}$$ The kinematical rearrangements of the variables produce the appearance of the $\cos\phi$ and $\cos2\phi$ terms in the second line of eq. (\[th:SIDIS-L1\]), that is, there are contributions to the structure functions $F_{UU}^{\cos\phi}$ and $F_{UU}^{\cos2\phi}$, see also [@Anselmino:2005nn]. Similarly, the convolution of lepton tensor with the spin-1 part $$\begin{aligned}
\label{th:SIDIS-L2}
{\left(}g_T^{\mu\nu}-2\frac{q_T^\mu q_T^\nu}{q_T^2}{\right)}L_{\mu\nu}&=&\frac{2Q^2}{1-\varepsilon}\Big[\varepsilon \cos(2\phi)
{\left(}1-\frac{{\bm{p}}_\perp^2 \gamma^2 }{2 z^2 Q^2(1-\varsigma^2)}{\right)}\\{\nonumber}&&
-\cos\phi \frac{\sqrt{2\varepsilon(1+\varepsilon){\bm{p}}_\perp^2}}{zQ}\frac{\sqrt{1-\varsigma_\perp^2}}{1-\varsigma^2}
+\frac{{\bm{p}}_\perp^2}{Q^2 z^2}\frac{\varepsilon-\frac{\gamma^2}{2}}{1-\varsigma^2}
\Big].\end{aligned}$$ produces also contribution to the $\cos\phi$ and $\cos2\phi$ parts.
The terms $\sim {\bm{p}}_\perp^2/Q^2$ presented in eqs.(\[th:SIDIS-L1\],\[th:SIDIS-L2\]) are to be modified by the power corrections to the TMD factorization. Moreover, these terms could be artifacts of the violation of transversality of hadron tensor (\[th:QED-ward\]) at the leading order of the TMD factorization. The status of these corrections could not be resolved without the computation of the power corrections.
### SIDIS cross-section in TMD factorization
Combining together the expressions for the hadronic tensor in eq. (\[SIDIS:Wmunu-J\]), the differential phase-space volume in eq. (\[th:phase-elem-1\]), the leptonic tensor in eq. (\[th:SIDIS-L1\], \[th:SIDIS-L2\]), the cross-section in eq. (\[def:xSec-0\]), and integrating over the azimuthal angles we obtain $$\begin{aligned}
\label{SIDIS:xSec}
&&\frac{d\sigma}{dxdz dQ^2 d{\bm{p}}_\perp^2}=\frac{\pi}{\sqrt{1-\varsigma_\perp^2}}\frac{\alpha_{\text{em}}^2}{Q^4}\frac{y^2}{1-\varepsilon}\frac{z_S}{z}
\\{\nonumber}&&
\qquad\times \sum_{f}e_f^2\Big[
{\left(}1+\frac{{\bm{q}}_T^2}{Q^2}\frac{\varepsilon-\frac{\gamma^2}{2}}{1+\gamma^2}{\right)}W_{f_1D_1}^f(Q,\sqrt{{\bm{q}}^2_T},x_S,z_S)
+\frac{{\bm{q}}_T^2}{Q^2}\frac{\varepsilon-\frac{\gamma^2}{2}}{1+\gamma^2}W_{h_1^\perp H_1^\perp}^f(Q,\sqrt{{\bm{q}}^2_T},x_S,z_S)
\Bigg],\end{aligned}$$ where $x_S$, $z_S$ and ${\bm{q}}^2_T$ are the functions of ${\bm{p}}_\perp^2$, $x$, and $z$ defined in eq. (\[def:SIDIS-x1\]), (\[def:SIDIS-z1\]) and eq. (\[def:qT<->pT\]), correspondingly. The functions $W$ are defined in eq. (\[def:WfD\], \[def:WhH\]).
The final expression for the cross section in eq. (\[SIDIS:xSec\]) explicitly shows that a part of power corrections has a kinematical origin, and thus are independent of the factorization theorem and can be taken into account with the present formalism without contradictions. Examples are, the factor $\sqrt{1-\varsigma_\perp^2}$ that is a part of the phase-space element, and the difference between $z_S$ and $z$ that is a consequence of the TMDFF definition. The separation between kinematical power corrections and higher orders in the power expansion of the cross-section is however not neat, because a detailed study of the factorization theorem correction is still not complete. The admixture of these effects can be seen in the second line of eq. (\[SIDIS:xSec\]), which is the present status of our understanding. In the fit of the presented later we omit the contribution $W_{h_1^\perp H_1^\perp}$ in eq. (\[SIDIS:xSec\]) and perform a check of the importance of mass-corrections for the agreement with experimental data in sec. \[sec:SIDIS-power-corr\].
DY cross-section
----------------
The Drell-Yan pair production (or DY for shortness) is defined by the process $$\begin{aligned}
h_1(P_1)+h_2(P_2)\to l(l)+l'(l')+X,\end{aligned}$$ where $l$, $l'$ are the lepton pair, $h_1$, $h_2$ are the colliding hadrons, and the symbols in brackets denote the momentum of each particle. In the following, we include the hadron masses and we neglect the lepton masses: $$\begin{aligned}
P_1^2=M_1^2,\qquad P_2^2=M_2^2,\qquad l^2=l'^2=m_l^2\simeq0.\end{aligned}$$ The energies of the DY experiments are higher than the SIDIS ones, and the interference of electro-weak (EW) bosons must be included. Approximating the interactions of leptons and hadrons by a single EW-gauge boson exchange one obtains the following expression for the differential cross-section $$\begin{aligned}
\label{th:DY-xsec0}
d\sigma=\frac{2\alpha^2_{\text{em}}}{s} \frac{d^3l}{2E}\frac{d^3l'}{2E'}\sum_{GG'} L^{GG'}_{\mu\nu}W_{GG'}^{\mu\nu}\Delta_G(q)\Delta^*_{G'}(q).\end{aligned}$$ where $q=l+l'$, $\alpha_{\text{em}}=e^2/4\pi$ is the QED coupling constant and the index $G$ runs over gauge bosons $\gamma$, $Z$. Here, we have approximated the exact flux factor $[(s-(M_1-M_2)^2)(s-(M_1+M_2)^2)]^{-1/2}$ with $1/s$ because the corrections of order $M^2/s$ are negligibly small for any considered data set. The function $\Delta_G(q)$ is defined as $$\begin{aligned}
\Delta_G(q)=\frac{1}{q^2+i0}\delta_{G\gamma}+\frac{1}{q^2-M_Z^2+i \Gamma_Z M_z}\delta_{GZ},\end{aligned}$$ with $M_Z=91.188$GeV and $\Gamma_Z=2.495$GeV [@Olive:2016xmw]. Finally, $L_{GG'}^{\mu\nu}$ and $W_{GG'}^{\mu\nu}$ are the leptonic and hadronic tensors that are defined as $$\begin{aligned}
\label{DY:L1}
L^{GG'}_{\mu\nu}&=&e^{-2}\langle 0|J^G_{\mu}(0)|l, l'\rangle \langle l, l'|J^{G'\dagger}_{\nu}(0)|0\rangle,
\\
W_{\mu\nu}&=&e^{-2}\int \frac{d^4x}{(2\pi)^4}e^{-i(x\cdot q)}\sum_X\langle P_1,P_2|J^{G\dagger}_{\mu}(x)|X\rangle \langle X|J^{G'}_{\nu}(0)|P_1,P_2\rangle,\end{aligned}$$ where $e$ is the lepton charge, and $J^G_\mu$ is the current for the production of EW gauge boson $G$.
Integrating the cross-section over a lepton momentum one finds $$\begin{aligned}
\label{th:DY-xsec1}
d\sigma=\frac{2\alpha^2_{\text{em}}}{s}d^4q\sum_{GG'}\widehat L^{GG'}_{\mu\nu}W_{GG'}^{\mu\nu}\Delta_G(q)\Delta^*_{G'}(q),\end{aligned}$$ where $q$ is the momentum of the EW-gauge boson. The new lepton tensor is $$\begin{aligned}
\label{DY:Lhat}
\widehat L^{GG'}_{\mu\nu}=\int \frac{d^3l}{2E}\frac{d^3l'}{2E'} \delta^{(4)}(l+l'-q)L^{GG'}_{\mu\nu}.\end{aligned}$$
### Factorization for hadronic tensor {#sec:DY:hadron-tensor}
The factorization for DY hadronic tensor is totally analogous to the SIDIS case. The vectors $n$ and $\bar n$ are defined by hadrons, $$\begin{aligned}
\label{def:DY-nn}
P_1^\mu&=&P_1^+ \bar n^\mu+\frac{M_1^2}{2P_1^+}n^\mu\simeq P_1^+ \bar n^\mu,\qquad P_2^\mu =P_2^- n^\mu+\frac{M_2^2}{2P_2^-}\bar n^\mu\simeq P_1^- n^\mu,\end{aligned}$$ where on r.h.s. the small contributions $\sim M^2/s$ are neglected. The inclusion of weak-boson exchange requires the consideration of a more general current. To this purpose we define $$\begin{aligned}
\label{def:J-EW}
J^\mu_G(x)=\bar q(x)[g^G_R\gamma^\mu(1+\gamma^5)+g^G_L\gamma^\mu(1-\gamma^5)]q(x),\end{aligned}$$ with the EW coupling constants $$\begin{aligned}
\label{def:gRgL}
g^\gamma_R=g^\gamma_L=\frac{e_f}{2},\qquad g_R^Z=\frac{-e_fs_W^2}{2s_Wc_W},\qquad g^Z_L=\frac{T_3-e_f s_W^2}{2s_Wc_W},\end{aligned}$$ where $e_f$ is the electric charge of a particle (in units of $e$), $T_3$ is the third projection of weak isospin, $s_W=\sin(\theta_W)$, $c_W=\cos(\theta_W)$.
Collecting all this, the unpolarized part of the factorized hadronic tensor reads $$\begin{aligned}
\label{DY:Wmunu}
W^{\mu\nu}_{GG'}&=&\sum_{f}|C_V(-Q^2,\mu^2)|^2\int \frac{d^2b}{(2\pi)^2}e^{-i(bq_T)} \Big[
\\{\nonumber}&& -2g_T^{\mu\nu}(g^G_Rg^{G'}_R+g^G_Lg^{G'}_L){\left(}f_{1,f{\leftarrow}h_1}f_{1,\bar f{\leftarrow}h_2}+f_{1,\bar f{\leftarrow}h_1}f_{1,f{\leftarrow}h_2}{\right)}\\{\nonumber}&&
-\frac{g_T^{\mu\nu}b^2-2b^\mu b^\nu}{2}M_1M_2(g^G_Rg^{G'}_R+g^G_Lg^{G'}_L){\left(}h^\perp_{1,f{\leftarrow}h_1}h^\perp_{1,\bar f{\leftarrow}h_2}+h^\perp_{1,\bar f{\leftarrow}h_1}h^\perp_{1,f{\leftarrow}h_2}{\right)}\\{\nonumber}&&
-2i\epsilon_T^{\mu\nu}(g^G_Rg^{G'}_R-g^G_Lg^{G'}_L){\left(}f_{1,f{\leftarrow}h_1}f_{1,\bar f{\leftarrow}h_2}-f_{1,\bar f{\leftarrow}h_1}f_{1,f{\leftarrow}h_2}{\right)}\\{\nonumber}&&
+i(\epsilon_T^{\mu\alpha} b_\alpha b^\nu+\epsilon_T^{\nu\alpha} b_\alpha b^\mu)\frac{M_1M_2}{2}(g^G_Rg^{G'}_L-g^G_Rg^{G'}_L){\left(}h^\perp_{1,f{\leftarrow}h_1}h^\perp_{1,\bar f{\leftarrow}h_2}-h^\perp_{1,\bar f{\leftarrow}h_1}h^\perp_{1,f{\leftarrow}h_2}{\right)}\Big]
\\{\nonumber}&& + O{\left(}\frac{q_T^2}{Q^2}{\right)},\end{aligned}$$ where $f$ runs through all quark flavours. The functions $f_{1,f{\leftarrow}h_1}$ are the TMDPDF and the functions $h^\perp_{1,f{\leftarrow}h_1}$ are the Boer-Mulders functions, defined in eq. (\[def:f1\], \[def:h1\]). In eq. (\[DY:Wmunu\]) we have omitted arguments of TMD distributions for brevity, however this can be included with e.g. $$\begin{aligned}
f_{1,f{\leftarrow}h_1}f_{1,\bar f{\leftarrow}h_2}\to f_{1,f{\leftarrow}h_1}(x_1,b;\mu,\zeta_1)f_{1,\bar f{\leftarrow}h_2}(x_2,b;\mu,\zeta_2),\end{aligned}$$ and similarly for all products of TMD distributions. The variables $x_1$ and $x_2$ measure the collinear fractions of parton momenta, $$\begin{aligned}
\label{def:x1x2}
x_1=\frac{q^+}{P_1^+},\qquad x_2=\frac{q^-}{P_2^-}.\end{aligned}$$ The flavor indices $f$ runs though all flavors of quarks and anti-quarks. Here, the flavor index $\bar f$ refers corresponding anti-parton. Note that, in the case of W-boson, the constants $g_{L/R}^W$ mix the flavors of quarks.
In the factorized hadronic tensor in eq. (\[DY:Wmunu\]) contributions are not equally important. In fact, the fifth line of eq. (\[DY:Wmunu\]) vanishes identically due to the peculiar combination of $g$-constants that is null for any electro-weak channel. The forth line can contribute only to $ZZ$ and $Z\gamma$-channels, that have an anti-symmetric part of the leptonic tensor. However, the resulting expression is anti-symmetric in the rapidity parameter, and thus it vanishes when the rapidity is measured/integrated on symmetric intervals. In principle, this part can contribute to a cross-section when experiments perform very asymmetric kinematic cuts on the detected leptons (e.g. at LHCb). However, even in this case the resulting integral is suppressed as $q_T^2/Q^2 e^{-2|y|}$ and it is numerically very small, e.g in some bins it can give a $10^{-6}-10^{-8}$-size relative to the leading contribution. Thus, in the following we do not consider contributions of the last two lines in eq. (\[DY:Wmunu\]).
Performing the integration over angles we obtain a result formally similar to the SIDIS case in eq. (\[SIDIS:Wmunu-J\]), $$\begin{aligned}
\label{DY:Wmunu-J}
W^{\mu\nu}_{GG'}&=&\frac{1}{2\pi}\sum_{f}2(g^G_Rg^{G'}_R+g^G_Lg^{G'}_L)\Big[-g_T^{\mu\nu} W^f_{f_1f_1}(Q,|q_T|,x_1,x_2)
\\{\nonumber}&& \qquad\qquad +{\left(}g_T^{\mu\nu}-2\frac{q_T^\mu q_T^\nu}{q_T^2}{\right)}W^f_{h_1^\perp h_1^\perp}(Q,|q_T|,x_1,x_2) + O{\left(}\frac{q_T^2}{Q^2}{\right)},\end{aligned}$$ where $$\begin{aligned}
\label{def:Wff}
W^f_{f_1f_1}(Q,q_T,x_1,x_2)&=&|C_V(-Q^2,\mu^2)|^2
\\{\nonumber}&& \qquad\int_0^\infty db\,bJ_0(bq_T)f_{1,f{\leftarrow}h_1}(x_1,b;\mu,\zeta_1)f_{1,\bar f{\leftarrow}h_2}(x_2,b;\mu,\zeta_2),
\\\label{def:Whh}
W^f_{h_1^\perp h_1^\perp}(Q,q_T,x_1,x_2)&=&\frac{M_1M_2}{4}|C_V(-Q^2,\mu^2)|^2
\\{\nonumber}&&\qquad\times
\int_0^\infty db\,b^3J_2(bq_T)h^\perp_{1,f{\leftarrow}h_1}(x_1,b;\mu,\zeta_1)h^\perp_{1,\bar f{\leftarrow}h_2}(x_2,b;\mu,\zeta_2).\end{aligned}$$
### Kinematic variables
The relevant kinematic variable in DY read $$\begin{aligned}
s=(P_1+P_2)^2,\qquad q^2=Q^2,\qquad y=\frac{1}{2}\ln\frac{q^+}{q^-}.\end{aligned}$$ The transverse components are projected by a tensor $g_T^{\mu\nu}$, that is orthogonal to $P_1^\mu$ and $P_2^\mu$, identically to the SIDIS case eq. (\[def:gT-SIDIS\]),$$\begin{aligned}
g_T^{\mu\nu}=g^{\mu\nu}-\frac{2}{s}{\left(}P_1^\mu P_2^\nu+P_2^\mu P_1^\nu{\right)},\end{aligned}$$ and we have dropped the negligible corrections of order of $M^2/s$. In this limit, the factorization theorem is expressed in the center-of-mass frame, the components of momenta are $P_1^+=P_2^-=\sqrt{s/2}$ and the variables $x_{1,2}$ in eq. (\[def:x1x2\]) are $$\begin{aligned}
x_1=\sqrt{\frac{Q^2+{\bm{q}}_T^2}{s}}e^{+y},\qquad x_2=\sqrt{\frac{Q^2+{\bm{q}}_T^2}{s}}e^{-y}.\end{aligned}$$ The differential phase-space element reads $$\begin{aligned}
\label{DY:dQ}
d^4q=\frac{1}{2}dQ^2dyd^2q_T=\frac{1}{4}dQ^2dy d{\bm{q}}_T^2 d\varphi,\end{aligned}$$ where $\varphi$ is the azimuthal angle of the vector boson.
### Lepton tensor and fiducial cuts {#sec:DY-fiducial}
In experiments not all final state leptons are collected in the measurements and fiducial cuts are for instance performed at LHC. We use the same implementation of cuts as in [@Scimemi:2017etj; @Bertone:2019nxa]. However, here we give a more general discussion to see how they affect the power suppressed parts of the cross section.
The lepton tensor of unpolarized DY formally written in eq. (\[DY:L1\]) is $$\begin{aligned}
L_{\mu\nu}^{GG'}&=&8\Big[(l^\mu l'^\nu+l^\nu l'^\mu-g^{\mu\nu}(ll')){\left(}g_{G}^Rg_{G'}^R+g_{G}^Lg_{G'}^L{\right)}+ i \epsilon^{\mu\nu \alpha\beta}l_\alpha l'_\beta {\left(}g_{G}^Rg_{G'}^R-g_{G}^Lg_{G'}^L{\right)}\Big],\end{aligned}$$ where $g_G^{R}$($g_G^{L}$) are the couplings of right (left) components of a lepton field to EW current as in eq. (\[def:J-EW\]). In the case of $W$ boson, these couplings also carry flavor indices. As discussed in sec. \[sec:DY:hadron-tensor\], the anti-symmetric part does not contribute visibly to the unpolarized cross-section even in the presence of asymmetric fiducial cuts.
The DY cross-section contains the lepton tensor integrated over the lepton momenta with $l+l'=q$, in eq. (\[DY:Lhat\]), and this gives $$\begin{aligned}
\label{DY:lG}
(-g_T^{\mu\nu})\hat{L}^{GG'}_{\mu\nu}&=16{\left(}g_{G}^Rg_{G'}^R+g_{G}^Lg_{G'}^L{\right)}\int \frac{d^3l}{2E}\frac{d^3l'}{2E'}\delta^{(4)}(l+l'-q)((ll')-(ll')_T)
\\{\nonumber}&={\left[}2{\left(}g_{G}^Rg_{G'}^R+g_{G}^Lg_{G'}^L{\right)}{\right]}\frac{4\pi}{3}Q^2{\left(}1+\frac{{\bm{q}}_T^2}{2Q^2}{\right)},\\
\label{DY:lGG}
(g_T^{\mu\nu}-2\frac{q_T^\mu q_T^\nu}{q_T^2})\hat{L}^{GG'}_{\mu\nu}&= -32{\left(}g_{G}^Rg_{G'}^R+g_{G}^Lg_{G'}^L{\right)}\\{\nonumber}&\times\int \frac{d^3l}{2E}\frac{d^3l'}{2E'}\delta^{(4)}(l+l'-q)
\frac{2l_T^2l'_T+(ll')_Tl_T^2+(ll')_T{l'}^2_T}{q_T^2}
\\{\nonumber}&={\left[}2{\left(}g_{G}^Rg_{G'}^R+g_{G}^Lg_{G'}^L{\right)}{\right]}\frac{4\pi}{3}Q^2\frac{{\bm{q}}_T^2}{Q^2}.\end{aligned}$$ The cuts on the lepton pair at LHC are usually reported as $$\begin{aligned}
\eta_{\text{min}}<\eta,\eta'<\eta_{\text{max}},\qquad l_T^2>p_1^2,\qquad {l'}^2_T>p_2^2,\end{aligned}$$ where $\eta$ and $\eta'$ are pseudo-rapidity of the leptons. In the presence of these cuts the integration volume of the leptonic tensor can be done only numerically. To account this effect we introduce cut factors as $$\begin{aligned}
\label{DY:P1}
\mathcal{P}_1&=&\int \frac{d^3l}{2E}\frac{d^3l'}{2E'}\delta^{(4)}(l+l'-q)((ll')-(ll')_T)\theta(\text{cuts})\Big/{\left[}\frac{\pi}{6}Q^2{\left(}1+\frac{{\bm{q}}_T^2}{2Q^2}{\right)}{\right]}^{-1},
\\\label{DY:P2}
\mathcal{P}_2&=&\frac{12}{\pi}\int \frac{d^3l}{2E}\frac{d^3l'}{2E'}\delta^{(4)}(l+l'-q)(2l_T^2l'^2_T+(ll')_Tl_T^2+(ll')_T{l'}^2_T)\theta(\text{cuts}).\end{aligned}$$ These factors are equal to one in the absence of cuts. The impact of these cuts at LHC is extremely important and depends on the rapidity interval and the value of the vector boson transverse momentum. We show $\mathcal{P}_{1,2}$ for ATLAS experiment in fig. \[fig:cut-factors\]. One can see that the factor $\mathcal{P}_2$ is enhanced at smaller $q_T$ and in general these factors are very different from 1.
![\[fig:cut-factors\] Plot of cut-factors $\mathcal{P}_{1,2}$ (\[DY:P1\],\[DY:P2\]) versus $q_T$\[GeV\] in the case of fiducial cuts of ATLAS and CMS Z-boson measurement [@Aad:2014xaa; @Aad:2015auj; @Chatrchyan:2011wt; @Khachatryan:2016nbe], at $Q=91.$GeV and different values of $y$. The lines with $y=0.0$ and $y=0.5$ are very close to each other.](Figures/P1.pdf "fig:"){width="40.00000%"} ![\[fig:cut-factors\] Plot of cut-factors $\mathcal{P}_{1,2}$ (\[DY:P1\],\[DY:P2\]) versus $q_T$\[GeV\] in the case of fiducial cuts of ATLAS and CMS Z-boson measurement [@Aad:2014xaa; @Aad:2015auj; @Chatrchyan:2011wt; @Khachatryan:2016nbe], at $Q=91.$GeV and different values of $y$. The lines with $y=0.0$ and $y=0.5$ are very close to each other.](Figures/P2.pdf "fig:"){width="40.00000%"}
### DY cross-section in TMD factorization
Collecting the expressions for the hadronic tensor eq. (\[DY:Wmunu-J\]), the differential phase-space element in eq. (\[DY:dQ\]), the leptonic tensor (\[DY:lG\], \[DY:lGG\]) with the fiducial cuts in eq. (\[DY:P1\], \[DY:P2\]), we obtain the final cross-section in the TMD factorization. For the case of neutral vector boson (i.e. Z- and $\gamma$- bosons) it reads $$\begin{aligned}
\label{DY:xSec}
&&\frac{d\sigma}{dQ^2dy d{\bm{q}}_T^2}=\frac{2\pi}{3N_c}\frac{\alpha^2_{\text{em}}}{sQ^2}\sum_{f}\Big[{\left(}1+\frac{{\bm{q}}_T^2}{2Q^2}{\right)}\mathcal{P}_1W_{f_1f_1}^f(Q,\sqrt{{\bm{q}}_T^2})+\frac{{\bm{q}}_T^2}{Q^2}\mathcal{P}_2W_{h_1^\perp h_1^\perp}^f(Q,\sqrt{{\bm{q}}_T^2})\Big]
\\{\nonumber}&& \qquad
\times \Big[z^{\gamma\gamma}_lz_f^{\gamma\gamma}+z^{\gamma Z}_lz_f^{\gamma Z}\frac{2Q^2(Q^2-M_Z^2)}{(Q^2-M_Z^2)^2+\Gamma_Z^2M_Z^2}+
z^{ZZ}_lz_f^{ZZ}\frac{Q^4(Q^2-M_Z^2)}{(Q^2-M_Z^2)^2+\Gamma_Z^2M_Z^2}\Big],\end{aligned}$$ where functions $W^f$ are defined in (\[def:Wff\], \[def:Whh\]), $M_Z$ and $\Gamma_Z$ are mass and width of Z-boson. The factors $z$ are the combinations of couplings $g^{R,L}$ for quarks and for leptons (\[def:gRgL\]): $$\begin{aligned}
z_f^{\gamma\gamma}&=&e_f^2,
\\
z_f^{\gamma Z}&=&\frac{T_3-2e_fs_W^2}{2s_W^2c_W^2},
\\
z_f^{Z Z}&=&\frac{(1-2|e_f|s_W^2)^2+4e_f^2s_W^4}{8s_W^2c_W^2}.\end{aligned}$$ The term $W_{h_1^\perp h_1^\perp}^f$ describes the contributions of the Boer-Mulders functions. We omit this term in the rest of the paper leaving its discussion to future work.
Structure functions $W$
-----------------------
The TMD factorization provides the cross section for DY and SIDIS in terms of 4 structure functions $W_{XY}$ defined in eq. (\[def:WfD\], \[def:WhH\], \[def:Wff\], \[def:Whh\]) and each of them is a Hankel convolution of two TMD distributions times a hard coefficient function. We remark that the TMD include all the non-perturbative information of the process, and it is different from the one contained in a collinear PDF. The structure functions $W_{h_1^\perp H_1^\perp }$ and $W_{h_1^\perp h_1^\perp }$ are formally of higher twist with respect to the others. While higher twist contributions are formally accompanied by ${\bm{q}}_T^2/Q^2$ factors, the complex kinematics of the experiments (especially in the SIDIS case) makes it hard to distinguish purely non-perturbative higher-twist effects from the kinematical ones. For instance, the azimuthal angles measured in the lab frames and in the Breit frame for SIDIS are different and some non-perturbative QCD effects can be overlooked when we pass from one frame to the other. The only way to solve this problem would be a complete inclusion of higher power corrections to the cross section, which goes beyond the scope of the present work. For this reason, while we consider the exact kinematics, as described in the previous section, we also put $$\begin{aligned}
W_{h_1^\perp H_1^\perp }^f(Q,q_T,x,z)=0,\qquad W_{h_1^\perp h_1^\perp }^f(Q,q_T,x,x')=0.\end{aligned}$$ The effect of this assumption must be very small at ${\bm{q}}_T^2\ll Q^2$, and this justifies the conservative data sets used in the present fit (see sec. \[sec:data\]).
The $Q$ dependence of $W_{XY}$ is dictated by the TMD evolution, and it is discussed in the next section \[sec:evolution\]. The asymptotic limit of high $q_T$ allows for a perturbative matching of TMD distributions to collinear ones and it is discussed in sec. \[sec:matching\]. The non-perturbative inputs on top of the large-$q_T$ asymptotic limit are discussed in sec. \[sec:ansatzNP\]. Finally, we summarize all theoretical inputs in sec. \[sec:summary-theory\].
### TMD evolution and optimal TMD distributions {#sec:evolution}
While the differential evolution equations for TMD are fixed by the factorization theorem, the boundary conditions of their solution are a matter of choice. They clearly determine the convergence of the perturbative series and the success of the theoretical description of DY and SIDIS spectrum. In this paper, we work with the so-called $\zeta$-prescription described in [@Scimemi:2018xaf], and including the improvement found in [@Vladimirov:2019bfa]. The prescription consists in defining the TMD distribution on a null-evolution line. The null-evolution line has the defining property of keeping the evolution factor for a TMD distributions equal to one for all values of the impact parameter $b$. Because of this property, the $\zeta$-prescription is conceptually different from other popular prescriptions, where the reference scales do not belong to a null-evolution line. In this case, the resulting (reference) TMD distribution has an admixture of a perturbative evolution factor between different values of $b$. Therefore, the $\zeta$-prescription has an important advantage, that a TMD distribution is independent of any perturbative parameter, i.e. it is completely non-perturbative and one can freely parameterize a distribution without any respect to perturbation order. For a detailed description and analyses of TMD evolution and the $\zeta$-prescription we refer to [@Scimemi:2018xaf], whereas here we present only the final expressions without derivation.
The system of TMD evolution equations is $$\begin{aligned}
\label{def:TMD_ev_UV}
\mu^2 \frac{d}{d\mu^2} F(x,b;\mu,\zeta)&=&\frac{\gamma_F(\mu,\zeta)}{2}F(x,b;\mu,\zeta),
\\\label{def:TMD_ev_RAP}
\zeta\frac{d}{d\zeta}F(x,b;\mu,\zeta)&=& -\mathcal{D}(\mu,b)F(x,b;\mu,\zeta),\end{aligned}$$ where $F$ is any TMD distribution ($f_1$ or $D_1$ in the present case). The TMD evolution equations are not sensitive to the flavor of a parton[^1] and thus we omit flavor indices in this section for simplicity. The eq. (\[def:TMD\_ev\_UV\]) is a standard renormalization group equation, which comes from the renormalization of the ultraviolet divergences, with the function $\gamma_F(\mu,\zeta)$ being the anomalous dimension. The eq. (\[def:TMD\_ev\_RAP\]) results from the factorization of rapidity divergences. The function $\mathcal{D}(\mu,b)$ is called the rapidity anomalous dimension (RAD). The RAD is a generic non-perturbative function that can be computed at small values of $b$ in the perturbation theory. The perturbative expression for the RAD and $\gamma_F$ can be found in the literature (e.g. see appendix of ref. [@Echevarria:2016scs]). In this work we use the resummed version of RAD [@Echevarria:2012pw]. The resummed expressions are also given in appendix \[app:RAD\] (see also appendix B in ref. [@Bizon:2018foh]).
The scales $\mu$ and $\zeta$ have independent origins, and this has important consequences. To start with, the TMD evolution takes place in the plane $(\mu,\zeta)$. The solution of equations eq. (\[def:TMD\_ev\_UV\], \[def:TMD\_ev\_RAP\]) for the evolution from a point $(\mu_f,\zeta_f)$ to a point $(\mu_i,\zeta_i)$ is $$\begin{aligned}
\label{def:exp[evol]}
F(x,b;\mu_f,\zeta_f)=\exp{\left[}\int_P {\left(}\gamma_F(\mu,\zeta)\frac{d\mu}{\mu}-\mathcal{D}(\mu,b)\frac{d\zeta}{\zeta}{\right)}{\right]}F(x,b;\mu_i,\zeta_i)\end{aligned}$$ where $P$ is any path in $(\mu,\zeta)$-plane that connects initial $(\mu_i,\zeta_i)$ and final points $(\mu_f,\zeta_f)$. The value of evolution is (in principle) independent on the path, thanks to integrability condition (also known as Collins-Soper (CS) equation [@Collins:1981va]) $$\begin{aligned}
\label{def:integrability}
-\zeta\frac{d\gamma_F(\mu,\zeta)}{d\zeta}=\mu\frac{d\mathcal{D}(\mu,b)}{d\mu}=\Gamma_{\text{cusp}}(\mu),\end{aligned}$$ where $\Gamma_{\text{cusp}}(\mu)$ is the cusp anomalous dimension. This equation dictates the logarithmic structure of anomalous dimensions. In particular, the TMD anomalous dimension is $$\begin{aligned}
\gamma_F(\mu,\zeta)=\Gamma_{\text{cusp}}(\mu)\ln{\left(}\frac{\mu^2}{\zeta}{\right)}-\gamma_V(\mu).\end{aligned}$$ The formal path-independence of eq. (\[def:exp\[evol\]\]) is violated at any fixed order of perturbation theory. The penalty term is proportional to the area surrounded by paths, and can be huge in the case of very separated scales. Nevertheless, the path dependence decreases with the increase of the perturbative order and it is numerically small at N$^3$LO [@Scimemi:2018xaf].
The final scales of the evolution are binded to the hard scale of factorization such that $\mu^2_f\sim Q^2$ and $\zeta_{1f}\zeta_{2f}=Q^4$. In particular, we choose the symmetric point $$\begin{aligned}
\mu^2_f= Q^2,\qquad \zeta_{1f}=\zeta_{2f}=Q^2.\end{aligned}$$ The TMD initial (or defining) scale is chosen with the $\zeta$-prescription and deserves some explanation. In the $\zeta$-prescription the scales $\mu$ and $\zeta$ belong to a null-evolution line, that we parameterize as $(\mu,\zeta_\mu(b))$. To find the null-evolution line, we recall that the system of eq. (\[def:TMD\_ev\_UV\], \[def:TMD\_ev\_RAP\]) is a two-dimensional gradient equation ($\pmb\nabla F=\mathbf{E}F$) with the field $\mathbf{E}=(\gamma_F(\mu,\zeta)/2,-\mathcal{D}(\mu,b))$. Therefore, the null-evolution line is simply an equipotential line of the field $\mathbf{E}$. It provides the equation that define $\zeta_\mu(b)$ such that $$\begin{aligned}
\label{th:special-line}
\Gamma_{\text{cusp}}(\mu)\ln{\left(}\frac{\mu^2}{\zeta_\mu(b)}{\right)}-\gamma_V(\mu)=2\mathcal{D}(\mu,b)\frac{d\ln \zeta_\mu(b)}{d\ln \mu^2},\end{aligned}$$ A TMD distribution does not evolve between scales belonging to the same equipotential line by definition.
Among equipotential lines there is a special line that passes though the saddle point $(\mu_0,\zeta_0)$ of the field $\mathbf{E}$. The values $(\mu_0,\zeta_0)$ are defined as $$\begin{aligned}
\label{th:saddle-point}
\mathcal{D}(\mu_0,b)=0,\qquad \gamma_F(\mu_0,\zeta_0)=0.\end{aligned}$$ The special equipotential line is favored for the definition of defining TMD scales for two important reasons. First, there is only one saddle point in the evolution field, and thus, the special null-evolution line is unique. Second, the special null-evolution line is the only null-evolution line, which has finite $\zeta$ at all values of $\mu$ (bigger than $\Lambda_\text{QCD}$). These properties follow from its definition and they are very useful. In fig. \[fig:zeta-line\] we show the force-lines of the evolution field $\mathbf{E}$ (in grey, with arrows), null-evolution lines, (thick grey lines, orthogonal to the force-lines), and the lines that cross at the saddle point (in red) at different values of $b$. In this figure the special line is the one that goes from left to right in each panel.
The concept of $\zeta$ prescription has been introduced in ref. [@Scimemi:2017etj] and elaborated in [@Scimemi:2018xaf]. Presently we use a form slightly different from the original version of refs. [@Scimemi:2017etj; @Scimemi:2018xaf]. Here we follow the updated realization introduced in ref. [@Vladimirov:2019bfa] that has been used for the description of the pion-induced DY process. In refs. [@Scimemi:2017etj; @Bertone:2019nxa] the $\zeta$-lines has been taken perturbative for all ranges of $b$ (with slight deformations due to the Landau pole). Notwithstanding, such definition introduces an undesired correlation between the non-perturbative parts of the TMD distribution and RAD. In ref. [@Vladimirov:2019bfa] a new simple solution has been found for the values of special null-evolution line at large $b$ that accurately incorporates non-perturbative effects, without adding new parameters to the fit. In appendix \[app:zeta-line\] we present the expression for the special line as it is used in this fit.
![\[fig:zeta-line\] In the $(\zeta,\mu)$ plane we show the lines of force of the TMD evolution field $\mathbf{E}$ at different values of $b$ (in grey, with arrows). The thick continuous gray lines are null-evolution (equipotential) lines. Red lines are the equipotential lines that define the saddle point. The red line which crosses each panel from left to right is the special evolution curve where the TMD are defined. The blue dashed lines in each plot correspond to the final scale choice ($\mu_f, \zeta_f$) for typical experimental measurements. From left panel to right panel we have chosen respectively $Q=5$, $91$ and $150$ GeV. Black dashed lines with arrows are path of evolution implemented in eq. (\[def:TMD-evolved\]). ](Figures/zetaLine.pdf){width="95.00000%"}
A TMD distribution $F(x,b;\mu,\zeta_\mu)$ with $\zeta_\mu$ belonging to the special line is called optimal TMD distribution, and denoted by $F(x,b)$ (without scale arguments), to emphasize its uniqueness and independence on scale $\mu$. The exact independence of optimal TMD distribution on scale $\mu$, allows us to select the simplest path for the evolution exponent in eq. (\[def:exp\[evol\]\]), that is, the path at fixed value of $\mu=Q$ along $\zeta$ from the value $\zeta_f=Q^2$ down to any point of $\zeta_i=\zeta_Q(b)$. In fig. \[fig:zeta-line\] this path is visualized by black-dashed lines. The resulting expression for the evolved TMD distributions is exceptionally simple $$\begin{aligned}
\label{def:TMD-evolved}
F(x,b;Q,Q^2)={\left(}\frac{Q^2}{\zeta_Q(b)}{\right)}^{-\mathcal{D}(b,Q)}F(x,b).\end{aligned}$$ We recall that this expression is same for all (quark) TMDPDFs and TMDFF. Substituting (\[def:TMD-evolved\]) into the definition of structure functions $W$ we obtain, $$\begin{aligned}
\label{def:Wff-final}
W_{f_1f_1}^f(Q,q_T;x_1,x_2)&=&|C_V(-Q^2,Q^2)|^2
\\{\nonumber}&&\qquad\times \int_0^\infty db\, bJ_0(bq_T)f_{1,f{\leftarrow}h}(x_1,b)f_{1,\bar f{\leftarrow}h}(x_2,b){\left(}\frac{Q^2}{\zeta_Q(b)}{\right)}^{-2\mathcal{D}(b,Q)},
\\\label{def:WfD-final}
W_{f_1D_1}^f(Q,q_T;x_S,z_S)&=&|C_V(Q^2,Q^2)|^2
\\{\nonumber}&&\qquad\times \int_0^\infty db\, bJ_0(bq_T)f_{1,f{\leftarrow}h}(x_S,b)D_{1,f\to h}(z_S,b){\left(}\frac{Q^2}{\zeta_Q(b)}{\right)}^{-2\mathcal{D}(b,Q)}.\end{aligned}$$ These are the final expressions used to extract NP-functions.
The simplicity of expressions (\[def:Wff-final\],\[def:WfD-final\]) is also accompanied by a good convergence of the cross section. In fig. \[fig:convergence\] we show the comparison of curves for DY and SIDIS cross-section at typical energies. In the plot the TMD distributions and the NP part of the evolution are held fixed while the perturbative orders are changed. The perturbative series converges very well, and the difference between NNLO and N$^3$LO factorization is of order of percents. This is an additional positive aspect of the $\zeta$-prescription, which is due to fact that all perturbative series are evaluated at $\mu=Q$.
![\[fig:convergence\] The cross-section at different orders of TMD factorization and for different boson energies. The legend of the perturbative orders means that N$^k$LO (N$^k$LL) incorporates $a_s^k$-order ($a_s^{k-1}$-order) of the coefficient function, $a_s^{k}$-order of anomalous dimensions with $a_s^{k+1}$-order of $\Gamma_{\text{cusp}}$. The TMD distributions and the NP part of the evolution is the same for all cases.](Figures/order-DY-91.pdf "fig:"){width="32.00000%"} ![\[fig:convergence\] The cross-section at different orders of TMD factorization and for different boson energies. The legend of the perturbative orders means that N$^k$LO (N$^k$LL) incorporates $a_s^k$-order ($a_s^{k-1}$-order) of the coefficient function, $a_s^{k}$-order of anomalous dimensions with $a_s^{k+1}$-order of $\Gamma_{\text{cusp}}$. The TMD distributions and the NP part of the evolution is the same for all cases.](Figures/order-DY-8.pdf "fig:"){width="32.00000%"} ![\[fig:convergence\] The cross-section at different orders of TMD factorization and for different boson energies. The legend of the perturbative orders means that N$^k$LO (N$^k$LL) incorporates $a_s^k$-order ($a_s^{k-1}$-order) of the coefficient function, $a_s^{k}$-order of anomalous dimensions with $a_s^{k+1}$-order of $\Gamma_{\text{cusp}}$. The TMD distributions and the NP part of the evolution is the same for all cases.](Figures/order-SIDIS-4.pdf "fig:"){width="32.00000%"}
### Matching of TMD distribution to collinear distributions {#sec:matching}
The TMD are generic non-perturbative functions that depend on the parton fraction $x$ and the impact parameter $b$. A fit of a two-variable function is a hopeless task due to the enormous parametric freedom. This freedom can be essentially reduced by the matching of a $b\to0$ boundary of a TMD distribution to the corresponding collinear distribution. In the asymptotic limit of small-$b$ one has $$\begin{aligned}
\label{def:F-match}
\lim_{b\to0}f_{1,f{\leftarrow}h}(x,b)&=&\sum_{f'}\int_x^1 \frac{dy}{y} C_{f{\leftarrow}f'}{\left(}\frac{x}{y},\mathbf{L}_{\mu_{\text{\tiny OPE}}},a_s(\mu_{\text{\tiny OPE}}){\right)}f_{1,f'{\leftarrow}h}(y,\mu_{\text{\tiny OPE}}),
\\\label{def:D-match}
\lim_{b\to0}D_{1,f\to h}(z,b)&=&\sum_{f'}\int_z^1 \frac{dy}{y} \mathbb{C}_{f\to f'}{\left(}\frac{z}{y},\mathbf{L}_{\mu_{\text{\tiny OPE}}},a_s(\mu_{\text{\tiny OPE}}){\right)}\frac{d_{1,f'\to h}(y,\mu_{\text{\tiny OPE}})}{y^2},\end{aligned}$$ where $f_1(x,\mu)$ and $d_1(x,\mu)$ are collinear PDF and FF, the label $f'$ runs over all active quarks, anti-quarks and a gluon, and $$\begin{aligned}
\mathbf{L}_\mu=\ln{\left(}\frac{{\bm{b}}^2 \mu^2}{4\exp^{-2\gamma_E}}{\right)},\qquad a_s(\mu)=\frac{g^2(\mu)}{(4\pi)^2},\end{aligned}$$ with $\gamma_E$ being the Euler constant and $g$ being QCD coupling constant. The extra factor $y^{-2}$ in eq. (\[def:D-match\]) is present due to the normalization difference of the TMD operator in eq. (\[def:D1\]) and the collinear operator, see e.g. [@Collins:2011zzd; @Echevarria:2015usa]. The coefficient functions $C$ and $\mathbb{C}$ can be calculated with operator product expansion methods (for a general review see ref. [@Scimemi:2019gge]) and in the case of unpolarized distributions the coefficient functions are known up to NNLO [@Gehrmann:2014yya; @Echevarria:2015usa; @Echevarria:2016scs; @Luo:2019hmp]. The coefficient function $C$ has the general form $$\begin{aligned}
\label{th:match-coeff}
&&C_{f{\leftarrow}f'}(x,\mathbf{L}_{\mu},a_s)=\delta(\bar x)\delta_{ff'}+a_s{\left(}-\mathbf{L}_\mu P^{(1)}_{f{\leftarrow}f'}+C_{f{\leftarrow}f'}^{(1,0)}{\right)}\\{\nonumber}&&\qquad\qquad+a_s^2\Big[\frac{P^{(1)}_{f{\leftarrow}k}\otimes P^{(1)}_{k{\leftarrow}f'}-\beta_0 P_{f{\leftarrow}f'}^{(1)}}{2}\mathbf{L}^2_\mu
-\mathbf{L}_\mu {\left(}P^{(2)}_{f{\leftarrow}f'}+C_{f{\leftarrow}k}^{(1,0)}\otimes P^{(1)}_{k{\leftarrow}f'}-\beta_0 C_{f{\leftarrow}k}^{(1,0)}{\right)}\\{\nonumber}&& \qquad\qquad\qquad+C_{f{\leftarrow}f'}^{(2,0)}+\frac{d^{(2,0)}\gamma_1}{\Gamma_0}\delta(\bar x)\delta_{ff'}\Big]+O(a_s^3),\end{aligned}$$ where $\bar x=1-x$, the symbol $\otimes$ denotes the Mellin convolution, and a summation over the intermediate flavour index $k$ is implied. In eq. (\[th:match-coeff\]) we have omitted argument $x$ of functions on left-hand-side for brevity. The functions $P^{(n)}(x)$ are the coefficients of the PDF evolution kernel $P(x)=\sum_n a_s^n P^{(n)}(x)$ (DGLAP kernel), which can be found f.i. in ref. [@Moch:1999eb]. The functions $C_{f{\leftarrow}f'}^{(n,0)}(x)$ are given in [@Gehrmann:2014yya; @Echevarria:2015usa; @Echevarria:2016scs; @Luo:2019hmp]. In particular, the NLO terms are $$\begin{aligned}
C_{q{\leftarrow}q}^{(1,0)}(x)=C_F{\left(}2\bar x-\delta(\bar x)\frac{\pi^2}{6}{\right)},\qquad C_{q{\leftarrow}g}^{(1,0)}(x)=2x\bar x.\end{aligned}$$ The last term in the square brackets of eq. (\[th:match-coeff\]) is the consequence of the boundary condition of eq. (\[th:saddle-point\]), and it consists of some coefficients of the anomalous dimension defined in eq. (\[app:def:dnk\], \[app:gammaV\]).
In the case of TMDFF the matching coefficient $\mathbb{C}$ follows the same pattern as in eq. (\[th:match-coeff\]) with the replacement of the PDF DGLAP kernels $P^{(n)}_{f{\leftarrow}f'}(x)$ by the FF DGLAP kernels $P^{(n)}_{f{\leftarrow}f'}(z)$ (they can be found f.i. in ref. [@Stratmann:1996hn]), and $C_{f{\leftarrow}f'}^{(n,0)}(x)$ by $\mathbb{C}_{f\to f'}^{(n,0)}(z)$ [@Echevarria:2015usa; @Echevarria:2016scs]. In TMDFF case, the NLO terms are $$\begin{aligned}
\mathbb{C}_{q\to q}^{(1,0)}(z)=\frac{C_F}{z^2}{\left(}2\bar z+\frac{4(1+z^2)\ln z}{1-z}-\delta(\bar z)\frac{\pi^2}{6}{\right)},
&&
\mathbb{C}_{q\to g}^{(1,0)}(z)=\frac{2C_F}{z^2}{\left(}z+2(1-\bar z^2)\frac{\ln z}{z}{\right)}.\;\end{aligned}$$ In the $\zeta$-prescription, the scale $\mu_{\text{OPE}}$ is entirely encapsulated inside the convolutions in eq. (\[def:F-match\], \[def:D-match\]) and it has no connection to the scales of the TMD evolution, as it happens in the case of $b^*$-prescription [@Collins:2011zzd; @Aybat:2011zv]. This fact gives an enormous advantage to achieve a complete decorrelation of RAD from TMD distributions (we will be more quantitative about this point in later sections). The scale $\mu_{\text{OPE}}$ has to be selected such that on one hand, it minimizes the logarithm contributions at $b\to 0$, and on another hand, it does not hit the Landau pole at large-$b$. We use the following value $$\begin{aligned}
\mu_{\text{OPE}}=\frac{2e^{-\gamma_E}}{b}+2\text{GeV}.\end{aligned}$$
Let us not that in the $\zeta$-prescription, the coefficient functions of small-$b$ matching in eq. (\[th:match-coeff\]) does not contain a double-logarithm contribution. For that reason the perturbative convergence, as well as, the radius of convergence improves. Both these facts make the $\zeta$-prescription highly advantageous.
### Ansatzes for NP functions {#sec:ansatzNP}
In this work we deal with *three independent non-perturbative functions* in total. These are the unpolarized (optimal) TMDPDF, $f_1(x,b)$, the unpolarized (optimal) TMDFF, $D_1(x,b)$, and the RAD, $\mathcal{D}(b,\mu)$. The amount of perturbative and non-perturbative contributions to each function depends on the value of the impact parameter $b$. Namely, at small values of $b$ the perturbative approximation is good and the TMD distributions can be matched onto collinear functions as in eq. (\[def:F-match\], \[def:D-match\]). In the case of the RAD the small-$b$ limit is given in appendix \[app:RAD\]. The small-$b$ perturbative expressions gains power corrections in even powers ${\bm{b}}^{2n}$ [@Scimemi:2016ffw]. Therefore, with the increase of $b$ the perturbative approximation becomes less and less correct, and must be replaced by some generic function.
The phenomenological ansatzes for TMD distributions that satisfy this picture, can be written as following: $$\begin{aligned}
\label{def:phen-f1}
f_{1,f{\leftarrow}h}(x,b)&=&\int_x^1 \frac{dy}{y}\sum_{f'}C_{f{\leftarrow}f'}{\left(}y,\mathbf{L}_{\mu_{\text{OPE}}},a_s(\mu_{\text{\tiny OPE}}){\right)}f_{1,f'{\leftarrow}h}{\left(}\frac{x}{y},\mu_{\text{\tiny OPE}}{\right)}f_{\text{NP}}(x,b),
\\\label{def:phen-D1}
D_{1,f\to h}(z,b)&=&\frac{1}{z^2}\int_z^1 \frac{dy}{y}\sum_{f'} y^2\mathbb{C}_{f\to f'}{\left(}y,\mathbf{L}_{\mu_{\text{\tiny OPE}}},a_s(\mu_{\text{\tiny OPE}}){\right)}d_{1,f'\to h}{\left(}\frac{z}{y},\mu_{\tiny \text{\tiny OPE}}{\right)}D_{\text{NP}}(z,b),\end{aligned}$$ where functions $f_{\text{NP}}$ and $D_{\text{NP}}$ are non-perturbative functions. Note, that in our ansatz we do not modify the value of $b$ within the coefficient function. Therefore, at large-$b$ the logarithm part of the coefficient function grows unrestrictedly. This growth is suppressed by the non-pertrubative functions.
Generally, the functions $f_{\text{NP}}$ and $D_{\text{NP}}$ depend also on parton flavor $f$ and hadron type $h$. However, *in the present work we use the approximation that $f_{\text{NP}}$ and $D_{\text{NP}}$ are flavor and hadron-type independent.* All hadron- and flavor dependence is driven by the collinear PDFs and FFs (see also sec. \[sec:nuclear\]). Given such an ansatz the only requirement for NP functions is that they are even-functions of $b$ that turn to unity for $b\to 0$. We use the following parameterizations $$\begin{aligned}
\label{def:fNP}
f_{NP}(x,b)&=&\exp{\left(}-\frac{\lambda_1(1-x)+\lambda_2 x+x(1-x)\lambda_5}{\sqrt{1+\lambda_3 x^{\lambda_4} {\bm{b}}^2}}{\bm{b}}^2{\right)},
\\\label{def:DNP}
D_{NP}(x,b)&=&\exp{\left(}-\frac{\eta_1 z+\eta_2 (1-z)}{\sqrt{1+\eta_3({\bm{b}}/z)^2}}\frac{{\bm{b}}^2}{z^2}{\right)}{\left(}1+\eta_4 \frac{{\bm{b}}^2}{z^2}{\right)},\end{aligned}$$ and we extract $\lambda_i$ and $\eta_i$ from our fit. The functional form of $f_{NP}$ has been already used in [@Bertone:2019nxa]. It has five free parameters which grant a sufficient flexibility in $x$-space as needed for the description of the precise LHC data. The form of $D_{\text{NP}}$ has been suggested in [@Bacchetta:2017gcc] (albeit there are more parameters in [@Bacchetta:2017gcc]). In both cases the function has exponential or Gaussian form depending on the relative size of $\lambda_{1,2,5}/\lambda_3$, and $\eta_{1,2}/\eta_3$. There are natural restrictions on the parameter space $\lambda_{1,2,3}>0$, $\eta_{1,2,3}>0$, $\lambda_5\gtrsim-2(\lambda_1+\lambda_2)$, due to demand that TMD distribution is null $b\to \infty$.
The ansatz for the non-perturbative part of the RAD has different form, because one expects a different behavior at large-$b$. We use the following expression suggested in [@Bertone:2019nxa], $$\begin{aligned}
\label{NP:RAD}
\mathcal{D}(\mu,b)=\mathcal{D}_{\text{resum}}(\mu,b^*(b))+c_0 bb^*(b),\end{aligned}$$ where $$\begin{aligned}
b^*(b)=\frac{b}{\sqrt{1+{\bm{b}}^2/B^2_{\text{NP}}}}.\end{aligned}$$ The function $\mathcal{D}_{\text{resum}}$ is the resummed perturbative expansion of RAD [@Echevarria:2012pw; @Scimemi:2018xaf] reported in the appendix \[app:RAD\]. At LO it reads $$\begin{aligned}
\label{NP:RAD:LO}
\mathcal{D}^{\text{LO}}_{\text{resum}}=-\frac{\Gamma_0}{2\beta_0}\ln{\left(}1-\beta_0 a_s(\mu)\mathbf{L}_\mu{\right)}.\end{aligned}$$ The higher order expressions (up to N$^3$LO) are given in eq. (\[app:RAD:resum\]). The parameters $c_0$ and $B_{\text{NP}}$ are free positive parameters, in principle totally uncorrelated from the rest of non-perturbative parameters.
The resummed expression for RAD explicitly has the singularity in $b$ (see e.g. eq. (\[NP:RAD:LO\])). The singularity designates the convergence radius of the perturbative expression. Consequently, the perturbative behavior must be turned off well before $b$ approaches the singularity. In the ansatz in eq. (\[NP:RAD\]), this is achieved freezing the perturbative part at $b\sim B_{\text{NP}}$. The singularity is located at $\beta_0a_s(\mu)\mathbf{L}_\mu=1$ and thus, the value of $B_{\text{NP}}$ is restricted from above by: $B_{\text{NP}}\lesssim 2e^{-\gamma_E}\Lambda_{\text{QCD}}^{-1}\approx 4$GeV$^{-1}$.
At large-$b$ the non-perturbative expression for RAD in eq. (\[NP:RAD\]) is asymptotically linear in $b$, $\mathcal{D}\sim c_0B_{\text{NP}} b$. Such a behavior is different from the typical quadratic form. The linear behavior is suggested by model calculations of the RAD [@Tafat:2001in; @SAVA], and the coefficient $c_0$ can be related to the QCD string tension.
The special null-evolution line can be incorporated both at perturbative and non-perturbative level. In [@Scimemi:2017etj] and [@Bertone:2019nxa] the special null-evolution line included only its perturbative part for simplicity. This part is the most important one because it guarantees the cancellation of double-logarithms in the matching coefficient. However, at large-$b$, the non-perturbative corrections to the RAD are large and cannot be ignored: in [@Scimemi:2017etj] they can be seen as a part of the non-perturbative model, at the price of introducing an undesired correlation between $f_{NP}$ and $\mathcal{D}$. In order to adjust the null-evolution curve with a non-perturbative RAD one has to solve eq. (\[th:special-line\]) including the RAD in the full generality. Such solution can be found in principle, but its numerical implementation is problematic at very small-$b$, because it is very difficult to obtain the exact numerical cancellation of the *perturbative* series of logarithms with an *exact* solution. To by-pass this problem we use the perturbative solution at very small $b$, (and hence cancel all logarithm exactly) and turn it to an exact solution at larger $b$. This is realized by $$\begin{aligned}
\label{NP:zeta}
\zeta_\mu(b)=\zeta^{\text{pert}}_\mu(b)e^{-\frac{b^2}{B^2_{\text{NP}}}}+\zeta^{\text{exact}}_\mu(b){\left(}1-e^{-\frac{b^2}{B^2_{\text{NP}}}}{\right)},\end{aligned}$$ that is, for $b^2\ll B^2_\text{NP}$ we have the [*perturbative*]{} solution, and one turns to the [*exact*]{} for larger $b$. Since the RAD is entirely perturbative at small-$b$, the numerical difference between eq. (\[NP:zeta\]) and $\zeta^{\text{exact}}_\mu(b)$ is negligibly small.
Summary on theory input {#sec:summary-theory}
-----------------------
In this subsection we summarize the relevant points needed for a description of the DY and SIDIS differential cross section in the TMD formalism. The cross-section of DY and SIDIS are given by eq. (\[DY:xSec\], \[SIDIS:xSec\]). They contain four structure functions, from which in the present work, we consider only $W_{f_1D_1}$ and $W_{f_1f_1}$. We drop structure functions $W_{h_1^\perp H_1^\perp }$ and $W_{h_1^\perp h_1^\perp }$ since mostly probably they are a part of the power correction to TMD factorization. The functions $W_{f_1D_1}$ and $W_{f_1f_1}$ are considered instead without restrictions.
The eq. (\[DY:xSec\], \[SIDIS:xSec\]) contain a variety of power suppressed contributions, which have different sources:
- The power corrections to TMD factorization. These corrections appear during the factorization procedure for the hadronic tensors, see eq. (\[SIDIS:Wmunu\], \[DY:Wmunu\]). One can distinguish two kinds of power corrections: The corrections that are proportional to the leading structure functions $W_{XY}$, and mixes with the so-called Wandzura–Wilczek terms (in the case of SIDIS, this part of cross-section has been studied recently in [@Bastami:2018xqd]); And the corrections that involve genuine “twist-3” TMD distributions (some part of these corrections is discussed in ref. [@Balitsky:2017gis]).
- The mass and ${\bm{q}}_T^2$ dependence within the momentum fraction variables $(x_1,x_2)$(DY) and $(x_S,z_S)$(SIDIS), see eq. (\[def:x1x2\], \[def:x1z1\]). Despite the correction in the momentum fraction can be interpreted as a part of the power correction to TMD factorization (contributing to the Wandzura–Wilczek terms), we consider them on their own. These corrections come from the field-modes separation and the definition of the scattering plane, and they can be seen as the “Nachtmann-variable for TMD factorization”.
- Fiducial cuts for DY. The cut factors for the DY lepton tensor in eq. (\[DY:P1\], \[DY:P2\]) are a source of power corrections and mixture between different structure functions. They are accumulated in a separate factors, and have totally auxiliary nature. They must be accounted for the proper description of LHC data.
- Mismatch between factorization and laboratory frames in SIDIS. The azimuthal angles and transverse planes are defined differently in the factorization and laboratory frames see eq. (\[def:gT-SIDIS\], \[def:gPerp-SIDIS\]). This introduces target-mass, produced-mass, and $q_T$-corrections. A good example is the ${\bm{p}}_\perp$-linear contribution to the structure function $F_{UU}^{\cos\phi}$ (\[th:SIDIS-L1\], \[th:SIDIS-L2\]), which is a purely a frame-dependence effect.
- Cross-section phase-space volume in SIDIS. In the case of a non-negligible mass for the detected particle, the phase-volume contains power corrections. They are accumulated in a universal factor in eq. (\[th:phase-elem-1\]), and are a part of definition of the observable.
Some of the power corrections of this list can be accounted exactly (e.g. the corrections to the phase-space, the collinear momentum fractions, the relation between ${\bm{q}}^2_T$ and ${\bm{p}}_\perp^2$), while some are absolutely unknown (i.e. the power correction to the TMD factorization). The main problem is that currently the TMD factorization does not allow for a systematic separation of power corrections. The reason is the non-transversality of the hadron tensor in TMD factorization: $$\begin{aligned}
\label{th:QED-ward}
q_\mu W^{\mu\nu}_{\text{TMD fact.}}\neq 0.\end{aligned}$$ This violation generates non-physical power-suppressed contributions $q_\mu W^{\mu\nu}\sim q_T$, and mixes together kinematic and higher-twist corrections.
*In the present work, we keep all power suppressed factors as in eq. (\[DY:xSec\], \[SIDIS:xSec\])* (with $W_{h_1^\perp H_1^\perp}=W_{h_1^\perp h_1^\perp}=0$). Simultaneously, we apply conservative cuts on the data such that $q_T$ is small. In sec. \[sec:DY:x12\] and \[sec:SIDIS-power-corr\], we test the influence of the power corrections to the fit quality. In some sense, this test provide us an estimation of the systematic error due to the presence of unknown power correction.
The structure functions $W_{f_1D_1}$ and $W_{f_1f_1}$ are evaluated according to eq. (\[def:Wff-final\], \[def:WfD-final\]). The phenomenological ansatzes for the optimal unpolarized TMDPDF and TMDFF are defined in eq. (\[def:phen-f1\], \[def:phen-D1\], \[def:fNP\], \[def:DNP\]). At small-$b$ TMD distributions are matched to corresponding collinear distributions. We have found that the result of our fit highly depends on the set of collinear distributions. The study of this effect are in sec. \[sec:PDF\],\[sec:SIDIS-FF\]. The phenomenological ansatz for the RAD is given in eq. (\[NP:RAD\]). In table \[tab:pert\] we list the perturbative orders used in each factor of the cross section. The N$^3$LO perturbative composition used here is equivalent to the one used in [@Bizon:2018foh; @Bizon:2019zgf] on the resummation side. A total of 11 phenomenological parameters are determined by the fit procedure. Two of these parameters describe the RAD, 5 are for the unpolarized TMDPDF, and 4 are for the unpolarized TMDFF.
[|c|c||c|c|c|c|c|c V[4]{} c|]{}Evolution & Acronym in&$C_V$ & $\Gamma_{\text{cusp}}$ & $\gamma_V$ & $\mathcal{D}_{\text{resum}}$ & $\zeta_\mu^{\text{pert}}$ & $\zeta_\mu^{\text{exact}}$& $C$, $\mathbb{C}$\
+matching &present work&&&&&&&\
NNLO+NNLO & NNLO &$\alpha_s^2$ & $\alpha_s^3$ ($\Gamma_2$) & $\alpha_s^2$ ($\gamma_2$) & $\alpha_s^2$ ($d_2$) & $\alpha_s^1$ ($v_1$) & $\alpha_s^1$ ($g_2$)& $\alpha_s^2$\
N$^3$LO+NNLO & N$^3$LO&$\alpha_s^{3}$ & $\alpha_s^4$ ($\Gamma_3$) & $\alpha_s^3$ ($\gamma_3$) & $\alpha_s^3$ ($d_3$) & $\alpha_s^2$ ($v_2$) & $\alpha_s^2$ ($g_3$)& $\alpha_s^2$\
Data overview {#sec:data}
=============
In the present work, we consider the extraction of unpolarized TMD in DY and SIDIS data, extending so the analysis of ref. [@Bertone:2019nxa] and including the theoretical improvements described in the previous sections. The selection of data is crucial for a proper TMD extraction, because of the limits imposed by the factorization theorem. These constraints are here discussed for both type of reactions.
SIDIS data
----------
[|c||c|c|c|c|c|]{} Experiment & Reaction & ref. & Kinematics &
--------------
$N_{\rm pt}$
after cuts
--------------
: \[tab:SIDIS-data\]Summary of the SIDIS data included in the fit. For each data set we report reference, reaction, kinematic region, and number of points that are left after the application of consistency cuts in eq. (\[SIDIS-data:Q>2\], \[SIDIS-data:qT<0.25 Q\]).
\
& $p\to \pi^+$ & & & 24\
& $p\to \pi^-$ &&& 24\
& $p\to K^+$ &&& 24\
& $p\to K^-$ &&& 24\
& $D\to \pi^+$ &&& 24\
& $D\to \pi^-$ &&& 24\
& $D\to K^+$ &&& 24\
& $D\to K^-$ &&& 24\
& $d\to h^+$ & & & 195\
& $d\to h^-$ &&& 195\
&&&&\
Total &&&& 582\
In the current literature, one can find several measurements of the unpolarized SIDIS [@Derrick:1995xg; @Adloff:1996dy; @Asaturyan:2011mq; @Airapetian:2012ki; @Adolph:2013stb; @Aghasyan:2017ctw] and a total of some thousands of data points. We restrict out attention only to those data whose kinematical features are compatible with the energy scaling of the TMD factorization theorem. The first constraint comes from the di-lepton invariant mass ($Q$) and in general from the energy scale of the processes. Most of SIDIS reactions have been measured at fixed target experiments, that are typically run at low energies. Unfortunately much of these data do not accomplish the QCD factorization request of a high $Q$ to separate field modes. To secure our analysis (but still leave some data) we have used a restriction on the average $Q$ of a data point, namely $$\begin{aligned}
\label{SIDIS-data:Q>2}
\langle Q \rangle \geq 2 \text{GeV}.\end{aligned}$$ Here, $\langle Q \rangle$ is the value of $Q$ averaged over the multiplicity value in a bin, see fig. \[fig:bining\]. The restriction in eq. (\[SIDIS-data:Q>2\]) quite reduces the pool of data. In particular, eq. (\[SIDIS-data:Q>2\]) completely discards the JLAB measurement published in [@Asaturyan:2011mq], and cuts out the most part of HERMES data in ref. [@Airapetian:2012ki].
The second constraint comes from the TMD factorization assumptions. Namely, the TMD factorization regime is fully consistent only for low values of $q_T/Q$ and receives quadratic power corrections of order $(q_T/Q)^2$, see eq. (\[SIDIS:Wmunu-J\]) and eq. (\[SIDIS:xSec\]). We consider data such that $$\begin{aligned}
\label{SIDIS-data:qT<0.25 Q}
\delta\equiv\frac{\langle q_T \rangle}{\langle Q\rangle}<0.25,\end{aligned}$$ where the value $0.25$ was deduced in [@Scimemi:2017etj]. From the $q_T$ interval of eq. (\[SIDIS-data:qT<0.25 Q\]), one can expect a $\sim4-6\%$ influence of the power corrections, which is well inside the uncertainties of the data. In sec. \[sec:TMD-limit\], we have tested the cutting condition eq. (\[SIDIS-data:qT<0.25 Q\]) considering the data at different $\delta$, and found eq. (\[SIDIS-data:qT<0.25 Q\]) sufficient.
![\[fig:bining\] Illustration for bin shapes in HERMES kinematics. Three bins are shown ($0.12<x<0.2$, $0.2<x<0.35$, $0.35<x<0.6$). Solid lines are boundaries of the fiducial region. Color density demonstrates the distribution of multiplicity value within a bin. Crosses show the averaged $(x,Q)$ over multiplicities in a bin. The bin $0.12<x<0.2$ is not included in the fit since it has $\langle Q\rangle <2$GeV.](Figures/bining.pdf){width="50.00000%"}
It should not pass unobserved that eq. (\[SIDIS-data:qT<0.25 Q\]) is written in terms of ${\bm{q}}^2_T$, that is the natural variable of the TMD factorization approach, whereas the data are presented in terms of ${\bm{p}}_\perp^2$. These variables are related by ${\bm{q}}_T^2\simeq {\bm{p}}_\perp^2/z^2$, see eq. (\[def:qT<->pT\]). Thus, the cut in eq. (\[SIDIS-data:qT<0.25 Q\]) puts also a restriction on $z$. Altogether it makes the allowed values of ${\bm{p}}_\perp^2$ even smaller, ${\bm{p}}_\perp\lesssim 0.25 z Q$. In particular, we have to completely discard the measurements of H1 and ZEUS collaborations [@Derrick:1995xg; @Adloff:1996dy] that are made at very small values of $z$, despite the relatively high values of $Q$.
After the application of eq. (\[SIDIS-data:Q>2\], \[SIDIS-data:qT<0.25 Q\]) we are left with the data taken by HERMES and COMPASS[^2] collaborations [@Airapetian:2012ki; @Aghasyan:2017ctw]. For HERMES we have selected the `zxpt-3D`-binning set due to the finer bins in $p_T$. The COMPASS data includes the subtraction of vector-boson channel, and thus we also select the subtracted HERMES data (`.vmsub` set). In total we have 582 points that cover the region of $1.5\simeq Q\simeq 9$ GeV, $10^{-2}\simeq x<0.6$, $0.2<z<0.8$. The summary of the considered data is reported in table \[tab:SIDIS-data\].
DY data
-------
[|c||c|c|c|c|c|c|]{} Experiment & ref. &$\sqrt{s}$ \[GeV\]& $Q$ \[GeV\] & $y$/$x_F$ &
----------
fiducial
region
----------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
&
--------------
$N_{\rm pt}$
after cuts
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
\
E288 (200) & [@Ito:1980ev] & 19.4 &
----------------
4 - 9 in
1 GeV bins$^*$
----------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& $0.1<x_F<0.7$ & -& 43\
E288 (300) & [@Ito:1980ev] & 23.8 &
----------------
4 - 12 in
1 GeV bins$^*$
----------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& $-0.09<x_F<0.51$ & - & 53\
E288 (400) & [@Ito:1980ev] & 27.4 &
----------------
5 - 14 in
1 GeV bins$^*$
----------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& $-0.27<x_F<0.33$ & - & 76\
E605 & [@Moreno:1990sf] & 38.8 &
------------
7 - 18 in
5 bins$^*$
------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& $-0.1<x_F<0.2$ & -& 53\
E772 & [@McGaughey:1994dx] & 38.8 &
------------
5 - 15 in
8 bins$^*$
------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& $0.1<x_F<0.3$ & - & 35\
PHENIX & [@Aidala:2018ajl] & 200 & 4.8 - 8.2 & $1.2<y<2.2$ & - & 3\
CDF (run1) & [@Affolder:1999jh] & 1800 & 66 - 116 & - & - & 33\
CDF (run2) & [@Aaltonen:2012fi] & 1960 & 66 - 116 & - & - & 39\
D0 (run1) & [@Abbott:1999wk] & 1800 & 75 - 105 & - & - & 16\
D0 (run2) & [@Abazov:2007ac] & 1960 & 70 - 110 & - & - & 8\
D0 (run2)$_\mu$ & [@Abazov:2010kn] & 1960 & 65 - 115 & $|y|<1.7$ &
--------------
$p_T>15$ GeV
$|\eta|<1.7$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 3\
ATLAS (7TeV) & [@Aad:2014xaa] & 7000 & 66 - 116 &
-------------
$|y|<1$
$1<|y|<2$
$2<|y|<2.4$
-------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
&
--------------
$p_T>20$ GeV
$|\eta|<2.4$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 15\
ATLAS (8TeV) & [@Aad:2015auj] & 8000 & 66 - 116 &
-----------
$|y|<2.4$
in 6 bins
-----------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
&
--------------
$p_T>20$ GeV
$|\eta|<2.4$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 30\
ATLAS (8TeV) & [@Aad:2015auj] & 8000 & 46 - 66 & $|y|<2.4$ &
--------------
$p_T>20$ GeV
$|\eta|<2.4$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 3\
ATLAS (8TeV) & [@Aad:2015auj] & 8000 & 116 - 150 & $|y|<2.4$ &
--------------
$p_T>20$ GeV
$|\eta|<2.4$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 7\
CMS (7TeV) & [@Chatrchyan:2011wt] & 7000 & 60 - 120 & $|y|<2.1$ &
--------------
$p_T>20$ GeV
$|\eta|<2.1$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 8\
CMS (8TeV) & [@Khachatryan:2016nbe] & 8000 & 60 - 120 & $|y|<2.1$ &
--------------
$p_T>20$ GeV
$|\eta|<2.1$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 8\
LHCb (7TeV) & [@Aaij:2015gna] & 7000 & 60 - 120 & $2<y<4.5$ &
--------------
$p_T>20$ GeV
$2<\eta<4.5$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 8\
LHCb (8TeV) & [@Aaij:2015zlq] & 8000 & 60 - 120 & $2<y<4.5$ &
--------------
$p_T>20$ GeV
$2<\eta<4.5$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 7\
LHCb (13TeV) & [@Aaij:2016mgv] & 13000 & 60 - 120 & $2<y<4.5$ &
--------------
$p_T>20$ GeV
$2<\eta<4.5$
--------------
: Summary table for the data included in the fit.\[tab:data\]. For each data set we report: the reference publication, the centre-of-mass energy, the coverage in $Q$ and $y$ or $x_F$, possible cuts on the fiducial region, and the number of data points that survive the cut in eq. (\[DY-data:cuts\]).
& 9\
Total & & & & & & 457\
Bins with $9\lesssim Q \lesssim 11$ are omitted due to the $\Upsilon$ resonance.
The DY data are selected following the same principles as the SIDIS data, eq. (\[SIDIS-data:qT<0.25 Q\]) (the rule (\[SIDIS-data:Q>2\]) makes no sense now, because DY processes are measured at sufficiently high-energies) with only small modifications. The changes consist in the cutting some extra higher-$q_T$ data points for several specific data sets (this concerns mainly ATLAS measurements of Z-boson production). The reason for it is that the estimated size of power corrections at $q_T/Q\sim 0.25$ is of order of $5\%$, however, some highly precise data are measured with much better accuracy. So, given a data point $p\pm \sigma$, with $p$ being the central value and $\sigma$ its uncorrelated relative uncertainty, corresponding to some values of $q_T$ and $Q$, we include it in the fit only if $$\begin{aligned}
\label{DY-data:cuts}
\delta\equiv \frac{\langle q_T\rangle}{\langle Q\rangle } <0.1,\qquad \text{or}\qquad \delta<0.25\quad \text{if}\quad \delta^2<\sigma.\end{aligned}$$ In other words, if the (uncorrelated) experimental uncertainty of a given data point is smaller than the theoretical uncertainty associated to the expected size of power corrections, we drop this point from the fit. This is the origin of the second condition in eq. (\[DY-data:cuts\]).
The resulting data set contains 457 data points, and spans a wide range in energy, from $Q=4$ GeV to $Q=150$ GeV, and in $x$, from $x\sim0.5\cdot10^{-4}$ to $x\sim 1$. Table \[tab:data\] reports a summary of the full data set included in our fit. This selection of the data is the same as the one considered in our earlier work [@Bertone:2019nxa]. In the current fit, we compare the absolute values of the cross-section, whenever they are available. The only data set that require normalization factors are all CMS data, ATLAS at 7 TeV, and DO run2 measurements. For these sets we have normalized the integral of the theory prediction to corresponding integral over the data (see explicit expression in ref. [@Scimemi:2017etj]).
Summary of the data set {#sec:summary-data}
-----------------------
In total for the extraction of unpolarized TMD distribution we analyze 1039 data points that are almost equally distributes between SIDIS (582 points) and DY (457 points) processes. All these points contribute to the determination of the TMD evolution kernel $\mathcal{D}$ and unpolarized TMDPDF $f_1$. The determination of unpolarized TMDFF is based only on SIDIS data. In addition, we recall that a single DY data point is simultaneously sensitive to a larger and a smaller value of $x$. This is because the cross section is given by a pair of TMDPDFs, eq. (\[def:Wff\]), computed at $x_1$ and $x_2$ such that $x_1x_2 \simeq Q^2/s$. So, the statistical weight of a DY point in the determination of TMDPDF is effectively doubled.
The kinematic region in $x$ and $Q$ covered by the data set and thus contributing to the determination of TMDPDF is shown in fig. \[fig:dataPoints\]. The boxes enclose the sub-regions covered by the single data sets. Looking at fig. \[fig:dataPoints\], it is possible to distinguish two main clusters of data: the “low-energy experiments”, *i.e.* E288, E605, E772, PHENIX, COMPASS and HERMES that place themselves at invariant-mass energies between 1 and 18 GeV, and the “high-energy experiments”, *i.e.* all those from Tevatron and LHC, that are instead distributed around the $Z$-peak region. From this plot we observe that, kinematic ranges of SIDIS and DY data do not overlap.
![\[fig:dataPoints\] Density of data in the plane $(Q,x)$ (a darker color corresponds to a higher density).](Figures/TMDPDF-data.pdf){width="60.00000%"}
As a final comment of this section let us mention that our data selection is particularly conservative because it drops points that could potentially be described by TMD factorization (see e.g. ref. [@Bacchetta:2017gcc] where a less conservative choice of cuts is used). However, our fitted data set guarantees that we operate well within the range of validity of TMD factorization. In sec. \[sec:GLOBAL-fit\] we show that unexpectedly our extraction can describe a larger set of data as well.
Fit procedure {#sec:fitprocedure}
=============
The experimental data are usually provided in a form specific for each setup. In order to extract valuable information for the TMD extraction, one has to detail the methodology that has been followed, and this is the purpose of this section. Finally, we also provide a suitable definition of the $\chi^2$ that allows for a correct exploitation of experimental uncertainties.
Treatment of nuclear targets and charged hadrons {#sec:nuclear}
------------------------------------------------
The data from E288, E605 (*Cu*), E772, COMPASS, (part of) HERMES (isoscalar targets) come from nuclear target processes. In these cases, we perform the iso-spin rotation of the corresponding TMDPDF that simulates the nuclear-target effects. For example, we replace u-, and d-quark distributions by $$\begin{aligned}
f_{1,u{\leftarrow}A}(x,b)&=&\frac{Z}{A}f_{1,u{\leftarrow}p}(x,b)+\frac{A-Z}{A}f_{1,d{\leftarrow}p}(x,b),
\\
f_{1,d{\leftarrow}A}(x,b)&=&\frac{Z}{A}f_{1,d{\leftarrow}p}(x,b)+\frac{A-Z}{A}f_{1,u{\leftarrow}p}(x,b),\end{aligned}$$ where A(Z) is atomic number(charge) of a nuclear target. In principle, for E288, E605 data extracted from very heavy targets one should also incorporate the nuclear modification factor that depends on $x$. However, we found it unnecessary since the systematic errors of these experiments are much larger than a possible nuclear effect.
The measurements of SIDIS are made in a number of different channels. The HERMES data include $\pi^\pm$ and $K^\pm$, and COMPASS data are for charged hadrons, $h^\pm$. Pions and kaons are described by an individual TMDFFs. However, charged hadron are a composition of different TMDFFs. According eq. (\[def:D1\]) the TMDFF for charged hadrons is a direct sum of TMDFFs for individual hadrons: $$\begin{aligned}
\label{def:h+-}
D_{1,f\to h^\pm}(x,b)=\sum_{h\in h^\pm}D_{1,f\to h}(x,b)=D_{1,f\to \pi^\pm}(x,b)+D_{1,f\to K^\pm}(x,b)+...~,\end{aligned}$$ where dots denote the higher-mass hadron states. At COMPASS energies, this sum is dominated by the pion ($65-75\%$), and the kaon ($15-20\%$) contributions. The residual term is lead by proton/antiproton contribution ($2-5\%$). The contribution of other particles is smaller (for discussion and references see [@Bertone:2018ecm; @Bertone:2017tyb]). Thus, in our study we use the first two terms of eq. (\[def:h+-\]) to simulate the charged hadron fragmentation.
The SIDIS measurements in refs. [@Airapetian:2012ki; @Aghasyan:2017ctw] are given in form of multiplicities. The SIDIS multiplicity is defined as $$\begin{aligned}
\label{def:multiplicity}
\frac{dM^h(x,Q^2,z,{\bm{p}}_\perp^2)}{dz d {\bm{p}}_\perp^2}={\left(}\frac{d\sigma}{dxdz dQ^2 d{\bm{p}}_\perp^2}{\right)}\Big/{\left(}\frac{d\sigma_{\text{DIS}}}{dxdQ^2}{\right)},\end{aligned}$$ where $d\sigma_{\text{DIS}}$ is the differential cross-section for DIS. It reads $$\begin{aligned}
\frac{d\sigma_{\text{DIS}}}{dxdQ^2}=\frac{4\pi \alpha_{\text{em}}^2}{xQ^4}\Big[{\left(}1-y-\frac{y^2\gamma^2}{4}{\right)}F_2(x,Q^2)+xy^2 F_1(x,Q^2)\Big],\end{aligned}$$ where $F_1$ and $F_2$ are DIS structure functions. The DIS cross-section cannot be computed starting from TMD factorization, but it is described by the collinear factorization theorem. In order to evaluate the multiplicity we have pre-computed the DIS cross-section (integrated over the given bin) by the `APFEL`-library [@Bertone:2013vaa], and then divided the TMD prediction according to eq. (\[def:multiplicity\]).
Bin integration in SIDIS and DY
-------------------------------
The majority of SIDIS data is measured at relatively low-Q and in large bins. The cross-section value changes greatly within a bin, and so, binning effects are known to be strong. For a measured cross-section $d\sigma/dx dz dQ^2d {\bm{p}}_\perp^2$, a bin is specified by $\{x_{\text{min}},x_\text{max}\}$, $\{z_{\text{min}},z_\text{max}\}$, $\{Q_{\text{min}},Q_\text{max}\}$ and $\{{\bm{p}}_{\text{min}},{\bm{p}}_\text{max}\}$. The binning constraints impose certain cuts on the measured phase space. Typically, these cuts are given as intervals of the variable $y$ and of the invariant mass of photon-target system $W^2=(P+q)^2$, which belong to ranges $\{y_{\text{min}},y_\text{max}\}$ and $\{W^2_\text{min},W^2_\text{max}\}$. Both these variables are connected to $x$ and $Q^2$, $$\begin{aligned}
W^2=M^2+Q^2 \frac{1-x}{x},\qquad y=\frac{Q^2}{x(s-M^2)}.\end{aligned}$$ where $s$ is the Mandelshtam variable $s=(P+l)^2$. So, in the presence of fiducial cuts in SIDIS the bin boundaries are $$\begin{aligned}
\hat x_{\text{min}}(Q)&=&\max\{x_{\text{min}},\frac{Q^2}{y_{\text{max}}(s-M^2)},\frac{Q^2}{Q^2+W^2_{\text{max}}-M^2}\},
\\
\hat x_{\text{max}}(Q)&=&\min\{x_{\text{max}},\frac{Q^2}{y_{\text{min}}(s-M^2)},\frac{Q^2}{Q^2+W^2_{\text{min}}-M^2}\},
\\
\hat Q^2_{\text{min}}&=&\max\{Q^2_{\text{min}},x_{\text{min}}y_{\text{min}}(s-M^2),\frac{x_{\text{min}}}{1-x_{\text{min}}}(W^2_{\text{min}}-M^2)\},
\\
\hat Q^2_{\text{max}}&=&\min\{Q^2_{\text{max}},x_{\text{max}}y_{\text{max}}(s-M^2),\frac{x_{\text{max}}}{1-x_{\text{max}}}(W^2_{\text{max}}-M^2)\}.\end{aligned}$$ An example of effects of cuts in the bins is shown in fig. \[fig:bining\]. In the case of multiplicity measurements the bin effects are taken into account with the cross-section $$\begin{aligned}
&&\frac{dM^h(x,Q^2,z,{\bm{p}}_\perp^2)}{dz d {\bm{p}}_\perp^2}\Bigg|_{\text{bin}}=(z_\text{max}-z_{\text{min}})^{-1}({\bm{p}}^2_\text{max}-{\bm{p}}^2_{\text{min}})^{-1}
\\{\nonumber}&&\qquad \times\int_{{\bm{p}}_{\text{min}}}^{{\bm{p}}_{\text{max}}}2{\bm{p}}_\perp d {\bm{p}}_\perp \int_{z_{\text{min}}}^{z_{\text{max}}} dz \int_{\hat Q_\text{min}}^{\hat Q_\text{max}}2Q dQ \int_{\hat x_{\text{min}}(Q)}^{\hat x_{\text{max}}(Q)}dx \frac{d\sigma}{dxdz dQ^2 d{\bm{p}}_\perp^2}
\\{\nonumber}&&\qquad \Big/\int_{\hat Q_\text{min}}^{\hat Q_\text{max}}2Q dQ \int_{\hat x_{\text{min}}(Q)}^{\hat x_{\text{max}}(Q)}dx \frac{d\sigma_{\text{DIS}}}{dxdQ^2 },\end{aligned}$$ where the expression in the first line is the volume of $(z,{\bm{p}}_\perp^2)$-bin.
In the case of DY the binning effects are also extremely important. The difference in the value of the cross section between center-of-bin and the averaged/integrated value can reach tenth of percents, especially, for very low-energy bins (where the change in $Q$ is rapid), and for very wide bins (such as Z-boson measurement). We have used the definition $$\begin{aligned}
\frac{d\sigma}{dQ^2dy{\bm{q}}_T^2}\Big|_\text{bin}=(Q^2_{\text{max}}-Q^2_\text{min})^{-1}({\bm{q}}^2_{\text{max}}-{\bm{q}}^2_\text{min})^{-1}(y_{\text{max}}-y_\text{min})^{-1}
\\{\nonumber}\times \int_{{\bm{q}}_\text{min}}^{{\bm{q}}_\text{max}} 2{\bm{q}}_T d{\bm{q}}_T \int_{Q_\text{min}}^{Q_\text{max}} 2Q dQ
\int_{y_\text{min}}^{y_\text{max}} dy \frac{d\sigma}{dQ^2dy{\bm{q}}_T^2}.\end{aligned}$$
Definition of $\chi^2$-test function and estimation of uncertainties
--------------------------------------------------------------------
To test the theory prediction against the experimental measurement we compute the $\chi^2$-test function $$\label{eq:chi2cov}
\chi^2=\sum_{i,j=1}^{n}{\left(}m_i-t_i\right)V_{ij}^{-1}{\left(}m_j-t_j{\right)},$$ where $m_i$ is the central value of $i$’th measurement, $t_i$ is the theory prediction for this measurement and $V_{ij}$ is the covariance matrix. An accurate definition of the covariance matrix is essential for a correct exploitation of experimental uncertainties. In order to build the covariance matrix we distinguish, uncorrelated and correlated uncertainties. For example, a typical data point has the structure $$m_i\pm \sigma_{i,\text{stat}} \pm \sigma_{i,\text{unc}} \pm \sigma_{i,\text{corr}}^{(1)}\pm\dots \pm \sigma_{i,\text{corr}}^{(k)},$$ where $m_i$ the reported central value, $\sigma_{i,\text{stat}}$ is (uncorrelated) statistical uncertainty, $\sigma_{i,\text{unc}}$ is uncorrelated systematic uncertainty, and $\sigma_{i,\text{corr}}^{(k)}$ are correlated systematic uncertainties. Uncorrelated uncertainties give an estimate of the degree of knowledge of a particular data point irrespective of the other measurements of the data set. Instead, correlated uncertainties provide an estimate of the correlation between the statistical fluctuations of two separate data points of the same data set. With this information at hand, one can construct the covariance matrix $V_{ij}$ as follows (for more detailed discussion on this definition see refs. [@Ball:2008by; @Ball:2012wy]): $$\label{eq:covmat}
V_{ij}={\left(}\sigma_{i,\text{stat}}^2 +\sigma_{i,\text{unc}}^2{\right)}\delta_{ij} + \sum_{l=1}^{k}\sigma_{i,\text{corr}}^{(l)}\sigma_{j,\text{corr}}^{(l)}.$$ Equipped by this definition of covariance matrix the $\chi^2$-test in eq. (\[eq:chi2cov\]) takes into account the nature of the experimental uncertainties leading to a faithful estimate of the agreement between data and theoretical predictions.
To estimate the error propagation from the experimental data to the extracted values of TMD distributions we have used the replica method. This method is described in details in ref. [@Ball:2008by]. It consists in the generation of $N$ replicas of the pseudo data, and minimization of the $\chi^2$ on each replica. The resulting set of $N$ vectors of NP parameters is distributed in accordance to the distribution law of the data. And thus, it represents a Monte Carlo sample that is used to evaluate mean values, standard deviation and correlations of the NP parameters. For the estimation of error propagation we consider $N=100$ replicas. The procedure of $\chi^2$-minimization for each replica is the most computationally heavy part of the fit.
The proper treatment of correlated uncertainties is essential in global analysis. The presence of sizable correlated uncertanties could result into a misleading visual disagreement between theory prediction and the (central values of) data points. Namely, the theory prediction for a data set could be globally shifted by significant amount, that is nonetheless in agreement with correlated experimental uncertainty. To quantify the effects of correlated shifts we use the nuisance parameter method presented in [@Ball:2008by; @Ball:2012wy]. Within the nuisance parameter method one is able to determine the shift $d_i$ of a theory prediction $t_i$ for the $i$’th data point, such that $\bar t_i=t_i+d_i$ contributes only to the uncorrelated part of the $\chi^2$-value. The value $d_i$ is interpreted as a shift caused by the correlated uncertainties. It is computed as $$\begin{aligned}
d_i=\sum_{l,m=1}^k \sigma_{i,\text{corr}}^{(l)}\,A^{-1}_{lm}\,\rho_m,\end{aligned}$$ where $$\begin{aligned}
A_{lm}=\delta_{lm}+\sum_{i=1}^n\frac{\sigma_{i,\text{corr}}^{(l)}\sigma_{i,\text{corr}}^{(m)}}{\sigma_{i,\text{stat}}^2 +\sigma_{i,\text{unc}}^2},\qquad
\rho_l=\sum_{i=1}^n \frac{m_i-t_i}{\sigma_{i,\text{stat}}^2 +\sigma_{i,\text{unc}}^2}\sigma_{i,\text{corr}}^{(l)}~.\end{aligned}$$ It also instructive to check the average systematic shift, which we define as $$\begin{aligned}
\label{def:d/sigma}
\langle d/\sigma \rangle = \frac{1}{n}\sum_{i=1}^n \frac{d_i}{m_i}.\end{aligned}$$ It shows a general deficit/excess of the theory with respect to the data for a given data set.
Let us note that the multiplicities of the SIDIS are convenient as an experimental object due to the fact that systematic uncertainties related to measurement efficiency and the beam luminosity cancel in the ratio. However, theoretically multiplicities are worse defined, since the denominator and the numerator of multiplicity ratio (\[def:multiplicity\]) need a completely different theoretical treatment. In order to account this effect, we have computed the uncertainty of theory prediction for DIS cross-section for each bin and added it as a fully correlated error for each data set. We should admit that the theory uncertainty for DIS cross-section is negligibly small (typically, $0.1-2.0\%$) in comparison to systematic uncertainties of experiment. Thus, this addition affects the resulting values of $\chi^2$ very little on the level of $\pm10^{-2}$ per point.
Artemide
--------
The computation of the cross-section is made with the code `artemide` that is developed by us. `Artemide` is organized as a package of Fortran 95 modules, each devoted to evaluation of a single theory construct, such as the TMD evolution factor, a TMD distribution, or their combinations such as structure functions $W$ and cross-sections. The `artemide` also evaluates all necessary procedures needed for the comparison with the experimental data, such as bin-integration routines and cut factors. For simplicity of data analysis `artemide` is equipped by a python interface, called `harpy`. The `artemide` package together with the `harpy` is available in the repository [@web].
The module organization of `artemide` allows for flexible use. In particular, it gives to a user a full access to non-perturbative ansatzes and models. Although `artemide` is based on the $\zeta$-prescription, it also includes other strategies for TMD evolution, such as CSS evolution [@Aybat:2011zv], $\gamma$-improved evolution [@Scimemi:2018xaf] and their derivatives. The user has full control on the perturbative orders, and can set each individual part to a particular (known) order. Currently, `artemide` can evaluate unpolarized TMD distributions, and linearly polarized gluon distributions together with the related cross-sections, such as DY, SIDIS, Higgs-production (for application see [@Gutierrez-Reyes:2019rug]), etc. In future, we plan to include more processes and distributions.
The evaluation of a single cross-section point that is to be compared with the experimental one, implies the evaluation of a number of integrals: two Mellin convolution for small-$b$ matching eq. (\[def:phen-f1\], \[def:phen-D1\]), the Hankel-type integral for the structure function $W$ eq. (\[def:WfD-final\], \[def:Wff-final\]), and 3(in DY case)/4(in SIDIS case) bin-integrations. Note, that in the $\zeta$-prescription one does not need to evaluate integrations for TMD evolution, which is its additional positive point. Altogether, it makes the evaluation of TMD cross-section rather expansive in terms of computing time. `Artemide` uses adaptive integration routines to ensure the required computation accuracy. To speed-up the evaluation, `artemide` precomputes the tables of Mellin convolutions for TMD distributions that are the most difficult integrations. Nonetheless, the computation cost is rather high. In particular, it takes about 4.5 (3.2) minutes to evaluate a single $\chi^2$ value for the full data set of DY and SIDIS given in sec. \[sec:data\] on an average 8-core (12-core) processor (2.5GHz) depending on the NP-values. Therefore, the minimization $\chi^2$ and especially the computation of error-propagation are especially long. Due to that we are restricted in certain important directions of studies (e.g. error-propagation of PDF sets, and flavour dependence).
Fit of DY {#sec:DY-fit}
=========
The data-set and the functional input for the DY fit is inherited from our earlier study [@Bertone:2019nxa]. The only modification is the update of the functional form of the special null-evolution line in eq. (\[NP:zeta\]), which in the present case matches the exact solution at large-$b$. This update leads relatively minor formal changes, while some values of the model parameter are changed as a result of the fit. The value of $\chi^2$ (per 457 points) is reduced from $1.174\; {\text{\cite{Bertone:2019nxa}}}\to1.168\; ({\text{this work}})$. The main impact takes place at low-energies. In particular, the typical deficit in the cross-section for low-energy experiments is reduced by 5-6% (compare table 3 in [@Bertone:2019nxa] with table \[tab:final\]), which however does not significantly affects the $\chi^2$ values due to the large correlated uncertainties of fixed-target DY measurements.
In this section, we present the fit of DY data-set only. Since the general picture is similar to ref. [@Bertone:2019nxa], we concentrate on the sources of systematic uncertainties of our approach. We discuss the dependence on the collinear PDF, that serves as a boundary for TMDPDF, and the effects of $q_T$ corrections in the definitions of $x_{1,2}$.
Dependence on PDF {#sec:PDF}
-----------------
Short name Full name Ref. LHAPDF id.
------------ ------------------------- ------------------------- ------------
NNPDF31 NNPDF31\_nnlo\_as\_0118 [@Ball:2017nwa] 303600
HERA20 HERAPDF20\_NNLO\_VAR [@Abramowicz:2015mha] 61230
MMHT14 MMHT2014nnlo68cl [@Harland-Lang:2014zoa] 25300
CT14 CT14nnlo [@Dulat:2015mca] 13000
PDF4LHC PDF4LHC15\_nnlo\_100 [@Butterworth:2015oua] 91700
: \[tab:PDFs\] List of collinear PDF used as the boundary for unpolarized TMDPDF.
The collinear PDF is an important part of our model for TMDPDF, e.g. eq. (\[def:phen-f1\]). The small-$b$ matching essentially reduces the number of NP parameters for TMDPDF and guarantees the asymptotic agreement of the TMDPDF with the collinear observables. The small-$b$ part of the Hankel integral gives the dominant contribution to the cross-section, especially for $q_T\sim 10$-$20$ GeV. Therefore, the quality of our fit and the values of the extracted NP parameter are robustly correlated with the collinear PDF set. This observation has been made earlier, e.g. see discussion in [@Signori:2013mda; @Bacchetta:2017gcc; @Bertone:2019nxa], but it has not been systematically studied. Ideally, the PDF set and TMDPDF are to be coherently extracted in a global fit of collinear and TMD observables. Meanwhile, we treat a collinear input as an independent parameter which we cannot control, thus test various values available from the literature.
![\[fig:f1-param\] Comparison of NP parameter for unpolarized TMDPDF extracted in different fits. The values above the red-dashed line are extracted in fit of DY data, see table \[tab:DY-PDF-chi\]. The values below the red-dashed line are extracted in global fit of DY and SIDIS data, see table \[tab:NP-param-final\]. The vertical dashed lines and gray boxes correspond to average mean and standard deviation of the results of the global fit. The blue points and their error-bars correspond to the estimation of the uncertainties from the collinear PDF (see sec. \[sec:PDFuncert\]). The input collinear distinctions are marked in the right column. For CT14 and MMHT14 some NP parameters are beyond the plot region.](Figures/TMD-Ev-Comparison.pdf){width="50.00000%"}
![\[fig:f1-param\] Comparison of NP parameter for unpolarized TMDPDF extracted in different fits. The values above the red-dashed line are extracted in fit of DY data, see table \[tab:DY-PDF-chi\]. The values below the red-dashed line are extracted in global fit of DY and SIDIS data, see table \[tab:NP-param-final\]. The vertical dashed lines and gray boxes correspond to average mean and standard deviation of the results of the global fit. The blue points and their error-bars correspond to the estimation of the uncertainties from the collinear PDF (see sec. \[sec:PDFuncert\]). The input collinear distinctions are marked in the right column. For CT14 and MMHT14 some NP parameters are beyond the plot region.](Figures/TMD-PDF-Comparison.pdf){width="100.00000%"}
There is an enormous amount of available PDF sets. We have tested some of the most popular sets that are recently extracted at NNLO accuracy, see table \[tab:PDFs\]. All sets have LHAPDF interface [@Buckley:2014ana]. For each PDF set we have performed the full fit procedure with the estimation of the error-propagation. In the fit, the central value of PDFs are used, see sec. \[sec:PDFuncert\] for the discussion on the possible impact of PDF uncertainties. The values of the $\chi^2/(N_{pt}=457)$ and the NP parameters are reported in table \[tab:DY-PDF-chi\] for each PDF set in table \[tab:PDFs\]. The visual comparison of the parameter values is shown in fig. \[fig:DNP-param\] and \[fig:f1-param\]. The parameters of RAD ($B_{\text{NP}}$ and $c_0$) are rather stable with respect to input PDF, and in agreement with each other (note, that $B_{\text{NP}}$ and $c_0$ are anti-correlated, see sec. \[sec:NP-RAD-final\]).
Contrary to the RAD, the parameters $\lambda_i$ show a significant dependence on the collinear PDF (see fig. \[fig:f1-param\]). It is expected, since a different collinear PDF dictates a different shape in $x$, while the $Q$-dependence is not changed. The parameters $\lambda_1$ and $\lambda_2$ do not change significantly with different PDFs, while a bigger change is provided by the parameters $\lambda_{3,4,5}$. This is because parameters $\lambda_{1,2}$ dictate the main shape of $f_{NP}$ at middle values of $b$, whereas other parameters are responsible for the large-$b$ tale ($\lambda_{3,4}$) or fine-tuning of $x$-shape $(\lambda_5)$.
In table \[tab:DY-PDF-chi\], fits are ordered according to the $\chi^2/N_{pt}$ value obtained in the DY fit. The distribution of the values of $\chi^2$ between experiments changes for different PDFs. For example, NNPDF31 demonstrates some tension between ATLAS and LHCb subsets (see table 3 in ref. [@Bertone:2019nxa], and also table \[tab:final\]). In the case of HERA20 this tension reduces. The value of $\chi^2/N_{pt}$ for ATLAS measurements is practically the same in both cases, we find $2.02_{\text{NNPDF}}$ vs. $1.99_{\text{HERA}}$ for $N_{pt}=55$ (note, that the bin-by-bin distribution of $\chi^2$ changes between the sets). On contrary, the value of $\chi^2/N_{pt}$ for LHCb measurement undoubtedly differ in the two sets of PDF, as we find $2.93_{\text{NNPDF}}$ vs. $1.24_{\text{HERA}}$ for $N_{pt}=24$. The main part of the improvement happens due to the general normalization, that is lower by 3-5% in NNPDF case, and almost exact in HERA case.
The TMD distributions with NNPDF31 and HERA20 show a $\chi^2$ value better than all the other, e.g. table \[tab:DY-PDF-chi\]. These PDFs have also less tension between high- and low-energy data. For this reason, in the next sections we will consider only PDFs from these extractions. Nonetheless, we preferably select NNPDF31 set in the global SIDIS and DY analysis. The reason is that NNPDF31 distribution is extracted from the global pool of data, whereas HERA20 uses exclusively data from HERA. At the same time, we must admit that HERA20 distribution provides a spectacularly low values of $\chi^2$ in our global fit.
[|l V[4]{} c V[4]{} l|ll|]{} PDF set & $\chi^2/N_{pt}$ & Parameters for $\mathcal{D}$& Parameters for $f_1$ &\
HERA20 & 0.95 &
----------------------------------
$B_\text{NP}=2.29\pm 0.43$
$c_0=(2.22\pm0.93)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.324\pm0.029$
$\lambda_2=13.2\pm2.9$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(3.56\pm 1.59)\cdot 10^2$
$\lambda_4=2.05\pm0.26$
$\lambda_5=-10.4\pm 3.5$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
HERA20(N$^3$LO) & 0.96 &
----------------------------------
$B_\text{NP}=1.94\pm 0.41$
$c_0=(3.35\pm0.68)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.326\pm0.024$
$\lambda_2=10.1\pm1.6$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(2.73\pm 0.91)\cdot 10^2$
$\lambda_4=1.70\pm0.19$
$\lambda_5=-6.5\pm 2.4$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
NNPDF31(N$^3$LO) & 1.01 &
----------------------------------
$B_\text{NP}=1.62\pm 0.24$
$c_0=(3.42\pm1.04)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.282\pm0.017$
$\lambda_2=9.7\pm1.3$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(3.17\pm 0.83)\cdot 10^2$
$\lambda_4=2.42\pm0.13$
$\lambda_5=-6.1\pm 1.6$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
NNPDF31 & 1.17 &
----------------------------------
$B_\text{NP}=1.86\pm 0.30$
$c_0=(2.96\pm1.04)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.253\pm0.032$
$\lambda_2=9.0\pm3.0$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(3.47\pm 1.16)\cdot 10^2$
$\lambda_4=2.48\pm0.15$
$\lambda_5=-5.7\pm 3.4$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
MMHT14 & 1.35 &
----------------------------------
$B_\text{NP}=1.55\pm 0.29$
$c_0=(4.70\pm1.77)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.198\pm0.040$
$\lambda_2=26.4\pm4.9$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(26.8\pm 13.2)\cdot 10^2$
$\lambda_4=3.01\pm0.17$
$\lambda_5=-23.4\pm 5.4$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
PDF4LHC & 1.56 &
----------------------------------
$B_\text{NP}=1.93\pm 0.47$
$c_0=(3.66\pm2.09)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.218\pm0.041$
$\lambda_2=17.9\pm4.5$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
--------------------------------------
$\lambda_3=(9.26\pm 8.38)\cdot 10^2$
$\lambda_4=2.54\pm0.17$
$\lambda_5=-15.5\pm 4.7$
--------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
CT14 & 1.63 &
----------------------------------
$B_\text{NP}=2.35\pm 0.61$
$c_0=(2.27\pm1.33)\cdot 10^{-2}$
----------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
---------------------------
$\lambda_1=0.277\pm0.029$
$\lambda_2=24.9\pm2.9$
---------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
&
-------------------------------------
$\lambda_3=(12.4\pm 3.2)\cdot 10^2$
$\lambda_4=2.67\pm0.13$
$\lambda_5=-23.8\pm 2.9$
-------------------------------------
: \[tab:DY-PDF-chi\] Values of $\chi^2$ and NP parameters obtained obtained in the fit of DY set of the data with different PDF inputs.
\
Impact of exact values for $x_{1,2}$ and power corrections {#sec:DY:x12}
----------------------------------------------------------
As discussed in sec. \[sec:summary-theory\], the factorization formula eq. (\[DY:xSec\]) for DY contains three types of power corrections. The corrections related to TMD factorization cannot be tested, without extra modeling. The corrections due to fiducial cuts must be included without restrictions. Thus it is possible to test only the power correction due to presence of $q_T/Q$ in the exact values of $x_{1,2}$, eq. (\[def:x1x2\]). The amount of this correction is obtained comparing the fits of the DY data with $$\begin{aligned}
{\nonumber}(\text{exact})x_{1,2}=\sqrt{\frac{Q^2+{\bm{q}}_T^2}{s}}e^{\pm y}\qquad \text{vs.}\qquad (\text{approx.}) x_{1,2}=\frac{Q}{\sqrt{s}}e^{\pm y}.\end{aligned}$$ The approximate values for $x_{1,2}$ lead to higher values of $\chi^2$. In particular, with the approximate $x_{1,2}$ for the NNPDF31 set we have obtained $\chi^2/N_{pt}=1.35$ and $1.27$ at NNLO and N$^3$LO respectively. In the case of HERA20 set, we obtain $\chi^2/N_{pt}=1.03$ and $1.03$. Comparing these values to the ones reported in table \[tab:DY-PDF-chi\] (1.17 and 1.01; 0.95 and 0.96, respectively), we conclude that the quality of fit is worse.
The deterioration of the fit quality takes place in both high- and low- energy parts of the data. In the ATLAS experiment (that is the most precise set at our disposal, with $N_{pt}=55$), we observe the changes in $\chi^2/N_{pt}$: $1.82\to 2.83$ for NNPDF31 and $1.90\to 2.27$ for HERA20. For the fixed target experiments we have $\chi^2/N_{pt}$: $0.91\to 1.31$ for NNPDF31 and $0.71\to 0.97$ for HERA30 (here $N_{pt}=260$). We have also observed that the value of $\chi^2$ worsens mainly due the change in the shape of cross-section, whereas the normalization part slightly reduces the $\chi^2$. The values NP parameters varies within the error-bands and the change in the central values is not significant.
Therefore, we conclude that exact values of $x_{1,2}$ (\[def:x1x2\]) considerably improve the quality of the fit. This conclusion is in agreement with the theory expectations presented in sec. \[sec:summary-theory\].
Uncertainties due to collinear PDFs {#sec:PDFuncert}
-----------------------------------
The model in eq. (\[def:phen-f1\]) is not sensitive to changes of the NP parameters at small-$b$. For this reason, the error-band on the TMD distribution vanishes for $b\lesssim 0.5$GeV$^{-1}$. The only way to modify the TMD distribution in this region is to vary the values of collinear PDF. In sec. \[sec:PDF\] we have demonstrated that the quality of the fit, as well as the values of extracted NP parameters, essentially depend on the collinear PDF and in our extraction we have used the central values of PDF sets, ignoring the uncertainties of PDF determination. These uncertainties are however large and could cover the gap among different TMD fits if taken into account. Unfortunately, the incorporation of the PDF uncertainties into the analysis is extremely demanding in terms of computer time, especially for the full data set. In order to provide a quantitative estimate in this section we consider only the NNPDF31 data set with NNLO TMD evolution for the fit of DY data.
Thus, we have performed a fit for each one of the 100 replicas of the NNPDF31 collinear distributions. The minimization of the $\chi^2$ is done with a simplified procedure in order to speed up the computation, because for many replicas the search of $\chi^2$-minimum took much longer time in comparison to the central value minimization. It appears that the data is very demanding on the collinear PDF input. So, for some (distant from the central) replicas the fit does not converge (yielding $\chi^2/N_{pt}>5$) or produces extreme values of NP parameters (e.g. $B_{\text{NP}}<0.7$GeV). The values of NP parameters that run into the boundary of the allowed phase space region were discarded (almost $30\%$ of total replicas). The resulting distribution of NP parameters gives an estimate of the sensitivity for PDF distribution. The NP parameters and their uncertainties that we have obtained are the following $$\begin{aligned}
&&B_{\text{NP}}=1.7\pm 0.30,\qquad c_0=0.297\pm 0.006,
\\
&&\lambda_1=0.266\pm0.066,\qquad \lambda_2=10.6\pm 3.1,\qquad \lambda_3=158.\pm 133.,
\\{\nonumber}&&\lambda_4=2.55\pm 0.91,\qquad \lambda_5=-7.12\pm.3.92.\end{aligned}$$ These value are compatible with the typical values for NP parameters presented in table \[tab:DY-PDF-chi\], see also fig. \[fig:DNP-param\] and fig. \[fig:f1-param\].
In fig. \[fig:PDFunc-TMD\] we show the comparison of error-bands on the TMDPDF, obtained from the error-propagation from the experiment to NP parameters (blue band), and from the PDF uncertainty (red band), as described above. The main difference in these bands is that the PDF-uncertainty band is sizable already at $b=0$, and for larger $b$ these bands expand in the similar amount. The PDF-uncertainty band is different for different flavors, and larger for non-valence partons. The resulting estimation for the (predicted) cross-section is shown in fig. \[fig:PDFunc-sigma\]. For the high energy case, the uncertainty is of order of $1\%$, while at low energies it reaches 20-40%.
The bands that we show here certainly do not accurately represent the uncertainties of TMDPDF, since many of PDF replicas do not fit the data. It implies that the TMD distributions can be used as a tool for the restriction of collinear PDFs together with the standard collinear observables. At the current stage, we can only conclude that the uncertainties of TMDPDF at small-$b$ (that are out of control in the current model) are sizable. For an accurate estimation of these errors one has to apply more sophisticated techniques, such as reweighing of PDF values [@Ball:2010gb] by TMD extraction, or even joint fits of TMD distributions and collinear distributions, which are beyond of the present work.
![\[fig:PDFunc-TMD\] The size of the uncertainty bands for unpolarized TMDPDFs due to collinear PDF uncertainty (red band) and due to experimental uncertainties (blue band), for d and s-quarks at different values of $x$. Both bands are weighted to the TMPDF obtained with NNPDF31 collinear distribution.](Figures/PDFunc-TMDs.pdf){width="80.00000%"}
![\[fig:PDFunc-sigma\] The size of the uncertainty bands for predicted cross-section due to collinear PDF uncertainty (red band) and due to experimental uncertainties (blue band), at low and high-energies. Both bands are weighted to the prediction obtained with NNPDF31 collinear distribution.](Figures/PDFunc-LE-sigma-ratio.pdf "fig:"){width="40.00000%"} ![\[fig:PDFunc-sigma\] The size of the uncertainty bands for predicted cross-section due to collinear PDF uncertainty (red band) and due to experimental uncertainties (blue band), at low and high-energies. Both bands are weighted to the prediction obtained with NNPDF31 collinear distribution.](Figures/PDFunc-HE-sigma-ratio.pdf "fig:"){width="40.00000%"}
Fit of SIDIS {#sec:SIDIS-fit}
============
In this section, we use the unpolarized TMDPDF and TMD evolution, extracted in the fit of DY data, to fit the SIDIS data. The main aim is to test the universality of the TMD evolution, and TMDPDF. Namely, the SIDIS data should be easily fitted adjusting only the parameters of TMDFF. Indeed, we have find that the TMDFF in eq. (\[def:phen-D1\]) (with a 4-parameter ansatz) together with the TMDPDF and $\mathcal{D}$ (extracted from DY data) provide a very good description of the available SIDIS data. *It is one of the main results of the present work that demonstrates the complete universality of TMD factorization functions.* Its universality has been already tested in [@Vladimirov:2019bfa], where it was applied to the fit of pion-induced DY, and it has been used in studies of the TMD distributions with jets [@Gutierrez-Reyes:2019msa; @Gutierrez-Reyes:2019vbx; @Gutierrez-Reyes:2018qez]. To our best knowledge, the test presented here is made for the first time, because in the previous studies DY and SIDIS cases were considered or independently or simultaneously [@Bacchetta:2017gcc]. Also we discuss the dependence on the collinear unpolarized FF, and the impact of power corrections.
Dependence on FF {#sec:SIDIS-FF}
----------------
[|l V[4]{} c V[4]{} ll|]{} PDF & FF sets & $\chi^2/N_{pt}$ & Parameters for $d_1$ &\
HERA20 & DSS & 0.76 &
------------------------
$\eta_1=0.290\pm0.014$
$\eta_2=0.469\pm0.016$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
------------------------
$\eta_3=0.459\pm0.027$
$\eta_4=0.496\pm0.027$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
HERA20 & DSS (N$^3$LO) & 0.88 &
------------------------
$\eta_1=0.282\pm0.010$
$\eta_2=0.466\pm0.012$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
------------------------
$\eta_3=0.468\pm0.021$
$\eta_4=0.504\pm0.025$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
HERA20 & JAM19 & 0.93 &
-------------------------
$\eta_1=0.164\pm 0.012$
$\eta_2=0.286\pm 0.016$
-------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
-------------------------
$\eta_3=0.223\pm 0.027$
$\eta_4=0.341\pm 0.018$
-------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
NNPDF31 & DSS & 1.00 &
------------------------
$\eta_1=0.257\pm0.009$
$\eta_2=0.480\pm0.010$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
------------------------
$\eta_3=0.455\pm0.017$
$\eta_4=0.540\pm0.020$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
NNPDF31 & DSS (N$^3$LO) & 1.31 &
------------------------
$\eta_1=0.245\pm0.011$
$\eta_2=0.475\pm0.011$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
------------------------
$\eta_3=0.463\pm0.020$
$\eta_4=0.556\pm0.019$
------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
NNPDF31 & JAM19 & 1.65 &
-------------------------
$\eta_1=0.141\pm 0.012$
$\eta_2=0.293\pm 0.017$
-------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
&
-------------------------
$\eta_3=0.224\pm 0.028$
$\eta_4=0.373\pm 0.018$
-------------------------
: \[tab:SIDIS-PDF-FF-chi\] Values of $\chi^2$ and NP parameters obtained in the fit of SIDIS data with different FF inputs. The TMD evolution parameters and TMDPDF parameters are fixed from the fit of DY data (see table \[tab:DY-PDF-chi\]), and labeled by the PDF set. The visual presentation of this table is given in fig. \[fig:d1-param\].
\
![\[fig:d1-param\] Comparison of NP parameter for unpolarized TMDFF extracted in different fits. The values above the red-dashed line are extracted in fit of SIDIS data with fixed TMD evolution and TMDPDF, see table \[tab:SIDIS-PDF-FF-chi\]. The values below the red-dashed line are extracted in global fit of DY and SIDIS data, see table \[tab:NP-param-final\]. The vertical dashed lines and gray boxes correspond to average mean and standard deviation of the results of the global fit. The input collinear distinctions are marked in the right column.](Figures/TMD-FF-Comparison.pdf){width="83.30000%"}
In contrast to collinear PDFs, there are not too many extraction of collinear FFs. We have considered three sets of collinear FFs. Namely, DSS set[^3](that is a composition of pion FFs from [@deFlorian:2014xna] (DSS14) and kaon FFs from [@deFlorian:2017lwf] (DSS17)), the JAM19 set [@Sato:2019yez] and the NNFF10 set [@Bertone:2017tyb]. All these extractions are made with NLO collinear evolution ($a_s^2$).
The comparison of fits with different FFs, and some of TMDPDF (together with TMD evolution) extracted in the previous section are presented in table \[tab:SIDIS-PDF-FF-chi\]. The NNFF set is not presented in the table due to the low quality of the predictions, as it is described below. As it is seen from table \[tab:SIDIS-PDF-FF-chi\], the TMD factorization perfectly describes the low-$q_T$ SIDIS data with TMDPDF and TMD evolution fixed by DY measurements. It is one of the main result of the present analysis.
The values of $\chi^2/N_{pt}$ are rather small (e.g. 0.76 for combination of HERA20 & DSS), which may indicate an over-fit problem. However, this is not the case for the following reason. The main source of low-$\chi^2$ is the COMPASS data set. The COMPASS data have very large uncorrelated systematic uncertainty for a great amount of points. Here, the systematic uncertainty is (much) larger than the statistical uncertainty, and therefore, the COMPASS data points form smooth lines with huge uncorrelated uncertainty band. As a result, the contribution of each point to the $\chi^2$-value is small.
The values of $\chi^2$ depend on the input TMDPDF and TMD evolution (compare NNLO and N$^3$LO cases) in a reasonable amount. This is mainly due to the different values of $c_0$ constant in these cases. We recall that the SIDIS measurements are made at much lower energy in comparison to DY, and thus they are more sensitive to $\mathcal{D}$ at large-$b$. Later in sec. \[sec:GLOBAL-fit\] we show that in the joint fit of SIDIS and DY data, the uncertainty of the evolution factor at large-$b$ is reduced.
The difference between DSS and JAM collinear FFs sets is of minor importance. It is due to the fact low-energy data are less sensitive to the small-$b$ part of the TMD distributions (and thus to collinear distributions). Given in addition that the data are not very precise, the uncertainty in FF sets are compensated by the NP function $D_{NP}$. The effect of compensation is clear from the very different values of $\eta_i$ constants for DSS and JAM19 set. Note, that in all cases we obtain a positive and sizable $b^2$-term in $D_{NP}$ (parameter $\eta_4$). It could indicate a hidden issue in the values of collinear FFs. However, we conclude that contrary to DY case, the SIDIS TMD data are not very restrictive on the values of collinear FFs.
The NNFF distributions are not able to fit the data with a $\chi^2/N_{pt}$ better than $\sim 6.8$. The reason of such an enormous discrepancy is obvious. The NNFF1.0 extraction is made from the $ee$-annihilation data only [@Bertone:2017tyb], and thus is sensitive only to particular combinations of quark-flavors. The flavour separation is thus made a posteriori assuming exact iso-spin symmetry. As a result, the FF for sea quarks have very small (and even negative) values. In the processes where the production of a hadron is dominated by the sea-quark channel, the cross-section obtained with NNFF10 collinear FF is much smaller then the experimental one. A crystal clear example is the process $p\to K^-$, where both valence quarks of $K^-$, $\bar u s$, are sea-quarks for the proton, and thus the dominant channel is the production of $K^-$ from $u$ and $d$ quarks. However, FF for $u$ and $d$-quarks in $K^-$ are negative in NNFF extraction, and the resulting cross-section appears to be negative as well. The situation improves, if we select only the processes with dominant valence channel, e.g. $d\to\pi^\pm$, in this case we obtain $\chi^2/N_{pt}\sim 2.2$. The COMPASS measurement can be also considered separately with the NNFF1.1 set of FF for charged hadrons [@Bertone:2018ecm], in this case we obtain $\chi^2/N_{pt}\sim 1.6$. In any case, we have found that NNFF sets of FF are not suitable for the description of SIDIS data.
The uncertainties on NP parameters presented in table \[tab:SIDIS-PDF-FF-chi\] are *unrealistically small*. Given the fact that the data is not very accurate, it indicates a significant underestimation of the uncertainty for TMDFF. We guess that the underestimation of uncertainties is caused mainly by the function bias of $D_{NP}$. To resolve the situation one could use a more flexible ansatz, e.g. by inclusion of more NP-parameters. Unfortunately, this strategy is not very efficient. Already with the current set of parameters we have very low $\chi^2$, and the increase of the number of parameters could lead to an over-fit problem. Also, the computation time with a bigger number of parameters increases.
Impact of power corrections {#sec:SIDIS-power-corr}
---------------------------
Considering the expression for the SIDIS cross-section eq. (\[SIDIS:xSec\]) we distinguish four types of power corrections: *(m/Q)* the corrections due to non-zero produced mass, *(M/Q)* the corrections due to non-zero target mass, *($q_T$/Q)* the $q_T/Q$-terms in the expression for cross-section and *($x_Sz_S$)* the $q_T/Q$-terms in the expressions for $x_S$ and $z_S$, eq. (\[def:x1z1\]). In order to test the impact of these corrections, we have performed the (central value) fits including corrections in different combinations. The resulting values of $\chi^2/N_{pt}$ are reported in table \[tab:SIDIS-power\].
------------------------------------- ------ -------- -------- -------- -------- --------
include *(m/Q)* yes **no** yes yes **no** **no**
include *(M/Q)* yes yes **no** yes **no** **no**
include *($q_T$/Q)* in kinematics yes yes yes **no** **no** **no**
include *($q_T$/Q)* in $x_S$, $z_S$ yes yes yes yes yes **no**
$\chi^2/N_{pt}$ 1.00 1.00 1.09 1.06 1.16 1.31
------------------------------------- ------ -------- -------- -------- -------- --------
: \[tab:SIDIS-power\] Comparison of results of the fit with different combination of power suppressed terms. The fit is made only for the central values, with fixed TMD evolution and TMDPDF as in NNPDF3.1, with DSS collinear FF.
Let us summarize the observations:
- *Produced mass corrections*. The produced mass-corrections are not necessary extremely small, as it is typically assumed. These corrections appear in the ratio with other kinematic variables through the variable $\varsigma^2$, eq. (\[def:mass-var\]). In most part of data bins the value of $\varsigma^2$ is negligible, $\varsigma^2\sim 10^{-3}$, but for some low-energy and low-z bins it can reach $\varsigma^2\sim 10^{-2}$. For example, the HERMES bin with $0.2<z<0.4$, $0.2<x<0.35$ with produced kaon has $\varsigma^2\sim 0.04$. As it is clear from table \[tab:SIDIS-power\], current data are not sensitive to these corrections. The difference in $\chi^2/N_{pt}$ is of the order $10^{-3}$.
- *Target mass corrections*. The target mass corrections appear through the variable $\gamma^2$ in eq. (\[def:mass-var\]) and at low Q it has a rather significant size, e.g. for some bins in HERMES data $\gamma^2\sim 0.13$, for some bins in COMPASS data $\gamma^2\sim 0.06$. Therefore, one can expect up $10\%$ impact of $\gamma^2$ for certain bins. Note, that the dependence on $\gamma^2$ is non-linear and is different in different edges of the bin. Checking the values in table \[tab:SIDIS-power\], we observe that the target mass correction produces a small but visible effect on the fit quality especially for HERMES data where the change in $\chi^2/N_{pt}$ is $1.09\to 1.24$.
- *$q_T/Q$ correction in kinematics*. This correction cannot be large due to the cuts on the data sets that we have performed. For $q_T\sim 0.25 Q$ which is the highest value of $q_T$ that we have considered, we can have $(q_T/Q)^2\sim 0.06$. In addition, the first correction of this type to the cross section is linear in $(q_T/Q)^2$ and it can be easily compensated by a change of the non-perturbative parameters in $D_{\text{NP}}$. Indeed, we observe that the impact on the $\chi^2$ is small.
- *$q_T/Q$ correction in $x_S$ and $z_S$*. For $q_T\sim 0.25 Q$ (which is the maximum considered $q_T$ ), the difference between exact $x_S$ and $x$ is $\sim 0.06$, and much smaller between $z_S$ and $z$. Nonetheless, this correction changes the shape of the cross-section in a way that is difficult to compensate by NP parameters. Thus, the inclusion of this correction visibly improves the agreement. Let us note that the same conclusion has been made for the DY case, in sec. \[sec:DY:x12\].
We conclude that the impact of each individual correction is rather small, but the inclusion of any of them improves the agreement between theory and data. Most relevant effects are the target mass correction and the ones due to $x_S$ and $z_S$. Accounting of all effects simultaneously leads to a qualitative improvement in $\chi^2$-values.
We also admit that the inclusion of power correction considerably affect the values of parameters $\eta$ (especially $\eta_{1,4}$). The values of parameters $\eta_{1,4}$ varies in the range $(-10,+25)\%$. The values of parameters $\eta_{2,3}$ varies in the range $(-4,+8)\%$. It shows that our estimation of uncertainties on parameters presented in table \[tab:SIDIS-PDF-FF-chi\] are extremely underestimated. Possibly, the main source of underestimation is the bias of our model, which is not surprising since we have only 4 parameters for all partons flavors and particle kinds. *The tests of power corrections suggest that the real error-band on the extracted TMDFF is an order of magnitude larger.*
Limits of TMD factorization for SIDIS {#sec:TMD-limit}
-------------------------------------
In ref. [@Scimemi:2017etj] we tested the limits for TMD factorization using the DY data, showing that the natural limit of the leading power TMD factorization is $\delta\simeq 0.2-0.25$, where $\delta =q_T^{max}/Q$ and $q_T^{max}$ is the maximum value of the transverse momentum in the data sets included in the fit. We have tested the same boundary using the SIDIS data and the result of the global fit (presented in the next section) evaluating the value of $\chi^2$ (without minimization) for different selections of SIDIS data. We have considered two possible cuts on data selection $\langle Q\rangle>1$ and $\langle Q\rangle>2$ , eq. (\[SIDIS-data:Q>2\]), and the result is shown in fig. \[fig:chi2\].
The values of $\chi^2/N_{pt}$ grow when $\delta>0.25$. The same effect has been observed in ref. [@Scimemi:2017etj] for DY. Therefore, we conclude that our earlier estimation for limits of TMD factorization as $\delta\lesssim 0.2-0.25$ holds also in the SIDIS case. It is interesting to observe that the channel with the fastest grow of $\chi^2/N_{pt}$ is $d\to K^-$ (and the next is $p\to\pi^+$), which could indicate a possible tension in the description of this reaction.
The inclusion of data at $\langle Q\rangle<2$GeV almost doubles the values of $\chi^2/N_{pt}$ (e.g. $\chi^2/N_{pt}=1.19$ for $\delta=0.25$). Taking into account the large uncertainties of the COMPASS measurement, it shows that the factorization is broken down at such low values of $Q$. This is an expected result, since in this region the power corrections dominate the cross-section. In sec. \[sec:agreement\], we show data and our predictions including the low-Q bins and up to $\delta=0.4$.
![\[fig:chi2\]The value of $\chi^2/N_{pt}$ depending on the cuts of the data for SIDIS. The theory prediction is calculated with NNPDF31 & DSS set at NNLO. The numbers on vertical lines shows the number of points in the cut data set.](Figures/chi2-SIDIS-num.pdf){width="50.00000%"}
Global fit of DY and SIDIS data {#sec:GLOBAL-fit}
===============================
The fit of SIDIS data shows the perfect universality of the TMD distributions and a good agreement between theory and experiment. Performing a global fit of DY and SIDIS data we essentially reduce the uncertainty for $\mathcal{D}$. The resulting sets of TMDPDF, TMDFF and RAD extracted from the global fit represent the SV19 TMD distributions (at NNLO and N$^3$LO). As an input for this set we have used NNPDF31 and DSS collinear distributions, because these sets are in good agreement with the global set of collinear observables, and show the best values of $\chi^2$ (see discussions in the previous sections).
Agreement between theory and data {#sec:agreement}
---------------------------------
Table \[tab:final\] shows the distribution of the $\chi^2$-values per individual experiments. In total we have considered 1039 points, 457 for DY and 582 for SIDIS. They form three large subsets: DY at high energy, DY at low energy, and SIDIS (at low energy). The worst $\chi^2$ values are concentrated in the high energy DY subset, because of the very high precision of Z-boson production data measured at LHC. Simultaneously, these data robustly restrict the values of TMD distributions (TMDPDF and RAD) at $b\lesssim 1$GeV$^{-1}$. The lowest $\chi^2$ is for SIDIS data and especially for COMPASS data ($\chi^2/N_{pt}=0.65$, with $N_{pt}=390$ that is more than the third part of the total data), due to large uncorrelated systematic uncertainty (see discussion in sec. \[sec:SIDIS-FF\]).
[|l | c V[4]{} c | c V[4]{} c |c V[4]{}]{} & &\
Data set & $N_{pt}$ & $\chi^2/N_{pt}$ & $\langle d/\sigma \rangle$ & $\chi^2/N_{pt}$ & $\langle d/\sigma \rangle$\
CDF run1 & 33 & 0.66 & 8.4% & 0.67 & 7.8%\
CDF run2 & 39 & 1.28 & 2.8% & 1.41 & 2.1%\
D0 run1 & 16 & 0.72 & 0.1% & 0.78 & -0.5%\
D0 run2 & 8 & 1.38 & - & 1.64 & -\
D0 run2 ($\mu$) & 3 & 0.62 & - & 0.69 & -\
Tevatron total & 99 & 0.97 & & 1.06 &\
ATLAS 7TeV 0.0<|y|<1.0 & 5 & 1.66 & - & 0.81 & -\
ATLAS 7TeV 1.0<|y|<2.0 & 5 & 5.94 & - & 4.09 & -\
ATLAS 7TeV 2.0<|y|<2.4 & 5 & 1.49 & - & 1.26 & -\
ATLAS 8TeV 0.0<|y|<0.4 & 5 & 2.51 & 3.5% & 3.40 & 2.8%\
ATLAS 8TeV 0.4<|y|<0.8 & 5 & 2.95 & 3.5% & 3.03 & 2.7%\
ATLAS 8TeV 0.8<|y|<1.2 & 5 & 1.30 & 3.7% & 1.45 & 2.9%\
ATLAS 8TeV 1.2<|y|<1.6 & 5 & 2.03 & 4.2% & 1.53 & 3.4%\
ATLAS 8TeV 1.6<|y|<2.0 & 5 & 1.47 & 4.9% & 0.70 & 4.1%\
ATLAS 8TeV 2.0<|y|<2.4 & 5 & 2.64 & 5.6% & 2.10 & 4.8%\
ATLAS 8TeV 46<Q<66GeV & 3 & 0.31 & 1.1% & 0.31 & 0.2%\
ATLAS 8TeV 116<Q<150GeV & 7 & 0.84 & 1.9% & 0.97 & 1.2%\
ATLAS total & 55 & 2.12 & & 1.82 &\
CMS 7TeV & 8 & 1.25 & - & 1.24 & -\
CMS 8TeV & 8 & 0.77 & - & 0.76 & -\
CMS total & 16 & 1.01 & & 1.00 &\
LHCb 7TeV & 8 & 2.68 & 5.8% & 2.37 & 5.2%\
LHCb 8TeV & 7 & 4.81 & 5.8% & 4.16 & 5.1%\
LHCb 13TeV & 9 & 0.91 & 6.4% & 0.81 & 5.7%\
LHCb total & 24 & 2.63 & & 2.31 &\
**High energy DY total** & 194 & 1.51 & & 1.42 &\
PHE200 & 3 & 0.28 & 0.2% & 0.29 & -0.3%\
E228-200 & 43 & 1.00 & 35.7% & 1.12 & 35.0%\
E228-300 & 53 & 0.90 & 29.2% & 1.01 & 28.3%\
E228-400 & 76 & 0.86 & 20.6% & 0.96 & 19.5%\
E772 & 35 & 1.84 & 9.5% & 1.91 & 8.5%\
E605 & 53 & 0.57 & 21.3% & 0.60 & 20.1%\
**Low energy DY total** & 263 & 0.96 & & 1.04 &\
HERMES ($p\to \pi^+$) & 24 & 2.20 & 1.7% & 3.06 & 2.2%\
HERMES ($p\to \pi^-$) & 24 & 1.12 & 0.6% & 1.45 & 0.9%\
HERMES ($p\to K^+$) & 24 & 0.71 & -0.1% & 0.66 & 0.0%\
HERMES ($p\to K^-$) & 24 & 0.69 & 0.0% & 0.66 & 0.0%\
HERMES ($d\to \pi^+$) & 24 & 0.57 & 0.3% & 0.78 & 0.8%\
HERMES ($d\to \pi^-$) & 24 & 0.74 & 0.5% & 0.96 & 0.7%\
HERMES ($d\to K^+$) & 24 & 0.52 & -0.1% & 0.53 & 0.0%\
HERMES ($d\to K^-$) & 24 & 1.27 & 0.0% & 1.17 & 0.1%\
HERMES total & 192 & 0.98 & & 1.16&\
COMPASS ($d\to h^+$) & 195 & 0.61 & 3.3% & 0.76 & 5.1%\
COMPASS ($d\to h^-$) & 195 & 0.68 & -2.3% & 0.92 & -0.5%\
COMPASS total & 390 & 0.65 & & 0.84 &\
**SIDIS total** & 582 & 0.76 & & 0.95 &\
**Total** & **1039** & **0.95** & & **1.06** &\
Altogether we obtain the global value of $\chi^2/N_{pt}=0.95$ and $1.06$ for NNLO and N$^3$LO respectively. These values can be compared to $1.55$ (for $N_{pt,\rm{total}}=8059$) and $1.02$ (for $N_{pt;\rm{SIDIS}}=477$ that is close to our data selection) obtained in the global fit of DY and SIDIS in ref. [@Bacchetta:2017gcc]. The increase of $\chi^2/N_{pt}$ between NNLO and N$^3$LO cases does not indicate a reduction of the fit quality. This change in $\chi^2$ happens mainly because of COMPASS data, for which the $\chi^2$-value increase $0.65\to 0.85$. On contrary the $\chi^2$-value for ATLAS data reduces $2.12\to1.82$ (mostly due to the improvement in the total normalization). Therefore, we conclude that both NNLO and N$^3$LO fits are in agreement, although N$^3$LO shows a better agreement with high-energy data.
In table \[tab:final\] we also present the values of the difference in the normalization between theory and data due to the correlated shift (see definition in (\[def:d/sigma\])). The measurements in the table \[tab:final\] without this value (e.g. CMS) are normalized to the total cross-section. Note, that the shift value is common to the full data subset (e.g. for all 195 point of COMPASS $d\to h^+$).
Some final consideration on each data sets are:
- *The high energy DY* data have a common deficit of 2-5% in the normalization, which has been already observed in [@Bertone:2019nxa]. It can be caused by different sources, being the main ones the collinear PDF (e.g. in the case of HERA20 PDF the deficit is much smaller, 0-3%). Another source is the presence of corrections linear in $q_T$ due to fiducial cuts, discussed in sec. \[sec:DY-fiducial\]. This deficit is responsible for a larger value of $\chi^2$ for this sub-set. The nuisance parameter decomposition for high energy DY is $1.51=1.28+0.23$, where the last number is the penalty contribution to $\chi^2$ due correlated uncertainties.
- *The low energy DY* data is significantly underestimated by the TMD factorization formula. However, this underestimation is within the expected correlated systematic uncertainties of the data. This is a known issue of fixed target experiments. The underestimation has been also observed for the pion-induced DY [@Vladimirov:2019bfa] (E615 and E537 experiments), and for the same low-energy DY experiments (E228 and E605) in ref. [@Bacchetta:2019tcu]. Note, that in ref. [@Bacchetta:2019tcu] the high-$q_T$ part of the measurements has been considered (in collinear factorization), and the observed discrepancy is an order of magnitude larger. Also, the present fit has somewhat lesser deficit in the normalization (by $5-6\%$) in comparison to previous one [@Bertone:2019nxa]. We connect it to the corrected shape of $\zeta$-line at large-$b$.
- *SIDIS data* do not show any problem with the total normalization. This statement is in some contradiction to the literature. In [@Bacchetta:2017gcc] the authors report a significant contribution of normalization to $\chi^2$ from the HERMES data (the COMPASS data was normalized exactly). In ref. [@Gonzalez-Hernandez:2018ipj] an enormous discrepancy between theory and data in the collinear factorization limit has been observed too.
In figures \[fig:ATLAS\]-\[fig:COMPASS3\] we present all the data points used in the fit together with the theory prediction lines. In these figures we also show the data points that were not included in the fit due to the cutting conditions eq. (\[SIDIS-data:Q>2\], \[SIDIS-data:qT<0.25 Q\], \[DY-data:cuts\]) in order to demonstrate the behavior of TMD factorization beyond its limits.
![\[fig:ATLAS\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS at different values of $y$ and $s$. The solid lines are absolute prediction. The dashed lines are the theory prediction shifted by $\langle d/\sigma\rangle$ that is indicated on each case together with the values of $\chi^2/_{np}$ for given data set. Blue (red) color corresponds to the theory prediction at NNLO (N$^3$LO). The ratio boxes shows same plot weighted by the shifted theory prediction at NNLO. Vertical dashed lines show the part of the data included in the fit (to the left of the line).](Figures/ATLAS.pdf){width="100.00000%"}
![\[fig:CMS+LHCb\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS, CMS, LHCb and PHENIX at different values of $s$ and Q. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/CMS.pdf){width="68.00000%"}
![\[fig:CMS+LHCb\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS, CMS, LHCb and PHENIX at different values of $s$ and Q. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/LHCb.pdf){width="1.\textwidth"}
![\[fig:CMS+LHCb\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS, CMS, LHCb and PHENIX at different values of $s$ and Q. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/ATLAS-46-66.pdf "fig:"){width="32.00000%"} ![\[fig:CMS+LHCb\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS, CMS, LHCb and PHENIX at different values of $s$ and Q. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/ATLAS-116-150.pdf "fig:"){width="32.00000%"} ![\[fig:CMS+LHCb\] Differential cross-section for the $Z/\gamma^*$ boson production measured by ATLAS, CMS, LHCb and PHENIX at different values of $s$ and Q. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/PHENIX.pdf "fig:"){width="32.00000%"}
![\[fig:tevatron\] Differential cross-section for the $Z/\gamma^*$ boson production measured by CDF and D0 at different values of $s$. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/Tevatron_1.pdf){width="100.00000%"}
![\[fig:tevatron\] Differential cross-section for the $Z/\gamma^*$ boson production measured by CDF and D0 at different values of $s$. The figure elements are the same as in fig. \[fig:ATLAS\]](Figures/Tevatron_2.pdf){width="68.00000%"}
![\[fig:E228\] Differential cross-section of DY process measured by E288 at different values of $s$ and $Q$. The solid (dashed) lines are the theory prediction at NNLO (N$^3$LO) shifted by the average systematic shift (see table \[tab:final\]). Filled (empty) point were (not) included in the fit of NP parameters.](Figures/E228.pdf){width="100.00000%"}
![\[fig:E772+E605\] Differential cross-section of DY process measured by E605 and E772 at different values of $s$ and $Q$. The solid (dashed) lines are the theory prediction at NNLO (N$^3$LO) shifted by the average systematic shift (see table \[tab:final\]). Filled (empty) point were (not) included in the fit of NP parameters. For clarity the data of E772 is multiplied by the factors indicated in the plot.](Figures/E605.pdf "fig:"){width="33.00000%"} ![\[fig:E772+E605\] Differential cross-section of DY process measured by E605 and E772 at different values of $s$ and $Q$. The solid (dashed) lines are the theory prediction at NNLO (N$^3$LO) shifted by the average systematic shift (see table \[tab:final\]). Filled (empty) point were (not) included in the fit of NP parameters. For clarity the data of E772 is multiplied by the factors indicated in the plot.](Figures/E772.pdf "fig:"){width="32.00000%"}
![\[fig:HERMES-pi\] Unpolarized SIDIS multiplicities (multiplied by $z^2$) for production of pions off proton/deuteron measured by HERMES in different bins of $x$, $z$ and $p_T$. Solid (dashed) lines show the theory prediction at NNLO (N$^3$LO). Filled (empty) point were (not) included in the fit of NP parameters. On the top of the table the value of $\chi^2/N_{pt}$ for each channel is presented, the value in brackets being the $\chi^2/N_{pt}$ for shown set of the data (empty and filled points together). For clarity each $p_T$ bin is shifted by an offset indicated in the legend.](Figures/HERMES_pi.pdf){width="100.00000%"}
![\[fig:HERMES-K\] Unpolarized SIDIS multiplicities (multiplied by $z^2$) for production of kaons off proton/deuteron measured by HERMES in different bins of $x$, $z$ and $p_T$. Solid (dashed) lines show the theory prediction at NNLO (N$^3$LO). Filled (empty) point were (not) included in the fit of NP parameters. On the top of the table the value of $\chi^2/N_{pt}$ for each channel is presented, the value in brackets being the $\chi^2/N_{pt}$ for shown set of the data (empty and filled points together). For clarity each $p_T$ bin is shifted by an offset indicated in the legend.](Figures/HERMES_K.pdf){width="100.00000%"}
![\[fig:COMPASS1\] Unpolarized SIDIS multiplicities (multiplied by $z^2$) for production of positively charged hadrons off deuteron measured by COMPASS in different bins of $x$, $z$, $Q$ and $p_T$. Solid (dashed) lines show the theory prediction at NNLO (N$^3$LO). Filled (empty) point were (not) included in the fit of NP parameters. For clarity each $p_T$ bin is shifted by an offset indicated in the legend. The continuation of the picture is in fig. \[fig:COMPASS3\].](Figures/COMPASS_H+_1.pdf){width="100.00000%"}
![\[fig:COMPASS2\] Unpolarized SIDIS multiplicities (multiplied by $z^2$) for production of negatively charged hadrons off deuteron measured by COMPASS in different bins of $x$, $z$, $Q$ and $p_T$. Solid (dashed) lines show the theory prediction at NNLO (N$^3$LO). Filled (empty) points were (not) included in the fit of NP parameters. For clarity each $p_T$ bin is shifted by an offset indicated in the legend. The continuation of the picture is in fig. \[fig:COMPASS3\].](Figures/COMPASS_H-_1.pdf){width="100.00000%"}
![\[fig:COMPASS3\] Continuation of the plots \[fig:COMPASS1\] and \[fig:COMPASS2\].](Figures/COMPASS_H+_2.pdf "fig:"){width="45.00000%"} ![\[fig:COMPASS3\] Continuation of the plots \[fig:COMPASS1\] and \[fig:COMPASS2\].](Figures/COMPASS_H-_2.pdf "fig:"){width="45.00000%"}
Values of NP parameters
-----------------------
The extracted values of NP parameters are in the table \[tab:NP-param-final\]. The central values of parameters do not shift much with respect to individual fits of DY and SIDIS data. The main effect of the global fit is the reduction of uncertainties for RAD and TMDPDF by $\sim 40-50\%$. In figures \[fig:DNP-param\], \[fig:f1-param\] and \[fig:d1-param\] we show the values of NP parameters obtained in all fits of this work. Generally, the parameters obtained in different fits are in agreement, except $\lambda_{3,4,5}$ that mainly serve for the fine-tune of TMDPDF to LHC data.
![\[fig:correlation\]The correlation matrices for NP parameters obtained in the global fit of DY and SIDIS. Numbers indicate the values of matrix elements with correlation higher then $0.3$.](Figures/correlation-NNLO.pdf "fig:"){width="40.00000%"} ![\[fig:correlation\]The correlation matrices for NP parameters obtained in the global fit of DY and SIDIS. Numbers indicate the values of matrix elements with correlation higher then $0.3$.](Figures/correlation-N3LO.pdf "fig:"){width="40.00000%"} ![\[fig:correlation\]The correlation matrices for NP parameters obtained in the global fit of DY and SIDIS. Numbers indicate the values of matrix elements with correlation higher then $0.3$.](Figures/correlation-scale.pdf "fig:"){width="5.67000%"}
All NP parameters are correlated. The correlation matrices for NP parameters are shown in fig. \[fig:correlation\]. Ideally, one would expect the complete independence of NP parameters contributing to RAD, TMDPDF and TMDFF. In this case the correlation matrices would have a block-diagonal form. In reality, we observe the correlations between the blocks related to independent functions. In the case of NNLO these correlations are not large, and the block-diagonal structure is evident. The biggest (anti-)correlation is between $c_0$ and $\lambda_1$, with the correlation matrix element $-0.67$, with the rest being much smaller. The source of this correlation is evident – it is due to the precise Z-boson production measurements by LHC. In the N$^3$LO case the correlation are much stronger. The biggest (anti-)correlation again is between $c_0$ and $\lambda_1$, with correlation matrix element $-0.82$, with some other elements reaching $\pm 0.5$ and it indicates a possible tension in our description of the data at N$^3$LO.
$\chi^2/N_{pt}$
----------------- ----- ---------------------------- ------------------------------------- ------------------------
RAD $B_\text{NP}=1.93\pm0.17$ $c_0=(3.91 \pm 0.63)\times 10^{-2}$
$\lambda_1=0.198\pm 0.019$ $\lambda_2=9.30\pm0.55$ $\lambda_3=431.\pm96.$
$\lambda_4=2.12\pm0.09$ $\lambda_5=-4.44\pm1.05$
$\eta_1=0.260\pm0.015$ $\eta_2=0.476\pm0.009$
$\eta_3=0.478\pm0.018$ $\eta_4=0.483\pm0.030$
RAD $B_\text{NP}=1.93\pm0.22$ $c_0=(4.27 \pm 1.05)\times 10^{-2}$
$\lambda_1=0.224\pm 0.029$ $\lambda_2=9.24\pm0.46$ $\lambda_3=375.\pm89.$
$\lambda_4=2.15\pm0.19$ $\lambda_5=-4.97\pm1.37$
$\eta_1=0.233\pm0.018$ $\eta_2=0.479\pm0.025$
$\eta_3=0.472\pm0.041$ $\eta_4=0.511\pm0.040$
: \[tab:NP-param-final\] Values of $\chi^2$ and NP parameters obtained obtained in the global fit of DY and SIDIS data. The collinear distributions are NNPDF31 and DSS.
Comments on the extracted TMD distributions {#sec:final}
===========================================
![\[fig:RAD\] (left) Comparison of NNLO RAD extracted in DY fit (NNPDF31), and global fit of DY and SIDIS (NNPDF31& DSS). Shaded area shows the $1\sigma$-uncertainty band. The dashed lines show the extraction made in refs.[@Bacchetta:2017gcc] and [@Scimemi:2017etj] at LO and NNLO of RAD correspondingly. (right) Distribution of replica points in different fits of RAD. Dashed lines show the mean values of RAD extracted in the global fit of DY and SIDIS.](Figures/RAD_comparison22.pdf "fig:"){width="44.00000%"} ![\[fig:RAD\] (left) Comparison of NNLO RAD extracted in DY fit (NNPDF31), and global fit of DY and SIDIS (NNPDF31& DSS). Shaded area shows the $1\sigma$-uncertainty band. The dashed lines show the extraction made in refs.[@Bacchetta:2017gcc] and [@Scimemi:2017etj] at LO and NNLO of RAD correspondingly. (right) Distribution of replica points in different fits of RAD. Dashed lines show the mean values of RAD extracted in the global fit of DY and SIDIS.](Figures/RAD_distribution.pdf "fig:"){width="54.00000%"}
The non-perturbative distribution extracted in this work show several features that are interesting for theory investigations. For instance, the RAD that measures the properties of the soft gluon exchanges and that is inclusively sensitive to the QCD vacuum structure. The factorization theorem ensures that the values of $B_{\text{NP}}$ and $c_0$ are totally uncorrelated from the rest of TMD parameters, because they are of complete different origins. As we have an extraction of these parameters from data we can expect that a certain correlation is re-introduced in the fitting process. In fig. \[fig:correlation\] we check this statement on the present global fit and we find that it is qualitatively verified in our DY+SIDIS fit. In the figure the only non-perturbative parameters which show a higher (anti)correlation with the RAD are to $c_0$ and $\lambda_1$ in the TMDPDF. Apart from this, the independence of the RAD parameters from the rest of TMD is certainly a success of the $\zeta$-prescription, which allows a clear separation of all these effects. In the rest of this section we report some considerations specifically for each of the functions that we have extracted.
Non-perturbative RAD {#sec:NP-RAD-final}
--------------------
In fig. \[fig:RAD\] (left) we plot the RAD as a function of $b$ with its uncertainty band. We present only the RAD extracted with NNPDF31 fits, but the picture does not change significantly for all other PDF sets. In this figure we can test the universality of the RAD looking at its extraction in DY and DY+SIDIS. At small $b$ the perturbative structure of the RAD dominates and we find practically no difference in its behavior as coming from different fits. The difference between these two cases happens at large $b$ and it is at most of 10%. The $1\sigma$-uncertainty bands of DY and global fit do not strictly overlap, which possibility indicates their underestimation.
In the same fig. \[fig:RAD\] (left) we also compare our RAD with the one obtained in [@Bacchetta:2017gcc] and [@Scimemi:2017etj]. In refs. [@Bacchetta:2017gcc; @Scimemi:2017etj] a different shape of NP ansatz for RAD has been used, with a quadratic behavior at large-$b$. Such an ansatz has been used often, and (as we have also checked) it is able to describe the data. Nonetheless we disregard it because the global $\chi^2/N_{pt}$ is worse ($1.11$ and $1.34$ at NNLO and N$^3$LO, correspondingly), with much larger correlation between parameters. Additionally, the linear asymptotic behavior used in our ansatz is supported by non-perturbative models. Possibly, the uncertainty band is biased by this model, and the realistic band is larger by a factor two at most.
In fig. \[fig:RAD\] (right) we show the scattering of replicas in ($B_{\text{NP}},c_0$)-plane collected from all fits. It is clear that the parameters $B_{\text{NP}}$ and $c_0$ are strongly anti-correlated (see also fig. \[fig:correlation\]) and this is a consequence of the non-perturbative model, since the variation of $c_0$ can be compensated by a variation of $B_{\text{NP}}$ up to $b^4$-corrections. The replicas of the global fit (orange points) are scattered in a much smaller area and this provides a $\sim 40\%$ smaller error-bands on parameters. Generally, the inclusion of the SIDIS data drastically constraints the values of $B_{\text{NP}}$, and for that reason they are very important for the determination of RAD. We conclude that the RAD extracted in the global fit is more reliable, in comparison to the one done using DY data only.
The RAD that we have extracted is valid for all distributions and it has been used also to describe the pion-induced DY [@Vladimirov:2019bfa]. For further reduction of the uncertainty of the RAD one should consider more precise low- and intermediate-energy processes, such as up-coming JLab12 measurements, and the future EIC.
TMD distributions {#sec:final-TMD}
-----------------
![\[fig:TMDs\] Example of extracted (optimal) unpolarized TMD distributions. The color indicates the relative size of the uncertainty band](Figures/uPDF.pdf "fig:"){width="48.00000%"} ![\[fig:TMDs\] Example of extracted (optimal) unpolarized TMD distributions. The color indicates the relative size of the uncertainty band](Figures/uFF.pdf "fig:"){width="48.00000%"}
The quark TMDPDF and TMDFF are extracted simultaneously including high QCD perturbative orders for the first time to our knowledge. The non-perturbative parameters obtained using the PDF set NNPDF31 and the fragmentation set DSS are reported in table \[tab:NP-param-final\]. Within one set of PDF the error induced from the PDF replicas dominates the experimental error of TMD. Thus, the error that we have reported on TMD parameters is certainly underestimated. To determine a realistic uncertainty band , one must invent a flexible ansatz for NP-part of TMD distributions that does not contradict the known theory. It appears to be a non-trivial task, which we leave for a future study.
The TMD distributions show a non-trivial intrinsic structure. An example of distributions in $(x,b)$-plane is presented in fig. \[fig:TMDs\]. Depending on $x$ the $b-$behavior apparently changes. We observe (the same observation has been made in ref. [@Bacchetta:2017gcc]) that the unpolarized TMDFF gain a large $b^2$-term in the NP part. It could indicate a non-trivial hadronisation physics, or a tension between colinear and TMD distributions. The study of its origin should be addressed by future studies.
Conclusion
==========
Standing the TMD factorization of DY and SIDIS cross-section, one identifies at least three non-perturbative QCD distributions in each cross-section – two TMD parton distributions and a non-perturbative rapidity anomalous dimension (RAD). These functions should be extracted from the experimental data. Given such a large number of phenomenological functions, their universality plays a crucial role. In this work, we have shown that the TMD distributions and RAD are indeed universal functions.
In order to confirm the universality statement, we have firstly extracted the RAD ($\mathcal{D}$) and the unpolarized TMDPDF ($f_1$) from the DY data, and secondly we have used them to describe the SIDIS data (extracting in addition the unpolarized TMDFF, $D_1$). To our best knowledge, this is the first clear-cut demonstration of the universality of the TMD non-perturbative components. This demonstration is the main result of this work. The subsidiary results are the values of extracted unpolarized TMD distributions and RAD, that could be used to predict and describe the low-$q_T$ spectrum of current (LHC, COMPASS, RHIC) and future (EIC, LHeC) experiments.
The sets of data included in this analysis contain in total 1039 points (almost equally distributed between SIDIS, 582 points, and DY, 457 points). We have the data from fixed target DY measurements, Tevatron, RHIC, LHC, COMPASS, and HERMES. Unfortunately, only low-energy measurements are available for SIDIS data. At the moment, we have not included any data from HERA multiplicities because they do not accomplish the kinematical requirements for the TMD factorization. Contrary to some observations in the literature [@Anselmino:2013lza; @Bacchetta:2017gcc], we have not found any problem with the normalization of HERMES and COMPASS data, although the systematic experimental errors quit precision to the final result.
The data analysis is made with the current theory state-of-art, including all known perturbative QCD orders, i.e. N$^3$LO for the hard part and the evolution, and NNLO for the collinear matching. The NNLO and N$^3$LO predictions are very close to each other, which is a good signal indicating that the perturbative part of the cross-section is saturated. We have also collected all recent modifications and updates of the TMD factorization approach, such as target-mass corrections, frame-corrections, and exact evolution solution at large-$b$. Individually these aspects are subtle, however, cumulatively, they are sizable. In sec. \[sec:2\] we have presented a comprehensive collection of theory expressions used in this work. Let us also mention that the N$^3$LO evolution, as well as a non-trivial QCD matching for TMDFF (NNLO vs. LO) is used here for the first time.
The scales definition and the evolution/modeling separation is done according to $\zeta$-prescription. The $\zeta$-prescription is equivalent to the popular CSS-scheme since it satisfies the same set of differential equations. Nonetheless, this equivalence is strict only within an all-order perturbation theory and it is numerically violated for any truncated series. The origin for this discrepancy is well-understood [@Scimemi:2018xaf] – it comes from spurious contributions in the CSS formalism that vanish in the exact perturbation theory. At LO and NLO, the numerical value of spurious contributions is large, but it is tiny at N$^3$LO [@Scimemi:2018xaf]. Therefore, the $\zeta$-prescription provides a faster convergence and the better stability of the perturbative series that is shown in fig. \[fig:convergence\]. Additionally, but not less important, the $\zeta$-prescription grants a strict separation of perturbative and non-perturbative pieces and thus allows a stronger universality of the phenomenological functions, fig. \[fig:correlation\]. In particular, the RAD extracted here can be used in the analysis of the jet-production [@Gutierrez-Reyes:2019msa; @Gutierrez-Reyes:2019vbx; @Gutierrez-Reyes:2018qez]. Preliminary lattice results are also in qualitative agreement with the RAD in $\zeta$-prescription [@ilattice]. The success of the present global fit confirms the reliability of the $\zeta$-prescription.
Many points of the TMD phenomenology are discussed quantitatively for the first time (to our best knowledge). We critically consider each detail of the factorization that have a disputable nature, f.i. power corrections to collinear variables. We demonstrated that the inclusion of these details improves the agreement between theory and the data. A particularly important check made here for the first time is the test of the limit of the TMD factorization approximation for SIDIS. In the DY case, the phenomenological limit of TMD factorization is $q_T\lesssim 0.25 Q$, as it has been shown in ref. [@Scimemi:2017etj]. We have found that SIDIS also obeys this rule. It is important information since it opens the door for reliable predictions for SIDIS cross-section.
The estimation of the uncertainty for extracted distributions is made by the replica method that gives a reliable error-propagation of experimental errors. On top of it one should include the uncertainty of other theoretical ingredients, and in particular the collinear PDF error. We have checked that the prediction of the TMD factorization is crucially sensitive to the values of collinear PDFs. It indicates that our extraction has a considerable additional uncertainty due to the uncertainty of the collinear input. However, we were not able to accurately quantify the size of this uncertainty band, due to the high computational costs of such analysis. We leave this study for the future.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Valerio Bertone, Gunnar Schnell, Pia Zurita and Elke Aschenauer for stimulation discussions and correspondence. I.S. is supported by the Spanish MECD grant FPA2016-75654-C2-2-P. This project has received funding from the European Union Horizon 2020 research and innovation program under grant agreement No 824093 (STRONG-2020). The research was supported by DFG grant N.430824754 as a part of the Research Unit FOR 2926.
Expression for $|C_V|$
======================
The hard coefficient for TMD factorization formula is a product of hard matching coefficient for quark current $|C_V|^2$. At NLO hard matching coefficient reads $$\begin{aligned}
C_V(q,\mu)=1+a_s C_F{\left(}-\ln^2{\left(}\frac{-q^2}{\mu^2}{\right)}+3\ln{\left(}\frac{-q^2}{\mu^2}{\right)}-8+\frac{\pi^2}{6}{\right)}+O(a_s^2).\end{aligned}$$ The expression for NNLO can be found f.i. in ref. [@Matsuura:1988sm; @Kramer:1986sg], and at N$^3$LO in [@Gehrmann:2010ue].
The hard coefficient for SIDIS and DY kinematics are different only by the sign of $q^2$ that is space-like (times-like) for DY (SIDIS). Since in our work $\mu^2=Q^2$ the expression simplifies. In particular, in the DY kinematics the logarithms turns to $\ln(-1)=\pm i\pi$, whereas in the case of SIDIS logarithm vanish. The NNLO expression for DY hard coefficient is $$\begin{aligned}
|C_V(-Q^2,Q^2)|^2&=&1+a_s C_F{\left(}-16+\frac{7\pi^2}{3}{\right)}+a_s^2 C_F\Big[C_F{\left(}\frac{511}{4}-\frac{83\pi^2}{3}-60\zeta_3+\frac{67\pi^4}{30}{\right)}\\{\nonumber}&&+C_A{\left(}-\frac{51157}{324}+\frac{1061\pi^2}{54}+\frac{626\zeta_3}{9}-\frac{8\pi^4}{45}{\right)}+N_f{\left(}\frac{4085}{162}-\frac{91\pi^2}{27}+\frac{4\zeta_3}{9}{\right)}\Big],\end{aligned}$$ where $C_F(=4/3)$ and $C_A(=3)$ are quadratic Casimir eigenvalues of fundamental and adjoint representation of SU(3). $N_f$ is the number of active quark flavors. The NNLO expression for SIDIS hard coefficient is $$\begin{aligned}
|C_V(Q^2,Q^2)|^2&=&1+a_s C_F{\left(}-16+\frac{\pi^2}{3}{\right)}+a_s^2 C_F\Big[C_F{\left(}\frac{511}{4}-\frac{13\pi^2}{3}-60\zeta_3+\frac{13\pi^4}{30}{\right)}\\{\nonumber}&&+C_A{\left(}-\frac{51157}{324}-\frac{337}{54}+\frac{626\zeta_3}{9}+\frac{22\pi^4}{45}{\right)}+N_f{\left(}\frac{4085}{162}+\frac{23\pi^2}{27}+\frac{4\zeta_3}{9}{\right)}\Big].\end{aligned}$$ One can see that the difference between SIDIS and DY hard coefficients is $$\begin{aligned}
&&|C_V(Q^2,Q^2)|^2-|C_V(-Q^2,Q^2)|^2=-2\pi^2 a_sC_F
\\{\nonumber}&&\qquad +\pi^2a_s^2C_F{\left[}C_F{\left(}32-\frac{8\pi^2}{3}{\right)}+C_A{\left(}-\frac{233}{9}+\frac{2\pi^4}{3}{\right)}+N_f\frac{38}{9}{\right]}+O(a_s^3).\end{aligned}$$ These corrections are known as $(\pi^2a_s)^n$-corrections. They could be resummed to all orders [@Ahrens:2008qu]. However, in the case of vector-boson production the correction coming from $(\pi^2a_s)^n$ is not significant (of order of next-to-given order correction [@Ahrens:2008qu]).
Perturbative expression for $\mathcal{D}$ {#app:RAD}
=========================================
The rapidity anomalous dimension (RAD) $\mathcal{D}(\mu,b)$ is generally non-perturbative function, which can be computed perturbatively only at small-b, see e.g.[@Vladimirov:2016dll; @Echevarria:2015byo] for NNLO and N$^3$LO computations. RAD satisfies the integrability condition (\[def:integrability\]) which can be seen as renormalization group equation, $$\begin{aligned}
\label{app:RAD:RGE}
\mu^2 \frac{d \mathcal{D}(\mu,b)}{d\mu^2}=\frac{\Gamma_{\text{cusp}}(\mu)}{2}.\end{aligned}$$ Consequently, the $a_s^n$-order of RAD contains logarithms up to order $\mathbf{L}_\mu^{n}$. We define $$\begin{aligned}
\label{app:def:dnk}
\mathcal{D}_{\text{pert}}(\mu,b)=\sum_{n=1}^\infty a_s^n(\mu)\sum_{k=0}^n \mathbf{L}_\mu^k d^{(n,k)},\end{aligned}$$ where $d^{(n,k)}$ are numbers, and $$\begin{aligned}
\label{app:Lmu}
\mathbf{L}_\mu=\ln{\left(}\frac{\mu^2{\bm{b}}^2}{4e^{-2\gamma_E}}{\right)}.\end{aligned}$$ The values for $d^{(n,k)}$ for $k>0$ can be computed from (\[app:RAD:RGE\]) in the terms of $\Gamma_i$, $\beta_i$ and $d^{(n,0)}$. Here, and in the following we define beta-function, and coefficients of $\Gamma_{\text{cusp}}$ as $$\begin{aligned}
\label{app:def-beta-Gamma}
\mu^2 \frac{d a_s(\mu)}{d\mu^2}=-\beta(a_s)=-\sum_{n=0}^\infty a_s^{n+2}(\mu)\beta_n,\qquad \Gamma_{\text{cusp}}(\mu)=\sum_{n=0}^\infty a_s^{n+1}(\mu)\Gamma_n.\end{aligned}$$ The leading terms are $\beta_0=\frac{11}{3}C_A-\frac{2}{3}N_f$, $\Gamma_0=4C_F$, and $d^{(1,0)}=0$. The values of $d^{(2,0)}$ and $d^{(3,0)}$ were computed in [@Becher:2010tm; @Echevarria:2015byo] and [@Li:2016ctv; @Vladimirov:2016dll] correspondingly. The values of $\beta$-function coefficient are known up to $\beta_4$ [@Baikov:2016tgj], the values of $\Gamma_i$ are known up to $\Gamma_3$-order for the quark case, see [@Moch:2017uml; @Moch:2018wjh; @Lee:2019zop] and references within.
The series (\[app:RAD:RGE\]) has a small convergence radius since the expansion variable $(a_s\mathbf{L}_\mu)$ is fastly became bigger then 1 with the increase of $b$. To improve the convergence properties of RAD we use the resummed expression [@Echevarria:2012pw; @Scimemi:2018xaf; @Bizon:2018foh]. In this case we write $$\begin{aligned}
\label{app:RAD:resum}
\mathcal{D}_{\text{resum}}(\mu,b)=\sum_{n=0}^\infty a_s^n(\mu)d_n(X),\qquad X\equiv\beta_0a_s(\mu)\mathbf{L}_\mu.\end{aligned}$$ The functions $d_n$ satisfy the set of equations $$\begin{aligned}
\label{app:RAD:eqn_dn}
\beta_0 d'_n(X)-\sum_{k=0}^n \beta_k ((n-k)d_{n-k}(X)+Xd'_{n-k}(X))=\frac{\Gamma_n}{2},\end{aligned}$$ where $d_n'(X)=\partial d_n(X)/\partial X$. The boundary conditions are $d_n(X=0)=d^{(n,0)}$. These equations are be solved recursively starting from the equation at $n=0$. The solutions of (\[app:RAD:eqn\_dn\]) are $$\begin{aligned}
\label{app:d0}
d_0(X)&=&-\frac{\Gamma_0}{2\beta_0}\ln(1-X),
\\
d_1(X)&=&\frac{1}{2\beta_0(1-X)}\Big[-\frac{\beta_1\Gamma_0}{\beta_0}(\ln(1-X)+X)+\Gamma_1X\Big],
\\
d_2(X)&=&\frac{1}{(1-X)^2}\Big[\frac{\Gamma_0\beta_1^2}{4\beta_0^3}{\left(}\ln^2(1-X)-X^2{\right)}+\frac{\beta_1\Gamma_1}{4\beta_0^2}{\left(}X^2-2X-2\ln(1-X){\right)}\\{\nonumber}&&+\frac{\Gamma_0\beta_2}{4\beta_0^2}X^2-\frac{\Gamma_2}{4\beta_0}X(X-2)+d^{(2,0)}\Big],
\\{\nonumber}d_3(X)&=&\frac{1}{(1-X)^3}\Big[
-\frac{\Gamma_0\beta_1^3}{6\beta_0^4}{\left(}\ln^3(1-X)-\frac{3}{2}\ln^2(1-X)-3X\ln(1-X)+X^3-\frac{3}{2}X^2{\right)}\\ &&
+\frac{\beta_1^2\Gamma_1}{2\beta_0^3}{\left(}\ln^2(1-X)+\frac{X^3}{3}-X^2{\right)}-\frac{\beta_2\beta_1\Gamma_0}{2\beta_0^3}{\left(}X\ln(1-X)+\frac{2}{3}X^3-X^2{\right)}\\{\nonumber}&&-\frac{\beta_1\Gamma_2}{2\beta_0^2}{\left(}\ln(1-X)+\frac{X^3}{3}-X^2+X{\right)}+\frac{X^2}{12\beta_0^2}{\left(}\beta_3\Gamma_0(3-2X)+2\beta_2\Gamma_1(3-X){\right)}\\{\nonumber}&&+\frac{\Gamma_3}{6\beta_0}X(3-3X+X^2)-\frac{2\beta_1 d^{(2,0)}}{\beta_0}\ln(1-X)+d^{(3,0)}\Big]\end{aligned}$$ At $X\to1$ this expression has a pole, that is equivalent to Landau pole. This show the convergence radius of this expansion ${\bm{b}}\simeq 2e^{-\gamma_E}\Lambda_{QCD}^{-1}\approx 4$GeV$^{-1}$.
Expression for $\zeta_\mu$ {#app:zeta-line}
==========================
The concept of the special null-evolution line plays the central role in $\zeta$-prescription. The $\zeta$-prescription, the double evolution and properties of TMD evolution have been elaborated in ref. [@Scimemi:2018xaf]. In this appendix, we present the expressions for the special null-evolution line $\zeta_{\text{pert}}$ and $\zeta_{\text{NP}}$ that were used in the fit.
The definition of the special null-evolution line is discussed in the sec. \[sec:evolution\]. Parameterizing an equipotential line as $(\mu,\zeta_\mu(b))$, one finds the following equation $$\begin{aligned}
\label{app:sp-line}
\Gamma_{\text{cusp}}(\mu)\ln{\left(}\frac{\mu^2}{\zeta_\mu(b)}{\right)}-\gamma_V(\mu)=2\mathcal{D}(\mu,b) \frac{d \ln \zeta_\mu(b)}{d \ln \mu^2}.\end{aligned}$$ The special null-evolution line is the line that passes thorough the saddle point $(\mu_0,\zeta_0)$ of the evolution field. The saddle point is defined as $$\begin{aligned}
\label{app:sp_boundary}
\mathcal{D}(\mu_0,b)=0,\qquad \gamma_F(\mu_0,\zeta_0)=0.\end{aligned}$$ Note, that this boundary condition guarantees the finiteness of $\zeta_\mu(b)$ at all non-singular values of $\mu$.
Originally the $\zeta$-prescription has been implemented in the perturbative regime only [@Scimemi:2017etj]. This part is the most important since it gives the cancellation of double-logarithms in the matching coefficient. However, at large-$b$, non-perturbative corrections to RAD become large and could not be ignored (although they can be seen as a part of NP model, but it introduces an undesired correlation between $f_{NP}$ and $\mathcal{D}$). Therefore, one have to solve equation (\[app:sp-line\]) with a generic non-perturbative RAD. Such solution can be found but its numerical implementation is problematic at very small-$b$. The problem is that it is very difficult to obtain the cancellation of *perturbative* logarithms at *exact* solution because at $b\to0$ the numerical values of logarithms are huge. Therefore, a good practice is to use the perturbative solution at very small-$b$, (and hence cancel all logarithm exactly) and turn to the exact solution at larger $b$. This is implemented in the ansatz eq. (\[NP:zeta\]).
In the following sections we provide expressions for $\zeta_\mu^{\text{exact}}$ and $\zeta_\mu^{\text{pert}}$ that were used in the fit procedure.
Perturbative expression
-----------------------
The perturbative solution eq. (\[app:sp-line\]) is conveniently written as $$\begin{aligned}
\zeta_\mu^{\text{pert}}(b)=\frac{\mu}{b}2e^{-\gamma_E}e^{v(\mu,b)},\end{aligned}$$ where $$\begin{aligned}
\label{app:zeta-pert}
v(\mu,b)=\sum_{n=0}^\infty a_s^n(\mu) v_n(\mathbf{L}_\mu),\end{aligned}$$ with $\mathbf{L}_\mu$ defined in eq. (\[app:Lmu\]). The general expression for $v_n$ can be found in [@Scimemi:2018xaf] (see eq. (5.14)). We apply the boundary condition eq. (\[app:sp\_boundary\]), which in the perturbative regime turns into requirement of finiteness of $v_n$ at $\mathbf{L}_\mu\to0$. The values of $v_n$ up to NNLO are $$\begin{aligned}
v_0(\mathbf{L}_\mu)&=&\frac{\gamma_1}{\Gamma_0},
\\
v_1(\mathbf{L}_\mu)&=&\frac{\beta_0}{12}\mathbf{L}_\mu^2-\frac{\gamma_1\Gamma_1}{\Gamma_0^2}+\frac{\gamma_2+d^{(2,0)}}{\Gamma_0},
\\
v_2(\mathbf{L}_\mu)&=&\frac{\beta_0^2}{24}\mathbf{L}_\mu^3+{\left(}\frac{\beta_1}{12}+\frac{\beta_0\Gamma_1}{\Gamma_0}{\right)}\mathbf{L}_\mu^2+
{\left(}\frac{\beta_0\gamma_2}{2\Gamma_0}+\frac{4\beta_0d^{(2,0)}}{3\Gamma_0}-\frac{\beta_0\gamma_1\Gamma_1}{2\Gamma_0^2}{\right)}\mathbf{L}_\mu
\\{\nonumber}&&\qquad+\frac{\gamma_1\Gamma_1^2}{\Gamma_0^3}-\frac{\gamma_1\Gamma_2+\gamma_2\Gamma_1+d^{(2,0)}\Gamma_1}{\Gamma_0^2}+\frac{\gamma_3+d^{(3,0)}}{\Gamma_0}.\end{aligned}$$ The definition of perturbative coefficients is given in eq. (\[app:def:dnk\], \[app:def-beta-Gamma\]) and $$\begin{aligned}
\label{app:gammaV}
\gamma_V(\mu)=\sum_{n=1}^\infty a_s^n(\mu) \gamma_n.\end{aligned}$$ Similarly, to RAD the $\zeta_\mu^{\text{pert}}(b)$ can be resummed in terms of $a_s\mathbf{L}_\mu$ (see appendix A in [@Scimemi:2018xaf]). However, it is not necessary in our ansatz eq. (\[NP:zeta\]) because the perturbative expression turns into exact much before the problems with convergence occur.
Exact expression
----------------
The evolution field, the equipotential line $\zeta_\mu$ and the position of the saddle point $(\mu_0,\zeta_0)$, depends on values of $b$, which is treated as a free parameter. It causes certain problems in the implementation of the $\zeta$-prescription exactly. The lesser problem is that additional numerical computations are required to determine the position of saddle-point and the values of the line for different non-perturbative models of $\mathcal{D}$. The greater problem is that at larger $b$ the value of $\mu_0$ decreases and at some large value of $b$ (typically $b\sim 3$GeV$^{-1}$) $\mu_0$ is smaller than $\Lambda_{QCD}$. Due to this behavior, it is impossible to determine the special null-evolution line at large-$b$ numerically. However, the special null-evolution line is still uniquely defined by the continuation from smaller values of $b$.
In ref. [@Vladimirov:2019bfa] a simple solution of this problem has been found. The central idea is to use the non-perturbative RAD as a generalized coordinates $(a_s,\mathcal{D})$ instead of the natural arguments $(\mu,b)$. We introduce the function $g$ as $$\begin{aligned}
\zeta_\mu^{\text{exact}}(b)=\mu^2 e^{-g(a_s(\mu),\mathcal{D}(\mu,b))/\mathcal{D}(\mu,b)}.\end{aligned}$$ It satisfies the following linear equation in partial derivatives $$\begin{aligned}
\label{app:g-}
2\mathcal{D}+2\beta(a_s)\frac{\partial g(a_s,\mathcal{D})}{\partial a_s}-\Gamma_{\text{cusp}}(a_s)\frac{\partial g(a_s,\mathcal{D})}{\partial \mathcal{D}}+\gamma_V(a_s)=0.\end{aligned}$$ The saddle point boundary condition eq. (\[app:sp\_boundary\]) turns into $$\begin{aligned}
\label{app:g-boundary}
g(a_s,0)=0.\end{aligned}$$ The equation (\[app:g-\]) can be solved exactly, but the application of boundary condition eq. (\[app:g-boundary\]) requires the solution of functional equation with transcendental functions.
On another hand, the values of $a_s$ used in the $\zeta$-prescription are always small, since they are evaluated at the hard scale $Q$, see (\[def:TMD-evolved\]). Therefore, it is natural and numerically accurate to consider the expansion in $a_s$. Note, that such an expansion already incorporates the non-perturbative corrections exactly. Denoting $$\begin{aligned}
\label{app:g}
g(a_s,\mathcal{D})=\frac{1}{a_s}\frac{2\beta_0^2}{\Gamma_0}\sum_{n=0}^\infty a_s^n g_n(\mathcal{D}),\end{aligned}$$ We find $$\begin{aligned}
\label{app:g0}
g_0&=&e^{-p}-1+p,
\\
g_1&=&\frac{\beta_1}{\beta_0}{\left(}e^{-p}-1+p-\frac{p^2}{2}{\right)}-\frac{\Gamma_1}{\Gamma_0}(e^{-p}-1+p)+\frac{\beta_0\gamma_1}{\gamma_0}p,
\\\label{app:g2}
g_2&=&{\left(}\frac{\Gamma_1^2}{\Gamma_0^2}-\frac{\Gamma_2}{\Gamma_0}{\right)}(\cosh(p)-1)+{\left(}\frac{\beta_1\Gamma_1}{\beta_0\Gamma_0}-\frac{\beta_2}{\beta_0}{\right)}(\sinh(p)-p)
\\{\nonumber}&& \qquad\qquad +{\left(}\frac{\beta_0\gamma_2}{\Gamma_0}-\frac{\beta_0\gamma_1\Gamma_1}{\Gamma_0^2}{\right)}(e^p-1),
\\{\nonumber}g_3&=&{\left(}\frac{\beta_1(\beta_1\Gamma_1-\beta_2\Gamma_0)}{12\beta_0^2\Gamma_0}-\frac{\beta_3\Gamma_0-2\beta_2\Gamma_1+\beta_1\Gamma_2}{12\beta_0\Gamma_0}-
\frac{\Gamma_1^3}{3\Gamma_0}+\frac{\Gamma_1\Gamma_2}{2\Gamma_0^2}-\frac{\Gamma_3}{6\Gamma_0}{\right)}(e^{2p}+2e^{-p}-3)
\\{\nonumber}&&+{\left(}\frac{\Gamma_1^3}{\Gamma_0^3}-\frac{\Gamma_1\Gamma_2}{\Gamma_0^2}-\frac{\beta_2\Gamma_1}{\beta_0\Gamma_0}{\right)}(\cosh(p)-1)
+\frac{\beta_0}{\Gamma_0}{\left(}\frac{\gamma_1\Gamma_1^2}{\Gamma_0^2}-\frac{\gamma_2\Gamma_1}{\Gamma_0}-\frac{\gamma_1\Gamma_2}{\Gamma_0}+\gamma_3{\right)}e^p(e^p-1)
\\\label{app:g3} &&+{\left(}\frac{\beta_1\gamma_2}{\Gamma_0}-\frac{\beta_1\gamma_1\Gamma_1}{\Gamma_0^2}-\frac{\beta_0\gamma_3}{\Gamma_0}+\frac{\beta_0\gamma_1\Gamma_2}{\Gamma_0^2}{\right)}\frac{(e^p-1)^2}{2}+\frac{\beta_3}{2\beta_0}(e^{-p}+p-1)
\\{\nonumber}&& + \frac{\beta_1}{2\beta_0}{\left(}\frac{\beta_2}{\beta_0}-\frac{\beta_1\Gamma_1}{\beta_0\Gamma_0}+\frac{\Gamma_2}{\Gamma_0}{\right)}(e^p-p-1).\end{aligned}$$ where $p=2\beta_0 \mathcal{D}/\Gamma_0$. The expressions (\[app:g0\]-\[app:g3\]) provide a very accurate approximation, since $a_s$ is evaluated at $\mu=Q$ and typically $a_s=g^2/(4\pi)^2\sim 10^{-2}$. Most importantly this expression is valid for all values of $b$, even when the saddle point is below $\Lambda_{QCD}$.
Let us mention that $g_2$ and $g_3$ exponentially grow at large-$\mathcal{D}$. It demonstrates that at large$-\mathcal{D}$ the series is an asymptotic series. However, this effect takes a place when $\mathcal{D}\sim 3-5$ which corresponds to typical values for $b\sim 20-25$GeV$^{-1}$. It does not cause any problem but effectively cuts the Hankel integral (practically, the integration converges much earlier).
[^1]: The TMD evolution is sensitive to the color-representation. Since in this work we deal only with quark channels, we do not write the corresponding labels.
[^2]: We do not consider the data from [@Adolph:2013stb] since they have large systematic errors, and fully replaced by [@Aghasyan:2017ctw].
[^3]: We are thankful to R. Sassot for providing us the actual grids for DSS FFs.
|
---
abstract: |
Gisin’s argument against deterministic nonlinear Schrödinger equations is shown to be valid for every (formally) nonlinearizable case of the general Doebner-Goldin 2-particle equation in the following form:
The time-dependence of the position probability distribution of a particle ‘behind the moon’ may be instantaneously changed by an arbitrarily small instantaneous change of the potential ‘inside the laboratory’. [*PACS:*]{} 03.65.Bz\
[*Keywords:*]{} Nonlinear Schrödinger equations, Nonlocality
author:
- |
W. Lücke[^1]\
Arnold Sommerfeld Institute for Mathematical Physics\
Technical University of Clausthal\
D-38678 Clausthal, Germany
date: 'October 10, 1997'
title: ' Gisin Nonlocality of the Doebner-Goldin 2-Particle Equation'
---
= -8 true mm = -8 true mm
= cmss10 at 5pt = cmss10 at 7pt = cmss10 === \#1 =cmbxti10 scaled 1 \#1[[\#1]{}]{} \#1[.]{} \#1 \#1[[\#1]{}]{} \#1[| \#1 |]{} \#1[ \#1 ]{}
22.5cm
Introduction
============
Several years ago N. Gisin pointed out that for every (deterministic, scalar) nonlinear 2-particle Schrödinger theory there is an initial wave function $\Psi_0(\xv_1,\xv_2)\in
L^2(\bbbr^3)\otimes L^2(\bbbr^3)$ and a self-adjoint operator $\OA\otimes \Op 1$ such that the ‘expection value’ of $\OA\otimes \Op
1$ may be almost instantaneously influenced by performing measurements on particle 2 [@Gisin2; @Gisin3]. As shown in [@LueckeNL] the existence of such a does not depend on Gisin’s questionable assumptions concerning the measuring process. More precisely, the following holds:
There is an initial wave function $\Psi_0(\xv_1,\xv_2)\in
L^2(\bbbr^3)\otimes L^2(\bbbr^3)$ and a self-adjoint operator $\OA\otimes \Op 1$ such that $$\left\langle \Psi^{V_t}_t\mid\OA\otimes \Op 1\,\Psi^{V_t}_t \right\rangle$$ depends nontrivially on $V_t\,$, where $ \Psi^{V_t}_t$ denotes the solution of the corresponding initial value problem for the nonlinear 2-particle Schrödinger equation[^2] $$\label{GNLS}
i\partial_t\Psi_t(\xv_1,\xv_2) = \left(-\frac{1}{2} \Delta +
V_t(\xv_2)\right) \Psi_t(\xv_1,\xv_2)
+F_t[\Psi_t](\xv_1,\xv_2)\;,\quad\Delta=\Delta_(\xv_1,\xv_2)\,,$$ even if the latter is formally local in the sense that the nonlinearity $F_t$ is a [**local**]{} (non-linear) functional: $$F_t[\Psi](\xv_1,\xv_2) = F_t[\Phi](\xv_1,\xv_2)
\quad\forall\,(\xv_1,\xv_2) \notin\supp(\Psi-\Phi)\,,$$
However, in a nonlinear quantum theory one cannot consider all linear self-adjoint operators as physical observables. Therefore such Gisin effects may be completely irrelevant as in the linearizable case of the general Doebner-Goldin equation [@LueckeNL]. In order to stress this we call a Gisin effect if the corresponding operator $\OA\otimes\Op 1$ is a physical observable. If $\OA$ is a multiplier in $L^2(\bbbr^3)$ this is certainly the case due to the fundamental assumption of nonlinear quantum mechanics: $$\modulus{\Psi_t(\xv_1,\xv_2)}^2 = \left\{\begin{array}[c]{l}
\mbox{probability density for localization of particle 1 around } \xv_1\\
\mbox{and particle 2 around }\xv_2 \mbox{ at time }t\,.
\end{array}\rstop\}$$
The purpose of the present paper is to show that there are relevant Gisin effects for every case, except $c_2= -2c_5\,$, of the 2-particle [@DoGo; @DoGoGen]. Using the short-hand notation $$\rho_t \stackrel{\rm def}{=} \modulus{\Psi}^2\;, \quad \jv_t
\stackrel{\rm def}{=} \Im \left(\overline{\Psi_t} \Nv \Psi_t\right)\;,
\quad \Nv = \Nv_{(\xv_1,\xv_2)}\,,$$ this nonlinear Schrödinger equation,[^3] up to some [*nonlinear gauge transformation*]{} $$\Psi \longmapsto e^{i\lambda
\ln\modulus{\Psi}} \Psi\;,\quad\lambda\in\bbbr\,,$$ is given by (\[GNLS\]) and $$\label{DG}
F[\Psi] = \left(c_1 \frac{\Nv\cdot
\jv}{\rho} + c_2 \frac{\Delta\rho}{\rho} + c_3 \frac{\jv\,\,^2}{\rho^2}+
c_4 \frac{\jv\cdot \Nv\rho}{\rho^2} + c_5
\frac{(\Nv\rho)^2}{\rho^2}\right) \Psi\,,$$ with real parameters $c_1,\ldots,c_5$ [@DoGoPR; @DoGoNa].
Previous Results
================
We assume that there are sufficiently many, well-behaved, solutions of (\[GNLS\]),(\[DG\]) – at least locally in time – which are physically acceptable in the following sense:[^4]
> Switching on $V_t$ instantaneously does not cause an instantaneous change of the wave function.
Then we have a relevant Gisin effect whenever there is a sufficiently well-behaved initial wave function $\Psi_0$ and some integer $k$ for which the function $$\label{instGE}
\left(\left(\frac{\partial}{\partial t}\right)^k \int
\rho_t(\xv_1,\xv_2) \,{\rm d}\xv_2\right)_{|_{t=0}}$$ of $\xv_1$ depends nontrivially on $V_t=V\,$. That such instantaneous Gisin effects exist unless $$\label{Werner}
c_3=c_1+c_4=0$$ was first shown by R. Werner [@Werner]. Since Werner’s Ansatz, using entangled Gauß solutions for oscillator potentials, was too special the $V$-dependent part of (\[instGE\]) was calculated in [@LuNa] for general initial conditions and $k\leq
3\,$. This only confirmed Werner´s result. We will show, however, that (\[Werner\]) does not exclude nontrivial $V$-dependence of (\[instGE\]) for $k=4\,$.
Additional Gisin Effects
========================
Since (\[instGE\]) becomes very complicated for $k>3$ we calculate $
\left(\left(\frac{\partial}{\partial t}\right)^k \int x_1^1
\rho_t(\xv) \,{\rm d}\xv\right)_{|_{t=0}}\,,
$ instead, assuming $\Psi_t$ to be sufficiently well behaved.
Using the continuity equation $$\partial_t \rho_t + \Nv\cdot\jv_t = 0$$ we get by partial integration w.r.t. $x_1^1$ the first Ehrenfest relation $$\partial_t\int x_1^1\,\rho_t(\xv)\,{\rm d}\xv = \Im \int
\overline{\Psi_t(\xv)}\,\partial_1 \Psi_t(\xv) \,{\rm d}\xv\,,$$ where $$\partial_t\stackrel{\rm def}{=}\frac{\partial}{\partial t}\;,\quad
\partial_1 \stackrel{\rm def}{=}\frac{\partial}{\partial x_1^1}\;,
\quad \xv \stackrel{\rm def}{=}(\xv_1,\xv_2)\,.$$ Further differentiation w.r.t. $t$ yields $$\left(\partial_t\right)^2 \int x^1\,\rho_t(\xv)\,{\rm d}\xv = -\int
\rho_t(\xv)\, \partial_1 R[\Psi_t](\xv)\,{\rm d}\xv$$ (note that since $\partial_1 V=0$), $$\label{ess3}
\left(\partial_t\right)^3 \int
x^1\,\rho_t(\xv)\,{\rm d}\xv = \int \Nv\cdot\jv(\xv,t)\,\partial_1
R[\Psi_t](\xv)\,{\rm d}\xv - \int
\rho_t(\xv)\, \partial_1 \partial R[\Psi_t](\xv)\,{\rm d}\xv\,,$$ and finally $$\label{ess4}
\begin{array}[c]{rcl}
\displaystyle \left(\partial_t\right)^4 \int x^1\,\rho_t(\xv)\,{\rm
d}\xv &=& \displaystyle \int \left( \partial_t\Nv\cdot \vec\jmath_t(\xv)
\right)\,\partial_1 R[\Psi_t](\xv)\,{\rm d}\xv\\
&&\displaystyle\phantom{y}+ 2\int \left( \Nv\cdot \vec\jmath_t(\xv)
\right)\,\partial_1\partial_t R[\Psi_t](\xv)\,{\rm d}\xv\\
&&\displaystyle \phantom{y} - \int \rho_t(\xv)\, \partial_1\partial_t^2
R[\Psi_t](\xv)\,{\rm d}\xv\,,
\end{array}$$ where $$F[\Psi_t](\xv) = R[\Psi_t](\xv)\,\Psi_t(\xv)\,.$$ Since the $V$-dependent part of $\left(\partial_t \Nv\cdot
\vec\jmath_t(\xv)\right)_{|_{t=0}}$ is[^5] $$\label{ess-dot-diff-j}
\begin{array}[c]{rcl}
\displaystyle {\rm ess\,}\left(\partial_t \Nv\cdot
\vec\jmath_t(\xv)\right)_{|_{t=0}} &=&\displaystyle
\Nv\cdot\Re\left(\overline{\Psi_0(\xv)}[V(\xv_2),\Nv]_-
\Psi_0(\xv)\right)\\
&=& \displaystyle -\Nv\cdot\left( \rho_0(\xv)\Nv V(\xv_2)\right)\,,
\end{array}$$ (\[ess3\]) and (\[ess4\]) depend linearly on $R$ and therefore the contributions of the different terms in (\[DG\]) may be checked separately.
Let us first consider the special case $c_4=-c_1\ne 0\,,\;c_\nu=0\,$ else, i.e. $$\label{case1}
R[\Psi](\xv) = c_1\left( \frac{\Nv\cdot \jv}{\rho} - \frac{\jv\cdot
\Nv\rho}{\rho^2}\right) = c_1\,\Delta \arg\left(\Psi(\xv)\right)\,.$$ Here $$\partial_t R[\Psi_t](\xv) = -c_1\Delta \Re\left( \frac{i\partial_t
\Psi_t(\xv)}{\Psi_t(\xv)}\right)$$ and therefore $${\rm ess\,}\left(\partial_tR[\Psi_t](\xv)\right)_{|_{t=0}} =
-c_1\,\Delta V(\xv_2)\,.$$ Due to $$\begin{array}[c]{rcl}
\partial_1{\rm ess}\left(\partial_t^2 R[\Psi_t(\xv)]\right)_{|_{t=0}}
&=& \displaystyle \frac{c_1}{2}\,\partial_1\Delta \, {\rm ess}\left(
\Re \left( \partial_t \frac{\Delta \Psi_t(\xv)}{\Psi_t(\xv)} \right)
\right)_{|_{t=0}}\\
&=& \displaystyle \frac{c_1}{2}\,\partial_1\Delta \,
\Im\left(\frac{[\Delta,V(\xv_2)]_-\Psi_0(\xv)}{\Psi_0(\xv)} \right)\\
&=& -c_1\,\partial_1\Delta \left(\left(\Nv V(\xv_2)\right)\cdot\Nv
\arg\left(\Psi_0(\xv)\right)\right)
\end{array}$$ the $V$-dependent part of (\[ess4\]) at $t=0$ is $$\label{result1}
\begin{array}[c]{l}
\displaystyle {\rm ess}\left(\left(\partial_t\right)^4
\int x^1\,\rho_t(\xv)\,{\rm d}\xv\right)_{|_{t=0}}\\
\displaystyle = -c_1\int \left(\Nv\cdot\left(\rho_0(\xv)\Nv
V(\xv_2)\right)\right) \Delta\partial_1 \arg(\Psi_0(\xv))\,{\rm d}\xv\\
\displaystyle \phantom{=} -c_1\int \rho_0(\xv)\,\partial_1
\Delta\left( (\Nv V(\xv_2))\cdot \Nv \arg(\Psi_0(\xv))\right)\,{\rm
d}\xv\\
\displaystyle = -c_1\int \rho_0(\xv)\, [\Delta,\Nv V(\xv_2)]_- \cdot \Nv
\partial_1 \arg(\Psi_0(\xv))\,{\rm d}\xv\,.
\end{array}$$ Obviously, (\[result1\]) does not always vanish. Hence there are relevant Gisin effects for (\[case1\]).
Next, let us consider the case $c_2\ne 0\,,\; c_\nu=0\,$ else, i.e. $$\label{case2}
R[\Psi](\xv) = c_2\,\frac{\Delta \rho(\xv)}{\rho(\xv)}\,.$$ Then, since $${\rm ess}\left(\partial_t^2 \frac{\Delta
\rho_t(\xv)}{\rho_t(\xv)}\right)_{|_{t=0}} =\left(\frac{\Delta
\rho_0(\xv)}{\rho_0(\xv)^2} -\frac{1}{\rho_0(\xv)}\Delta\right) {\rm
ess\,}\left(\partial_t \Nv\cdot \vec\jmath_t(\xv)\right)_{|_{t=0}}\,,$$ the $V$-dependent part of (\[ess4\]) at $t=0$ is $$\begin{array}[c]{l}
\displaystyle {\rm ess}\left(\left(\partial_t\right)^4
\int x^1\,\rho_t(\xv)\,{\rm d}\xv\right)_{|_{t=0}}\\
= \displaystyle -c_2\int \left(\Nv\cdot\left(\rho_0(\xv)\Nv
V(\xv_2)\right)\right) \partial_1
\frac{\Delta \rho_0(\xv)}{\rho_0(\xv)}\,{\rm d}\xv\\
\displaystyle\phantom{=}+c_2 \int \rho_0(\xv)\partial_1\left(\left(
\frac{\Delta \rho_0(\xv)}{\rho_0(\xv)^2} -\frac{1}{\rho_0(\xv)}
\Delta\right)\left( \Nv\cdot\left(\rho_0(\xv)\Nv
V(\xv_2)\right)\right)\right){\rm d}\xv\\
= \displaystyle c_2\int \left(\left[\Delta,
\frac{1}{\rho_0(\xv)}\right]_-
\partial_1\rho_0(\xv)\right)
\Nv\cdot\left(\rho_0(\xv)\Nv
V(\xv_2)\right){\rm
d}\xv\,.
\end{array}$$ This is nonzero, for example, when $$V(\xv_2) = (x_2^1)^3\;,\quad\rho_0(\xv)= \exp\left(-\norm{\xv}^2 -
x_1^1\,x_2^1\right)\,.$$ Hence there are relevant Gisin effects for (\[case2\]), too.
Finally, let us consider the case[^6] $$\label{final}
c_1=c_3=c_4=0=c_2+2c_5\,,$$ i.e. $$\label{ec}
R[\Psi](\xv) = c_2\left(\frac{\Delta \rho(\xv)}{\rho(\xv)} -\frac 12
\left(\frac{\Nv \rho(\xv)}{\rho(\xv)}\right)^2\right)\,.$$ In this case (\[GNLS\]),(\[DG\]) fulfills also the second Ehrenfest relation [@NattD]. Then already $\left(\partial_t\right)^2 \int
x^1\,\rho_t(\xv)\,{\rm d}\xv$ vanishes for all $t\,$, hence (\[ec\]) does [**not**]{} contribute to (\[ess4\]).
Conclusion {#S-Concl}
==========
We already know from [@Werner; @LuNa] — or easily rederive from (\[ess3\]) — that for (\[DG\]) there are relevant Gisin effects whenever Werner’s condition $$c_3=c_1+c4=0$$ is violated. If this condition is fulfilled, $R[\Psi] = F[\Psi]/\Psi$ may be written in the form $$\begin{array}[c]{rcl}
R &=& c_1\, \left( R_1 -R_4\right) + c_2\,R_2 + c_5\,R_5\\
&=& c_1\, \left( R_1 -R_4\right) + (c_2+2c_5)\,R_2 + c_5\,\left(-2R_2 +
R_5\right)\,.
\end{array}$$ where $$R_1[\Psi]\stackrel{\rm def}{=} \frac{\Nv\cdot
\jv}{\rho}\;, \quad R_2[\Psi]\stackrel{\rm def}{=}
\frac{\Delta\rho}{\rho}\;, \quad R_4[\Psi]\stackrel{\rm def}{=}
\frac{\jv\cdot \Nv\rho}{\rho^2}\;, \quad R_5[\Psi]\stackrel{\rm def}{=}
\frac{(\Nv\rho)^2}{\rho^2}\,.$$ As shown in the previous section, the contributions to ${\rm
ess}\left(\left(\partial_t\right)^4 \int x^1\,\rho_t(\xv)\,{\rm
d}\xv\right)_{|_{t=0}}$ by $R_1-R_4$ and $R_2$ are functionally independent while $-2R_2 +R_5$ does not contribute. We conclude that there are relevant instantaneous Gisin effects for the general Doebner-Goldin equation (\[GNLS\]),(\[DG\]) whenever condition (\[final\]) is violated. Due to translation invariance this represents a serious locality problem:
> The time-dependence of the position probability distribution of a particle ‘behind the moon’ may be instantaneously changed by an arbitrarily small instantaneous change of the potential ‘inside the laboratory’.
Since the change of the potential may be arbitrarily small, such superluminal effects are unacceptable in spite of the nonrelativistic character of the theory.
One might try to exclude relevant Gisin effects by changing the coupling of nonlinear quantum mechanical systems. However, the coupling should be
- the same for uncorrelated subsystems,
- invariant under nonlinear gauge transformations, and
- mathematically respectable for sufficiently many wave functions.
Unfortunately, no modification fulfilling these requirements is known up to now.
Of course, the results presented in this paper do not indicate any problem for the general Doebner-Goldin equation if it is interpreted as a 1-particle equation, as originally suggested [@DoGo; @DoGoGen].
Acknowledgement {#acknowledgement .unnumbered}
===============
I am indebted to H.D. Doebner for strong encouragement.
[BBM76]{}
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H.-D. Doebner and G.A. Goldin, Properties of nonlinear [S]{}chr[ö]{}dinger equations associated with diffeomorphism groups representations. J. Phys. A 27 (1994) 1771–1780.
H.-D. Doebner and G.A. Goldin, Introducing nonlinear gauge transformations in a family of nonlinear [S]{}chr[ö]{}dinger equations. Phys. Rev. A 54 (1996) 3764–3771.
H.-D. Doebner, G.A. Goldin, and P. Nattermann, Gauge transformations in quantum mechanics and the unification of nonlinear [S]{}chr[ö]{}dinger equations. ASI-TPA/21/96, quant-ph/9709036.
N. Gisin, Weinberg’s non-linear quantum mechanics and superluminal communications. Physics Letters A 143 (1990) 1–2.
N. Gisin, Relevant and irrelevant nonlinear [S]{}chr[ö]{}dinger equations. In: Nonlinear, deformed and irreversible quantum systems, eds. H.-D. Doebner, V.K. Dobrev, and P. Nattermann (World Scientific, 1995) p. 109–124.
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W. L[ü]{}cke and P. Nattermann, Nonlinear quantum mechanics and locality, ASI-TPA/12/97, quant-ph/9707055. To appear in: B. Gruber and M. Ramek, editors, [*Symmetry in Science X*]{}, Plenum Press, New York, 1998.
P. Nattermann, Struktur und [E]{}igenschaften einer [F]{}amilie nichtlinearer [S]{}chr[ö]{}dingergleichungen, Diplomarbeit, TU Clausthal (1993).
P. Nattermann, Solutions of the general Doebner-Goldin equation via nonlinear transformations. In: Proceedings of the XXVI Symposium on Mathematical Physics, Toruń, December 7-10, 1993, p. 47.
P. Nattermann, Dynamics in [B]{}orel-quantization: [N]{}onlinear [S]{}chr[ö]{}dinger equations vs. master equations, Dissertation, TU Clausthal (1997).
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R. Werner, private communication.
[^1]: E-mail: aswl@pt.tu-clausthal.de
[^2]: We use units in which $\hbar=1$ and $m=1\,$.
[^3]: For simplicity we consider only the special case $V_t(\xv_1,\xv_2) = V_t(\xv_2)\,$.
[^4]: This assumption is well known to be fulfilled for the linear Schrödinger equation and should follow along the lines discussed in [@Teismann] for the Doebner-Goldin equation.
[^5]: We always denote by $
{\rm ess\,}(X)$ the partial sum of $V$-dependent items of $X\,$.
[^6]: Note that (\[GNLS\]),(\[DG\]) is (formally) linearizable in this case [@NattT93].
|
---
abstract: 'In this paper we introduce the concept of quantum private channel within the continuous variables framework (CVPQC) and investigate its properties. In terms of CVPQC we naturally define a “maximally" mixed state in phase space together with its explicit construction and show that for increasing number of encryption operations (which sets the length of a shared key between Alice and Bob) the encrypted state is arbitrarily close to the maximally mixed state in the sense of the Hilbert-Schmidt distance. We bring the exact solution for the distance dependence and give also a rough estimate of the necessary number of bits of the shared secret key (i.e. how much classical resources are needed for an approximate encryption of a generally unknown continuous-variable state). The definition of the CVPQC is analyzed from the Holevo bound point of view which determines an upper bound of information about an incoming state an eavesdropper is able to get from his optimal measurement.'
author:
- Kamil Brádler
title: Continuous variable private quantum channel
---
Introduction
============
The task of quantum state encryption [@quant_ver_pad; @PQC; @encryption] (quantum Vernam cipher, quantum one-time pad) is defined as follows. Let’s suppose that there are two communicating parties, Alice and Bob, and Alice wants to send an arbitrary unknown quantum state to Bob (the state needn’t to be known either Alice or Bob). They are connected through the quantum channel which is accessible to Eve’s manipulations. To avoid any possible information leakage from the state to Eve (via some kind of generalized measurement or state estimation) Alice and Bob share a secret and random string (classical key) of bits by which Alice chooses a unitary operation from the given (and publicly known) set. The transformed state is sent to Bob via the quantum channel who applies the same operation to decrypt the state. For an external observer (e.g. Eve) the state on the channel is a mixture of all possible transformations that Alice can create because she doesn’t know the secret string. If, moreover, the mixture is independent on the state to be encrypted then Eve has no chance to deduce any information on the state. We say that the state was perfectly encrypted. After the formalization of this procedure we get the definition of [*private quantum channel – PQC*]{} [@PQC]. As the mixture it is advantageous to choose a maximally mixed state – identity density matrix.
It is natural to ask how many operations are needed for encryption of a $k$-qubit input state. Based on thoughts on entropy it was shown [@encryption] and later generalized [@PQC] for PQC with ancilla that $2k$ classical bits are sufficient (i.e. $2^{2k}$ operations). Generally, for perfect encryption of $d$-dimensional quantum state $d^{2}$ unitary operations are needed. Thus, the length of participant’s secret and random key must have at least $2\log d$ bits [^1]. After weakening the security definition an approximate secrecy was stated in [@approx_encryption] defining of [*approximate private quantum channel – aPQC*]{}. Then, it was shown that asymptotically just $d\log d$ operations are needed for approximate quantum state encryption. Next progress in this research topic was achieved in [@approx_encryption1] where a polynomial algorithm for constructing a set of encryption/decryption operations suitable for aPQC was presented.
In this paper we investigate the possibility of quantum state encryption for continuous variables (CV) [@CV]. Under CV we understand two conjugate observables such as e.g. position and momentum of a particle. Especially, we concentrate on coherent states states which Wigner function has the form of the normalized Gaussian distribution. Gaussian states are the most important class of CV states used in quantum communication and computation [@CV_prehled]. The most of intriguing processes and algorithms discovered for discrete $d$-level quantum systems were also generalized for CV. Among others, let’s mention CV quantum state teleportation [@CV_tele1; @CV_tele2], CV quantum state cloning [@CV_clone] and quantum computation with CV [@CV_comp]. Importantly, a great progress was made in quantum key distribution (QKD) based on CV where theoretical groundwork was laid [@CV_crypto1; @CV_crypto2; @CV_crypto3; @CV_crypto4; @CV_crypto5] and experiments were performed [@CV_crypto_exp1; @CV_crypto_exp2].
After brief introducing into the questions of distances used in quantum information theory in Sec. \[sec\_intro\] the main part follows in Sec. \[sec\_main\] and Sec. \[sec\_threats\]. There we present the notion of CV quantum state encryption and in the later we define a continuous-variable private quantum channel (CVPQC). This, foremost, consists of defining the “maximally” mixed state within the context of continuous variables and estimating the length of a secret shared key between Alice and Bob for a given secrecy (by secrecy it is meant the HS distance between “maximally" mixed and the investigated state). In Sec. \[sec\_threats\] we will touch the question of “maximality” of the mixed state (from now on, let’s omit the apostrophes) in the context of bosonic channels [@bos_chan1; @bos_chan2] and their generalized lossy [@bos_chan_lossy] and Gaussian relatives [@bos_gauss_chan_mem]. We will also discuss important differences between discrete and CV encryption from the viewpoint of eavesdropping followed by the calculation of the Holevo bound limiting information accessible to Eve from the encrypted channel. Necessary Appendices come at the end of the paper. In Appendix \[app\_exact\_value\] we give a derivation of the exact formula for the HS distance. The object of Appendix \[app\_guess\] is to inference the mentioned estimate of the HS distance.
Measures of quantum states closeness {#sec_intro}
====================================
Quantum states can be distinguished in the sense of their mutual distance. The distance is usually induced by a norm defined on the space of quantum states. This is the case of Schatten $p$-norm $$\label{schatten}
\|A\|_p
=\left(
{\mathop{{\mathrm{Tr}}_{}}}\left(\left|A\right|^p\right)
\right)^\frac{1}{ p}
=\left(
{\mathop{{\mathrm{Tr}}_{}}}\left(A^p\right)
\right)^\frac{1}{ p},$$ where $|A|=\sqrt{A^\dagger A}$ and the last equation is valid for $A=A^\dagger$. If $A=\varrho_1-\varrho_2$ the Hermiticity of $A$ is still preserved and for $p=2$ we get the Hilbert-Schmidt (HS) distance $$\label{HSdist}
D_{HS}(\varrho_1,\varrho_2)
=\|\varrho_1-\varrho_2\|_2=\sqrt{{\mathop{{\mathrm{Tr}}_{}}}\left((\varrho_1-\varrho_2)^2\right)}.$$ On the other hand, there is a whole family of distances based on Uhlmann’s fidelity [@uhlmann] $$\label{fid_uhl}
F(\rho_{1},\rho_{2})={\mathop{{\mathrm{Tr}}_{}}}\left(\left(\sqrt{\rho_{1}}\rho_{2}
\sqrt{\rho_{1}}\right)^\frac{1}{2}\right).$$ One of them is Bures distance [@bures] $$\label{buresdist}
D_B(\varrho_1,\varrho_2)
=\left(2-2{\mathop{{\mathrm{Tr}}_{}}}\sqrt{\sqrt{\varrho_1}\varrho_2\sqrt{\varrho_1}}\right)^\frac{1}{2}$$ which coincides with the HS distance if $\varrho_1,\varrho_2$ are pure states.
There is a certain equivocation which distance is more suitable for a given task in QIT where a general problem of quantum state distinguishability or closeness has to be resolved. Both distances have many useful properties and they are subject of detailed investigation [@SommZyc]. For example, output “quality" of a quantum state in the problem of universal quantum-copying machine was first analyzed with the help of the HS distance [@UQCM_HS] and later revisited from the viewpoint of the Bures and trace distance (Schatten $p$-norm (\[schatten\]) for $p=1$) [@UQCM_Bures]. Other problem, among others, where closeness of two quantum states plays a significant role is quantum state estimation [@quantstateest]. Here the closeness of estimated states is often measured by the HS distance [@quantstateest_HS].
The motivation for using the HS distance in our calculation is two-fold. First, the security criterion for approximate quantum state encryption [@approx_encryption] is based on the operator distance induced by operator norm ($p\to\infty$ in (\[schatten\])) while some other results therein are calculated for the trace distance which is in some sense weaker than the operator distance (the reason is computational difficulty). Second, for our purpose we need to calculate distance between two infinitely-dimensional mixed states what is difficult in the case of all fidelity based distances. However, we suppose that the HS distance is a good choice for coherent states and provides an adequate view on measure of the closeness of two states.
CV state encryption and its analysis {#sec_main}
====================================
Let’s define the task of CV state encryption. As in the case of discrete variable quantum state encryption, Alice and Bob are interconnected via a quantum channel which is fully accessible to Eve’s manipulations. Both legal participants share a secret string of random bits (key) which sufficient length is, among others, subject of this paper. The key indexes several unitary operations which Alice/Bob chooses to encrypt/decrypt single-mode coherent states. The purpose of the encryption is to secure these states from leakage of any information about them to an eavesdropper (Eve). The way to achieve this task is to “randomize" every coherent state to be close as much as possible to a maximally mixed state (maximally in the sense specified next). The randomization is performed with several publicly known unitary operations (it is meant that the set from which participants choose is known but the particular sequence of operations from the set is given by the key which is kept secret). So, suppose that Alice generates or gets an arbitrary and generally unknown single-mode coherent state. The only public knowledge about the state is its appearance somewhere inside the circle of radius $b$ in phase space with the given distribution probability. Here we suppose that states occur with the same probability for all $r\leq
b$ and with zero probability elsewhere.
Suppose for a while that Alice encrypts a vacuum state. This situation will be immediately generalized with the help of the HS distance properties for an unknown coherent state within the circle of radius $b$ as stated in the previous paragraph. The first problem we have to tackle is the definition of a maximally (or completely) mixed state. Here the situation is different from, generally, $d$-dimensional discrete Hilbert space where a normalized unit matrix is considered as the maximally mixed state. This is inappropriate in phase space nevertheless we may inspire ourselves in a way the discrete maximally mixed state is generated. In fact, we get the maximally mixed state (in case of qubits $\varrho_V=\varrho_{r\vartheta\varphi}$) by integrating out over all density matrix populating the Bloch sphere $\openone\propto\int\varrho_V{\rm d}V$ (irrespective of the fact that the finite number of unitary operations suffice for this task as the theory of PQC learns). Similarly, as a maximally mixed state [^2] we can naturally choose an integral performed over all possible single-mode states within the circle of radius $r\leq b$ $$\begin{aligned}
\label{integral_0b}
\openone_b
& =\frac{1}{ C}\int{\rm d}^2\alpha{| \alpha\rangle\!\langle \alpha |}
=\frac{1}{ C}\sum_{m,n=0}^\infty{| m\rangle\!\langle n |}\int_0^{2\pi}
{\rm d}\vartheta\int_0^{b}{\rm d}r
\ e^{-r^2}\frac{r^{m+n+1}}{\sqrt{m!n!}}e^{i(m-n)\vartheta}
=\frac{2\pi}{ C}\sum_{n=0}^\infty\frac{{| n\rangle\!\langle n |}}{ n!}
\int_0^{b}{\rm d}r\ e^{-r^2}r^{2n+1}{\nonumber}\\
& =\frac{\pi}{ C}\sum_{n=0}^\infty\frac{{| n\rangle\!\langle n |}}{ n!}
\int_0^{b^2}{\rm d}x\ e^{-x}x^n
=\frac{\pi}{ C}\exp\left(-b^2\right)\sum_{n=0}^\infty
\left(\exp\left(b^2\right)-\sum_{k=0}^n\frac{b^{2k}}{ k!}\right){| n\rangle\!\langle n |}.\end{aligned}$$ ${\mathop{\left|\alpha\right>}\nolimits}=D(\alpha){\mathop{\left|0\right>}\nolimits}
=\exp(-|\alpha|^2/2)\sum_{n=0}^\infty\frac{\alpha^n}{\sqrt{n!}}{\mathop{\left|n\right>}\nolimits}$ is a coherent state represented as a displaced vacuum via the displacement operator $D(\alpha)$ (from now on it will not be mentioned explicitly that $|\alpha|\leq b$ for all displacement operators used in this paper and for given $b$) and $C=\pi b^2$ is a normalization constant. Note that calculation (\[integral\_0b\]) for $b\to\infty$ without the normalization is nothing else than well known resolving of unity giving the evidence of spanning the whole Hilbert space.
Having defined the maximally mixed state let’s investigate which operations Alice uses for encryption. This transformation should be as close as possible to the maximally state in the HS distance sense and the closeness will depend on the number of used operations. Beforehand, we will note how to facilitate forthcoming tedious calculations on a sample example. Suppose that Alice has e.g. four encryption operations, which displace the vacuum state to the same distance $r_4$ from the origin but under four different angles (from symmetrical reason these angles are multiples of $\frac{\pi}{2}$ in this case). Alice chooses these operations with the same probability. Now, if we write down the overall mixture from the states, it can be shown that there exists a computationally advantageous “conformation" when the mixture reads $$\label{mix_on_circle__example}
\varrho_4=\frac{1}{4}\sum_{q=1}^4{| \alpha_{4q}\rangle\!\langle \alpha_{4q} |}
=\exp(-r_4^2)\sum_{m,n=0}^\infty\frac{r_4^{m+n}}{\sqrt{m!n!}}
{| m\rangle\!\langle n |}\,\delta_m^{m'},$$ where $m'=m+4l$, $l=1,\dots,\infty$ and $\delta_m^{m'}=1$ for all $m$ (occupation number) else $\delta_m^{m'}=0$. Informally, (\[mix\_on\_circle\_\_example\]) is always a real density matrix with off-diagonal non-zero “stripes" separated from main diagonal and from a neighbouring stripe by three zero off-diagonals. $\varrho_4$ acquires relatively simple form (\[mix\_on\_circle\_\_example\]) if $\alpha_{4q}=r_4e^{iq\vartheta_{4q}}$ for $\vartheta_{4q}=\frac{\pi}{4}+(q-1)\frac{\pi}{2}$. This can be generalized for arbitrary number of states dispersed on a circle with given radius $r_p$ $$\label{mix_on_circle__general}
\varrho_p=\frac{1}{ p}\sum_{q=1}^p{| \alpha_{pq}\rangle\!\langle \alpha_{pq} |}
=\exp(-r_p^2)\sum_{m,n=0}^\infty\frac{r_p^{m+n}}{\sqrt{m!n!}}
{| m\rangle\!\langle n |}\,\delta_m^{m'},$$ where $p\in{{\mathbb Z}}^+$,$m'=m+pl$ and $\delta_m^{m'}$ is defined as before. Favorable “$p$-conformations” (\[mix\_on\_circle\_\_general\]) occur when $\alpha_{pq}=r_pe^{iq\vartheta_{pq}}$ for $\vartheta_{pq}=\frac{\pi}{ p}(2q-1)$. Now, we may proceed to the mixture characterizing all Alice’s encryption operations. She chooses $N\geq1$ and defines $r_p=pb/N$ for $p=1,\dots,N$. Then $$\label{mix_on_circle}
\Phi_N=\frac{1}{M}\sum_{p=1}^Np\varrho_p
=\frac{1}{
M}\sum_{p=1}^N\sum_{q=1}^pD(\alpha_{pq}){| 0\rangle\!\langle 0 |}D^\dagger(\alpha_{pq})$$ with normalization $M=\frac{N(N+1)}{2}$. To sum up the protocol, Alice and Bob share a secret and random key which indexes their operations. So, Alice equiprobably chooses from the set of $M$ displacement operators $D(\alpha_{pq})=\exp(\alpha_{pq}a^\dagger-\alpha_{pq}^*a)$ where just one operator creates a coherent state on the circle of radius $r_1$, two operators generate two states on the circle of radius $r_2$ and so forth up to $N$. Mixtures of the states on particular circles are in favourable form (\[mix\_on\_circle\_\_general\]) and the whole state is (\[mix\_on\_circle\]). The rest of the protocol is the same as in discrete state encryption. Alice sends the encrypted state through a quantum channel towards Bob who makes Alice’s inverse operation to decrypt the state.
The keystone in quantum state encryption both discrete and CV is the fact that an unknown and arbitrary state can be encrypted. If the state was known we would’t need this procedure at all because Alice could just send information about the preparation of the state to Bob. So, it would suffice to encrypt this classical information with the Vernam cipher. Also, our definition of CVPQC must be independent on the state which is to be encrypted. To provide this we will find useful unitary invariance of the HS distance $D_{HS}(\varrho,\sigma)=D_{HS}(U\varrho
U^\dagger,U\sigma U^\dagger)$ for an arbitrary unitary matrix $U$. This invariance is, however, necessary but not sufficient condition for our purpose. The second important issue is due to advantageous algebraic properties of displacement operators. Let’s demonstrate it on Alice’s encryption algorithm which stays the same as before. If she gets an arbitrary coherent state ${\mathop{\left|\beta\right>}\nolimits}$ ($|\beta|\leq b$) she randomly chooses one from $M$ displacement operators $D(\alpha_{pq})$ (as the shared key with Bob dictates). Then, even if two displacement operators generally do not commute we may write a general encryption TPCP (trace-preserving completely positive) map $$\label{map_Alice}
\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})=\frac{1}{
M}\sum_{p=1}^N\sum_{q=1}^pD(\alpha_{pq})D(\beta){| 0\rangle\!\langle 0 |}D^\dagger(\beta)D^\dagger(\alpha_{pq})
=\frac{1}{
M}\sum_{p=1}^N\sum_{q=1}^pD(\beta)D(\alpha_{pq}){| 0\rangle\!\langle 0 |}D^\dagger(\alpha_{pq})D^\dagger(\beta)
=D(\beta)\Phi_ND^\dagger(\beta).$$ It is obvious that generally $\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})\not=\Phi_N$ (and similarly $\int
D(\alpha){| \beta\rangle\!\langle \beta |}D^\dagger(\alpha){\rm
d}^2\alpha\propto
D(\beta)\openone_bD^\dagger(\beta)=\openone_b^{\beta}\not
=\openone_b$) but their HS distances are equivalent, i.e. $D_{HS}(\openone_b^{\beta},\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits}))=D_{HS}(\openone_b,\Phi_N)$. Thus, we are henceforth entitled to make all calculations of the HS distance between $\openone_b^{\beta}$ and $\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})$ with explicit forms (\[integral\_0b\]) and (\[mix\_on\_circle\]). After some calculations we will see (Appendix \[app\_guess\]) that $$\label{HSdist_guess}
D_{HS}^2\left(\openone_b^{\beta},\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})\right)
\approx\left(\frac{1}{
N+1}\right)^2+\mathcal{O}\left(N^{-4}\right),$$ which holds for all for all input coherent states ${\mathop{\left|\beta\right>}\nolimits}$. The guess (\[HSdist\_guess\]) is far from optimal (e.g. independent on $b$) and is not even an inequality. But this doesn’t matter because the exact form can be derived (for details see Appendix \[app\_exact\_value\]). Its only problem is relative complexity so we cannot easily deduce the number of operations for a given level of secrecy. Nevertheless, (\[HSdist\_guess\]) asymptotically approaches to the exact expression (as is shown in Appendix \[app\_guess\]). Notice that in spite of the derivation of (\[HSdist\_guess\]) (or, next, analytical expression (\[HSdist\_explicit\])) we still cannot reasonably define a CVPQC. We will do so in section \[sec\_threats\] after presenting another assumptions regarding eavesdropping on our private quantum channel.
If we consider the described unitary invariance of the HS distance we write down lhs of (\[HSdist\_guess\]) $$\label{HSdist_into_traces}
D_{HS}^2(\openone_b,\Phi_N)
={\mathop{{\mathrm{Tr}}_{}}}\left((\openone_b-\Phi_N)^2\right)
={\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b^2\right)-2{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\Phi_N\right)
+{\mathop{{\mathrm{Tr}}_{}}}\left(\Phi_N^2\right).$$ After some calculations (see mentioned Appendix \[app\_exact\_value\]) we get a little bit lengthy expression $$\begin{aligned}
\label{HSdist_explicit}
D_{HS}^2(\openone_b,\Phi_N)
& =\frac{1}{ b^2\exp(2b^2)}
\left(
\exp(2b^2)-I_0\left(2b^2\right)-I_1\left(2b^2\right)
\right)
-\frac{4\exp(-b^2)}{ b^2N(N+1)}
\sum_{p=1}^N\frac{p}{\exp\left(r_p^2\right)}
\sum_{k=1}^\infty\left(\frac{b}{ r_p}\right)^kI_k(2r_pb){\nonumber}\\
& +\left(\frac{2}{ N(N+1)}\right)^2
\left[\sum_{p=1}^N\frac{p^2}{\exp\left(2r_p^2\right)}
\left(
I_0\left(2r_p^2\right)+2\sum_{k=1}^\infty I_{pk}\left(2r_p^2\right)
\right)
\right.{\nonumber}\\
& +\sum_{{\genfrac{}{}{0pt}{}{p_1,p_2=1}{p_1\not=p_2}}}^N\frac{p_1p_2}{\exp\left(r_{p_1}^2+r_{p_2}^2\right)}
\left.
\left(
I_0(2R_{12})+2\sum_{k=1}^\infty I_{p_1p_2k}(2R_{12})
\right)
\right],\end{aligned}$$ where $r_p=pb/N,r_{p_l}=p_lb/N,R_{12}=p_1p_2\left(b/N\right)^2$ and $I_n(x)$ is a modified Bessel function of the first kind of order $n$ $$\label{bessfce}
I_n(x)=\sum_{s=0}^\infty{\left(\frac{x}{2}\right)^{n+2s}}\frac{1}{(n+s)!s!}.$$ However, as we declared, for a rough estimation of the number of operations for a given secrecy we will employ (\[HSdist\_guess\]). Using $M=\frac{N(N+1)}{2}\approx\frac{(N+1)^2}{2}$ (and then again $M\approx\frac{N^2}{2}$) we find that the dependence of the number of bits on the HS distance (\[HSdist\_explicit\]) is $n=\log
M=-2(1+\log D_{HS})+\log \left(1+\sqrt{1+CD_{HS}^2}\right)
\approx-1-2\log D_{HS}$ because the guess is in particular precise for $D_{HS}\ll1$. $C$ is a constant (polynomial) bounding $N^{-4}$ in (\[HSdist\_guess\]) from above. By the numerical simulations based on (\[HSdist\_explicit\]) for different $b$ and $M$ we see that the guess is accurate and for higher $M$ even serves as a relatively good upper bound.
Simplified encryption {#simplified-encryption .unnumbered}
---------------------
Now, let’s see what is going to happen if we simplify our encryption protocol. The arrangement is the same, i.e. Alice’s task is to encipher single-mode coherent states and her only knowledge is that the states are somewhere inside the circle of radius $r\leq b$. However, Alice’s technology is limited and we will tend to replace technologically demanding displacements by simple phase shifts given by the well known time evolution ${\mathop{\left|\alpha(t)\right>}\nolimits}=\exp(-iHt/\hbar){\mathop{\left|\alpha\right>}\nolimits}$ with the Hamiltonian $H=\hbar\omega(n+\frac{1}{2})$. The resulting state is ${\mathop{\left|\alpha(t)\right>}\nolimits}=\exp(-i\omega t/2){\mathop{\left|\alpha\exp(-i\omega t)\right>}\nolimits}$ so the state undertakes the rotation about $\Theta=\omega t$ regardless the distance (which stays preserved) from the phase space origin. Note that unitary operator $U=\exp(-iHt/\hbar)$ has the form $$\label{unitary_evolution}
U=\exp\left(-\frac{i\omega t}{2}\right)\sum_{n=0}^\infty\exp(-i\omega t n)
{| n\rangle\!\langle n |}.$$ Next, suppose that Alice is equipped with $p$ encryption operations where she can rotate the state about $q$ multiples of $2\pi/p$. So, $\Theta_q=q2\pi/p$ where $q=1,\dots,p$. Then, for someone without any information about the angle of rotation chosen by Alice (e.g. Eve) the state leaving Alice is in the form $$\label{ro_p_tilde}
\tilde\varrho_p=\frac{1}{
p}\sum_{q=1}^{p}U_q{| \alpha\rangle\!\langle \alpha |}U^{-1}_q=\frac{1}{
p}\sum_{q=1}^{p}\left({| \alpha\rangle\!\langle \alpha |}\right)_q=\frac{1}{
p}\sum_{q=1}^{p}{| \alpha e^\frac{q2\pi}{p}\rangle\!\langle \alpha
e^\frac{q2\pi}{ p} |}$$ where $U_q$ is (\[unitary\_evolution\]) with $\Theta_q=\omega
t_q=q2\pi/p$. Because Alice doesn’t know where exactly the state ${\mathop{\left|\alpha\right>}\nolimits}$ is placed we cannot write an explicit form of $\tilde\varrho_q$. Apparently, however, for arbitrary $p$ it can be transformed to our favourite state (\[mix\_on\_circle\_\_general\]), again with the help of unitary operation (\[unitary\_evolution\]) $$\label{ro_p}
\varrho_p(r_p)=U\tilde\varrho_pU^{-1}
=\exp(-r_p^2)\sum_{m,n=0}^\infty\frac{r_p^{m+n}}{\sqrt{m!n!}}
{| m\rangle\!\langle n |}\,\delta_m^{m'},$$ where $m'=m+pl$, $l=1,\dots,\infty$. It remains to show that $\openone_b=U\openone_bU^{-1}$. This can be readily seen from the fact that (\[unitary\_evolution\]) is diagonal. This is vital. Now we can easily calculate the HS distance of $\varrho_p(r_p)$ and $\openone_b$ and regarding the unitary invariance of the HS distance the result will be valid for any arbitrary input coherent state (this trick is akin to that one used in (\[map\_Alice\])). After the calculation we get the same result as in (\[HSdist\_explicit\]) but for fixed $p$ and without the summation over $p_1,p_2$ $$\begin{aligned}
\label{HSdist_simple_explicit}
D_{HS}^2(\openone_b,\varrho_p)
& =\frac{1}{ b^2\exp(2b^2)}
\left(
\exp(2b^2)-I_0\left(2b^2\right)-I_1\left(2b^2\right)
\right)
-\frac{2}{\exp\left(b^2\right)b^2}
\frac{1}{\exp\left(r_p^2\right)}
\sum_{k=1}^\infty\left(\frac{b}{ r_p}\right)^kI_k(2r_pb){\nonumber}\\
& +\frac{1}{\exp\left(2r_p^2\right)}
\left(
I_0\left(2r_p^2\right)+2\sum_{k=1}^\infty I_{pk}\left(2r_p^2\right)
\right).\end{aligned}$$
The behaviour of this quantity is completely different compared to (\[HSdist\_explicit\]) describing the original case as can be seen in Fig. \[figs\_saturated\_min\] (A). As an example we put $b=2$ and we see that there exists $r_{min}<b$ for which the HS distance between (\[integral\_0b\]) and (\[ro\_p\]) reaches its minimum for sufficiently high $p$. In order to find a radius minimizing the HS distance of these density matrices for various $b$ (we put $p\to\infty$ and thus $2\sum_{k=1}^\infty
I_{pk}\left(2r_p^2\right)$ vanishes – we seek for, say, a saturated minimum which is independent on the number of operations on a given circle) we can calculate $$\label{min_D_HS}
\frac{\partial\left(D_{HS}^2(\openone_b,\varrho_p)\right)}{\partial r}
\stackrel{p\to\infty}{=}\frac{4}{\exp\left(2r_{min}^2\right)}
\left(
r_{min}I_1\left(2r_{min}^2\right)-r_{min}I_0\left(2r_{min}^2\right)
+\frac{\exp\left(2r_{min}^2\right)}{b\exp\left(b^2\right)}I_1\left(2r_{min}b\right)
\right)=0.$$ The solution is the root of the expression in parentheses. Unfortunately, an analytical solution wasn’t found so numerical calculation of $r_{min}$ for several $b\in\left(0,7\right.\rangle$ is depicted in Fig. \[figs\_saturated\_min\] (B).
What are the physical consequences of this whole simplification? From Fig. \[figs\_saturated\_min\] (A) we see that a maximally mixed state $\openone_b$ can be in the distance sense substituted with a mixture of coherent state on a circle (with radius $r_{min}$ where the diversion from $\openone_b$ is smallest) and the mixing requires relatively few encryption operations (low $p$ for the distance saturation). It holds not only for $b=2$ what is the case in the picture. So, now we can make the process of encryption/decryption technologically easier. We equip Alice/Bob with mentioned phase shifts leading to (\[ro\_p\]) and just one displacement operator $D(r_{min})$. Every incoming coherent state is first displaced and then encrypted by choosing a phase shift (indexed by a secret key). The role of Fig. \[figs\_saturated\_min\] (B) is in a proper choice of $r_{min}$ for given $b$. Finally, Alice examines Fig. \[figs\_saturated\_min\] (A) to select a sufficient number $p$ of phase shifts for which the minimum of HS distance (\[HSdist\_simple\_explicit\]) stays essentially the same.
The important point is that incoming coherent states come to Alice randomly and equiprobably for given $b$ so after this simplified encryption (and for many encryption instances) they are as close as possible to full-fashioned encrypted states as they would be if processed by using (\[mix\_on\_circle\]), i.e. the case of fully technologically equipped Alice and Bob, respectively. The overall advantage is the use of just one technologically demanding operation which doesn’t need to be tuned for given $b$ (it stays fixed in the encryption protocol described at the beginning of chapter \[sec\_main\]).
Eavesdropping on encrypted CV states {#sec_threats}
====================================
Quantum channels are TPCP maps on quantum states. Following the classical channel coding theorem [@shannon48] one may ask how much information is a quantum channel able to convey. This question naturally leads to the definition of quantum channel capacity (for a nice survey on various quantum channel capacities see Ref. [@shor_capa]) as the maximum of the accessible information over the probability distributions of the input states ensemble $\{p_i,\sigma_i\}$ entering the channel (after classical-quantum coding onto input quantum states $x_i\rightleftharpoons\sigma_i$ the ensemble of classical messages is indicated by the variable $X=\{p(x_i),x_i\}$ where $p(x_i)\equiv p_i$). The accessible information $I_{acc}$ itself is the maximization over all measurement of the mutual information $I(X;Y)$ with the variable $Y=\{p\,(y_j),y_j\}$ giving the probability $p\,(y_j)$ of the result $y_j$ (an output alphabet) of a measurement on the channel output. Sometimes it might be difficult to calculate the accessible information to determine the channel capacity. Then, we can estimate the accessible information from above by the Holevo bound $$\label{holevo_bound}
\chi\left(\{p_i,\sigma_i\}\right)
=S\biggl(\sum_ip_i\sigma_i\biggr)-\sum_ip_iS(\sigma_i),$$ for which was proved [@holevo_bound] that $\chi\geq I_{acc}$ ($S(\varrho)$ is the von Neumann entropy) reaching equality if all $\sigma_i$ commute.
In our case we try to get as close as possible to our maximally mixed state (\[integral\_0b\]) with TPCP map (\[mix\_on\_circle\]) (or, generally, to $\openone_b^\beta$ with (\[map\_Alice\])) which both belong to the class of bosonic channels [@bos_chan1; @bos_chan2]. However, our task is quite different from reaching the highest quantum channel capacity (what means “tuning” of $p_i$). Now, the variable $X$ is fixed. For our purpose the Holevo bound can tell us what is the upper bound on information which is a third party (Eve) able to learn from her very best (that is optimal) measurement on the quantum channel. The question now is of what state we should calculate the Holevo bound to find Eve’s maximum of attainable information on [*incoming*]{} states. Unlike the discrete case (perfect and approximate encryption) the condition of perfect and acceptable closeness of an encrypted state to the maximally mixed state is not in this case sufficient. Let’s demonstrate the reason for this difference on the Bloch sphere (i.e. perfect qubit encryption). If Alice encrypts an unknown qubit (with a PQC compound from e.g. four Pauli matrices) she gets a maximally mixed two-qubit state (normalized unity matrix). It means that Eve cannot construct any measurement giving her information on the state. Moreover, even if she had a priori information on the state (in the sense that Alice gets and encrypts many copies of the same unknown state) she wouldn’t be able to use any method of unknown quantum states reconstruction [@quantstateest]. She wouldn’t get any clue where to find the state because all are transformed (encrypted) to the maximally mixed state dwelling in the center of the sphere. Unfortunately, this is not our case. Here, if Alice encrypts many identical coherent states (howbeit unknown) Eve could in principle reconstruct in which part of the phase space the encryption had been carried out. Because we next suppose that encryption operations are publicly known (naturally not the secret key sequence itself) Eve then could be able to deduce the original state from many variant ciphers of the same state. So we will suppose that incoming state do not exhibit such statistics and, moreover, they are distributed equiprobably and randomly in the whole region of considering (i.e. within the circle of radius $r\leq b$) [^3]. Less restrictive requirement could be a distribution independent on the phase and changing only with the distance from the origin (rotationally invariant) what will be discussed at the end of this chapter.
Based on theses thoughts, we may finally define CVPQC. We call the object $\{\mathcal{B},P_{\mathcal{B}},\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits}),\openone_b^{\beta}\}$ as CVPQC where $\mathcal{B}$ is the set of all coherent states ${\mathop{\left|\beta\right>}\nolimits}$ ($|\beta|\leq b$) with distribution $P_{\mathcal{B}}$ around the origin of phase space, $\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})$ is the TPCP map defined in (\[map\_Alice\]) and $\openone_b^{\beta}$ is maximally mixed state centered around an input state ${\mathop{\left|\beta\right>}\nolimits}$ (displaced (\[integral\_0b\])). Let’s note that the definition of CVPQC is valid for all coherent mixed states of the general form $\varrho_\beta=\sum_ip_i({| \beta\rangle\!\langle \beta |})_i$ with $\sum_ip_i=1$ and $|\beta|_i\leq b,\,\forall i$ because of convexity of HS norm. The situation is similar as in [@approx_encryption] for the operator norm.
Considering the assumptions from the previous paragraph let’s focus on the problem of security on the channel. For our purpose we employ an integral version of (\[holevo\_bound\]). First note that we are now interested in the limiting case when Alice uses infinitely many encryption operations [^4]. More importantly, it remains to correctly answer the question posed above. That is, what actually calculate as the Holevo bound? We know that Eve will measure on the encrypted channel to get any information on incoming states (states before the encryption). To learn something about position of a coherent state before encryption Eve must try to distinguish among objects which preserve some information about input states location in phase space. In other words Eve has to choose a correct “encoding” $\sigma_i$ from Eq. (\[holevo\_bound\]) but now for a continuous index. Due to the used encryption operation (\[map\_Alice\]) (which in our case $N\to\infty$ transforms to $\openone_b^\beta$ as is proved in Appendix \[app\_exact\_value\]) a state $\openone_b^\beta$ is that object which preserves the information on placement of an incoming state ${\mathop{\left|\beta\right>}\nolimits}$ because is centered around it. Then, a continuous version of (\[holevo\_bound\]) reads $$\label{holevo_bound_int1}
\chi\bigl(\{P(\beta),\openone_b^\beta\}\bigr)
=S\bigl(\tilde\Lambda_{2b}\bigr)
-\int{\rm
d^2}\beta\,P(\beta)S\bigl(\openone_b^\beta\bigr),$$ where $\tilde\Lambda_{2b}$ is a total state leaving Alice’s apparatus (normalized Eq. (\[Lambda\])), $P(\beta)$ is an input distribution from the CVPQC definition and $\openone_b^\beta=D(\beta)\openone_bD^\dagger(\beta)$. Since $S(D(\beta)\openone_bD^\dagger(\beta))=S(\openone_b)$ for all $\beta$ it immediately follows from (\[holevo\_bound\_int1\]) $$\label{holevo_bound_int2}
\chi\left(\{P(\beta),\openone_b^\beta\}\right)
=S\bigl(\tilde\Lambda_{2b}\bigr)-S\bigl(\openone_b\bigr).$$
To calculate the Holevo bound (\[holevo\_bound\_int2\]) it remains to find the total state after the encryption procedure provided that Eve doesn’t know neither which particular coherent state ${\mathop{\left|\beta\right>}\nolimits}$ comes to Alice nor the key used for encryption. Thus, because the incoming distribution $P(\beta)\equiv P(x,\vartheta)=1$ (no need to normalize it here) is publicly known then Alice quite accidentally experiences $\openone_b$ also as a total [*incoming*]{} state. So, the state [*leaving*]{} Alice is in the form $$\begin{aligned}
\label{Lambda}
\Lambda_{2b}
& = \int{\rm d^2}\alpha D(\alpha)
\left(\int{\rm d^2}\beta\,{| \beta\rangle\!\langle \beta |}\right)D^\dagger(\alpha)
=\int_{0}^{b}\int_{0}^{2\pi}{\rm d^2}\beta
\left(
\int_{0}^{b}\int_{0}^{2\pi}{\rm d^2}\alpha\,
D(\alpha){| \beta\rangle\!\langle \beta |}D^\dagger(\alpha)
\right){\nonumber}\\
& =\int_{0}^{b}x{\rm d}x\int_{0}^{b}y{\rm d}y\int_{0}^{2\pi}{\rm
d}\varphi
\left(
\int_{0}^{2\pi}{\rm d}\vartheta\,{| \beta+\alpha\rangle\!\langle \beta+\alpha |}
\right)
=2\pi\sum_{n=0}^{\infty}\frac{{| n\rangle\!\langle n |}}{n!}
\int_{0}^{b}\int_{0}^{b}\int_{0}^{2\pi}{\rm d}x{\rm d}y{\rm d}\varphi
\ e^{-R^2}R^{2n+1}xy,\end{aligned}$$ where $R^2=x^2+y^2-2xy\cos\varphi\leq4b^2$, ${\rm d^2}\alpha=y{\rm
d}y{\rm d}\varphi$, ${\rm d^2}\beta=x{\rm d}x{\rm d}\vartheta$ and $D(\alpha)$ are encryption displacements (again uniformly distributed as the CVPQC definition dictates and without a normalization). Shortly said, by changing the order of integration we see that $\Lambda_{2b}$ is diagonal. Concretely, the term in the parentheses in the reordered integral (second row in (\[Lambda\])) is an ordinary coherent state ${\mathop{\left|\beta+\alpha\right>}\nolimits}$ ($r$ is its distance from the origin and abscissae formed by points in the phase space $(\alpha,\beta)$ and $(\beta,0)$ contain a relative angle $\varphi$) and is integrated out over the phase. This gives a diagonal matrix and next integrating just changes the statistics on the diagonal. Note that $\Lambda_{2b}$ “covers” the area of radius $2b$ in phase space. Unfortunately, it is difficult to obtain an analytical solution of (\[Lambda\]). Nevertheless, based on (\[Lambda\]) we may generally write (tilde indicates normalization) $\tilde\Lambda_{2b}=\sum_n\lambda^{(b)}_n{| n\rangle\!\langle n |}$, where $\lambda^{(b)}_n$ are needed to be determined numerically. Inserting $\tilde\Lambda_{2b}=\sum_n\lambda^{(b)}_n{| n\rangle\!\langle n |}$ and (\[integral\_0b\]) into Eq. (\[holevo\_bound\_int2\]) we find the desired Holevo bound what is depicted in Fig. \[fig\_holevo\_bound\]. It is interesting to note that the convergence of the Holevo bound is tightly connected with the energy constraint $\int{\mathop{{\mathrm{Tr}}_{}}}\left(\varrho_\beta a^\dagger
a\right){\rm d}P_{\mathcal{B}}\leq\bar n_b$ which is automatically satisfied through the finiteness of $b$ in our CVPQC. For a general distribution $P(\beta)$ it is satisfied due to the natural condition $\int{\rm d^2}\beta P(\beta)=1$. The importance of the constraint has been already recognized for capacities of bosonic channels [@bos_chan1; @bos_chan2; @bos_chan_lossy; @bos_gauss_chan_mem].
From Fig. \[fig\_holevo\_bound\] we see that there is no chance for Eve to find which coherent state actually passes on the channel. This is not so surprising considering the non-discrete character of the input distribution $P(\beta)$. But Eve cannot divide the appropriate area at least roughly to approximately devise the position of a particular encrypted state $\openone_b^\beta$ (and from this to derive the desired position of an incoming state ${\mathop{\left|\beta\right>}\nolimits}$). From the informational point of view the Holevo bound results could be interpreted such that for a given $b$ there doesn’t exist any optimal measurement enabling Eve to divide the circle of the radius $2b$ in more than $2^{\chi}$ sections and tells her in which one the encrypted state occurs.
Having fixed the input distribution $P(\beta)$ of coherent states coming to Alice another way how to next decrease the Holevo bound might be a different definition of an encryption distribution in Eq. (\[Lambda\]) where a uniform distribution is implicitly used. If this distribution was symmetrical (i.e. rotationally invariant, for example Gaussian one) then we would find Eq. (\[Lambda\]) diagonal as well and thus we could easily calculate (\[holevo\_bound\_int2\]). However, a big challenge is proving the optimality of such distribution function. Anyhow, it means that by minimizing the Holevo bound (\[holevo\_bound\_int2\]) within the context of the CVPQC definition we could tackle the previously mentioned problem of maximality of our mixed state by putting this more suitable distribution into Eq. (\[integral\_0b\]).
Conclusion
==========
In this work we have opened the problem of continuous variable encryption of unknown quantum states and introduced the concept of private quantum channels (CVPQC) into this area. For the start we have restricted ourselves on coherent states which belong to the important class of states with the Gaussian distribution function. A particular continuous variable private quantum channel was proposed and we have studied its properties. Firstly, it means that we have established the notion of so called maximally mixed state with regard to its non-discrete (continuous variable) nature. For this kind of mixture we were interested how many encryption operations are sufficient to consider an incoming coherent state to be secure. This quantity was determined by calculating the Hilbert-Schmidt distance between the mixture and the encrypted state for an arbitrary number of encryption operations. Next, we have studied the possibility of eavesdropping on the quantum channel. We have supposed that Eve is able to perform an optimal measurement to get the maximum information on the state (which is limited by the calculated Holevo bound) or is able to use some quantum state estimation methods. The second possibility (which requires Eve’s a priori information on the statistics of the states in the sense that she knows that Alice gets many copies of the same unknown coherent state to encrypt) restricts the CVPQC definition. The way how to avoid this kind of attack is the most desired direction of next research.
Beside the above mentioned topic we have addressed many more intriguing questions which can stimulate another research in this area and improve the existing protocol. Among many topics let’s name the problem of either universal distribution of incoming state or a universal distribution of encryption operations or both. This is tightly connected with the freedom in choice of the definition of maximally mixed state. We have seen that there exists a certain ambiguity in the definition of what we call here the maximally mixed state in phase space. Its relevance is measured by the accessible information (or eventually limited from above by the Holevo bound) Eve can get and the question is what kind of definition of the maximally mixed state is the most appropriate for a given incoming distribution of states. Together with this topics another generalization presents itself. It is the possibility to encrypt another Gaussian states, especially squeezed states.
The author is grateful to L. Mišta, R. Filip and J. Fiurášek for useful discussions in early stage of this work, M. Dušek for reading the manuscript and T. Holý for granting the computational capacity. The support from the EC project SECOQC (IST-2002-506813) is acknowledged.
{#app_exact_value}
Eq. (\[HSdist\_into\_traces\]) consists of three parts. Let’s calculate them step by step. If we rewrite (\[integral\_0b\]) in a more suitable form we get diagonal elements in the form $$\label{app_integral0b}
\openone_b(n,n)=\frac{1}{b^2\exp(b^2)}
\left(\sum_{m=n+1}^\infty\frac{b^{2m}}{m!}\right){| n\rangle\!\langle n |}.$$ Now, we may easily calculate $$\begin{aligned}
{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b^2\right)
& = \frac{1}{b^4\exp(2b^2)}
\sum_{n=1}^\infty\left(\sum_{m=n}^\infty\frac{b^{2m}}{ m!}\right)^2
= \frac{1}{b^4\exp(2b^2)}\left(\sum_{n=1}^\infty\frac{b^{4n}}{(n!)^2}n
+ 2\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{b^{4n+2k}}{ n!(n+k)!}n\right){\nonumber}\\
& = \frac{1}{b^4\exp(2b^2)}\left(\sum_{n=0}^\infty\frac{b^{4(n+1)}}{ n!(n+1)!}
+ 2\sum_{k=1}^\infty\sum_{n=0}^\infty\frac{b^{4(n+1)+2k}}{n!(n+k+1)!}\right)
= \frac{1}{b^2\exp(2b^2)}\left(I_1\left(2b^2\right)
+ 2\sum_{k=1}^\infty I_{k+1}\left(2b^2\right)\right){\nonumber}\\
& =
\frac{1}{b^2\exp(2b^2)}\left(\exp(2b^2)-I_0\left(2b^2\right)-I_1\left(2b^2\right)\right),\end{aligned}$$ where $I_s$ is a modified Bessel function of the first kind of order $s$. The last row is due to identity $$\label{bess_ident}
I_0(x)+2\sum_{k=1}^\infty I_k(x)=\exp(x).$$ For the calculation of the cross element ${\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\Phi_N\right)$ we use (\[app\_integral0b\]) again. But for the sake of clarity we will calculate only $p{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\varrho_p\right)$. The overall trace is given by summation ${\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\Phi_N\right)=1/M
\sum_{p=1}^Np{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\varrho_p\right)$ what follows from (\[mix\_on\_circle\]). From (\[mix\_on\_circle\_\_general\]) we get a general diagonal element $$\label{prhop}
p\varrho_p(n,n)=p\exp\left(-r_p^2\right)\frac{r_p^{2n}}{ n!}
{| n\rangle\!\langle n |}$$ and then $$\begin{aligned}
p{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\varrho_p\right)
&= \frac{1}{ b^2\exp(b^2)}\frac{p}{\exp(r_p^2)}
\sum_{n=0}^\infty\sum_{m=n+1}^\infty\frac{r_p^{2n}}{ n!}\frac{b^{2m}}{ m!}
= \frac{1}{ b^2\exp(b^2)}\frac{p}{\exp(r_p^2)}
\sum_{k=1}^\infty\sum_{n=0}^\infty\frac{b^{2(n+k)}}{ n!(n+k)!}r_p^{2n}{\nonumber}\\
&= \frac{1}{ b^2\exp(b^2)}\frac{p}{\exp(r_p^2)}\sum_{k=1}^\infty
\left(\frac{b}{ r_p}\right)^kI_k(2r_pb).\end{aligned}$$ Overall, we have $$\label{cross}
{\mathop{{\mathrm{Tr}}_{}}}\left(\openone_b\Phi_N\right)
=\frac{2}{ N(N+1)}\frac{1}{ b^2\exp(b^2)}
\sum_{p=1}^Np\exp\left(-r_p^2\right)\sum_{k=1}^\infty
\left(\frac{b}{ r_p}\right)^kI_k(2r_pb).$$ The third part can be written $$\label{Phi_n^2}
{\mathop{{\mathrm{Tr}}_{}}}\left(\Phi_N^2\right)=\left(\frac{2}{N(N+1)}\right)^2
{\mathop{{\mathrm{Tr}}_{}}}\left(\sum_{i,j=1}^{N}p_ip_j\varrho_{p_{i}}\varrho_{p_{j}}\right)$$ where particular summands have the form $$\label{rhoirhoj}
{\mathop{{\mathrm{Tr}}_{}}}\left(p_ip_j\varrho_{p_{i}}\varrho_{p_{j}}\right)
=\frac{p_ip_j}{\exp\left(r_{p_i}^2+r_{p_j}^2\right)}
\left(
I_0(2R_{ij})+2\sum_{k=1}^\infty I_{p_ip_jk}(2R_{ij})
\right),$$ what can be easily seen if we substitute (\[mix\_on\_circle\_\_general\]) into a slight generalization of (\[bess\_ident\]) $$\label{bess_ident2}
I_0(2xy)+2\sum_{k=1}^\infty I_k(2xy)=\exp(x^2)\exp(y^2),$$ where for our purpose $x=r_{p_i}=p_ib/N,y=r_{p_j}=p_jb/N,R_{ij}=p_ip_j(b/N)^2$.
{#app_guess}
When performing a limit transition it can be shown that $\lim_{N\to\infty}|\Phi_N-\openone_b|$=0. The only nonzero elements of $\Phi_N$ stays on its diagonal and are equal to diagonal elements of $\openone_b$. Let’s demonstrate it on first diagonal element ($n=0$). From (\[mix\_on\_circle\_\_general\]) and (\[mix\_on\_circle\]) we see that $$\label{Phi_N00}
\Phi_N(0,0)=\frac{2}{ N(N+1)}\sum_{p=1}^Np\exp(-r_p^2){| 0\rangle\!\langle 0 |}
=\frac{2}{ N(N+1)}\sum_{p=1}^N\sum_{n=0}^\infty(-1)^n{b^{2n}}{ n!}
\frac{p^{2n+1}}{ N^{2n}}{| 0\rangle\!\langle 0 |}.$$ Assuming that $$\label{powersum}
\sum_{p=1}^Np^d=\frac{1}{
d+1}\sum_{p=1}^{d+1}(-1)^{d+1-p}\binom{d+1}{p}B_{d+1-p}N^p,$$ where $B_n$ are Bernoulli numbers, inserting the highest polynomial $\frac{N^{d+1}}{ d+1}$ (lower polynomials subsequently tends to zero) from (\[powersum\]) into (\[Phi\_N00\]) and finally letting $N\to\infty$ we get (omitting ketbra) $$\label{PhiN00_trans}
\lim_{N\to\infty}\Phi_N(0,0)=2\lim_{N\to\infty}\sum_{n=0}^\infty
(-1)^n\frac{b^{2n}}{ n!}\frac{N^{2(n+2)}}{ N^{2n+1}(N+1)}\frac{1}{2(n+1)}
=\sum_{n=1}^\infty(-1)^n\frac{b^{2n}}{ n!}={1-\exp(-b^2)}{
b^2}\equiv\openone_b(0,0)$$ as can be seen from (\[app\_integral0b\]). In this way we could continue for all diagonal elements of $\Phi_N(n,n)$. But turn our attention elsewhere. If we realize that a general diagonal element is of the form $$\label{PhiNnn}
\Phi_N(n,n)=\frac{2}{ N(N+1)}\frac{1}{n!}
\sum_{p=1}^Np\exp(-r_p^2)r_p^{2n}{| n\rangle\!\langle n |},$$ we see that the expression for the HS distance (\[HSdist\_explicit\]) can be simplified in the following way $$\begin{aligned}
\label{HS_simple}
D_{HS}^2(\openone_b,\Phi_N)& =|\openone_b-\Phi_N|^2\\
& +
2\left(\frac{2}{N(N+1)}\right)^2
\left[\sum_{k=1}^N\frac{k^2}{\exp\left(2r_k^2\right)}
\left(
\sum_{n=1}^\infty I_{kn}\left(2r_k^2\right)
\right)
+\sum_{{\genfrac{}{}{0pt}{}{k_1,k_2=1}{k_1\not=k_2}}}^N\frac{k_1k_2}{\exp\left(r_{k_1}^2+r_{k_2}^2\right)}
\left(
\sum_{n=1}^\infty I_{k_1k_2n}(2R_{12})
\right)
\right].{\nonumber}\end{aligned}$$ This form will help us in deriving (\[HSdist\_guess\]) $$\label{guess}
|\openone_b-\Phi_N|=\sum_{k=0}^\infty|\openone_b(k,k)-\Phi_N(k,k)|
\approx\sum_{k=0}^\infty\left|\openone_b(k,k)\left(1-\frac{N}{ N+1}\right)\right|
=\frac{1}{ N+1}$$ and hence $|\openone_b-\Phi_N|^2\approx\left(\frac{1}{N+1}\right)^2$. We again put the highest polynomial $\frac{N^{d+1}}{d+1}$ into $\Phi_N(k,k)$ and used the normalization condition $\sum_{k=0}^\infty\openone_b(k,k)=1$. Second part of (\[HS\_simple\]) tends to zero even faster. We can see it with the help of (\[bess\_ident2\]) from which follows that $\sum_{n=1}^\infty
I_{kn}\left(2r_k^2\right)\ll\exp\left(2r_k^2\right)$ especially for higher $k$ (and, of course, similarly for the second summand in square brackets of expression (\[HS\_simple\])). From these considerations (\[HSdist\_guess\]) follows and, not surprisingly, asymptotically approaches the exact value.
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[^1]: Throughout this paper, all logarithms have base two.
[^2]: In fact, Eq. (\[integral\_0b\]) belongs to the class of bosonic channels later discussed in section \[sec\_threats\]. We will address the problem of the “measure of maximality” of the mixed state in the context of the calculated Holevo bound on the channel.
[^3]: To avoid misunderstanding we distinguish in this chapter between distribution and statistics. As usually, distribution means a probability of occurrence for incoming states or, eventually, for encryption operations. By contrast, statistics means correlation or better relationship between incoming states what may be publicly known, e.g. information that several incoming states will be the same. This has no influence on the distribution.
[^4]: Of course, in reality for every ${\mathop{\left|\beta\right>}\nolimits}$ Alice produces a “finite” mixture $\mathcal{E}_N({\mathop{\left|\beta\right>}\nolimits})$ from (\[map\_Alice\]) which should be as close as possible to $\openone_b^\beta=D(\beta)\openone_bD^\dagger(\beta)$. For the examination how these states are close to each other and its connection to the number of bits of the key we have derived expression (\[HSdist\_guess\]).
|
---
abstract: 'We propose a new approach to calculate perturbatively the effects of a particular deformed Heisenberg algebra on energy spectrum. We use this method to calculate the harmonic oscillator spectrum and find that corrections are in agreement with a previous calculation. Then, we apply this approach to obtain the hydrogen atom spectrum and we find that splittings of degenerate energy levels appear. Comparison with experimental data yields an interesting upper bound for the deformation parameter of the Heisenberg algebra.'
address: 'Université de Mons-Hainaut, B-7000 Mons, BELGIQUE'
author:
- 'F. Brau[^1]'
title: Minimal Length Uncertainty Relation and Hydrogen Atom
---
Introduction {#sec:intro}
============
Study of modified Heisenberg algebra, by adding certain small corrections to the canonical commutation relations, arouse a great interest for some years (see for example [@kemp95; @hinr96; @kemp97b; @kemp97; @brou99]). These modifications yield new short distance structure characterized by a finite minimal uncertainty $\Delta x_0$ in position measurement. The existence of this minimal observable length has been suggested by quantum gravity and string theory [@gros88; @amat89; @magg93; @amel97; @haro98]. In this context, the new short distance behavior would arise at the Planck scale, and $\Delta x_0$ would correspond to a fundamental quantity closely linked with the structure of the space-time [@kemp98]. This feature constitutes a part of the motivation to study the effects of this modified algebra on various observables.
Recently, it has been suggested that this formalism could be also used to describe, as an effective theory, non-pointlike particles, e.g. hadrons, quasi-particles or collective excitations [@kemp97b]. In this case, $\Delta x_0$ is interpreted as a parameter linked with the structure of particles and their finite size. In the work [@kemp97b] the $d$-dimensional isotropic harmonic oscillator was solved, in the context of a nonvanishing $\Delta x_0$, with a special interest to the 3-dimensional case. This calculation shows that splittings of usual degenerate energy levels appear, leaving only the degeneracy due to the independence of the energy on the azimuthal quantum number, $m$. It has been also indicated that application to the hydrogen atom should yield the relation between the scale of a non-pointlikeness of the electron and the scale of the caused effects on the hydrogen spectrum. Indeed, the high precision of the experimental data for the transition $1S-2S$ [@udem97], for example, can yield an interesting upper bound for the possible, in the sense studied here, finite size of the electron.
The purpose of this work is to continue to investigate whether the Ansatz concerning the deformation of the Heisenberg algebra, with suitably adjusted scale, may also serve for an effective low energy description of non-pointlike particles. In this way, we calculate corrections to the hydrogen spectrum using the minimally modified Heisenberg algebra, i.e. which preserves the commutation relations between position operators. To perform this calculation we propose a new approach which allow to solve the Schrödinger equation in the position representation. This method leads to the correct harmonic oscillator spectrum found in Ref. [@kemp97b]. Application to hydrogen atom shows that splittings of the usual degenerate energy levels are also present and that these corrections cannot be seen experimentally if $\Delta x_0$ is smaller than $0.01$ fm.
Method {#sec:theory}
======
The modified Heisenberg algebra studied here, as it has been done in Ref. [@kemp97b], is defined by the following commutation relations ($\hbar
= c = 1$) $$\begin{aligned}
\label{eq1}
\nonumber
\left[\hat{X}_i,\hat{P}_j\right] &=& i\left(\delta_{ij}+\beta \delta_{ij}
\hat{P}^2
+\beta' \hat{P}_i \hat{P}_j\right), \\
\left[\hat{P}_i,\hat{P}_j\right] &=& 0,\end{aligned}$$ where $\hat{P}^2 = \sum_{i=1}^{3} \hat{P}_i \hat{P}_i$ and where $\beta,
\beta' > 0$ are considered as small quantities of the first order. In this paper, we study only the case $\beta'=2\beta$, which leaves the commutation relations between the operators $\hat{X}_i$ unchanged [@kemp97b], i.e. $\left[\hat{X}_i,\hat{X}_j\right] = 0$. This constitutes the minimal extension of the Heisenberg algebra and is thus of a special interest.
To calculate a spectrum for a given potential, we must find a representation of the operators $\hat{X}_i$ and $\hat{P}_i$, involving position variables $x_i$ and partial derivatives with respect to these position variables, which satisfies Eqs. (\[eq1\]), and solve the corresponding Schrödinger equation: $$\label{eq2}
\left[ \frac{\hat{P}^2}{2m} + V\left(\vec{\hat{X}}\right)\right]\,
\Psi(\vec{x}\,) = E\, \Psi(\vec{x}\,).$$ It is straightforward to verify that the following representation fulfill the relations (\[eq1\]), in the first order in $\beta$, $$\begin{aligned}
\label{eq3}
\nonumber
\hat{X}_i\ \Psi(\vec{x}\,) &=& x_i \Psi(\vec{x}\,), \\
\hat{P}_i\, \Psi(\vec{x}\,) &=& p_i \left(1+\beta \vec{p}\,^2\right)
\Psi(\vec{x}\,) \quad \rm{with} \quad p_i=\frac{1}{i}
\frac{\partial}{\partial x_i}.\end{aligned}$$ Neglecting terms of order $\beta^2$, the Schrödinger equation (\[eq2\]) takes the form $$\label{eq4}
\left[ \frac{\vec{p}\,^2}{2m} +\frac{\beta}{m} \vec{p}\,^4 +
V(\vec{x}\,)\right]\,
\Psi(\vec{x}\,) = E\, \Psi(\vec{x}\,).$$ This is the ordinary Schrödinger equation with an additional term proportional to $\vec{p}\,^4$. As this correction is assumed to be small, we calculate its effects on energy spectra in the first order of perturbations. The evaluation of the spectrum to the first order in the deformation parameter $\beta$ leads to $$\label{eq5}
E_{k}=E_{k}^0+\Delta E_{k},$$ where $k$ denotes the set of quantum numbers which labels the energy level, and where $\Delta E_{k}$ are the eigenvalues of the matrix $$\label{eq6}
\frac{\beta}{m} \langle\Psi_{k}^0(\vec{x}\,)|\,\vec{p}\,^4\,
|\Psi_{k'}^0(\vec{x}\,)
\rangle \equiv
\frac{\beta}{m} \langle k| \vec{p}\,^4 |k'\rangle,$$ where $\Psi_{k}^0(\vec{x}\,)$ are solutions of (\[eq4\]) with $\beta=0$. This matrix is computed with all the wave functions corresponding to the unperturbed energy level $E_{k}^0$. This is a $g \times g$ matrix where $g$ is the multiplicity of the state $E_{k}^0$ considered. In general, $\Delta E_{k}$ takes $f$ ($f \leq g$) different values which removes the degeneracy of some energy levels. For an arbitrary interaction $V(\vec{x}\,)$ used in the Schrödinger equation, the matrix (\[eq6\]) is non-diagonal. But, since we know the action of $\vec{p}\,^2$ (from Eq. (\[eq4\])) on the unperturbed wave functions, the expression of the matrix elements, for a central potential, can be written as $$\label{eq6b}
4\beta m \left(\left(E^0_{n,\ell}\right)^2 \delta_{n
n'}-(E_{n,\ell}^0+E_{n',\ell}^0)\langle
n\ell m| V(r) |n'\ell m\rangle +\langle n\ell m| V(r)^2 |n'\ell m\rangle
\right)\, \delta_{\ell \ell'} \delta_{m m'},$$ and, in the cases studied here, there are no degenerate states with equal values of the angular momentum $\ell$ and the azimuthal quantum number $m$ which have different values of radial quantum number $n$. Thus the matrix (\[eq6\]) is diagonal and the correction to the spectrum can be written a $$\label{eq7}
\Delta E_{n,\ell}= 4\beta m \left(\left(E_{n,\ell}^0\right)^2-2E_{n,\ell}^0\,
\langle n\ell m|
V(r)|n\ell m
\rangle+\langle n\ell m|V(r)^2|n\ell m \rangle\right).$$ This nice relation can be simplified if one considers power-law central potential, $V(r)\sim r^p$. In this case, the virial theorem gives $$\label{eq8}
\langle n\ell m| V(r)|n\ell m \rangle= \frac{2}{p+2}\, E_{n,\ell}^0,$$ which leads to the following form for the expression of the energy level shift in the first order in $\beta$: $$\label{eq9}
\Delta E_{n,\ell}= 4\beta m \left(\left(E_{n,\ell}^0\right)^2
\left(\frac{p-2}{p+2}\right)+\langle
n \ell m|V(r)^2|n\ell m \rangle\right).$$ This simple expression will allow us to find the correction of the harmonic oscillator and hydrogen spectra just by calculating the mean value of the square of the potential.
Harmonic Oscillator {#subsec:firstf}
===================
For this potential, the energy level shift is only given by the mean value of the square of the potential. The normalized unperturbed wave function of the harmonic oscillator reads $$\label{eq10}
\Psi^0_{n \ell m}(\vec{r}\,)= \lambda^{3/2} \sqrt{\frac{2\
n!}{\Gamma(n+\ell+3/2)}}\ (\lambda r)^\ell\, e^{-(\lambda r)^2/2}\,
L^{\ell+1/2}_{n}\left((\lambda r)^2\right)\, Y_{lm}(\theta,\varphi),$$ where $\lambda=\sqrt{m\omega}$ and $L^{\alpha}_n(x)$ are Laguerre polynomials [@grad80 p. 1037]. $n$ is the radial quantum number. Using the change of variable $x=(\lambda
r)^2$, the energy shift is found to be: $$\label{eq11}
\Delta E_{n,\ell}= \frac{4\beta m (n!) k^2}{\lambda^4 \Gamma(n+\ell+3/2)} \
\int_0^{\infty} x^{\ell+5/2}\, e^{-x}\,
\left[L^{\ell+1/2}_{n}(x)\right]^2\,dx,$$ where $2k=m\omega^2$ is the strength of the oscillator force. The calculation of the remaining integral is straightforward. Knowing the following relations concerning the Laguerre polynomials [@grad80 p. 1037, p. 844] $$\begin{aligned}
\label{eq12}
L^{\alpha-1}_{n}(x)&=&L^{\alpha}_{n}(x)-L^{\alpha}_{n-1}(x),\\
\label{eq13}
\int_0^{\infty} e^{-x}\,
x^{\alpha}\,L^{\alpha}_{n}(x)\,L^{\alpha}_{m}(x)\, dx &=&
\frac{\Gamma(\alpha+n+1)}{n!}\, \delta_{nm},\end{aligned}$$ we obtain the expression of the harmonic oscillator spectrum for the modified Heisenberg algebra (\[eq1\]): $$\label{eq14}
E_{n,\ell}= \omega (2n+\ell+3/2)+ (\Delta x_0)^2 \frac{m\omega^2}{5}
(6n^2+9n+6nl+\ell^2+4\ell+15/4),$$ where $\Delta x_0=\sqrt{5\beta}$. This formula reproduces exactly the splittings calculated in Ref. [@kemp97b] using another approach. Because the dependence on quantum numbers of the correction term is not of the form $f(2n+\ell)$, we obtain splittings of degenerate levels. But the energy does not depend on the azimuthal quantum number $m$ and each level remains $(2\ell+1)$-fold degenerate.
This example shows the usefulness of this approach which provides, with simple calculations, an analytical expression of the energy shift. The main interest of this method is that it can easily be used to solve analytically or numerically other problems, such as the Coulomb problem which is solved in the next section.
Hydrogen Atom {#subsec:secondf}
=============
As we mentioned in the Introduction, the evaluation of corrections of the energy spectrum can provide information concerning, in the sense studied here, an assumed finite size of electrons. The method used here to describe non-pointlike particles neglects the internal structure degree of freedom. But obviously these effects have much smaller order of magnitude and thus can be omitted.
The normalized unperturbed wave function of the hydrogen atom reads $$\label{eq15}
\Psi^0_{n \ell m}(\vec{r}\,)= (2\gamma_n)^{3/2}
\sqrt{\frac{(n-\ell-1)!}{2n(n+\ell)!}}\ (2\gamma_n r)^\ell\, e^{-\gamma_n r}\,
L^{2\ell+1}_{n-\ell-1}(2\gamma_n r)\, Y_{lm}(\theta,\varphi),$$ where $\gamma_n=m\alpha/n$ and $\alpha$ is the fine structure constant. $n$ is the principal quantum number and $\ell$ varies between $0$ and $n-1$. The change of variable $x=2\gamma_n r$ allow to write the energy shift as $$\label{eq16}
\Delta E_{n,\ell}= -12\beta m \left(E^0_{n,\ell}\right)^2+8\beta m \gamma_n^2
\alpha^2\,
\frac{(n-\ell-1)!}{n(n+\ell)!} \
\int_0^{\infty} x^{2\ell}\, e^{-x}\,
\left[L^{2\ell+1}_{n-\ell-1}(x)\right]^2\,dx.$$ Like for the harmonic oscillator problem, the evaluation of this integral is quite simple. Indeed, using the following relation for Laguerre polynomials [@grad80 p. 1038] $$\label{eq17}
\sum_{m=0}^{n} L_m^{\alpha}(x) =L_n^{\alpha+1} (x),$$ with the relation (\[eq13\]) and the following summation formula $$\label{eq18}
\sum_{p=0}^{b} \frac{(p+a)!}{p!} =\frac{(a+b+1)!}{(1+a) b!},$$ the expression of the hydrogen spectrum, in the first order in the deformation parameter $\beta$, reads $$\label{eq19}
E_{n,\ell}=-\frac{m\alpha^2}{2n^2}+\left(\Delta x_0\right)^2 \, \frac{m^3
\alpha^4}{5}\,
\frac{(4n-3(\ell+1/2))}{n^4 (\ell+1/2)}.$$ This formula shows that the corrections to the spectrum are always positive. The value of this additional term is maximum for the ground state and for each value of $n$, the maximal contribution is obtained for $\ell=0$ levels. Like in the harmonic oscillator case, the correction term, which depends explicitly on $\ell$, lifts the degeneracy of energy levels which remain, however, $(2\ell+1)$-fold degenerate.
The accuracy concerning the measurement of the frequency of the radiation emitted during the transition $1S-2S$ is about 1 kHz [@udem97]. Thus the energy difference between this two levels is determined with a precision about $10^{-12}$ eV. Then, if we suppose that effects of finite size of electrons cannot yet be seen experimentally, we find $$\label{eq20}
\Delta x_0 \leq 0.01\ \rm{fm}.$$ But corrections calculated here could already play a role in the theoretical description of the hydrogen atom since the accuracy of theoretical calculations is less good than the precision of experimental data. The main theoretical error is the determination of the proton charge radius. Thus, at this moment, confrontation between experimental data and standard theoretical calculations cannot exclude the effects studied in this paper.
Nevertheless, the upper bound (\[eq20\]) seems to be reasonable. Moreover, a naive argument can give an order of magnitude of an “experimental" upper bound for the finite size of the electron. Indeed, a lower bound for the mass of an excited state of the electron is about 85 GeV [@pdg98]. Thus a photon with an energy of about 85 GeV cannot excited an electron. In a first approximation, this means that the resolution obtained with this photon is not sufficient to detect a finite size of electrons. The wave length of a such photon could constitute an upper bound for the size of electrons, $$\label{eq21}
\Delta x_0 \leq \lambda \sim 0.015\ \rm{fm}.$$ This naive argument applied to the nucleon and its first radial excitation N(1440) yields a size of about 2.5 fm which is the correct order of magnitude.
Thus in a (very?) near future, with improvement of the accuracy of experimental data and above all improvement of the precision of standard theoretical calculations, it could be possible either to lower down the upper bound (\[eq20\]) or detect the existence of a non-vanishing $\Delta x_0$.
Summary {#sec:summary}
=======
We have proposed a new formulation of the Schrödinger equation which takes into account the deformation of the Heisenberg algebra in the first order in the deformation parameter $\beta$. This modified algebra introduces a minimal observable length in the uncertainty relations. It has been proposed in Ref. [@kemp97b] that this framework could be used to describe non-pointlike particles as an effective low energy theory, neglecting their internal structure degree of freedom. The minimal length $\Delta x_0$ would be then linked with the non-pointlikeness of particles.
In Sec. \[subsec:firstf\], we have calculated, with the new approach, the corrections to the harmonic oscillator spectrum which are in agreement with those derived in a previous calculation using another approach [@kemp97b]. Note that this method can be generalized to other dimensions. In particular, we have verify that the spectrum of the 1-dimensional harmonic oscillator is in agreement with that found in Ref. [@kemp95]. Moreover, the wave function in the position space can also be calculated, in the first order in $\beta$, just as various observables associated to the systems studied.
In Sec. \[subsec:secondf\], we have used this method to obtained the corrections to the hydrogen atom spectrum. Comparison with the experimental data for the transition $1S-2S$ [@udem97] yields a plausible upper bound for the non-pointlikeness $\Delta x_0$ of the electron which is about $0.01$ fm.
The formulation of the Schrödinger equation proposed here could prove to be very useful to study properties of some systems and their various associated observables in the context of the deformed Heisenberg algebra studied here.
We thank Professor F. Michel for stimulating discussions, and Professor Y. Brihaye for reading the manuscript. We would like to thank IISN for financial support.
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[^1]: E-mail: fabian.brau@umh.ac.be
|
[**Comment on “Dislocation Structure and Mobility in hcp $^4$He"**]{}\
In their Letter [@Borda], Borda, Cai, and de Koning report the results of [*ab initio*]{} simulations of dislocations responsible for the giant plasticity [@Balibar]. The authors claim key insights into the recent interpretations of (i) the giant plasticity and (ii) the mass flow junction experiments. The purpose of this Comment is clarifying the role of dislocations in the mass flow in conjunction with explaining that the part (ii) of the claim is misleading.
Borda [*et al.*]{} find that their dislocations do not have superfluid cores. This fact, however, is [*not*]{} crucial for the interpretation of the mass supertransport, including the effect of giant isochoric compressibility (aka the syringe effect) [@Hallock2009]. Furthermore, the fact is not even new.
The first-principle theoretical results showing that certain (!) screw and edge dislocations feature superfluid cores were reported in Refs. [@screw] and [@sclimb], respectively. Also, the behavior of the edge dislocation of Ref. [@sclimb] was found to be consistent with the phenomenon of superclimb. In its turn, the superclimb remains the only known underlying mechanism behind the syringe effect. Apart from being fundamentally interesting on its own, the syringe effect is central for the liquidless supertransport setups [@Beamish2016]. Thus, any theory aiming at explaining the supertransport through solid must account for the syringe effect too. The results of Ref. [@screw; @sclimb] provide a consistent, and up to now unique, first-principle basis for interpreting all known supertransport-related phenomena in solid 4 [@Hallock2009; @Hallock2012; @Hallock2014; @Beamish2016].
As is known, dislocations are characterized by orientation of their core and the Burgers vector. The dislocations studied in Ref. [@Borda]—with the core and Burgers vector both [*along*]{} basal plane—are different from those found to have superfluid core [@screw; @sclimb]—with the Burgers vector [*perpendicular*]{} to the basal plane and core either perpendicular to the plane [@screw] or along the plane [@sclimb]. Hence, the statement that (Qt) “the interpretation of recent mass flow experiments in terms of a network of 1D Luttinger-liquid systems in the form of superfluid dislocation cores does not involve basal-plane dislocations" made in Ref. [@Borda] is misleading.
Moreover, that cores of dislocations with the Burgers vector along basal plane (studied in Ref. [@Borda]) are not superfluid has been emphasized in Refs. [@GB; @screw; @stress]: in the caption to Fig. 6 in Ref. [@GB]; on p.1 at the end of the 1st and beginning of the 2nd column and on p.3 of Ref. [@screw], (Qt) “\[In the case of edge dislocations, this protocol leads to an insulating ground state.\]"; in the last paragraph on p.3 of Ref. [@stress]. Likewise, the effect of splitting into partials—claimed in Ref. [@Borda] as a new and crucial observation—has been reported in Refs. [@stress; @sclimb] for the edge dislocations of both types—with (in the section “Numerical results" on p.3 of Ref. [@sclimb]) and without superfluid cores (in the the last paragraph on p.3 of Ref. [@stress]). Furthermore, the energy of the structural fault found in Ref. [@sclimb] to be much smaller than any other typical energy scale in solid 4 implies large splitting of dislocations with core in the basal plane. Thus, Ref. [@Borda] neither negates the results of Refs. [@GB; @screw; @stress; @sclimb] nor provides new insights into superfluidity of dislocations.
Finally, it is worth noting that the observation of no superfluidity in Ref. [@Borda] lacks not only novelty but, most likely, also the control of the numerical data. While dealing with large systems, the authors do not use worm updates [@WA]. In such a case, PIMC algorithm is known to be notoriously prone to non-ergodicity in the worldline winding number space, so that the absence of macroscopic permutation cycles could merely reflect the non-ergodicity of the scheme rather than the absence of superfluidity.
We thank Nikolay Prokof’ev for useful discussions and acknowledge support from the National Science Foundation under the grants PHY-1314469 and PHY-1314735.\
[A. B. Kuklov$^1$ and B. V. Svistunov$^2$\
$^1$Department of Engineering Science and Physics, CUNY, Staten Island, NY 10314.\
$^2$Department of Physics, University of Massachusetts, Amherst, MA 01003.]{}
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---
abstract: 'On the basis of Dupont’s work, we exhibit a cocycle in the simplicial de Rham complex which represents the Chern character. We also prove the related conjecture due to Brylinski. This gives a way to construct a cocycle in a local truncated complex.'
author:
- Naoya Suzuki
title: |
The Chern Character in the Simplicial\
de Rham Complex
---
Introduction
============
It is well-known that there is one-to-one correspondence between the characteristic classes of $G$-bundles and the elements in the cohomology ring of the classifying space $BG$. So it is important to investigate $H^*(BG)$ in research on the characteristic classes. However, in general $BG$ is not a manifold so we can not adapt the usual de Rham theory on it. To overcome this problem, a total complex of a double complex ${\Omega}^{*} (NG(*)) $ which is associated to a simplicial manifold $\{ NG(*) \}$ is often used. In brief, $\{ NG(*) \}$ is a sequence of manifolds $\{ NG(p) =G^p \}_{ p=0,1, \cdots }$ together with face operators ${\varepsilon}_{i} : NG(p) \rightarrow NG(p-1) $ for $ i= 0, \cdots , p $ satisfying relations ${\varepsilon}_{i}{\varepsilon}_{j} ={\varepsilon}_{j-1}{\varepsilon}_{i}$ for $i<j$ (The standard definition also involves degeneracy operators but we do not need them here). The cohomology ring of ${\Omega}^{*} (NG(*)) $ is isomorphic to $H^*(BG)$ so we can use this complex as a candidate of the de Rham complex on $BG$.
In [@Dup], Dupont introduced another double complex $A^{*,*}(NG)$ on $NG$ and showed the cohomology ring of its total complex $A^*(NG)$ is also isomorphic to $H^*(BG)$. Then he used it to construct a homomorphism from $I^* (G)$, the $G$-invariant polynomial ring over Lie algebra $\mathcal{G}$, to $H^*(BG)$ for a classical Lie group $G$.
The images of this homomorphism in ${\Omega}^{*} (NG(*)) $ are called the Bott-Shulman-Stasheff forms. The main purpose of this paper is to exhibit these cocycles precisely when they represent the Chern characters.
In addition, we also show that the conjecture due to Brylinski in [@Bry] is true. This gives a way to construct a cocycle in a local truncated complex $[\sigma_{<p}\Omega^{*} _{\rm loc}(NG)]$ whose cohomology class is mapped to the cohomology class of the Bott-Shulman-Stasheff form in a local double complex by a boundary map. His original motivation to introduce these complexes and the conjecture is to study the local cohomology group of the gauge group ${\rm Map}(X,G)$ and the Lie algebra cohomology of its Lie algebra. Actually, as a special case $X=S^1$, he constructed the standard Kac-Moody $2$-cocycle for a loop Lie algebra by using the cocycle in the local truncated complex $[\sigma_{<2}\Omega^{3} _{\rm loc}(NG)]$.
The outline of this paper is as follows. In section 2, we briefly recall the universal Chern-Weil theory due to Dupont. In section 3, we obtain the Bott-Shulman-Stasheff form in ${\Omega}^{*} (NG(*)) $ which represents the Chern character ${\rm ch}_p $. In section 4, we introduce some result about the Chern-Simons forms. In section 5, we prove Brylinski’s conjecture.
Review of the universal Chern-Weil Theory
=========================================
In this section we recall the universal Chern-Weil theory following [@Dup2]. For any Lie group $G$, we have simplicial manifolds $NG$, $N \bar{G}$ and simplicial $G$-bundle $\gamma : N \bar{G} \rightarrow NG$ as follows:\
$NG(q) = \overbrace{G \times \cdots \times G }^{q-times} \ni (h_1 , \cdots , h_q ) :$\
face operators ${\varepsilon}_{i} : NG(q) \rightarrow NG(q-1) $ $${\varepsilon}_{i}(h_1 , \cdots , h_q )=\begin{cases}
(h_2 , \cdots , h_q ) & i=0 \\
(h_1 , \cdots ,h_i h_{i+1} , \cdots , h_q ) & i=1 , \cdots , q-1 \\
(h_1 , \cdots , h_{q-1} ) & i=q.
\end{cases}$$
$N \bar{G} (q) = \overbrace{ G \times \cdots \times G }^{q+1 - times} \ni (g_1 , \cdots , g_{q+1} ) :$\
face operators $ \bar{\varepsilon}_{i} : N \bar{G}(q) \rightarrow N \bar{G}(q-1) $ $$\bar{{\varepsilon}} _{i} (g_0 , \cdots , g_{q} ) = (g_0 , \cdots , g_{i-1} , g_{i+1}, \cdots , g_{q}) \qquad i=0,1, \cdots ,q.$$
We define $\gamma : N \bar{G} \rightarrow NG $ as $ \gamma (g_0 , \cdots , g_{q} ) = (g_0 {g_1}^{-1} , \cdots , g_{q-1} {g_{q}}^{-1} )$.\
For any simplicial manifold $X = \{ X_* \}$, we can associate a topological space $\parallel X \parallel $ called the fat realization. Since any $G$-bundle $\pi : E \rightarrow M$ can be realized as a pull-back of the fat realization of $\gamma $, $\parallel \gamma \parallel$ is the universal bundle $EG \rightarrow BG$ [@Seg].
Now we construct a double complex associated to a simplicial manifold.
For any simplicial manifold $ \{ X_* \}$ with face operators $\{ {\varepsilon}_* \}$, we define a double complex as follows: $${\Omega}^{p,q} (X) := {\Omega}^{q} (X_p)$$ Derivatives are: $$d' := \sum _{i=0} ^{p+1} (-1)^{i} {\varepsilon}_{i} ^{*} , \qquad d'' := (-1)^{p} \times {\rm the \enspace exterior \enspace differential \enspace on \enspace }{ \Omega ^*(X_p) }.$$
For $NG$ and $N \bar{G} $ the following holds [@Bot2] [@Dup2] [@Mos].
There exist ring isomorphisms $$H({\Omega}^{*} (NG)) \cong H^{*} (BG ), \qquad H({\Omega}^{*} (N \bar{G})) \cong H^{*} (EG ).$$ Here ${\Omega}^{*} (NG)$ and ${\Omega}^{*} (N \bar{G})$ mean the total complexes.
For example, the derivative $d'+d'':\Omega ^{p}(NG) \rightarrow \Omega ^{p+1}(NG)$ is given as follows:
$$\begin{CD}
{\Omega}^{p} (G ) \\
@AA{-d}A \\
{\Omega}^{p-1} (G )@>{{\varepsilon}_{0} ^{*} - {\varepsilon}_{1} ^{*} +{\varepsilon}_{2} ^{*} }>>{\Omega}^{p-1} (NG(2))\\
@.@AAdA\\
@.{\Omega}^{p-2} (NG(2))\\
@.@. \ddots \\
@.@.@.{\Omega}^{1} (NG(p)) \\
@.@.@.@AA{(-1)^p d }A\\
@.@.@.{\Omega}^{0} (NG(p))@>{ \sum _{i=0} ^{p+1} (-1)^{i} {\varepsilon}_{i} ^{*}}>> {\Omega}^{0} (NG(p+1))
\end{CD}$$
Let $\pi : P \rightarrow M$ be a principal $G$-bundle and $\{ g_{\alpha \beta}:U_{\alpha \beta} \rightarrow G \}$ be the transition functions of it. Then we can pull-back the cocycle in $\Omega ^*(NG)$ to the Čech-de Rham complex of $M$ by $\{g_{\alpha \beta} \}$. When $\kappa$ is the characteristic class which corresponds to the cocycle in $\Omega ^*(NG)$, the image of $ g_{\alpha \beta} ^*$ in $H^* _{\check{C}ech-de Rham} (M)$ is the characteristic class $\kappa (P)$ of $\pi : P \rightarrow M$. For more details, see for instance [@Mos].
There is another double complex associated to a simplicial manifold.
A simplicial $n$-form on a simplicial manifold $ \{ {X}_{p} \} $ is a sequence $ \{ {\phi}^{(p)} \}$ of $n$-forms ${\phi}^{(p)}$ on ${\Delta}^{p} \times {X}_{p} $ such that $${({\varepsilon}^{i} \times id )}^{*} {\phi}^{(p)} = {(id \times {\varepsilon}_{i} )}^{*} {\phi}^{(p-1)}.$$ Here ${\varepsilon}^{i}$ is the canonical $i$-th face operator of ${\Delta}^{p}$.\
[Let]{} $A^{k,l} (X)$ be the set of all simplicial $(k+l)$-forms on ${\Delta}^{p} \times {X}_{p} $ which are expressed locally of the form $$\sum { a_{ i_1 \cdots i_k j_1 \cdots j_l } (dt_{i_1 } \wedge \cdots \wedge dt_{i_k } \wedge dx_{j_1 } \wedge \cdots \wedge dx_{j_l })}$$ where $(t_0, t_1, \cdots, t_p)$ are the barycentric coordinates in ${\Delta}^{p} $ and $x_j $ are the local coordinates in $ {X}_{p} $. We call these forms $(k,l)$-form on ${\Delta}^{p} \times {X}_{p} $ and define derivatives as: $$d' := {\rm the \enspace exterior \enspace differential \enspace on \enspace } {\Delta}^{p}$$ $$d'' := (-1)^{k} \times {\rm the \enspace exterior \enspace differential \enspace on \enspace } {X_p }.$$ Then $(A^{k,l} (X) , d' , d'' )$ is a double complex.\
Let $A^{*} (X)$ denote the total complex of $A^{*,*}(X)$. We define a map $I_{\Delta} : A^{*} (X) \rightarrow {\Omega}^{*} (X) $ as follows: $$I_{ \Delta }( \alpha ) := \int_{{\Delta }^{p}} ( { \alpha } |_{{ \Delta }^{p} \times {X}_{p} } ).$$ Then the following theorem holds [@Dup].
$ I_{ \Delta } $ [ induces a natural ring isomorphism]{} $$I_{ \Delta } ^{*} : H( A^{*} (X)) \cong H({\Omega}^{*} (X)).$$
Let $\mathcal{G}$ denote the Lie algebra of $G$. A connection on a simplicial $G$-bundle $\pi : \{ E_p \} \rightarrow \{ M_p \} $ is a sequence of $1$-forms $\{ \theta \}$ on $\{ E_p \}$ with coefficients $\mathcal{G}$ such that $\theta $ restricted to ${\Delta}^{p} \times {E}_{p} $ is a usual connection form on a principal $G$-bundle ${\Delta}^{p} \times {E}_{p} \rightarrow {\Delta}^{p} \times {M}_{p} $.
Dupont constructed a canonical connection $\theta \in A^1 (N \bar{G} )$ on ${\gamma} : N \bar{G} \rightarrow NG $ in the following way:
$${\theta } |_{{ \Delta }^{p} \times N \bar{G} (p)} := t_0 {\theta }_0 + \cdots + t_{p} {\theta }_{p}.$$
Here ${\theta }_i $ is defined by ${\theta }_i = {\rm pr}_i ^{*} \bar {\theta } $ where ${\rm pr}_i : { \Delta }^{p} \times N \bar{G} (p) \rightarrow G $ is the projection into the $i$-th factor of $ N \bar{G} (p) $ and $\bar {\theta }$ is the Maurer-Cartan form of $G$. We also obtain its curvature $\Omega \in A^2 (N \bar{G} )$ on ${\gamma }$ as: $$\Omega |_{{ \Delta }^{p} \times N \bar{G} (p) }= d \theta |_{{ \Delta }^{p} \times N \bar{G} (p) }
+ \frac{1}{2} [ \theta |_{{ \Delta }^{p} \times N \bar{G} (p) } , \theta |_{{ \Delta }^{p} \times N \bar{G} (p) } ].$$
Let ${\rm I}^{*} (G)$ denote the ring of $G$-invariant polynomials on $\mathcal{G}$. For $P \in I^* (G)$, we restrict $P( \Omega ) \in A^{*} (N \bar{G} )$ to each ${\Delta}^{p} \times N \bar{G} (p)
\rightarrow {\Delta}^{p} \times NG(p) $ and apply the usual Chern-Weil theory then we have a simplicial $2k$-form $P( \Omega )$ on $NG$.
Now we have a canonical homomorphism $${w}:{\rm I}^{*} (G) \rightarrow H({\Omega}^{*} (NG))$$ which maps $P \in I^* (G)$ to ${w}(P)=[I_{\Delta } ( P({\Omega}) )]$.
The Chern character in the double complex
=========================================
In this section we exhibit a cocycle in $\Omega ^{*,*}(NG) $ which represents the Chern character. Throughout this section, $G= GL(n ; \mathbb{C} )$ and $ {\rm ch }_p$ means the $p$-th Chern character.
Note that the diagram below is commutative, since $I_{\Delta }$ acts only on the differential forms on ${ \Delta }^{*} $, and so does ${\gamma}^{*}$ on differential forms on each $N G (*)$. $$\begin{CD}
A^{*,*}(N \bar{G} )@>{I_{\Delta }}>>{\Omega}^{*,*}(N \bar{G} )\\
@A{\gamma}^{*}AA@AA{\gamma}^{*}A\\
A^{*,*}(N G )@>{I_{\Delta }}>>{\Omega}^{*,*}(N G )
\end{CD}$$
We first give the cocycle in ${\Omega}^{p+q}(N \bar{G}(p-q)) (0 \leq q \leq p-1)$ which corresponds to the $p$-th Chern character by restricting $ ({1}/{p! } )\thinspace {\rm tr} \left( \left( { - \Omega }/{2 \pi i } \right) ^p \right)
\in A^{2p}(N \bar{G} ) $ to $A^{p-q,p+q} ( {\Delta}^{p-q} \times N \bar{G}(p-q))$ and integrating it along ${\Delta}^{p-q}$. Then we give the cocycle in ${\Omega}^{p+q}(N {G}(p-q))$ which hits to it by ${\gamma}^{*}$.
Since $[\theta _i , \theta _j ]= \theta _i \wedge \theta _j + \theta _j \wedge \theta _i$ for any $i,j$, $$\Omega |_{{ \Delta }^{p-q} \times N \bar{G} (p-q) }
= -\sum _{i=1} ^{p-q} dt_i \wedge (\theta _0 - \theta _{i} )
- \sum _{ 0 \leq i < j \leq p-q } t_i t_j ( \theta _i - \theta _j ) ^{2}.$$
Now $$dt_i \wedge (\theta _0 -\theta _{i} ) = dt_i \wedge \{ (\theta _0 -\theta _{1} ) + (\theta _{1} -\theta _{2} ) +
\cdots + (\theta _{i-1} -\theta _{i} ) \}$$
and for any ${\mathcal{G} }$-valued differential forms $ \alpha , \beta , \gamma $ and any integer $ 0 \leq \forall x \leq p-q-1 $, the equation $ \alpha \wedge (dt_i \wedge ( \theta _x - \theta _{x+1} )) \wedge \beta \wedge (dt_j \wedge ( \theta _x - \theta _{x+1} )) \wedge \gamma = - \alpha \wedge (dt_j \wedge ( \theta _x - \theta _{x+1} )) \wedge \beta \wedge (dt_i \wedge ( \theta _x - \theta _{x+1} )) \wedge \gamma $ holds, so the terms of the forms above cancel with each other in $ \left( - \Omega |_{{ \Delta }^{p-q} \times N \bar{G} (p-q) } \right) ^{p} $. Then we see:
$$\left( - \Omega |_{{ \Delta }^{p-q} \times N \bar{G} (p-q) } \right) ^{p}
= \left( \sum _{i=1} ^{p-q} dt_i \wedge (\theta _{i-1} - \theta _{i} )
+ \sum _{ 0 \leq i < j \leq p-q } t_i t_j ( \theta _i - \theta _j ) ^{2} \right) ^{p}.$$
Now we obtain the following theorem.
[We set:]{} $$\bar{S}_{p-q}=\sum _{\sigma \in \mathfrak{S} _{p-q-1 } }({\rm sgn} ( \sigma ) ) (\theta _{ \sigma (1)} - \theta _{ \sigma (1)+1 } )
\cdots (\theta _{ \sigma (p-q-1) } - \theta _{ \sigma (p-q-1)+1 } )$$ [Then the cocycle in ]{} $ {\Omega}^{p+q} ( N \bar{G} (p-q) ) \ (0 \leq q \leq p-1) $ [ which corresponds to the $p$-th Chern character ${\rm ch}_p$ is]{}
$$\frac{1}{p! } \left(\frac{1}{2 \pi i } \right)^p (-1)^{(p-q)(p-q-1)/2 } \times \hspace{18em}$$ $$\mathrm{tr} \sum \left( (p(\theta _0 - \theta _{1})) \wedge \bar{H}_q (\bar{S}_{p-q}) \times \int _{{\Delta}^{p-q}} \prod_{i<j} (t_i t_j)^{a_{ij}(\bar{H}_q (\bar{S}_{p-q}))} dt_1 \wedge \cdots \wedge dt_{p-q} \right).$$ Here $\bar{H}_q (\bar{S}_{p-q})$ means the terms that $ (\theta _{i} - \theta _{j} )^2 \enspace (1 \leq i < j \leq p-q+1 ) $ [ are put ]{}$q$[ -times between ]{}$(\theta _{k-1} - \theta _{k} )$ [ and ]{}$(\theta _{l} - \theta _{l+1} )$ [ in $ \bar{S}_{p-q}$ permitting overlaps; $a_{ij}(\bar{H}_q (\bar{S}_{p-q}))$ means the number of ]{}$(\theta _{i} - \theta _{j} )^2 $[ in it. $\sum$ means the sum of all such terms]{}.
The cocycle in $ {\Omega}^{p+q} ( N \bar{G} (p-q) ) $ which corresponds to ${\rm ch}_p$ is given by $$\int_{{\Delta}^{p-q}} \frac{1}{p!} {\rm tr} \left( \left( \frac{- \Omega |_{{ \Delta }^{p-q} \times N \bar{G} (p-q) } }{2 \pi i} \right) ^p \right) \hspace{15em}$$ $$= \frac{1}{p!} \left(\frac{1}{2 \pi i } \right)^p \int_{{\Delta}^{p-q}} {\rm tr} \left( \left( \sum _{i=1} ^{p-q} dt_i \wedge (\theta _{i-1} - \theta _{i} )
+ \sum _{ 0 \leq i < j \leq p-q } t_i t_j ( \theta _i - \theta _j ) ^{2} \right) ^{p} \right).$$ By calculating this equation, we can check that the statement of Theorem 3.1 is true.
For the purpose of getting the differential forms in $\Omega ^{*,*}(NG) $ which hit the cocycles in Theorem 3.1 by ${\gamma}^{*}$, we set $$\varphi_s:=h_1 \cdots h_{s-1}dh_s h^{-1} _s \cdots h^{-1} _1.$$ Here $h_i$ is the $i$-th factor of $NG(*)$.
A straightforward calculation shows that $${\gamma}^{*} {\rm{tr}} (\varphi_{i_1} \varphi_{i_2} \cdots \varphi_{i_{p-1}} \varphi_{i_p} ) =\mathrm{tr}(\theta _{i_1-1} - \theta _{i_1 } ) (\theta _{i_2-1} - \theta _{i_2 } ) \cdots (\theta _{i_p-1} - \theta _{i_p } ).$$
From the above, we conclude:
We set: $$R_{ij} = (\varphi_{i}+ \varphi_{i+1} + \cdots + \varphi_{j-1})^{2} \qquad (1 \leq i<j \leq p-q+1 )$$ $$S_{p-q} =\sum _{\sigma \in \mathfrak{S} _{p-q-1 } } {\rm sgn} ( \sigma ) \varphi_{ \sigma (1)+1 } \cdots \varphi_{ \sigma (p-q-1)+1 }.$$ Then the cocycle in $ \Omega ^{p+q} (NG(p-q)) \ (0 \leq q \leq p-1)$ which represents the $p$-th Chern character ${\rm ch}_p $ is $$\frac{1}{(p-1)! } \left( \frac{1}{2 \pi i } \right)^p (-1)^{(p-q)(p-q-1)/2 } \times \hspace{17em}$$ $$\mathrm{tr} \sum \left( \varphi_1 \wedge {H_q} ({S}_{p-q})
\times \int _{{\Delta}^{p-q}} \prod_{i<j} (t_{i-1} t_{j-1})^{a_{ij}({H_q}({S}))} dt_1 \wedge \cdots \wedge dt_{p-q} \right).$$ Here ${H_q} ({S}_{p-q})$ means the term that $ R_{ij} \enspace (1 \leq i < j \leq p-q+1 ) $ are put $q$ -times between $\psi_k$ and $\psi_l $ in ${S}_{p-q} $ permitting overlaps; $a_{ij}({H}_q({S}_{p-q}))$ means the number of $R_{ij} $ in it. $\sum$ means the sum of all such terms.
We can easily check that the cocycle in Theorem 3.2 is mapped to the cochain in Theorem 3.1 by $\gamma^* : \Omega ^{p+q} (NG(p-q))
\rightarrow \Omega ^{p+q} (N \bar{G}(p-q)) $. The statement folllows from this.
The coefficients in theorem 3.2 are calculated using the following famous formula. $$\int _{{\Delta}^{r}}t_{0} ^{b_0} t_1 ^{b_1} \cdots t_{r} ^{b_{r}}dt_1 \wedge \cdots \wedge dt_{r} =
\frac{b_0 ! ~ b_1 ! \cdots b_r !}{(b_0 + b_1 + \cdots + b_r +r)!}.$$
The cochain $\omega_{p}$ in $ \Omega ^{2p-1} (NG(1)) $ which corresponds to the $p$-th Chern character is given as follows: $$\omega_{1}= \frac{1}{p! } \left( \frac{1}{2 \pi i } \right)^p \frac{1}{ {}_{2p-1}C_{p-1}}{\rm tr }(h^{-1} dh )^{2p-1}.$$
The cochain $\omega_{p}$ in $ \Omega ^{p} (NG(p)) $ which corresponds to the $p$-th Chern character is given as follows: $$\omega_{p}= (-1)^{p(p-1)/2 } \frac{1}{p!(p-1)! } \left( \frac{1}{2 \pi i } \right)^p \mathrm{tr} \left( \varphi_1 \wedge \sum _{\sigma \in \mathfrak{S} _{p-1 } } {\rm sgn} ( \sigma ) \varphi_{ \sigma (1)+1 } \cdots \varphi_{ \sigma (p-1)+1 } \right).$$
The cocycle which represents the second Chern character ${\rm ch}_2 $ in $ \Omega ^{4} (NG) $ is the sum of the following $C_{1,3}$ and $C_{2,2}$: $$\begin{CD}
0 \\
@AA{d''}A \\
C_{1,3} \in {\Omega}^{3} (G )@>{d'}>>{\Omega}^{3} (NG(2))\\
@.@AA{d''}A\\
@.C_{2,2} \in {\Omega}^{2} (NG(2))@>{d'}>> 0
\end{CD}$$ $$C_{1,3} = \left( \frac{1}{2 \pi i} \right) ^2 \frac{1}{6}{\rm tr}({h^{-1}dh} )^3 ,
\qquad C_{2,2} = \left( \frac{1}{2 \pi i} \right) ^2 \frac{-1}{2}{\rm tr}( dh_1 dh_2 h_2 ^{-1} h_1 ^{-1} ).$$
The cocycle which represents the second Chern class ${\rm c}_2 $ in $ \Omega ^{4} (NG) $ is the sum of the following $c_{1,3}$ and $c_{2,2}$: $$\begin{CD}
0 \\
@AA{d''}A \\
c_{1,3} \in {\Omega}^{3} (G )@>{d'}>>{\Omega}^{3} (NG(2))\\
@.@AA{d''}A\\
@.c_{2,2} \in {\Omega}^{2} (NG(2))@>{d'}>> 0
\end{CD}$$ $$c _{1,3} = \left( \frac{1}{2 \pi i} \right) ^2 \frac{-1}{6}{\rm tr}({h^{-1}dh} )^3 \hspace{17em}$$ $$c _{2,2} = \left( \frac{1}{2 \pi i} \right) ^2 \frac{1}{2}{\rm tr}( dh_1 dh_2 h_2 ^{-1} h_1 ^{-1})- \left( \frac{1}{2 \pi i} \right) ^2 \frac{1}{2}{\rm tr}( h_1 ^{-1} dh_1){\rm tr}( h_2 ^{-1}dh_2 ).$$
The cocycle which represents the $3$rd Chern character ${\rm ch}_3 $ in $ \Omega ^{6} (NG) $ is the sum of the following $C_{1,5} , C_{2,4}$ and $C_{3,3}$: $$\begin{CD}
0 \\
@AA{d''}A \\
{ C_{1,5} \in {\Omega}^{5} (G )}@>{d'}>>{\Omega}^{5} (NG(2))\\
@.@AA{d''}A\\
@.{ C_{2,4} \in {\Omega}^{4} (NG(2))}@>{d'}>> {\Omega}^{4} (NG(3)) \\
@.@.@AA{d''}A\\
@.@.{ C_{3,3} \in {\Omega}^{3} (NG(3))}@>{d'}>> 0
\end{CD}$$
$$C_{1,5} = \frac{1}{3!} \left( \frac{1}{2 \pi i} \right) ^3 \frac{1 }{10}{\rm tr}({h^{-1}dh} )^5 \hspace{18em}$$
$$C_{2,4}= \frac{-1}{3!} \left( \frac{1}{2 \pi i} \right) ^3 (
\frac{1 }{2}{\rm tr} ( dh_1 {h_1}^{-1}dh_1 {h_1}^{-1}dh_1 dh_2 {h_2}^{-1}{h_1}^{-1}) \hspace{7em}$$ $$+ \frac{1 }{4} {\rm tr} ( dh_1 dh_2 {h_2}^{-1}{h_1}^{-1}dh_1 dh_2 {h_2}^{-1}{h_1}^{-1})$$ $$\hspace{9em} + \frac{1 }{2} {\rm tr} ( dh_1 dh_2 {h_2}^{-1}dh_2 {h_2}^{-1}dh_2 {h_2}^{-1}{h_1}^{-1}))$$
$$C_{3,3} =\frac{-1}{3!} \left( \frac{1}{2 \pi i} \right) ^3 (\frac{1 }{2}{\rm tr} ( dh_1 dh_2 dh_3 {h_3}^{-1}{h_2}^{-1}{h_1}^{-1} ) \hspace{10em}$$ $$\hspace{10em} -\frac{1 }{2} {\rm tr} (dh_1 h_2 dh_3 {h_3}^{-1}{h_2}^{-1} dh_2 {h_2}^{-1}{h_1}^{-1})).$$
The Chern-Simons form
======================
We briefly recall the notion of the Chern-Simons form in [@Chern].
Let $\pi : E \rightarrow M$ be any principal $G$-bundle and $\theta$, $\Omega$ denote its connection form and the curvature. For any $ P \in {\rm I}^{k} (G) $, we define the $(2k-1)$-form $TP( \theta )$ on $E$ as:\
$$TP( \theta ) := k \int_{0}^{1} P(\theta \wedge { \phi }_t ^{k-1})dt.$$ Here ${ \phi }_t := t \Omega + \frac{1}{2}t(t-1) [ \theta , \theta ]$. Then the equation $ d(TP( \theta )) = P( { \Omega }^k ) $ holds and $TP( \theta )$ is called the Chern-Simons form of $P( { \Omega }^k ) $. When the bundle is flat, its curvature vanishes and hence $ d(TP( \theta )) = P( { \Omega }^k ) = 0 $.
Now we put the simplicial connection into $TP$ and using the same argument in section 3, then we obtain the Chern-Simons form in $\Omega ^{2p-1}(N \bar{G}) $.
The Chern-Simons form in ${\Omega}^{3}(N \overline{U(n)} )$ which corresponds to the second Chern class $c_2$ is the sum of the following $Tc_{0,3}$, $Tc_{1,2}$: $$\begin{CD}
0 \\
@AA{d''}A \\
Tc_{0,3} \in {\Omega}^{3} (U(n) )@>{d'}>>{\Omega}^{3} (N \overline{U(n)}(1))\\
@.@AA{d''}A\\
@.Tc_{1,2} \in {\Omega}^{2} (N \overline{U(n)}(1))@>{d'}>> {\Omega}^{2} (N \overline{U(n)}(2))
\end{CD}$$ $$Tc_{0,3} = \left( \frac{1}{2 \pi i} \right) ^2 \frac{1}{6}{\rm tr}({g^{-1}dg} )^3 \hspace{15em}$$ $$Tc_{1,2} = \left( \frac{1}{2 \pi i} \right) ^2 \left( \frac{1}{2}{\rm tr}( g_0 ^{-1} dg_0 g_1 ^{-1}dg_1 )
- \frac{1}{2}{\rm tr}( g_0 ^{-1} dg_0 ){\rm tr}(g_1 ^{-1}dg_1 )\right).$$
The term $\left( \frac{1}{2 \pi i} \right) ^2 \frac{1}{2}{\rm tr}( g_0 ^{-1} dg_0 ){\rm tr}(g_1 ^{-1}dg_1 )$ vanishes when we restrict it to $SU(n)$.
Formulas for a cocycle in a truncated complex
=============================================
In this section, we prove the conjecture due to Brylinski in [@Bry].
At first, we introduce the filtered local simplicial de Rham complex.
The filtered local simplicial de Rham complex $F^p\Omega_{{\rm loc}} ^{*,*}(NG)$ over a simplicial manifold $NG$ is defined as follows:\
$$F^p \Omega^{r,s} _{\rm loc}(NG)=\begin{cases}
\underrightarrow{\rm lim}_{1 \in V \subset G^r} ~~\Omega ^s(V) & ~{\rm if}~ s \ge p \\
0 & {\rm otherwise}.
\end{cases}$$
Let $F^p \Omega^{*}(NG)$ be a filtered complex $$F^p \Omega^{r,s} (NG)=\begin{cases}
\Omega ^s(NG(r)) & ~{\rm if}~ s \ge p \\
0 & {\rm otherwise}
\end{cases}$$ and $[\sigma_{<p}\Omega ^{*}({NG})]$ a truncated complex $$[\sigma_{<p}\Omega ^{r,s}({NG})]=\begin{cases}
0 & ~{\rm if}~ s \ge p \\
\Omega ^s(NG(r)) & ~{\rm otherwise}.
\end{cases}$$ Then there is an exact sequence: $$0 \rightarrow F^p \Omega^{*}(NG) \rightarrow \Omega^{*}(NG) \rightarrow [\sigma_{<p}\Omega^{*}({NG}) ] \rightarrow 0$$ which induces a boundary map $$\beta:H^l(NG,[\sigma_{<p}\Omega_{\rm loc} ^{*}]) \rightarrow H^{l+1}(NG,[F^p \Omega^{*} _{\rm loc}]).$$
Let $\omega_1 + \cdots + \omega_p$, $\omega_{p-q} \in \Omega^{p+q}(NG(p-q))$ be the cocycle in $\Omega^{2p}(NG)$ which represents the $p$-th Chern character. By using this cocycle, Brylinski constructed a cochain $\eta$ in $[\sigma_{<p}\Omega ^{*} _{\rm loc}({NG})]$ in the following way.
We take a contractible open set $U \subset G$ containing $1$. Using the same argument in [@Dup2 Lemma 9.7], we can construct mappings $\{ \sigma_l:{ \Delta }^l\times U^l \rightarrow U \}_{0 \le l}$ inductively with the following properties:\
(1) $\sigma_0(pt)=1$;\
(2) $${\sigma}_l({\varepsilon}^{j}(t_0, \cdots , t_{l-1});h_1, \cdots, h_l)=\begin{cases}
{\sigma}_{l-1}( t_0, \cdots, t_{l-1};{\varepsilon}_{j}(h_1, \cdots , h_l)) & ~{\rm if}~j \ge 1 \\
h_1 \cdot {\sigma}_{l-1}(t_0, \cdots, t_{l-1};h_2, \cdots , h_l) & ~{\rm if}~j=0.
\end{cases}$$ Then we define mappings $\{f_{m,q}: { \Delta }^q \times U^{m+q-1} \rightarrow G^m \}$ by $$f_{m,q}(t_0, \cdots , t_q;h_1, \cdots, h_{m+q-1}):=(h_1, \cdots , h_{m-1}, {\sigma}_q(t_0, \cdots , t_q;h_m, \cdots , h_{m+q-1})).$$ We can check $f_{m,q} \circ {\varepsilon}_{j} = f_{m,q+1} \circ {\varepsilon}^{j-m+1}:{ \Delta }^q \times U^{m+q} \rightarrow G^m $ if $m \le j \le m+q$ and $f_{m,q} \circ {\varepsilon}_{j} = {\varepsilon}_{j} \circ f_{m+1,q}$ if $m-1 \ge j \ge 0$ and ${\varepsilon}_{m} \circ f_{m+1,q} = f_{m,q+1}
\circ {\varepsilon}^0$ holds.
We define a $(2p-m-q)$-form $\beta_{m,q}$ on $U^{m+q-1}$ by $\beta_{m,q}=(-1)^m \int_{{ \Delta }^q}f_{m,q} ^* {\omega}_m$. Then the cochain $\eta$ is defined as the sum of following $\eta_l$ on $U^{2p-1-l}$ for $0 \le l \le p-1$: $$\eta _l:=\sum_{m+q=2p-l,~ m \ge 1}\beta _{m,q}.$$
Now we are ready to state the theorem whose statement is conjectured by Brylinski [@Bry].
$\eta := \eta _0+ \cdots + \eta_{p-1}$ is a cocycle in $[\sigma_{<p}\Omega_{\rm loc} ^{*}(NG)]$ whose cohomology class is mapped to $[\omega_1 + \cdots + \omega_p]$ in $H^{2p}(NG,[F^p \Omega^{*} _{\rm loc}])$ by a boundary map $\beta : H^{2p-1}(NG,[\sigma_{<p}\Omega_{\rm loc} ^{*}]) \rightarrow H^{2p}(NG,[F^p \Omega^{*} _{\rm loc}])$.
To prove this, it suffices to show the equation below holds true for any $l$ which satisfies $0 \le l \le 2p-1$ since $\omega_{2p-l}=0$ if $0 \le l \le p-1$: $$\sum_{i=0} ^{2p-l}(-1)^i {\varepsilon}_{i} ^{*} \eta_l = (-1)^{2p-l+1}d \eta_{l-1} + \omega_{2p-l}.$$ The left side of this equation is equal to $$\sum_{m+q=2p-l,~ m \ge 1} (-1)^m \left( \int_{{ \Delta }^q} \sum_{i=0} ^{m-1} (-1)^i(f_{m,q} \circ {\varepsilon}_{i} )^* {\omega}_m + \int_{{ \Delta }^q} \sum_{i=m} ^{m+q} (-1)^i(f_{m,q} \circ {\varepsilon}_{i} )^* {\omega}_m \right).$$ We can check that $$\sum_{i=0} ^{m-1}(-1)^i (f_{m,q} \circ {\varepsilon}_{i} )^* {\omega}_m = f_{m+1,q} ^* ( \sum_{i=0} ^{m-1} (-1)^i {\varepsilon}_{i} ^* {\omega}_m)$$ hence by using the cocycle relation $\sum_{i=0} ^{m+1} (-1)^i {\varepsilon}_{i} ^* {\omega}_m = (-1)^m d \omega_{m+1}$, we can see the following holds:
$$\int_{{ \Delta }^q} \sum_{i=0} ^{m-1} (-1)^i(f_{m,q} \circ {\varepsilon}_{i} )^* {\omega}_m
= \int_{{ \Delta }^q}(-1)^m d f_{m+1,q} ^* \omega _{m+1} \hspace{10em}$$ $$\hspace{5em} - \left( (-1)^m \int_{{ \Delta }^q}( {\varepsilon}_{m} \circ f_{m+1,q})^* {\omega}_m
+ (-1)^{m+1} \int_{{ \Delta }^q}( {\varepsilon}_{m+1} \circ f_{m+1,q})^* {\omega}_m\right).$$
Note that $ \int_{{ \Delta }^q}( {\varepsilon}_{m+1} \circ f_{m+1,q})^* {\omega}_m = 0 $ for $q \ge 1$ and $\int_{{ \Delta }^q}( {\varepsilon}_{m+1} \circ f_{m+1,q})^* {\omega}_m = \omega_{2p-l}$ if $q=0$.
We can also check that $$\int_{{ \Delta }^q} \sum_{i=m} ^{m+q} (-1)^i (f_{m,q} \circ {\varepsilon}_{i} )^* {\omega}_m
= \int_{{ \Delta }^q} \sum_{i=m} ^{m+q} (-1)^i(f_{m,q+1} \circ {\varepsilon}^{i-m+1} )^* {\omega}_m.$$ We set $j=i-m+1$, then we see that $\int_{{ \Delta }^q} \sum_{i=m} ^{m+q} (-1)^i(f_{m,q+1} \circ {\varepsilon}^{i-m+1} )^* {\omega}_m$ is equal to $$\sum_{j=0} ^{q+1} \left( (-1)^{j+m-1} \int_{{ \Delta }^q}(f_{m,q+1} \circ {\varepsilon}^{j} )^* {\omega}_m \right) -
(-1)^{m-1} \int_{{ \Delta }^q}({\varepsilon}_{m} \circ f_{m+1,q})^* {\omega}_m$$ since ${\varepsilon}_{m} \circ f_{m+1,q} = f_{m,q+1} \circ {\varepsilon}^0$.
From above, we can see that $\sum_{i=0} ^{2p-l}(-1)^i {\varepsilon}_{i} ^{*} \eta_l$ is equal to $$\omega_{2p-l}+ \sum_{m+q=2p-l,~ m \ge 1} \left( \int_{{ \Delta }^q}d f_{m+1,q} ^* \omega _{m+1} + \sum_{j=0} ^{q+1} (-1)^{j-1} \int_{{ \Delta }^q}(f_{m,q+1} \circ {\varepsilon}^{j} )^* {\omega}_m \right).$$ On the other hand, for any $(m',q')$ which satisfies $m'+q'=2p-(l-1)$ the following equation holds: $$(-1)^{q'}d\int_{{ \Delta }^{q'}} f_{m',q'} ^* \omega _{m'}= \int_{{ \Delta }^{q'}} df_{m',q'} ^* \omega _{m'}
-\sum_{j=0} ^{q'} \int_{{ \Delta }^{q'-1}} (-1)^j {{\varepsilon}^{j}} ^* f_{m',q'} ^* \omega _{m'}.$$ Therefore $(-1)^{2p-l+1}d \eta_{l-1}$ is equal to $$\sum_{m'+q'=2p-l+1,~ m' \ge 1}\left(\int_{{ \Delta }^{q'}} df_{m',q'} ^* \omega _{m'}
-\sum_{j=0} ^{q'} \int_{{ \Delta }^{q'-1}} (-1)^j {{\varepsilon}^{j}} ^* f_{m',q'} ^* \omega _{m'}\right).$$ This completes the proof.
Let me explain Brylinski’s motivation in [@Bry] to introduce these complexes and the conjecture briefly. Let $LU$ be the free loop group of a contractible open set $U \subset G$ containing $1$ and ${\rm ev}:LU \times S^1 \to U$ be the evaluation map, i.e. for $\gamma \in LU$ and $\theta \in S^1$, ${\rm ev}(\gamma , \theta)$ is defined as $\gamma (\theta)$. Then $\int _{S^1} {\rm ev}^*$ maps $\eta_{1} \in \Omega ^1(U^{2p-2})$ to a cochain in $\Omega ^0(LU^{2p-2})$. This cochain defines a cohomology class in local cohomology group $H^{2p-2} _{\rm loc}(LU, {\mathbb C})$. Brylinski constructed a natural map from $H^{2p-2} _{\rm loc}(LU, {\mathbb C})$ to the the Lie algebra cohomology $H^{2p-2}(L \mathcal{G}, {\mathbb C})$. Then as a special case $p=2$, he used the cocycle in the local truncated complex $[\sigma_{<2}\Omega^{3} _{\rm loc}(NG)]$ to construct the standard Kac-Moody $2$-cocycle. He treated not only the free loop group but also the gauge group ${\rm Map}(X,G)$ for a compact oriented manifold $X$.
[**Acknowledgments.**]{}\
I am indebted to Professor H. Moriyoshi for helpful discussion and good advice. I would like to thank the referee for his/her several suggestions to improve this paper.
[99]{} R. Bott, On the Chern-Weil homomorphism and the continuous cohomology of the Lie group, Adv. in Math. 11 (1973), 289-303. R. Bott, H. Shulman, J. Stasheff, On the de Rham Theory of Certain Classifying Spaces, Adv. in Math. 20 (1976), 43-56. J-L. Brylinski, Differentiable cohomology of gauge groups, math.DG/0011069. S.S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. 99 (1974) 48-69. J.L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Top. Vol 15(1976),233-245, Perg Press. J.L. Dupont, Curvature and Characteristic Classes, Lecture Notes in Math. 640, Springer Verlag, 1978. M. Mostow and J. Perchick, Notes on Gel’fand-Fuks Cohomology and Characteristic Classes (Lectures by Bott).In Eleventh Holiday Symposium. New Mexico State University, December 1973. G. Segal, Classifying spaces and spectral sequences. Inst.Hautes Études Sci.Publ.Math.No.34 1968 105-112.
Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi-ken, 464-8602, Japan.\
e-mail: suzuki.naoya@c.mbox.nagoya-u.ac.jp
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---
abstract: 'Defects usually play an important role in tuning and modifying various properties of semiconducting or insulating materials. Therefore we study the impact of point and line defects on the electronic structure and optical properties of MoS$_2$ monolayers using density-functional methods. The different types of defects form electronic states that are spatially localized on the defect. The strongly localized nature is reflected in weak electronic interactions between individual point or line defect and a weak dependence of the defect formation energy on the defect concentration or line defect separation. In the electronic energy spectrum the defect states occur as deep levels in the band gap, as shallow levels very close to the band edges, as well as levels in-between the bulk states. Due to their strongly localized nature, all states of point defects are sharply peaked in energy. Periodic line defects form nearly dispersionless one-dimensional band structures and the related spectral features are also strongly peaked. The electronic structure of the monolayer system is quite robust and it is well preserved for point defect concentrations of up to 6%. The impact of point defects on the optical absorption for concentrations of 1% and below is found to be very small. For higher defect concentrations molybdenum vacancies were found to quench the overall absorption and sulfur defects lead to sharp absorption peaks below the absorption edge of the ideal monolayer. For line defects, we did not find a considerable impact on the absorption spectrum. These results support recent experiments on defective transition metal chalcogenides.'
author:
- 'J. Kunstmann$^1$'
- 'T.B. Wendumu$^{1,2}$'
- 'G. Seifert$^1$'
bibliography:
- 'references.bib'
title: 'Localized defect states in MoS$_2$ monolayers: electronic and optical properties'
---
Introduction
============
Defects usually play an important role in tuning and modifying various properties of semiconducting or insulating materials. Substitutional defects in semiconductors are employed to increase the electrical conductivity of the material. The defects create impurity states (in-gap states) in the band gap near the valence (p-type) or conduction (n-type) band edge that, when thermally populated or depopulated, create free charge carriers and thus enhance the conductivity. In ionic crystals certain types of defects form luminescent centers, such as color centers (electrons trapped in anion vacancy sites) or paramagnetic impurities (d- or f-element cations). The electronic states related to the localized electrons of the impurities form in-gap states within the large intrinsic band gap of ionic crystals. The localized states of color centers and d-element impurities couple strongly to the continuum of phonon states and form broad vibronic bands in the absorption or emission spectra. The resulting optical properties are observed as the color of gemstones (e.g. ruby, emerald) and find application in solid-state lasers, white LEDs, luminescent light bulbs or phosphors. Defects in crystals are also of interest as potential single photon emitters. Individual photons can be emitted from quantum systems with localized states such as cold atoms, molecules, semiconductor quantum dots or color centers [@Lounis2005a].
With the recent rise of 2D materials there has also been a considerable research interest in transition metal dichalcogenides (TMDs) monolayer structures such as molybdenum disulphide (MoS$_2$). A MoS$_2$ monolayer (ML) consists of two atomic layers of close-packed S atoms separated by one close packed Mo atomic layer [@Chhowalla2013] and it is a semiconductor with an direct optical band gap of 1.9 eV, [@Splendiani2010; @Mak2010] whereas its bulk counterpart has an indirect optical band gap of 1.3 eV [@Kam1982a]. Compared to the bulk a ML exhibits stronger photoluminescence and reduced screening leads to strong excitonic effects. [@Splendiani2010; @Mak2010] Because of such advantageous optical and electronic properties, MoS$_2$ is believed to be a promising building block for future applications in nanoelectronics and optoelectronics [@Radisavljevic2011; @Wang2012a].
Combining the interesting physics of defects with the unique properties of MoS$_2$ seems very promising and various experimental studies of point defects have been reported [@Komsa2013b; @Zhou2013; @Qiu2013b; @Hong2015]. Among many other insights, these works highlight the importance and abundance of sulfur vacancy defects. Theoretical studies of various point defects [@Yuan2014a; @Noh2014; @Komsa2015; @Haldar2015; @Pandey2016] consistently explain this with a low formation energy of this defect. Sulfur vacancy defects are currently believed to be the main reason for the low mobility, observed in back gated field effect transistors using MoS$_2$, grown by chemical vapor deposition.[@Pandey2016] Line defects and grain boundaries (GB) within ML-MoS$_2$ have also been studied experimentally [@Komsa2013b; @Najmaei2013; @Zhou2013; @VanderZande2013; @Tongay2013] and theoretically [@Zou2012; @Enyashin2013; @Ghorbani-Asl2013a; @Zou2015]. Both point and line defects introduce in-gap states in semiconducting ML-MoS$_2$. Due to their one-dimensional nature, certain line defects are believed to form metallic wires [@Liu2014d; @Zou2015; @Gibertini2015]. Similarly, the edge states of triangular MoS$_2$ platelets are metallic-like [@Bollinger2001] and exhibit bright photoluminescence, and recent theoretically studies [@Wendumu2014; @Joswig2015] explain this result. Recently, the emission of single photons from WSe$_2$ MLs was reported. [@Srivastava2014a; @He2015; @Koperski2015; @Chakraborty2015; @Kumar2015] All authors attribute the single photon emission to localized states caused by intrinsic defects. However the detailed nature of these defect states remains to be determined.
In view of these interesting results, this paper aims at a deeper understanding of electronic and optical properties of defective MoS$_2$ ML. Because the localized nature of defect states is crucial for single-photon emission, this paper focuses on the degree of localization and studies point defects as function of the defect concentration and 8-4 type line defects as function of their separation length $d$. Analysis of highly defective MoS$_2$ samples in the transition electron microscope determined the area density of defects to be as high as $10^{13}$ cm$^{-2}$.[@Qiu2013b] This corresponds to defect concentrations of the order 1% (see Table \[tab:concentration\]). The concentration dependence of the optical and transport properties of random point defects in the limit of small concentration (1% and smaller) were studied within an empirical tight-binding approach by Yuan et al.[@Yuan2014a] Here we are mostly interested in the localization of individual defects and therefore study larger concentrations.
Our results for sulfur mono and divacancies and Mo and MoS$_2$ vacancies and mirror and tilt grain boundaries (line defects) show that defects states in MoS$_2$ ML are strongly localized on the defect. The strongly localized nature is reflected in weak electronic interactions between individual point or line defect and a weak dependence of the defect formation energy on the defect concentration or line defect separation. Most significantly, these point and line defects create sharp, characteristic peaks within the band gap, but the electronic bulk properties of the ML are robust for defect concentrations of up to about 6%. The impact of point defects on the optical absorptions for concentrations of 1% and below is found to be very small. Similarly, the considered line defects are found to have a almost no impact on the absorption spectrum.
Computational Details
=====================
For all electronic structure calculations and structural optimization of systems with point defects, we have utilized the density-functional tight-binding (DFTB) method because it allows to study large systems. [@Seifert1986; @Porezag1995] DFTB is based on the density functional theory of Hohenberg and Kohn [@Hohenberg1964] in the formulation of Kohn and Sham [@Kohn1965]. The single-particle Kohn-Sham eigenfunctions are expanded with a set of localized atom-centered basis functions. These functions are determined by self-consistent LDA density functional calculations of isolated atoms employing a large set of Slater-type basis functions. The effective one-electron potential in the Kohn-Sham Hamiltonian is approximated as a superposition of the potentials of neutral atoms. Additionally, only one- and two-center integrals are calculated to set up the Hamilton matrix. We have taken a minimal valence basis, including the 5s, 5p, and 4d orbitals for molybdenum, and the 3s and 3p orbitals for sulfur. States below these levels were treated within a frozen-core approximation.
Moreover, we have used time-dependent density functional response theory within the DFTB formulation (TD-DFRT-TB) for the calculations of all excitation spectra [@Niehaus2001]. This is also referred to a linear response theory. [@Casida1996] To obtain the excitation energies, the coupling matrix, which gives the response of the potential with respect to a change in the electron density, has to be built. In our scheme, we approximate the coupling matrix in the so-called $\gamma$-approximation, [@Niehaus2001; @Frauenheim2002] which allows for an efficient calculation of the excitation energies and the required oscillator strengths within the dipole approximation. The $\gamma$-approximation is based on the adiabatic LDA approximation which does not include electron-hole interactions. Therefore excitonic effects are not described. The proper description of excitonic effects goes well beyond the scope of this work and is currently only possible for systems with only a few atoms per unit cell.[@Qiu2013a] The TD-DFRT-TB method, however, allows to study systems with thousands of atoms.
DFTB and TD-DFRT-TB calculations were performed with the deMon computer code[@Koster2011] using periodic boundary conditions. The k-space was sampled at the $\Gamma$-point, only. Therefore all calculations were done in sufficiently large supercells.
The line defects were structurally optimized with density functional theory (DFT) calculations using the Vienna Ab-initio Simulation Package (VASP).[@Kresse1996; @Kresse1996a] These optimization runs were performed with the PBE generalized gradient approximation for the exchange-correlation functional,[@Perdew1996] employing the PAW method [@Blochl1994a] and a plane-wave basis set with a kinetic energy cutoff of 364 eV. For the k-point sampling of the Brillouin zone, $\Gamma$-point centered grids and an in-plane sampling density of 0.1/[Å]{}$^2$ were used. The k-space integration was carried out with a Gaussian smearing width of 0.05 eV for all calculations.
In all calculations we simulate infinite 2D layers and use periodic boundary conditions. The unit cells were built with at least 14 [Å]{} separation between replicas in the perpendicular direction to achieve negligible interaction. All systems were fully structurally optimized until all inter-atomic forces were below 0.01 eV/[Å]{}.[^1] Spin-orbit interactions (SOI) in ML-MoS$_2$ induce a small band splitting of ca. 0.15 eV near the valence band maximum but they have negligible influence on the geometry, heat of formation, and the localization of in-gap states.[@Molina-Sanchez2013] Therefore we neglected SOI, because these SOI effects are very small compared to the effects that we are discussing below.
Results and Discussion
======================
defect structures
-----------------
Figure \[fig:PD-struct\](a) shows the structure of the four vacancy point defects within ML-MoS$_2$ that are considered here. The defect $V_S$ corresponds to a single S atom vacancy on one side of the ML, in $V_{2S}$ two S atoms are removed, one from the upper and one from the lower S layer, V$_{Mo}$ is a Mo atom vacancy and V$_{MoS_2}$ represents the absence of a MoS$_2$ unit. Experimental [@Komsa2013b; @Zhou2013; @Hong2015] and theoretical studies [@Yuan2014a; @Noh2014; @Komsa2015; @Haldar2015] of point defects highlight the importance of sulfur vacancy defects V$_S$ and V$_{2S}$, which are abundant because of their low defect formation energy. Therefore our work focuses on vacancy-type defects. The point defects are placed in the center of hexagonal supercells of varying cell sizes, as shown in Fig. \[fig:PD-struct\](b). So we study hexagonal arrangements of point defects of varying defect concentration. If the supercell contains one defect, then the defect concentration is $1/N_{cells}$, where $N_{cells}$ the number of primitive unit cells per supercell. Table \[tab:concentration\] lists defect concentrations, defect separations and area densities of the considered systems.
![(a) The structures of the point defects models, considered in this study: single sulfur vacancy V$_S$, double sulfur vacancy V$_{2S}$, single molybdenum vacancy V$_{Mo}$, and MoS$_2$ vacancy V$_{MoS_2}$. (b) shows the hexagonal supercells (gray lines) of monolayer MoS$_2$, used to simulate a defect concentration of 11.1%. In all panels brown and yellow balls represents molybdenum and sulfur atoms, respectively. []{data-label="fig:PD-struct"}](pics/PD-structure){width="15cm"}
[defect concentration (%)]{} 25.0 11.1 6.3 1.2
------------------------------------------------ ------- ------- ------- -------
[defect separation (Å)]{} 6.56 9.84 13.12 29.52
[area density ($10^{13}$ defects/cm$^{-2}$)]{} 26.83 11.93 6.708 1.325
: Parameters, characterizing MoS$_2$ monolayers with different concentrations of $V_S$, $V_{2S}$, V$_{Mo}$ and V$_{MoS_2}$ point defects. []{data-label="tab:concentration"}
![(a) The atomic structure and the unit cell (gray lines) of the 8-4 mirror grain boundary model in a MoS$_2$ monolayer. $d$ is the separation between the individual GBs. Due to the periodic boundary conditions the simulated systems correspond to arrays of parallel, aligned GBs. (b) 4-8-8 and 8-4-4 mirror GB structures and (c) 5-7 tilt GB structure. The left and right parts of the GBs are represented by magenta and gray color, respectively. Brown and yellow balls represents Mo and S-atoms, respectively. []{data-label="fig:LD-struc"}](pics/LD-structure){width="12cm"}
The second type of systems, studied here, are periodic line defects or grain boundaries (GBs) as shown in Fig. \[fig:LD-struc\]. We study three types of mirror grain boundaries, 8-4, 4-4-8 [@VanderZande2013] and 8-8-4 and one 5-7 tilt GB with a tilt angle of 38.2$^\circ$. The tilt angle is the misorientation angle of adjacent grains of ML-MoS$_2$ and the numbers (e.g. 8-4 or 5-7) correspond to the sequence of topological “polygon” defects that create the GB. Mind that the defects are only polygons in the planar projection and that their actual structure is three-dimensional. The tilt angle of mirror GBs is 60$^\circ$ (or 180$^\circ$ or 300$^\circ$). Furthermore there are different ways of constructing 5-7 line defects. Our 5-7 defects include Mo-Mo bonds in the 7-rings and S-S bonds in the 5-rings. They are structurally identical on the left and on the right side. Different types of 5-7 GB were previously studied. [@Zou2012] Since we use periodic boundary conditions, the tilt, induced by one GB, has to be compensated by a second GB with an opposite tilt to create a periodic system. Therefore all our supercells contain two GBs, one on the left and one on the right. The supercells, containing two GBs and the separation length $d$ between the GBs are illustrated in Fig. \[fig:LD-struc\].
\[sec:localization\] strongly localized defect states
-----------------------------------------------------
![ The strongly localized nature of (a) point and (b) line defect states. The defect state is localized on the shell of atoms directly surrounding the defect (a atoms), the second shell (b atoms) has very small contributions and atoms beyond that (c,d atoms) do not significantly contribute to the defect state. (a) Panels, from top to bottom: the local density of states (LDOS) of the Mo atoms a,b,c,d near the V$_{2S}$ point defect (1.2% defect concentration). (b) Similar results for the LDOS of the 8-4 grain boundary (GB separation is $d=67$ Å). The last panel in (a) and (b) shows the total DOS of monolayer (ML) MoS$_2$.[]{data-label="fig:PD-localization"}](pics/localization){width="15cm"}
First, we discuss the observation that both point and line defect states in MoS$_2$ are strongly localized. This is demonstrated by the local density of states (LDOS) of Mo atoms along a line starting on the defect (a atoms) and moving away from it (atoms b,c,d) in Fig.\[fig:PD-localization\]. The LDOS is the contribution of an individual atom to the total DOS. We analyzed the LDOS of all considered systems and the result was always as it is shown in Fig.\[fig:PD-localization\] for the V$_{2S}$ point defect in (a) and the 8-4 GB in (b): the defect state is localized on the a atoms, i.e. the shell of Mo atoms directly surrounding the point defect or, for GBs, the atoms involved in the “polygon” defect. The second shell of Mo atoms (b atoms) contributes very little to the defect state and atoms beyond that shell are basically bulk atoms with no significant contributions. Another manifestation of the strong localization of the defect states is the relative insensitivity of the defect formation energy of point defects on the defect concentration, even in the limit of high concentrations. We used DFTB to calculate the defect formation energy as described in the appendix and listed in table \[tab:Eform\]. The relative increase of the defect formation energies of V$_S$ and V$_{2S}$ defects from concentrations of 1.2% to 6.3%, 11.1%, and 25% on the average is only 3% and 7%, 25%, respectively. This result is particularly significant for systems with 25% defects, where every fourth unit cell has a defect and still the defect formation energy for sulfur vacancies increases only by 25% as compared to the dilute limit (1.2%). The strongly localized nature of the defect states leads to very small interactions between individual defects, even if they are separated by only a few nanometers (also see the defect separations in Tab. \[tab:concentration\]). This result explains why sulfur vacancy defects can be found in high concentrations. [@Qiu2013b] For simplicity, our defect structures are symmetrically arranged and homogeneously distributed. This is certainly an idealized situation. However, the strong localization of the defect states and the small electronic interactions between them, leads us to conjecture that the insights of this article will be largely independent of the specific arrangement of the point defects.
Similar calculations of the defect formation energy of the 8-4 GBs for different separations $d$ ranging between 13 and 67 [Å]{} did not show any separation dependence. This shows the absence of long range stress fields and also agrees with the electronic structure analysis that shows that the defect states are strongly localized on a few shells of atoms near the actual defect and do not interact with other line defects that are nearby ($d > 13$ [Å]{}).
characterization of the in-gap states
-------------------------------------
![The ’fingerprints’ of point defects: characteristic, defect induced in-gap states of the considered point defects shown by the total density of states (DOS). The in-gap states are highlighted in color and the bulk states (and ideal monolayer MoS$_2$) are shown in black. []{data-label="fig:in-gap-PD"}](pics/PD-TDOS-ingap){width="9cm"}
Figure \[fig:in-gap-PD\] indicates in color the in-gap states induced by different types of point defects. The highlighted states are ’deep levels’. Additionally, the defects also induce ’shallow levels’ that are close to the valence or condition band edges or levels that lie deep within these bands. In Fig. \[fig:in-gap-PD\] the latter states are not explicitly indicated.
Single and double sulfur vacancies V$_{S}$, V$_{2S}$ are characterized by a single peak near the center of the band gap. The peak corresponds to two unoccupied states (this counting neglects the spin-degeneracy). [@Noh2014; @Zhou2013; @Pandey2016; @Haldar2015] The difference between V$_{S}$ and V$_{2S}$ is that the level shifts to lower energies. Mo point defects induce three deep levels. The two levels at lower energies are two-fold degenerate and the last one is a non-degenerate state. [@Noh2014; @Haldar2015] Finally the V$_{MoS_2}$ defect has four levels in the gap. The second level (at about -3 eV) is two-fold degenerate the other three are single levels. A comparision with full DFT calculations indicates that DFTB tends to place the in-gap states closer to the band edges, while full DFT places the levels more in the center of the gap. However, neither DFT nor DFTB are suitable to precisely determine the position of the in-gap states. This requires quasiparticle-based electronic structure methods that are computationally very demanding and go beyond the scope of the present work. The strong degree of localization combined with sharply peaked in-gap states could potentially explain the emission of single photons from point defects in ML WSe$_2$ (that is very similar to MoS$_2$). [@Srivastava2014a; @He2015; @Koperski2015; @Chakraborty2015; @Kumar2015] Single photon emission is only possible if isolated quantum system with localized states are excited. The strong localization of the point defects creates atom-like states with sharply peaked eigenstates. However electronic excitations between deep levels would result in photons with lower energies than the absorption edge of the monolayer. But experimentally the frequency of the single-photons is observed at energies only slightly lower than this edge. Furthermore, electron-phonon coupling should broaden the localized in-gap states into vibronic bands as it is frequently observed in luminescent centers.[@Lounis2005a] The recent reports however found sharply peaked optical spectra. Further analysis about the nature of the single photon emission in TMDs is therefore necessary.
![The ’fingerprints’ of line defects: characteristic, defect induced in-gap states of the line defects shown by the band structure and the local density of states (LDOS) of a Mo atom at the defect. For the 488+844 system the LDOS is given for the individual line defects ($a$ is an atom at a 4-unit and a$^*$ is at a 8-unit of the 844 line defect). The monolayer band gap is indicated by dashed horizontal lines. []{data-label="fig:in-gap-LD"}](pics/LD-ingap.pdf){width="\columnwidth"}
Also line defects induce in-gap states as shown for all considered GBs in Fig. \[fig:in-gap-LD\]. Here we consider periodic line defects as shown in Fig. \[fig:LD-struc\]. Such structures are actually an idealization. An analysis of real line defects in the transmission electron microscope reveal that real structures can rarely be considered as ideal and periodic.[@VanderZande2013] They are rather irregular with occasional periodic fragments. For the theoretical analysis such periodic fragments, however, offer a good model to analyze the basic change of the electronic structure due to GBs.
The fact that our line defects are one-dimensional, periodic structures leads to the formation of a defect band structure that are related to states that are delocalized along the direction of the GB. However, as demonstrated in the previous section, the states do not spread out much into the direction perpendicular to the GB. The cosine-like dispersion of the defect bands leads to a broader energy signature in the density of states (Fig. \[fig:in-gap-LD\]) as compared to point defects. The most dispersive defect bands occur as typical double peaks in the DOS (e.g. 84-GB or 488-GB).
Similar to point defects also line defects create a characteristic ’fingerprint’ of in-gap states.[@Zou2012; @VanderZande2013; @Zhou2013; @Zou2015] The 8-4 GB induces a single ’deep’ defect band.[^2] Its small dispersion creates two peaks in the DOS just below -3 eV in Fig. \[fig:in-gap-LD\]. The peak at about -3.5 eV corresponds to the onset of a shallow defect band. The 5-7 GB introduces 3 relatively dispersionless defect bands within the band gap. The band structure of the 488+844 system exhibits multiple defect bands where the distinction between shallow and deep levels is not always easy to make. The LDOS indicates which GB creates with states. The 488 GB generates the defect band around -3 eV (occuring as double-peak) and further levels above and below the valence band edge (VBE). The states of the 844 GB can be distinguished between states related the structural 4-units ($a^*$ atoms) and the 8-units ($a$ atoms). The $a^*$ states are near the VBE and two bands near the conduction band edge (CBE) are related to $a$ atoms. A comparison with DFT results shows again that DFTB places the in-gap states closer to the band edges.
dependence of electronic structure on point defect concentration
----------------------------------------------------------------
![The electronic density of states (DOS) for different point defect models (V$_S$ and V$_{Mo}$) and point defect concentrations, varying between 1.2% and 25.0%, and the DOS of the pristine MoS$_2$ layer. The dashed lines represent the Fermi-energy. Point defects in concentrations of up to 6% (i) leave the bulk properties of MoS$_2$ nearly intact, (ii) create sharp in-gap states whose intensity merely increases with the concentration but their energy position is constant. []{data-label="fig:PD-DOS"}](pics/PD-DOS-concentr){width="10cm"}
The influence of varying defect concentrations on the electronic structure is depicted in Fig. \[fig:PD-DOS\] for the examples of V$_{2S}$ and V$_{Mo}$. For defect concentration of up to 6% the bulk electronic structure is left almost intact. With increasing concentration the in-gap states grow in intensity due to a stronger relative weight as compared to the ideal ML. Due to their strongly localized nature, defects do not interact much and therefore remain quasi-isolated objects. As a consequence the energy position of the in-gap states remains constant and is not changing as the concentration increases. Only for concentrations of 10% and more, the modifications of the general electronic structure are significant and for 25% even the bulk states are strongly influenced, as it to be expected for such high defect concentrations. The same trends were found for the other considered point defects (V$_S$ and V$_{MoS_2}$, not shown).
impact on optical absorption
----------------------------
(a)![The impact point and line defects on the optical absorption spectrum of monolayer MoS$_2$. (a) The impact of the considered point defects for different concentrations. The defect-free system is shown as black line. For defect concentrations of 1.2% (and below) the influence on the spectra is very small for all systems. (b) The impact of line defects on the absorption spectrum. As indicated by the comparison with the monolayer spectrum, the presence of the line defects does not have a significant influence on the optical absorption. []{data-label="fig:PD-absorpt"}](pics/PD-absorption "fig:"){width="9cm"} (b) ![The impact point and line defects on the optical absorption spectrum of monolayer MoS$_2$. (a) The impact of the considered point defects for different concentrations. The defect-free system is shown as black line. For defect concentrations of 1.2% (and below) the influence on the spectra is very small for all systems. (b) The impact of line defects on the absorption spectrum. As indicated by the comparison with the monolayer spectrum, the presence of the line defects does not have a significant influence on the optical absorption. []{data-label="fig:PD-absorpt"}](pics/LD-absorption "fig:"){width="4.5cm"}
The concentration dependence of the optical and transport properties of random point defects in the limit of small concentration (1% and smaller) was studied previously within an empirical tight-binding approach by Yuan et al.[@Yuan2014a] The general impact on the transport and optical properties at these concentrations was found to be rather small and our study confirms these results. Here, we study the impact of point defects in the limit of higher defect concentrations (more than 1.2%). Figure \[fig:PD-absorpt\](a) shows our results which are based on the explicit calculations of the oscillator strength using TD-DFRT-TB which takes into account the symmetry of the orbitals, the dipole transition matrix elements as well-as many-body energy renormalizations of the transition energies. Such approach is necessary because the intensity of specific optical transitions cannot be inferred from pure electronic structure analysis as done in the previous section. For the defect-free ML the (’bulk’) absorption edge is found to be at 1.9 eV, which agrees well with experimental findings.[@Splendiani2010; @Mak2010] Increasing defect concentrations for all defects (i) lower the absorption at energies greater than 1.9 eV and (ii) introduce absorption peaks at energies smaller than 1.9 eV. Sulfur defects at high concentrations (above 10%) introduce intense defect peaks, while for V$_{Mo}$ and V$_{MoS_2}$ the absorption is generally low. So the creation of V$_{Mo}$ defects in high concentrations would be a way to quench the the overall absorption in MoS$_2$, while samples with high concentrations of $V_{S}$ or $V_{2S}$ defects introduce intense spectral features below the bulk absorption edge. However, at defect concentrations of 1.2% and below the overall impact on the absorption spectrum for all type of defects is very small and the absorption spectrum nearly coincides with that of the defect-free ML.
Figure \[fig:PD-absorpt\](b) shows the absorption spectrum of the 8-4 GB. Although line defects introduce in-gap states (see Fig. \[fig:in-gap-LD\]) the related oscillator strength for the optical transitions is so small that it does not have a significant influence on the absorption and the spectrum is almost indistinguishable from the defect-free ML. For the other GB systems the result is the same (and therefore not explicitly shown). We also checked the influence of the GB separation on the absorption but did not find any significant influence in the range ($d =32 \dots 67$ [Å]{}). A comparison with table \[tab:concentration\] shows that these are defects separations where in the case of point defects electronic interactions could be neglected and our results for the GBs are consistent with this picture.
Conclusions
===========
In view of recent experiments that explore physical phenomena of defective transition metal chalcogenides (doping, optical properties, single-photon emission) our study aimed at a deeper understanding of such systems. We employed electronic structure methods based on density functional theory to study the electronic and optical properties of single layer MoS$_2$ with different types of point and line defects. For all types of defects we found that the electronic states that are induced by the defects are strongly localized on atoms that are forming the defect and on the shell of atoms surrounding the defect and do not significantly reach out much further in space. The strongly localized nature is further reflected in a weak dependence of the defect formation energy on the defect concentration or line defect separation. In the electronic energy spectrum the defect states occur as deep levels in the bulk band gap, as shallow levels very close to the band edges, as well as in-between the bulk states. Due to their strongly localized nature, all states of point defects are sharply peaked in energy. Periodic line defects form nearly dispersionless one-dimensional band structures and the related spectral features are also strongly peaked. Electronic structure analysis reveals that point defects in concentrations of up to 6% leave the bulk properties of MoS$_2$ nearly intact and create sharp in-gap states whose intensity merely increases with the concentration but their energy position is constant. The impact of point defects on the optical absorption for concentrations of 1% and below is found to be very small. For higher defect concentrations molybdenum vacancies quench the overall absorption and sulfur defects lead to sharp absorption peaks below the absorption edge of the ideal ML. The considered line defects have such a low oscillator strength that there is practically no impact on the absorption spectrum.
This work was financially supported by International Max Planck Research School “Dynamical Processes in Atoms, Molecules and Solids” and the computational resources for this project were provided by ZIH Dresden. We thank Dr. Igor Baburin for fruitful discussions.
\[sec:appendix\]Defect Formation energy
=======================================
The defect formation energy as discussed in Sec. \[sec:localization\] is defined as $$E_\mathrm{form} = [E_\mathrm{system} - N_\mathrm{Mo} \mu_\mathrm{Mo} - N_\mathrm{S} \mu_\mathrm{S}]/N_\mathrm{defect},$$ where $E_\mathrm{system}$ is the total energy of a supercell with $N_\mathrm{defect}$ defects, $N_\mathrm{Mo,S}$ is the number of Mo or S atoms in the supercell, $\mu_\mathrm{Mo,S}$ is the chemical potential of Mo or S, respectively. Using standard relations of thermodynamics this expression can be transformed to [@Groß2009] $$E_\mathrm{form} = [E_\mathrm{system} - (N_\mathrm{Mo} \ \mu_\mathrm{MoS_2}^\mathrm{bulk}) -
(2 N_\mathrm{Mo} - N_\mathrm{S}) \mu_\mathrm{S}]/N_\mathrm{defect},$$ where $\mu_\mathrm{MoS_2}^\mathrm{bulk}$ is the chemical potential of bulk MoS$_2$. The potential $\mu_\mathrm{S}$ varies between $$\frac{1}{2} \Delta_R H + \mu_\mathrm{S}^\mathrm{bulk} \le \mu_\mathrm{S} \le \mu_\mathrm{S}^\mathrm{bulk},$$ the Mo-rich limit (left-hand side) and the S-rich limit (right-hand side). $\Delta_R H$ is the heat of formation of MoS$_2$, and $\mu_\mathrm{S}^\mathrm{bulk}$ is the chemical potential of bulk sulfur (alpha-S). To obtain the formation energies in table \[tab:Eform\] the total energies $E_\mathrm{system}$ and $\mu_\mathrm{MoS_2}^\mathrm{bulk} = -176.191$ eV were calculated with DFTB, the parameters $\Delta_R H = -2.99$ eV, $\mu_\mathrm{S}^\mathrm{bulk} = -6.13$ eV, that merely define the range of the chemical potential $\mu_\mathrm{S}$, were adjusted to DFT values of $E_\mathrm{form}$ according to reference. [@Zhou2013] The last line of table \[tab:Eform\] provides a comparison of our DFTB formation energies with values obtained by DFT calculations. We obtain good agreement and the small deviation are within standard errors of the DFTB approximation.
------------------------------ ------------- ------------ ------------- ------------
defect conc.(%) [Mo-rich]{} [S-rich]{} [Mo-rich]{} [S-rich]{}
25.0 2.54 4.03 4.41 7.40
11.1 2.11 3.61 3.61 6.61
6.3 2.02 3.52 3.41 6.40
1.2 1.97 3.46 3.26 6.26
literature [@Zhou2013] (3 %) 1.5 3.0 3.0 6.0
------------------------------ ------------- ------------ ------------- ------------
: The concentration dependence of the formation energies of sulfur vacancy defects.[]{data-label="tab:Eform"}
[^1]: Mind that DFT and DFTB find slightly different Mo-S equilibrium bond lengths of 2.41 [Å]{} and 2.45 [Å]{}, respectively. In the electronic properties these differences are reflected in different band gaps of systems with point defect (optimized with DFTB) and line defects (optimized with DFT)
[^2]: As our unit cells contain two GBs, the number of in-gap states is actually doubled.
|
---
author:
- Peter Ballett
- ', Matheus Hostert'
- ', Silvia Pascoli'
- ', Yuber F. Perez-Gonzalez'
- ', Zahra Tabrizi'
- and Renata Zukanovich Funchal
bibliography:
- 'trident.bib'
title: Neutrino Trident Scattering at Near Detectors
---
Introduction {#sec:intro}
============
The Standard Model (SM) has been confronted with a variety of experimental data and has so far emerged as an impressive phenomenological description of nature, except in the neutrino sector. The observation of neutrino flavour oscillations by solar, atmospheric, reactor and accelerator neutrino experiments over the last 50 years has revealed the existence of neutrino mass and flavour mixing, making necessary the first significant extension of the SM.
The precise determination of the neutrino mixing parameters as well as the search for the neutrino mass ordering and leptonic CP violation drive both present and future accelerator neutrino experiments. To accomplish these tasks, these experiments rely on state-of-the-art near detectors, made of heavy materials, located a few hundred meters downstream of the neutrino source and subjected to a high intensity beam. Their main purpose is to ensure high precision measurements at a far detector by reducing the systematic uncertainties related to neutrino fluxes, charged-current (CC) and neutral-current (NC) cross sections and backgrounds. The high beam luminosity they are subjected to (about $10^{21}$ protons on target) and their relatively large fiducial mass of high-$Z$ materials (typically 100 ton) make these detectors ideal places to investigate rare neutrino-nucleus interactions ($\sigma\lesssim 10^{-44}$ cm${}^2$), such as neutrino trident scattering.
Trident events are processes predicted by the SM as the result of (anti)neutrino-nucleus scattering with the production of a charged lepton pair [@Czyz:1964zz; @Lovseth:1971vv; @Fujikawa:1971nx; @Brown:1971qr; @Koike:1971tu], ${\accentset{(-)}{\nu}}_{\alpha}+{\cal{H}} \to {\accentset{(-)}{\nu}}_{\alpha \,{\rm{or}} \, \kappa(\beta)} + \ell_{\beta}^- + \ell_\kappa^+ +{\cal{H}}$, $\{\alpha,\beta,\kappa\}\in \{e,\mu,\tau\}$[^1] where $\cal{H}$ denotes a hadronic target. Depending on the (anti)neutrino and charged lepton flavours in the final-state, the process will be mediated by the $Z^0$ boson, $W$ boson or both. Coherent interactions between (anti)neutrinos and the atomic nuclei are expected to dominate these processes as long as the momentum transferred $Q$ is significantly smaller than the inverse of the nuclear size [@Czyz:1964zz]. For larger momentum transfers diffractive and deep-inelastic scattering become increasingly relevant [@Magill:2016hgc]. Although this process exists for all combinations of same-flavour or mixed flavour charged-lepton final-states, to this day only the $\nu_\mu$-induced dimuon mode, ${\accentset{(-)}{\nu}}_\mu + {\cal{H}} \to {\accentset{(-)}{\nu}}_\mu + \mu^+ + \mu^- + {\cal{H}}$, has been observed. The first measurement of this trident signal performed by CHARM II [@Geiregat:1990gz] is also the one with the largest statistics: 55 signal events in a beam of neutrinos and antineutrinos with $\langle E_\nu \rangle \approx 20$ GeV. Other measurements by CCFR [@Mishra:1991bv] and NuTeV [@Adams:1998yf] at larger energies soon followed.
As the measurement of trident events may provide a sensitive test of the weak sector [@Brown:1973ih] as well as placing constraints on physics beyond the SM [@Mishra:1991bv; @Gaidaenko:2000hg; @Altmannshofer:2014pba; @Kaneta:2016uyt; @Ge2017; @Magill:2017mps; @Falkowski:2018dmy] it is relevant to investigate how to probe the other modes. This was recognized by the authors of Ref. [@Magill:2016hgc] who calculated the cross sections for trident production in all possible flavour combinations and estimated the number of events expected for the DUNE and SHiP experiments. They used the Equivalent Photon Approximation (EPA) [@Belusevic:1987cw] to compute the cross section in the coherent and diffractive regimes of the scattering. The EPA, however, is known to breakdown for final state electrons [@Kozhushner:1962aa; @Shabalin:1963aa; @Czyz:1964zz] leading, as we will demonstrate here, to an overestimation of the cross section that in some cases is by more than 200%.
In this work, we present a unified treatment of the coherent and diffractive trident calculation beyond the EPA for all modes. We then compute the number and distribution of events expected in each mode at various near detectors, devoting particular attention to the case of liquid argon (LAr) detectors, as they are expected to lead the field of precision neutrino scattering measurements over the next few decades thanks to their excellent tracking and calorimetry capabilities. Finally, we address the likely backgrounds that may hinder these experimental searches — a question that we believe to be of utmost importance given the rarity of the process, and one that has been omitted in earlier sensitivity studies [@Magill:2016hgc; @Altmannshofer:2014pba].
This paper is organized as follows. In Sec. \[sec:xsec\], we explain how to correctly calculate the trident SM cross sections, comparing our results to the EPA and explicitly showing the breakdown of this approximation. In Sec. \[sec:LAr\], we discuss the trident event rates and kinematic distributions at the near detectors of several present and future neutrino oscillation experiments based on LAr technology: the three detectors of the Short-Baseline Neutrino (SBN) Program at Fermilab [@SBNproposal] and the near detector for the long-baseline Deep Underground Neutrino Experiment (DUNE) [@Acciarri:2016ooe; @DUNECDRvolII], also located at Fermilab. We also consider the potential gains from an optimistic future facility: a 100 t LAr detector subject to the novel low-systematics neutrino beam of the Neutrinos from STORed Muons ($\nu$STORM) project [@Soler:2015ada; @nuSTORM2017]. We discuss the sources of background events at these facilities, providing a GENIE-level analysis [@Andreopoulos2009] of how to reduce these backgrounds and assessing the impact they are expected to have on the trident measurement. In Sec. \[sec:others\], we discuss other near detectors that use more conventional technologies: the Interactive Neutrino GRID (INGRID) [@Abe:2011xv; @Abe:2015biq; @Abe:2016fic; @Abe:2016tez], the on-axis iron near detector for T2K at J-PARC, as well as three detectors at the Neutrino at the Main Injector (NuMI) beamline at Fermilab, the one for the Main INjector ExpeRiment $\nu$-A (MINER$\nu$A) [@Altinok:2017xua; @MINERvA:2017] and the near detectors for the Main Injector Oscillation Search (MINOS) [@Adamson:2014pgc; @AlpernBoehm] and the Numi Off-axis $\nu_e$ Appearance (NO$\nu$A) experiment [@Wang:Biao; @sanchez_mayly_2018_1286758]. Finally, in Sec. \[sec:conc\] we present our last remarks and conclusions.
Trident Production Cross Section {#sec:xsec}
================================
In this section we consider neutrino trident production in the SM, defined as the process where a (anti)neutrino scattering off a hadronic system ${\cal H}$ produces a pair of same-flavour or mixed flavour charged leptons $${\accentset{(-)}{\nu}}_{\alpha}(p_1) \,+\, {\cal H}(P) \,\to\, {\accentset{(-)}{\nu}}_{\alpha\, {\rm or}\, \kappa(\beta)} (p_2) \,+\, \ell_\beta^- (p_4) \,+\, \ell_\kappa^+ (p_3) \,+\, {\cal H}(P^\prime),
\label{eq:indices}$$ where $\beta (\kappa)$ corresponds to the flavour index of the negative (positive) charged lepton in both neutrino and antineutrino cases. Neutrino trident scattering can be divided into three regimes depending on the nature of the hadronic target: coherent, diffractive and deep inelastic, when the neutrino scatters off the nuclei, nucleons and quarks, respectively. At the energies relevant for neutrino oscillation experiments, the deep inelastic scattering contribution amounts at most to 1% of the total trident production cross section [@Magill:2016hgc] and we will not consider it further.
The cross section for trident production has been calculated before in the literature, both in the context of the $V-A$ theory [@Czyz:1964zz; @Lovseth:1971vv; @Fujikawa:1971nx] and in the SM [@Brown:1973ih], while the EPA treatment was developed in Refs. [@Kozhushner:1962aa; @Shabalin:1963aa; @Belusevic:1987cw]. Most calculations have focused on the coherent channels [@Czyz:1964zz; @Lovseth:1971vv; @Fujikawa:1971nx; @Brown:1973ih; @Belusevic:1987cw] but the diffractive process has been considered in [@Czyz:1964zz; @Lovseth:1971vv]. More recently, calculations using the EPA have been performed for coherent scattering with a dimuon final-state [@Altmannshofer:2014pba], and for all combinations of hadronic targets and flavours of final-states in [@Magill:2016hgc]. While the EPA is expected to agree reasonably well with the full calculation for coherent channels with dimuon final-states, the assumptions of this approximation are invalid for the coherent process with electrons in the final-state [@Kozhushner:1962aa; @Shabalin:1963aa; @Czyz:1964zz]. For this reason, we perform the full $2\to 4$ calculation without the EPA in a manner applicable to any hadronic target, following a similar approach to Refs. [@Czyz:1964zz; @Lovseth:1971vv]. Our treatment of the cross section allows us to quantitatively assess the breakdown of the EPA in both coherent and diffractive channels for all final-state flavours, an issue we come back to in Sec. \[sec:EPAbreakdown\].
We write the total cross section for neutrino trident production off a nucleus ${\cal N}$ with $Z$ protons and $(A-Z)$ neutrons as the sum $$\sigma_\mathrm{\nu {\cal N}} = \sigma_\mathrm{\nu c} +\sigma_\mathrm{\nu d}\, ,$$ where $\sigma_\mathrm{\nu c}$ ($\sigma_\mathrm{\nu d}$) is the coherent (diffractive) part of the cross section.
The relevant diagrams for these processes in the coherent or diffractive regimes involve the boson $Z^0$, $W$ or both mediators, depending on the particular mode. In the four-point interaction limit, depicted in [Fig. \[fig:Tdiagrams\]]{}, these reduce to only two contributions, one where the photon couples to the negatively and one to the positively charged lepton. In Table \[tab:tridentmodes\] we present the processes that will be considered in this work as well as the SM contributions present in each. Although our formalism applies also to processes with final-state $\tau$ leptons, the increased threshold makes them irrelevant for the experiments of interest in this study and we do not consider them further. The trident amplitude for a coherent (${\rm X=c}$) or diffractive (${\rm X=d}$) scattering regime can be written as $$i \mathcal{M} = \mathrm{L}^\mu (\{p_i\},q) \, \frac{-ig_{\mu \nu}}{q^2} \, \mathrm{H}_{\rm X}^{\nu}(P,P^{\prime})\, ,$$ where $\{p_i\}=\{p_2,p_3,p_4\}$ is the set of outgoing leptonic momenta. $ \mathrm{L}^\mu (\{p_i\},q)$ is the total leptonic amplitude $$\begin{aligned}
\mathrm{L}^\mu & \equiv - \frac{ie G_F}{\sqrt{2}}[\bar{u}(p_2)\gamma^\tau(1-\gamma_5)u(p_1)] \times
\bar{u}(p_4)\left[\gamma_\tau(V_{\alpha\beta\kappa}-A_{\alpha\beta\kappa}\gamma_5)\frac{1}{(\slashed{q}-\slashed{p}_3-m_3)}\gamma^\mu \right . \nonumber \\
& \left . + \gamma^\mu \frac{1}{(\slashed{p}_4-\slashed{q}-m_4)} \gamma_\tau (V_{\alpha\beta\kappa}-A_{\alpha\beta\kappa}\gamma_5) \right] v(p_3)\, ,
\label{eq:Lmu}\end{aligned}$$ and $\mathrm{H}_{\rm X}^{\nu}(P,P^{\prime})$ is the total hadronic amplitude $$\begin{aligned}
H_{\rm X}^\nu &\equiv \langle {\cal H}(P) \vert J_\mathrm{E.M.}^\nu (q^2)\vert {\cal H}(P^\prime)\rangle\, ,
\label{eq:Hmu}\end{aligned}$$ with $q \equiv P - P^\prime$ denoting the transferred momentum, $m_3$ ($m_4$) the positively (negatively) charged lepton mass, $V_{\alpha\beta\kappa}(A_{\alpha\beta\kappa})\equiv g_{V}^{\beta}(g_A^{\beta})\delta_{\beta\kappa}+\delta_{\alpha\beta} \,(\beta=\alpha \, \mathrm{or} \; \kappa)$ the vector (axial) couplings, depending on the channel and have labels in accordance to Eq. (\[eq:indices\]), and ${J}^\nu_{\rm{E.M.}} (q^2)$ the electromagnetic current for the hadronic system ${\cal H}$ (a nucleus or a nucleon).
------------------------------------------------------------------------------------------------------------- --
**(Anti)Neutrino & **SM Contributions\
${\accentset{(-)}{\nu}}_\mu\, {\cal H} \to {\accentset{(-)}{\nu}}_\mu\, \mu^- \mu^+\, {\cal H}$ & CC + NC\
${\accentset{(-)}{\nu}}_\mu\, {\cal H} \to {\accentset{(-)}{\nu}}_e\, e^\pm \mu^\mp\, {\cal H}$ & CC\
${\accentset{(-)}{\nu}}_\mu\, {\cal H} \to {\accentset{(-)}{\nu}}_\mu\, e^- e^+\, {\cal H}$ & NC\
${\accentset{(-)}{\nu}}_e\, {\cal H} \to {\accentset{(-)}{\nu}}_e\, e^- e^+\, {\cal H}$ & CC + NC\
${\accentset{(-)}{\nu}}_e\, {\cal H} \to {\accentset{(-)}{\nu}}_\mu\, \mu^\pm e^\mp\, {\cal H}$ & CC\
${\accentset{(-)}{\nu}}_e\, {\cal H} \to {\accentset{(-)}{\nu}}_e\, \mu^- \mu^+\, {\cal H}$ & NC\
****
------------------------------------------------------------------------------------------------------------- --
: \[tab:tridentmodes\] (Anti)Neutrino trident processes considered in this paper.
We can write the differential cross section as $$\begin{aligned}
\frac{\dd^2 \sigma_{\nu {\rm X}}}{\dd Q^2 \dd \hat{s}}= \frac{1}{32 \pi^2(s-M_{\cal H}^2)^2}\frac{\mathrm{H}_{\rm X}^{\mu\nu}\mathrm{L}_{\mu\nu}}{Q^4}\, ,\end{aligned}$$ where $s = (p_1 + P)^2$, $\hat{s} \equiv 2\,(p_1 \vdot q)$, $Q^2 = -q^2$ and $M_{\cal H}$ is the mass of the hadronic target. We have also introduced the hadronic tensor $\mathrm{H}_{\rm X}^{\mu \nu}$ $$\begin{aligned}
\mathrm{H}_{\rm X}^{\mu\nu} &\equiv \overline{\sum_{\rm{spins}}} \left(\mathrm{H}_{\rm X}^\mu\right)^* \mathrm{H}_{\rm X}^\nu.
$$ The two scattering regimes in which the hadronic tensor is computed will be discussed in more detail in Sec. \[sec:had\]. The leptonic tensor, $\mathrm{L}^{\mu \nu}$, integrated over the phase space of the three final-state leptons, $\dd^{3} \Pi \left(p_1 + q; \{p_i\}\right)$, and merely summed over final and initial spins is given by $$\mathrm{L}^{\mu \nu} (p_1, q) \equiv \int \dd ^{3} \Pi \left(p_1 + q; \{p_i\}\right) \left( \sum_{\rm{spins}} \left( \mathrm{L}^\mu \right)^* \mathrm{L}^\nu \right)\, .
\label{eq:Lmunu}$$ We can use $\mathrm{L}^{\mu \nu}$ to define two scalar functions, one related to the longitudinal ($\mathrm{L}_{\mathrm{L}}$) and the other to the transverse ($\mathrm{L}_{\mathrm{T}}$) polarization of the exchanged photon $$\mathrm{L}_{\mathrm{T}} = -\frac{1}{2}\left( g^{\mu \nu} - \frac{4Q^2}{\hat{s}^2} p_1^\mu p_1^\nu \right) \mathrm{L}_{\mu \nu}, \quad \mathrm{and} \quad \mathrm{L_{L}} = \frac{4Q^2}{\hat{s}^2} p_1^\mu p_1^\nu \mathrm{L}_{\mu \nu}\, .\label{eq:LT_LL}$$ This allows us to write the differential cross section as a sum of a longitudinal and a transverse contribution [@Hand:1963bb] as follows $$\begin{aligned}
\frac{\dd^2 \sigma_{\nu {\rm X}}}{\dd Q^2 \dd \hat{s}} &= \frac{1}{32 \pi^2} \frac{1}{\hat{s}\,Q^2} \left [ h_{\rm X}^\mathrm{T}(Q^2, \hat{s}) \, \sigma^\mathrm{T}_{\nu \gamma} (Q^2, \hat{s}) + h_{\rm X}^\mathrm{L}(Q^2, \hat{s}) \, \sigma^\mathrm{L}_{\nu \gamma} (Q^2, \hat{s}) \right] \, ,\label{eq:full_diff_xsec}\end{aligned}$$ where we have defined two functions for the flux of longitudinal and transverse virtual photons
\[eq:splitting\_function\] $$\begin{aligned}
h_{\rm X}^\mathrm{T}(Q^2, \hat{s}) &\equiv \frac{2}{(E_\nu M_{\cal H})^2} \left[ p_{1 \mu} p_{1\nu} -\frac{\hat{s}^2}{4 Q^2} \, g_{\mu \nu} \right]\mathrm{H}_{\rm X}^{\mu \nu} , \quad \text{and} \\ \quad
h_{\rm X}^\mathrm{L}(Q^2, \hat{s}) &\equiv \frac{1}{(E_\nu M_{\cal H})^2} \, p_{1\mu} p_{1\nu}\, \mathrm{H}_{\rm X}^{\mu \nu}\, ,
$$
and two leptonic neutrino-photon cross sections associated with them[^2] $$\sigma^\mathrm{T}_{\nu \gamma} (Q^2, \hat{s}) = \frac{\mathrm{L_T}}{2 \hat{s}}\, , \quad \mathrm{and} \quad
\sigma^\mathrm{L}_{\nu \gamma} (Q^2, \hat{s}) = \frac{\mathrm{L_L}}{\hat{s}}\, .$$ The kinematically allowed region in the $(Q^2,\hat{s})$ plane can be obtained by considering the full four-body phase space, as in [@Czyz:1964zz; @Lovseth:1971vv; @Fujikawa:1971nx]. The limits for such physical region are given by
\[eq:qslimts\] $$\begin{aligned}
Q_{\rm min}^2&=\frac{M_{\cal H} \hat{s}^2}{2E_\nu(2E_\nu M_{\cal H}-\hat{s})},&\
Q_{\rm max}^2&=\hat{s}-m_L^2,\label{eq:qlimts}\\
\hat{s}_{\rm min}&=\frac{E_\nu}{2E_\nu + M_{\cal H}}\left[m_L^2+2E_\nu M_{\cal H} -\Delta\right]&\
\hat{s}_{\rm max}&=\frac{E_\nu}{2E_\nu + M_{\cal H}}\left[m_L^2+2E_\nu M_{\cal H} +\Delta\right],\label{eq:shatlimts}
\end{aligned}$$
with $m_L\equiv m_3+m_4$, and $$\begin{aligned}
\Delta \equiv \sqrt{(2E_\nu M_{\cal H}-m_L^2)^2-4M_{\cal H}^2 m_L^2}\,.\end{aligned}$$ Let us emphasize that [Eq. (\[eq:full\_diff\_xsec\])]{} is an exact decomposition, and does not rely on any approximation of the process. In the following section, we will show how to calculate the flux functions $h_{\rm X}^\mathrm{T}$ and $h_{\rm X}^\mathrm{L}$ from Eq. \[eq:splitting\_function\] in different scattering regimes. The total cross section for the process can then be computed by finding $\sigma^\mathrm{L}_{\nu \gamma}$ and $\sigma^\mathrm{T}_{\nu \gamma}$ from Eqs. (\[eq:Lmu\]), (\[eq:Lmunu\]) and (\[eq:LT\_LL\]) and integrating over all allowed values of $Q^2$ and $\hat{s}$. Note that $\sigma^\mathrm{L}_{\nu \gamma}$ and $\sigma^\mathrm{T}_{\nu \gamma}$ are universal functions for a given leptonic process and need only to be computed once.
Hadronic Scattering Regimes {#sec:had}
---------------------------
Depending on the magnitude of the virtuality of the photon, $Q = \sqrt{-q^2}$, the hadronic current can contribute in different ways to the trident process. Thus, given the decomposition in [Eq. (\[eq:full\_diff\_xsec\])]{}, the change in the hadronic treatment translates to computing the flux factors $h_{\rm X}^\mathrm{T}$ and $h_{\rm X}^\mathrm{L}$ for each scattering regime. From those flux factors, $\sigma_{\nu\mathrm{c}}$ and $\sigma_{\nu\mathrm{d}}$ can be calculated.
### Coherent Regime (${\rm H}^{\mu \nu}_{\rm c}$)
In the coherent scattering regime the incoming neutrino interacts with the whole nucleus without resolving its substructure. For this to occur frequently, we need small values of $Q$. Despite the relatively large neutrino energies in contemporary neutrino beams, this is still allowed for trident.
In this regime, the hadronic tensor $\mathrm{H}^{\mu\nu}_\mathrm{c}$ for a ground state spin-zero nucleus of charge $Z e$ can be written in terms of the nuclear electromagnetic form factor $F(Q^2)$, discussed in more detail in Appendix \[app:formfactors\], as $$\mathrm{H}^{\mu \nu}_\mathrm{c} = 4Z^2 e^2 \left| F (Q^2)\right|^2 \left(P^\mu - \frac{q^\mu}{2}\right) \left(P^\nu - \frac{q^\nu}{2}\right).$$ From Eq. \[eq:splitting\_function\], we find that the transverse and longitudinal flux functions for the coherent regime are
\[eq:hcoh\] $$\begin{aligned}
h^\mathrm{T}_\mathrm{c}(Q^2, \hat{s}) &= 8 Z^2 e^2 \left(1 - \frac{\hat{s}}{2E_\nu M} - \frac{\hat{s}^2}{4 E_\nu^2 Q^2}\,\right) |F (Q^2)|^2\, , \\
h^\mathrm{L}_\mathrm{c}(Q^2, \hat{s}) &= 4 Z^2 e^2 \left(1 - \frac{\hat{s}}{4E_\nu M}\right)^2 |F (Q^2)|^2\, ,\end{aligned}$$
where $E_\nu$ is the energy of the incoming neutrino and $M$ is the nuclear mass. For a fixed value of $\hat{s}$ in the physical region, the $h^{\rm T}_{\rm c}$ flux function becomes zero at $Q_{\rm min}$ while the longitudinal component does not. This different behaviour can be seen explicitly in their definitions, Eqs. , as the terms in the parenthesis in $h^{\rm T}_{\rm c}$ cancel each other at $Q_{\rm min}$. This does not occur for $h^{\rm L}_{\rm c}$ since the physical values of $\hat{s}$ are always smaller than $E_\nu M$ in this hadronic regime. Due to this fact, $Q_{\rm min}$, which according to Eq. depends on both the neutrino energy and target material, can be approximated to $$\begin{aligned}
Q_{\rm min} \approx \frac{\hat{s}}{2E_{\nu}},\end{aligned}$$ which only depends on the incoming neutrino energy. On the other hand, as $Q$ becomes large, the flux functions $h^{T,L}$ become quite similar, $h^{\rm T}_{\rm c}\approx 2 h^{\rm L}_{\rm c}$, and favour small values of $\hat{s}$. After some critical value of the virtuality $Q$, $h^{\rm T, L}_{\rm c}$ become negligible due to the nuclear form factor. The $Q$ value at which this happens depends on the target material, but not on the incoming neutrino energy. For instance, in the case of an Ar target the flux functions basically vanish for $Q\gtrsim 250$ MeV.
The final cross sections for coherent neutrino trident production on Argon can be seen in [Fig. \[fig:coh\_xsec\]]{}. Despite thresholds being important for the behaviour of these cross sections for GeV neutrino energies, we can see that mixed channels quickly become the most important due to their CC nature. At large energies one can then rank the cross sections from largest to smallest as CC, CC+NC, and NC only channels. Nevertheless, one must be aware of the fact that the cross sections are dominated by low $Q^2$ even at large energies, leading to large effects due to the final-state lepton masses as discussed in [@Magill:2016hgc].
![Cross sections for coherent neutrino trident production on $^{40}$Ar (left) and $^{208}$Pb (right) normalized to $\sigma_0 = Z^2\, 10^{-44}$ cm$^2$. The full (dashed) lines correspond to the scattering of an incoming $\nu_\mu$ ($\nu_e$) produced by the NC (light-blue), CC (purple), and CC+NC (orange) SM interactions. \[fig:coh\_xsec\]](figs/Xsec_4PS_coh.pdf){width="\textwidth"}
### Diffractive Regime ($\mathrm{H}^{\mu\nu}_\mathrm{d}$)
At larger $Q^2$, the neutrino interacts with the individual nucleons of the nucleus. In this diffractive regime $\mathrm{H}^{\mu\nu}_\mathrm{d}$ is given by the sum of the contributions of the two types of nucleons: protons ($\mathrm{N=p}$) and neutrons ($\mathrm{N=n}$), so $$\mathrm{H}^{\mu \nu}_\mathrm{d} (P, P^\prime) = Z\, \mathrm{H}^{\mu \nu}_\mathrm{p} (P, P^\prime)+
(A-Z)\,\mathrm{H}^{\mu \nu}_\mathrm{n}(P, P^\prime)\, ,$$ where each $\mathrm{H}^{\mu \nu}_\mathrm{N}$ is the square of the matrix element of the nucleon electromagnetic current summed over final and averaged over initial spins. Neglecting second class currents, the matrix elements take the form $$\bra{\mathrm{N}(P^\prime)} {J}^\mu_{\rm{E.M.}} (Q^2) \ket{\mathrm{N} (P) } = e \, \overline{u}_\mathrm{N} (P^\prime) \left[ \gamma^\mu F^\mathrm{N}_1(Q^2) - i \frac{\sigma^{\mu \nu} q_{\nu}}{2 M_{\rm N}} F^\mathrm{N}_2(Q^2) \right] u_\mathrm{N}(P)\, ,$$ with $F^\mathrm{N}_{1,2}(Q^2)$ the Dirac and Pauli form factors, respectively. The hadronic tensors are then given by [@Kniehl:1990iv] $$\mathrm{H}^{\mu \nu}_\mathrm{N} = e^2 \left[ 4 \, H_1^\mathrm{N}(Q^2) \left(P^\mu - \frac{q^\mu}{2}\right)\left(P^\nu - \frac{q^\nu}{2}\right) - H_2^\mathrm{N}(Q^2) \left( Q^2 g^{\mu \nu} + q^\mu q^\nu \right) \right]\, ,$$ where the $H_1^\mathrm{N}(Q^2)$ and $H_2^\mathrm{N}(Q^2)$ form factors, functions of $F^\mathrm{N}_{1,2}(Q^2)$, are given in Appendix \[app:formfactors\]. The flux functions in the diffractive regime can then be calculated as
\[eq:dcoh\] $$\begin{aligned}
h^\mathrm{T}_\mathrm{N}(Q^2, \hat{s}) &= 8 \, e^2 \left[ \left(1 - \frac{\hat{s}}{2E_\nu M_{\rm N}} - \frac{\hat{s}^2}{4 E_\nu^2 Q^2 }\,\right) H_1^\mathrm{N}(Q^2) + \frac{\hat{s}^2}{8E_\nu^2 M_{\rm N}^2} H_2^\mathrm{N}(Q^2)\right ]\, ,\label{eq:hTdiff}\\
h^\mathrm{L}_\mathrm{N}(Q^2, \hat{s}) &= 4 e^2 \, \left[ \left(1-\frac{\hat{s}}{4 E_\nu M_{\rm N}} \right)^2 H_1^\mathrm{N}(Q^2) - \frac{\hat{s}^2}{16 E_\nu^2 M_{\rm N}^2} H_2^\mathrm{N}(Q^2)\right]\, .\label{eq:hLdiff}\end{aligned}$$
In the case of the proton, the flux functions $h^{\rm T, L}_{\rm p}$ have some unique features given the presence of both electric and magnetic contributions. Specifically, the transverse function is non-zero at $Q=Q_{\rm min}$ for a fixed $\hat{s}$, due to the additional term proportional to $H_2^{\rm p}$. Indeed, for large values of $\hat{s}$, the $H_2^{\rm p}$ term dominates the transverse function. An opposite behaviour occurs for the longitudinal component. There, the $H_1^{\rm p}$ term dominates over the second term for all physical values of $\hat{s}$, $Q$, and for any incoming neutrino energy. On the other hand, the flux functions of the neutron, which have only the magnetic moment contribution, have somewhat different characteristics. While $h^{\rm T}_{\rm n}$ behaves similarly to $h^{\rm T}_{\rm p}$, that is, it is dominated by the second term for large values of $\hat{s}$, $h^{\rm L}_{\rm n}$ is zero at $Q_{\rm min}$ due to the exact cancellation between the $H_{1,2}^{\rm n}$ terms. This cancellation is not evident from Eq. ; however, simplifying the longitudinal component for the neutron case, one finds $$\begin{aligned}
h^\mathrm{L}_\mathrm{n}(Q^2, \hat{s}) &=4 e^2 \left(1+\frac{Q^2}{4M_{\rm n}^2}\right)\frac{Q^2}{4 M_{\rm N}^2}\left( 1 - \frac{\hat{s}}{2 E_\nu M_{\rm N}} - \frac{\hat{s}^2}{4 E_\nu^2 Q^2} \right) \left| F^\mathrm{n}_2(Q^2) \right|^2,\end{aligned}$$ which is zero for $Q=Q_{\rm min}$. Also, this shows why $h^{\rm L}_{\rm p}$ does not vanish at $Q_{\rm min}$ since there we have the additional contribution of the electric component.
When the neutrino interacts with an individual nucleon inside the nucleus, one must be aware of the nuclear effects at play. One such effect is Pauli blocking, a suppression of neutrino-nucleon interactions due to the Pauli exclusion principle. Modelling the nucleus as an ideal Fermi gas of protons and neutrons, one can take Pauli blocking effects into account by requiring that the hit nucleon cannot be in a state which is already occupied [@Brown:1971qr]. This requirement is implemented in our calculations by a simple replacement of the differential diffractive cross section $$\begin{aligned}
\frac{\dd^2 \sigma_{\nu \mathrm{d}}}{\dd Q^2 \dd \hat{s}}\to f (|\vec{q}|) \, \frac{\dd^2 \sigma_{\nu \mathrm{d}}}{\dd Q^2 \dd \hat{s}},\end{aligned}$$ where $|\vec{q}|$ is the magnitude of the transferred three-momentum in the lab frame. In particular, following [@Brown:1971qr], assuming an equal density of neutrons and protons, we have $$f (|\vec{q}|) = \begin{cases} \displaystyle
\frac{3}{2} \frac{|\vec{q}|}{2 \, k_F} - \frac{1}{2} \left( \frac{|\vec{q}|}{2 \, k_F} \right)^3 ,\, &\mathrm{if }\;\; |\vec{q}| < 2\, k_F\, ,\\
1,\, &\mathrm{if }\;\; |\vec{q}| \geq 2 \, k_F\, ,
\end{cases}$$ where $k_F$ is the Fermi momentum of the gas, taken to be $235$ MeV. This is a rather low value of $k_F$ and the assumption of equal density of neutrons and protons must be taken with care for heavy nuclei. We refrain from trying to model any additional nuclear effects as we believe that this is the dominant effect on the total diffractive rate, particularly when requiring no hadronic activity in the event. The net result is a reduction of the diffractive cross section by about $50\%$ for protons and $20\%$ for neutrons.
Our final cross sections for this regime can be seen in [Fig. \[fig:dif\_xsec\]]{}. One can clearly see that the neutron contribution is subdominant, and that, up to factors of $Z^2$, the proton one is comparable to the coherent cross section. Note that now the typical values of $Q^2$ are much larger than in the coherent regime and the impact of the final-state lepton masses is much smaller.
![Cross sections for diffractive neutrino trident production on neutrons (left) and protons (right), including Pauli blocking effects as described in the text, normalized to $\sigma_0 = 10^{-44}$ cm$^2$. The full (dashed) lines correspond to the scattering of an incoming $\nu_\mu$ ($\nu_e$) produced by the NC (light-blue), CC (purple), and CC+NC (orange) SM interactions. \[fig:dif\_xsec\]](figs/Xsec_4PS_diff.pdf){width="\textwidth"}
Breakdown of the EPA \[sec:EPAbreakdown\]
-----------------------------------------
In order to understand the breakdown of the EPA in the neutrino trident case, let us first remind briefly the reader about the Weizsäcker–Williams method of equivalent photons in Quantum Electrodynamics (QED) [@vonWeizsacker:1934nji; @Williams:1934ad], and the main reason for its validity in that theory. The EPA, first introduced by E. Fermi [@Fermi:1924tc], is based on a simple principle: when an ultra-relativistic particle $P_i$ approaches a charged system $C_s$, like a nucleus, it will perceive the electromagnetic fields as nearly transverse, similar to the fields of a pulse of radiation, [*i.e.*]{}, as an on-shell photon. Therefore, it is possible to obtain an approximate total cross section for the inelastic scattering process producing a set of final particles $P_f$, $\sigma_{\rm t}(P_i + C_s \to P_f + C_s)$, by computing the scattering of the incoming particle with a real photon integrated over the energy spectrum of the off-shell photons, $$\begin{aligned}
\sigma_{\rm t}(P_i + C_s \to P_f + C_s)\approx\int\, dP(Q^2,\hat{s})\,\sigma_\gamma(P_i + \gamma \to P_f; \hat{s}, Q^2 = 0),\end{aligned}$$ where the photo-production cross section for the process $P_i + \gamma \to P_f$, $\sigma_\gamma(P_i + \gamma \to P_f; \hat{s}, Q^2 = 0)$, depends on the center-of-mass energy of the $P_i$–photon system, $\sqrt{\hat{s}}$. Here $dP(Q^2,\hat{s})$ corresponds to the energy spectrum of the virtual photons, that is, the probability of emission of a virtual photon with transferred four-momentum $Q^2$ resulting in an center-of-mass energy $\sqrt{\hat{s}}$. For trident scattering off a nuclear target, this probability can be approximated by [@Belusevic:1987cw; @Altmannshofer:2014pba] $$\begin{aligned}
\label{eq:GenEPA}
dP(Q^2,\hat{s})=\frac{Z^2e^2}{4\pi^2}|F (Q^2)|^2\,\frac{d\hat{s}}{\hat{s}}\,\frac{dQ^2}{Q^2}\, .\end{aligned}$$ A crucial fact in QED is that the cross section $\sigma_\gamma^{\rm QED}(P_i + \gamma \to P_f; \hat{s},0)$ is inversely proportional to $\hat{s}$, $$\begin{aligned}
\sigma_\gamma^{\rm QED}(P_i + \gamma \to P_f; \hat{s},0) \propto \frac{1}{\hat{s}}\,.\end{aligned}$$ We see clearly that small values of $\hat{s}$ and consequently of the transferred four-momentum $Q^2$ dominate the cross section. Hence, the on-shell contribution is much more significant than the off-shell one, so the EPA will be valid and give the correct cross section estimate for any QED process.
Now, let us consider the case of neutrino trident production. In this case, the equivalent-photon cross section in the four-point interaction limit has a completely opposite dependence on the center-of-mass energy; it is *proportional* to $\hat{s}$, $$\begin{aligned}
\sigma_\gamma^{\rm FL}(P_i + \gamma \to P_f; \hat{s}, 0)\propto G_{\rm F}^2\, \hat{s}\, .\end{aligned}$$ This dependence is a manifestation of the unitarity violation in the Fermi theory. Therefore, we can see that for weak processes larger values of $\hat{s}$, and, consequently, larger values of $Q^2$ are more significant [@Kozhushner:1962aa; @Shabalin:1963aa]. The EPA is then generally not valid for the neutrino trident production, as the virtual photon contribution dominates over the real one. Nevertheless, one may wonder if there is a situation in which the EPA can give a reasonable estimate for a neutrino trident process. As noticed in the early literature [@Kozhushner:1962aa; @Shabalin:1963aa], the presence of the nuclear form factor introduces a cut in the transferred momentum which, in turn, makes the EPA applicable for the specific case of the dimuon channel in the coherent regime. Let us discuss this in more detail.
Recalling our exact decomposition, [Eq. (\[eq:full\_diff\_xsec\])]{}, it is necessary to consider two assumptions for implementing the EPA [@Kozhushner:1962aa]:
1. The longitudinal polarization contribution to the cross section can be neglected, i.e., $\sigma_{\nu\gamma}^\mathrm{L}(Q^2,\hat{s})\approx 0$;
2. The transverse polarization contribution to the cross section can be taken to be on-shell, i.e., $\sigma^\text{T}_{\nu\gamma}(Q^2,\hat{s}) \approx \sigma^\text{T}_{\nu\gamma}(0,\hat{s})$.
Assuming for now that these approximations hold, we can find a simplified expression for the coherent neutrino-target process, described by Eqs. (\[eq:full\_diff\_xsec\]) and (\[eq:hcoh\]), in terms of the photon-neutrino cross section[^3]: $$\begin{aligned}
\sigma_\text{EPA} = \frac{Z^2e^2}{4\pi^2}\int_{m_L^2}^{\hat{s}_{\rm max}} \frac{d\hat{s}}{\hat{s}}\,
\sigma^\mathrm{T}_{\nu\gamma}(0,\hat{s})
\int_{(\hat{s}/2E_\nu)^2}^{Q^2_{\rm max}}\frac{|F (Q^2)|^2}{Q^4} \left[ Q^2(1-y) - M_{\cal H}^2y^2\right]dQ^2\, , \end{aligned}$$ where we introduced the fractional change of the nucleus energy $y$, defined as $\hat{s} = (s-M_{\cal H}^2)y$, and the integration limits can be obtained from after considering that $m_L^2\ll E_\nu M_{\cal H}$. Keeping only the leading terms in the small parameter $y$ [@Belusevic:1987cw], we recover the EPA applied to the neutrino trident case $$\begin{aligned}
\label{eq:EPA_bad}
\sigma_\text{EPA} = \int \sigma^\mathrm{T}_{\nu\gamma}(0,\hat{s}) \, dP(Q^2,\hat{s})\, ,\end{aligned}$$ where $dP(Q^2,\hat{s})$ is given in Eq. (\[eq:GenEPA\]). The EPA in the form of [Eq. (\[eq:EPA\_bad\])]{} has been used in trident calculations for the coherent dimuon channel [@Altmannshofer:2014pba] as well as for coherent mixed- and electron-flavour trident modes and diffractive trident modes [@Magill:2016hgc]. Using our decomposition, we can explicitly compute both $\sigma^\mathrm{L}_{\nu \gamma}$ and $\sigma^\mathrm{T}_{\nu \gamma}$ and verify if the EPA conditions are satisfied for any channel and, if they are not, quantify the error introduced by making this approximation. For that purpose, we will compare the results of the full calculation, [Eq. (\[eq:full\_diff\_xsec\])]{}, with the EPA results, [Eq. (\[eq:EPA\_bad\])]{}, by computing the following ratios in the physical region of the $(Q,\hat{s})$ plane, $$\begin{aligned}
\label{eq:ratios}
\frac{\sigma^{\rm L}(Q^2,\hat{s})\,h_{\rm c}^{\rm L}(Q^2,\hat{s})}{\sigma^{\rm T}(Q^2,\hat{s})\,h_{\rm c}^{\rm T}(Q^2,\hat{s})}\, , \quad \frac{\sigma^\mathrm{T}_{\nu\gamma}(Q^2,\hat{s})}{\sigma^\mathrm{T}_{\nu\gamma}(0,\hat{s})}\, .\end{aligned}$$ The first ratio in Eq. will indicate where the longitudinal contribution can be neglected compared to the transverse one; while, the second ratio will show where the transverse contribution behaves as an on-shell photon.
As an illustration of the general behaviour, we show in Fig. \[fig:4PSvsEPA\] those ratios of cross sections for an incoming $\nu_\mu$ of fixed energy $E_\nu=3$ GeV colliding coherently with an $^{40}$Ar target, for the dielectron (left panels), mixed (middle panels) and dimuon (right panels) channels. On the top panels of Fig. \[fig:4PSvsEPA\] we see that the longitudinal component can be neglected for $Q\lesssim m_\alpha$, for the dielectron and dimuon channels, $\alpha=e,\mu$, while in the mixed case there is a much less pronounced hierarchy between the transverse and longitudinal components. On the bottom panels we have the comparison between on-shell and off-shell transverse photo-production cross sections. Again, we find that the EPA is only valid for $Q \lesssim m_\alpha$ for the dielectron and dimuon channels. For the mixed case, there is only a very small region in $Q < 10^{-2}$GeV for which the off-shell transverse cross section is comparable to the on-shell one. This relative suppression of the off-shell cross section can be understood by noticing that $Q$ enters the lepton propagators, suppressing the process for $Q \gtrsim m_\alpha$. For mixed channels it is then the smallest mass scale ($m_e$) that dictates the fall-off of the matrix element in $Q$, whilst the heaviest mass ($m_\mu$) defines the phase space boundaries, rendering most of this phase space incompatible with the EPA assumptions.
![\[fig:4PSvsEPA\] Comparison between the full calculation of the trident production coherent cross section and the EPA in the kinematically allowed region of the $(Q,\hat{s})$ plane for an incoming $\nu_\mu$ with fixed energy $E_\nu=3$ GeV colliding with an $^{40}$Ar target. The left, middle and right panels correspond to the dielectron, mixed and dimuon final-states, respectively. The top panels correspond to the comparison between the longitudinal and transverse contributions while the bottom ones show the ratio between the transverse cross sections computed for an specific value of $Q$ with the cross section for an on-shell photon. The thick black dashed lines correspond to the cut in the $Q^2$ integration at $\Lambda_{\rm QCD}^2/ A^{2/3}$, and the shadowed region around these lines account for a variation of $20\%$ in the value of this cut. The purple dashed lines are for $Q=m_\alpha$, $\alpha=e,\mu$ for the unmixed cases.](figs/4PS_vs_EPA.pdf){width="\textwidth"}
These results explicitly show that the EPA is, in principle, not suitable for any neutrino trident process as it can overestimate the cross section quite substantially by treating the photo-production cross section at large $Q^2$ as on-shell. However, as previously mentioned, in the coherent regime the nuclear form factor introduces a strong suppression for large values of $Q^2$. In general, this dominates the behaviour of the cross sections for values of $Q^2$ smaller than the purely kinematic limit, $Q^2_{\rm max}$, and of the order of $\Lambda_{\rm QCD}/ A^{1/3}\approx 0.06$ GeV for coherent scattering on $^{40}$Ar. In the dimuon case, the latter scale happens to be smaller than the charged lepton masses, implying that the region where the EPA breaks down is heavily suppressed due to the nuclear form factor. The same cannot be said about coherent trident channels involving electrons, as the nuclear form factor suppression happens for much larger values of $Q$ than the EPA breakdown. Furthermore, for diffractive scattering the nucleon form factors suppress the cross sections only for much larger $Q$ values, $Q\approx 0.8$ GeV. The effective range of integration then includes a significant region where the EPA assumptions are invalid, leading to an overestimation of the diffractive cross section for every process regardless of the flavours of their final-state charged leptons.
![\[fig:comparison4PS\_EPA\] Ratio $\mathcal{R}$ of the trident cross section calculated using the EPA to the full four-body calculation. Left panel: Ratio in the coherent regime on $^{40}$Ar. The full curves correspond to the central value of $Q_{\rm cut}$, and the upper (lower) boundary corresponds to a choice 100 times larger ($20\%$ smaller). Right panel: Ratio in the diffractive regime for scattering on protons, where the full curves corresponds to the central value of $1.0$ GeV, and the upper (lower) boundary corresponds to a choice 100 times larger ($20\%$ smaller); we have taken the lower limit in the integration on $Q$ to match the choice of the coherent regime. A guide to the eye at $\mathcal{R} = 1$ is also shown.](figs/XSec_ratio.pdf){width="\textwidth"}
In some calculations, artificial cuts have been imposed on the range of $Q^2$, affecting the validity of the EPA. In Ref. [@Magill:2016hgc], it is claimed that to avoid double counting between different regimes, an artificial cut must be imposed, lowering the upper limit of integration in $Q^2$. Ref. [@Magill:2016hgc] chooses a value of $Q^{\rm cut}_{\rm max} = \Lambda_{\rm QCD}/ A^{1/3}$ in the coherent regime (black thick dashed lines in Fig. \[fig:4PSvsEPA\]), and $Q^{\rm cut}_{\rm min}= {\rm max}\left( \Lambda_{\rm QCD}/ A^{1/3}, \hat{s}/2E_\nu\right)$ and $Q^{\rm cut}_{\rm max} = 1.0$ GeV in the diffractive regime. We believe that no such cut is required on physical grounds[^4], and their presence will impact the EPA cross section quite dramatically. Let us first consider the dimuon case in the coherent regime, where the EPA assumptions hold reasonably well in the relevant parts of phase space. By introducing a value for $Q^{\rm cut}_{\rm max}$ we would be decreasing the total relevant phase space for the process, reducing the total cross section. Therefore, despite the EPA tendency to overestimate the cross section in this channel, an artificial cut in $Q^2$ can actually lead to an underestimation of the cross section. In the electron channels, where the EPA breakdown is much more dramatic, we can expect that the overestimation of the cross section by the EPA is reduced by the cut $Q^{\rm cut}_{\rm max}$. In fact, one way to improve the EPA for the dielectron channel is to artificially cut on the $Q^2$ integral around the region where the approximation breaks down [@Frixione:1993yw]. This cut does then improve the coherent EPA calculation by decreasing the overestimation of the cross section. However, an energy independent cut cannot provide a good estimate of the cross section over all values of $E_\nu$. To illustrate our point and to quantify the errors induced by the EPA, we show on the left panel of [Fig. \[fig:comparison4PS\_EPA\]]{} the ratio $\mathcal{R}$ of the trident cross section calculated using the EPA with an artificial cut at $Q^2_\text{cut}$, as performed in [@Magill:2016hgc], to the full calculation used in this work as a function of the incoming neutrino energy: $$\mathcal{R} = \frac{\sigma_{\rm EPA} (E_\nu) \vert_{Q_{\rm cut}}}{\sigma_{\rm 4PS} (E_\nu)}\,.$$ In this plot we vary the artificial cut on $Q^2$ around the choice of [@Magill:2016hgc] (shown as the central dashed line) in two ways. First we reduce it by $20 \%$, and then increase it by a large factor, recovering the case with no $Q^2$ cut. From this, our conclusions about the validity of the approximation are confirmed, and it becomes evident that the trident coherent cross section is very sensitive to the choice of $Q^2_{\mathrm{cut}}$. In particular, the EPA with all the assumptions that lead to [Eq. (\[eq:EPA\_bad\])]{} and the absence of a $Q^2$ cut can lead to an overestimation of all trident channels, including the dimuon one. Once the cut is implemented, however, the approximation becomes better for the dimuon channel, but still unacceptable for the electron ones. It is also clear that an energy independent cut cannot give the correct cross section at all energies. This is particularly troublesome for detectors subjected to a neutrino flux covering a wide energy range such as the near detectors for DUNE and MINOS or MINER$\nu$A. Moreover, [Eq. (\[eq:EPA\_bad\])]{} fails at low energies, and generally, overestimates the coherent cross sections by at least 200%. At these energies, one must be wary of the additional approximations in [Eq. (\[eq:EPA\_bad\])]{} regarding the integration limits and the small $y$ limit.
On the right panel of [Fig. \[fig:comparison4PS\_EPA\]]{} we illustrate what happens in the diffractive regime, where the nucleon form factors impact the cross section at much larger values of $Q^2$ and have a slower fall-off. We see that the diffractive cross section is dramatically overestimated over the full range of $E_\nu$ considered and for any trident mode. The discrepancy is particularly important for $E_\nu \lesssim$ 5 GeV and larger than in the coherent regime by at least an order of magnitude[^5]. We also see that the cuts on $Q^2$ impact the EPA calculation much less dramatically, and that its use is unlikely to yield the correct result.
Given these problems with both coherent and diffractive cross section calculations due to the breakdown of the EPA for trident production, in what follows we will use the complete four-body calculation.
Coherent versus Diffractive Scattering in Trident Production {#subsec:cohdiff}
------------------------------------------------------------
![\[fig:RatioCDvsT\] On the left (right) panel we show the ratio of the coherent (full lines) and the diffractive (dashed lines) contributions to the total trident cross section for an incoming flux of $\nu_\mu$($\nu_e$) as a function of $E_\nu$ for an $^{40}$Ar target.](figs/Ratio_CDvsT.pdf){width="\textwidth"}
Let us now comment on the significance of the coherent and diffractive contributions to the total cross for the different trident channels. In Fig. \[fig:RatioCDvsT\] we present the ratio of the coherent and the diffractive scattering cross sections to the total cross section for an $^{40}$Ar target for an incoming $\nu_\mu$ (left) and $\nu_e$ (right) neutrino. We can see that the coherent regime dominates at all neutrino energies when there is an electron in the final-state, especially in the dielectron case. This can be explained by noting that the $Q^2$ necessary to create an electron pair is smaller than the one needed to create a muon; thus, coherent scattering is more likely to occur for this mode. Conversely, as one needs larger momentum transferred to produce a muon (either accompanied by an electron or another muon) the diffractive regime becomes more likely in these modes, as we can explicitly see in Fig. \[fig:RatioCDvsT\]. Because of this effect the diffractive contribution is $\lesssim$ 10%, except for the dimuon channel where it can be between $30$ and $40$% in most of the energy region. Furthermore, when we compare the two incoming types of neutrinos, we see that for an incoming $\nu_\mu$ the diffractive contribution is larger than the coherent one in the range $0.3\ {\rm GeV}\lesssim E_\nu \lesssim 0.8$ GeV, while for an incoming $\nu_e$ this never happens. This difference can be explained by the fact that CC and NC contributions are simultaneously present for the scattering of an initial $\nu_\mu$ creating a muon pair, whereas for an initial $\nu_e$ creating a muon pair, we will only have the NC contribution, see Table \[tab:tridentmodes\].
An important difference between the coherent and diffractive regimes will be in their hadronic signatures in the detector. Neutrino trident production is usually associated with zero hadronic energy at the vertex, a feature that proved very useful in reducing backgrounds in previous measurements. Whilst this is a natural assumption for the coherent regime, it need not be the case in the diffractive one. In fact, in the latter it is likely that the struck nucleon is ejected from the nucleus in a significant fraction of events with $Q$ exceeding the nuclear binding energy [^6]. Since the dominant diffractive contribution comes from scattering on protons, these could then be visible in the detector if their energies are above threshold. On the other hand, the struck nucleon is subject to many nuclear effects which may significantly affect the hadronic signature, such as interactions of the struck nucleon in the nuclear medium as well as reabsorption. Our calculation of Pauli blocking, for example, shows large suppressions ($\sim 50\%$) precisely in the low $Q^2$ region, usually associated with no hadronic activity. This then raises the question of how well one can predict the hadronic signatures of diffractive events given the difficulty in modelling the nuclear environment. We therefore do not commit to an estimate of the number of diffractive events that would have a coherent-like hadronic signature, but merely point out that this might introduce additional uncertainties in the calculation, especially in the $\mu^+ \mu^-$ channel where the diffractive contribution is comparable to the coherent one. Finally, from now on we will refer to the number of trident events with no hadronic activity as coherent-like, where this number can range from coherent only to coherent plus all diffractive events.
Trident Events in LAr Detectors {#sec:LAr}
===============================
In this section we calculate the total number of expected trident events for some present and future LAr detectors with different fiducial masses, total exposures and beamlines. In Table \[tab:LAr\] we specify the values used for each set-up and in Fig. \[fig:LAr\] we show the total production cross section for each neutrino trident mode of Table \[tab:tridentmodes\] as well as the neutrino fluxes as a function of $E_\nu$ at the position of each experiment.
Event Rates {#subsec:rates}
-----------
The total number of trident events, $N^{\text{\Neptune}}_{\rm X}$, expected for a given trident mode at any detector is written as $$\begin{aligned}
\label{eq:nevents}
N^{\text{\Neptune}}_{\rm X}={\rm Norm}\times\int dE_\nu \, \sigma_{\nu {\rm X}}(E_\nu) \frac{d\phi_{\nu}(E_\nu)}{dE_\nu}\epsilon(E_\nu)\,,\end{aligned}$$ where $\sigma_{\nu {\rm X}}$ can be the trident total (${\rm X}={\cal N}$), coherent ($\mathrm{X=c}$) or diffractive ($\mathrm{X=d}$) cross sections for a given mode, $\phi_{\nu}$ is the flux of the incoming neutrino and $\epsilon(E_\nu)$ is the efficiency of detection of the charged leptons. In the calculations of this section, we assume an efficiency of $100\%$[^7]. The normalization is calculated as $${\rm Norm}= {\rm{Exposure}}~[{\rm{POT}}] \times \frac{{\rm{Fiducial~Detector~Mass}\times N_A}}{m_{\rm T}} \left[{\rm{target~particles}}\right],$$ where $m_{\rm T}$ is the molar mass of the target particle and $N_A$ is Avogadro’s number. Two features of the cross sections are important for the event rate calculation: threshold effects, especially for channels involving muons in the final-state, and cross section’s growth with energy. In particular, we expect higher trident event rates for experiments with higher energy neutrino beams.
We start our study with the three detectors of the SBN program, one of which, $\mu$BooNE, is already installed and taking data at Fermilab. These three LAr time projection chamber detectors are located along the Booster Neutrino Beam line which is by now a well-understood source, having the focus of active research for over 15 years. Although the number of trident events expected in these detectors is rather low, they may offer one of the first opportunities to study trident events in LAr, as well as to better understand their backgrounds in this medium and to devise improved analysis techniques. After that we study the proposed near detector for DUNE. This turns out to be the most important LAr detector for trident production since it will provide the highest number of events in both neutrino and antineutrino modes. Finally, having in mind the novel flavour composition of neutrino beams from muon facilities, we investigate trident rates at a 100 t LAr detector for the $\nu$STORM project. This last facility could offer a very well understood neutrino beam with as many electron neutrinos as muon antineutrinos from muon decays, creating new possibilities for trident scattering measurements.
![\[fig:LAr\]Energy distribution of the neutrino fluxes at the position of the LAr detectors DUNE (top left, [@DUNE:flux]), SBND (top right,[@SBNproposal]) and $\nu$STORM (bottom left, [@nuSTORM2017]) and of the cross sections for the various trident modes (bottom right). The fluxes at $\mu$BooNE and ICARUS are similar to the one shown for SBND when normalized over distance.](figs/LAr_f+xsec.pdf){width="1.\textwidth"}
### The SBN Program {#subsubsec:SBND}
The SBN Program at Fermilab is a joint endeavour by three collaborations ICARUS, $\mu$BooNE and SBND to perform searches for eV-sterile neutrinos and study neutrino-Ar cross sections [@SBNproposal]. As can be seen in Tab. \[tab:LAr\], SBND has the shortest baseline (110 m) and therefore the largest neutrino fluxes (shown in Fig. \[fig:LAr\] and taken from Fig. 3 of [@SBNproposal]). The largest detector, ICARUS, is also the one with the longest baseline (600 m) and consequently subject to the lowest neutrino fluxes. The ratio between the fluxes at the different detectors are $\phi_{\mu\rm{BooNE}}/\phi_{\rm{SBND}}=5$% and $\phi_{\rm{ICARUS}}/\phi_{\rm{SBND}}=3$%. The neutrino beam composition is about 93% of $\nu_\mu$, 6% of $\overline\nu_\mu$ and $1\%$ of $\nu_e+\overline{\nu}_e$.
Considering the difference in fluxes and the total number of targets in each of these detectors, one can estimate the following ratios of trident events: ${N^\text{\Neptune}_{\mu\rm{BooNE}}}/{N^\text{\Neptune}_{\rm{SBND}}}\sim 8$% and ${N^\text{\Neptune}_{\rm{ICARUS}}}/{N^\text{\Neptune}_{\rm{SBND}}}\sim 10$%. Unfortunately, since the fluxes are peaked at a rather low energy ($E_\nu \lesssim 1$ GeV), where the trident cross sections are still quite small ($\lesssim 10^{-42}$ cm$^2$) we expect very few trident events produced. The exact number of trident events for those detectors according to our calculations is presented in Tab. \[tab:LArrates\]. For each trident channel the first (second) row shows the number of coherent (diffractive) events. As expected, less than a total of 20 events across all channels can be detected by SBND, and a negligible rate of events is expected at $\mu$BooNE and ICARUS.
### DUNE Near Detector {#subsubsec:DUNE}
The DUNE experiment will operate with neutrino as well as antineutrino LBNF beams produced by directing a 1.2 MW beam of protons onto a fixed target [@Acciarri:2016ooe; @DUNECDRvolII]. The design of the near detector is not finalised, but the current designs favour a mixed technology detector combining a LAr TPC with a larger tracker module. In this work, we will assume that DUNE ND is a LAr detector located at $574$ m from the target with a fiducial mass of 50 t [@WeberTalk]. As the trident event rate scales with the density of the target, any tracker module will not significantly influence the total event rate, and does not feature in our estimates; although, its presence is assumed to improve reconstruction of final-state muons. Our estimates can be easily scaled for the final design by using [Eq. (\[eq:nevents\])]{}.
For the first 6 years of data taking (3 years in the neutrino plus 3 years in the antineutrino mode) the collaboration expects $1.83\times 10^{21}$ POT/year with a plan to upgrade the beam after the 6th year for 2 extra years in each beam mode with double exposure, making a total of $1.83 \times(3+2\times2)\times 10^{21}~{\rm{POT}}$ for each mode [@DUNE:exposure]. We will assume the total 10-year exposure in our calculations. We use the optimized 3-horn fluxes for a beam of 62.4 GeV protons taken from Ref. [@DUNE:flux] as the relevant fluxes at the DUNE ND location (see Fig. \[fig:LAr\]). The beam composition of the neutrino (antineutrino) beam is about 96% $\nu_\mu$ ($\overline\nu_\mu$), 4% $\overline\nu_\mu$ ($\nu_\mu$) and 1% $\nu_e+\overline\nu_e$.
The number of trident events for DUNE ND can be found in Tab. \[tab:LArrates\]. The numbers in parentheses correspond to antineutrino beam mode. Note that although the trident cross sections are the same for neutrinos and antineutrinos, the fluxes are a bit lower for the antineutrino beam, as a consequence we predict a lower event rate for this beam[^8]. Due to the much higher energy and wider energy range of the neutrino fluxes at DUNE ND, as compared to the SBN detectors, DUNE can observe a considerable number of trident events, about 300 times the number of trident events expected for SBND just in the neutrino mode. Moreover, the subdominant component of each beam mode will also contribute to the signal. For example, we expect to observe $2051$ trident events in the $\overline{\nu}_\mu\to\overline{\nu}_e e^- \mu^+$ channel in the antineutrino mode. However, we also expect $235$ events in the $\nu_\mu\to\nu_e e^+ \mu^-$ channel produced by the subdominant component of $\nu_\mu$ in the antineutrino beam. We have considered 100% detection efficiency here, however, we will see in Sec. \[subsec:bck\] that after implementing hadronic vetos, detector thresholds and kinematical cuts to substantially reduce the background we expect an efficiency of about 47%-65% on coherent tridents, depending on the channel (see Tab. \[tab:DUNE\_ND\_NU\_BG\]).
The mixed flavour trident channel is the one with the highest statistics (more than 6500 events adding neutrino and antineutrino beam modes), 18% of which are produced by diffractive scattering. The dielectron channel comes next with a total of a bit more than 2000 events, 12% of which are produced by diffractive scattering. Although the dimuon channel is the less copious one, with only about 840 events produced, almost 41% of these events are produced by a diffractive process. This can be understood by recalling our discussions in Sec. \[subsec:cohdiff\].
Finally, we note that a dedicated high-energy run at DUNE has been mooted, to be undertaken after the full period of data collecting for the oscillation analysis. Thanks to the higher energies of the beam, this has the potential to see a significant number of neutrino tridents, provided it can collect enough POTs.
### $\nu$STORM {#subsubsec:nuSTORM}
In this section we study the trident rates for a possible LAr detector for the proposed $\nu$STORM experiment [@Soler:2015ada; @nuSTORM2017]. The $\nu$STORM facility is based on a neutrino factory-like design and has the goal to search for sterile neutrinos and study neutrino nucleus cross sections [@Adey:2014rfv]. Although this proposal is in its early days, $\nu$STORM has the potential to make cross section measurements with unprecedented precision. In its current design, $120$-GeV protons are used to produce pions from a fixed target with the pions subsequently decaying into muons and neutrinos. The muons are captured in a storage ring and during repeated passes around the ring they decay to produce neutrinos. Consequently, the storage ring is an intense source of three types of neutrino flavours: $\nu_\mu$ from $\pi^+$ and $K^+$ decays, which will be more than $99\%$ of the total flux, $\nu_e$ and $\overline\nu_\mu$ from recirculated muon decays which will comprise less than $1\%$ of the total flux. An important point, however, is that the neutrinos coming from the pion and kaon decays can be separated by event timing from the ones produced by the stored muons. This distinction allows the $\nu_\mu$ flux to be studied almost independently from the $\overline{\nu}_\mu$ and $\nu_e$ flux. In addition, it implies after the initial flash of meson-derived events, that the flux consists of as many electron neutrinos as muon antineutrinos. We will assume a LAr detector for $\nu$STORM at a baseline of 50m with 100t of fiducial mass with an exposure of $10^{21}$ POT. The neutrino fluxes, assuming a central $\mu^+$ momentum of $3.8$ GeV/c in the storage ring, are taken from Ref. [@nuSTORM2017] and are shown in Fig. \[fig:LAr\].
In Tab. \[tab:LArrates\], we show the results of our calculations for $\nu$STORM. More than $97\%$ of the events from the incoming $\nu_\mu$ are from pion decays and only less than $3\%$ from kaon decays. Since we only consider the decay of mesons with positive charges and we expect neutral and wrong charge contamination to be small, we do not have trident events from incoming $\overline\nu_e$. The total number of mixed flavour, dielectron and dimuon channel events is, respectively, 230, 125 and 20, much less than what can be achieved at the larger neutrino energies available at the DUNE ND. The novel flavour structure of the beam does enhance the contribution of $\nu_e$ induced tridents with respect to the ${\accentset{(-)}{\nu}}_\mu$ ones, but this contribution only becomes dominant for the $e^+e^-$ tridents in the muon decay events. Finally, we emphasize that the experimental design parameters for $\nu$STORM are far from definite. Increasing the energy of stored muons and the size of the detector are both viable options which could significantly enhance the rates we present.
Kinematical Distributions at DUNE ND {#subsec:kine}
------------------------------------
In this section we explore the trident signal in more detail, showing some relevant kinematical distributions for coherent and diffractive events. For concreteness, and due to its large number of events, we choose to focus on the DUNE ND, only commenting slightly on the signal at the lower energies of SBN and $\nu$STORM. The observables we calculate are the invariant mass of the charged leptons $m^2_{\ell^+ \ell^-}$, their separation angle $\Delta \theta$ and their individual energies $E_\pm$. The flux convolved distributions of these observables are shown for the DUNE ND in neutrino mode in [Fig. \[fig:DUNE\_ND\_dist\]]{}. In these plots, we sum all trident channels with a given undistinguishable final-state proportionally to their rates, although $\nu_\mu$ initiated processes always dominate. The coherent and diffractive contributions are shown separately and on the same axes, but we do not worry about their relative normalization. Other potentially interesting quantities are the angle between the cone formed by the two charged leptons and the beam, $\alpha_C$, and the angle of each charged lepton with respect to the beam direction, $\theta_\pm$. These additional observables are explored in [Appendix \[app:distributions\]]{}. We also report the distributions of the momentum transfer to the hadronic system, $Q^2$. Although this is not a directly measurable quantity, it is a strong discriminant between the coherent and diffractive processes. We do not present the antineutrino distributions here, but they are qualitatively similar.
Perhaps one of the most valuable tools for background suppression in the measurement of the $\mu^+\mu^-$ trident signal at CHARM II, CCFR and NuTeV [@Geiregat:1990gz; @Mishra:1991bv; @Adams:1998yf] was the smallness of the invariant mass $m^2_{\ell^+ \ell^-}$. This feature, shown here on the top row of [Fig. \[fig:DUNE\_ND\_dist\]]{}, is also present at lower energies, where the distributions become even more peaked at lower values; although, the diffractive events tend to be have a more uniform distribution in this variable. This is also true for the angular separation $\Delta \theta$, where coherent dimuon tridents tends to be quite collimated, with $90\%$ of events having $\Delta \theta < 20^\circ$, whilst diffractive ones are less so, with only $47\%$ of events surviving the cut. This difference is much less pronounced for mixed and dielectron channels, where only half of our coherent events obey $\Delta \theta < 20^\circ$, when $37\%$ of diffractive events do so.
An interesting feature of same flavour tridents induced by a neutrino (antineutrino) is that the negative (positive) charged lepton tends to be slightly more energetic than its counterpart, whilst for mixed tridents muons tend to carry away most of the energy. These considerations are also reflected in the angular distributions. The most energetic particle is also the more forward one. For instance, in mixed neutrino induced tridents, $\sim 80 \%$ of the $\mu^-$ are expected to be within $10^\circ$ of the beam direction, whilst only $\sim 35 \%$ of their $e^+$ counterparts do so (see [Appendix \[app:distributions\]]{} for additional distributions).
![Flux convolved neutrino trident production distributions for DUNE ND in neutrino mode. In purple we show the coherent contribution in $^{40}$Ar and in blue the diffractive contribution from protons as targets only (including Pauli blocking). The coherent and diffractive distributions are normalized independently. The relative importance of each contribution as a function of $E_\nu$ can be seen in Fig. \[fig:RatioCDvsT\]. \[fig:DUNE\_ND\_dist\]](figs/DUNE_nu_3horn_mll_theta_E.pdf){width="\textwidth"}
Finally, we mention that detection thresholds can also be important for trident channels with electrons in the final-state. Assuming, for example, a detection threshold for muons and electromagnetic (EM) showers of 30 MeV in LAr, we end up with efficiencies of (99%, 71%, 77%, 86%) for ($\mu^+ \mu^-$, $e^+ e^-$, $e^+ \mu^-$, $e^- \mu^+$) coherent tridents. These efficiencies become (96%, 91%, 93%, 96%) for diffractive tridents, dropping for $\mu^+\mu^-$ and increasing for all others. For comparison, at the lower neutrino energies of SBND and assuming the same detection thresholds, the efficiencies for coherent and diffractive tridents are slightly lower, (97%, 57%, 67%, 77%) and (90%, 81%, 85%, 90%) respectively.
Background Estimates for Neutrino Trident in LAr {#subsec:bck}
------------------------------------------------
The study of any rare process is a struggle against both systematic uncertainties in the event rates and unavoidable background processes. True dilepton signatures are naturally rare in neutrino scattering experiments, but with modest rates of particle misidentification a non-trivial background arises. In this section we estimate the background to trident processes in LAr and its impact on the trident measurement. We perform our analysis only for DUNE ND, in neutrino and antineutrino mode, but our results are expected to be broadly applicable to other LAr detectors. We have generated a sample of $1.1 \times 10^6$ background events using GENIE [@Andreopoulos2009] for incident electron and muon flavour neutrinos and antineutrinos. It is worth noting, however, that this event sample will in fact be smaller than the total number of neutrino interactions expected in the DUNE ND. Our goal, therefore, will be to demonstrate that with modest analysis cuts background levels can be suppressed significantly such that they become comparable to or smaller than the signals we are looking for. In the absence of events that satisfy our background definition, we argue that the frequency of that type of event is less than one in $1.1\times 10^6$ interactions of the corresponding initial neutrino.
To account for misreconstruction in the detector, we implement resolutions as a gaussian smear around the true MC energies and angles. We assume relative energy resolutions as $\sigma/E = 15\%/\sqrt{E}$ for $e/\gamma$ showers and protons, and $6\%/\sqrt{E}$ for charged pions and muons. Angular resolutions are assumed to be $1^\circ$ for all particles (proton angles are never smeared in our analysis). The detection thresholds are a crucial part of the analysis, since for many channels one ends up with very soft electrons. We take thresholds to be $30$ MeV for muons and $e/\gamma$ showers kinetic energy, $21$ MeV for protons and $100$ MeV for $\pi^{\pm}$ [@DUNECDRvolII].
### Background Candidates {#subsubsec:misID}
We focus on three final-state charged lepton combinations: $\mu^+\mu^-$, $\mu^\pm e^\mp$ and $e^+e^-$. Genuine production of these states is possible in background processes, but usually rare, deriving from meson resonances or other prompt decays. The majority of the background is expected to be from particle misidentification (misID). We assume that protons can always be identified above threshold and that neutrons leave no detectable signature in the detector. In addition, we require no charge ID capabilities from the detector and assume that the interaction vertex can always be reconstructed. Under these assumptions, we have incorporated three misidentifications which will affect our analysis, and give our naive estimates for their rates in Tab. \[tab:misIDlist\]. Any other particle pairs are assumed to be distinguishable from each other when needed.
-------------------------------- --
**misID & **Rate\
$\gamma$ as $e^\pm$ & 0.05\
& 0.1 (w/ vertex)\
& 1 (no vertex + overlapping)\
$\pi^\pm$ as $\mu^\pm$ & 0.1\
****
-------------------------------- --
: \[tab:misIDlist\] Assumed misID rates for various particles in a LAr detector. We take these values to be constant in energy.
The requirement of no hadronic activity helps constrain the possible background processes, but one is still left with significant events with invisible hadronic activity and other coherent neutrino-nucleus scatterings. These are then reduced by choosing appropriate cuts on physical observables, exploring the discrepancies between our signal and the background. In our GENIE analysis, we include all events that have final-states identical to trident, or that could be interpreted as a trident final-state considering our proposed misID scenarios. Our dominant sources of background for $\mu^+ \mu^-$ tridents are $\nu_\mu$-initiated charged-current events with an additional charged pion in the final-state ($\nu_\mu$CC$1\pi^\pm$). For $e^+e^-$ tridents, the most important processes are neutral current scattering with a $\pi^0$ (NC$\pi^0$), while for mixed $e^\pm \mu^\mp$ tridents, the $\nu_\mu$-initiated charged-current events with a final-state $\pi^0$ (CC$\pi^0$) dominate the backgrounds. In each case, the pion is misidentified to mimic the true trident final-state. Other relevant topologies include charm production, CC$\gamma$ and $\nu_e$CC$\pi^\pm$. For a detailed discussion of these backgrounds processes we refer the reader to [Appendix \[app:backgrounds\]]{}.
### \[sec:DUNE\_bg\_rates\]Estimates for the DUNE ND
In this section we provide estimates for the total background for each trident final-state for the DUNE ND. The number of total inclusive CC interactions in the 50 t detector due to neutrinos of all flavours is calculated to be $5.18 \times 10^8$. We scale our background event numbers to match this, and argue that one has to reach suppressions of order $10^{-6} - 10^{-5}$ to have a chance to observe trident events. Whenever our cuts remove all background events from our sample, we assume the true background rate is one event per $1.1\times10^6$ $\nu$ interactions and scale it to the appropriate number of events in the ND, applying the misID rate whenever relevant. Within our framework, this provides a conservative estimate as the true background is expected to be smaller.
Our estimates are shown in [Table \[tab:DUNE\_ND\_NU\_BG\]]{}. We start with the total number of background candidates $\rm N_B^{\mathrm{misID}}$, using only the naive misID rates shown in [Table \[tab:misIDlist\]]{}. These are much larger than the trident rates we expect, by at least 2 orders of magnitude. Next, we veto any hadronic activity at the interaction vertex, obtaining $\rm N_B^{\mathrm{had}}$. We emphasize that this veto also affects the diffractive tridents in a non-trivial way, and therefore we remain agnostic about the hadronic signature of these. Finally, one can look at the kinematical distributions of coherent trident in [Section \[subsec:kine\]]{} and try to estimate optimal one dimensional cuts for the DUNE ND based on the kinematics of the final-state charged leptons. This is a simple way to explore the striking differences between the peaked nature of our signal and the smoother background. In a real experimental setting it is desirable to have optimization methods for isolating signal from background, preferably with a multivariate analyses. However, even in our simple analysis, cutting on the small angles to the beamline and the low invariant masses of our trident signal can achieve the desired background suppressions. For the $\mu^+\mu^-$ tridents we show the effect of our cuts in [Fig. \[fig:bkg\_flow\]]{}. The cuts are defined to be $m^2_{\mu^+ \mu^-} < 0.2 \ \mathrm{GeV}^2$, $\Delta \theta < 20^\circ$, $\theta_\pm < 15^\circ$. The kinematics is very similar in the other trident channels, with slightly less forward distributions for electrons. For the $e^+ e^-$ channel we take $m^2_{e^+ e^-} < 0.1 \ \mathrm{GeV}^2$, $\Delta \theta < 40^\circ$ and $\theta_\pm < 20^\circ$. The asymmetry between the positive and negative charged leptons is visible in the distributions, where the latter tends to be more energetic. This feature was not explored in our cuts, as it is not significant enough to further improve background discrimination. In the mixed flavour tridents, however, one sees a much more pronounced asymmetry. The muon tends to carry most of the energy and be more forward than the electron, which can make the search for this channel more challenging due to the softness of the electron in the high energy event. Nevertheless, the low invariant masses and forward profiles can still serve as powerful tool for background discrimination, provided the event can be well reconstructed. We assume that is the case here and use the following cuts on the background: $m^2_{e^\pm \mu^\mp} < 0.1 \ \mathrm{GeV}^2$, $\Delta \theta < 20^\circ$, $\theta_e < 40^\circ$ and $\theta_\mu < 20^\circ$. When performing kinematical cuts, we also include the effects of detection thresholds after smearing. For a discussion on the impact of these thresholds on the trident signal see [Section \[subsec:kine\]]{}. The resulting signal efficiencies due to our cuts and thresholds are shown in the last two columns of [Table \[tab:DUNE\_ND\_NU\_BG\]]{}. One can see that these are all $ \approx 50\%$ or greater for our coherent samples, whilst all background numbers remain much below the trident signal. The diffractive samples are also somewhat more affected by our cuts than the coherent ones. If one is worried about the contamination of coherent events by diffractive ones, then the kinematics of the charged leptons alone can help reduce this, independently of the hadronic energy deposition of the events. For instance, in the case where all $\mu^+\mu^-$ diffractive events appear with no hadronic signature, then after our cuts the diffractive contribution is reduced from $41\%$ to $15\%$ of the total trident signal. This reduction is, however, also subject to large uncertainties coming from nuclear effects. In summary, the set of results above are encouraging, suggesting that the signal of coherent-like trident production is sufficiently unique to allow for its search at near detectors despite naively large backgrounds.
Finally, we comment on some of the limitations of our analysis. The low rate of trident events calls for a more careful evaluation of other subdominant processes that could be easily be overlooked. For channels involving electrons, it is possible that de-excitation photons and internal bremsstrahlung become a source of background, as these also produce very soft EM showers, none of which are implemented in GENIE. The question of reconstruction of these soft EM showers, accompanied either by a high energy muon or by another soft EM shower also would have to be addressed, especially in the latter case where a trigger for these soft events would have to be in place. A more complete analysis is also needed for treating the decay products of charged pions and muons produced in neutrino interactions, as well as rare meson decay channels (like the Dalitz decay of neutral pions $\pi^0 \to \gamma e^+ e^-$). Cosmic ray events are not expected to be a problem due to the requirement of a vertex and a correlation with the beam for trident events. Perhaps even more exotic processes with three final-state charged leptons, like the radiative trimuon production [@Smith:1977nx], could also behave as a background when a single particle is undetected. We are not aware of any estimates for the rate of processes of the type $\nu_{\alpha} (\overline{\nu}_{\alpha}) + \mathcal{H} \to \ell_\alpha^- (\ell_\alpha^+) + \ell_\beta^+ + \ell_{\beta}^- + \mathcal{H^\prime}$ at the DUNE ND, but note that its rate is comparable to neutrino trident production at energies above $30$ GeV [@Albright:1978mg]. Improvements on our analysis should come from the collaboration’s sophisticated simulations, allowing for a better quantification of hadronic activity, more realistic misID rates and more accurate detector responses.
Trident Events in Other Near Detector Facilities {#sec:others}
================================================
The search for neutrino trident production events certainly benefits from the capabilities of LAr technologies but need not be limited to it. In this section we study neutrino trident production rates at non-LAr experiments which have finished data taking or are still running: the on-axis near detector of T2K (INGRID), the near detectors of MINOS and NO$\nu$A and the MINER$\nu$A experiment. We calculate the total number of trident events as in Eq. (\[eq:nevents\]), taking into account the fact that some detectors are made of composite material. We summarize in Tab. \[tab:others\] the details of all non-LAr detectors considered in this section. We limit ourselves to a discussion of the total rates in the fiducial volume, but remark that a careful consideration of each detector is needed in order to assess their true potential to detect a trident signal. For instance, requirements about low energy EM shower reconstruction, hadronic activity measurements and event containment would have to be met to a good degree in order for the detector to be competitive.
INGRID {#subsec:INGRID}
------
INGRID, the on-axis near detector of the T2K experiment, is located 280 m from the beam source. It consists of 14 identical iron modules, each with a mass of $7.1$ t, resulting in a total fiducial mass of $99.4$ t [@Abe:2011xv]. The modules are spread over a range of angles between $0^\circ$ and $1.1^\circ$ with respect to the beam axis. The currently approved T2K exposure is $(3.9+3.9)\times 10^{21}$ POT in neutrino + antineutrino modes (T2K-I), with the goal to increase it to a total exposure of $(1+1)\times 10^{22}$ POT in the second phase of the experiment (T2K-II) [@Abe:2016tez]. Hence we expect approximately $2.6$ times more trident events for T2K-II.
We use the on-axis neutrino mode flux spectra at the INGRID module-3 from Ref. [@Abe:2015biq], as shown on the top of the first panel of Fig. \[fig:others\]. The flux contribution for each neutrino flavour and energy range is listed in Table 1 of Ref. [@Abe:2015biq]. The total neutrino flux flavour composition at module-3 is 92.5% $\nu_\mu$, 5.8% $\overline\nu_\mu$, 1.5% $\nu_e$ and 0.2% $\overline\nu_e$. We assume here that the fluxes at the other 13 modules are the same as at module-3. Although this is not exactly correct it should provide a reasonable estimate of the total rate.
Under these assumptions we show the total number of trident events we calculated for INGRID in the first (second) column of Tab. \[tab:otherrates\] for T2K-I (T2K-II) exposure. We predict about 660 (1700) events for the mixed, 300 (770) events for the dielectron and 50 (130) events for the dimuon channel for T2K-I (T2K-II). These numbers, although less than those expected at the DUNE ND, are already very significant and worth further consideration. We expect, however, that the main challenge will be the reconstruction of final state electrons in these iron detectors.
![\[fig:others\] Energy distribution of the neutrino fluxes at the position of the detector (top plot) and corresponding total trident production cross sections (bottom plot) for: INGRID [@Abe:2015biq] (first panel), MINOS ND [@fluxes:nonLAr] (second panel), NO$\nu$A ND[@fluxes:nonLAr] (third panel) and MINER$\nu$A[@fluxes:nonLAr] (fourth panel). The cross sections show here for the composite detectors are normalized by the total number of atoms.](figs/NLAr_f+xsec.pdf){width="\textwidth"}
MINOS/MINOS+ Near Detector {#subsec:MINOS}
--------------------------
The MINOS near detector is a magnetized, coarse-grained tracking calorimeter, made primarily of steel and plastic scintillator. Placed $1.04$ km away from the NuMI target at Fermilab [@Aliaga:2016oaz], it weighs $980$ t and is similar to the far detector in design. In our analysis, we assume a similar fiducial volume cut to the standard $\nu_\mu$ CC analyses, namely a fiducial mass of $28.6$ t made of $80\%$ of iron and $20\%$ of carbon [@Boehm:2009zz].
The experiment ran from 2005 till 2012 in the low energy (LE) configuration of the NuMI beam ($E_\nu^{\rm peak} \approx 3$ GeV) and collected $10.56\times 10^{20} ~(3.36\times 10^{20})$ POT in the neutrino (antineutrino) beam [@Aurisano]. The successor to MINOS, MINOS+, ran with the same detectors subjected to the medium energy (ME) configuration of the NuMI beam ($E_\nu^{\rm peak} \approx 7$ GeV) from 2013 to 2016, and has collected $9.69\times 10^{20}$ POT in the neutrino mode. To calculate the trident event rates we use the fluxes taken from Ref. [@fluxes:nonLAr]. The flavour composition at MINOS ND is 89% (18%) $\nu_\mu$ and 10% (81%) $\overline \nu_\mu$ for the neutrino (antineutrino) beam and about 1% $\nu_e+\overline\nu_e$ for either beam mode. We assume that the MINOS+ neutrino flux is identical to the one at the MINER$\nu$A experiment (see section \[subsec:MINERvA\]). These fluxes and total trident production cross sections are shown on the second panel of Fig. \[fig:others\].
Due to the multi-component material of the detector, the corresponding cross sections that enter in Eq. (\[eq:nevents\]) are: $$\sigma_{\rm \nu X}^{\rm{MINOS}}= \sum_{i=\rm Fe,C} f_i \, \sigma_{\rm \nu X}^{i}\,,$$ where $f_i$ is the number of nuclei $i$ over the total number of nuclei in the detector. As a reference, the weighted cross sections, normalized by the total number of atoms, is also shown in Fig. \[fig:others\].
We report the total number of trident events for MINOS ND in Tab. \[tab:otherrates\]. Although the cross section for iron is about two times larger than for argon and the neutrino fluxes similar, the number of trident events at MINOS ND is much smaller than the expected one at DUNE ND due to a lower exposure and fiducial mass. We predict that about 270 (68) mixed, 70 (17) dielectron and 40 (9) dimuon trident events were produced at this detector with the neutrino (antineutrino) LE NuMI beam. The rates are expected to be larger for MINOS+, as it benefits from the larger energies of the ME NuMI beam configuration and has similar number of POT to MINOS in neutrino mode. In total, we predict about 880 mixed, 70 dielectron and 135 dimuon trident events.
The stringent cut on the fiducial volume assumed here implies a reduction from the $980$ t near detector bulk mass to $28.6$ t. This cut can be relaxed, depending on the signature considered, and may significantly enhance the rates we quote. A careful analysis of trident signatures outside the fiducial volume would be necessary, but we point out that our rates can increase by at most a factor of $\approx30$.
NO$\nu$A Near Detector {#subsec:NOvA}
----------------------
The NO$\nu$A near detector is a fine grained low-Z liquid-scintillator detector placed off-axis from the NuMI beam at a distance of $1$ km. Its total mass is $330$ t, with almost $70$% of it active mass ($231$ t). In this analysis we assume all of this active mass to also be fiducial. The detector is mainly made of 70% mineral oil (CH$_2$) and 30% of PVC (${\rm{C_2H_3Cl}}$) [@Wang:Biao]. A total exposure of $8.85 \, (6.9)\times10^{20}$ POT has been collected in the neutrino (antineutrino) beam mode prior to 2018 [@sanchez_mayly_2018_1286758].
The NO$\nu$A ND neutrino fluxes (taken from Ref. [@fluxes:nonLAr]) peak at slightly lower energies than the MINOS or MINER$\nu$A ones, $E_\nu^{\rm peak} \approx 2$ GeV, and are shown in the third panel of Fig. \[fig:others\]. The flavour composition is 91% (11%) $\nu_\mu$ and 8% (88%) $\overline \nu_\mu$ in the neutrino (antineutrino) mode and about 1% $\nu_e+\overline\nu_e$ in each mode.
Here the cross sections entering in Eq. (\[eq:nevents\]) are calculated as $$\begin{aligned}
\sigma^{\rm{NO\nu A}}_{\rm \nu X}=\sum_{i=\rm C, Cl, H} f_i \sigma_{\rm \nu X}^{i} \,,\end{aligned}$$ where $f_i$ is the number of nuclei $i$ over the total number of nuclei in the detector. As a reference, the weighted cross sections, normalized by the total number of atoms, is shown in Fig. \[fig:others\].
In Tab. \[tab:otherrates\] we show our predictions for the number of trident events at NO$\nu$A ND. Comparing NO$\nu$A and MINOS, we see that while NO$\nu$A ND has a fiducial mass almost 8 times larger, the flux times total cross section at MINOS ND is at least two orders of magnitude larger than at NO$\nu$A ND, especially above 4 GeV (see Fig. \[fig:others\]), making the rates at MINOS ND larger than the rates at NO$\nu$A ND. NO$\nu$A is planning to collect a total exposure of $36\,(36)\times10^{20}$ POT in the neutrino (antineutrino) mode (NO$\nu$A-II) [@sanchez_mayly_2018_1286758; @NovaII], making the expected rates almost $4.1 (5.2)$ times larger (shown in Tab. \[tab:otherrates\]). In this case the expected dimuons and mixed events at MINOS+ would be at least two times larger than NO$\nu$A-II. On the other hand, for NO$\nu$A-II there will be two times more dielectron events given the much higher exposure.
MINER$\nu$A {#subsec:MINERvA}
-----------
The multi-component MINER$\nu$A detector was mainly designed to measure neutrino and antineutrino interaction cross sections with different nuclei in the 1-20 GeV range of energy [@MINERvA:2017]. The detector is located at $1.035$ km from the NuMI target. We assume a fiducial mass of about $8$ t, with a composition of 75% CH, 9% Pb, 8% Fe, 6% H$_2$O and $2\%$ C. The experiment has collected $12\times10^{20}$ POT in the neutrino mode and is planning to reach the same exposure in the antineutrino mode by 2019, both using the medium energy flux of NuMI beam configuration (shown in fourth panel of Fig. \[fig:others\]). We do not include the low energy runs, as these have lower number of POT and lower neutrino energies. The neutrino (antineutrino) beam is composed of 95% (7%) $\nu_\mu$ and 4% (92%) $\overline \nu_\mu$, both beams have about $1\%$ of $\nu_e+\overline\nu_e$.
For MINER$\nu$A the cross sections in Eq. (\[eq:nevents\]) are calculated as $$\begin{aligned}
\sigma^{\rm MINER \nu A}_{\rm \nu X}=\sum_{i=\rm C, Cl, H, Pb, Fe,O} f_i \, \sigma_{\rm \nu X}^{i} \,,\end{aligned}$$ where $f_i$ is the number of nuclei $i$ over the total number of nuclei in the detector. As a reference, the weighted cross sections, normalized by the total number of atoms, is shown in Fig. \[fig:others\].
The total number of trident events we estimate for MINER$\nu$A are listed in Tab. \[tab:otherrates\]. As expected, these are lower than MINOS+, as the latter has a larger fiducial mass. MINER$\nu$A, however, benefits from its fine grained technology and its dedicated design for cross section measurements.
Conclusions {#sec:conc}
===========
Neutrino trident events are predicted by the SM, however, only $\overline{\nu}_\mu$ initiated dimuon tridents have been observed in small numbers, typically fewer than 100 events. This will change in the near future thanks to the current and future generations of precision neutrino scattering and oscillation experiments, which incorporate state-of-the-art detectors located at short distances from intense neutrino sources. In this work we discuss the calculation of the neutrino trident cross section for all flavours and hadronic targets, and provide estimates for the number and distributions of events at 9 current or future neutrino detectors: five detectors based on the new LAr technology (SBND, $\mu$BooNE, ICARUS, DUNE ND and $\nu$STORM ND) as well as four more conventional detectors (INGRID, MINOS ND, NO$\nu$A ND and MINER$\nu$A).
We have stressed the need for a full four-body phase space calculation of the trident cross sections without using the EPA. This approximation has been employed in recent calculations and can lead to overestimations of the cross section by 200% or more at the peak neutrino energies relevant for many accelerator neutrino experiments. Moreover, we show why the EPA is not applicable for computing trident cross sections, and provide the first quantitative assessment of this breakdown for coherent and diffractive hadronic regimes. We find that the breakdown of the approximation is most severe for processes with electrons in the final-state and for diffractive scattering of all final state flavours. For coherent dimuon production, the approximation can give a reasonable result at large neutrino energies. This is due to the nuclear form factors that serendipitously suppress those regions of phase space where the EPA is least applicable. We also demonstrated that the best results in this channel are achieved when applying artificial cuts to the phase space. However, even in this case, at energies relevant for the above experiments, the EPA can artificially suppress the coherent scattering contribution and increase the diffractive one giving rise to an incorrect rate and distributions of observable quantities. For instance, the invariant mass of the charged lepton pair $m^2_{\ell \ell}$ and their angular separation $\Delta \theta$ are more uniformly distributed for diffractive than for coherent trident scattering. Using the correct distributions is crucial to correctly disentangle the signal from the background by cutting on these powerful discriminators.
Our calculations show that DUNE ND is the future detector with the highest neutrino trident statistics, more than 6500 mixed events, 18% produced by diffractive scattering, more than 2000 dielectron events, 12% produced by diffractive scattering and about 840 dimuon events, almost 41% of those produced by a diffractive process. Making use of our efficiencies (see [Table \[tab:DUNE\_ND\_NU\_BG\]]{}), assuming an ideal background suppression and neglecting systematic uncertainties, we quote the statistical uncertainty on the coherent-like flux averaged cross section for the DUNE ND. We do this for coherent only events and, in brackets, for coherent plus diffractive events, yielding $$\frac{\delta \langle \sigma^{e^\pm\mu^\mp} \rangle}{\langle \sigma^{e^\pm\mu^\mp} \rangle} = 1.8\% \, (1.6\%), \quad \frac{\delta \langle \sigma^{e^+e^-} \rangle}{\langle \sigma^{e^+e^-} \rangle} = 3.4\% \,(3.3\%) \quad \mathrm{and} \quad \frac{\delta \langle\sigma^{\mu^+\mu^-} \rangle}{\langle \sigma^{\mu^+\mu^-} \rangle} = 5.5\% \,(5.1\%).$$ In this optimistic framework we expect the true statistical uncertainty on coherent-like tridents to lie between the two numbers quoted, depending on how many diffractive events contribute to the coherent-like event sample. This impressive precision would provide unprecedented knowledge of the trident process and the nuclear effects governing the interplay between coherent and diffractive regimes. We emphasize, however, that given these small values for the relative uncertainties, the trident cross section will likely be dominated by systematic uncertainties from detector response and backgrounds which are not modelled here.
For DUNE ND, we have studied the distribution of observables which could help distinguish trident events from the background. We have estimated the background for each trident channel via a Monte Carlo simulation using GENIE, and identified the dominant contributions arising primarily from particle misidentification. We conclude that reaching background rates of the order ${\cal O}(10^{-6}-10^{-5})$ times the CC rate is necessary to observe trident events at DUNE ND, and given the distinctive kinematic behaviour of the trident signal a simple cut-based GENIE-level analysis suggests that this is an attainable goal in a LAr TPC.
Existing facilities may also be able to make a neutrino trident measurement at their near detectors. Despite not including reconstruction efficiencies nor an indication of the impact of backgrounds, we find that the largest trident statistics is available at INGRID, the T2K on-axis near detector. We predict about 660 (1700) events for the mixed flavour, 300 (770) events for the dielectron and 50 (130) events for the dimuon channel for T2K-I (T2K-II). The more fine-grained near detector of MINOS and MINOS+ is also expected to have collected a significant numbers of events during its run. As such, the very first measurement of neutrino trident production of mixed and dielectron channels may be at hand.
The authors would like to thank TseChun Wang for his involvement during the initial stages of this project.
MH would like to thank Alberto Gago, José Antonio Becerra Aguilar and Kate Scholberg for useful discussions regarding the detection of trident events. YFPG and MH would like to thank Gabriel Magill and Ryan Plestid for helpful discussions on the cross section computation considering the EPA. ZT appreciates the useful discussions with Maxim Pospelov and Joachim Kopp. RZF would like to thank Thomas J Carroll for discussions about the MINOS near detector
This work was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and Conselho Nacional de Ciência e Tecnologia (CNPq). This project has also received support from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 690575 (RISE InvisiblesPlus) and No. 674896 (ITN Elusives) SP and PB are supported by the European Research Council under ERC Grant “NuMass” (FP7-IDEAS-ERC ERC-CG 617143). SP acknowledges partial support from the Wolfson Foundation and the Royal Society.
\[app:formfactors\]Form Factors
===============================
In the coherent regime, we use a Woods-Saxon (WS) form factor due to its success in reproducing the experimental data [@Fricke:1995zz; @Jentschura2009]. The WS form factor is the Fourier transform of the nuclear charge distribution, defined as $$\rho(r) = \frac{\rho_0}{1+\exp\left(\dfrac{r - r_0}{a}\right)} \, ,$$ where we take $r_0 = 1.126 \, A^{1/3}$ fm and $a = 0.523$ fm. One can then calculate the WS form factor as $$F(Q^2) = \frac{1}{\int \rho(r) \, \dd^3r} \int \rho(r) \, \exp\left( -i \vec{q} \vdot \vec{r}\right) d^3r \, .$$ Here we use an analytic expression for the symmetrized Fermi function [@Anni1994; @Sprung1997] instead of calculating the WS form factor numerically. This symmetrized form is found to agree very well with the full calculation and reads $$F(Q^2) = \frac{3 \pi a}{r_0^2 + \pi^2 a^2} \frac{\pi a \coth{(\pi Q a)} \sin{(Q r_0)} - r_0 \cos{(Q r_0)} }{Q r_0 \sinh{(\pi Q a)}}\, .$$ In the diffractive regime, we work with the functions $H_1^{\rm N}(Q^2)$ and $H_2^{\rm N}(Q^2)$, which depend on the Dirac and Pauli form factors of the nucleon ${\rm N}$ as follows $$H_1^{\rm N}(Q^2)= |F_1^{\rm N}(Q^2)|^2 - \tau |F_2^{\rm N}(Q^2)|^2\, , \quad \mathrm{and} \quad H_2^{\rm N}(Q^2) = \left| F_1^{\rm N}(Q^2) + F_2^{\rm N}(Q^2)\right|^2 \, ,$$ where $\tau = -Q^2/4M^2$. The form factors $F_1^{\rm N}(Q^2)$ and $F_2^{\rm N}(Q^2)$ can be related to the usual Sachs electric $G_\mathrm{E}$ and magnetic $G_{\mathrm{M}}$ form factors. These have a simple dipole parametrization $$\begin{aligned}
G^{\rm N}_E(Q^2) =& F^{\rm N}_1 (Q^2) + \tau F^{\rm N}_2 (Q^2) = \begin{cases}
0,\, &\mathrm{if }\, {\rm N} = n,\\
G_D(Q^2),\, &\mathrm{if }\, {\rm N} = p,
\end{cases} \\
G^{\rm N}_M(Q^2) =& F^{\rm N}_1 (q^2) + F^{\rm N}_2 (Q^2) = \begin{cases}
\mu_n \, G_D(Q^2),\, &\mathrm{if }\, {\rm N} = n,\\
\mu_p \, G_D(Q^2),\, &\mathrm{if }\, {\rm N} = p,
\end{cases}\end{aligned}$$ where $\mu_{p,n}$ is the nucleon magnetic moment in units of the nuclear magneton and $G_D(Q^2) = (1 + Q^2/M_V^2)^{-2}$ is a simple dipole form factor with $M_V = 0.84$ GeV.
Kinematical Distributions \[app:distributions\]
===============================================
![Flux convolved neutrino trident production distributions for DUNE ND in neutrino mode in additional variables. In purple we show the coherent contribution in $^{40}$Ar and in blue the diffractive contribution from protons as targets only (including Pauli blocking). The coherent and diffractive distributions are normalized independently. \[fig:other\_dists\]](figs/DUNE_nu_3horn_aC_Q2_thetapm.pdf){width="92.00000%"}
In this section, we show additional distributions in different observables for neutrino trident production, also focused on the DUNE ND in neutrino mode. While trident events are generally quite forward going, their angular behaviour is quite interesting. We consider here the angle between the charged lepton cone and the neutrino beam, $\alpha_C$, defined as $$\cos{\alpha_C} = \frac{ (\vec{p}_3 + \vec{p}_4) \vdot \vec{p}_1}{ |\vec{p}_3 + \vec{p}_4| |\vec{p}_1| }\, ,$$ and in the individual angle of the charged lepton to the neutrino beam, $\theta$. For same flavour tridents we define $\theta$ for each charge of the visible final-state, whilst for mixed tridents we use their flavour. We also show the distribution in $Q^2 = {-q^2}$, where $q = (P - P^\prime)$, which is of particular interest when considering coherency and the impact of form factors.
Individual Backgrounds {#app:backgrounds}
======================
Here we discuss backgrounds to trident final-states in more detail. We start by motivating our misID rates shown in [Table \[tab:misIDlist\]]{}, and then discuss the dominant background processes individually.
In LAr photons can be distinguished from a single electron if their showers start displaced from the vertex (if present). Photons have a conversion length in LAr of around 18 cm, meaning $5$–$10\%$ could be expected to convert quickly enough to hinder electron-photon discrimination by this means if the resolution on the gap is from $1$–$2$ cm [@Acciarri:2016sli]. Once pair conversion happens, photons can be distinguished from a single electron purely by $\dd E/\dd x$ measurements in the first 1–2 cm of their showers. Motivated by the success of this method as shown at ArgoNeuT [@Acciarri:2016sli] and based on projections for DUNE [@Acciarri:2016ooe], we assume that $5\%$ of photons would be taken as $e^\pm$ with perfect efficiency, without the need for an event vertex. Needless to say that a dedicated study for trident topologies would be necessary for a more complete study. It is worth noting that our remarks concern only the misID of a single photon for a single electron, whilst the distinction between a photon and an overlapping $e^+e^-$ pair without a vertex can be much more challenging. For this reason we take the misID rate between an overlapping $e^+e^-$ pair and a photon to be 1 in the absence of a vertex.
Charged pions are notorious for faking long muon tracks. We estimate this misID rate as arising from through-going pions, which do not exhibit the decay kink used in their identification. We assume an interaction length of around $1$ m, meaning that about $5\%$ of particles travel $\sim3$ meters and escape the fiducial volume. Assuming that this is the most likely way a pion can spoof a muon, we estimate a naive suppression rate of $10^{-2}$. In a more complete study, it is desirable to explore the length of the muon and pion tracks inside the detector as a function of energy. The length of the contained tracks can also be an important tool for background suppression which we leave to future studies.
Pion Production
---------------
Coherent pion production in its charged ($\nu + A \to \ell^\mp + \pi^\pm + A$) and neutral ($\nu + A \to \nu + \pi^0 + A$) current version is very abundant at GeV energies. The cross section for these processes is modelled in GENIE using a modern version of the Rein-Sehgal model [@REIN198329; @Rein:2006di]. The charged current version serves mainly as a background to $\mu^+ \mu^-$ tridents, but can also appear as a background for $e^\pm \mu^\mp$ tridents for incoming electron neutrinos or antineutrinos. It has been studied before at MiniBooNE [@AguilarArevalo:2010xt], MINER$\nu$A [@Higuera:2014azj; @Mislivec:2017qfz], T2K [@Abe:2016fic; @Abe:2016aoo], and for the first time in LAr at ArgoNeuT [@Acciarri:2014eit]. This process has a very distict low 4-momentum transfer to the nucleus $|t|$ [@Higuera:2014azj], but a much flatter distribution in invariant mass if compared to trident. The neutral current version of coherent pion production serves as a background to $e^+e^-$ tridents. This process has been studied before by the MiniBooNE [@AguilarArevalo:2009ww], SciBooNE [@Kurimoto:2010rc] and in LAr by the ArgoNeuT collaboration [@Acciarri:2015ncl]. There are two possibilities for these events to fake an $e^+e^-$ trident: when one of the gammas produced in the $\pi^0$ decay is missed and the other is misIDed for an overlapping $e^+e^-$ pair, and when both photons are each misIDed for a single electron. This signature also comes with low hadronic activity, but for separated visible photons the invariant mass is a natural discriminator, as in the detector $m_{\gamma \gamma} \approx m_{\pi^0}$.
Resonant pion production can also contribute to trident backgrounds in the absence of any reconstructed protons. Resonant pion production can be larger than its coherent counterpart and is modelled in GENIE by the Rein-Sehgal model [@Rein:1980wg]. Its CC version was measured by MiniBooNE [@AguilarArevalo:2010xt], K2K [@Mariani:2010ez], MINOS [@Adamson:2014pgc], and MINER$\nu$A [@Altinok:2017xua]. In the latter measurement one can clearly see the large number of events with undetected protons. The misIDed photon and the charged lepton invariant mass are once more flatter than the trident ones, allowing for a kinematical discrimination whenever a single photon is undetected. It is worth noting that these are some of the dominant underlying processes for pion production in GENIE, but all events leading to topologies relevant for trident are included in our analysis.
Charm Production
----------------
Since the first observation of dimuon pairs from charm production in neutrino interaction by the HPWF experiment in 1974 [@Benvenuti:1975ru], a lot has been learned about these processes (see [@Lellis:2004yn] for a review) in neutrino experiments. Particularly, the production of charm quarks and their subsequent weak decays into muons or electrons have been identified as a major source of background for early trident searches. At the lower neutrino energies at DUNE, however, this is expected to be a smaller yet non-negligible contribution. From our GENIE samples, we estimate that a charmed state is produced at a rate of around $10^{-4}(N_\text{CC}+N_\text{NC})$. Most of these produce either D mesons, $\Lambda_c$ or $\Sigma_c$ baryons. These particles decay in chains, emitting a muon with a branching ratio of around $0.1$, and are always accompanied by pions or other hadronic particles. We therefore expect these rates to be negligible with a hadronic veto, and do not consider them further. We hope, however, that future studies will address these channels in more detail.
CC$\gamma$ and NC$\gamma$
-------------------------
The emission of a single photon alongside a CC process could be a background for $\mu e$ tridents if the photon is misIDed as a single electron. When the photon is produced in a NC event, it can be a background to overlapping $e^+e^-$ tridents. In GENIE, these topologies arise mainly due to resonance radiative decays and from the intra-nuclear processes. For this reason, it usually comes accompanied with extra hadronic activity. For hadronic resonances, we have simulated CC processes in GENIE and estimated the multiplicities: $0.5\%$ single photon and $1\%$ double photon emission from CC rates. Radiative photon production from the charged lepton, on the other hand, does not need to come accompanied by hadrons. It is phase space and $\alpha\approx1/137$ suppressed with respect to CCQE rates, and therefore could occur at appreciable rates compared to our signal. This contribution, however, is not included in GENIE and is absent from our samples. The rates of internal photon bremsstrahlung have been estimated before, particularly for T2K where a low-energy photon is an important background for electron appearance searches [@Efrosinin:2009zz], and as a background to the low energy events at MiniBooNE [@Bodek:2007wb]. De-excitation gammas from the struck nuclei can also generate CC$\gamma$ or NC$\gamma$ topologies [@PhysRevLett.108.052505]. These contributions for Ar are not included in GENIE, but are expected to come with a distinct energy profile, which can be tagged on.
[^1]: Throughout the manuscript we will consider ${\alpha,\beta, \kappa}$ as flavour indexes.
[^2]: Note that we include a factor of $1/2$ in $\sigma^\mathrm{T}_{\nu \gamma}$ to match the polarization averaging of the on-shell cross section: $\sigma_{\nu \gamma}^{\rm on-shell} = \frac{1}{2 \hat{s}} \left( \overline{\sum}_r (\epsilon_r^\mu)^* \epsilon^\nu_r \, {\rm L}_{\mu\nu} \right) \big\vert_{Q^2=0} = \frac{1}{4 \hat{s}} \left( - g^{\mu\nu} L_{\mu\nu}\right) \big\vert_{Q^2=0} = \frac{{\rm L_T}}{2 \hat{s}}\big\vert_{Q^2=0} = \sigma_{\nu\gamma}^\text{T}(0,\hat{s})$.
[^3]: An analogous expression can be obtained for the diffractive regime from Eq. (\[eq:dcoh\]).
[^4]: It should be noted that the coherent and diffractive regimes have different phase space boundaries and that the form factors should guarantee their independence.
[^5]: There are some differences in the treatment of the hadronic system between the EPA calculation in [@Magill:2016hgc] and the one presented here. However, these differences are of the order 10% to 20%. Note also that we do not implement any Pauli blocking when calculating $\mathcal{R}$ to avoid ambiguities over the choice of the range of $Q^2$.
[^6]: The peak of our diffractive $Q^2$ distributions happens at around $Q \approx 300$ MeV, much beyond the typical binding energy for Ar (see [Appendix \[app:distributions\]]{}). Without Pauli suppression, however, we expect this value to drop.
[^7]: See [Section \[subsec:kine\]]{} for a discussion on the detection efficiencies for trident events and backgrounds.
[^8]: A similar difference will apply to the processes constituting the background to the trident process, although there is an additional suppression in many channels due to the lower antineutrino cross sections.
|
---
abstract: 'Let $X$ be a Hamiltonian vector field defined on a symplectic manifold $(M,\omega)$, $g$ a nowhere vanishing smooth function defined on an open dense subset $M^0$ of $M$. We will say that the vector field $Y=gX$ is *conformally Hamiltonian*. We prove that when $X$ is complete, when $Y$ is Hamiltonian with respect to another symplectic form $\omega_2$ defined on $M^0$, and when another technical condition is satisfied, then there is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega)$, which maps each orbit to itself and is equivariant with respect to the flows of the vector fields $Y$ on $M^0$ and $X$ on $M$. This result explains why the diffeomorphism of the phase space of the Kepler problem restricted to the negative (resp. positive) values of the energy function, onto an open subset of the cotangent bundle to a three-dimensional sphere (resp. two-sheeted hyperboloid), discovered by Györgyi (1968) [@gyorgyi], re-discovered by Ligon and Schaaf (1976) [@ligonschaaf], is a symplectic diffeomorphism. Cushman and Duistermaat (1997) [@cushmanduistermaat] have shown that the Györgyi-Ligon-Schaaf diffeomorphism is characterized by three very natural properties; here that diffeomorphism is obtained by composition of the diffeomorphism given by our result about conformally Hamiltonian vector fields with a (non-symplectic) diffeomorphism built by a variant of Moser’s method [@moser]. Infinitesimal symmetries of the Kepler problem are discussed, and it is shown that their space is a Lie algebroid with zero anchor map rather than a Lie algebra.'
title: 'A property of conformally Hamiltonian vector fields; application to the Kepler problem'
---
<span style="font-variant:small-caps;">Charles-Michel Marle</span>
(Communicated by the associate editor name)
Introduction
============
I am very glad to submit a paper for the special issue of the Journal of Geometric Mechanics in honour of Tudor Ratiu. I followed his scientific work for several years; I specially praise the superb book he wrote with Juan-Pablo Ortega [@ortegaratiu]. I hope that he will find some interest in the present work.
The Kepler problem is a completely integrable Hamiltonian system with important applications in the physical world: it provides a very accurate model of the motion of planets in the solar system, and its quantized equivalent provides a good model of the hydrogen atom. Moreover, several features of the Kepler problem make it very interesting for the mathematician: some of its motions (those in which there is a collision of the moving point with the attractive centre) are not defined for all times, but the system can be regularized, *i.e.* mapped into a new Hamitonian system whose motions are defined for all times; the Lie algebra of infinitesimal symmetries of a given energy level of the phase space depends on that energy level.
In Section \[conformallyfields\] some results in symplectic geometry are presented. These results will be used in Section \[kepler\] to explain why a diffeomorphism between the phase space of the Kepler problem, restricted to negative values (resp. positive values) of the energy, and an open subset of the cotangent space to a 3-dimensional sphere (resp., a 3-dimensional two-sheeted hyperboloid) is symplectic. That diffeomorphism was discovered by Györgyi [@gyorgyi] (1968), re-discovered by Ligon and Schaaf [@ligonschaaf] (1976) and discussed by Cushman and Duistermaat [@cushmanduistermaat] (1997), who have shown that it is characterized by the three very natural properties:
1. It maps the set of points in the phase space of the Kepler problem where the energy is negative (resp., positive) onto the tangent bundle of the $3$-sphere (resp., the two-sheeted $3$-dimensional revolution hyperboloid) with its zero section removed.
2. It intertwines the Kepler and Delaunay vector fields (a rescaling of the geodesic vector field on the $3$-sphere, or on the $3$-dimensional hyperboloid).
3. It intertwines the $so(4)$-momentum mappings of the Kepler and Delauney vector fields.
We will see that the remarkable properties of that diffeomorphism appear as very natural consequences of the results presented in Section \[conformallyfields\].
We will also discuss the weak infinitesimal symmetries of the Kepler problem, and we will show that their set is a Lie algebroid with a zero anchor map, rather than a Lie algebra.
After completion and submission of the present paper, the work of Gert Heckman and Tim de Laat [@heckmandelaat], recently posted on arXiv, was indicated to us. The method used by these authors to explain the properties of the Györgyi-Ligon-Schaaf diffeomorphism rests on the same ideas as ours. We also learnt that conformally Hamiltonian vector fields were used, in the theory of bi-Hamiltonian vector fields, by A.J. Maciejewski, M. Prybylska and A.V. Tsiganov [@MaciePryTsi].
Conformally Hamiltonian vector fields {#conformallyfields}
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Notations and conventions {#notations}
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Unless another assumption is explicitly stated, all manifolds, functions, applications, vector fields and differential forms considered in this work will be assumed to be smooth, *i.e* differentiable of class $C^\infty$.
\[flow\] Let $X$ be a vector field on a manifold $M$. The *differential equation determined by* $X$ is the ordinary differential equation $$\frac{d\varphi(t)}{dt}=X\bigl(\varphi(t)\bigr)\,.$$ The *flow* of that equation is the map $\Phi_X$, defined on an open subset $D_X$ of ${{\mathbb{R}}}\times M$, with values in $M$, such that, for each $x\in M$, the parametrized curve $t\mapsto \Phi_X(t,x)$ is the maximal solution $\varphi$ of the above differential equation which takes the value $x$ for $t=0$. It means that $$\frac{\partial\Phi(t,x)}{\partial t}=X\bigl(\Phi(t,x)\bigr)
\quad\text{for each\ }(t,x)\in D_X\,,\quad \Phi_X(0,x)=x\quad\text{for each\ }x\in M\,,$$ and that the open subset $D_X$ of ${{\mathbb{R}}}\times M$ is such that, for each $x\in M$ $$I_x=\bigl\{t\in{{\mathbb{R}}}\bigm|(t,x)\in D_x\bigr\}$$ is the largest open interval of ${{\mathbb{R}}}$ on which a solution $\varphi$ of the differential equation determined by $X$ satisfying $\varphi(0)=x$ can be defined.
\[partial flow\] The map $\Phi_X$ is sometimes called *partial flow* to distinguish it from the *full flow* $\Psi_X$, which is the map defined on an open subset of ${{\mathbb{R}}}\times{{\mathbb{R}}}\times M$, with values in $M$, such that, for each $t_0\in {{\mathbb{R}}}$ and $x \in M$, the parametrized curve $t\mapsto \Psi_X(t,t_0,x)$ is the maximal solution of the above differential equation which takes the value $x_0$ for $t=t_0$. The vector field $X$ being time-independent, we have $\Psi_X(t,t_0 ,x_0)=\Phi_X(t-t_0,x_0)$, so the full flow is determined by the partial flow. It is no longer true for ordinary differential equations determined by a time-dependent vector field; the full flow should then be used instead of the partial flow.
Change of independent variable in ordinary differential equations {#changevar}
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On a manifold $M$, let $X$ be a vector field and $g$ a nowhere vanishing function. Consider the two ordinary differential equations $$\begin{aligned}
\frac{d\varphi(t)}{dt}&=g\bigl(\varphi(t)\bigr)X\bigl(\varphi(t)\bigr)\,,\tag{$*$}\\
\frac{d\psi(s)}{ds}&=X\bigl(\psi(s)\bigr)\,.\tag{$**$}
\end{aligned}$$ Let us assume that there exists a smooth function $\sigma:{{\mathbb{R}}}\times M\to{{\mathbb{R}}}$ such that for each solution $\varphi$ of $(*)$ $$\frac{d}{dt}\sigma\bigl(t,\varphi(t)\bigr)=g\bigl(\varphi(t)\bigr)\,.\tag{${*}{*}{*}$}$$ For each solution $\varphi:I_\varphi\to M$ of $(*)$ defined on the open interval $I_\varphi$ of ${{\mathbb{R}}}$, let $$\sigma_\varphi(t)=\sigma\bigl(t,\varphi(t))\,.$$ Then the function $\sigma_\phi$ is a diffeomorphism from $I_\varphi$ onto another open interval $\sigma_\varphi(I_\varphi)$ of ${{\mathbb{R}}}$, and the map $$\psi:\sigma_\varphi(I_\varphi)\to M\,,\quad s\mapsto \psi(s)=\varphi\circ\sigma_\varphi^{-1}(s)$$ is a solution of the ordinary differential equation $(**)$.
The derivative of the function $\sigma_\varphi$, at each poit $t\in I_\varphi$, is $\displaystyle g\bigl(\varphi(t))$, which never vanishes. Therefore $\sigma_\varphi$ is a diffeomorphism. The chain rule shows that $\psi$ is a solution of $(**)$.
For a general smooth vector field $X$ and a general smooth nowhere vanishing function $g$ given on $M$, there may be no globally defined function $\sigma:{{\mathbb{R}}}\times M\to{{\mathbb{R}}}$ verifying $({*}{*}{*})$. In the Kepler problem, that function exists, as we will see in subsection \[levi-civitaparameter\], and is affine in the variable $t$. The following proposition gives some information about the existence of the map $\sigma$.
On a manifold $M$, let $X$ be a vector field and $g:M\to{{\mathbb{R}}}\backslash\{0\}$ a nowhere vanishing function. We denote by $\Phi_X:D_X\to M$ the flow of $X$ (\[flow\]). For each $t_0\in{{\mathbb{R}}}$, there exists a smooth function $\sigma$, defined on an open neighbourhood $W_{t_0}$ of $\{t_0\}\times M$ in ${{\mathbb{R}}}\times M$, such that for each solution $\varphi:I_\varphi\to M$ of the differential equation determined by $X$ defined on an open interval $I_\varphi$ containing $t_0$, and each $t\in I_\varphi$, $$\frac{d}{dt}\sigma\bigl(t,\varphi(t)\bigr)=g\bigl(\varphi(t)\bigr)\,.\tag{${*}$}$$ The function $\sigma$ is not unique: any smooth function defined on $M$ can be chosen for its restriction to $\{t_0\}\times M$. The function $\sigma$ is affine with respect to the variable $t$ if and only if its restriction $\sigma_{t_0}:M\to {{\mathbb{R}}}$, $\sigma_{t_0}(x)=\sigma(t_0,x)$ is such that the Lie derivative with respect to the vector field $X$ of the function $g-\sigma_{t_0}$ is constant along each integral curve of $X$. When the preceding condition is satisfied, the affine extension of $\sigma$ to the whole ${{\mathbb{R}}}\times M$, still denoted by $\sigma$, satisfies $(*)$ for all solutions $\varphi:I\to M$ of the differential equation determined by $X$ and all $t\in I$.
Let us choose any $t_0\in {{\mathbb{R}}}$. The map $(\theta,x)\mapsto(t_0-\theta,x)$ is a diffeomorphism of ${{\mathbb{R}}}\times M$ onto itself, which sends the open subset $D_X$ on which the flow $\Phi_X$ is defined onto the subset $W_{t_0}=\bigl\{\,(t,x)\in{{\mathbb{R}}}\times M\bigm|(t_0-t,x)\in D_X\}$. Therefore $W_{t_0}$ is an open subset of ${{\mathbb{R}}}\times M$ which contains $\{t_0\}\times M$. Let $\sigma_{t_O}:M\to{{\mathbb{R}}}$ be any smooth function. The formula $$\sigma(t,x)=\sigma_{t_0}\circ\Phi_X(t_0-t,x)+\int_{t_0}^tg\circ\Phi_X(\tau-t,x)\,d\tau$$ defines a smooth function $\sigma:W_{t_0}\to{{\mathbb{R}}}$ whose restriction to $\{t_0\}\times M$ is $(t_0,x)\mapsto
\bigl(t_0,\sigma_{t_0}(x)\bigr)$. For each $(t,x_0)\in{{\mathbb{R}}}\times M$ such that $(t-t_0,x_0)\in D_X$, we have $$\begin{aligned}
\sigma\bigl(t,\Phi_X(t-t_0,x_0)\bigr)&=\sigma_{t_0}\circ\Phi_X\bigl(t_0-t,\Phi_X(t-t_0,x_0)\bigr)\\
&\quad\quad +\int_{t_0}^tg\circ\Phi_X\bigl(\tau-t,\Phi_X(t-t_0,x_0)\bigr)\,d\tau\\
&=\sigma_{t_0}(x_0)+\int_{t_0}^tg\circ\Phi_X(\tau-t_0,x_0)\,d\tau\,.
\end{aligned}$$ Therefore $$\frac{d}{dt}\sigma\bigl(t,\Phi(t-t_0,x_0)\bigr)=g\bigl(\Phi_X(t-t_0,x_0)\bigr)\,.$$ Since $t\mapsto\Phi_X(t-t_0,x_0)$ is the maximal solution of the differential equation determined by $X$ which takes the value $x_0$ for $t=t_0$, we see that the map $\sigma$ satisfies condition $(*)$ of the statement above.
The map $\sigma$ is affine with respect to $t$ if and only if its partial derivative with respect to $t$ does not depend on $t$. We have $$\begin{aligned}
\frac{\partial\sigma(t,x)}{\partial t}
&=-\bigl\langle d\sigma_{t_0}\circ\Phi_X(t_0-t,x),X\circ\Phi_X(t_0-t,x)\bigr\rangle\\
&\quad\quad +g(x)+\int_{t_0}^t\frac{\partial}{\partial t}\bigl(g\circ\Phi_X(\tau-t,x)\bigr)\,d\tau\,.
\end{aligned}$$ Taking into account $$\frac{\partial}{\partial t}\bigl(g\circ\Phi_X(\tau-t,x)\bigr)
=-\frac{\partial}{\partial \tau}\bigl(g\circ\Phi_X(\tau-t,x)\bigr)$$ we obtain $$\begin{aligned}
\frac{\partial\sigma(t,x)}{\partial t}
&=-\bigl\langle d\sigma_{t_0}\circ\Phi_X(t_0-t,x),X\circ\Phi_X(t_0-t,x)\bigr\rangle\\
&\quad\quad +g(x)-\Bigl(\bigl\langle dg\circ\Phi_X(\tau-t,x),X\circ\Phi_X(\tau-t,x)\bigr\rangle\Bigr)
\Bigm|_{\tau=t_0}^{\tau=t}\,.
\end{aligned}$$ Setting $\theta=t_0-t$, we see that $\sigma$ is affine with respect to $t$ if and only if, for all $(\theta,x)\in D_X$, $$\bigl\langle d(g-\sigma_{t_0})\circ\Phi_X(\theta,x),X\circ\Phi_X(\theta,x)\bigr\rangle
=\bigl\langle d(g-\sigma_{t_0})(x),X(x)\bigr\rangle\,,$$ which can also be written as $${\mathcal L}(X)(g-\sigma_{t_0})\bigl(\Phi_X(\theta,x)\bigr)
={\mathcal L}(X)(g-\sigma_{t_0})(x)\,,$$ where ${\mathcal L}(X)(g-\sigma_{t_0})$ is the Lie derivative of the function $g-\sigma_{t_0}$ with respect to the vector field $X$. This equality expresses the fact that ${\mathcal L}(X)(g-\sigma_{t_0})$ is constant along each integral curve of $X$. When the preceding condition is satisfied, the function $\sigma$ can be uniquely extended into a function, affine with respect to the variable $t$, defined on ${{\mathbb{R}}}\times M$, and one easily check that it satisfies condition $(*)$ of the statement above for all solutions $\varphi:I\to M$ of the differential equation determined by $X$ and all $t\in I$.
Hamiltonian and conformally Hamiltonian vector fields {#confhamfields}
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Let $(M,\omega)$ be a smooth symplectic manifold and $H:M\to{{\mathbb{R}}}$ a smooth function. The unique vector field $X_H$ such that $i({X}_H)\omega=-dH$ is called the *Hamiltonian vector field* associated to $H$, and $H$ is called a *Hamiltonian* for $X_H$. Let $g:M\to {{\mathbb{R}}}\backslash\{0\}$ be a nowhere vanishing function. The vector field $Y=g\,{X}_H$ will be called a *conformally Hamiltonian vector field*, with $H$ as *Hamiltonian* and $g$ as *conformal factor*. The vector field $Y$ satisfies $$i(Y)\omega=gi({X_H})\omega=-g\,dH\,.$$
\[result1\] Let $(M,\omega_1)$ be a symplectic manifold, $H:M\to{{\mathbb{R}}}$ a smooth Hamiltonian, $X$ the associated Hamiltonian vector field. We assume that $X$ is complete; in other words, its flow $\Phi_X$ is defined on the whole of ${{\mathbb{R}}}\times M$. Let $M^0$ be an open dense subset of $M$, $g:M^0\to{{\mathbb{R}}}\backslash\{0\}$ be a smooth, nowhere vanishing function and $Y=gX$ be the conformally Hamiltonian vector field on $M^0$, with Hamiltonian $H$ and conformal factor $g$. Its flow will be denoted by $\Phi_Y$. Let $\sigma:{{\mathbb{R}}}\times M^0\to {{\mathbb{R}}}$ be a smooth function such that for each maximal solution $\varphi$ of the differential equation determined by $Y$, $$\frac{d\sigma\bigl(t,\varphi(t)\bigr)}{dt}=g\bigl(\varphi(t)\bigr)\,.$$ We assume that there exists on $M^0$ another symplectic form $\omega_2$ such that $$i(Y)\omega_2= -dH\,.$$ In other words, the vector field $Y$ is both Hamiltonian with respect to $\omega_2$ with $H$ as Hamiltonian and conformally Hamiltonian with respect to $\omega_1$ with the same $H$ as Hamiltonian and with $g$ as conformal factor.
Under these assumptions, the map $$\Xi:M^0\to M\,,\quad x\mapsto\Xi(x)=\Phi_X\bigl(-\sigma(0,x),x\bigr)$$ is a symplectic diffeomorphism from $(M^0,\omega_2)$ onto an open subset of $(M,\omega_1)$, equivariant with respect to the flow of $Y$ on $M^0$ and the flow of $X$ on $M$, that is $$\Xi^*\omega_1=\omega_2$$ and for each $(t,x)$ in the open subset of ${{\mathbb{R}}}\times M^0$ on which $\Phi_Y$ is defined $$\Phi_X\bigl(t,\Xi(x)\bigr)=\Xi\bigl(\Phi_Y(t,x)\bigr)\,.$$
The maximal integral curve of the differential equation determined by $Y$, which takes the value $ x_0$ for $t=t_0$, is $$t\mapsto\Phi_Y(t-t_0, x_0)\,.$$ The same geometric curve in $M^0$, parametrized by $s=\sigma(t,x)$ instead of $t$, is an integral curve of the differential equation determined by the vector field $X$. The values of the parameter $s$ which correspond to $(t_0,x_0)$ and to $\bigl(t,\Phi(t-t_0,x_0))$ are, respectively, $$s_0=\sigma(t_0,x_0)\quad\hbox{and}\quad
s=\sigma\bigl(t, \Phi(t-t_0,x_0)\bigr)\,.$$ Since $\Phi_X$ is the flow of the vector field $X$, we have $$\Phi_Y(t-t_0,x_0)=\Phi_X\Bigl(\sigma\bigl(t,\Phi(t-t_0,x_0)\bigr)-\sigma(t_0,x_0)\,,\,x_0\Bigr)\,.$$ Let $\Upsilon:{{\mathbb{R}}}\times M^0\to{{\mathbb{R}}}\times M$ be the map $$(t,x)\mapsto \Upsilon(t,x)=\Bigl(t,\Phi_X\bigl(t-\sigma(t,x),x\bigr)\Bigr)\,.$$ We are going to prove that for all $t, t_0, x_0$ such that $\Phi_Y(t-t_0,x_0)$ is defined, $$\Upsilon\bigl(t,\Phi_Y(t-t_0,x_0)\bigr)=\Bigl(t, \Phi_X\bigl(t-\sigma(t_0,x_0),x_0\bigr)\Bigr)\,.$$ The above formula expresses the fact that $\Upsilon$ maps the graph of the integral curve $t\mapsto\Phi_Y(t-t_0,x_0)$ of the vector field $Y$ which takes the value $x_0$ for $t=t_0$, into the graph of the integral curve $t\mapsto\Phi_X\bigl(t-\sigma(t_0,x_0),x_0\bigr)$ which takes the value $x_0$ for $t=\sigma(t_0,x_0)$. Observe that $\Upsilon$ associates, to the point of the integral curve of $Y$ reached for the value $t$ of the parameter, the point of the integral curve of $X$ reached for the same value $s=t$ of the parameter.
Replacing $x$ by $\Phi_Y(t-t_0, x_0)$ in the formula which defines $\Upsilon$, we get $$\Upsilon\bigl(t,\Phi_Y(t-t_0,x_0)\bigr)=\Biggl(t,\Phi_X\Bigl(t-\sigma\bigl(t,\Phi_Y(t-t_0,x_0)\bigr),\Phi_Y(t-t_0,x_0)
\Bigr)\Biggr)\,.$$ But we have shown that $$\Phi_Y(t-t_0,x_0)=\Phi_X\Bigl(\sigma\bigl(t,\Phi_Y(t-t_0,x_0)\bigr)-\sigma(t_0,x_0)\,,\,x_0\Bigr)\,.$$ Therefore $$\begin{aligned}
&\Phi_X\Bigl(t-\sigma\bigl(t,\Phi_Y(t-t_0,x_0)\bigr),\Phi_Y(t-t_0,x_0)
\Bigr)\\
&\quad=\Phi_X\Biggl(t-\sigma\bigl(t,\Phi_Y(t-t_0,x_0)\bigr),
\Phi_X\Bigl(\sigma\bigl(t,\Phi_Y(t-t_0,x_0)\bigr)-\sigma(t_0,x_0)\,,\,x_0\Bigr)
\Biggr)\\
&\quad=\Phi_X\bigl(t-\sigma(t_0,x_0),x_0\bigr)\,.
\end{aligned}$$ We have proven that $\Upsilon\bigl(t,\Phi(t-t_0,x_0)\bigr)=\Bigl(t, \Psi\bigl(t-\sigma(t_0,x_0),x_0\bigr)\Bigr)$. Since $Y$ is a Hamiltonian vector field on $(M^0,\omega_2)$, with $H$ as Hamiltonian, the kernel of the closed $2$-form on ${{\mathbb{R}}}\times M^0$ $$\widetilde \omega_2=\omega_2-dH\wedge dt$$ is the rank-one completely integrable distribution generated by the nowhere vanishing vector field $$\frac{\partial}{\partial t}+ Y\,,$$ where $t$ denotes the coordinate function on the factor ${{\mathbb{R}}}$. We have indeed $$i\left(\frac{\partial}{\partial t}+ Y\right)(\omega_2-dH\wedge dt)=dH-dH=0\,.$$ Similarly, since $X$ is a Hamiltonian vector field on $(M,\omega_1)$, with $H$ as Hamiltonian, the kernel of the closed $2$-form on ${{\mathbb{R}}}\times M$ $$\widetilde \omega_1=\omega_1-dH\wedge ds$$ is the rank-one completely integrable distribution generated by the nowhere vanishing vector field $$\frac{\partial}{\partial s}+ X\,,$$ where the coordinate function on the factor ${{\mathbb{R}}}$ is now denoted by $s$. We have indeed $$i\left(\frac{\partial}{\partial s}+ X\right)(\omega_1-dH\wedge ds)=dH-dH=0\,.$$ We recall that $\Upsilon:{{\mathbb{R}}}\times M^0\to{{\mathbb{R}}}\times M$ maps injectively each leaf of the foliation of ${{\mathbb{R}}}\times M^0$ into a leaf of the foliation of ${{\mathbb{R}}}\times M$. On the manifold ${{\mathbb{R}}}\times M^0$, the 2-forms $\widetilde\omega_2$ and $\Upsilon^*\widetilde\omega_1$ both have the same kernel, since their kernels determine the same foliation. Each of these $2$-forms is therefore the product of the other by a nowhere vanishing function. This function is in fact the constant $1$, because $$\Upsilon^*s=t\,,\quad, \Upsilon^*H=H\,,\quad\hbox{so}\quad\Upsilon^*(dH\wedge ds)=dH\wedge dt\,,$$ and we have $\Upsilon^*\widetilde{\omega_1}=\widetilde{\omega_2}$.
Restricted to $\{0\}\times M^0$, the map $\Upsilon$ becomes $$(0,x)\mapsto\Upsilon(0,x)=\Bigl(0, \Psi\bigr(-\sigma(0,x),x\bigr)\Bigr)=\bigl(0,\Xi(x)\bigr)\,.$$ Since $\omega_2$ and $\omega_1$ are the forms induced, respectively, by $\widetilde\omega_2$ on $\{0\}\times M^0$ and by $\widetilde\omega_1$ on $\{0\}\times M$, we have $$\Xi^*\omega_1=\omega_2\,.$$ The $2$-forms $\omega_1$ and $\omega_2$ being nondegenerate, that proves that the map $\Xi$ is open. But its geometric interpretation proves that $\Xi$ is injective: for each $x_0\in M^0$, $\Xi(x_0)$ is the point of $M$ reached, for the value $s=0$ of the parameter $s$, by the integral curve $\psi$ of $X$ whose value for $s=s_0=\sigma(0,x_0)$ is $\psi(s_0)=x_0$; it always exists, because $X$ is assumed to be complete. Being open and injective, $\Xi$ is a diffeomorphism of $M^0$ onto an open subset of $M$, and more precisely, since $\Xi^*\omega_1=\omega_2$, a symplectic diffeomorphism of $(M^0,\omega_2)$ onto an open subset of $(M,\omega_1)$.
We have seen that $\Upsilon$ is equivariant with respect to the flows of the vector fields $\displaystyle\frac{\partial}{\partial t}+Y$ on ${{\mathbb{R}}}\times M^0$ and $\displaystyle\frac{\partial}{\partial s}+X$ on ${{\mathbb{R}}}\times M$. By projection on the second factor, we see that $\Xi$ is equivariant with respect to the flows of $Y$ on $M^0$ and $X$ on $M$.
Application to the Kepler problem {#kepler}
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The equations of motion of the Kepler problem {#equationsofmotion}
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In the physical space, mathematically modelled by an Euclidean affine 3-dimensional space $\mathcal E$, with associated Euclidean vector space $\vect{\mathcal E}$, let $P$ be a material point of mass $m$ subjected to the gravitational field created by an attractive centre $O$. The physical time is mathematically modelled by a real independent variable $t$. Let $$\vect r = \vect{OP}\,;\quad r=\Vert \vect r\Vert\,;
\quad \vect p = m\frac{d\vect r}{dt}\,;\quad p=\Vert\vect p\Vert\,.$$ The force $\vect f$ exerted on $P$ is $$\vect f = -\frac{km\vect r}{r^3}\,,$$ where $k$ is the constant which characterizes the acceleration field created by $O$.
The equations of motion are $$\frac{d\vect r}{dt} =\frac{\vect p}{m}\,,\quad
\frac{d\vect p}{dt}= -\frac{km\vect r}{r^3}\,.$$ The second equation above becomes singular when $r=0$, so we will assume that the massive point $P$ evolves in ${\mathcal E}\backslash\{O\}$. That space will be called the *configuration space* of the Kepler problem.
First integrals {#firstintegrals}
---------------
The above equations of motion are those determined by a Hamiltonian, time-independent vector field on a symplectic 6-dimensional manifold, called the *phase space* of the Kepler problem: it is the cotangent bundle $T^*\bigl({\mathcal E}\backslash\{O\}\bigr)$ to the configuration space. We will identify the tangent and cotangent bundles by means of the Euclidean scalar product. The Hamiltonian, whose physical meaning is the total energy (kinetic plus potential) of $P$, is $$E=E(\vect r, \vect p)=\frac{p^2}{2m}-\frac{mk}{r}\,.$$ The energy $E$ is a first integral of the motion, *i.e.*, it is constant along each integral curve.
The group ${\rm SO}(3)$ acts on the configuration space ${\mathcal E}\backslash\{O\}$ by rotations around $O$; the canonical lift of that action to the cotangent bundle leaves invariant the Liouville $1$-form and its exterior derivative, the canonical symplectic form of $T^*\bigl({\mathcal E}\backslash\{O\}\bigr)$. It also leaves invariant the Hamiltonian $E$. The lifted ${\rm SO}(3)$-action is Hamiltonian, and the corresponding momentum map is an ${\frak so}(3)^*$-valued first integral. Since the Euclidean vector space $\vect{\mathcal E}$ is of dimension $3$, once an orientation of that space has been chosen, we can identify the Lie algebra ${\frak so}(3)$ and its dual space ${\frak so}(3)^*$ with the vector space $\vect{\mathcal E}$ itself. With that identification the bracket in the Lie algebra ${\frak so}(3)$ becomes the *vector product*, denoted by $(\vect u, \vect v)\mapsto\vect u\times \vect v$; the coupling between spaces in duality becomes the *scalar product* $(\vect u, \vect v)\mapsto\vect u\cdot \vect v$; the momentum map of the Hamiltonian action of ${\rm SO}(3)$ is (up to a sign change) the well known *angular momentum* $$\vect L=\vect r\times\vect p\,.$$ In addition to the energy $E$ and the angular momentum $\vect L$, the equations of motion of the Kepler problem have as a first integral the *eccentricity vector* $\vect{\varepsilon}$ discovered by *Jakob Herman (1678–1753)* [@herman; @bernoulli], improperly called the *Laplace vector*, or the *Runge-Lenz vector*, whose origin remained mysterious for a long time: $$\vect \varepsilon=-\frac{\vect r}{r}+\frac{\vect p\times\vect L}{m^2k}
= \left(\frac{p^2}{m^2k}-\frac{1}{r}\right)\,\vect r
-\frac{\vect p\cdot\vect r}{m^2k}\,\vect p\,.$$ Using the well-known fact (the *first Kepler’s law*) that the orbit (the curve described by $P$ as a function of time) is a conic section with $O$ as one of its foci, the eccentricity vector has a very simple geometric meaning: it is a dimensionless vector, parallel to the major axis of the orbit, directed from the attracting centre towards the perihelion, of length numerically equal to the eccentricity of the orbit.
The hodograph {#hodograph}
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We will not recall in full how the equations of motion of the Kepler problem can be solved, since it is done in several excellent texts [@anosov; @cushmanbates; @feynman; @milnor]. Let us however briefly indicate the proof, due to William Rowan Hamilton (1805–1865) [@hamilton], of an important fact: the hodograph[^1] of each motion of the Kepler problem is a circle or an arc of a circle.
Let us first look at solutions for which at the initial time $t_0$, $\vect r$ and $\vect p$ are not collinear, so $\vect L(t_0)\neq 0$. Since $\vect L$ is time-independent, $\vect r$ and $\vect p$ are never collinear. We choose an orthonormal frame, positively oriented, with $O$ as origin and unit vectors $\vect{e_x}$, $\vect{e_y}$ and $\vect{e_z}$ with $\vect L$ parallel to $\vect{e_z}$. We have $$\vect L= L \vect{e_z}\,,\quad\hbox{where $L$ is a constant.}$$ The vectors $\vect r$ et $\vect p$ remain for all times parallel to the plane $xOy$. Let $\theta$ be the polar angle made by $\vect r$ with $\vect{e_x}$. We have $$\begin{split}
{\vect r}&=r\cos\theta \vect{e_x}+r \sin\theta \vect{e_y}\,,\\
{\vect p}&=m\left(\frac{dr}{dt}\cos\theta-r\frac{d\theta}{dt}\sin\theta\right)\vect{e_x}
+m\left(\frac{dr}{dt}\sin\theta + r\frac{d\theta}{dt}\cos\theta\right)\vect{e_y}\,\\
\vect L&=mr^2\frac{d\theta}{dt}\vect{e_z}\,.
\end{split}$$ Therefore $$L=mr^2\frac{d\theta}{dt}=\hbox{Constant}\,.$$ This is the *second Kepler law*, also called *law of areas*, since $\displaystyle\frac{L}{2m}$ is the area swept by the straight segment $OP$ during an unit time, with sign $+$ if $\theta$ increases with time and $-$ if it decreases. Observe that $\theta$’s variation is strictly monotonic. So we can take $\theta$ as independent variable, instead of time $t$. We may write $$\frac{d\vect p}{d\theta}=\frac{d\vect p}{dt}\,\frac{dt}{d\theta}=
-\frac{m^2k}{L}(\cos\theta\vect{e_x}+\sin\theta\vect{e_y})\,.$$ This ordinary differential equation for the unknown $\vect p$, which no more involves $\vect r$, can be readily integrated: $$\vect p=\frac{m^2k}{L}(-\sin\theta \vect{e_x}+\cos\theta\vect{e_y})+\vect c\,,$$ where $\vect c$ is a (vector) integrating constant. We will choose $\vect{e_y}$ such that $$\vect c = c \vect{e_y}\,,$$ where $c$ is a numeric constant of the same sign as $L$.
With $O$ as origin let us draw two vectors in the plane $xOy$, the first one (constant) being equal to $\vect c$, and the second one (which varies with $\theta$) equal to $\vect p$. The end point of that second vector moves on a circle whose centre is the end point of the vector equal to $\vect c$, and whose radius is $${\mathcal R}=\frac{m^2k}{\vert L\vert}\,.$$ This circle (or part of a circle) is (up to multiplication by $m$) the *hodograph* of the Kepler problem.
A short calculation leads to the following very simple relation between the energy $E$ of a motion, the radius $\mathcal R$ of its hodograph and the distance $\vert c\vert$ from the attracting centre $O$ to the centre of the hodograph: $$2mE=c^2-{\mathcal R}^2\,.$$ Observe that the right-hand side $c^2-{\mathcal R}^2$ is the *power*[^2] of $O$ with respect to the hodograph.
The Levi-Civita parameter {#levi-civitaparameter}
-------------------------
Let $\sigma$ be the function, defined on the product with ${{\mathbb{R}}}$ of the phase space of the Kepler problem, $$\sigma(t,\vect r, \vect p)=\frac{1}{mk}(\vect p\cdot\vect r-2 E(\vect r, \vect p)t\bigr)\,.$$ A short calculation using the equations of motion shows that for any solution $t\mapsto \bigl(\vect{r(t)}, \vect{p(t)}\bigr)$ of the Kepler problem, $$\frac{d\sigma\bigl(\vect{r(t)},\vect{p(t)}\bigr)}{dt}=\frac{1}{r(t)}\,.$$ With $s(t)=\sigma\bigl(\vect{r(t)},\vect{p(t)})$ as the new independent variable, instead of the time $t$, the equations of motion become $$\frac{d\vect{r(s)}}{ds}=\frac{r(s)\vect{p(s)}}{m}\,,\quad
\frac{d\vect{p(s)}}{ds}=-\frac{mk\vect{r(s)}}{r^2(s)}\,.$$ I will call $s$ the *Levi-Civita parameter*: it was introduced, as far as I know for the first time, by Tullio Levi-Civita in [@levicivita], by the differential relation which expresses $ds$ as a fuction of $dt$ and $r(t)$. The integrated formula which gives $\sigma(t,\vect r,\vect p)$ is in the paper [@souriautorino] by Jean-Marie Souriau, and in Exercise 8, chapter II of the book by R. Cushman and L. Bates [@cushmanbates]. This formula has probably been known before for a long time in the Celestial Mechanics community, but I do not know who found it for the first time.
With the Levi-Civita parameter as new independent variable, the system is no longer Hamiltonian, but rather *conformally Hamiltonian*. The Levi-Civita parameter will be used in subsection \[symplecticdiff\] in a slightly different context: we will define a diffeomorphism $S$ from the phase space of the Kepler problem restricted to negative (resp. positive) values of the energy, onto an open dense subset of the cotangent bundle to a three-dimensional sphere (resp., to one sheet of a two-sheeted three-dimensional hyperboloid). On this new phase space equipped with its canonical symplectic form, the image of the vector field which determines the equations of motion of the Kepler problem will be conformally Hamiltonian instead of Hamiltonian, while the image of the vector field transformed by the use of the Levi-Civita parameter as independent variable will be Hamiltonian.
The Györgyi-Ligon-Schaaf symplectic diffeomorphism {#glsdiffeo}
--------------------------------------------------
G. Györgyi [@gyorgyi] gave the expression of a symplectic diffeomorphism from the phase space of the Kepler problem restricted to the negative (resp. positive) values of the energy, onto an open subset of the cotangent bundle to a three-dimensional sphere (resp. two-sheeted hyperboloid). He obtained this diffeomorphism in two steps. First, following the ideas of Fock [@fock], employed two years later by Moser in his well known paper [@moser], he arranged the components of the first integrals $E$, $\vect L$ and $\vect\varepsilon$ of the Kepler problem into a $4\times4$ matrix, and used that matrix to define two $4$-dimensional vectors $\vect\rho$ and $\vect\pi$ (formulae 2.15 and 2.16 of Györgyi’s 1968 paper [@gyorgyi]). I believe that the map $(\vect r, \vect p)\mapsto (\vect\rho,\vect\pi)$ obtained by Györgyi after difficult to follow calculations, is the map, maybe rescaled, I have called $S^{-1}$ in Subsection \[smap\] below. Second, in Section entiteled Getting back the time $t$; Bacry’s generators of his 1968 paper, Györgyi composes the map $(\vect r, \vect p)\mapsto (\vect\rho,\vect\pi)$ with a suitably chosen map deduced from the flow of the Kepler vector field. He did not prove that the map he finally obtained is symplectic. This result was proven by Ligon and Schaaf [@ligonschaaf] who, eight years later, rediscovered the same diffeomorphism. Twenty years later, R. H. Cushman and J. J. Duistermaat [@cushmanduistermaat] discussed the properties of that diffeomorphism, gave new proofs of its remarkable properties and stated the nice characterization indicated in the Introduction.
For negative values of the energy, the Györgyi-Ligon-Schaaf diffeomorphism is the map which, to each element $(\vect r, \vect p)$ of the phase space of the Kepler problem such that $E(\vect r, \vect p)<0$, associates the element $\bigl((\xi_0,\vect \xi),(\eta_0,\vect\eta)\bigr)$ of ${{\mathbb{R}}}^4\times{{\mathbb{R}}}^4$ $$\begin{aligned}
\xi_0&=\frac{\sqrt{-2mE(\vect r, \vect p)}}{mk}\,\vect r.\vect p\sin\varphi
+\left(\frac{rp^2}{mk}-1\right)\cos\varphi\,,\\
\vect \xi&=\left(\frac{\vect r}{r}-\frac{\vect r.\vect p}{mk}\,\vect p\right)\sin\varphi
+\frac{\sqrt{-2mE(\vect r, \vect p)}}{mk}\,r\vect p\cos\varphi\,,\\
\eta_0&=-\vect r.\vect p\cos\varphi
+\frac{mk}{\sqrt{-2mE(\vect r,\vect p)}}\left(\frac{rp^2}{mk}-1 \right)\sin\varphi\,,\\
\vect\eta&=-\frac{mk}{\sqrt{-2mE(\vect r, \vect p)}}\left(\frac{\vect r}{r}-\frac{\vect r.\vect p}{mk}\,
\vect p\right) \cos\varphi +r\vect p\sin\varphi\,.\\\end{aligned}$$ In these formulae $\varphi$ is the angle given, as a function of $(\vect r, \vect p)$, by $$\varphi(\vect r,\vect p)=\frac{\sqrt{-2mE(\vect r,\vect p)}}{mk\vect r.\vect p}\,.$$ The quantities $\vect \xi=(\xi_1,\xi_2, \xi_3)$ and $\vect\eta=(\eta_1,\eta_2, \eta_3)$ are vectors of ${{\mathbb{R}}}^3$, while $(\xi_0,\vect\xi)=(\xi_0,\xi_1,\xi_2, \xi_3)$ and $(\eta_0,\vect\eta)=(\eta_0,\eta_1,\eta_2,\eta_3)$ are vectors of ${{\mathbb{R}}}^4$, such that $$\begin{aligned}
\xi_0^2+\Vert\vect\xi\Vert^2&=\xi_0^2+\xi_1^2+\xi_2^2+\xi_3^2=1\,,\\
\xi_0\eta_0+\vect{\mathstrut\xi}.\vect{\mathstrut\eta}&=\xi_0\eta_0+\xi_1\eta_1+\xi_2\eta_2+\xi_3\eta_3=0\,,\\
\eta_0^2+\Vert\vect\eta\Vert^2&=\eta_0^2+\eta_1^2+\eta_2^2+\eta_3^2>0\,.\end{aligned}$$ In other words, $(\xi_0,\vect\xi)$ is a point of the sphere $S^3$ of radius 1 centered at the origin, embedded in ${{\mathbb{R}}}^4$ (endowed with its usual Euclidean metric), and $(\eta_0,\vect\eta)$ is a non-zero vector tangent to that sphere at point $(\xi_0,\vect\xi)$. Using the usual scalar product of ${{\mathbb{R}}}^4$ to identify vectors and covectors, we may consider $\bigl((\xi_0,\vect\xi),(\eta_0,\vect\eta)\bigr)$ as an element of the cotangent bundle to the sphere $S^3$ minus its zero section.
For positive values of the energy, the Györgyi-Ligon-Schaaf diffeomorphism is given by similar formulae, the trigonometric functions $\sin$ and $\cos$ being replaced by the hyperbolic functions $\sinh$ and $\cosh$ and the Euclidean metric on ${{\mathbb{R}}}^4$ by the Lorentz pseudo-Euclidean metric.
We will show in what follows that the Györgyi-Ligon-Schaaf symplectic diffeomorphism can be easily obtained, and that all its properties can be proven, by an adaptation of Moser’s method for the regularization of the Kepler problem and application of Theorem \[result1\]. That method rests on the fact that hodographs of the Kepler problem are circles or arcs of circles, and that the stereographic projection maps circles into circles. We do not need to know in advance that the eccentricity vector $\vect\varepsilon$ is a first integral of the Kepler problem, since this property is an easy consequence of Moser’s method.
Stereographic projection {#stereoproj}
------------------------
In 1935, V. A. Fock [@fock] used an inverse stereographic projection for the determination of the energy levels of hydrogen atoms in quantum mechanics. More recently J. Moser [@moser] used the same idea for the regularization of the Kepler problem. His method yields a correspondence between oriented hodographs of motions of the Kepler problem with a fixed, negative energy, and great oriented circles of a three-dimensional sphere. As explained by D. V. Anosov [@anosov] and J. Milnor [@milnor], the method can easily be adapted for motions with a fixed, positive energy: instead of a three-dimensional sphere, one should use one sheet of a two-sheeted three-dimensional hyperboloid of revolution, and one obtains a correspondence between oriented hodographs of motions of the Kepler problem with the chosen positive energy, and connected components of great hyperbolas drawn on that sheet of hyperboloid (that means intersections of that sheet of hyperboloid with a plane containing the symmetry centre).
To deal with cases $E<0$ and $E>0$ simultaneously, we introduce the auxiliary quantity $$\zeta =
\begin{cases}
1 &\hbox{if}\quad E<0\,,\cr
-1 &\hbox{if}\quad E>0\,.\cr
\end{cases}$$ Let $(O, \vect{e_x},\vect{e_y},\vect{e_z})$ be an orthonormal frame of $\mathcal E$. with $O$ as origin. We add to the basis $(\vect{e_x},\vect{e_y},\vect{e_z})$ of $\vect{\mathcal E}$ a fourth vector $\vect{e_h}$ and we denote by $h$ the corresponding coordinate. So we get a 4-dimensional affine space $\mathcal F$. The physical space $\mathcal E$ will be identified with the affine subspace of $\mathcal F$ determined by the equation $h=0$.
In $\mathcal F$, for each $\rho>0$, let $Q_\rho$ be the quadric defined by $$h^2+\zeta(x^2+y^2+z^2)=\rho^2\,.$$ It is a sphere if $\zeta=1$, a two-sheeted hyperboloid if $\zeta=-1$. We will see that there is a value of $\rho$ particularly suited for each value of the energy $E$.
Let $N$ be the point with coordinates $(x=y=z=0,\quad h=\rho)$. The *stereographic projection* (usual if $\zeta=1$, generalized if $\zeta=-1$) of the quadric $Q_\rho$ minus point $N$ on the space $\mathcal E$ is the map which associates, with each point $M\in Q_\rho\backslash\{N\}$, the intersection point $m$ of the straight line which joins $N$ and $M$, with $\mathcal E$. See Figure \[projstereo\]. If $\zeta=1$ that map is a diffeomorphism from $Q_\rho\backslash\{N\}$ onto $\mathcal E$. If $\zeta=-1$ it is a diffeomorphism from $Q_\rho\backslash \{N\}$ onto the open subset of $\mathcal E$ complementary to the 2-sphere of centre $O$ and radius $\rho$. The upper sheet ($h>0$) of the hyperboloid (minus point $N$) is mapped onto the outside of that sphere and the lower sheet ($h<0$) onto its inside.
![Stereographic projection and its generalization[]{data-label="projstereo"}](keplerratiufig.1 "fig:")\
Cotangent lift {#cotangentlift}
--------------
Since the (maybe generalized) stereographic projection is a diffeomorphism, it can be uniquely lifted to the cotangent bundles in such a way that the pull-back of the Liouville form on $T^*{\mathcal E}$ is equal to the Liouville form on $T^*(Q_\rho\backslash\{N\})$. We apply this construction with $\vect{Om}=\vect p$, the linear momentum of a Keplerian motion of energy $E\neq 0$.
Each $1$-form on $\mathcal E$ can be written $\vect r\cdot d\vect{Om}$, where $\vect r$ is a vector field on $\mathcal E$, or, since $\vect{Om}=\vect p$, $$\vect r\cdot d\vect p=r_x\,dp_x+r_y\,dp_y+r_z\,dp_z\,.$$ Recall that $\zeta=1$ if $E<0$ and $\zeta =-1$ if $E>0$. The equation of $Q_\rho$ is $$\vect{OM}\cdot \vect{OM}= x^2+y^2+z^2+\zeta h^2=\zeta \rho^2\,.$$ Each $1$-form on $Q_\rho$ can be written $$\vect W\cdot d\vect{OM}= W_x\, dx+W_y\,dy+W_z\,dz+\zeta W_h\,dh\,,$$ where $\vect W$ is a vector field defined on $Q_\rho$, tangent to that quadric. Therefore, $$\vect W\cdot \vect{OM}=W_x x +W_y y +W_z z+\zeta W_h h=0\,.$$
The cotangent lift of the (maybe generalized) stereographic projection will be denoted by $S_\rho$. It maps each pair $(M, \vect W)$ made by a point $M\in Q_\rho\backslash\{N\}$ and a vector $\vect W$ tangent to $Q_\rho$ at that point, onto a pair $(\vect r, \vect p)$ of two vectors of $\vect{\mathcal E}$, in such a way that $$\vect W\cdot d\vect{OM}=\vect r\cdot d\vect p\,.$$ The formulae which give $(\vect p, \vect r)$ as functions of $(\vect{OM},\vect W)$ are easily obtained after some calculations: $$\left\{
\begin{aligned}
\vect p&=\frac{\rho}{\rho-h}\,\vect {O\mu}\,,\quad\hbox{with}\quad h^2+\zeta\Vert \vect{O\mu}\Vert^2=\rho^2\,,\\
\vect r&=\frac{\rho-h}{\rho}\,\vect {W_3}+\frac{W_h}{\rho}\,\vect{O\mu}\,,
\end{aligned}
\right.$$ where we have set $$\vect{OM} =\vect{O\mu}+h\vect{e_h}\,,\quad \vect W=\vect{W_3}+W_h\vect{e_h}\,,$$ The formulae for the inverse transform are: $$\left\{
\begin{aligned}
\vect{O\mu}&=\frac{2\rho^2}{\rho^2+\zeta p^2}\vect p\,,\\
h&=\rho\,\frac{p^2-\zeta\rho^2}{p^2+\zeta\rho^2}\,,
\quad\hbox{with}\quad p=\Vert\vect p\Vert=\sqrt{p_x^2+p_y^2+p_z^2}\,,\\
\vect{W_3}&=\frac{\rho^2+\zeta p^2}{2\rho^2}\vect r-\frac{\zeta\vect r.\vect p}{\rho^2}\vect p\,,\\
W_h&=\frac{\zeta\vect r.\vect p}{\rho}\,.
\end{aligned}
\right.$$ These formulae prove that if $\vect{p_1}=\vect{Om_1}$ and $\vect{p_2}=\vect{Om_2}$ are two collinear vectors in $\vect{\mathcal E}$ such that $$\vect{Om_1}.\vect{Om_2}=-\zeta \rho^2\,,$$ their images, by the inverse stereographic projection, are two points $M_1$ and $M_2$ symmetric of each other with respect to $O$. That property may also be proven without calculation as a consequence of properties of the transformation by inversion.
We recall that if a straight line through $O$ meets the hodograph of a Keplerian motion at two points $m_1$ and $m_2$, $\vect{Om_1}.\vect{Om_2}$ is the power (see the second footnote in Subsection \[hodograph\]) of $O$ with respect to the hodograph, and is equal to $2mE$, where $E$ is the energy of the motion. Therefore the above results shows that the inverse streographic projection maps the hodographs of motions of energy $\displaystyle -\frac{\zeta\rho^2}{2m}$ on curves, drawn on the quadric $Q_\rho$, which are symmetric with respect to $O$. These curves are great circles of the sphere $Q_\rho$ if $\zeta=1$, and great hyperbolas of the hyperboloid $Q_\rho$ if $\zeta=-1$. The name great hyperbola is used by analogy with great circle: it means the intersections of $Q_\rho$ with a plane containing the symmetry centre $O$. As a consequence, the cotangent lift $S_\rho^{-1}$ of the inverse stereographic projection maps the hodographs of motions of the Kepler problem of energy $\displaystyle E=-\frac{\zeta\rho^2}{2m}$ on geodesics of the quadric $Q_\rho$. The hodographs of motions with a different value of the energy are mapped on circles (if $\zeta=1$) or hyperbolas (if $\zeta=-1$) drawn on the quadric $Q\rho$, but which are not *great* circles, or *great* hyperbolas.
So we see that for each value of the energy $E$, there is a particularly well suited value of $\rho$: it is the unique $\rho>0$ such that $\displaystyle E=-\frac{\zeta\rho^2}{2m}$, with $\zeta=1$ if $E<0$ and $\zeta=-1$ if $E>0$.
\[antisymp\] We have defined the mapping $S_\rho:T^*(Q_\rho\backslash\{N\})\to T^*{\mathcal E}$ so that $$(S_\rho)^*(\vect r\cdot d\vect{p})=\vect W\cdot d\vect{OM}\,.$$ Since the symplectic form of $T^*{\mathcal E}$ is $d(\vect p\cdot d\vect{r})$ and that of $T^*(Q_\rho\backslash\{N\})$ $d(\vect W\cdot d\vect{OM})$, the map $S_\rho$ is an *anti-symplectic* diffeomorphism, rather than a symplectic diffeomorphism.
The transformed Hamiltonian $(S_\rho)^* E=E\circ S_\rho$ is $$E\circ S_\rho =\frac{-\zeta\rho^2}{2m} +\frac{\zeta\rho^3}{m(\rho-h)\Vert \vect W\Vert}
\left(\Vert\vect W\Vert-\frac{km^2}{\rho^2}\right)\,,$$ where we have set $$\Vert\vect W\Vert=\sqrt{W_x^2+W_y^2+W_z^2+\zeta W_h^2}\,.$$ The expression of $E\circ S_\rho$ shows that $\displaystyle E\left(\vect r,\vect p\right)=\frac{-\zeta\rho^2}{2m}$ if and only if $S_\rho^{-1}\left(\vect r, \vect p\right)=\left(\vect{OM}, \vect W\right)$ is such that $\displaystyle\Vert \vect W\Vert=\frac{km^2}{\rho^2}$. By differentiation we obtain $$d(E\circ S_\rho) =\frac{\zeta\rho^3}{m(\rho-h)\Vert \vect W\Vert}\,d
\left(\Vert\vect W\Vert\right)+\left(\Vert\vect W\Vert-\frac{km^2}{\rho^2}\right)
d\left(\frac{\zeta\rho^3}{m(\rho-h)\Vert \vect W\Vert}\right)\,.$$ For a motion of energy $\displaystyle E=-\frac{\zeta\rho^2}{2m}$, the second term of the right hand side of the above equality vanishes, since $\displaystyle \left(\Vert\vect W\Vert-\frac{km^2}{\rho^2}\right)=0$, and we see that $S_\rho^{-1}$ transforms the Hamiltonian vector field of the Kepler problem into a conformally Hamiltonian vector field on $T^*(Q_\rho\backslash\{N\})$ minus the zero section, with Hamiltonian $\Vert \vect W\Vert$ and conformal factor $\displaystyle \frac{-\zeta\rho^3}{m(\rho-h)\Vert \vect W\Vert}$. We introduced a minus sign in the conformal factor to account for the fact that $S_\rho^{-1}$ is anti-symplectic (see Remark \[antisymp\]).
Motions of zero energy {#zeroenergy}
----------------------
For Keplerian motions of energy zero the stereographic projection should be replaced by an inversion with pole $O$ and ratio $ l$. Formulae for its cotangent lift are $$\left\{
\begin{aligned}
\vect{p}&=\frac{ l}{\Vert\vect{OM}\Vert^2}\,\vect{OM}\,,\\
\vect r&=\frac{\Vert\vect{OM}\Vert^2}{ l}\,\vect W-\frac{2\vect W\cdot \vect{OM}}{ l}\,\vect{OM}\,.
\end{aligned}
\right.$$ The correspondence $(\vect p, \vect r)\mapsto(\vect{OM}, \vect W)$ being involutive, the formulae for the inverse transformation are $$\left\{
\begin{aligned}
\vect{OM}&=\frac{ l}{p^2}\,\vect p\,,\\
\vect W&=\frac{p^2}{ l}\,\vect r-\frac{2\vect r\cdot \vect p}{ l}\,\vect p\,.
\end{aligned}
\right.$$ Denoting by $S_0$ the cotangent lift of the inversion of pole $O$ and ratio $ l$, we easily obtain the transformed Hamiltonian $$E\circ {S_0}=\frac{ l^2}{2m\Vert \vect W\Vert\,\Vert\vect{OM}\Vert^2}\,
\left(\Vert \vect W\Vert - \frac{2m^2k}{ l} \right)\,.$$ Now $\vect W$ is a vector of $\vect{\mathcal E}$, with 3 components, so we have set $$\Vert\vect W\Vert=\sqrt{W_x^2+W_y^2+W_z^2}\,.$$ As above this result proves that $E\left(\vect r,\vect p\right)=0$ if and only if $S_0^{-1}\left(\vect r, \vect p\right)=\left(\vect{OM}, \vect W\right)$ is such that $\displaystyle\Vert\vect W\Vert=\frac{2km^2}{ l}$.
By differentiation we obtain $$d(E\circ {S_0})=\frac{ l^2}{2m\Vert \vect W\Vert\,\Vert\vect{OM}\Vert^2}\,
d\left(\Vert \vect W\Vert\right)+\left(\Vert \vect W\Vert - \frac{2m^2k}{ l} \right)
d\left(\frac{ l^2}{2m\Vert \vect W\Vert\,\Vert\vect{OM}\Vert^2}
\right)\,.$$ For a motion of energy $\displaystyle E=0$, the second term of the right hand side of the above equality vanishes, since $\displaystyle \left(\Vert\vect W\Vert-\frac{2km^2}{ l}\right)=0$, and we see that $S_0^{-1}$ transforms the Hamiltonian vector field of the Kepler problem into a conformally Hamiltonian vector field on $T^*({\mathcal E}\backslash\{0\})$ minus the zero section, with Hamiltonian $\Vert \vect W\Vert$ and conformal factor $\displaystyle \frac{- l^2}{2m\Vert \vect W\Vert\,\Vert\vect{OM}\Vert^2}$. We introduced a minus sign in the conformal factor to account for the fact that $S_0$ is anti-symplectic (Remark \[antisymp\] applies to $S_0$ as well as to $S_\rho$).
Infinitesimal symmetries {#infinitesimalsymmetries}
------------------------
Let us first recall some properties about infinitesimal symmetries. Consider a vector field $X$ on a manifold $M$. Another vector field $Z$ on $M$ is said to be an *infinitesimal symmetry* of the differential equation determined by $X$ (or, in short, of $X$) if $[Z,X]=0$; it is said to be a *weak infinitesimal symmetry* of $X$ if at each point $x\in M$, $[Z,X](x)$ and $X(x)$ are collinear. If $Z$ is an infinitesimal symmetry or a weak infinitesimal symmetry of $X$ and $h$ a smooth function, $Z$ is a weak infinitesimal symmetry of $hX$, since $$[Z, hX]= h[Z,X]+\bigl({\mathcal L}(Z)h\bigr)\,X\,,$$ whichs shows that at each point $x\in M$, $[Z,hX](x)$ and $X(x)$ are collinear. When $X=X_H$ is a Hamiltonian vector field on a symplectic manifold $(M,\omega)$, with a function $H$ as Hamiltonian, a vector field $Z$ on $M$ such that ${\mathcal L}(Z)\omega=0$ and ${\mathcal L}(Z)H=0$ is an infinitesimal symmetry of $X_H$ and a weak infinitesimal symmetry of $hX_H$ for any smooth function $h$. When in addition $Z$ is Hamiltonian, its Hamiltonian is a first integral of $X_H$ and of $gX_H$. Conversely, if $f$ and $g$ are two smooth first integrals of the Hamiltonian vector field $X_H$, the vector field $gX_f$, $X_f$ being the Hamiltonian vector field with Hamiltonian $f$, is an infinitesimal symmetry of $X_H$ and a weak infinitesimal symmetry of $hX_H$ for any smooth function $h$. We have indeed $$\begin{aligned}
[gX_f,hX_H]&=gh[X_f,X_H]+g\bigl(i(X_f)h\bigr)\,X_H-h\bigl(i(X_H)g\bigr)\,X_f\\
&=ghX_{\{f,H\}}+g\{f,h\}\,X_H-h\{H,g\}\,X_f=g\{f,h\}\,X_H\,,
\end{aligned}$$ since $\{f,H\}=\{H,g\}=0$.
For the vector field which determines the equations of motion of the Kepler problem, we already know a three-dimensional vector space of infinitesimal symmetries: it is made by the canonical lifts to the cotangent bundle of infinitesimal rotations of the configuration space ${\mathcal E}\backslash\{O\}$ around the attractive centre $O$. These canonical lifts are Hamiltonian vector fields, and their Hamiltonians (linear combinations of the components of the angular momentum $L$) are first integrals of the equations of motion. The results obtained in Subsections \[cotangentlift\] and \[zeroenergy\] show that other weak infinitesimal symmetries exist. In these subsections, for each value $e$ of the energy $E$, we have built a symplectic diffeomorphism[^3] of the phase space of the Kepler problem onto an open subset of another symplectic manifold: the cotangent space to a three-dimensional sphere when $e<0$, the cotangent space to $\mathcal E$ when $e=0$ and the cotangent space to a three-dimensional hyperboloid when $e>0$. That sphere, affine space $\mathcal E$ or hyperboloid will be called the *new configuration space*, and its cotangent bundle (with the zero section removed) the *new phase space*.
For each energy level $e$, the new phase space has a 6-dimensional Lie group of global symmetries: ${\rm SO}(4)$ when $e<0$, the group ${\rm SE}(3)$ of orientation-preserving isometries (rotations and translations) of $\mathcal E$ when $e=0$ and the Lorentz group ${\rm SO}(3,1)$ when $e>0$. The direct image of the Hamiltonian vector field of the Kepler problem, restricted to each energy level of the phase space, is equal to the restriction, to the corresponding submanifold of the new phase space, of a conformally Hamiltonian vector field, whose Hamiltonian is invariant by the action of the $6$-dimensional Lie group of symmetries of the new phase space. Therefore, on each energy level of the phase space of the Kepler problem, six linearly independent weak infinitesimal symmetries exist, of which three are the already known linearly independent infinitesimal symmetries associated with the cotangent lifts of infinitesimal rotations of $\mathcal E$ around three non-coplanar axes through $O$. The other three correspond to the additional symmetries of the considered energy level of the new phase space. We will see that they depend smoothly on the value of the energy. Therefore, on the phase space of the Kepler problem considered as a whole, there exist $6$ linearly independent vector fields, three of them being infinitesimal symmetries and the other three weak infinitesimal symmetries. We will see that these six vector fields, which of course are tangent to each energy level, are Hamiltonian and are all infinitesimal symmetries of the Kepler vector field. So their Hamiltonians are first integrals of the equations of motion.
In Subsection \[energymomentum\] we will see that these infinitesimal symmetries do not span a Lie algebra, but rather a fibered space in Lie algebras [@douadylazard] over the base ${{\mathbb{R}}}$, the value of the energy running over that base, in other words a *Lie algebroid* [@cannasweinstein; @mackenzie] of a special kind, with a zero anchor map.
Let us recall [@libermannmarle; @ortegaratiu] that if $\Phi:G\times M\to M$ is a left action of a Lie group $G$ on a manifold $M$, its canonical lift to the cotangent bundle is a Hamiltonian action $\widehat\Phi:G\times T^*M\to T^*M$ with an equivariant momentum map $J:T^*M\to{\mathcal G}^*$ given by $$\langle J(\xi), X\rangle = \bigl\langle \xi, X_M\bigl(\pi_M(\xi)\bigr)\bigr\rangle\,.$$ In this formula, $\pi_M:T^*M\to M$ is the canonical projection; $X\in{\mathcal G}$, the Lie algebra of $G$; $\xi\in{\mathcal G}^*$, the dual space of $\mathcal G$; and $X_M$ is the fundamental vector field on $M$ associated to $X$, defined by $$X_M(x)=\frac{d\Bigl(\Phi\bigl(\exp(tX),x\bigr)\Bigr)}{dt}\Biggm|_{t=0}\quad\text{for each}\quad x\in M\,.$$ The momentum map $J$ is a first integral of any Hamiltonian vector field $X_H$ or conformally Hamiltonian vector field $gX_H$ on $T^*M$ whose Hamiltonian $H$ is invariant under $\widehat \Phi$. For each $X\in {\mathcal G}$ the Hamiltonian vector field on $T^*M$ whose Hamiltonian is $\xi\mapsto\langle J(\xi),X\rangle$ is an infinitesimal symmetry of $X_H$.
We apply these results, with for $\widehat\Phi$ the action on the new phase space of its symmetry group. As a vector space, the Lie algebra of the symmetry group is canonically isomorphic to $\vect{\mathcal E}\times\vect{\mathcal E}$, but with a Lie algebra bracket $[\ ,\ ]_e$ which depends on the energy level $e$. That bracket will be determined in Subsection \[energymomentum\]. Two kinds of infinitesimal symmetries of the new phase space exist. The first kind is made by the fundamental vector fields which correspond to elements $(\vect{u_1},0)$ of $\vect{\mathcal E}\times\vect{\mathcal E}$. These vector fields are the canonical lifts to the cotangent bundle of the infinitesimal rotation of $\mathcal E$ around the axis through $O$ parallel to $\vect{u_1}$. The second kind is made by the fundamental vector fields which correspond to elements $(0,\vect{u_2})$ of $\vect{\mathcal E}\times\vect{\mathcal E}$. These vector fields are the canonical lifts to the cotangent bundle of the following infinitesimal transformations of the new configuration space:
- when $e<0$, the restriction to the sphere $Q_\rho$, with $\rho^2=-2me$, of the infinitesimal rotation of $\vect{\mathcal F}=\vect{\mathcal E}\times({{\mathbb{R}}}\times\vect{e_h})$ in which the plane spanned by $\vect{u_2}$ and $\vect{e_h}$ rotates, while the vector subspace of $\vect{\mathcal E}$ orthogonal to $\vect{u_2}$ remains fixed;
- when $e=0$, the infinitesimal translation of $\mathcal E$ parallel to $\vect{u_2}$;
- when $e>0$, the restriction to the hyperboloid $Q_\rho$, with $\rho^2=2me$, of the infinitesimal Lorentz transformation of $\vect{\mathcal F}=\vect{\mathcal E}\times({{\mathbb{R}}}\times\vect{e_h})$ in which the plane spanned by $\vect{u_2}$ and $\vect{e_h}$ is transformed, while the vector subspace of $\vect{\mathcal E}$ orthogonal to $\vect{u_2}$ remains fixed.
The space $\vect{\mathcal E}\times\vect{\mathcal E}$ will be identified with its dual space by using the scalar product on each factor. The momentum map, which will be denoted by $K_e$, can be written $K_e=(K_{1e},K_{2e})$, each component taking its values in $\vect{\mathcal E}$.
In the three cases $e<0$, $e=0$ and $e>0$, the Hamiltonian $H$ has the same expression $$(\vect{OM},\vect W)\mapsto H(\vect{OM},\vect W)=\Vert \vect W\Vert\,.$$ It is equivalent to the classical Hamiltonian $\displaystyle \frac{\Vert\vect W\Vert^2}{2}$ of a particle moving freely on $Q_\rho$ when $e\neq 0$ or on $\mathcal E$ when $e=0$. It is indeed invariant under the action $\widehat \Phi$.
The first component $K_{1e}$ of the momentum map is $$K_{1e}(\vect{OM},\vect{W})=
\begin{cases}
\vect{O\mu}\times\vect{W_3}&\text{when\quad $e\neq 0$,}\\
\vect{OM}\times\vect{W}&\text{when\quad $e=0$,}
\end{cases}$$ and its second component is $$K_{2e}(\vect{OM},\vect{W})=
\begin{cases}
\zeta(h\vect{W_3} - W_h\vect{O\mu})&\text{when\quad $e\neq 0$,}\\
\vect W&\text{when\quad $e=0$.}
\end{cases}$$ As above, we have set when $e\neq 0$ $$\vect{OM}=\vect{O\mu}+h\vect{e_h}\,,\quad \vect{W}=\vect{W_3}+W_h\vect{e_h}\,.$$ Let us now come back to the phase space of the Kepler problem. The composed map $J_e=K_e\circ S_\rho^{-1}$, which has two components $J_{1e}=K_{1e}\circ S_\rho^{-1}$ and $J_{2e}=K_{2e}\circ S_\rho^{-1}$, is the momentum map of the Lie algebra action of infinitesimal symmetries. Using the formulae which give $\vect {OM}$ and $\vect W$ as functions of $\vect r$ and $\vect p$, we obtain for the first component, in the three cases $e<0$, $e=0$ and $e>0$ $$J_{1e}=\vect p\times \vect r\,.$$ Its expression does not depend on $e$. Up to a change of sign, it is the angular momentum $\vect L$. For the second component we obtain $$J_{2e}=
\begin{cases}
\displaystyle
\frac{p^2-\zeta\rho^2}{2\rho}\,\vect r-\frac{\vect r.\vect p}{\rho}\,\vect p&\text{when\quad $e\neq0$,}\\
\displaystyle
\frac{p^2}{l}\,\vect r-\frac{2\vect r.\vect p}{l}\,\vect p&\text{when\quad $e=0$.}
\end{cases}$$ Using the value of the energy $\displaystyle E=\frac{p^2}{2m}-\frac{mk}{r}$ and, when $e\neq 0$, the equality $\zeta \rho^2=-2mE$, these formulae become $$J_{2e}=
\begin{cases}
\displaystyle
\frac{m^2k}{\sqrt{-2\zeta m e}}\,\vect\varepsilon&\text{when\quad $e\neq0$,}\\
\displaystyle
\frac{2m^2k}{l}\,\vect \varepsilon&\text{when\quad $e=0$,}
\end{cases}$$ where $\vect \varepsilon$ is the eccentricity vector. Since the energy $E(\vect r, \vect p)$ is a first integral, we see that by choosing $l=2m^2k$ and, for $e\neq 0$, by multiplying the weak infinitesimal symmetries by the smooth function $\displaystyle \frac{\sqrt{-2\zeta m E(\vect r, \vect p)}}{m^2k}$, we can arrange things so that the second component of the momentum map becomes $$J_{2e}=\vect\varepsilon\,,$$ which no longer depends on the energy level $e$. Therefore, although we have defined them separately on each energy level, the three additional weak infinitesimal symmetries smoothly depend on the energy level, since they are the Hamiltonian vector fields whose Hamiltonians are the components of $\vect \varepsilon$; they are true (not only weak) infinitesimas symmetries since they are Hamiltonian vector fields whose Hamiltonians are first integrals of the Kepler equations of motion.
The energy-momentum space and map {#energymomentum}
---------------------------------
The Poisson brackets of the components of the angular momentum $\vect L$ and of the eccentricity vector $\vect\varepsilon$ in an orthonormal frame $(\vect{e_x},\vect{e_y},\vect{e_z})$ of $\vect{\mathcal E}$ are $$\begin{aligned}
\{L_x,L_y\}&=-L_z\,,&\{L_y,L_z\}&=-L_x\,,& \{L_z,L_x\}&=-L_y\,;\\
\{L_x,\varepsilon_x\}&=0\,,& \{L_x,\varepsilon_y\}&=-\varepsilon_z\,,& \{L_x,\varepsilon_z\}&=\varepsilon_y\,;\\
\{L_y,\varepsilon_x\}&=\varepsilon_z\,,& \{L_y,\varepsilon_y\}&=0\,,& \{L_y,\varepsilon_z\}&=-\varepsilon_x\,;\\
\{L_z,\varepsilon_x\}&=-\varepsilon_y\,,& \{L_z,\varepsilon_y\}&=\varepsilon_x\,,& \{L_z,\varepsilon_z\}&=0\,;\\
\{\varepsilon_x,\varepsilon_y\}&=\frac{2E}{m^3k^2}L_z\,,&
\{\varepsilon_y,\varepsilon_z\}&=\frac{2E}{m^3k^2}L_x\,,&
\{\varepsilon_z,\varepsilon_x\}&=\frac{2E}{m^3k^2}L_y\,.
\end{aligned}$$ The family of fuctions on the phase space of the Kepler problem spanned by the components of $\vect L$ and of $\vect\varepsilon$ is not a Lie algebra, because the function $E$ appears in the right hand sides of equalities in the last line. Of course, if we replace $\vect\varepsilon$ by $\displaystyle\frac{\vect\varepsilon}{\sqrt{-\zeta E}}$, we get a Lie algebra isomorphic to $so(4)$ when $E<0$ and to $so(3,1)$ when $E>0$. But in our opinion, this is not a good idea, because if we do this the energy level $E=0$ is lost and the true geometric nature of the family of infinitesimal symmetries, on the whole phase space of the Kepler problem, is hidden. The above formulae make up the *bracket table* of smooth functions defined on the dual of a Lie algebroid (with zero anchor map), whose base is ${{\mathbb{R}}}$ (spanned by the coordinate $E$) and standard fibre ${{\mathbb{R}}}^6={{\mathbb{R}}}^3\times{{\mathbb{R}}}^3$ (spanned by the components of $\vect L$ and $\vect\varepsilon$ as coordinate functions).
The *energy-momentum map* is the map $\mathcal J$, defined on the phase space of the Kepler problem, with values in ${{\mathbb{R}}}\times\vect{\mathcal E}\times\vect{\mathcal E}$, $$(\vect r, \vect p)\mapsto{\mathcal J}(\vect r, \vect p)=\bigl(E(\vect r, \vect p),
\vect L(\vect r, \vect p), \vect\varepsilon(\vect r,\vect p)\bigr)\,.$$ The space ${{\mathbb{R}}}\times\vect{\mathcal E}\times\vect{\mathcal E}$ will be called the *energy-momentum space*. As seen above, that space is the dual of a Lie algebroid; it should be considered as fibered over its first factor ${{\mathbb{R}}}$, the fibre $\{e\}\times\vect{\mathcal E}\times\vect{\mathcal E}$ over each point $e\in{{\mathbb{R}}}$ being equipped with a linear Poisson structure which smoothly depends on $e$. With this structure, that fibre is the dual space of a Lie algebra which, as a vector space, can be identified with $\vect{\mathcal E}\times\vect{\mathcal E}$, with a Lie algebra bracket $[\ ,\ ]_e$ which smoothly depends on $e$. That bracket is easily deduced from the Poisson brackets of the components of $\vect L$ and $\vect\varepsilon$ indicated above: for each energy level $e$, the map which associates to $(\vect{u_1},\vect{u_2})\in\left(\vect{\mathcal E}\times\vect{\mathcal E},[\ ,\ ]_e\right)$ the Hamiltonian vector field with Hamiltonian $$(\vect r,\vect p)\mapsto \bigl\langle J(\vect r, \vect p), (\vect{u_1},\vect{ u_2})\bigr\rangle\,,$$ restricted to the energy level $e$ of the phase space, must be a Lie algebras homomorphism. This remark leads to the formula $$\bigl[(\vect{u_1},\vect{u_2}),(\vect{v_1},\vect{v_2}) \bigr]_e=
\left(-\vect{u_1}\times\vect{v_1}+\frac{2e}{m^3k^2}\vect{u_2}\times\vect{v_2}\,,\,-\vect{u_1}\times\vect{v_2}
+\vect{u_2}\times\vect{v_1}\right)\,.$$ By gluing together all the fibres $\vect{\mathcal E}\times\vect{\mathcal E}$ over all points $e\in {{\mathbb{R}}}$, we get on ${{\mathbb{R}}}\times \vect{\mathcal E}\times\vect{\mathcal E}$ a Lie algebroid structure, whose dual is the energy-momentum space. The above formula gives the bracket of two smooth sections $e\mapsto \bigl(\vect{u_1(e)},\vect{u_2(e)}\bigr)$ and $e\mapsto \bigl(\vect{v_1(e)},\vect{v_2(e)}\bigr)$ of that Lie algebroid. For each $e\in{{\mathbb{R}}}$, $\bigl[(\vect{u_1},\vect{u_2}),(\vect{v_1},\vect{v_2}) \bigr](e)$ only depends on the values taken by $(\vect{u_1},\vect{u_2})$ and $(\vect{v_1},\vect{v_2})$ at point $e$, because the considered Lie algebroid has a zero anchor map.
Action of a Lie algebroid on a symplectic manifold {#liealgebroidaction}
--------------------------------------------------
Let $\pi_A:A\to B$ be a Lie algebroid [@cannasweinstein; @mackenzie] whose base is a smooth manifold $B$ and whose anchor map is denoted by $\rho:A\to TB$. The bracket of two smooth sections $s_1:B\to A$ and $s_2:B\to A$ of $\pi_A$ is denoted by $\{s_1,s_2\}$. Let $\pi_M:M\to B$ be a surjective submersion of a smooth manifold $M$ onto the base $B$. An *action* of the Lie algebroid $\pi_A:A\to B$ on the fibered manifold $\pi_M:M\to B$ is a map $s\mapsto X_s$ which associates, to each smooth section $s:B\to A$ of $\pi_A$, a vector field $X_s$ on $M$, in such a way that for each smooth section $s$ of $\pi_A$ and each smooth function $f:B\to{{\mathbb{R}}}$, $$X_{fs}=(f\circ\pi_M)X_s\,,\quad T\pi_M\circ X_s=\rho\circ s\circ\pi_A\,,$$ and that for each pair $(s_1,s_2)$ of smooth sections of $\pi_A$, $$X_{s_1+s_2}=X_{s_1}+X_{s_2}\,,\quad[X_{s_1},X_{s_2}]=X_{\{s_1,s_2\}}\,.$$ Let us now assume that the manifold $M$ is endowed with a symplectic form $\omega$. The vector fields $X_s$ on $M$ associated to smooth sections $s$ of the Lie algebroid $\pi_A:A\to B$ cannot all be Hamiltonian, because if for some choice of the section $s$, $X_s$ is Hamiltonian, the vector field $X_{fs}$, associated to the section $fs$, where $f:B\to{{\mathbb{R}}}$ is a smooth function, will not in general be Hamiltonian. So one may wonder what should be a reasonable definition of a Lie algebroid Hamiltonian action on a symplectic manifold. The answer is suggested by the example of the Kepler problem.
In the last subsection, we described the Lie algebroid $\pi_A:A\to B$ of infinitesimal symmetries of the Kepler problem: $A={{\mathbb{R}}}\times\vect{\mathcal E}\times\vect{\mathcal E}$, $B={{\mathbb{R}}}$, $\pi_A:A\to B$ is the projection on the first factor, the anchor map is the zero map $A\to TB$. The vector bundle $\pi_A:A\to B$ being trivial, each pair of vectors $(\vect{u_1},\vect{u_2})\in\vect{\mathcal E}\times\vect{\mathcal E}$ can be considered as a (constant) section $s_{(\vect{u_1},\vect{u_2})}$ of our Lie algebroid. We defined the action of our Lie algebroid on the phase space of the Kepler problem by taking, as vector field $X_{s_{(\vect{u_1},\vect{u_2})}}$ associated to the section $s_{(\vect{u_1},\vect{u_2})}$, the Hamiltonian vector field with Hamiltonian $$(\vect r,\vect p)\mapsto\bigl\langle{\mathcal J}(\vect r,\vect p), s_{(\vect{u_1},\vect{u_2})}\bigr\rangle\,.$$ This condition unambiguously defines our Lie algebroid action (and, simultaneously, the bracket composition law of its sections) since the module of its smooth sections is spanned by the constant sections.
It would be interesting to see whether such a construction can be extended for more general Lie algebroids, with a nonzero anchor map.
The $\mathbf S$ map {#smap}
-------------------
The main disadvantage of Moser’s regularization method is that it handles separately each energy level. This disadvantage may be partially removed by the following procedure, which allows to handle together all negative (resp., all positive) energy levels. However, negative, positive and zero energy levels still cannot be handled together with that procedure.
Let us consider the quadric $Q_\rho$, defined by $$h^2+\zeta(x^2+y^2+z^2)=\zeta\rho^2\,,$$ where $\rho$ may take any positive value, and let $Q_R$ be the quadric defined by $$h^2+\zeta(x^2+y^2+z^2)=\zeta R^2\,,$$ where $R$ is a fixed positive quantity. To each point $M_R\in Q_R$, we associate the point $M_\rho\in Q_\rho$ such that $$\vect{OM_\rho}=\frac{\rho}{R}\,\vect{OM_R}\,.$$ We lift this diffeomorphism $Q_R\mapsto Q_\rho$ to the cotangent bundles, and we get a symplectic diffeomorphism ${\mathcal T}_\rho:T^*Q_R\to T^*Q_\rho$. This symplectic diffeomorphism associates to each pair $(M_R, \vect{W_R})$ made by a point $M_R\in Q_R$ and a vector $\vect{W_R}$ tangent to $Q_R$ at that point, the pair $(M_\rho, \vect{W_\rho})$ made by a point $M_\rho\in Q_\rho$ and a vector $\vect{W_\rho}$ tangent to $Q_\rho$ at that point: $$\vect{OM_\rho}=\frac{\rho}{R}\,\vect{OM_R}\,,\quad \vect{W_\rho}=\frac{R}{\rho}\,\vect{W_R}\,.$$ For each $\rho>0$, we compose the symplectic diffeomorphism ${\mathcal T}_\rho: T^*Q_R\to T^*Q_\rho$ with the symplectic diffeomorphism $S_\rho:T^*(Q_\rho\backslash\{N_\rho\})\to T^*{\mathcal E}$ built in Subsection \[cotangentlift\]. We obtain a family, indexed by $\rho>0$, of symplectic diffeomorphisms $S_{\rho,R}=S_\rho\circ{\mathcal T}_\rho:T^*(Q_R\backslash\{N_R\})\to T^*{\mathcal E}$: $$\left\{
\begin{aligned}
\vect p&=\frac{\rho}{R-h_R}\,\vect {O\mu_R}\,,\quad\hbox{with}\quad h_R^2+\zeta\Vert \vect{O\mu_R}\Vert^2=R^2\,,\\
\vect r&=\frac{R-h_R}{\rho}\,\vect {W_{3R}}+\frac{W_{hR}}{\rho}\,\vect{O\mu_R}\,,
\end{aligned}
\right.$$ where we have set $$\vect{OM_R} =\vect{O\mu_R}+h_R\vect{e_h}\,,\quad \vect{W_R}=\vect{W_{3R}}+W_{hR}\vect{e_h}\,.$$ The formulae for the inverse transformation are $$\left\{
\begin{aligned}
\vect{O\mu_R}&=\frac{2R\rho}{\rho^2+\zeta p^2}\vect p\,,\\
h_R&=R\,\frac{p^2-\zeta\rho^2}{p^2+\zeta\rho^2}\,,
\quad\hbox{with}\quad p=\Vert\vect p\Vert=\sqrt{p_x^2+p_y^2+p_z^2}\,,\\
\vect{W_{3R}}&=\frac{\rho^2+\zeta p^2}{2R\rho}\vect r-\frac{\zeta\vect r.\vect p}{R\rho}\vect p\,,\\
W_{hR}&=\frac{\zeta\vect r.\vect p}{R}\,.
\end{aligned}
\right.$$ The diffeomorphism $S_{\rho,R}^{-1}$ sends the subset of $T^*\bigl({\mathcal E}\backslash\{O\}\bigr)$ on which the energy is $\displaystyle E(\vect r, \vect p)=-\frac{\zeta \rho^2}{2m}$ into the subset of $T^*Q_R$ on which $\displaystyle\Vert\vect W_R\Vert=\frac{km^2}{R\rho}$. Therefore, if $\rho_1$ and $\rho_2$ are two distinct possible values of $\rho$, the images by $S_{\rho_1,R}^{-1}$ of the energy level $\displaystyle E(\vect r, \vect p)=-\frac{\zeta \rho_1^2}{2m}$, and by $S_{\rho_2,R}^{-1}$ of the energy level $\displaystyle E(\vect r, \vect p)=-\frac{\zeta \rho_2^2}{2m}$, are disjoint. By restricting each map $S_{\rho,R}^{-1}$ to the subset of $T^*\bigl({\mathcal E}\backslash\{O\}\bigr)$ on which the energy is $\displaystyle E(\vect r, \vect p)=-\frac{\zeta \rho^2}{2m}$, and by gluing together these restricted maps for all possible values of $\rho$, we obtain a unique diffeomorphism $S^{-1}$ from the open subset of the cotangent bundle $T^*\bigl({\mathcal E}\backslash\{O\}\bigr)$ minus the zero section on which the energy $E(\vect r, \vect p)$ is negative if $\zeta=1$, positive if $\zeta=-1$, onto $T^*(Q_R\backslash\{N_R\})$ minus the zero section. This diffeomorphism, built by gluing together pieces of the symplectic diffeomorphisms $S_{\rho,R}^{-1}$ for all values of $\rho>0$, is no longer symplectic! It is given by the formulae, in which we no longer write the subscript $R$, $$\left\{
\begin{aligned}
\vect{O\mu}&=\zeta\,\frac{R\sqrt{\zeta r(2m^2k-rp^2)}}{m^2k}\,\vect p\,,\\
h&=\frac{R(rp^2-m^2k)}{m^2k}\,,\\
\vect{W_{3}}&=\zeta\,\frac{m^2k\vect r-r(\vect r.\vect p)\vect p}{R\sqrt{\zeta r(2m^2k-rp^2)}}\,,\\
W_{h}&=\zeta\,\frac{\vect r.\vect p}{R}\,.
\end{aligned}
\right.$$ We have set $$\vect{OM}=\vect{O\mu}+h\vect{e_h}\,,\quad \vect{W}=\vect{W_3}+W_h\vect{e_h}\,.$$ The transformed Hamiltonian $(S^{-1})^*E=E\circ S$ is $$H=E\circ S=-\frac{\zeta k^2m^3}{2R^2}\,\frac{1}{\Vert\vect W\Vert^2}\,.$$ Up to the constant factor $\displaystyle\frac{\zeta k^2m^3}{R^2}$, it is the *Delaunay Hamiltonian* [@cushmanbates; @cushmanduistermaat], *i.e.*, the Kepler Hamiltonian in Delaunay coordinates. This property is related to the fact that Delaunay variables are action-angle variables for the Kepler problem.
In the rest of the paper we write $Q$ for $Q_R$. The north pole of $Q$ is denoted by $N$ instead of $N_R$.
Let $X_E$ be the Hamiltonian vector field on $T^*({\mathcal E}\backslash\{0\})$ with Hamiltonian $E$, *i.e.* the Hamiltonian vector field of the Kepler problem, and let $X_H$ be the Hamiltonian vector field, on $T^*\bigl(Q\backslash\{N\}\bigr)$ minus the zero section, whose Hamiltonian $H=E\circ S$ is given by the above formula. The direct image $(S^{-1})_*(X_E)$ of the vector field $X_E$ by the diffeomorphism $S^{-1}$ is not equal to $X_H$, since $S^{-1}$ is not symplectic. A short calculation leads to its expression, $$(S^{-1})_*(X_E)= g\,X_H\,,\quad\text{with}\quad g=\frac{R}{h-R}\,.$$ The function $g$ is smooth on $T^*\bigl(Q\backslash\{N\}\bigr)$ minus the zero section, and becomes singular when $h=R$, *i.e.* on the fibre over the north pole $N$. We see that the map $S^{-1}$ sends the Hamiltonian vector field of the Kepler problem to a conformally Hamiltonian vector field, with $H$ as Hamiltonian and with $g$ as conformal factor.
The flow of $\mathbf {X_H}$ {#flowofXH}
---------------------------
Unlike the conformal factor $g$, which becomes singular on the cotangent space to $Q$ at the north pole, the Hamiltonian $H$ is smoothly defined on the whole cotangent bundle $T^*Q$ minus the zero section. The associated Hamiltonian vector field $X_H$ is complete. Its flow, defined on ${{\mathbb{R}}}\times \bigl(T^*Q\backslash\{\text{zero section}\}\bigr)$, $$\Phi_{X_H}\left(s,\bigl(\vect{OM(0)},\vect{W(0)}\bigr)\right)=\bigl(\vect{OM(s)},\vect{W(s)}\bigr)$$ has slightly different expressions for $\zeta=1$ (negative energy) and for $\zeta=-1$ (positive energy). Observe that $\Vert\vect W\Vert$ is a first integral of $X_H$. $$\text{For\ }\zeta=1\ (E<0)\,,\
\left\{
\begin{aligned}
\vect{OM(s)}&=\cos(\lambda s)\,\vect{OM(0)}
+\frac{R}{\Vert\vect{W(0)}\Vert}\,\sin(\lambda s)\,\vect{W(0)}\,,
\\
\vect{W(s)}&=-\frac{\Vert\vect{W(0)}\Vert}{R}\,\sin(\lambda s)\,\vect{OM(0)}+\cos(\lambda s)\,\vect{W(0)}\,.
\end{aligned}
\right.$$ $$\text{For\ }\zeta=-1\ (E>0)\,,\
\left\{
\begin{aligned}
\vect{OM(s)}&=\cosh(\lambda s)\,\vect{OM(0)}
-\frac{R}{\Vert\vect{W(0)}\Vert}\,\sinh(-\lambda s)\,\vect{W(0)}\,,
\\
\vect{W(s)}&=-\frac{\Vert\vect{W(0)}\Vert}{R}\,\sinh(-\lambda s)\,\vect{OM(0)}+\cosh(\lambda s)\,\vect{W(0)}\,.
\end{aligned}
\right.$$ We have set $\displaystyle\lambda=\frac{k^2m^3}{R^3\Vert\vect{W(0)}\Vert^3}$.
A symplectic diffeomorphism {#symplecticdiff}
---------------------------
At the end of Subsection \[smap\], we have seen that the image by $S^{-1}$ of the Hamiltonian vector field $X_E$ of the Kepler problem is a conformally Hamiltonian vector field $gX_H$ defined on an open dense subset of $T^*Q\backslash\{\text{zero section}\}$, whose Hamiltonian $H$ is smoothly defined on the whole $T^*Q\backslash\{\text{zero section}\}$; the corresponding Hamiltonian vector field $X_H$ is complete. Moreover the conformally Hamiltonian vector field $gX_H$ is also Hamiltonian, with Hamiltonian $H$, not for the canonical symplectic form of $T^*Q$, but for the pull-back by $S$ of the canonical symplectic form of $T^*{\mathcal E}$. We have seen in Subsection \[levi-civitaparameter\] that for every solution $t\mapsto\bigl(\vect{r(t)},\vect{p(t)}\bigr)$ of the equations of motion of the Kepler problem, $$\frac{d}{dt}\left(\frac{\vect{p(t)}.\vect{r(t)}-2E\bigl(\vect{r(t)},\vect{p(t)}\bigr)t}{mk}\right)
=\frac{1}{r(t)}\,.$$ The conformal factor $g$, whose expression is given at the end of Subsection \[smap\], may also be expressed as $$g=\frac{mk}{2}\,S^*\left(\frac{1}{rE(\vect r, \vect p)}\right)\,.$$ Therefore $$\frac{d}{dt}\left(\frac{\vect{p(t)}.\vect{r(t)}-2E\bigl(\vect{r(t)},\vect{p(t)}\bigr)t}
{2E\bigl(\vect{r(t)},\vect{p(t)}\bigr)}\right)=g\circ S^{-1}\,.$$ All the integral curves of the conformally Hamiltonian vector field $gX_H$ can be written $$t\mapsto S^{-1}\circ\bigl(\vect{r(t)},\vect{p(t)}\bigr)\,,$$ where $t\mapsto\bigl(\vect{r(t)},\vect{p(t)}\bigr)$ is a solution of the equations of motion of the Kepler problem. We see that the pull-back by $(\id_{{\mathbb{R}}},S)$ of the function, defined on the product with ${{\mathbb{R}}}$ of the open subset of the phase space of the Kepler problem on which the energy $E$ is negative if $\zeta=1$, positive if $\zeta=-1$, $$\frac{\vect p\cdot\vect r-2E(\vect r, \vect p)\,t}{2E(\vect r, \vect p)}$$ has the properties of the function $\sigma$ of Theorem \[result1\] of Subsection \[confhamfields\]. That Theorem can therefore be applied. It proves that by composing the symplectic diffeomorphism $S^{-1}$ with the flow $\Phi_{X_H}$ of the Hamiltonian vector field $X_H$, for suitably chosen values of the independent variable $s$, we get a symplectic diffeomorphism from the phase space of the Kepler problem, restricted either to negative, or to positive energies, onto an open subset of $T^*Q$. The explicit expression of this symplectic diffeomorphism is $$\left(\vect p,\vect r\right)\mapsto\left(\vect{OM(s)},\vect{W(s)}\right)\,,\quad\text{with}
\quad s=\frac{-\vect p\cdot\vect r}{2E\left(\vect r,\vect p\right)}\,.$$ The quantities $\bigl(\vect{OM(s)},\vect{W(s)}\bigr)$ are given, as functions of $\bigl(\vect{OM(0)},\vect{W(0)}\bigr)$, by the formulae at the end of Subsection \[flowofXH\]. We have to set $$\vect{OM(0)}=\vect{O\mu}+h\vect{e_h}\,,\quad \vect{W(0)}=\vect{W_3}+W_h\vect{e_h}\,.$$ The formulae at the end of Subsection \[smap\] give the expressions of $\vect{O\mu}$, $h$, $\vect{W_3}$ and $W_h$ as functions of $\vect{r}$ and $\vect{p}$. Finally we see that the symplectic diffeomorphism so obtained is that of Györgyi, Ligon and Schaaf. R. Cushman and L. Bates, in chapter II of the book [@cushmanbates], offer a detailed discussion of its properties, notably its behaviour with respect to the Liouville $1$-forms of the cotangent spaces of the Kepler configuration manifold and of the $3$-dimensional sphere (or hyperboloid).
Conclusion and perspectives
===========================
The properties of the diffeomorphism from the phase space of the Kepler problem to the cotangent bundle of a sphere or of an hyperboloid are explained, in Subsection \[smap\], from a point of view other than that used by Cushman and Duistermaat [@cushmanduistermaat]. Our explanation is founded on a very natural property of conformally Hamiltonian vector fields. However, the procedure we used, as well as those used by Györgyi [@gyorgyi], Ligon and Schaaf [@ligonschaaf] remains unsatisfactory in the fact that it handles separately negative and positive energy levels. The space of motions of the Kepler problem is connected, since by varying slowly the energy level, an elliptic motion can be transformed into a parabolic, then into an hyperbolic motion. By completely different methods, J.-M. Souriau [@souriautorino] proposed a global description of the manifold of motions of the Kepler problem, of its regularization and of its global and infinitesimal symmetries. It should be possible to do something similar at the level of the phase space instead of at the level of the manifold of motions.
The fact that the Kepler vector field is Hamiltonian with respect to a symplectic form and conformally Hamiltonian with respect to another syplectic form is remarkable; it is probably related to the complete integrability of that vector field, as suggested by an anonymous referee. In [@MaciePryTsi], A. Maciejewski, M. Prybylska and A. Tsiganov have used conformally Hamiltonian vector fields within the theory of bi-Hamiltonian systems, to build completely integrable systems.
We have shown that the infinitesimal symmetries of the Kepler problem form a Lie algebroid rather than a Lie algebra. Actions of Lie algebroids on symplectic manifolds is a subject which seems to us interesting; it should be nice to have other examples of Hamiltonian vector fields whose infinitesimal symmetries form a Lie algebroid, maybe with a non-zero anchor map.
A. Douady and M. Lazard [@douadylazard] have shown that Lie algebroids with a zero anchor map are integrable into Lie groupoids; it would be interesting to write down explicitly the action of a Lie groupoid with the groups ${\rm SO}(4)$, ${\rm SE}(4)$ and ${\rm SO}(3,1)$ as isotropy groups on a regularized phase space of the Kepler problem.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am much indebted towards my colleagues and friends of the Centre International de Théories Variationnelles, the International Workshop on Differential Geometric Methods in Mechanics and the Seminar of Hamiltonian Geometry for helpful discussions, towards Alan Weinstein who, after listening to a presentation of a preliminary version of this work, made several enlighting observations, and towards Alain Guichardet and Alain Albouy who communicated me their unpublished papers on the Kepler problem. I address my thanks to the two anonymous referees whose judicious observations and suggestions were very helpful.
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Received xxxx 20xx; revised xxxx 20xx.
[^1]: The *hodograph* of the motion of a particle moving in an affine space $\mathcal E$ is the curve, drawn in the associated vector space $\vect{\mathcal E}$, by the velocity vector $\vect v$ of the particle, as a function of time.
[^2]: In plane Euclidean geometry, the *power*\[power\] of a point $O$ with respect to a circle $\mathcal C$ is the real number $\vect{OA}.\vect{OB}$, where $A$ and $B$ are the two intersection points of $\mathcal C$ with a straight line $\mathcal D$ through $O$. That number does not depend on $\mathcal D$ and is equal to $\Vert \vect{OC}\Vert^2-{\mathcal R}^2$, where $C$ is the centre and $\mathcal R$ the radius of $\mathcal C$.
[^3]: Or rather an anti-symplectic diffeomerphism, but by a change of sign it can be transformed into a symplectic diffeomorphism
|
---
abstract: 'We consider the unit ball $\Omega\subset {{\mathbb R}^N}$ ($N\ge2$) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal configurations with respect to the torsional rigidity functional. In particular we analyse the role played by the configuration obtained by putting a smaller concentric ball inside $\Omega$. In this case the stress function admits an explicit form which is radially symmetric: this allows us to compute the sign of the second order shape derivative of the torsional rigidity functional with the aid of spherical harmonics. Depending on the ratio of the conductivities a symmetry breaking phenomenon occurs.'
author:
- 'Lorenzo Cavallina [^1]'
title: 'Locally optimal configurations for the two-phase torsion problem in the ball'
---
[2010 [*Mathematics Subject classification.*]{} 49Q10]{}
[*Keywords and phrases: torsion problem, optimization problem, elliptic PDE, shape derivative*]{}
Introduction
============
We will start by considering the following two-phase problem. Let $\Omega\subset {\mathbb R}^N$ ($N\ge 2$) be the unit open ball centered at the origin. Fix $m\in (0, {{\rm Vol}}(\Omega))$, where here we denote the $N$-dimensional Lebesgue measure of a set by ${{\rm Vol}}(\cdot)$ . Let $\omega\subset\subset \Omega$ be a sufficiently regular open set such that ${{\rm Vol}}(\omega)=m$. Fix two positive constants $\sigma_-$, $\sigma_+$ and consider the following [*distribution of conductivities*]{}: $$\sigma:=\sigma_\omega:= \sigma_- \mathbbm{1}_\omega+\sigma_+ \mathbbm{1}_{\Omega\setminus {\mkern 1.5mu\overline{\mkern-1.5mu\omega\mkern-1.5mu}\mkern 1.5mu}},$$ where by $\mathbbm{1}_A$ we denote the characteristic function of a set $A$ (i.e. $\mathbbm{1}_A(x)=1$ if $x\in A$ and vanishes otherwise).
Consider the following boundary value problem: $$\label{pb}
\begin{cases}
-{{\rm{div}}}{\left(\sigma_\omega \nabla u \right)}= 1 \quad&\text{ in } \Omega, \\
u=0 \quad&\text{ on }\partial \Omega.
\end{cases}$$ We recall the weak formulation of : $$\label{weak}
\int_\Omega \sigma_\omega \nabla u \cdot \nabla \varphi = \int_\Omega \varphi \;\;\;\;\text{ for all } \varphi\in H^1_0(\Omega).$$
Moreover, since $\sigma_\omega$ is piecewise constant, we can rewrite as follows $$\label{pb2}
\begin{cases}
-\sigma_\omega \Delta u = 1 \quad&\text{ in } \omega\cup \left(\Omega\setminus{\mkern 1.5mu\overline{\mkern-1.5mu\omega\mkern-1.5mu}\mkern 1.5mu}\right), \\
\sigma_-\partial_n u_-= \sigma_+\partial_n u_+ \quad&\text{ on } \partial \omega,\\
u=0 \quad&\text{ on } \partial \Omega,
\end{cases}$$ where the following notation is used: the symbol ${{{\bf {n}}}}$ is reserved for the outward unit normal and ${\partial_n}:=\frac{\partial}{\partial {{{\bf {n}}}}}$ denotes the usual normal derivative. Throughout the paper we will use the $+$ and $-$ subscripts to denote quantities in the two different phases (under this convention we have $({\sigma}_\omega)_+={\sigma}_+$ and $({\sigma}_\omega)_-={\sigma}_-$ and our notations are “consistent" at least in this respect). The second equality of has to be intended in the sense of traces. In the sequel, the notation $[f]:=f_+-f_-$ will be used to denote the jump of a function $f$ through the interface $\partial \omega$ (for example, following this convention, the second equality in reads “${\left[{\sigma}{\partial_n}u \right]}=0$ on $\partial \omega$”).
We consider the following [*torsional rigidity functional*]{}: $$E(\omega):= \int_\Omega \sigma_\omega |\nabla u_\omega|^2=\int_\omega {\sigma}_- |{\nabla}u_\omega|^2+\int_{\Omega\setminus \bar{\omega}} {\sigma}_+ |{\nabla}u_\omega|^2=
\int_\Omega u_\omega,$$ where $u_\omega$ is the unique (weak) solution of .
Physically speaking, we imagine our ball $\Omega$ being filled up with two different materials and the constants ${\sigma}_{\pm}$ represent how “hard" they are. The second equality of , which can be obtained by integrating by parts after splitting the integrals in , is usually referred to as [*transmission condition*]{} in the literature and can be interpreted as the continuity of the flux through the interface ${{\partial \omega}}$.\
The functional $E$, then, represents the torsional rigidity of an infinitely long composite beam. Our aim is to study (locally) optimal shapes of $\omega$ with respect to the functional $E$ under the fixed volume constraint. The one-phase version of this problem was first studied by Pólya. Let $D\subset {{\mathbb R}^N}$ ($N\ge 2$) be a bounded Lipschitz domain. Define the following shape functional $$\mathcal{E}(D):= \int_D |{\nabla}u_D|^2,$$ where the function $u_D$ (usually called *stress function*) is the unique solution to $$\begin{cases}
-{\Delta}u = 1 \quad &\text{in } D,\\
u=0\quad &\text{on } \partial D.
\end{cases}$$
The value $\mathcal{E}(D)$ represents the torsional rigidity of an infinitely long beam whose cross section is given by $D$. The following theorem (see [@polya]) tells us that beams with a spherical section are the “most resistant”.
\[pol\] The ball maximizes $\mathcal{E}$ among all Lipschitz domains with fixed volume.
Inspired by the result of Theorem \[pol\] it is natural to expect radially symmetrical configurations to be optimizers of some kind for $E$ (at least in the local sense). From now on we will consider $\omega:= B_R$ (the open ball centered at the origin, whose radius, $0<R<1$, is chosen to verify the volume constraint $|B_R|=m$) and use shape derivatives to analyze this configuration. This technique has already been used by Conca and Mahadevan in [@conca] and Dambrine and Kateb in [@sensitivity] for the minimization of the first Dirichlet eigenvalue in a similar two-phase setting ($\Omega$ being a ball) and it can be applied with ease to our case as well.
A direct calculation shows that the function $u$, solution to where $\omega={{B_R}}$, has the following expression: $$\label{uexpl}
u(x)= \begin{cases}
\frac{1-R^2}{2N{\sigma}_+}+\frac{R^2-|x|^2}{2N{\sigma}_-} \quad &\text{for }|x|\in [0,R],\\[1ex]
\frac{1-|x|^2}{2N{\sigma}_+} \quad&\text{for } |x|\in [R,1].
\end{cases}$$ In this paper we will use the following notation for Jacobian and Hessian matrix respectively. $$(D{\bf v})_{ij}:=\frac{\partial v_i}{\partial x_j}, \quad (D^2 f)_{ij}= \frac{\partial^2 f}{\partial x_i\partial x_j},$$ for all smooth real valued function $f$ and vector field ${\bf v}=(v_1,\dots, v_N)$ defined on $\Omega$. We will introduce some differential operators from tangential calculus that will be used in the sequel. For smooth $f$ and $\bf v$ defined on $\partial \omega$ we set $$\label{diffop}
\begin{aligned}
{\nabla}_\tau f &:= {\nabla}\widetilde{f}-({\nabla}\widetilde{f}\cdot {{{\bf {n}}}}){{{\bf {n}}}}\quad &\text{( tangential gradient)}, \\
{{\rm{div}}}_\tau {\bf v}&:= {{\rm{div}}}\widetilde{{\bf v}}-{{{\bf {n}}}}\cdot \left( D \widetilde{{\bf v}}{{{\bf {n}}}}\right) &\text{ (tangential divergence)},
\end{aligned}$$ where $\widetilde{f}$ and $\widetilde{\bf v}$ are some smooth extensions on the whole $\Omega$ of $f$ and $\bf v$ respectively. It is known that the differential operators defined in do not depend on the choice of the extensions. Moreover we denote by $D_\tau {\bf v}$ the matrix whose $i$-th row is given by ${\nabla}_\tau v_i$. We define the (additive) mean curvature of $\partial \omega$ as $H:={{\rm{div}}}_\tau {{{\bf {n}}}}$ (cf. [@henrot; @SG]). According to this definition, the mean curvature $H$ of $\partial {{B_R}}$ is given by $(N-1)/R$. A first key result of this paper is the following.
\[UNO\] For all suitable perturbations that fix the volume (at least at first order), the first shape derivative of $E$ at $B_R$ vanishes.
An improvement of Theorem \[UNO\] is given by the following precise result (obtained by studying second order shape derivatives).
\[DUE\] Let ${\sigma}_-,{\sigma}_+>0$ and $R\in(0,1)$. If ${\sigma}_->{\sigma}_+$ then $B_R$ is a local maximizer for the functional $E$ under the fixed volume constraint.
On the other hand, if ${\sigma}_-<{\sigma}_+$ then $B_R$ is a saddle shape for the functional $E$ under the fixed volume constraint.
In section 2 we will give the precise definition of shape derivatives and introduce the famous Hadamard forumlas, a precious tool for computing them. In the end of section 2 a proof of Theorem \[UNO\] will emerge as a natural consequence of our calculations. Section 3 will be devoted to the computation of the second order shape derivative of the functional $E$ in the case $\omega={{B_R}}$. In Section 4 we will finally calculate the sign of the second order shape derivative of $E$ by means of the spherical harmonics. The last section contains an analysis of the different behaviour that arises when volume preserving transformations are replaced by surface area preserving ones.
Computation of the first order shape derivative:\
=================================================
We consider the following class of perturbations with support compactly contained in $\Omega$: $$\mathcal{A}:=\Big\{ \Phi \in {{\mathcal C}}^\infty\big([0,1)\times {{\mathbb R}^N},{{\mathbb R}^N}\big) \;\Big|\; \Phi(0,\cdot)= {\rm Id},\; \exists K\subset\subset \Omega \text{ s.t.}\;
\Phi(t,x)=x \; \forall t\in [0,1) ,\; \forall x\in {{\mathbb R}^N}\setminus K \Big\}.$$
For $\Phi \in \mathcal{A}$ we will frequently write $\Phi(t)$ to denote $\Phi(t,\cdot)$ and, for all domain $D$ in ${{\mathbb R}^N}$, we will denote by $\Phi(t)(D)$ the set of all $\Phi(t,x)$ for $x\in D$. We will also use the notation $D_t:=\Phi(t)(D)$ when it does not create confusion. In the sequel the following notation for the first order approximation (in the “time” variable) of $\Phi$ will be used. $$\label{whosh}
\Phi(t)={\rm Id} + t {{{\bf {h}}}}+o(t) \quad \text{as }t\to0,$$ where ${{{\bf {h}}}}$ is a smooth vector field. In particular we will write $h_n:={{{\bf {h}}}}\cdot {{{\bf {n}}}}$ (the normal component of ${{{\bf {h}}}}$) and ${{{\bf {h}}}}_\tau : = {{{\bf {h}}}}- {h_n}{{{\bf {n}}}}$ on the interface. We are ready to introduce the definition of shape derivative of a shape functional $J$ with respect to a deformation field $\Phi$ in $\mathcal{A}$ as the following derivative along the path associated to $\Phi$. $${{\left.\kern-\nulldelimiterspace \frac{d}{dt} J(D_t) \vphantom{|} \right|_{t=0} }} = \lim_{t\to 0} \frac{J(D_t)-J(D)}{t}.$$
This subject is very deep. Many different formulations of shape derivatives associated to various kinds of deformation fields have been proposed over the years. We refer to [@SG] for a detailed analysis on the equivalence between the various methods. For the study of second (or even higher) order shape derivatives and their computation we refer to [@SG; @structure; @simon; @new].
The structure theorem for first and second order shape derivatives (cf. [@henrot], Theorem 5.9.2, page 220 and the subsequent corollaries) yields the following expansion. For every shape functional $J$, domain $D$ and pertubation field $\Phi$ in $\mathcal{A}$, under suitable smoothness assumptions the following holds.
$$\label{expansion}
J(D_t)=J(D)+t \, l_1^J (D)(h_n)+\frac{t^2}{2}\left( l_2^J(D)({h_n},{h_n})+l_1^J(D)(Z) \right) + o(t^2) \quad \text{ as }t\to 0,$$
for some linear $l_1^J(D): {{\mathcal C}}^\infty (\partial D)\to {\mathbb R}$ and bilinear form $l_2^J(D): {{\mathcal C}}^\infty (\partial D)\times {{\mathcal C}}^\infty (\partial D)\to{\mathbb R}$ to be determined eventually. Moreover for the ease of notation we have set $$Z:=\left( V'+D{{{\bf {h}}}}{{{\bf {h}}}}\right)\cdot {{{\bf {n}}}}+ ((D_\tau {{{\bf {n}}}}) {{{{{\bf {h}}}}_\tau}})\cdot {{{{{\bf {h}}}}_\tau}}-2{\nabla}_\tau {h_n}\cdot{{{{{\bf {h}}}}_\tau}},$$ where $V(t,\Phi(t)):=\partial_t \Phi(t)$ and $V':=\partial_t V(t,\cdot)$. According to the expansion , the first order shape derivative of a shape functional depends only on its first order apporximation by means of its normal components. On the other hand the second order derivative contains an “acceleration” term $l_1^J(D)(Z)$. It is woth noticing that, (see Corollary 5.9.4, page 221 of [@henrot]) $Z$ vanishes in the special case when $\Phi={{{\rm Id}}}+t{{{\bf {h}}}}$ with ${{{\bf {h}}}}_\tau={{\bf 0}}$ on $\partial D$ (this will be a key observation to compute the bilinear form $l_2^J$).
We will state the following lemma, which will aid us in the computations of the linear and bilinear forms $l_1^J(D)$ and $l_2^J(D)$ for various shape functionals (cf. [@henrot], formula (5.17), page 176 and formulas (5.110) and (5.111), page 227).
\[nonloso\] Take $\Phi\in {{\mathcal{A}}}$ and let $f=f(t,x)\in {{\mathcal C}}^2([0,T), L^1({{\mathbb R}^N}))\cap {{\mathcal C}}^1([0,T), W^{1,1}({{\mathbb R}^N}))$. For every smooth domain $D$ in ${{\mathbb R}^N}$ define $
J(D_t):=\int_{D_t} f(t)
$ (where we omit the space variable for the sake of readability). Then the following identities hold: $$\label{deri1}
l_1^J(D)({h_n})= \int_D {{\left.\kern-\nulldelimiterspace \partial_t f \vphantom{|} \right|_{t=0} }} + \int_{\partial D} f(0) h_n,$$ $$\label{deri2}
l_2^J(D)({h_n},{h_n})=\int_D {{\left.\kern-\nulldelimiterspace \partial_{tt}^2 f \vphantom{|} \right|_{t=0} }}+\int_{\partial D}2{{\left.\kern-\nulldelimiterspace \partial_t f \vphantom{|} \right|_{t=0} }}h_n+\big(H f(0)+\partial_n f(0) \big)h_n^2.$$
Since we are going to compute second order shape derivatives of a shape functional subject to a volume constraint, we will need to restric our attention to the class of perturbations in ${{\mathcal{A}}}$ that fix the volume of $\omega$: $$\mathcal{B}(\omega):= \big\{ \Phi\in \mathcal{A} \;\big |\; {{\rm Vol}}\big(\Phi(t)(\omega)\big)={{\rm Vol}}(\omega)=m \text{ for all } t\in [0,1) \big\}.$$ We will simply write ${{\mathcal{B}}}$ in place of ${{\mathcal{B}}}({{B_R}})$. Employing the use of Lemma \[nonloso\] for the volume functional ${{\rm Vol}}$ and of the expansion , for all $\Phi\in{{\mathcal{A}}}$ we get $$\label{volex}
{{\rm Vol}}(\omega_t)={{\rm Vol}}(\omega)+ t \int_{\partial\omega} {h_n}+ \frac{t^2}{2} \left( \int_{\partial \omega} H h_n^2 + \int_{\partial\omega} Z \right) + o(t^2) \text{ as }t\to0.$$ This yields the following two conditions: $$\begin{aligned}
&\int_{\partial \omega} {h_n}=0, \quad\quad\quad&\text{($1^{\rm st}$ order volume preserving)} \label{1st}\\
&\int_{\partial \omega} H h_n^2+ \int_{\partial \omega}Z=0. &\text{($2^{\rm nd}$ order volume preserving)}\label{2nd}\end{aligned}$$
For every admissible perturbation field $\Phi={{{\rm Id}}}+t{{{\bf {h}}}}$ in ${{\mathcal{A}}}$, with ${{{\bf {h}}}}$ satisfying , we can find some perturbation field $\widehat{\Phi}\in{{\mathcal{B}}}$ such that $\widehat{\Phi}={{{\rm Id}}}+ t {{{\bf {h}}}}+ o(t)$ as $t\to 0$. For example, the following construction works just fine: $$\widehat{\Phi}(t,x)=\frac{\Phi(t,x)}{\eta(x)\left( \frac{{{\rm Vol}}(\Phi(t)(\omega))}{{{\rm Vol}}(\omega)} \right)^{1/N}+(1-\eta(x))},$$ where $\eta$ is a suitable smooth cutoff function compactly supported in $\Omega$ that attains the value $1$ on a neighbourhood of $\omega$.
We will now introduce the concepts of “shape" and “material" derivative of a path of real valued functions defined on $\Omega$. Fix an admissible perturbation field $\Phi\in{{\mathcal{A}}}$ and let $u=u(t,x)$ be defined on $[0,1)\times\Omega$ for some positive $T$. Computing the partial derivative with respect to $t$ at a fixed point $x\in\Omega$ is usually called [*shape derivative*]{} of $u$; we will write: $$u'(t_0,x):= \frac{\partial u}{\partial t} (t_0,x), \;\text{ for }x\in\Omega, t_0\in [0,1).$$ On the other hand differentiating along the trajectories gives rise to the [*material derivative*]{}: $$\dot{u}(t_0,x):= \frac{\partial v}{\partial t}(t_0,x), \;x\in\Omega, t_0\in [0,1);$$ where $v(t,x):=u(t, \Phi(t,x))$. From now on for the sake of brevity we will omit the dependency on the “time" variable unless strictly necessary and write $u(x)$, $u'(x)$ and $\dot{u}(x)$ for$u(0,x)$, $u'(0,x)$ and $\dot{u}(0,x)$. The following relationship between shape and material derivatives hold true: $$\begin{aligned}
\label{u'a} u'&=\dot{u}-{\nabla}u \cdot {{{\bf {h}}}}.
$$ We are interested in the case where $u(t,\cdot):=u_{(B_R)_t}$ i.e. it is the solution to problem when $\omega=\Phi(t)(B_R)$. In this case, since by symmetry we have ${\nabla}u = ({\partial_n}u) {{{\bf {n}}}}$, the formula above admits the following simpler form on the interface $\partial B_R$: $$\label{u'}
u'=\dot{u}-({\partial_n}u) {h_n}.
$$
It is natural to ask whether the shape derivatives of the functional $E$ are well defined. Actually, by a standard argument using the implicit function theorem for Banach spaces (we refer to [@conca; @sensitivity] for the details) it can be proven that the application mapping every smooth vector field ${\bf h}$ compactly supported in $\Omega$ to $E\left(({{{\rm Id}}}+{{{\bf {h}}}})(\omega)\right)$ is of class $\mathcal{C}^\infty$ in a neighbourhood of ${\bf h }={\bf 0}$. This implies the shape differentiability of the functional $E$ for any admissible deformation field $\Phi\in {{\mathcal{A}}}$. As a byproduct we obtain the smoothness of the material derivative $\dot{u}$.
As already remarked in [@sensitivity] (Remark 2.1), in contrast to material derivatives, the shape derivative $u'$ of the solution to our problem has a jump through the interface. This is due to the presence of the gradient term in formula (recall that the transmission condition provides only the continuity of the flux). On the other hand we will still be using shape derivatives because they are easier to handle in computations (and writing Hadamard formulas using them is simpler).
For any given admissible $\Phi\in{{\mathcal{A}}}$, the corresponding $u'$ can be characterized as the (unique) solution to the following problem in the class of functions whose restriction to both ${{B_R}}$ and $\Omega\setminus \overline{{{B_R}}}$ is smooth: $$\label{u'pb}
\begin{cases}
{\Delta}u' =0 \quad&\text{in } {{B_R}}\cup (\Omega\setminus \overline{{{B_R}}}),\\
{\left[{\sigma}{\partial_n}u' \right]}=0 \quad&\text{on }{{\partial B_R}},\\
{\left[u' \right]}=-{\left[{\partial_n}u \right]}{h_n}\quad&\text{on } {{\partial B_R}},\\
u'= 0 \quad&\text{on }\partial \Omega.
\end{cases}$$
Let us now prove that $u'$ solves . First we take the shape derivative of both sides of the first equation in at points away from the interface: $$\label{laplu'}
\Delta u' = 0 \text{ in }\omega \cup (\Omega \setminus \overline{{{B_R}}}).$$ In order to prove that ${\left[{\sigma}{\partial_n}u' \right]}$ vanishes on $\partial B_R$ we will proceed as follows. We performing the change of variables $y:=\Phi(t,x)$ in and set $\varphi(x)=:\psi\left( \Phi(t,x) \right)$. Taking the derivative with respect to $t$, bearing in mind the first order approximation of $\Phi$ given by yields the following. $$\begin{aligned}
{\int_\Omega}\sigma \left( - D{{{\bf {h}}}}\nabla u + \nabla \dot{u} \right)\cdot \nabla \psi - {\int_\Omega}{\sigma}{\nabla}u \cdot D{{{\bf {h}}}}{\nabla}\psi + {\int_\Omega}{\sigma}{\nabla}u \cdot {\nabla}\psi {{\rm{div}}}{ {{{\bf {h}}}}} = {\int_\Omega}\psi {{\rm{div}}}{{{\bf {h}}}}.
$$
Rearranging the terms yields: $${\int_\Omega}{\sigma}{\nabla}\dot{u}\cdot {\nabla}\psi +{\int_\Omega}{\sigma}\underbrace{\left( -D {{{\bf {h}}}}- D{{{\bf {h}}}}^T + ({{\rm{div}}}{{{\bf {h}}}}) I \right){\nabla}u \cdot {\nabla}\psi}_{\mathlarger{\mathlarger{\mathlarger{\circledast}}}} = -{\int_\Omega}{{{\bf {h}}}}\cdot {\nabla}\psi.$$
Let [**x**]{} and [**y**]{} be two sufficiently smooth vector fields in ${{\mathbb R}^N}$ such that $D {\bf x}= \left( D {\bf x}\right) ^T$ and $D {\bf y}= \left( D {\bf y}\right) ^T$. It is easy to check that the following identity holds: $$\left( -D {{{\bf {h}}}}- D{{{\bf {h}}}}^T + ({{\rm{div}}}{{{\bf {h}}}}) I \right) {\bf x}\cdot {\bf y} = {{\rm{div}}}\left(({\bf x}\cdot {\bf y}) {{{\bf {h}}}}\right) - {\nabla}({{{\bf {h}}}}\cdot {\bf x})\cdot {\bf y}- {\nabla}({{{\bf {h}}}}\cdot {\bf y})\cdot {\bf x}.$$
We can apply this identity with ${\bf x}={\nabla}u$ and ${\bf y}= {\nabla}\psi$ to rewrite $\mathlarger{\mathlarger{\mathlarger{\circledast}}}$ as follows: $$\left( -D {{{\bf {h}}}}- D{{{\bf {h}}}}^T + ({{\rm{div}}}{{{\bf {h}}}}) I \right){\nabla}u \cdot {\nabla}\psi=\underbrace{{{\rm{div}}}({\nabla}u \cdot {\nabla}\psi {{{\bf {h}}}})}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {1};}}} -\underbrace{{\nabla}({{{\bf {h}}}}\cdot {\nabla}u)\cdot {\nabla}\psi}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {2};}}} -\underbrace{{\nabla}({{{\bf {h}}}}\cdot {\nabla}\psi)\cdot {\nabla}u}_{{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {3};}}}.$$ Thus $$\begin{aligned}
{\int_\Omega}{\sigma}{\nabla}\dot{u}\cdot {\nabla}\psi - {\int_{{{\partial B_R}}}}{\left[{\sigma}{\nabla}u \cdot {\nabla}\psi \right]}{h_n}- {\int_\Omega}{\sigma}{\nabla}({{{\bf {h}}}}\cdot {\nabla}u)\cdot {\nabla}\psi+{\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_- {\left[{\partial_n}\psi \right]}{h_n}=0,
$$ where we have split the integrals and integrated by parts to handle the terms coming from ${\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {1};}}$ and ${\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {3};}}$.
Now, merging together the integrals on $\Omega$ in the left hand side by and exploiting the fact that ${\nabla}u = ({\partial_n}u) {{{\bf {n}}}}$ on ${{\partial B_R}}$, the above simplifies to $$\label{this}
{\int_\Omega}{\sigma}{\nabla}u' \cdot {\nabla}\psi = 0.$$
Splitting the domain of integration and integrating by parts, we obtain $$\nonumber
\begin{aligned}
0=-\int_{{B_R}}{\sigma}_- \Delta u_-' \psi + \int_{{{\partial B_R}}} {\sigma}_- {\partial_n}u_-' \psi -\int_{\Omega\setminus \overline{{{B_R}}}} {\sigma}_+ \Delta u_+' \psi -\int_{{{\partial B_R}}} {\sigma}_+{\partial_n}u_+' \psi-\int_{\partial \Omega} {\sigma}_+ {\partial_n}u_+' \psi \\
= {\int_{{{\partial B_R}}}}{\left[{\sigma}{\partial_n}u' \right]} \psi,
\end{aligned}$$ where in the last equality we have used and the fact that $\psi$ vanishes on $\partial\Omega$. By the arbitrariness of $\psi\in H_0^1(\Omega)$ we can conclude that ${\left[{\sigma}{\partial_n}u' \right]}=0$ on $\partial {{B_R}}$. The remaining conditions of problem are a consequence of .
To prove uniqueness for this problem in the class of functions whose restriction to both ${{B_R}}$ and $\Omega\setminus \overline{{{B_R}}}$ is smooth, just consider the difference between two solutions of such problem and call it $w$. Then $w$ solves $$\begin{cases}
{\Delta}w =0 \quad&\text{in } {{B_R}}\cup (\Omega\setminus \overline{{{B_R}}}),\\
{\left[{\sigma}{\partial_n}w \right]}=0 \quad&\text{on }{{\partial B_R}},\\
{\left[w \right]}=0\quad&\text{on } {{\partial B_R}},\\
w=0 \quad&\text{on }\partial \Omega;
\end{cases}$$ in other words, $w$ solves $$\begin{cases}
-{{\rm{div}}}({\sigma}{\nabla}w)=0 \quad&\text{in }\Omega,\\
w=0 \quad&\text{on }\partial\Omega.
\end{cases}$$ Since the only solution to the problem above is the constant function $0$, uniqueness for Problem is proven.
We emphasize that formulas and are valid only for $f$ belonging at least to the class ${{\mathcal C}}^2([0,T), L^1({{\mathbb R}^N}))\cap {{\mathcal C}}^1([0,T), W^{1,1}({{\mathbb R}^N}))$. We would like to apply them to $f(t)=u_t$ and $f(t)=\sigma_t |\nabla u_t|^2$, where ${\sigma}_t$ and $u_t$ are the distribution of conductivities and the solution of problem respectively corresponding to the case $\omega=({{B_R}})_t$. On the other hand, $u_t$ is not regular enough in the entire domain $\Omega$, despite being fairly smooth in both $\omega_t$ and $\Omega_t\setminus {\mkern 1.5mu\overline{\mkern-1.5mu\omega_t\mkern-1.5mu}\mkern 1.5mu}$: therefore we need to split the integrals in order to apply and (this will give rise to interface integral terms by integration by parts).
\[thm1\] For all $\Phi\in{{\mathcal{A}}}$ we have $$l_1^E(B_R)({h_n})=-{\int_{{{\partial B_R}}}}{\left[{\sigma}|{\nabla}u|^2 \right]} {h_n}.$$ In particular, for all $\Phi$ satisfying the first order volume preserving condition we get $l_1^E(B_R)({h_n})=0$.
We apply formula to $
E(\omega_t)=\int_\Omega u_t=\int_{\omega_t} u_t+\int_{{\Omega\setminus {\mkern 1.5mu\overline{\mkern-1.5mu\omega_t\mkern-1.5mu}\mkern 1.5mu}}} u_t
$ to get $$l_1^E(B_R)({h_n})= \int_{{B_R}}u_-'+\int_{{{\partial B_R}}}u_- h_n + \int_{\Omega\setminus \overline{B_R}} u_+'-\int_{{{\partial B_R}}} u_+ h_n.$$Using the jump notation we rewrite the previous expression as follows $$\label{sonoichi}
l_1^E(B_R)({h_n})= \int_\Omega u' - \int_{{{\partial B_R}}} [u h_n]=\int_\Omega u';$$ notice that the surface integral in vanishes as both $u$ and $h_n$ are continuous through the interface.\
Next we apply to $E(\omega_t)=\int_\Omega \sigma_t |\nabla u_t|^2$. $$\begin{aligned}
l_1^E(B_R)({h_n})= & 2\int_{{{B_R}}} \sigma_- \nabla u_-\cdot \nabla u_-' + \int_{{{\partial B_R}}} \sigma_- |\nabla u_-|^2 h_n+ \\
& 2\int_{{{\Omega\setminus \overline{B_R}}}} \sigma_+ \nabla u_+\cdot \nabla u_+' +\int_{\partial\Omega} \sigma_+ |\nabla u_+|^2 h_n - \int_{{{\partial B_R}}} \sigma_+ |\nabla u_+|^2 h_n.
$$ Thus we get the following: $$\label{sononi}
l_1^E(B_R)({h_n})= 2\int_\Omega \sigma \nabla u\cdot \nabla u' -{\int_{{{\partial B_R}}}}{\left[\sigma |\nabla u|^2 \right]}h_n.$$ Comparing (choose $\psi=u$) with gives $$\label{that}
l_1^E(B_R)({h_n})= -{\int_{{{\partial B_R}}}}{\left[{\sigma}|{\nabla}u|^2 \right]} {h_n}.$$ By symmetry, the term $ {\left[\sigma |\nabla u|^2 \right]}$ is constant on ${{\partial B_R}}$ and can be moved outside the integral sign. Therefore we have $$l_1^E(B_R)({h_n})= 0 \text{ for all $\Phi$ satisfying \eqref{1st}}.$$ This holds in particular for all $\Phi\in{{\mathcal{B}}}$.
Computation of the second order shape derivative
================================================
The result of the previous chapter tells us that the configuration corresponding to ${{B_R}}$ is a critical shape for the functional $E$ under the fixed volume constraint. In order to obtain more precise information, we will need an explicit formula for the second order shape derivative of $E$. The main result of this chapter consists of the computation of the bilinear form $l_2^E(B_R)({h_n},{h_n})$.
\[lduu\] For all $\Phi\in{{\mathcal{A}}}$ we have $${{l_2^E(B_R)({h_n}{h_n})}}=-2{\int_{{{\partial B_R}}}}{\sigma}_-{\partial_n}u_- {\left[{\partial_n}u' \right]}{h_n}-2{\int_{{{\partial B_R}}}}{\sigma}_-{\partial_n}u_- {\left[\partial_{nn}^2 u \right]}h_n^2-{\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_-{\left[{\partial_n}u \right]} H h_n^2.$$
Take $\Phi={{{\rm Id}}}+t{{{\bf {h}}}}$ in ${{\mathcal{A}}}$ with ${{{\bf {h}}}}={h_n}{{{\bf {n}}}}$ on ${{\partial B_R}}$. As remarked after , $Z$ vanishes in this case. We get $${{l_2^E(B_R)({h_n}{h_n})}}=\frac{d^2}{dt^2}{{\left.\kern-\nulldelimiterspace E(\Phi(t)(B_R)) \vphantom{|} \right|_{t=0} }}.$$ Hence, substituting the expression of the first order shape derivative obtained in Theorem \[thm1\] yields $$\nonumber
{{l_2^E(B_R)({h_n}{h_n})}}= -{{\left.\kern-\nulldelimiterspace \frac{d}{dt}\left( \int_{(\partial{{B_R}})_t} {\left[{\sigma}_t |{\nabla}u_t|^2 \right]}{h_n}\circ \Phi^{-1}(t) \right) \vphantom{|} \right|_{t=0} }}.$$ We unfold the jump in the surface integral above and apply the divergence theorem to obtain $$\label{ltuu}
{{l_2^E(B_R)({h_n}{h_n})}}=\frac{d}{dt}{{\left.\kern-\nulldelimiterspace \left( \int_{(B_R)_t} {{\rm{div}}}\left( {\sigma}_- |{\nabla}u_t|^2 {{{\bf {h}}}}\circ \Phi^{-1}(t) \right) - \int_{\Omega \setminus (B_R)_t} {{\rm{div}}}\left( {\sigma}_+ |{\nabla}u_t|^2 {{{\bf {h}}}}\circ \Phi^{-1}(t) \right) \right) \vphantom{|} \right|_{t=0} }}$$ We will treat each integral individually. By we have $\partial_t {{\left.\kern-\nulldelimiterspace \left( \Phi \right) \vphantom{|} \right|_{t=0} }}=-{{{\bf {h}}}}$, therefore $\partial_t{{\left.\kern-\nulldelimiterspace \left({{{\bf {h}}}}\circ \Phi^{-1} \right) \vphantom{|} \right|_{t=0} }}$=-D[[[**[h]{}**]{}]{}]{}[[[**[h]{}**]{}]{}]{}. Now set $f(t):={\sigma}_- |{\nabla}u_t|^2$. By we have $$\nonumber
\begin{aligned}
\frac{d}{dt}{{\left.\kern-\nulldelimiterspace \left(\int_{(B_R)_t} {{\rm{div}}}\left( f(t) {{{\bf {h}}}}\circ \Phi^{-1}(t) \right)\right) \vphantom{|} \right|_{t=0} }} = \underbrace{ \int_{{B_R}}\partial_t{{\left.\kern-\nulldelimiterspace \left( {{\rm{div}}}\left( f(t) {{{\bf {h}}}}\circ\Phi^{-1}(t) \right) \right) \vphantom{|} \right|_{t=0} }}}_{(A)}
+ \underbrace{ {\int_{{{\partial B_R}}}}{{\rm{div}}}(f(0){{{\bf {h}}}}){h_n}}_{(B)}.
\end{aligned}$$ We have $$\nonumber
\begin{aligned}
(A)=\int_{{B_R}}{{\rm{div}}}\left( \partial_t {{\left.\kern-\nulldelimiterspace f \vphantom{|} \right|_{t=0} }} {{{\bf {h}}}}+ f(0)\partial_t {{\left.\kern-\nulldelimiterspace ({{{\bf {h}}}}\circ \Phi^{-1}(t)) \vphantom{|} \right|_{t=0} }} \right) =
\int_{{B_R}}{{\rm{div}}}\left( \partial_t {{\left.\kern-\nulldelimiterspace f \vphantom{|} \right|_{t=0} }} {{{\bf {h}}}}\right) - \int_{{B_R}}{{\rm{div}}}\left( f(0)D{{{\bf {h}}}}{{{\bf {h}}}}\right) = \\
\int_{{B_R}}\partial_t {{\left.\kern-\nulldelimiterspace f \vphantom{|} \right|_{t=0} }} {h_n}- {\int_{{{\partial B_R}}}}f(0) {{{\bf {n}}}}\cdot D{{{\bf {h}}}}{{{\bf {h}}}}.
\end{aligned}$$ On the other hand $$(B)= {\int_{{{\partial B_R}}}}{{\rm{div}}}\left( f(0) {{{\bf {h}}}}\right){h_n}= {\int_{{{\partial B_R}}}}\left( {\nabla}f(0)\cdot {{{\bf {h}}}}+ f(0){{\rm{div}}}{{{\bf {h}}}}\right) {h_n}.$$ Using the fact that ${{{\bf {h}}}}={h_n}{{{\bf {n}}}}$ and ${{\rm{div}}}{{{\bf {h}}}}-{{{\bf {n}}}}\cdot D{{{\bf {h}}}}{{{\bf {n}}}}=: {{\rm{div}}}_\tau ({h_n}{{{\bf {n}}}})=H{h_n}$ (c.f. Equation (5.22), page 366 of [@SG]) we get $$(A)+(B)= {\int_{{{\partial B_R}}}}f' {h_n}+ {\int_{{{\partial B_R}}}}({\partial_n}f+ Hf)h_n^2.$$ Substituting $f(t)={\sigma}_-|{\nabla}u_t|^2$ yields $$(A)+(B)= 2 {\int_{{{\partial B_R}}}}{\sigma}_- {\nabla}u_- \cdot {\nabla}u_-' {h_n}+ 2{\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_- \partial_{nn}^2 u_- h_n^2 + {\int_{{{\partial B_R}}}}{\sigma}_- |{\nabla}u_-|^2 H h_n^2.$$ The calculation for the integral over $\Omega\setminus ({{B_R}})_t$ in is analogous. We conclude that $${{l_2^E(B_R)({h_n}{h_n})}}= -2 {\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_- {\left[{\partial_n}u' \right]}{h_n}-2{\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_- {\left[\partial_{nn}^2 u \right]}h_n^2- {\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}u_-{\left[{\partial_n}u \right]} H h_n^2.$$
Classification of the critical shape ${{B_R}}$:\
================================================
In order to classify the critical shape ${{B_R}}$ of the functional $E$ under the volume constraint we will use the expansion shown in . For all $\Phi\in{{\mathcal{B}}}$ and $t>0$ small, it reads $$\label{eex}
E\left( \Phi(t)({{B_R}})\right)= E({{B_R}})+ \frac{t^2}{2}\left( {{l_2^E(B_R)({h_n}{h_n})}}-{\int_{{{\partial B_R}}}}{\left[{\sigma}|{\nabla}u|^2 \right]} Z \right)+o(t^2).$$ Employing the use of the second order volume preserving condition and the fact that, by symmetry, the quantity ${\left[{\sigma}|{\nabla}u|^2 \right]}$ is constant on the interface ${{\partial B_R}}$ we have $$-{\int_{{{\partial B_R}}}}{\left[{\sigma}|{\nabla}u|^2 \right]} Z = {\int_{{{\partial B_R}}}}{\left[{\sigma}|{\nabla}u|^2 \right]} H h_n^2.$$ Combining this with the result of Theorem \[lduu\] yields $$E\left( \Phi(t)({{B_R}})\right)= E({{B_R}})+ {t^2}{\left\{ -{\int_{{{\partial B_R}}}}{\sigma}_-{\partial_n}u_-{\left[{\partial_n}u' \right]}{h_n}-{\int_{{{\partial B_R}}}}{\sigma}_-{\partial_n}u_- {\left[\partial_{nn}^2 u \right]}h_n^2 \right\}}+o(t^2).$$ We will denote the expression between braces in the above by $Q({h_n})$. Since $u'$ depends linearly on ${h_n}$ (see ), it follows immediately that $Q({h_n})$ is a quadratic form in ${h_n}$. Since both $u$ and $u'$ verify the transmission condition (see and ) we have $${\sigma}_-{\partial_n}u_-{\left[{\partial_n}u' \right]}={\left[{\sigma}{\partial_n}u{\partial_n}u' \right]}= {\sigma}_-{\partial_n}{u'}_-{\left[{\partial_n}u \right]} \text{ on }{{\partial B_R}}.$$ Using the explicit expression of $u$ given in , after some elementary calculation we write $$Q({h_n})=\frac{R}{N}\left(\frac{1}{{\sigma}_-}-\frac{1}{{\sigma}_+}\right)\left( -{\int_{{{\partial B_R}}}}{\sigma}_- {\partial_n}{u'}_- {h_n}+\frac{1}{N}{\int_{{{\partial B_R}}}}h_n^2 \right).$$
In the following we will try to find an explicit expression for $u'$. To this end we will perform the spherical harmonic expansion of the function ${h_n}:{{\partial B_R}}\to {\mathbb R}$. We set $$\label{hexp}
{h_n}(R\theta)=\sum_{k=1}^\infty \sum_{i=1}^{d_k} \alpha_{k,i} Y_{k,i}(\theta) \quad \text{ for all } \theta\in \partial B_1
.$$ The functions $Y_{k,i}$ are called [*spherical harmonics*]{} in the literature. They form a complete orthonormal system of $L^2(\partial B_1)$ and are defined as the solutions of the following eigenvalue problem: $$-{\Delta}_\tau Y_{k,i}=\lambda_k Y_{k,i} \quad \text{ on }\partial B_1,\\
$$ where ${\Delta}_\tau:= {{\rm{div}}}_\tau {\nabla}_\tau$ is the Laplace-Beltrami operator on the unit sphere. We impose the following normalization coniditon $$\label{normalization}
\int_{\partial B_1} Y_{k,i}^2=R^{1-N}.$$ The following expressions for the eigenvalues $\lambda_k$ and the corresponding multiplicities $d_k$ are also known: $$\label{lambdak}
\lambda_k= k(k+N-2), \quad \quad d_k= \binom{N+k-1}{k}-\binom{N+k-2}{k-1}.$$ Notice that the value $k=0$ had to be excluded from the summation in because ${h_n}$ verifies the first order volume preserving condition .
Let us pick an arbitrary $k\in\{1,2,\dots\}$ and $i\in\{1,\dots, d_k\}$. We will use the method of separation of variables to find the solution of problem in the particular case when ${h_n}(R\theta)=Y_{k,i}(\theta)$, for all $\theta\in \partial B_1$.
![How $\Phi(t)(B_R)$ looks like for small $t$ when ${h_n}(R\cdot)=Y_{k,i}$, in 2 dimensions.[]{data-label="sphar"}](sphar.png){width="80.00000%"}
Set $r:=|x|$ and, for $x\ne 0$, $\theta:=x/|x|$. We will be searching for solutions to of the form $u'=u'(r,\theta)=f(r)g(\theta)$. Using the well known decomposition formula for the Laplacian into its radial and angular components, the equation ${\Delta}u'=0$ in ${{B_R}}\cup (\Omega\setminus \overline{{{B_R}}})$ can be rewritten as $$0={\Delta}u'(x) = f_{rr}(r)g(\theta)+\frac{N-1}{r}f_r(r)g(\theta)+\frac{1}{r^2}f(r){\Delta}_{\tau}g(\theta) \;\text{for }r\in(0,R)\cup (R,1),\, \theta\in\partial B_1.$$ Take $g=Y_{k,i}$. Under this assumption, we get the following equation for $f$: $$\label{f}
f_{rr}+\frac{N-1}{r}f_r-\frac{\lambda_k}{r^2}f=0 \quad\text{in } (0,R)\cup (R,1).$$ It can be easily checked that the solutions to the above consist of linear combinations of $r^\eta$ and $r^\xi$, where $$\label{xieta}
\begin{aligned}
\eta&=\eta(k)=\frac{1}{2} \left( 2-N + \sqrt{ (N-2)^2+4\lambda_k } \right)=k,\\
\xi&=\xi(k)= \frac{1}{2} \left( 2-N - \sqrt{ (N-2)^2+4\lambda_k } \right)=2-N-k.
\end{aligned}$$ Since equation is defined for $r\in (0,R)\cup (R,1)$, we have that the following holds for some real constants $A$, $B$, $C$ and $D$; $$f(r)= \begin{cases}
Ar^{2-N-k}+Br^k \quad&\text{for } r\in(0,R),\\
Cr^{2-N-k}+Dr^k \quad&\text{for } r\in(R,1).
\end{cases}$$ Moreover, since ${2-N-k}$ is negative, $A$ must vanish, otherwise a singularity would occur at $r=0$. The other three constants can be obtained by the interface and boundary conditions of problem bearing in mind that $u'(r,\theta)=f(r)Y_{k,i}(\theta)=f(r)h_n(R \theta)$. We get the following system: $$\begin{cases}
C R^{2-N-k}+ DR^k-BR^k= -\frac{R}{N{\sigma}_-} + \frac{R}{N {\sigma}_+},\\
{\sigma}_- kB R^{k-1}= {\sigma}_+ {(2-N-k)} C R^{{2-N-k}} + {\sigma}_+ k D R^{k-1},\\
C+D=0.
\end{cases}$$ Although this system of equations could be easily solved completely for all its indeterminates, we will just need to find the explicit value of $B$ in order to go on with our computations. We have $$\label{B}
B=B_k=\frac{R^{1-k}}{N{\sigma}_-} \cdot \frac{k({\sigma}_--{\sigma}_+)R^k-(2-N-k)({\sigma}_--{\sigma}_+)R^{2-N-k}}{k({\sigma}_--{\sigma}_+)R^k+((2-N-k){\sigma}_+-k{\sigma}_-)R^{2-N-k}}.$$ Therefore, in the particular case when ${h_n}(R\,\cdot)=Y_{k,i}$, $${u'}_-={u'}_-(r,\theta)=B_k r^k Y_{k,i}(\theta), \quad r\in[0,R), \; \theta\in\partial B_1,$$ where $B_k$ is defined as in . By linearity, we can recover the expansion of ${u'}_-$ in the general case (i.e. when holds): $$\begin{aligned}
&{u'}_-(r,\theta)= \sum_{k=1}^\infty \sum_{i=1}^{d_k} \alpha_k B_k r^k Y_{k,i}(\theta), \quad r\in[0,R),\; \theta\in\partial B_1, \quad\text{and therefore} \\
&{\partial_n}{u'}_-(R,\theta)=\sum_{k=1}^\infty \sum_{i=1}^{d_k} \alpha_k B_k k R^{k-1} Y_{k,i}(\theta), \quad\quad\quad\theta\in\partial B_1.
\end{aligned}$$ We can now diagonalize the quadratic form $Q$, in other words we can consider only the case ${h_n}(R\,\cdot)=Y_{k,i}$ for all possible pairs $(k,i)$. We can write $Q$ as a function of $k$ as follows: $$\label{Q}
\begin{aligned}
Q({h_n})=Q(k)=\frac{R}{N}\left(\frac{{\sigma}_+-{\sigma}_-}{{\sigma}_+{\sigma}_-}\right)\left(-{\sigma}_-B_k k R^{k-1} +\frac{1}{N}\right)=\\
\frac{R}{N^2}\left( \frac{{\sigma}_+-{\sigma}_-}{{\sigma}_+{\sigma}_-}\right)\left(1- k \frac{k({\sigma}_--{\sigma}_+)R^k-(2-N-k)({\sigma}_--{\sigma}_+)R^{2-N-k}}{k({\sigma}_--{\sigma}_+)R^k+((2-N-k){\sigma}_+-k{\sigma}_-)R^{2-N-k}} \right).
\end{aligned}$$
The following lemma will play a central role in determining the sign of $Q(k)$ and hence proving Theorem \[mainthm\].
\[lemmone\] For all $R\in (0,1)$ and ${\sigma}_-,{\sigma}_+>0$, the function $k\mapsto Q(k)$ defined in is monotone decreasing for $k\ge 1$.
Let us denote by $\rho$ the ratio of the the conductivities, namely $\rho:= {\sigma}_-/{\sigma}_+$. We get $$Q(k)=\frac{R}{N^2}\left( \frac{1-\rho}{{\sigma}_-}\right)\left(1- k \frac{k(\rho-1)R^k-(2-N-k)(\rho-1)R^{2-N-k}}{k(\rho-1)R^k+((2-N-k)-k\rho)R^{2-N-k}} \right).$$ In order to prove that the map $k\mapsto Q(k)$ is monotone decreasing it will be sufficient to prove that the real function $$j(x):=x\frac{x-(2-N-x)R^{2-N-2x}}{(1-\rho)x+ \left(-2+N+x+\rho x \right)R^{2-N-2x}}$$ is monotone increasing in the interval $(1,\infty)$. Notice that this does not depend on the sign of $\rho-1$. From now on we will adopt the following notation: $$\nonumber
L:=R^{-1}>1,\quad M:=N-2\ge 0,\quad P=P(x):= L^{2x+M}.$$ Using the notation introduced above, $j$ can be rewritten as follows $$j(x)=\frac{x^2+(x^2+Mx)P}{(1-\rho)x+(x+M+\rho x)P}.$$ In order to prove the monotonicity of $j$, we will compute its first derivative and then study its sign. We get $$j'(x)=\frac{MP(MP+2Px+2x)+x^2(P+1)^2+\rho x^2P (P-1/P-4x\log(L)-2M\log(L))}{\left((1-\rho)x+(x+M+\rho x)P \right)^2}.$$ The denominator in the above is positive and we claim that also the numerator is. To this end it suffices to show that the quantity multiplied by $\rho x^2P$ in the numerator, namely $P-1/P-4x\log(L)-2M\log(L)$, is positive for $x\in (1,\infty)$ (although, we will show a stronger fact, namely that it is positive for all $x>0$). $$\frac{d}{dx}\left(P-\frac{1}{P}-4x\log(L)-2M\log(L)\right)= 2{\log(L)}{\left(P+\frac{1}{P}-2\right)}>0 \quad\text{for }x>0,$$ where we used the fact that $L>1$ and that $P\mapsto P+P^{-1}-2$ is a non-negative function vanishing only at $P=1$ (which does not happen for positive $x$). We now claim that $${{\left.\kern-\nulldelimiterspace \left(P-\frac{1}{P}-4x\log(L)-2M\log(L)\right) \vphantom{|} \right|_{x=0} }}= L^M-\frac{1}{L^M}-2M \log(L)\ge 0.$$ This can be proven by an analogous reasoning: treating $M$ as a real variable and differentiating with respect to it, we obtain $$\frac{d}{dM}\left( L^M-\frac{1}{L^M}-2M \log(L) \right) = \log(L)\left(L^M+\frac{1}{L^M}-2\right)\ge 0$$ (notice that the equality holds only when $M=0$), moreover, $${{\left.\kern-\nulldelimiterspace \left( L^M-\frac{1}{L^M}-2M \log(L) \right) \vphantom{|} \right|_{M=0} }}=0,$$ which proves the claim.
We are now ready to prove the main result of the paper.
\[mainthm\] Let ${\sigma}_-,{\sigma}_+>0$ and $R\in(0,1)$. If ${\sigma}_->{\sigma}_+$ then $$\frac{d^2}{dt^2}{{\left.\kern-\nulldelimiterspace E\big(\Phi(t)(B_R)\big) \vphantom{|} \right|_{t=0} }}<0 \quad \text{ for all }\Phi\in{{\mathcal{B}}}.$$
Hence, $B_R$ is a local maximizer for the functional $E$ under the fixed volume constraint. On the other hand, if ${\sigma}_-<{\sigma}_+$, then there exist some $\Phi_1$ and $\Phi_2$ in ${{\mathcal{B}}}$, such that $$\frac{d^2}{dt^2}{{\left.\kern-\nulldelimiterspace E\big(\Phi_1(t)(B_R)\big) \vphantom{|} \right|_{t=0} }}<0,\quad \frac{d^2}{dt^2}{{\left.\kern-\nulldelimiterspace E\big(\Phi_2(t)(B_R)\big) \vphantom{|} \right|_{t=0} }}>0.$$ In other words, $B_R$ is a saddle shape for the functional $E$ under the fixed volume constraint.
 \[q\]
We have $$\label{quuno}
Q(1)=\frac{R}{N^2}\left( \frac{1-\rho}{{\sigma}_-}\right) \frac{N\rho}{\rho(R^N+1) +N-R^N-1}.$$ Since $N\ge 2$, $R\in (0,1)$, we have $N-R^N-1>0$ and therefore it is immediate to see that $Q(1)$ and $1-\rho$ have the same sign.
If ${\sigma}_->{\sigma}_+$, then, by Lemma \[lemmone\], we get in particular that $Q(k)$ is negative for all values of $k\ge 1$. This implies that the second order shape derivative of $E$ at $B_R$ is negative for all $\Phi\in{{\mathcal{B}}}$ and therefore $B_R$ is a local maximizer for the functional $E$ under the fixed volume constraint as claimed.
On the other hand, if ${\sigma}_-<{\sigma}_+$, by we have $Q(1)>0$. An elementary calculation shows that, for all ${\sigma}_-,{\sigma}_+>0$, $$\lim_{k\to\infty} Q(k)=-\infty.$$ Therefore, when ${\sigma}_-<{\sigma}_+$, $B_R$ is a saddle shape for the functional $E$ under the fixed volume constraint.
The surface area preserving case
================================
The method employed in this paper can be applied to other constraints without much effort. For instance, it might be interesting to see what happens when volume preserving perturbations are replaced by surface area preserving ones. Is $B_R$ a critical shape for the functional $E$ even in the class of domains of fixed surface area? If so, of what kind? We set ${{\rm Per}}(D):=\int_{\partial D} 1$ for all smooth bounded domain $D\subset {{\mathbb R}^N}$. The following expansion for the functional ${{\rm Per}}$ can be obtained just as we did for : $$\nonumber{{\rm Per}}(\omega_t)={{\rm Per}}(\omega)+ t \int_{\partial\omega}H {h_n}+ \frac{t^2}{2} \left( l_2^{{\rm Per}}(\omega) ({h_n},{h_n}) + \int_{\partial\omega}H Z \right) + o(t^2) \text{ as }t\to0,$$ where, (cf. [@henrot], page 225) $$\label{l2per}
l_2^{{\rm Per}}(\omega)({h_n},{h_n})=\int_{\partial \omega} |{\nabla}_\tau {h_n}|^2 +\int_{\partial \omega} \left( H^2 - {\rm tr}\big((D_\tau {{{\bf {n}}}})^T D_\tau {{{\bf {n}}}}\big) \right) h_n^2.$$ We get the following first and second order surface area preserving conditions. $$\begin{aligned}
\label{1st2nd}
&\int_{\partial \omega}H {h_n}=0,
&\int_{\partial \omega} |{\nabla}_\tau {h_n}|^2 +\int_{\partial \omega} \left( H^2 - {\rm tr}\big((D_\tau {{{\bf {n}}}})^T D_\tau {{{\bf {n}}}}\big) \right) h_n^2+ \int_{\partial \omega}H Z=0. \end{aligned}$$
Notice that when $\omega=B_R$, the first order surface area preserving condition is equivalent to the first order volume preserving condition and therefore, by Theorem \[thm1\], $B_R$ is a critical shape for $E$ under the fixed surface area constraint as well.
The study of the second order shape derivative of $E$ under this constraint is done as follows. Employing the use of together with the second order surface area preserving condition in we get $$\frac{d^2}{dt^2}{{\left.\kern-\nulldelimiterspace E\big(\Phi(t)(B_R)\big) \vphantom{|} \right|_{t=0} }}= l_2^E(B_R)({h_n},{h_n}) + \frac{{\left[{\sigma}|{\nabla}u|^2 \right]}}{H} l_2^{{\rm Per}}(B_R)({h_n},{h_n}).$$ In other words, we managed to write the shape Hessian of $E$ as a quadratic form in ${h_n}$. We can diagonalize it by considering ${h_n}(R\cdot)=Y_{k,i}$ for all possible pairs $(k,i)$, where we imposed again the normalization . Under this assumption, by we get $$\int_{{\partial B_R}}|{\nabla}_\tau {h_n}|^2= \frac{\lambda_k}{R^2}=\frac{k(k+N-2)}{R^2}.$$ We finally combine the expression for $l_2^E(B_R)$ of Theorem \[lduu\] with that of $l_2^{{\rm Per}}$ to obtain $$E\big(\Phi(t)(B_R)\big)= E(B_R) + t^2 \,\widetilde{Q}(k) + o(t^2) \text{ as }t\to 0,$$ where $$\widetilde{Q}(k)=\frac{R}{N^2}\left( \frac{1-\rho}{{\sigma}_-}\right)\left(\frac{3}{2} - \frac{k(k+N-2)}{2(N-1)} -k\,\frac{k(\rho-1)R^k-(2-N-k)(\rho-1)R^{2-N-k}}{k(\rho-1)R^k+((2-N-k)-k\rho)R^{2-N-k}} \right).$$
It is immediate to check that $\widetilde{Q}(1)=Q(1)$ and therefore, $\widetilde{Q}(1)$ is negative for ${\sigma}_->{\sigma}_+$ and positive otherwise. On the other hand, $\lim_{k\to\infty} \widetilde{Q}(k)=\infty$ for ${\sigma}_->{\sigma}_+$ and $\lim_{k\to\infty} \widetilde{Q}(k)=-\infty$ for ${\sigma}_-<{\sigma}_+$. In other words, under the surface area preserving constraint $B_R$ is always a saddle shape, independently of the relation between ${\sigma}_-$ and ${\sigma}_+$.
 \[qtil\]
We can give the following geometric interpretation to this unexpected result. Since the case $k=1$ corresponds to deformations that coincide with translations at first order, it is natural to expect a similar behaviour under both volume and surface area preserving constraint. On the other hand, high frequency perturbations (i.e. those corresponding to a very large eigenvalue) lead to the formation of indentations in the surface of $B_R$. Hence, in order to prevent the surface area of $B_R$ from expanding, its volume must inevitably shrink (this is due to the higher order terms in the expansion of $\Phi$). This behaviour can be confirmed by looking at the second order expansion of the volume functional under the effect of a surface area preserving transformation $\Phi\in{{\mathcal{A}}}$ on the ball: $${{\rm Vol}}\big(\Phi(t)({{B_R}})\big)={{\rm Vol}}({{B_R}})+\frac{t^2}{2}\left( \frac{1}{R}-\frac{k(k+N-2)}{(N-1)R} \right) +o(t^2) \text{ as }t\to 0.$$ We see that the second order term vanishes when $k=1$, while getting arbitrarily large for $k\gg1$. Since this shrinking effect becomes stronger the larger $k$ is, this suggests that the behaviour of $E\big(\Phi(t)(B_R)\big)$ for large $k$ might be approximated by that of the extreme case $\omega=\emptyset$. For instance, when ${\sigma}_->{\sigma}_+$ we have that $E(B_R)<E(\emptyset)$ and this is coherent with what we found, namely $\widetilde{Q}(k)>0$ for $k\gg 1$.
Acknowledgments {#acknowledgments .unnumbered}
===============
This paper is prepared as a partial fulfillment of the author’s doctor’s degree at Tohoku University. The author would like to thank Professor Shigeru Sakaguchi (Tohoku University) for his precious help in finding interesting problems and for sharing his naturally optimistic belief that they can be solved. Moreover we would like to thank the anonymous referee, who suggested to study the surface area preserving case and helped us find a mistake in our calculations. Their detailed analysis and comments on the previous version of this paper, contributed to make the new version shorter and more readable.
[10]{}
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<span style="font-variant:small-caps;">Research Center for Pure and Applied Mathematics, Graduate School of Information Sciences, Tohoku University, Sendai 980-8579 , Japan.</span>\
[*Electronic mail address:*]{} cava@ims.is.tohoku.ac.jp
[^1]: This research was partially supported by the Grant-in-Aid for Scientific Research (B) (\#26287020) Japan Society for the Promotion of Science
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---
abstract: 'We show within a geometrical model developed in earlier papers that multiplicity distributions are cut off at large multiplicities. The position and motion of the cut-off point is related to geometrical- and KNO scaling and their violation, in particular by the rise of the ratio $\sigma_{el}/\sigma_{t}.$ At the LHC energies a change of the regime, connected with the transition from shadowing to antishadowing is expected.'
address: |
Bogolyubov Institute for Theoretical Physics\
National Academy of Sciences of Ukraine\
Metrologicheskaya str. 14b, 03143 Kiev, Ukraine.
author:
- 'L.L. JENKOVSZKY[^1] and B.V. STRUMINSKY'
title: On the Distribution of Secondaries at High Energies
---
The properties of multiplicity distribution of secondaries $P(n)$ at high values of $n$ remain among the topical problems in high-energy physics. As pointed out in a series of recent papers [@Soso], the underlying dynamics behind these rare processes may be quite different from the bulk of events.
Our knowledge about high-energy multiplicity distributions comes from the data collected at the $ISR, \ \ \ Sp\bar pS$ collider (UA1, UA2 and UA5 experiments) and the Tevatron collider (CDF and E735 experiments). In should be noticed that the recent results from the E335 Collaboration taken at the Tevatron [@E735] do not completely agree with those obtained by the UA5 Collaboration at comparable energies at the $Sp \bar p S$ collider [@UA5] (see Fig. 1).
Notice that the delicate features of $\Psi(z)$ at very large multiplicities, near the large-$z$ edge can be better seen if the variable $z$ is used instead of $n.$
On the theoretical side, it became common [@Dremin], [@GU], [@Kuvshin], [@Odessa] to approximate the observed distributions by the convolution of two binomial distributions, accounting for the general “bell-like” shape of $P(n)$ with the observed structures (“knee” and possible oscillations) superimposed.
One of the hottest issues in this field is the dynamics of very high multiplicities (VHM) [@Giokaris], close to the kinematical limit imposed by the phase space. The VHM events are very rare, making up only about $10^{-7}$ of the total cross-sections at the LHC energy, which makes their experimental identification very difficult. An intriguing question is the possible existence of a cut-of in the VHM region, beyond $z=n/<n>\approx 5,$ where $<n>$ is the mean multiplicity. In our opinion, a better understanding of the underlying physics can be inferred only in a model involving both elastic and inelastic scattering, related by unitarity. Such an approach has been advocated in a series of papers [@Aliev], [@Chikovani], [@Kvaratshelia], summarized in ref. [@EChAYa].
After a brief summary of the main ideas behind this approach, we analyze the relation of the distribution of secondaries and the behavior of the elastic and total cross sections with the possible transition from shadowing to antishadowing [@TT].
We show that the existence of a cut-off at high multiplicities in the distribution $\Psi(z)$ is related to the validity of GS and KNO scaling.
The basic idea of the geometrical approach to multiple production, used in the present paper, is that the number of the secondaries at a given impact parameter $\rho,$ $n(\rho,s)$ is proportional to the amount of the hadronic matter in the collision or the the overlap function $G(\rho, s)$ $$<n(\rho.s)>=N(s)G(\rho,s), \eqno(1)$$ where $N(s)$ is related to mean multiplicity, not specified in this approach, and $G(\rho,s)$ is the overlap function, related by unitarity to elastic scattering $$Im h(\rho,s)=|h(\rho,s)|^2+G(\rho,s). \eqno(2)$$ where $h(\rho,s)$ is the elastic amplitude in the impact parameter representation. Unitarity, a key issue in this approach, enters both in the definition of the elastic amplitude and of the inelastic one (the overlap function).
In the $u-$ matrix unitarization (see [@EChAYa] and references therein) $$G_{in}={\Im u\over{1+2 \Im u+|u|^2}}, \eqno(3)$$ where $u$ is the elastic amplitude (input, or the “Born term”).
We use a dipole (DP) model for the elastic scattering amplitude, exibiting geometrical features and fitting the data. After u-matrix unitarization, the elastic amplitude reads (see [@EChAYa]) $$h(\rho,s)={u\over{1-i u}}, \eqno(4)$$ where $u(y,s)=ige^{-y}, \ \ y={\rho^2\over{4\alpha' L}},$ and $L\equiv\ln s$.
Remarkably, the ratio of the elastic to total cross sections in this model, $${\sigma_{el}\over\sigma_{t}}=1-{g\over {(1+g)\ln(1+g)}} \eqno(5)$$ fixes the (energy-dependent) values of the parameter $g$. Typical values of $g$ for several representative energies are quoted in [@EChAYa].
Rescattering corrections to $G_{in}(\rho,s)$ here will be accounted for phenomenologically according to the following prescription (see [@EChAYa] and earlier reference therein). $$G_{in}(\rho,s)=|S(\rho, s)| \tilde G_{in}(\rho,s), \eqno(6)$$ where $S(\rho,s))$ is the $S$ is the elastic scattering matrix, related to the $u$ matrix by $$S(\rho,s)={1+iu(\rho,s)\over{1-iu(\rho,s)}}. \eqno(7)$$
This procedure is not unique. For example, it allows the following generalization (see [@EChAYa] and earlier reference therein) $$G_{in}(\rho,s)=|S(\rho,s)|^{\alpha}\tilde G^{\alpha}(\rho,s),
\eqno(8)$$ where $\alpha$ is a parameter, varying between 0 and 1.
We assume $$<n(\rho,s)>=N(s)\tilde G^{\alpha}_{in}(\rho,s). \eqno(9)$$
The moments are defined by (see [@EChAYa] and earlier references therein) $$<n^k(s)>={N^k(s)\int G_{in}(\rho,s)(G_{in}^{alpha}(\rho,s))^k
d^2\rho\over{\int G_{in}(\rho,s)d^2\rho}} \eqno(10)$$ Now we insert the expression for the DP with the u-matrix unitarization (4) into (10) to get $$<n^k(s)>={N^k(s)(1+g)\over
g}\int_0^g{dx\over{(1+x)^2}}\Biggl(\Bigl({1+x\over{1-x}}\Bigr)^{\alpha}{x\over{(1+x)^2}}\Biggr)^k.
\eqno (11)$$ The mean multiplicity $<n(s)>$ is defined as $$<n(s)>={N(s)(1+g)\over
g}\int_0^g{x dx\over{(1+x)^4}}\Bigl({1+x\over{1-x}}\Bigr)^{\alpha}={N(s)\over a}. \eqno (12)$$
For the distributions we have $$P(n)={1+g\over
g}\int_0^g{dx\over{(1+x)^2}}\delta\Biggl(n-N\Bigl({1+x\over{1-x}}\Bigr)^{\alpha}{x\over{(1+
x)^2}}\Biggr).
\eqno (13)$$
Integration in (13) gives $$\psi(z)=<n>P(n)={1+g\over g}{x(1-x)\over{z(1+x)[(1-x)^2+2\alpha
x}]},$$ where $z=n/<n>.$
Since the above integral is non-zero only when the argument of the $\delta$ function vanishes, $$n=N\Biggl({1+x\over{1-x}}\Biggr)^{\alpha}{x\over{(1+x)^2}},$$ one gets a remarkable relation $$z={ax \over (1+x)^{2-\alpha}(1-x)^{\alpha}}. \eqno(14)$$
To calculate the distribution $\Psi(z)$ one needs the solution of equation (14). It can be found explicitly for two extreme cases, namely $\alpha=0$ and $\alpha=1.$ Otherwise, it can be calculated numerically.
The maximal value of $z$, corresponding to $x=g$ ($x$ varies between $0$ and $g$), can be found as: $$z_{max}={ag\over{(1+g)^{2-\alpha}}|(1-g)|^{\alpha}}. \eqno(15)$$
It can be seen from (15) that $z_{max}$ is a constant if $g$ is energy independent. The experimentally observed ratio $\sigma_{el}/\sigma_t$ varies between 53 Gev and 900 Gev from 0.174 to 0.225, implying the variation of g from 0.489 to 0.702, uniquely determined by the above ratio. This monotonic increase of g(s) in its turn pushes $z_{max}(s)$ outwards, terminating when $g$ reaches unity (according to [@EChAYa] this will happen around 10 TeV, i.e. at the future LHC), whenafter the term $|1-g|$ in (15) will start rising again, pulling $z_{max}(s)$ back to smaller values. I.e. $z_{max}(s)$ has its own maximum in $s$ at $g=1.$
The unusual behavior of $z_{max}(s)$ is not the only interesting feature of the present approach. This effect can be related to the behavior of the ratio $\sigma_{el}/\sigma_t$. As argued by Troshin and Tyurin (see [@TT]), $\sigma_{el}/\sigma_t$ may pass the so-called black disc limit and continue rising in a new, “antishadowing” mode of the $u$-matrix unitarity approach (multiplicity distributions were not considered in that paper). According to the recent calculations [@DJS] the transition from shadowing to antishadowing will also occur in the LHC energy region.
To summarize, we found a regularity connecting the geometrical properties in high-energy dynamics (GS and KNO scaling) with the dynamics of the high-multiplicity processes. We showed, in particular, that exact geometrical, or KNO scaling, implying constant $g$ in our model, results in a cut-off at large $z$ of the distribution function $\Psi(z).$ Any departure from scaling (energy dependence of $g$ in our model) shifts the point $z_m$ according to eq. (15). Within the present accelerator energy domain (ISR, SPS, Tevatron) $g$ varies from about 0.5 to about 0.8. It will reach the critical value $g=1$ at LHC, where we predict a change of the regime: $z_{max}(s)$ will start decreasing and the black disc limit will be passed (which, as shown in [@TT] and [@DJS], is not equivalent to the violation of the unitarity limit, but means passage from shadowing to antishadowing [@TT] and [@DJS]).
Finally, it should be noted that we use many model assumptions, decreasing the predictive power of our calculations. These assumptions concern mostly the way absorption corrections are introduced and the assumption of the local ($\delta$ function) dependence of multiplicities on the impact parameter. Both assumptions, as well as others can be modified. As a result we quantitive rather than qualitative changes in the results.
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[^1]: E-mail: jenk@gluk.org
|
---
abstract: |
We first present four graphic surgery formulae for the degree $n$ part $Z_n$ of the Kontsevich-Kuperberg-Thurston universal finite type invariant of rational homology spheres.
Each of these four formulae determines an alternate sum of the form $$\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$$ where $N$ is a finite set of disjoint operations to be performed on a rational homology sphere $M$, and $M_I$ denotes the manifold resulting from the operations in $I$. The first formula treats the case when $N$ is a set of $2n$ Lagrangian-preserving surgeries (a [*Lagrangian-preserving surgery*]{} replaces a rational homology handlebody by another such without changing the linking numbers of curves in its exterior). In the second formula, $N$ is a set of $n$ Dehn surgeries on the components of a boundary link. The third formula deals with the case of $3n$ surgeries on the components of an algebraically split link. The fourth formula is for $2n$ surgeries on the components of an algebraically split link in which all Milnor triple linking numbers vanish. In the case of homology spheres, these formulae can be seen as a refinement of the Garoufalidis-Goussarov-Polyak comparison of different filtrations of the rational vector space freely generated by oriented homology spheres (up to orientation-preserving homeomorphisms).
The presented formulae are then applied to the study of the variation of $Z_n$ under a $p/q$-surgery on a knot $K$. This variation is a degree $n$ polynomial in $q/p$ when the class of $q/p$ in ${\mathbb{Q}}/{\mathbb{Z}}$ is fixed, and the coefficients of these polynomials are knot invariants, for which various topological properties or topological definitions are given.
[**Keywords:**]{} finite type invariants, 3-manifolds, Jacobi diagrams, clovers, Kontsevich-Kuperberg-Thurston configuration space invariant, claspers, Casson-Walker invariant, Goussarov-Habiro filtration, surgery formulae, $Y$-graphs\
[**2000 Mathematics Subject Classification:**]{} 57M27 57N10 57M25 55R80
---
\
Prépublication de l’Institut Fourier n$^{o}$ 697 (2007) http://www-fourier.ujf-grenoble.fr/prepublications.html
Introduction
============
In this article, we shall focus on the real finite type invariants of homology $3$-spheres in the sense of Ohtsuki, Goussarov and Habiro, and on the topological properties of the surgery formulae that these invariants satisfy.
M. Kontsevich proposed a topological construction for an invariant $Z$ of oriented rational homology $3$-spheres using configuration space integrals. G. Kuperberg and D. Thurston proved that $Z$ is a universal finite type invariant for homology $3$-spheres, in the sense of Ohtsuki, Goussarov and Habiro [@kt; @lesconst]. Like the LMO invariant, the Kontsevich-Kuperberg-Thurston invariant $Z=(Z_n)_{n \in {\mathbb{N}}}$ takes its values in a space of Jacobi diagrams ${{\cal A}}=\prod_{n \in {\mathbb{N}}}{{\cal A}}_n$, and any real degree $n$ invariant $\nu$ of homology $3$-spheres is obtained from the Kontsevich-Kuperberg-Thurston invariant $(Z_i)_{i \in {\mathbb{N}}}$ by a composition with a linear form that kills the $Z_i$, for $i>n$.
We shall first prove four formulae for $Z_n$, for $n\in {\mathbb{N}}$. Each of these four formulae will determine an alternate sum of the form $$\sum_{I \subset N} (-1)^{\sharp I}Z_n(M_I)$$ where $N$ is a set of disjoint operations to be performed on $M$, and $M_I$ denotes the manifold resulting of the operations in $I$. Our first formula, Theorem \[thmflag\], will be a mere alternative statement of the main theorem of [@sumgen] and will treat the case when $N$ is a set of $2n$ Lagrangian-preserving surgeries (a [*Lagrangian-preserving surgery*]{} replaces a rational homology handlebody by another such without changing the linking numbers of curves in its exterior). In our second formula, Theorem \[thmfboun\], $N$ will be a set of $n$ rational surgeries on the components of a boundary link. Our third formula, Theorem \[thmfas\], will deal with the case of $3n$ surgeries on the components of an algebraically split link. Our fourth formula, Theorem \[thmfasmu\], will be for $2n$ surgeries on the components of an algebraically split link in which all Milnor triple linking numbers vanish.
In the case of integral homology spheres, these results can be seen as refinements of the Garoufalidis-Goussarov-Polyak comparison of the filtrations of the vector space generated by homology spheres, with respect to algebraically split links, boundary links or Lagrangian-preserving surgeries [@ggp; @al].
As it was proved by Garoufalidis in [@garouf], a degree $n$ finite type invariant $\nu$ of homology spheres satisfies a surgery formula that describes its variation under $1/r$-surgery on a knot $K$ as $$\nu(M(K;1/r))-\nu(M)=\sum_{i=1}^n \nu^{(i)}(K \subset M) r^i$$ where $\nu^{(i)}$ is a finite type knot invariant in the Vassiliev sense for knots in $S^3$ as defined in [@bn].
Since all the real finite type invariants of homology $3$-spheres can be obtained from the universal LMO invariant by composition with a linear form on the space of Jacobi diagrams [@lmo; @le], and since the LMO invariant is defined using the Kontsevich integral and surgery presentations of manifolds, the knot invariants $\nu^{(i)}$ can be explicitly given in terms of the Kontsevich integral of surgery presentations of the knots. See also the [Å]{}rhus construction [@Aa1; @Aa2; @Aa3].
We seek for a better topological understanding of the invariants $\nu^{(i)}$, and we shall relate some of them to the topology of Seifert surfaces of the knots. For example, for any degree $n$ invariant $\nu$, we give a formula in terms of the entries of the Seifert matrix and the weight system of $\nu$ for the leading coefficient $\nu^{(n)}$ of the surgery polynomial. See Theorem \[thmpol\]. We shall also prove that $\nu^{(i)}$ is of degree less than $2n$ for any $i<n$. This specifies a result of Garoufalidis and Habegger who proved that $(\nu(M(K;1))-\nu(M))$ is a degree $2n$ knot invariant with the same weight system as a degree $2n$ knot invariant induced by the Alexander polynomial, by using the LMO invariant [@gh].
Some of the results proved in this article can be refined in the extensively studied case of the Casson-Walker invariant $\lambda=W_1 \circ Z_1$, where $W_1({ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}})=2$, that satisfies the well-known formula, for a knot $K$ in a homology sphere $M$, $$\lambda(M(K;p/q))-\lambda(M)=\frac{q}{p}\lambda^{\prime}(K) + \lambda(L(p,-q))$$ where $\lambda^{\prime}(K)$ is half the second derivative of the normalized Alexander polynomial of $K$ evaluated at $1$ and $L(p,-q)$ is the lens space obtained by $p/q$-surgery on the unknot. We shall prove some graphical formulae for $\lambda^{\prime}(K)$ and for its variation under surgeries on disjoint algebraically unlinked knots in Propositions \[propcasknot\], \[propcasknotvar\], \[propcasknottwo\].
Next, we shall concentrate on the case of the degree $2$ invariant $\lambda_2= W_2 \circ Z_2$, where $W_2({\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-}(0,0)(.6,0)(.3,.2)
\psline{*-*}(.6,0)(.3,.5)(.3,.2)
\psline{*-}(.3,.5)(0,0)(.3,.2)
\end{pspicture}})=1$ and $W_2({ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}}\;{ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}})=0$. The invariants $\lambda$ and $\lambda_2$ generate the vector space of real-valued invariants of degree lower than $3$ that are additive under connected sum. We shall prove that $\lambda_2$ satisfies the surgery formula $$\lambda_2(M(K;p/q))-\lambda_2(M)=\lambda_2^{\prime \prime}(K) \left(\frac{q}{p}\right)^2 + w_3(K) \frac{q}{p} + c(q/p) \lambda^{\prime}(K) +\lambda_2(L(p;-q))$$ for a knot $K$ in a homology sphere $M$, where $c(q/p)$ only depends on $q/p$ modulo ${\mathbb{Z}}$, $\lambda_2^{\prime \prime}$ is explicitly given in Theorem \[thmpol\] and $w_3$ is a knot invariant, for which we shall prove various properties. These properties include a crossing change formula, Proposition \[propvarwthree\], and a formula for genus one knots $K$, Proposition \[propgenusone\]. For knots in $S^3$, $w_3$ is the degree $3$ knot invariant that changes sign under mirror image, and that maps the chord diagram with three diameters to $(-1)$. In his thesis [@auc], Emmanuel Auclair independently obtained a formula for $w_3(K)$ in terms of topological invariants of curves of an arbitrary Seifert surface of $K$, that is fortunately equivalent to Proposition \[propgenusone\] in the genus one case.
The article is organized as follows. The main results are stated precisely without proofs from Section \[secstatelag\] to Section \[secstatewthree\]. The proofs occupy the following sections. Questions and expected generalizations of the proved results are mentioned at the end.
The Kontsevich-Kuperberg-Thurston universal finite type invariant $Z$
=====================================================================
Jacobi diagrams {#subjac}
---------------
Here, a [*[Jacobi diagram]{}*]{} $\Gamma$ is a trivalent graph $\Gamma$ without simple loop like $\begin{pspicture}[.2](0,0)(.6,.4)
\psline{-*}(0.05,.2)(.25,.2)
\pscurve{-}(.25,.2)(.4,.05)(.55,.2)(.4,.35)(.25,.2)
\end{pspicture}$. The set of vertices of such a $\Gamma$ will be denoted by $V(\Gamma)$, its set of edges will be denoted by $E(\Gamma)$. A [*[half-edge]{}*]{} $c$ of $\Gamma$ is an element of $$H(\Gamma)=\{c=(v(c);e(c)) | v(c) \in V(\Gamma); e(c) \in E(\Gamma);v(c) \in e(c)\}.$$ An [*automorphism*]{} of $\Gamma$ is a permutation $b$ of $H(\Gamma)$ such that for any $c,c^{\prime} \in H(\Gamma)$, $$v(c)=v(c^{\prime}) \Longrightarrow v(b(c))=v(b(c^{\prime}))\;\;\;\mbox{and} \;\;\;e(c)=e(c^{\prime}) \Longrightarrow e(b(c))=e(b(c^{\prime})).$$ The number of automorphisms of $\Gamma$ is denoted by $\sharp \mbox{Aut}(\Gamma)$. For example, $ \sharp \mbox{Aut}({ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}})=12$. [*An orientation*]{} of a vertex of such a diagram $\Gamma$ is a cyclic order of the three half-edges that meet at that vertex. A Jacobi diagram $\Gamma$ is [*oriented*]{} if all its vertices are oriented (equipped with an orientation). The [*degree*]{} of such a diagram is half the number of its vertices.
Let ${{\cal A}}_n={{\cal A}}_n(\emptyset)$ denote the real vector space generated by the degree $n$ oriented Jacobi diagrams, quotiented out by the following relations AS and IHX:
$${\rm AS :} \begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psarc[linewidth=.5pt](.5,.5){.2}{-70}{15}
\psarc[linewidth=.5pt](.5,.5){.2}{70}{110}
\psarc[linewidth=.5pt]{->}(.5,.5){.2}{165}{250}
\psline{*-}(.5,.5)(.5,0)
\psline{-}(.1,.9)(.5,.5)
\psline{-}(.9,.9)(.5,.5)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\pscurve{-}(.9,.9)(.3,.7)(.5,.5)
\pscurve[border=2pt]{-}(.1,.9)(.7,.7)(.5,.5)
\psline{*-}(.5,.5)(.5,0)
\end{pspicture}=0,\;\;\mbox{and IHX :}
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{-*}(.1,1)(.35,.2)
\psline{*-}(.5,.5)(.5,1)
\psline{-}(.75,0)(.5,.5)
\psline{-}(.25,0)(.5,.5)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{*-}(.5,.6)(.5,1)
\psline{-}(.8,0)(.5,.6)
\psline{-}(.2,0)(.5,.6)
\pscurve[border=2pt]{-*}(.1,1)(.3,.3)(.7,.2)
\end{pspicture}
+
\begin{pspicture}[.2](0,-.2)(.8,1)
\psset{xunit=.7cm,yunit=.7cm}
\psline{*-}(.5,.35)(.5,1)
\psline{-}(.75,0)(.5,.35)
\psline{-}(.25,0)(.5,.35)
\pscurve[border=2pt]{-*}(.1,1)(.2,.75)(.7,.75)(.5,.85)
\end{pspicture}
=0.$$ Each of these relations relate diagrams which can be represented by planar immersions that are identical outside the part of them represented in the pictures. Here, the orientation of vertices is induced by the counterclockwise order of the half-edges. For example, AS identifies the sum of two diagrams which only differ by the orientation at one vertex to zero. ${{\cal A}}_0(\emptyset)$ is equal to ${\mathbb{R}}$ generated by the empty diagram.
The Kontsevich-Kuperberg-Thurston universal finite type invariant $Z$
---------------------------------------------------------------------
Let $\Lambda$ be ${\mathbb{Z}}$, ${\mathbb{Z}}/2{\mathbb{Z}}$ or ${\mathbb{Q}}$. A [*$\Lambda$-sphere*]{} is a compact oriented 3-manifold $M$ such that $H_{\ast}(M;\Lambda)=H_{\ast}(S^3;\Lambda)$. A ${\mathbb{Z}}$-sphere is also called a [*homology sphere*]{} while a [*rational homology sphere*]{} is a ${\mathbb{Q}}$-sphere. Following Witten, Axelrod, Singer, Kontsevich, Bott and Cattaneo, Greg Kuperberg and Dylan Thurston constructed invariants $Z_n=(Z_{KKT})_n$ of oriented ${\mathbb{Q}}$-spheres valued in ${{\cal A}}_n(\emptyset)$ and they proved that their invariants have the following property [@kt]. See also [@lesconst].
\[thktone\] An invariant $\nu$ of ${\mathbb{Z}}$-spheres valued in a real vector space $X$ is of degree $\leq n$ if and only if there exist linear maps $$\phi_k(\nu): {{\cal A}}_k(\emptyset) \longrightarrow X,$$ for any $k \leq n$, such that $$\nu=\sum_{k=0}^n
\phi_k(\nu) \circ Z_k.$$
A [*real [finite type invariant]{}*]{} of ${\mathbb{Z}}$-spheres is a topological invariant of ${\mathbb{Z}}$-spheres valued in a real vector space $X$ which is of degree less than some natural integer $n$. Theorem \[thktone\] can be used as a definition of degree $\leq n$ real-valued invariants of ${\mathbb{Z}}$-spheres.
A degree $\leq n$ invariant $\nu$ is of [*degree*]{} $n$ if $\phi_n(\nu) \neq 0$. In this case, $\phi_n(\nu)$ is the [*[weight system]{}*]{} of $\nu$ and is denoted by $W_{\nu}$.
Let $p^c \colon {{\cal A}}_k(\emptyset) \rightarrow {{\cal A}}_k(\emptyset)$ be the canonical linear projection of ${{\cal A}}_k(\emptyset)$ onto its subspace ${{\cal A}}^c_k(\emptyset)$ generated by the connected diagrams, such that $p^c$ maps the non-connected diagrams to $0$ and the restriction of $p^c$ to ${{\cal A}}^c_k(\emptyset)$ is of course the identity. Then $Z_n^c=p^c \circ Z_n$ is additive under connected sum. Furthermore any real-valued degree $n$ invariant belongs to the algebra generated by the $(\phi_{k,i} \circ Z_k^c)_{k \leq n}$ for linear forms $\phi_{k,i}$ generating the dual of ${{\cal A}}^c_k(\emptyset)$.
The above definition coincides with the Ohtsuki definition of real finite type invariants [@oht]. The Ohtsuki degree (that is always a multiple of $3$) is three times the above degree. See [@oht; @ggp; @hab; @al] and references therein for more discussions about the various concepts of finite-type invariants.
Lagrangian-preserving surgeries
===============================
\[secstatelag\]
[**Conventions:**]{} Unless otherwise mentioned, manifolds are compact and oriented. Boundaries are always oriented with the outward normal first convention. The normal bundle $N(V)$ of an oriented submanifold $V$ in an oriented manifold $M$ is oriented so that the tangent bundle $T_xM$ of the ambient manifold $M$ at some $x \in V$ is oriented as $T_xM = N_xV \oplus T_xV$. If $V$ and $W$ are two oriented transverse submanifolds of an oriented manifold $M$, their intersection is oriented so that the normal bundle of $T_x(V \cap W)$ is the sum $N_xV \oplus N_xW$. If the two manifolds are of complementary dimensions, then the sign of an intersection point is $+1$ if the orientation of its normal bundle coincides with the orientation of the ambient space that is equivalent to say that $T_xM=T_xV \oplus T_xW$. Otherwise, the sign is $-1$. If $V$ and $W$ are compact and if $V$ and $W$ are of complementary dimensions in $M$, their algebraic intersection is the sum of the signs of the intersection points, it is denoted by $\langle V, W \rangle_M$.
Recall that the [*linking number*]{} $lk(J,K)$ of two disjoint knots $J$ and $K$ in a rational homology sphere $M$ is the algebraic intersection of $J$ with a surface $\Sigma_K$ bounded by $K$ if $K$ is null-homologous, that $lk(J,.)$ is linear on $H_1(M \setminus J)$, and that $lk(J,K)=lk(K,J)$.
The [*Milnor triple linking number*]{} $\mu(K_1,K_2,K_3)$ of three null-homologous knots $K_1, K_2, K_3$ that do not link each other algebraically in a rational homology sphere $M$ may be defined, as the algebraic intersection of three Seifert surfaces $\Sigma_2$, $\Sigma_1$, $\Sigma_3$ of these knots in the complement of the other ones. $$\mu(K_1,K_2,K_3)=-\langle \Sigma_1,\Sigma_2,\Sigma_3\rangle= -\langle \Sigma_1 \cap \Sigma_2, \Sigma_3\rangle=-lk(K_3,\Sigma_1 \cap \Sigma_2).$$
We now describe part of the behaviour of the $Z_n$ under the Lagrangian surgeries defined below.
A [*genus $g$ ${\mathbb{Q}}$-handlebody*]{} is an (oriented, compact) 3-manifold $A$ with the same homology with rational coefficients as the standard (solid) handlebody $H_g$ below. $$H_g = \begin{pspicture}[.4](0,-.5)(4.5,.95)
\psset{xunit=.5cm,yunit=.5cm}
\psecurve{-}(5.7,1.3)(5.2,1.3)(3.9,1.8)(2.6,1.3)(1.3,1.8)(.1,1)(1.3,.1)(2.6,.7)
(3.9,.1)(5.2,.7)(5.7,.7)
\pscurve{-}(.8,1.2)(1,.9)(1.3,.8)(1.6,.9)(1.8,1.2)
\pscurve{-}(1,.9)(1.3,1.2)(1.6,.9)
\rput[r](.9,-.2){$a_1$}
\psecurve{->}(1.6,.4)(1.3,.8)(1.05,.4)(1.3,.1)
\psecurve{-}(1.3,.8)(1.05,.4)(1.3,.1)(1.6,.4)
\psecurve[linestyle=dashed,dash=3pt 2pt](1,.4)(1.3,.8)(1.55,.4)(1.3,.1)(1,.4)
\pscurve{-}(3.4,1.2)(3.6,.9)(3.9,.8)(4.2,.9)(4.4,1.2)
\pscurve{-}(3.6,.9)(3.9,1.2)(4.2,.9)
\rput[r](3.5,-.2){$a_2$}
\psecurve{->}(4.2,.4)(3.9,.8)(3.65,.4)(3.9,.1)
\psecurve{-}(3.9,.8)(3.65,.4)(3.9,.1)(4.2,.4)
\psecurve[linestyle=dashed,dash=3pt
2pt](3.6,.4)(3.9,.8)(4.15,.4)(3.9,.1)(3.6,.4)
\rput(5.8,1.3){\dots}
\rput(5.8,.7){\dots}
\psecurve{-}(5.8,1.3)(6.4,1.3)(7.7,1.8)(8.9,1)(7.7,.1)(6.4,.7)(5.8,.7)
\pscurve{-}(7.2,1.2)(7.4,.9)(7.7,.8)(8,.9)(8.2,1.2)
\pscurve{-}(7.4,.9)(7.7,1.2)(8,.9)
\rput[l](6.8,-.2){$a_g$}
\psecurve{->}(8,.4)(7.7,.8)(7.45,.4)(7.7,.1)
\psecurve{-}(7.7,.8)(7.45,.4)(7.7,.1)(8,.4)
\psecurve[linestyle=dashed,dash=3pt
2pt](7.4,.4)(7.7,.8)(7.95,.4)(7.7,.1)(7.4,.4)
\end{pspicture}$$ Note that the boundary of such a ${\mathbb{Q}}$-handlebody $A$ is homeomorphic to the boundary $(\partial H_g =\Sigma_g)$ of $H_g$.
For a ${\mathbb{Q}}$-handlebody $A$, ${\cal L}_A$ denotes the kernel of the map induced by the inclusion: $$H_1(\partial A;{\mathbb{Q}}) \longrightarrow H_1( A;{\mathbb{Q}}).$$ It is a Lagrangian of $(H_1(\partial A;{\mathbb{Q}}),\langle,\rangle_{\partial A})$, we call it the [*[Lagrangian]{}*]{} of $A$. A [*Lagrangian-preserving surgery*]{} or [*LP–surgery*]{} $(A,A^{\prime})$ is the replacement of a ${\mathbb{Q}}$-handlebody $A$ embedded in a $3$-manifold by another such $A^{\prime}$ with identical (identified via a homeomorphism) boundary and Lagrangian.
There is a canonical isomorphism $$\partial_{MV} \colon H_2(A \cup_{\partial A}
-A^{\prime};{\mathbb{Q}}) \rightarrow {\cal L}_A$$ that maps the class of a closed surface in the closed $3$-manifold $(A \cup_{\partial A}
-A^{\prime})$ to the boundary of its intersection with $A$. This isomorphism carries the algebraic triple intersection of surfaces to a trilinear antisymmetric form ${{\cal I}}_{AA^{\prime}}$ on ${{\cal L}}_A$. $${{\cal I}}_{AA^{\prime}}(a_{i},a_{j},a_{k})=\langle \partial_{MV}^{-1}(a_i), \partial_{MV}^{-1}(a_j), \partial_{MV}^{-1}(a_k)\rangle_{A \cup
-A^{\prime}}$$
Let $(a_1,a_2,\dots,a_g)$ be a basis of ${{\cal L}}_A$, and let $z_1,\dots,z_g$ be homology classes of $\partial A$, such that $(z_1,\dots,z_g)$ is dual to $(a_1,a_2,\dots,a_g)$ with respect to $\langle,\rangle_{\partial A}$ ($\langle a_i,z_j \rangle_{\partial A}=\delta_{ij}$). Note that $(z_1,\dots,z_g)$ is a basis of $H_1(A;{\mathbb{Q}})$. Represent ${{\cal I}}_{AA^{\prime}}$ by the following combination $T({{\cal I}}_{AA^{\prime}})$ of tripods whose three univalent vertices form an ordered set: $$T({{\cal I}}_{AA^{\prime}})=\sum_{\left\{\{i,j,k\} \subset \{1,2,\dots ,g_A\} ; i <j <k\right\}}{{\cal I}}_{AA^{\prime}}(a_{i},a_{j},a_{k})
\begin{pspicture}[0.2](-.2,-.1)(.5,.7)
\psline{-}(0.05,.3)(.45,.6)
\psline{*-}(0.05,.3)(.45,.3)
\psline{-}(0.05,.3)(.45,0)
\rput[l](.55,0){$z_i$}
\rput[l](.55,.3){$z_j$}
\rput[l](.55,.6){$z_k$}
\end{pspicture}$$ When $G$ is a graph with $2n$ trivalent vertices and with univalent vertices decorated by disjoint curves of $M$, define its contraction $\langle \langle G \rangle \rangle_n$ as the sum that runs over all the ways $p$ of gluing the univalent vertices two by two in order to produce a vertex-oriented Jacobi diagram $G_p$ $$\langle \langle G \rangle \rangle_n=\sum_p \ell(G_p)[G_p]$$ where $\ell(G_p)$ is the product over the pairs of glued univalent vertices, with respect to the [*pairing*]{} $p$, of the linking numbers of the corresponding curves. The contraction $\langle \langle . \rangle \rangle$ is linearly extended to linear combination of graphs, and the disjoint union of combinations of graphs is bilinear.
A [*$k$–component Lagrangian-preserving surgery datum*]{} in a rational homology sphere $M$ is a datum $(M;(A_i,A_i^{\prime})_{i \in \{1,\dots,k\}})$ of $k$ disjoint ${\mathbb{Q}}$–handlebodies $A_i$, for $i \in \{1,\dots,k\}$, in $M$, and $k$ associated LP-surgeries $(A_i,A_i^{\prime})$.
\[thmflag\] Let $$(M;(A_i,A_i^{\prime})_{i \in \{1,\dots,2n\}})$$ be a $2n$–component Lagrangian-preserving surgery datum in a rational homology sphere $\;M$. For $I \subset \{1,\dots,2n\}$, let $M_I$ denote the manifold obtained from $M$ by replacing $A_i$ by $A_i^{\prime}$ for all $i\in I$. Then $$\sum_{I \subset \{1,\dots,2n\}} (-1)^{\sharp I}Z_n(M_I)=\langle \langle \bigsqcup_{i \in \{1,\dots,2n\}} T({{\cal I}}_{A_iA_i^{\prime}}) \rangle \rangle_n.$$
We shall prove that this formula is equivalent to the formula of [@sumgen] in Section \[secprooflag\].
Let ${{\cal F}}_0$ be the rational vector space freely generated by the oriented ${\mathbb{Q}}$-spheres viewed up to oriented homeomorphisms. For a $k$–component Lagrangian-preserving surgery datum $(M;(A_i,A_i^{\prime})_{i \in \{1,\dots,k\}})$ in a rational homology sphere $\;M$, define $$[M;(A_i,A_i^{\prime})_{i \in \{1,\dots,k\}}]=\sum_{I \subset \{1,\dots,k\}} (-1)^{\sharp I}M_I \in {{\cal F}}_0$$ and define ${{\cal F}}_k$ as the subspace of ${{\cal F}}_0$ generated by elements of ${{\cal F}}_0$ of the above form. Then, it easily follows from the above theorem that $Z_n({{\cal F}}_{2n+1})=0$ where $Z_n$ is linearly extended to ${{\cal F}}_0$. For two elements $x$ and $y$ of ${{\cal F}}_0$, we write $x \stackrel{n}{\equiv} y$ to say that $x-y \in {{\cal F}}_{2n+1}$. Thus, if $x \stackrel{n}{\equiv} y$, then $Z_n(x)=Z_n(y)$.
The intersection of this filtration with the rational vector space freely generated by the oriented ${\mathbb{Z}}$-spheres is the Goussarov-Habiro filtration. (The inclusion of the Goussarov-Habiro filtration $({{\cal F}}^{GH}_{k})_k$ in the intersection is obvious, the other one comes from the fact that ${{\cal F}}^{GH}_{k}$ is the intersection of the kernels of the $Z_i$ for $2i < k$ because of the universality of $Z$.)
Surgeries on algebraically split links
======================================
Let $L(p_i,-q_i)$ be the lens space obtained from $S^3$ by $p_i/q_i$-surgery on a trivial knot. (The standard conventions for surgery coefficients are recalled in the beginning of Section \[seccasstate\].) When $L=(K_i;p_i/q_i)_{i \in N}$ is a given link whose components are equipped with surgery coefficients in a rational homology sphere $\;M$, for $I\subset N$, let $$M_I=M_{(K_i;p_i/q_i)_{i \in I}}\sharp \sharp_{j \in N \setminus I} L(p_j,-q_j)$$ denote the connected sum of the manifold $M_{(K_i;p_i/q_i)_{i \in I}}=M\left((K_i;p_i/q_i)_{i \in I}\right)$ obtained from $M$ by surgery on $(K_i;p_i/q_i)_{i \in I}$ and the lens spaces $L(p_j,-q_j)$ for $j \notin I$.
Set $$[M;(K_i;p_i/q_i)_{i \in N}]=\sum_{I \subset N} (-1)^{\sharp I} M_I.$$
Note that the connected sums with lens spaces are trivial when the $p_i$ are $1$.
The invariant $Z_n$ is linearly extended to ${{\cal F}}_0$. By the additivity of the connected part $Z^c_n$ of $Z_n$ under connected sum, if $N$ has more than one element, $$Z^c_n\left([M;(K_i;p_i/q_i)_{i \in N}]\right)=Z^c_n\left(\sum_{I \subset N} (-1)^{\sharp I} M_{(K_i;p_i/q_i)_{i \in I}}\right)$$ and the connected sums with lens spaces do not appear in this case either.
In Section \[secproofboun\], we shall see how Theorem \[thmflag\] easily implies the following surgery formula on $n$-component boundary links.
\[thmfboun\] Let $n$ and $r$ be elements of ${\mathbb{N}}$. Consider a link $(K_1,K_2, \dots, K_r)$ where all the $K_i$ bound disjoint oriented surfaces $\Sigma^i$. Let $p_i/q_i$ be a surgery coefficient for $K_i$, and let $(x_j^i,y_j^i)_{j=1,\dots,g(\Sigma^i)}$ be a symplectic basis for the Seifert surface $\Sigma^i$. Define $$I(\Sigma^i)= \sum_{(j,k) \in \{1,2,\dots, g(\Sigma^i)\}^2}
\begin{pspicture}[0.2](-.5,-.1)(1.7,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(1,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(1,0.05)(1.15,.35)
\psline{-}(1,0.05)(.85,.35)
\rput[br](-.15,.5){$x_j^i$}
\rput[b](.15,.5){$y_j^i$}
\rput[b](.85,.5){$(y_k^i)^+$}
\rput[lb](1.25,.5){$(x_k^i)^+$}
\end{pspicture}.$$ Then $$\begin{array}{lll}Z_n\left([M;(K_i;p_i/q_i)_{i \in \{1,\dots,r\}}]\right)&=0&\mbox{if}\; r>n
\\&=\frac{1}{2^n}\langle \langle \bigsqcup_{i \in \{1,\dots,n\}} (-\frac{q_i}{p_i} I(\Sigma^i)) \;\;\rangle \rangle\;&\mbox{if}\; r=n.\end{array}$$
A link $L$ in a $3$-manifold is said to be [*algebraically split*]{} if any component of $L$ is null-homologous in the exterior of the other ones (i.e. if any component of $L$ bounds a surface in the complement of the other components of $L$).
An [*edge-labelled*]{} Jacobi diagram is a Jacobi diagram $\Gamma$ equipped with a bijection from $E(\Gamma)$ to $\{1,2,3,\dots, 3n\}$ for some integer $n$. Let $D_{e,n}$ be the set of unoriented edge-labelled Jacobi diagrams of degree $n$. Let $L=(K_i)_{i \in \{1,2,3,\dots, 3n\}}$ be a $3n$–component algebraically split link. Let $\Gamma \in D_{e,n}$, orient $\Gamma$. To any vertex of $\Gamma$, whose incoming edges are labeled by $i,j,k$ with respect to the cyclic order induced by the orientation, associate the Milnor triple number $\mu(K_i,K_j,K_k)$. Then define $\mu_{\Gamma}(L)$ as the product over all the vertices of $\Gamma$ of the corresponding Milnor numbers of $L$. Note that $\mu_{\Gamma}(L)[\Gamma]$ does not depend on the orientation of $\Gamma$. Let $\theta(\Gamma)$ be the number of components of $\Gamma$ homeomorphic to ${ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}}$.
\[thmfas\] Let $n$ and $r$ be elements of ${\mathbb{N}}$. Let $L=(K_i;p_i/q_i)_{i \in \{1,2,3,\dots, r\}}$ be a (rationally) framed $r$–component algebraically split link in a rational homology sphere $\;M$. Then with the notation above, $$\begin{array}{lll}Z_n\left([M;(K_i;p_i/q_i)_{i \in \{1,\dots,r\}}]\right)&=0&\mbox{if}\; r>3n
\\&= \prod_{i=1}^{3n}\frac{q_i}{p_i}\sum_{\Gamma \in D_{e,n}} \frac{\mu_{\Gamma}(L)}{2^{\theta(\Gamma)}}[\Gamma]&\mbox{if}\; r=3n.\end{array}$$
A [*$2/3$-labelled*]{} Jacobi diagram is a degree $n$ Jacobi diagram $\Gamma$ equipped with an injection $\iota$ from $\{1,2,3,\dots, 2n\}$ to $E(\Gamma)$ such that at each vertex two edges of the image of $\iota$ meet one edge outside the image of $\iota$. Let $D_{2/3,n}$ be the set of unoriented $2/3$-labelled Jacobi diagrams of degree $n$. Let $(F_i)_{i=1, \dots 2n}$ be a collection of transverse oriented surfaces that meet pairwise only inside their respective interiors, such that $\langle F_i, F_j, F_k \rangle=0$ for any $\{i,j,k\} \subset \{1,2,3,\dots, 2n\}$. Let $\Gamma \in D_{2/3,n}$, orient $\Gamma$. For a vertex of $\Gamma$, whose half-edges belong to edges labelled by $(i,j,$ nothing$)$, with respect to the cyclic order induced by the orientation, assign the intersection curve $F_i \cap F_j$ to the unlabelled half-edge. To any unlabelled edge $e$ that is now equipped with intersection curves $F_i \cap F_j$ and $F_k \cap F_{\ell}$ associate the linking number $\ell((F_i)_{i=1, \dots 2n};\Gamma;e)$ of $F_i \cap F_j$ and $F^+_k \cap F^+_{\ell}$, where $F^+_k$ and $F^+_{\ell}$ are parallel copies of $F_k$ and $F_{\ell}$.
Note that there is no need to push the intersection curves by using parallels if $F_i$, $F_j$, $F_k$ and $F_{\ell}$ are distinct, to define this linking number. If $\{i,j\}=\{k,\ell\}$, this linking number is the self-linking of the intersection curve that is framed by the surface, up to sign. Now, note that $lk(F_i \cap F_j, F^+_i \cap F^+_{\ell})= lk(F_i \cap F_j, F^+_i \cap F_{\ell})$ and that $$lk(F^+_i \cap F_j, F_i \cap F_{\ell})-lk(F_i \cap F_j, F^+_i \cap F_{\ell})=\pm \langle F_i, F_j, F_{\ell} \rangle.$$ Therefore if the cardinality of $\{i,j\} \cap \{k,\ell\}$ is $1$, the linking number $\ell((F_i)_{i=1, \dots 2n};\Gamma;e)$ is well-defined, too. Define $\ell((F_i)_{i=1, \dots 2n};\Gamma)$ as the product over all the unlabelled edges of $\Gamma$ of the $\ell((F_i)_{i=1, \dots 2n};\Gamma;e)$. Note that $\ell((F_i)_{i=1, \dots 2n};\Gamma)[\Gamma]$ is independent of the orientation of $\Gamma$. Let $\sharp \mbox{Aut}_{2/3}(\Gamma)$ be the number of automorphisms of $\Gamma$ that preserve its $2/3$-labelling.
\[thmfasmu\] Let $n$ and $r$ be elements of ${\mathbb{N}}$. Let $L=(K_i;p_i/q_i)_{i \in \{1,2,3,\dots, r\}}$ be a framed $r$–component algebraically split link in a rational homology sphere $\;M$ such that for any $\{i,j,k\} \subset \{1,2,3,\dots, r\}$, $\mu(K_i,K_j,K_k)=0$.
Let $(F_i)_{i \in \{1,2,3,\dots, r\}}$ be a collection of transverse Seifert surfaces for the $K_i$ where $F_i$ does not meet the $K_j$ for $i \neq j$.
Then with the notation above $$\begin{array}{lll}Z_n\left([M;(K_i;p_i/q_i)_{i \in \{1,\dots,r\}}]\right)&=0&\mbox{if}\; r>2n\\ &=\prod_{i=1}^{2n}\frac{q_i}{p_i}\sum_{\Gamma \in D_{2/3,n}} \frac{\ell((F_i)_{i=1, \dots 2n};\Gamma)}{\sharp \mbox{Aut}_{2/3}(\Gamma)}[\Gamma]&\mbox{if}\; r=2n\end{array}$$ where the sum runs over all $2/3$-labelled unoriented Jacobi diagrams $\Gamma$.
When $M$ is a ${\mathbb{Z}}$-sphere, when the $p_i$ are equal to $1$, and when $r$ is greater or equal, than $n$ for Theorem \[thmfboun\], than $2n$ for Theorem \[thmfasmu\], and than $3n$ for Theorem \[thmfas\], the left-hand sides of the equalities of these theorems are in ${{\cal F}}_{2n}^{GH}$. Since the degree $n$ part of the LMO invariant coincides with $Z_n$ on ${{\cal F}}_{2n}^{GH}$, these three theorems hold for the LMO invariant as well, in these cases.
Theorems \[thmfas\] and \[thmfasmu\] will be proved in Section \[secprooffas\]. Their proofs will rely on some clasper calculus performed in Section \[secclasper\], that will also lead to the following proposition.
\[proprealmil\] Let $L=(K_i)_{i \in \{1,2,3,\dots, r\}}$ be an $r$–component algebraically split link in a rational homology sphere $\;M$. Then there exist transverse Seifert surfaces $\Sigma_i$ in $M \setminus \left(\cup_{j \neq i}K_j \right)$ for each component $K_i$ of $L$, such that, for any triple $(K_i,K_j,K_k)$ of components of $L$, the geometric triple intersection of the surfaces $\Sigma_i$, $\Sigma_j$ and $\Sigma_k$ is made of $|\mu(K_i,K_j,K_k)|$ points.
Section \[secprooffas\] also contains an equivalent definition of the Matveev Borromean surgery (or surgery on a $Y$-graph), see Proposition \[propborlag\].
On the polynomial form of the knot surgery formula
==================================================
Recall that for any rational homology sphere $M$, $Z_0(M)=1$. Theorem \[thmfboun\] implies that for any knot $K$ that bounds a surface $F$ in a rational homology sphere $M$ and for any two coprime integers $p$ and $q$ such that $p\neq 0$, $$Z_1(M(K;\frac{p}{q}))-Z_1(M)=\frac12 \langle \langle I(F)\rangle \rangle \frac{q}{p} +Z_1(L(p,-q)).$$ We shall see in Section \[secproofpol\] that Theorem \[thmfboun\] also easily implies the following theorem. The first part of this theorem is essentially [@garouf Prop. 4.1].
\[thmpol\] Let $p$ and $q$ be coprime integers such that $p\neq 0$. Let $n \in {\mathbb{N}}$, $n\geq 1$. Let $K$ be a knot that bounds a Seifert surface $F$ in a rational homology sphere $M$. Let $F^i$ be parallel copies of $F$ for $i \in \{1,\dots,n\}$, and let $L_i$ denote the framed link made of the boundary components of $\cup_{j=1}^i F_j$, where each component is framed by $1$.
Then $$Z_n(M(K;\frac{p}{q+rp}))-Z_n(M)=\sum_{i=0}^n Y_{n,q/p}^{(i)}(K \subset M) (r+\frac{q}{p})^i$$ for any $r \in {\mathbb{Z}}$ where the coefficients $Y_{n,q/p}^{(i)}(K)$ satisfy the following properties.
- $$Y_{n,q/p}^{(n)}(K)=\frac{(-1)^n}{n!}Z_n([M;L_n])= \frac{1}{n!2^n} \langle \langle \bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\;\rangle \rangle\; ,$$
- if $n\geq 2$, $$Y_{n,q/p}^{(n-1)}(K)=\frac{(-1)^{n-1}}{(n-1)!}\left(\left(\frac{n-1}2+\frac{q}{p}\right) Z_n\left([M;L_n]\right) +Z_n\left([M(K;\frac{p}{q});L_{n-1}]\right)\right),$$
- if $n\geq 2$, $p^c(Y_{n,q/p}^{(n-1)})= Y_{n,q/p}^{(n-1)c}$ does not depend on $p$ and $q$,
- if $i \leq n-1$, $Y_{n,q/p}^{(i)}$ only depends on $q/p$ mod ${\mathbb{Z}}$,
- if $U$ bounds a disk in $M$, then $Y_{n,q/p}^{(i)}(U \subset M)=0$ if $i>0$ and $$Y_{n,q/p}^{(0)}(U \subset M)=Z_n(M \sharp L(p,-q))-Z_n(M),$$
- $$Y_{n,0}^{(0)}(K \subset M)=0,$$
- $$Y_{n,q/p}^{(i)}(K \subset M)=(-1)^{i+n}Y_{n,-q/p}^{(i)}(K \subset - M).$$
A [*singular knot*]{} is an immersion of $S^1$ in a $3$-manifold whose only multiple points are transverse double points like
(0,0)(.5,.4) (0.05,0.05)(0.45,0.45) (0.05,0.45)(0.45,0.05) (0.25,.25)
.
Such a double point can be removed in a positive way
(0,0)(.5,.4) (0.05,0.05)(0.45,0.45) (0.05,0.45)(0.45,0.05)
or in a negative way
(0,0)(.5,.4) (0.05,0.45)(0.45,0.05) (0.05,0.05)(0.45,0.45)
.
Let $K^s$ be a singular primitive knot in a rational homology sphere with $k$ double points. Fix a bijection from $\{1,\dots,k\}$ to its set of double points. For $I \subset \{1,\dots,k\}$, let $K_I$ be the desingularisation of $K^s$ such that the singular points in the image of $I$ have been removed in a negative way, and the singular points outside the image of $I$ have become positive. If $y$ is a knot invariant valued in an abelian group, set $$y(K^s)=\sum_{I \subset \{1,\dots,k\}}(-1)^{\sharp I}y(K_I).$$
It may happen that we do not know whether $Z_n(M(K_I;\frac{p}{q+pr}))$ is polynomial in $r$ for a given $I$, but that we know that $$\sum_{I \subset \{1,\dots,k\}}(-1)^{\sharp I}Z_n(M(K_I;\frac{p}{q+pr}))$$ is. Then the definition of $Y_{n,q/p}^{(i)}(K^s)$ extends in an obvious way.
\[propzsing\] For any singular knot $K^s$ in a rational homology sphere with $k$ double points, for any integers $n$, $i$, $q$ and $p$, with $0 \leq i \leq n$,\
$Y_{n,q/p}^{(i)}(K^s)=0$ if $k > 2n$,\
$Y_{n,q/p}^{(i)}(K^s)=0$ if $k > 2n-1$ and if $i < n$.\
In other words, $Y_{n,q/p}^{(i)}$ is a knot invariant of degree at most $2n$ with respect to the crossing changes, and if $i<n$, $Y_{n,q/p}^{(i)}$ is a knot invariant of degree at most $(2n-1)$ with respect to the crossing changes.
Two disjoint pairs of points in $S^1$ are said to be [*unlinked*]{} if they bound disjoint intervals in $S^1$. Otherwise, they are said to be [*linked.*]{} Two double points of a singular knot are said to be [*linked*]{} if their preimages are linked.
Associate a symmetric linking matrix $[\ell_{ij}(K^s)]_{i,j \in \{1,2,\dots,k\}}$ to a singular knot $K^s$ with $k$ pairwise unlinked double points numbered from $1$ to $k$ in the following way. Each double point $i$
(0,0)(.5,.5) (0.05,0.05)(0.45,0.45) (0.05,0.45)(0.45,0.05) (0.25,.25)
can be [*smoothed*]{} to transform the knot into two oriented singular knots $K^{s\prime}_i$ and $K^{s\prime\prime}_i$.
(0,0)(1,.7) (-0.05,-0.05)(0.05,0.05)(0.25,.15)(0.45,0.05)(0.55,-0.05) (-0.05,.55)(0.05,0.45)(0.25,.35)(0.45,0.45)(0.55,.55) (.55,.45)[$K^{s\prime}_i$]{} (.55,.05)[$K^{s\prime\prime}_i$]{}
Set $$\ell_{ii}(K^s)=lk(K^{s\prime}_i,K^{s\prime\prime}_i).$$ If $i$ and $j$ label two unlinked double points, let $K^{s,j}_i$ be the curve among $K^{s\prime}_i$ and $K^{s\prime\prime}_i$ that does not contain the double point labeled by $j$, then $\ell_{ij}(K^s)= lk(K^{s,j}_i,K^{s,i}_j)$ if $i \neq j$.
\[propzsingun\] For any singular knot $K^s$ in a rational homology sphere with $k$ pairwise unlinked double points, for any integers $n$, $i$, $q$ and $p$, with $0 \leq i \leq n$, and $n\geq 1$,\
if $k > n$, $Y_{n,q/p}^{(i)}(K^s)=0$,\
if $k=n$, $Y_{n,q/p}^{(i)}(K^s)=Y_{n,0}^{(i)}(K^s)$ is an explicit homogeneous polynomial of degree $i$ in the coefficients of the linking matrix of $K^s$, and $Y_{n,q/p}^{(0)}(K^s)=0$.
Proposition \[propvarztwo\] will give explicit examples of computations of the above homogeneous polynomials.
A few formulae for the Casson-Walker invariant
==============================================
\[seccasstate\]
Set $\lambda=W_1 \circ Z_1$ where $W_1({ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}})=2.$ According to [@sumgen], $\lambda$ is the Casson-Walker invariant as normalized by Casson for ${\mathbb{Z}}$–spheres (see [@akmc; @gm; @mar]), $\lambda$ is half the Walker invariant as normalized in [@wal], and $\lambda$ coincides with $\frac{\overline{\lambda}}{|H_1(M)|}$ where $|H_1(M)|$ denotes the cardinality of $H_1(M;{\mathbb{Z}})$ for ${\mathbb{Q}}$–spheres, and $\overline{\lambda}$ is the extension of $|H_1(M)|\lambda$ to oriented closed $3$-manifolds that is denoted by $\lambda$ in [@pup].
A [*rationally algebraically split link*]{} is a link whose components do not link each other. The following proposition gives formulae that generalize Theorem \[thmfboun\], Theorem \[thmfas\] and Theorem \[thmfasmu\] in the degree $1$ case ($n=1$) for rationally algebraically split links.
The [*order*]{} of a knot $K$ in a rational homology sphere is the smallest positive integer $O_K$ such that $O_K K$ is null-homologous. A [*primitive curve*]{} on a torus $S^1 \times S^1$ is a non-separating simple closed curve on the torus. A [*primitive satellite*]{} of a knot is a primitive curve on the boundary $\partial N(K)$ of its tubular neighborhood. A surgery on a knot $K$ is determined by a primitive satellite $\mu$ (oriented arbitrarily) of the knot that will bound a disk inside the surgered torus after surgery. If $m_K$ is the meridian of $K$, the isotopy class of such a curve is determined by the pair $$(p_K =lk(\mu,K),q_K =\langle m_K, \mu \rangle_{\partial N(K)})$$ and the [*surgery coefficient*]{} is $p_K/q_K$.
For any order $d$ component $K$ of a rationally algebraically split link $L$, there exists an embedded surface $\Sigma$ in the complement of $L$ whose boundary $\partial \Sigma$ is made of essential parallel curves of the boundary $\partial N(K)$ of the tubular neighborhood $N(K)$ of $K$ such that $\partial \Sigma$ is homologous to $d$ parallels of $K$ in $N(K)$. Let $H_1(\Sigma)/H_1(\partial \Sigma)$ denote the quotient of $H_1(\Sigma)$ by the image of $H_1(\partial \Sigma)$ under the map induced by the inclusion. Let $B_s=(x_i,y_i)_{i \in \{1,\dots, g\}}$ be a symplectic basis of $H_1(\Sigma)/H_1(\partial \Sigma)$, define $$I(\Sigma)= \sum_{(j,k) \in \{1,2,\dots, g\}^2}
\begin{pspicture}[0.2](-.5,-.1)(1.7,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(1,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(1,0.05)(1.15,.35)
\psline{-}(1,0.05)(.85,.35)
\rput[br](-.15,.5){$x_j$}
\rput[b](.15,.5){$y_j$}
\rput[b](.85,.5){$y_k^+$}
\rput[lb](1.25,.5){$x_k^+$}
\end{pspicture}.$$
If $W_n \colon {{\cal A}}_n \rightarrow {\mathbb{Q}}$ is a linear form, then $W_n\left( \langle \langle \cdot \rangle \rangle\right)$ will also be denoted by $\langle \langle \cdot \rangle \rangle_{W_n}$.
For example, $$\langle \langle I(\Sigma) \rangle \rangle_{W_1}= 2 \sum_{(j,k) \in \{1,2,\dots, g\}^2} \left(lk(x_j,x_k^+)lk(y_j,y_k^+)-lk(x_j,y_k^+)lk(y_j,x_k^+)\right).$$
\[propcasknot\] Let $n$ be an integer. Set $N=\{1,\dots,n\}$. Let $L=(K_i;p_i/q_i)_{i \in N}$ be a framed rationally algebraically split link in a rational homology sphere $M$. Let $d_i$ be the order of $K_i$ in $H_1(M)$, let $\Sigma_i$ be a surface of $M\setminus L$ whose boundary is made of essential parallel curves of $\partial N(K_i)$ and is homologous to $d_iK_i$ in $N(K_i)$. If $n=1$, assume that the ${\mathbb{Q}}/{\mathbb{Z}}$–self-linking number of $K_1$ is zero.
Then $$\sum_{I \subset N} (-1)^{\sharp I} \lambda \left(M_{(K_i;p_i/q_i)_{i \in I}}\sharp \sharp_{j \in N \setminus I} L(p_j,-q_j)\right)
=(-1)^n \left(\prod_{i=1}^n\frac{q_i}{p_i}\right) \lambda^{\prime}(L)$$ where $$\begin{array}{lll}\lambda^{\prime}(L)
&= \frac{\langle \langle I(\Sigma_1) \rangle \rangle_{W_1}}{2d_1^2}+ \frac{1}{12} - \frac{1}{12d_1^2} \; &\mbox{if}\; n=1\\
&=\frac{\langle \langle I(\Sigma_1) \subset M(K_2;1)\rangle \rangle_{W_1}}{2d_1^2} - \frac{\langle \langle I(\Sigma_1) \subset M \rangle \rangle_{W_1}}{2d_1^2}= - \frac{lk(\Sigma_1 \cap \Sigma_2,(\Sigma_1 \cap \Sigma_2)_{\parallel})}{d_1^2d_2^2} \; &\mbox{if}\; n=2\\&= \frac{\langle \Sigma_1,\Sigma_2,\Sigma_3 \rangle^2}{d_1^2d_2^2d_3^2} \; &\mbox{if}\; n=3\\
&=0 \; &\mbox{if}\; n \geq 4\end{array}$$ and, if $n>1$, $$\sum_{I \subset N} (-1)^{\sharp I} \lambda \left(M_{(K_i;p_i/q_i)_{i \in I}}\sharp \sharp_{j \in N \setminus I} L(p_j,-q_j)\right)= \sum_{I \subset N} (-1)^{\sharp I} \lambda \left(M_{(K_i;p_i/q_i)_{i \in I}}\right).$$
This proposition is proved in Section \[secproofcasone\]. Under its hypotheses, we then obviously have the following equalities $$\lambda^{\prime}(K_1 \subset M(K_2,p/q))-\lambda^{\prime}(K_1 \subset M)
=\frac{q}{p}\lambda^{\prime}(K_1,K_2)$$ and $$\lambda^{\prime}\left(K_1 \subset M((K_2;p_2/q_2),(K_3;p_3/q_3))\right)-\lambda^{\prime}\left(K_1 \subset M(K_2;p_2/q_2)\right) -\lambda^{\prime}\left(K_1 \subset M(K_3;p_3/q_3)\right)$$ $$+\lambda^{\prime}(K_1 \subset M) =\frac{q_2q_3}{p_2p_3}\lambda^{\prime}(K_1,K_2,K_3).$$ Then the variation of linking numbers under surgery recalled in Lemma \[lemvarlk\] easily implies the following proposition (see also the proof of Lemma \[lemvarlambdaprime\]).
\[propcasknotvar\] Let $(K_1,K_2,K_3)$ be a rationally algebraically split link in a rational homology sphere $M$. Let $d_i$ be the order of $K_i$ in $H_1(M)$, let $\Sigma_i$ be a surface of $M\setminus L$ whose boundary is made of essential parallel curves of $\partial N(K_i)$ and is homologous to $d_iK_i$ in $N(K_i)$. Then
$$\begin{array}{ll}\lambda^{\prime}(K_1,K_2)& = -\frac{1}{4} \langle \langle \frac{1}{d^2_1} I(\Sigma_1) \; \begin{pspicture}[0.2](-.7,-.2)(1.1,.6)
\psline{-}(.35,0.05)(-.05,.05)
\rput[rt](-.1,.25){$K_2$}
\rput[lt](.4,.25){$K_{2\parallel}$}
\end{pspicture} \rangle \rangle_{W_1}\\
& = -\frac{1}{4} \langle \langle \frac{1}{d^2_2} I(\Sigma_2) \; \begin{pspicture}[0.2](-.7,-.2)(1.1,.6)
\psline{-}(.35,0.05)(-.05,.05)
\rput[rt](-.1,.25){$K_1$}
\rput[lt](.4,.25){$K_{1\parallel}$}
\end{pspicture} \rangle \rangle_{W_1},
\end{array}$$
$$\lambda^{\prime}(K_1,K_2,K_3) = \frac{1}{8d_1^2} \langle \langle I(\Sigma_1) \; \begin{pspicture}[0.2](-.7,-.2)(2.8,.6)
\psline{-}(.35,0.05)(-.05,.05)
\rput[rt](-.1,.25){$K_2$}
\rput[lt](.4,.25){$K_{2\parallel}$}
\psline{-}(2.05,0.05)(1.65,.05)
\rput[rt](1.6,.25){$K_3$}
\rput[lt](2.1,.25){$K_{3\parallel}$}
\end{pspicture} \rangle \rangle_{W_1}.$$
[ ]{}
\[propcasknottwo\] If $K^s$ is a singular knot with one double point, then $$\lambda^{\prime}(K^s)=\ell_{11}(K^s).$$
The easy proof of this well-known proposition is also given in Section \[secproofcasone\].
On the knot surgery formula for the degree $2$ invariant $\lambda_2$
====================================================================
\[secstatewthree\] Consider the degree $2$ invariant $$\lambda_2= W_2 \circ Z_2^c$$ where $W_2\left(\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-}(0,0)(.6,0)(.3,.2)
\psline{*-*}(.6,0)(.3,.5)(.3,.2)
\psline{*-}(.3,.5)(0,0)(.3,.2)
\end{pspicture}\right)=1$ and therefore $W_2\left({\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.1,0)(.2,.2)(.1,.4)
\pscurve(.5,0)(.4,.2)(.5,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}}\right)=2$. The invariant $\lambda_2$ is invariant under orientation change and additive under connected sum.
\[thmw3\] There exists a function $c$ from ${\mathbb{Q}}/{\mathbb{Z}}$ to ${\mathbb{Q}}$ such that $c(0)=0$, $c(q/p)=c(-q/p)$ and the following assertions hold. Let $r=q/p\in {\mathbb{Q}}\setminus \{0\}$, where $p$ and $q$ are coprime integers. Let $K$ be a knot that bounds a Seifert surface $F$ in a rational homology sphere $M$. Let $F^1$ and $F^2$ be two parallel copies of $F$. Then $$\lambda_2(M(K;1/r))-\lambda_2(M)=\lambda_2^{\prime \prime}(K) r^2 + w_3(K)r + C(K;q/p)+ \lambda_2(L(p;-q))$$ where $$\lambda_2^{\prime \prime}(K)= \frac{1}{8} \langle \langle \bigsqcup_{i \in \{1,2\}} (I(F^i)) \;\;\rangle \rangle_{W_2}$$ $$w_3(K \subset M)=-w_3(K \subset (-M))$$ and $C(.;q/p)$ is an invariant of null-homologous knots that only depends on $q/p$ mod ${\mathbb{Z}}$, such that:\
$C(K;0)=0$, and, if $K$ bounds a surface whose $H_1$ vanishes in $H_1(M)$, then $C(.;q/p)=c(q/p)\lambda^{\prime}(K)$.
Furthermore, if $K^s$ is a singular knot with two unlinked double points, then $$w_3(K^s)=-\frac{\ell_{12}(K^s)}{2}\;\;\mbox{and}\;\; C(K^s;q/p)=0.$$
Like all the statements in this section, the above theorem will be proved in Section \[secprooflambdatwo\].
\[propvarwthree\] Let $K^s$ be a singular knot with one double point in a rational homology sphere. Let $K^+$ and $K^-$ be its two desingularisations, and let $K^{\prime}$ and $K^{\prime \prime}$ be the two knots obtained from $K^s$ by smoothing the double point. Assume that $K^{\prime}$ and $K^{\prime \prime}$ are null-homologous, then $$w_3(K^+)-w_3(K^-)=\frac{\lambda^{\prime}(K^{\prime}) + \lambda^{\prime}(K^{\prime \prime})}{2} -\frac{\lambda^{\prime}(K^+) + \lambda^{\prime}(K^-)+lk^2(K^{\prime},K^{\prime \prime})}{4}.$$
Let
(-.1,-.1)(.6,.6) (0,0)(.5,.5) (.25,.25)[$x$]{}
denote a two strand braid with $|x|$ vertical juxtapositions of the motive
(.65,.2)(1.35,.8) (1,.1)(1.15,.25)(1,.4)(.85,.55)(1,.7) (1,.7)(1.15,.55)(1,.4)(.85,.25)(1,.1)
if $x>0$ and $|x|$ vertical juxtapositions of the motive
(.65,.2)(1.35,.8) (1,.1)(.85,.25)(1,.4)(1.15,.55)(1,.7) (1,.7)(.85,.55)(1,.4)(1.15,.25)(1,.1)
if $x<0$.
Let $x$, $y$ and $z$ be three odd numbers. Let $K(x,y,z)$ be the following pretzel knot that bounds a genus one Seifert surface $\Sigma$ whose thickening $H$ coincides with the thickening of the twice punctured disk next to it. $H$ is a genus two handlebody whose boundary is equipped with curves $X$, $Y$ and $Z$ that bound disks in its exterior.
(-.2,-1.5)(3.2,1.5) (2.9,.8)(2.9,1.4)(.1,1.4)(.1,.8) (.5,.8)(.5,1)(1.3,.7)(1.3,.5) (1.7,.5)(1.7,.7)(2.5,1)(2.5,.8) (2.7,1.4)(1.5,1.4) (.7,.925)(.9,.85) (1.9,.775)(2.1,.85) (1.2,-.5)(1.8,.5) (0,-.8)(.6,.8) (2.4,-.8)(3,.8) (.3,0)[$y$]{} (1.5,0)[$z$]{} (2.7,0)[$x$]{} (2.9,-.8)(2.9,-1.4)(.1,-1.4)(.1,-.8) (.5,-.8)(.5,-1)(1.3,-.7)(1.3,-.5) (1.7,-.5)(1.7,-.7)(2.5,-1)(2.5,-.8) (2.7,-1.4)(1.5,-1.4) (.7,-.925)(.9,-.85) (1.9,-.775)(2.1,-.85) (1.5,-1.3)[$K(x,y,z)$]{}
(-.2,-1.5)(3.2,1.5) (0,-1.4)(3,-1.4)(3,1.4)(0,1.4) (.6,-.9)(1.2,-.6)(1.2,.6)(.6,.9) (2.4,-.9)(1.8,-.6)(1.8,.6)(2.4,.9) (1,.95)[$X$]{} (.9,.9)(.4,1.1)(.4,-1.1)(1.4,-.7)(1.4,.7)(.9,.9) (2,.95)[$Y$]{} (2.1,.9)(1.6,.7)(1.6,-.7)(2.6,-1.1)(2.6,1.1)(2.1,.9) (1.6,-1.1)[$Z$]{} (1.5,-1.25)(2.8,-1.25)(2.8,1.25)(.2,1.25)(.2,-1.25)(1.5,-1.25)
(-.2,-1.5)(3.2,1.5) (2.9,.8)(2.9,1.4)(.1,1.4)(.1,.8) (.5,.8)(.5,1)(1.3,.7)(1.3,.5) (1.7,.5)(1.7,.7)(2.5,1)(2.5,.8) (2.7,1.4)(1.5,1.4) (.7,.925)(.9,.85) (1.9,.775)(2.1,.85) (1.7,-.7)(1.7,-.5)(1.7,-.2)(1.5,0)(1.3,.2)(1.3,.5)(1.3,.7) (1.7,.7)(1.7,.5)(1.7,.2)(1.5,0)(1.3,-.2)(1.3,-.5)(1.3,-.7) (.3,0)(.5,.2)(.3,.4)(.1,.6)(.1,.8)(.1,1.4) (.5,1.4)(.5,.8)(.5,.6)(.3,.4)(.1,.2)(.3,0) (.3,-.4)(.5,-.2)(.3,0)(.1,.2)(.3,.4) (.3,.4)(.5,.2)(.3,0)(.1,-.2)(.3,-.4) (.5,-1.4)(.5,-.8)(.5,-.6)(.3,-.4)(.1,-.2)(.3,0) (.3,0)(.5,-.2)(.3,-.4)(.1,-.6)(.1,-.8)(.1,-1.4) (2.5,-1.4)(2.5,-.8)(2.5,-.2)(2.7,0)(2.9,.2)(2.9,.8)(2.9,1.4) (2.5,1.4)(2.5,.8)(2.5,.2)(2.7,0)(2.9,-.2)(2.9,-.8)(2.9,-1.4) (2.9,-.8)(2.9,-1.4)(.1,-1.4)(.1,-.8) (.5,-.8)(.5,-1)(1.3,-.7)(1.3,-.5) (1.7,-.5)(1.7,-.7)(2.5,-1)(2.5,-.8) (2.7,-1.4)(1.5,-1.4) (.7,-.925)(.9,-.85) (1.9,-.775)(2.1,-.85) (1.5,-1.3)[$K(-1,3,1)$]{}
Note that any genus one knot that bounds a genus one surface, whose $H_1$ goes to $0$ in $H_1(M)$, may be written as the image of $K(x,y,z)$ under an embedding $\phi$ of $H$ into $M$ that maps $X$ and $Y$ to $0$ in $H_1(M \setminus \phi(H))$.
\[propgenusone\] Let $\phi$ be an embedding of $H$ in a rational homology sphere such that $\phi(X)$ and $\phi(Y)$ are null homologous in the exterior of $\phi(H)$. Then $$w_3(\phi(K(x,y,z)))=w_3(K(x,y,z)) - \frac{x}{2} \lambda^{\prime}(\phi(X)) - \frac{y}{2}
\lambda^{\prime}(\phi(Y))
- \frac{z}{2} \lambda^{\prime}(\phi(Z))+\frac{3}{2}\lambda^{\prime}(\phi(X),\phi(Y))$$ and $$w_3(K(x,y,z))=\frac{x^2(y+z)+y^2(x+z)+z^2(x+y)}{32} +\frac{xyz}{8} +\frac{x +y +z}{16}$$ where $\lambda^{\prime}(\phi(X))$ and $\lambda^{\prime}(\phi(X),\phi(Y))$ are defined in several equivalent ways in Section \[seccasstate\].
Proof of the Lagrangian-preserving surgery formula
==================================================
\[secprooflag\] In this section, we prove Theorem \[thmflag\] by proving that its formula is equivalent to the formula of [@sumgen] (or [@al] for the case of integral homology spheres). We first rewrite the right-hand side of the formula of Theorem \[thmflag\].
Let $g(i)$ be the genus of $A_i$. Let $(a^i_1,a^i_2,\dots,a^i_{g(i)})$ be a basis of ${{\cal L}}_{A_i}$, and let $z^i_1,\dots,z^i_{g(i)}$ be homology classes of $\partial A_i$, such that $\langle a^i_j,z^i_k \rangle_{\partial A}=\delta_{jk}$. Let $F$ be the set of maps $f$ from $\{1,\dots,2n\} \times \{1,2,3\}$ to ${\mathbb{N}}$ such that $1 \leq f(i,1) < f(i,2) <f(i,3) \leq g(i)$. Let $P$ be the set of pairings $p$ of the disjoint union $G^0$ of the following $2n$ tripods, that pair a univalent vertex of some tripod to a univalent vertex of a different tripod.
(-.2,-.1)(.8,.7) (0.05,.3)(.45,.6) (0.05,.3)(.45,.3) (0.05,.3)(.45,0) (.55,0)[$1$]{} (.55,.3)[$2$]{} (.55,.6)[$3$]{} (-.1,.3)[$i$]{}
Let $p \in P$. The half-edges of $G^0_p$ are naturally labeled in $\{1,\dots,2n\} \times \{1,2,3\}$. Assume that some $(f \in F)$ is given. With a half-edge of $G^0_p$ labeled by $(i,j)$ that belongs to the tripod $i$, associate the curve $z_{f(i,j)}^i$ of $\partial A_i$. Then to an edge of $G^0_p$, associate the linking number of the curves associated to its two half-edges, and define $lk(p;f)$ as the product over the edges of these linking numbers. Set $$c(p;f)=lk(p;f) \prod_{i=1}^{2n}{{\cal I}}_{A_iA_i^{\prime}}(a_{f(i,1)}^i,a_{f(i,2)}^i,a_{f(i,3)}^i),$$ and $$c(p)=\sum_{f \in F}c(p;f).$$
Then $$\langle \langle \bigsqcup_{i \in \{1,\dots,2n\}} T({{\cal I}}_{A_iA_i^{\prime}}) \rangle \rangle_n= \sum_{p\in P}c(p)[G^0_p].$$ Let $D$ be the set of unoriented Jacobi diagrams of degree $n$. Consider a Jacobi diagram $\Gamma$ of $D$. Let $P(\Gamma)$ be the set of the pairings $p$ of $P$ such that $G^0_p$ is isomorphic to $\Gamma$ as an unoriented Jacobi diagram. Then $$\langle \langle \bigsqcup_{i \in \{1,\dots,2n\}} T({{\cal I}}_{A_iA_i^{\prime}}) \rangle \rangle_n= \sum_{\Gamma \in D} \sum_{p\in P(\Gamma)}c(p)[G^0_p].$$
Fix $\Gamma$ in $D$. Let $B(\Gamma)$ be the set of bijections $b$ from the set $H(\Gamma)$ of half-edges of $\Gamma$ to $\{1,\dots, 2n\} \times \{1,2,3\}$ that map any half-edge $c$ of a vertex $v(c)$ to three images with the same first coordinate $b_1(c)=b_1(v(c))$. An element $b$ of $B(\Gamma)$ determines a pairing $p(b)$ of $P(\Gamma)$, and the number of elements of $B(\Gamma)$ that determine the same pairing is the number of automorphisms of $\Gamma$.
$$\sum_{p\in P(\Gamma)}c(p)[G^0_p]= \sum_{b \in B(\Gamma)}\frac{c(p(b))}{\sharp \mbox{Aut}(\Gamma)}[G^0_{p(b)}]= \sum_{b \in B(\Gamma), f \in F}\frac{c(p(b);f)}{\sharp \mbox{Aut}(\Gamma)}[G^0_{p(b)}].$$ Let $G(\Gamma)$ be the set of injections $g$ from the set $H(\Gamma)$ of half-edges of $\Gamma$ to $$\{(i,j) \in \{1,\dots, 2n\} \times {\mathbb{N}}; 1 \leq j \leq g(i)\}$$ that map the three half-edges of a vertex to three images with the same first coordinate, and that induce a bijection from $V(\Gamma)$ to $\{1,\dots, 2n\}$. An injection $g$ of $G(\Gamma)$ provides a natural bijection $b(g)$ of $B(\Gamma)$ and a map $f(g)$ of $F$ such that $g(c)=(b_1(c),f(g) \circ b(g)(c))$. Furthermore, such a $g$ orders the three half-edges of a vertex, and hence provides an orientation $o(g)$ of $\Gamma$. $$\sum_{p\in P(\Gamma)}c(p)[G^0_p]=\sum_{g \in G(\Gamma)}\frac{c(p(b(g));f(g))}{\sharp \mbox{Aut}(\Gamma)}[(\Gamma,o(g))].$$ Let $g \in G(\Gamma)$, its first coordinate $b_1(g)$ induces a bijection from $V(\Gamma)$ to $\{1,\dots, 2n\}$. Number the three half-edges of any vertex $w$ of $\Gamma$ with a bijection $b(w) \colon v^{-1}(w) \rightarrow \{1,2,3\}$, arbitrarily. This orients $\Gamma$ and equips each injection $g \in G(\Gamma)$ with a sign that is $+1$ if $o(g)$ coincides with this orientation of $\Gamma$ (except for an even number of vertices) and $(-1)$ otherwise. Furthermore, $g$ provides summands of $${{\cal I}}(A_i,A_i^{\prime})=\sum_{g_i \colon\{1,2,3\} \rightarrow \{1,2, \dots, g(i)\}}{{\cal I}}_{A_iA_i^{\prime}}(a_{g_i(1)}^i,a_{g_i(2)}^i,a_{g_i(3)}^i)z_{g_i(1)}^i \otimes z_{g_i(2)}^i \otimes z_{g_i(3)}^i$$ where $g(b(b_1(g)^{-1}(i))^{-1}(j))=(i,g_i(j))$. Note that the sign of an injection $g$ is $+1$ if the number of vertices $b_1(g)^{-1}(i)$ where the cyclic order induced by $g_i$ does not coincide with the cyclic order induced by $b(b_1(g)^{-1}(i))$ is even, and $(-1)$, otherwise. This shows that for any bijection $\sigma$ from $V(\Gamma)$ to $\{1,\dots, 2n\}$, $$\sum_{g \in G(\Gamma);b_1(g)=\sigma}{c(p(b(g));f(g))}[(\Gamma,o(g))]=lk((A_i,A_i^{\prime})_{i=1, \dots, 2n}; \Gamma ;\sigma)[\Gamma]$$ with the notation of [@al] or [@sumgen]. [ ]{}
A direct proof of the formula for boundary links
================================================
\[secproofboun\]
A Lagrangian-preserving surgery associated to a Seifert surface {#sublagboun}
---------------------------------------------------------------
Let $\Sigma$ be an oriented Seifert surface of a knot $K$ in a manifold $M$. Consider an annular neighborhood $[-3,0] \times K$ of $(\{0\} \times K)=K =\partial \Sigma$ in $\Sigma$, a small disk $D$ inside $]-2,-1[ \times K$, and an open disk $d$ in the interior of $D$. Let $F=\Sigma \setminus d$. Let $h_{F}$ be the composition of the two left-handed Dehn twists on $F$ along $c=\partial D$ and $K_2=\{-2\} \times K$ with the right-handed one along $K_1=\{-1\} \times K$.
(-.5,-.3)(6.3,3.3) (4.2,.8)(5.8,1)(5.8,2)(4.2,2.2)(1.5,3)(0,1.5)(1.5,0)(4.2,.8)(5.8,1)(5.8,2) (.2,1.5)(1.5,.2)(4.2,1)(5.6,1.1)(5.6,1.9)(4.2,2)(1.5,2.8)(.2,1.5)(1.5,.2)(5.6,1) (2,.5)(3.6,1.5)(2,2.3)(.4,1.5)(2,.7)(3.6,1.5)(2,2.5) (4.9,1.5)[.2]{} (4.9,1.5)[d]{} (4.9,1.5)[.4]{}[0]{}[300]{} (4.9,1.5)[.4]{}[-60]{}[0]{} (1,1.2)(1.1,1.35)(1,1.6)(1.2,1.9)(1.4,1.6)(1.3,1.35)(1.4,1.2) (1.2,1.7)(1.25,1.6)(1.2,1.5) (1.25,1.8)(1.2,1.7)(1.15,1.6)(1.2,1.5)(1.25,1.4) (2.6,1.2)(2.7,1.35)(2.6,1.6)(2.8,1.9)(3,1.6)(2.9,1.35)(3,1.2) (2.8,1.7)(2.85,1.6)(2.8,1.5) (2.85,1.8)(2.8,1.7)(2.75,1.6)(2.8,1.5)(2.85,1.4) (2,1.5)[…]{} (3.65,1.5)[$K_2$]{} (5.35,1.5)[$c$]{} (5.85,.95)[$K$]{} (1.5,.3)[$K_1$]{} (2,2.5)[$\Sigma$]{}
See $F$ as $F \times \{0\}$ in the boundary of the handlebody $A_{F}=F \times [-1,0]$ of $M$. Extend $h_F$ to a homeomorphism $h_A$ of $\partial A_{F}$ by defining it as the identity outside $F \times \{0\}$.
Let $A_{F}^{\prime}$ be a copy of $A_{F}$. Identify $\partial A^{\prime}_F$ with $\partial A_{F}$ with $$h_{A} \colon \partial A^{\prime}_F \rightarrow \partial A_{F}.$$
Define the [*surgery associated to $\Sigma$*]{} as the surgery associated with $(A_{F}, A_{F}^{\prime})$ (or $(A_{F}, A_{F}^{\prime}; h_{A})$). If $\iota$ denotes the embedding from $\partial A_{F}$ to $M$. This surgery replaces $$M =\left( M \setminus \mbox{Int}(A_{F}) \right) \cup_{\iota} A_{F}$$ by $$M_F=\left(M \setminus \mbox{Int}(A_{F}) \right) \cup_{\iota h_{A}} A_{F}^{\prime}.$$
With the notation above, the surgery $(A_{F}, A_{F}^{\prime})$ associated to $\Sigma$ is a Lagrangian-preserving surgery with the following properties. There is a homeomorphism from $M_F$ to $M$
- that extends the identity of $$M \setminus \left([-3,0] \times K \times [-1,0] \right),$$
- that transforms a curve going through $d \times [-1,0]$ by a band sum with $K$,
- that transforms a $0$-framed meridian $m$ of $K$ going through $d \times [-1,0]$ into a $0$-framed copy of $K$ isotopic to the framed curve $h_A^{-1}(m)$ of the following figure.
(-.2,-1)(6.2,3.1) (3.85,.8)(5.55,1)(6.05,2)(4.55,2.2)(2.25,3)(0,1.5)(.75,0)(3.85,.8)(5.55,1)(6.05,2) (.2,1.5)(.85,.2)(3.95,1)(5.4,1.1)(5.8,1.9)(4.45,2)(2.15,2.8)(.2,1.5)(.85,.2)(3.95,1) (1,.7)(3.6,1.5)(3,2.3)(.4,1.5)(1,.7)(3.6,1.5)(3,2.3) (5.1,1.5)(5,1.7)(4.7,1.5)(4.8,1.3) (4.9,1.5)[d]{} (4.7,1.1)(5.3,1.5)(5.1,1.9)(4.5,1.5)(4.7,1.1) (5.1,1.9)(4.5,1.5)(4.7,1.1)(5.3,1.5)(5.1,1.9) (1,1.2)(1.1,1.35)(1,1.6)(1.2,1.9)(1.4,1.6)(1.3,1.35)(1.4,1.2) (1.2,1.7)(1.25,1.6)(1.2,1.5) (1.25,1.8)(1.2,1.7)(1.15,1.6)(1.2,1.5)(1.25,1.4) (2.6,1.2)(2.7,1.35)(2.6,1.6)(2.8,1.9)(3,1.6)(2.9,1.35)(3,1.2) (2.8,1.7)(2.85,1.6)(2.8,1.5) (2.85,1.8)(2.8,1.7)(2.75,1.6)(2.8,1.5)(2.85,1.4) (2,1.5)[…]{} (3.65,1.5)[$K_2$]{} (5.35,1.5)[$c$]{} (.9,.3)[$K_1$]{} (2.5,2.5)[$\Sigma$]{} (6.1,1.95)(6.1,1) (-.05,1.45)(-.05,.5) (5.1,.5)(5,.7)(4.7,.5)(4.8,.3) (4.7,1.5)(4.7,.5) (2.25,2)(-.05,.5)(.75,-1)(3.85,-.2)(5.55,0)(6.1,1)(4.55,1.2) (2.15,-.5)[$A_F$]{} (5.1,1.5)(5.1,.5) (5.1,1.5)(5.55,1)(5.55,.5) (5.55,.5)(5.55,0)(5.1,.5) (5.65,.6)[$m$]{}
(-1,-.8)(6.2,3.1) (3.85,.8)(5.55,1)(6.05,2)(4.55,2.2)(2.25,3)(0,1.5)(.75,0)(3.85,.8)(5.55,1)(6.05,2) (1,.7)(3.6,1.5)(3,2.3)(.4,1.5)(1,.7)(3.6,1.5)(3,2.3) (5.1,1.5)(5,1.7)(4.7,1.5)(4.8,1.3) (4.9,1.5)[d]{} (1,1.2)(1.1,1.35)(1,1.6)(1.2,1.9)(1.4,1.6)(1.3,1.35)(1.4,1.2) (1.2,1.7)(1.25,1.6)(1.2,1.5) (1.25,1.8)(1.2,1.7)(1.15,1.6)(1.2,1.5)(1.25,1.4) (2.6,1.2)(2.7,1.35)(2.6,1.6)(2.8,1.9)(3,1.6)(2.9,1.35)(3,1.2) (2.8,1.7)(2.85,1.6)(2.8,1.5) (2.85,1.8)(2.8,1.7)(2.75,1.6)(2.8,1.5)(2.85,1.4) (2,1.5)[…]{} (3.65,1.5)[$K_2$]{} (2.5,2.5)[$\Sigma$]{} (6.1,1.9)(6.1,1) (-.05,1.4)(-.05,.5) (5.1,.5)(5,.7)(4.7,.5)(4.8,.3) (4.7,1.5)(4.7,.5) (2.25,2)(-.05,.5)(.75,-1)(3.85,-.2)(5.55,0)(6.1,1)(4.55,1.2) (2.15,-.5)[$A_F$]{} (5.1,1.5)(5.1,.5) (5.1,1.5)(5.25,1.4)(4.7,1.2)(4.5,1.5)(5.1,1.9)(5.8,1.9)(4.45,2)(2.15,2.8)(.2,1.5)(.85,.2)(3.95,1)(5.4,1)(5.55,1) (5.55,1)(5.55,0)(5.1,.5) (4.8,.15)[$h_A^{-1}(m)$]{}
[[Proof: ]{}]{}Observe that $h_F$ extends to $\Sigma \times [-1,0]$ as $$\begin{array}{llll} h \colon &\Sigma \times [-1,0] & \rightarrow & \Sigma \times [-1,0]\\
&(\sigma,t) & \mapsto & h(\sigma,t)=(h_t(\sigma),t)
\end{array}$$ where $h_0$ is the extension of $h_F$ by the identity on $d$ that is isotopic to the identity,\
$h_{-1}$ is the identity of $\Sigma$,\
$h_t$ coincides with the identity outside $[-5/2,-1/2]\times K(S^1)$,\
and $h_t$ is defined as follows on $[-5/2,-1/2]\times K(S^1)$.\
$\bullet$ When $t \leq -1/2$, then $h_t$ describes the following isotopy between $(h_{-1}=\mbox{identity})$ and the composition $h_{-1/2}$ of the left-handed Dehn twist along $K_2$ located on $[-5/2,-2]\times K(S^1)$ and the right-handed Dehn twist along $K_1$ located on $[-1,-1/2]\times K(S^1)$, $$\begin{array}{lll}
h_t(u,K(z)) & = \left(u,K\left(z \exp\left(i(2t+2)(4\pi (u+5/2))\right)\right)\right) & \mbox{if} \; u \leq -2 \\
h_t(u,K(z)) & = \left(u,K\left(z \exp\left(i(2t+2)(2\pi)\right)\right)\right) & \mbox{if} \; -2 \leq u \leq -1 \\
h_t(u,K(z)) & = \left(u,K\left(z \exp\left(-i(2t+2)(4\pi (u+1/2))\right)\right)\right) & \mbox{if} \; u \geq -1.
\end{array}$$ $\bullet$ When $t \geq -1/2$, then $h_t$ coincides with $h_{-1/2}$ outside the disk $D$ whose elements will be written as $D(z\in {\mathbb{C}})$, with $|z|\leq 1$. The elements of $d$ will be the $D(z)$ for $|z|<1/2$. On $D$, $h_t$ will describe the isotopy between the identity and the composition $h_{0}$ of the left-handed Dehn twist along $\partial D$ located on $\{D(z);1/2 \leq |z| \leq 1\}$ and a negative twist of $d$. $$\begin{array}{lll}
h_t(u,K(z)) & = \left(u,K\left(z \exp\left(i(4\pi (u+5/2))\right)\right)\right) & \mbox{if} \; u \leq -2 \\
h_t(u,K(z)) & = \left(u,K(z)\right) & \mbox{if} \; -2 \leq u \leq -1, (u,K(z)) \notin D \\
h_t(u,K(z)) & = \left(u,K\left(z \exp\left(-i(4\pi (u+1/2))\right)\right)\right) & \mbox{if} \; u \geq -1 \\
h_t(z \in D) & = z \exp\left(i\pi(2t+1)4(|z|-1)\right) & \mbox{if} \; |z| \geq 1/2 \\
h_t(z \in D) & = z \exp\left(-2i\pi(2t+1)\right) & \mbox{if} \; |z| \leq 1/2.\\
\end{array}$$ Now, $M_F$ is naturally homeomorphic to $$\left(M \setminus \mbox{Int}(\Sigma \times [-1,0]) \right) \cup_{h_{|\partial(\Sigma \times [-1,0])}} (\Sigma \times [-1,0])$$ that maps to $M$ by the identity outside $\Sigma \times [-1,0]$ and by $h$ on $\Sigma \times [-1,0]$, homeomorphically. Therefore, we indeed have a homeomorphism from $M_F$ to $M$ that is the identity outside $[-3,0]\times K \times [-1,0]$ and that maps $d \times [-1,0]$ to a cylinder that runs along $K$ before being negatively twisted. In particular, looking at the action of the homeomorphism on a framed arc $x \times [-1,0]$ where $x$ is on the boundary of $d$ shows that the meridian $m$ with its framing induced by the boundary of $A_F$ is mapped to a curve isotopic to $h_A^{-1}(m)$ in a tubular neighborhood of $K$ with the framing induced by the boundary of $A_F$.
Now, $H_1(\partial A_{F})$ is generated by the generators of $H_1(\Sigma)\times \{0\}$, the generators of $H_1(\Sigma)\times \{-1\}$, and the homology classes of $c=\partial D$ and $m$. Among them, only the class of $m$ could be affected by $h_A$, and it is not. Therefore $h_A$ acts trivially on $H_1(\partial A_{F})$, and the defined surgery is an $LP$–surgery. [ ]{}
Let $F \times [-1,2]$ be an extension of the previous neighborhood of $F$, and let $B_F=F \times [1,2]$. Define the homeomorphism $h_B$ of $\partial B_F$ as the identity anywhere except on $F \times \{1\}$ where it coincides with the homeomorphism $h_F$ of $F$ with the obvious identification.
Let $B_{F}^{\prime}$ be a copy of $B_{F}$. Identify $\partial B^{\prime}_{F}$ with $\partial B_{F}$ with $$h_B \colon \partial B^{\prime}_{F} \rightarrow \partial B_{F}.$$
Define the [*inverse surgery associated to $\Sigma$*]{} as the surgery associated with $(B_{F}, B_{F}^{\prime})$ (or $(B_{F}, B_{F}^{\prime}; h_B)$). Note that the previous study can be used for this surgery by using the central symmetry of $[-1,2]$.
Then, we have the following obvious lemma that justifies the terminology.
With the notation above, performing the two surgeries $(B_{F}, B_{F}^{\prime})$ and $(A_{F}, A_{F}^{\prime})$ affects neither $M$ nor the curves in the complement of $F \times [-1,2]$, while performing one of them changes a $0$-framed meridian of $K$ going through $d \times [-1,2]$ into a $0$-framed copy of $\pm K$.
[ ]{}
\[lemtriplag\] Let $(x_i, y_i)_{i=1, \dots, g}$ be a symplectic basis of $\Sigma$, then the tripod combination $T({{\cal I}}_{A_FA_F^{\prime}})$ associated to the surgery $(A_{F}, A_{F}^{\prime})$ is $$T({{\cal I}}_{A_FA_F^{\prime}})=-\sum_{i=1}^g
\begin{pspicture}[0.4](-.05,-.1)(.9,.7)
\psline{-}(0.05,.3)(.45,.6)
\psline{*-}(0.05,.3)(.45,.3)
\psline{-}(0.05,.3)(.45,0)
\rput[l](.55,0){$c$}
\rput[l](.55,.3){$x_i$}
\rput[l](.55,.6){$y_i$}
\end{pspicture}.$$
For a curve $c$ of $F$, let $c^+$ denote $c \times \{1\}$. The tripod combination $T({{\cal I}}_{B_FB_F^{\prime}})$ associated to the surgery $(B_{F}, B_{F}^{\prime})$ is $$T({{\cal I}}_{B_FB_F^{\prime}})=\sum_{i=1}^g
\begin{pspicture}[0.4](-.05,-.1)(1.45,.7)
\psline{-}(0.05,.3)(.45,.6)
\psline{*-}(0.05,.3)(.85,.3)
\psline{-}(0.05,.3)(.45,0)
\rput[l](.55,0){$c^+$}
\rput[l](.95,.3){$x_i^+$}
\rput[l](.55,.6){$y_i^+$}
\end{pspicture}.$$
[[Proof: ]{}]{}For a curve $c$ of $F$, $c^-$ denotes $c \times \{-1\}$. Use the basis $\left(m,(x_i - x_i^-,y_i-y_i^-)_{i=1, \dots, g}\right)$ of the Lagrangian of $A_F$ to compute the intersection form of $(A_F \cup -A^{\prime}_F)$. Its dual basis is $\left(c,(y_i, - x_i)_{i=1, \dots, g}\right)$. Note that the only curve of the Lagrangian basis that is modified by $h_A$ is $m$, and that $h_A(m)=mK_2^{-1}$. The isomorphism $\partial_{MV}^{-1}$ from ${{\cal L}}_{A_F}$ to $H_2(A_F \cup -A^{\prime}_F)$ satisfies $$\begin{array}{ll}
\partial_{MV}^{-1}(x_i - x_i^-) &= S(x_i)=-(x_i \times [-1,0]) \cup (x_i \times [-1,0] \subset A^{\prime}_F)\\
\partial_{MV}^{-1}(y_i - y_i^-) &=S(y_i)=-(y_i \times [-1,0]) \cup (y_i \times [-1,0] \subset A^{\prime}_F) \\
\partial_{MV}^{-1}(m) &=S_A(m)=D_m -(\Sigma \setminus (]-2,0] \times K))\cup (-D_m \subset A^{\prime}_F)
\end{array}.$$
Since $x_i$ intersects only $y_i$, $S(x_i)$ intersects only $S(y_i)$ and $S_A(m)$. The algebraic intersection of $S(x_i)$, $S(y_i)$ and $S_A(m)$ is $-1$.
For the surgery $(B_{F}, B_{F}^{\prime})$, $S_B(m)=D_m + \Sigma \setminus (]-2,0] \times K) \cup (-D_m \subset B^{\prime}_F)$, and the algebraic intersection of $S(x_i)$, $S(y_i)$ and $S_B(m)$ is $1$. [ ]{}
Proof of Theorem \[thmfboun\]
-----------------------------
For this proof, I could also have used the strategy of Section \[secprooffas\]. But I prefer this self-contained proof.
First recall the following easy lemma that will be used several times.
\[lemvarlk\] The variation of the linking number of two knots $J$ and $K$ after a $p/q$-surgery on a knot $V$ in a rational homology sphere $M$ is given by the following formula. $$lk_{M_{(V;p/q)}}(J,K)=lk_M(J,K)-\frac{q}{p}lk_M(V,J)lk_M(V,K).$$
[ ]{}
Let $(K_1,K_2, \dots, K_n)$ be a link where all the $K_i$ bound disjoint oriented surfaces $\Sigma^i$. Consider an embedding of $\coprod_{i=1}^r \Sigma^i \times [-1,2]$. Let $N=\{1,2, \dots, n\}$. For $i\in N$, associate surfaces $F^i= \Sigma^i \setminus d^i$ and $LP$–surgeries $(A_i,A_i^{\prime})=(A_{F^i}, A_{F^i}^{\prime})$ and $(B_i,B_i^{\prime})=(B_{F^i}, B_{F^i}^{\prime})$ as in Subsection \[sublagboun\]. Let $U_i$ be a meridian of $K_i$ going through $d^i \times [-1,2]$, so that performing one of the two surgeries transforms $U_i$ into $\pm K_i$ and performing both or none of them leaves $U_i$ unchanged. Then $$\begin{array}{ll}[M;(K_i;p_i/q_i)]&=M_{(U_i;p_i/q_i)}-M_{(K_i;p_i/q_i)}\\
&=\frac12[M_{(U_i;p_i/q_i)};(A_i,A_i^{\prime}),(B_i,B_i^{\prime})].\end{array}$$ More generally, for $J \subset \{(A_i,A_i^{\prime}),(B_i,B_i^{\prime})\}_{i=1,\dots,n}$, $$(M_{(U_i;p_i/q_i)_{i \in N}})_J=M_{(K_i;p_i/q_i)_{i \in I(J)}} \sharp \sharp_{j \notin I(J)} L(p_j,-q_j)=M_{I(J)}$$ where $I(J)$ is the set of elements $i$ of $N$ such that $\sharp \left(J \cap \{(A_i,A_i^{\prime}),(B_i,B_i^{\prime})\}\right)$ is one. Note that $(-1)^{\sharp J}= (-1)^{\sharp I(J)}$ and that for any subset $I$ of $N$ there are $2^n$ subsets $J$ of the set of LP-surgeries such that $I(J)=I$. Thus $$[M;(K_i;p_i/q_i)_{i \in N}]=\frac1{2^n}[M_{(U_i;p_i/q_i)_{i \in N}};(A_i,A_i^{\prime})_{i \in N},(B_i,B_i^{\prime})_{i \in N}].$$ In particular, we can apply Theorem \[thmflag\] to compute $Z_n([M;(K_i;p_i/q_i)_{i \in N}])$.
According to Lemma \[lemtriplag\], the tripods associated to the surgery $(A_{F^i}, A_{F^i}^{\prime})$ and to the surgery $(B_{F^i}, B_{F^i}^{\prime})$ are $-\sum_{i=1}^{g^i}
\begin{pspicture}[0.4](-.05,-.1)(1.3,.8)
\psline{-}(0.05,.3)(.45,.6)
\psline{*-}(0.05,.3)(.85,.3)
\psline{-}(0.05,.3)(.45,0)
\rput[l](.55,0){$c^i$}
\rput[l](.95,.3){$x_j^i$}
\rput[l](.55,.6){$y_j^i$}
\end{pspicture}$ and $\sum_{i=1}^{g^i}
\begin{pspicture}[0.4](-.05,-.1)(1.45,.8)
\psline{-}(0.05,.3)(.45,.6)
\psline{*-}(0.05,.3)(.85,.3)
\psline{-}(0.05,.3)(.45,0)
\rput[l](.55,0){$c^{i+}$}
\rput[l](.95,.3){$x_j^{i+}$}
\rput[l](.55,.6){$y_j^{i+}$}
\end{pspicture}$, respectively. The only curve that links $c^i$ algebraically in $M_{(U_i;p_i/q_i)_{i\in N}}$ among those appearing in all the tripods is $c^{i+}$ with a linking number $-q_i/p_i$. Therefore, these two must be paired together with this coefficient. Theorem \[thmfboun\] follows when $r=n$. The case $r>n$ can be either deduced from the case $r=n$ or proved directly, it is easy. [ ]{}
Some clasper calculus
=====================
\[secclasper\]
The proofs of Theorems \[thmfas\] and \[thmfasmu\] will be given in Section \[secprooffas\]. They will rely on the current section, where we recall some known clasper calculus and where we show how to present algebraically split links $L=(K_i)_{i=1,\dots,n}$ by claspers so that the associated Seifert surfaces $\Sigma_i$ of the components $K_i$ in $M \setminus \left(\cup_{j \neq i}K_j \right) $ have minimal triple intersection, namely so that for any triple $(K_i,K_j,K_k)$ of components of $L$, the geometric triple intersection of the transverse surfaces $\Sigma_i$, $\Sigma_j$ and $\Sigma_k$ is made of $|\mu(K_i,K_j,K_k)|$ points. (This shows Proposition \[proprealmil\] that will be a direct corollary of Lemma \[lemintclas\] and Proposition \[propclaspmu\].)
Two ways of seeing surgeries on $Y$-graphs {#subsectwoways}
------------------------------------------
Let $\Lambda$ be the graph embedded in the surface $\Sigma(\Lambda)$ shown below. In the $3$–handlebody $(N=\Sigma(\Lambda) \times [-1,1])$, the edges of $\Lambda$ are framed by a vector field normal to $\Sigma(\Lambda)=\Sigma(\Lambda) \times\{0\}$. $\Sigma(\Lambda)$ is called a [*framing surface*]{} for $\Lambda$.
(-2,0)(4,2) (0,1.95)(-.32,1.82)(-.45,1.5)(.05,.8)(.05,.5)(.5,.05)(.95,.5)(.95,.8)(1.45,1.5)(1.32,1.82)(1,1.95)(.5,1.75) (.5,.5)[.15]{} (1,1.5)[.15]{} (0,1.5)[.15]{} (.5,.5)[.3]{} (1,1.5)[.3]{} (0,1.5)[.3]{} (.5,1)(.5,.8) (.21,1.29)(.5,1)(.79,1.29)
A [*$Y$-graph*]{} in $M$ is the isotopy class of an embedding $\phi$ of $N$ (or $\Sigma(\Lambda)$) into $M$. Such an isotopy class is determined by the framed image of the framed unoriented graph $\Lambda $ under $\phi$. A [*leaf*]{} of a $Y$-graph $\phi$ is the image under $\phi$ of a simple loop of our graph $\Lambda$. An [*edge*]{} of $\phi$ is an edge of $\phi(\Lambda)$ that is not a leaf. With this terminology, a $Y$-graph has three edges and three leaves.
(-2,0)(4,1.9) (.5,.4)[.3]{} (1,1.5)[.3]{} (0,1.5)[.3]{} (.5,1)(.5,.7) (.21,1.29)(.5,1)(.79,1.29) (.55,.85)[edge]{} (1.5,1.5)[leaf]{}
The surgery on such a $Y$-graph can be defined in several equivalent ways.
Originally, it was defined by Matveev in [@matv] and named [*Borromeo transformation*]{} as the effect of the surgery on the following $6$-component framed link in the framed neighborhood of the $Y$-graph.
(-2,-.2)(4,2.2) (-.2,2.25)(-.52,2.12)(-.65,1.8)(-.05,.85)(-.05,.4)(.5,-.15)(1.05,.4)(1.05,.85)(1.65,1.8)(1.52,2.12)(1.2,2.25)(.5,2) (.5,.4)[.15]{} (1.1,1.7)[.15]{} (-.1,1.7)[.15]{} (.5,.4)[.35]{}[90]{}[0]{} (1.1,1.7)[.35]{}[-145]{}[180]{} (-.1,1.7)[.35]{}[-30]{}[180]{} (.5,1)[.2]{}[0]{}[180]{} (.5,1.2)(.3,1)(.5,.65)(.7,1)(.5,1.2) (.5,1.45)(.45,1.125)(.6,1)(.95,1.55) (.5,.4)[.35]{}[0]{}[90]{} (1.1,1.7)[.35]{}[180]{}[225]{} (.5,1.45)(.55,1.125)(.3,1)(.05,1.55) (-.1,1.7)[.35]{}[180]{}[-30]{} (.5,1)[.2]{}[60]{}[90]{} (.5,1)[.2]{}[150]{}[180]{} (-.1,1.7)[.35]{}[170]{}[190]{} (1.1,1.7)[.35]{}[175]{}[190]{} (.5,1.2)(.3,1)(.5,.65)(.7,1) (.5,1.2)(.3,1)(.5,.65)(.7,1)
The framing of the link is induced by the framing of the surface.
We shall prove the following proposition.
\[propborlag\] The above surgery is equivalent to the surgery $(A_F,A_F^{\prime})$ associated to the following subsurface $F$ of $\Sigma(\Lambda) \times [-1,1]$, with respect to the notation of Subsection \[sublagboun\].
(-.1,-1)(3.7,3) (0,2.7)(0,.7)(.6,.2)(1.7,-.8)(2.8,.2)(3.4,.7)(3.4,2.7)(1.7,2.9) (1.7,.4)(.8,1.5) (1.7,.4)(2.6,1.5) (1.7,.4)(1.7,.2) (2.6,1.9)[.4]{} (.8,1.9)[.4]{} (2.6,1.9)[.2]{} (.8,1.9)[.2]{} (1.7,-.1)[.2]{} (1.7,-.1)[.3]{}
(-.3,-1)(3.7,3) (1.1,1.6)[1.1]{}[0]{}[180]{} (1.1,1.6)[.7]{}[0]{}[180]{} (1.1,1.6)[.9]{} (2.3,1.6)[1.1]{}[0]{}[180]{} (2.3,1.6)[.7]{}[0]{}[180]{} (2.3,1.6)[.9]{} (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (1.1,2.7)(0,1.6)(.33,.83)(.9,.4)(1.7,-.6)(2.5,.4)(3.07,.83)(3.4,1.6)(2.3,2.7) (0,2.7)(0,.7)(.6,.2)(1.7,-.8)(2.8,.2)(3.4,.7)(3.4,2.7)(1.7,2.9) (1.7,.4)(1.1,.7) (1.7,.4)(2.3,.7) (1.7,.4)(1.7,.2) (2.6,1.9)[.2]{} (.8,1.9)[.2]{} (1.7,-.1)[.2]{} (1.7,-.1)[.3]{} (2.5,1.45)[$F$]{}
(-.3,-1)(3.5,3) (1.1,1.6)[1.1]{}[0]{}[180]{} (1.1,1.6)[.7]{}[0]{}[180]{} (1.1,1.6)[1]{}[-5]{}[185]{} (1.1,1.6)[.8]{}[-6]{}[186]{} (1.1,1.6)[.9]{} (2.3,1.6)[1.1]{}[0]{}[180]{} (2.3,1.6)[.7]{}[0]{}[180]{} (2.3,1.6)[1]{}[-5]{}[185]{} (2.3,1.6)[.8]{}[-6]{}[186]{} (2.3,1.6)[.9]{} (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (.3,1.5)(1.3,1.5) (1.5,1.5)(1.9,1.5) (2.6,1.5)(2.1,1.5) (3.1,1.5)(2.6,1.5) (2.5,1.45)[$K_2$]{} (1.1,2.7)(0,1.6)(.33,.83)(.9,.4)(1.5,-.6)(1.9,-.6)(2.5,.4)(3.07,.83)(3.4,1.6)(2.3,2.7) (1.1,2.6)(.1,1.6)(.9,.6)(1.7,.6)(2.5,.6)(3.3,1.6)(2.3,2.6) (0,2.7)(0,.7)(.6,.2)(1.7,-.8)(2.8,.2)(3.4,.7)(3.4,2.7)(1.7,2.9) (1.7,.4)(1.1,.7) (1.7,.4)(2.3,.7) (1.7,.4)(1.7,.2) (2.6,1.9)[.2]{} (.8,1.9)[.2]{} (1.7,-.1)[.2]{} (1.7,-.1)[.3]{} (1.7,-.1)[.4]{}[60]{}[360]{} (1.7,-.1)[.4]{}[0]{}[60]{} (2.05,.25)[$c$]{}
Let $G\subset M$ be a A leaf $l$ of a $Y$-component of $G$ is [*trivial*]{} if $l$ bounds an embedded disc that induces the framing of $l$, in $M\setminus G$. It is easy to see that with both definitions, performing the surgery on a $Y$-graph with a trivial leaf does not change the ambient manifold. More precisely, the following lemma is proved in [@ggp], for the first definition.
\[lemysur\] Let $M$ be an oriented $3$–manifold (with possible boundary). Let $G$ be a $Y$-graph in $M$ with a trivial leaf that bounds a disc $D$ in $M\setminus G$. Then
- for any framed graph $T_0$ in $M\setminus G$ that does not meet $D$, the pair $(M_G,T_0)$ is diffeomorphic to the pair $(M,T_0)$.
- If $T$ is a framed graph in $M\setminus G$ that meets ${\mbox{\rm Int}}(D)$ at exactly one point, then the pair $(M_G,T)$ is diffeomorphic to the pair $(M,T_G)$, where $T_G$ is the framed graph in $M$ below.
(-.2,.2)(2.4,2) (0,.5)(2,.5) (1,.5)[.2]{}[-160]{}[160]{} (.5,1.6)[.3]{}[140]{}[95]{} (.5,1.6)[.3]{}[95]{}[140]{} (1.5,1.6)[.3]{}[85]{}[40]{} (1.5,1.6)[.3]{}[40]{}[85]{} (.5,1.3)(1,1)(1.5,1.3) (1,1)(1,.7) (1.1,.9)[$G$]{} (-.05,.55)[$T$]{}
$\longrightarrow$
(-2,.2)(2.4,2) (1.1,1.6)[.66]{}[0]{}[95]{} (1.1,1.6)[.66]{}[95]{}[140]{} (1.1,1.6)[.66]{}[140]{}[180]{} (1.1,1.6)[.42]{}[0]{}[95]{} (1.1,1.6)[.42]{}[95]{}[140]{} (1.1,1.6)[.42]{}[140]{}[180]{} (2.3,1.6)[.66]{}[0]{}[40]{} (2.3,1.6)[.66]{}[40]{}[85]{} (2.3,1.6)[.66]{}[85]{}[180]{} (2.3,1.6)[.42]{}[0]{}[40]{} (2.3,1.6)[.42]{}[40]{}[85]{} (2.3,1.6)[.42]{}[85]{}[180]{} (1.1,2.7)(0,1.6)(.33,.93)(1.1,.6)(.9,.5)(.3,.5)(-.2,.5) (2.3,2.7)(3.4,1.6)(3.07,.93)(2.3,.6)(2.5,.5)(3.1,.5)(3.7,.5) (.25,.55)[$T_G$]{} (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6)
Now, it is proved in [@al Proof of Lemma 4.6], that this property fully determines the surgery. Therefore, since this property is also true for the second definition, the two definitions coincide and Proposition \[propborlag\] is proved. In particular, the second definition has the same symmetries as the first one obviously has.
This definition does not depend on the orientation of $\Sigma(\Lambda)$. Nevertheless, we shall sometimes need orientations of our $Y$-graphs. An [*orientation*]{} of a $Y$-graph is an orientation of its three leaves, together with a cyclic order on the $3$–element set they form, induced by an orientation of $\Sigma(\Lambda)$ as in the figure (everything turns counterclockwise).
(-2,0)(3,2) (.95,.9)(1.45,1.5)(1.32,1.82)(1,1.95)(.5,1.75)(0,1.95)(-.32,1.82)(-.45,1.5)(.05,.9)(.05,.5)(.5,.05)(.95,.5)(.95,.9)(1.45,1.5)(1.32,1.82) (.5,.5)[.3]{}[-90]{}[90]{} (1,1.5)[.3]{}[-90]{}[90]{} (0,1.5)[.3]{}[-90]{}[90]{} (.5,.5)[.3]{}[90]{}[-90]{} (1,1.5)[.3]{}[90]{}[-90]{} (0,1.5)[.3]{}[90]{}[-90]{} (.5,.5)[.15]{} (1,1.5)[.15]{} (0,1.5)[.15]{} (.5,1)(.5,.8) (.21,1.29)(.5,1)(.79,1.29) (.5,.5)[1]{} (1,1.5)[2]{} (0,1.5)[3]{}
An [*$n$–component $Y$-link*]{} $G\subset M$ is an embedding of the disjoint union of $n$ copies of $N$ into $M$ up to isotopy. The $Y$-surgery along a $Y$-link $G$ is defined as the surgery along each $Y$-component of $G$. The resulting manifold is denoted by $M_G$.
Some clasper calculus
---------------------
Recall the following equivalences between surgeries inside handlebodies -that can be themselves embedded in any $3$-manifold in an arbitrary way-. The first one is move $Y_3$ in [@ggp], as rectified by Emmanuel Auclair in his thesis [@auc].
\[lemY3\] The surgeries on the following two $Y$-links are equivalent.
(-.5,-.2)(2.5,2.2) (0,0)(2,2) (.5,.5)[.15]{} (.5,1.5)[.15]{} (1.5,.5)[.15]{} (1.5,1.5)[.15]{} (.5,.5)[.3]{} (.5,1.5)[.3]{} (1.5,.5)[.3]{} (.5,.8)(.5,1.2) (.5,1)(1.2,1.6)(1.5,1.8)(1.8,1.5)(1.5,.8)
(-.5,-.2)(2.5,2.2) (0,0)(2,2) (.5,.5)[.15]{} (.5,1.5)[.15]{} (1.5,.5)[.15]{} (1.5,1.5)[.15]{} (1.5,1.5)[.25]{} (.5,.5)[.25]{} (.5,1.5)[.25]{} (1.5,.5)[.25]{} (1.5,1.25)(1.5,1.1) (1.5,1.1)(1.35,.8)(1.17,.57) (1.5,1.1)(1.7,1.1)(1.7,.6) (1.5,.5)[.4]{}[-155]{}[180]{} (.4,.7)(.4,1.3) (1.1,.5)[.1]{}[120]{}[60]{} (.4,1)(1.2,.8)
\[lemY4\] The surgeries on the following two $Y$-links are equivalent.
(-.2,-.5)(2.5,2.2) (0,0)(2,2) (.5,.5)[.15]{} (.5,1.5)[.15]{} (1.5,.5)[.15]{} (1.5,1.5)[.15]{} (.5,1.5)[.25]{} (.5,.5)[.25]{} (1.25,1)(1.2,1.5)(1.5,1.8)(1.8,1.5)(1.75,1)(1.8,.5)(1.5,.2)(1.2,.5) (1.25,1)(1,1) (.67,.67)(1,1)(.67,1.33)
(-.2,-.5)(2.5,2.2) (0,0)(2,2) (.5,1.5)[.15]{} (1.5,.5)[.15]{} (1.5,1.5)[.15]{} (1.5,1.5)[.25]{} (.5,.5)[.25]{} (.5,1.5)[.25]{} (1.5,.5)[.25]{} (.5,.5)[.35]{} (1.33,.67)(.85,1)(.67,1.33) (.5,1.5)[.35]{} (.85,1)(.67,.67) (.85,1.5)(1.5,1.1)(1.5,1.25) (1.5,1.1)(1.5,.95)(1.85,.5)(1.5,.15)(1.1,.4)(.85,.5) (.5,.5)[.15]{}
The two equivalences above easily imply the following one.
\[lemY5\] The surgeries on the following two $Y$-links are equivalent.
(-.5,-.2)(3,2.2) (0,0)(2.8,2) (.5,.5)[.15]{} (.5,1.5)[.15]{} (2.3,.5)[.15]{} (2.3,1.5)[.15]{} (.5,.5)[.3]{} (.5,1.5)[.3]{} (2.3,.5)[.3]{} (.5,.8)(.5,1.2) (.5,1)(1.6,1.6)(2.3,1.8)(2.6,1.5)(2.3,.8)
(-.2,-.2)(3,2.2) (0,0)(2.8,2) (2.3,1.1)(2.5,1.1)(2.5,.6) (2.3,1.1)(1.2,.9) (.6,1)(1.45,1)(1.4,.8) (2.3,.5)[.4]{} (.5,1.5)[.15]{} (2.3,.5)[.15]{} (2.3,1.5)[.15]{} (.85,1.5)(1.7,1.4)(1.7,.5) (.5,.5)[.35]{} (.6,1)(.5,1.25) (.5,1.5)[.35]{}[180]{}[320]{} (.5,1.5)[.35]{}[-60]{}[180]{} (.6,1)(.5,.75) (.85,.5)(1,.9)(1.3,.4)(1.7,.5)(1.95,.5) (2.3,.5)[.4]{} (1.2,.7)[.2]{}[180]{}[5]{} (1.2,.7)[.2]{}[45]{}[105]{} (1.4,.7)[.1]{}[-65]{}[240]{} (2.3,1.5)[.25]{} (2.3,.5)[.25]{} (.5,.5)[.25]{} (.5,1.5)[.25]{} (.5,.5)[.15]{} (2.3,1.25)(2.3,1.1)
[ ]{}
As a consequence of Lemma \[lemY4\], we also have the following lemma that provides an inverse for a $Y$-graph. A mark $\begin{pspicture}[.4](0,0)(.3,.3)
\psline(.15,0)(.15,.3)
\psline[border=1pt]{-}(.05,.05)(.25,.25)
\end{pspicture}$ on an edge indicates a positive half-twist of this edge.
\[lemY6\] The surgery on the following $Y$-link is trivial.
(-.2,-.2)(2.5,2.2) (1.35,1)(.85,1) (1.6,1)[.35]{} (.85,1)(.67,1.33) (.5,1.5)[.35]{} (.5,.5)[.35]{} (.85,1)(.67,.67) (1.1,1.5)(1.6,1.45)(2.05,1)(1.5,.5)(.85,.5) (1.1,1.5)(1.5,1.8)(1.6,1.35) (1.55,1.55)(1.65,1.75) (0,0)(2.3,2) (.5,1.5)[.15]{} (1.6,1)[.15]{} (1.6,1)[.25]{} (.5,.5)[.25]{} (.5,1.5)[.25]{} (.85,1.5)(1.1,1.5) (.5,.5)[.15]{} (1.1,1.5)
A clasper presentation of algebraically split links
---------------------------------------------------
A leaf $\ell$ of a $Y$-link $G$ is a [*meridional leaf*]{} or is a [*meridian*]{} of a link $L$, if it is trivial, and if it bounds a meridian disk of some link component whose interior intersects $G\cup L$ at exactly one point of $L$.
Say that a $Y$-link $G$ [*laces*]{} the trivial $r$-component link $U^{(r)}$ of a connected $3$-manifold if
- each of the $Y$-link components contains a meridional leaf of $U^{(r)}$,
- The components $U_i$ of $U^{(r)}$ bound disjoint disks $(D_i)_{i=1,\dots,r}$ ($U_i = \partial D_i$) so that $D_i \cap G$ is inside the meridional leaves of $U_i$ (and contains one point per meridional leaf of $U_i$),
- no component of $G$ contains more than one meridional leaf of a given component $U_i$.
Performing the surgery on such a $G$ transforms $U^{(r)}$ into the link $(K_1, \dots, K_r)=U^{(r)}_G$ in $M$ that is [*presented by $(G,U^{(r)})$*]{}.
Since any null-homologous knot bounds an oriented Seifert surface, by Lemma \[lemysur\], it is easy to see that any null-homologous knot is presented by a pair $(G,U_1)$, where $G$ is a $Y$-link that laces the trivial knot.
(0,-.7)(5,1.6) (2.5,-.5)(5,-.2)(3,.2)(2,.2)(0,-.2)(2.5,-.5)(5,-.2)(2.5,.1) (1,.2)[.2]{}[-160]{}[160]{} (.7,1.2)[.2]{}[180]{}[0]{} (.7,1.2)[.2]{}[0]{}[180]{} (1.3,1.2)[.2]{}[180]{}[0]{} (1.3,1.2)[.2]{}[0]{}[180]{} (.7,1)(1,.7)(1.3,1) (1,.7)(1,.4) (2.2,.2)[.2]{}[-160]{}[160]{} (1.9,1.2)[.2]{}[180]{}[0]{} (1.9,1.2)[.2]{}[0]{}[180]{} (2.5,1.2)[.2]{}[180]{}[0]{} (2.5,1.2)[.2]{}[0]{}[180]{} (1.9,1)(2.2,.7)(2.5,1) (2.2,.7)(2.2,.4) (5.1,-.2)[$U_1$]{} (3,.5)[$\dots$]{} (4,.2)[.2]{}[-160]{}[160]{} (3.7,1.2)[.2]{}[180]{}[0]{} (3.7,1.2)[.2]{}[0]{}[180]{} (4.3,1.2)[.2]{}[180]{}[0]{} (4.3,1.2)[.2]{}[0]{}[180]{} (3.7,1)(4,.7)(4.3,1) (4,.7)(4,.4)
In a connected oriented compact $3$-manifold $M$ such that $H_2(M;{\mathbb{Z}})=0$, the [*linking number of a null-homologous knot $K$ with a knot $C$ in its complement*]{} is well-defined as the algebraic intersection of $C$ with a surface bounded by $K$. The [*Milnor triple linking number*]{} $\mu(K_1,K_2,K_3)$ of three null-homologous knots $K_1, K_2, K_3$ that do not link each other is also well-defined, as the algebraic intersection of three Seifert surfaces of these knots in the complement of the other ones with the sign $\mu(K_1,K_2,K_3)=-\langle \Sigma_1,\Sigma_2,\Sigma_3\rangle$.
Let $G$ be a $Y$-link that laces the trivial link $U^{(r)}$ of $M$. Let $m_i$ denote the homology class of the oriented meridian of $U_i$. Say that a component of $G$ is [*of type $(\varepsilon_i m_i, \varepsilon_j m_j,f)$*]{} if its leaves are one meridian of $U_i$, one meridian of $U_j$, and another oriented framed leaf $f$ and if it can be oriented so that the homology classes of its oriented leaves read $\varepsilon_i m_i$, $\varepsilon_j m_j$ and $[f]$ with respect to the cyclic order induced by the orientation, with $\varepsilon_i, \varepsilon_j \in \{-1,1\}$. Similarly, say that a component of $G$ is [*of type $(\varepsilon_i m_i, \varepsilon_j m_j,\varepsilon_k m_k)$*]{} if its leaves are one meridian of $U_i$, one meridian of $U_j$, and one meridian of $U_k$, and if it can be oriented so that the homology classes of its oriented leaves read $\varepsilon_i m_i$, $\varepsilon_j m_j$ and $\varepsilon_k m_k$ with respect to the cyclic order induced by the orientation, with $\varepsilon_i, \varepsilon_j, \varepsilon_k \in \{-1,1\}$.
\[lemintclas\] Let $G$ be a $Y$-link that laces the trivial link $U^{(r)}$ of an oriented connected $3$–manifold $M$. Let $L=(K_1, \dots, K_r)=U^{(r)}_G$ be the link presented by $G$. Then $L$ is algebraically split, and the $K_i$ bound surfaces $\Sigma_i$ such that
- for any $\{i,j\} \subset \{1,2,\dots,r\}$, $\Sigma_i \cap \Sigma_j$ is the union over all the components [*of type $(\varepsilon_i m_i, \varepsilon_j m_j,f)$*]{} of the framed oriented leaves $\varepsilon_i\varepsilon_j f$,
- for any $\{i,j,k\} \subset \{1,2,\dots,r\}$, the oriented intersection $\Sigma_i \cap \Sigma_j \cap \Sigma_k$ is a union over all the components [*of type $(\varepsilon_i m_i, \varepsilon_j m_j,\varepsilon_k m_k)$*]{} of points with sign $\varepsilon_i\varepsilon_j \varepsilon_k$.
In particular, if $H_2(M;{\mathbb{Z}})=0$, then $\mu(K_i,K_j,K_k)$ is the sum over all the components [*of type $(\varepsilon_i m_i, \varepsilon_j m_j,\varepsilon_k m_k)$*]{} of the contributions $(-\varepsilon_i\varepsilon_j \varepsilon_k)$.
[[Proof: ]{}]{}Define the [*index*]{} of a component $Y$ of $G$ as the smallest $i$ such that $Y$ has a meridional leaf of $U_i$. Realize the surgeries on the components of index $i$ of $G$ by applying Lemma \[lemysur\] to the trivial meridional leaf $\ell$ of $U_i$ and to the part of $U_i$ going through $\ell$. These surgeries transform $U^{(r)}$ into $L$ and allow us to see each $K_i$ as the boundary of a surface $\tilde{\Sigma}_i$ whose $1$-handles are thickenings of the framed leaves that are not meridians of $U_i$ of the components of index $i$.
So far, $\tilde{\Sigma}_k$ may intersect the $K_i$ with $i<k$ (but not the $K_i$ with $i>k$). More precisely, if $i<k$, each component of index $i$ of type $(m_i,\varepsilon m_k, f)$ or $(m_i,f,-\varepsilon m_k)$ gives rise to an arc of intersection of $\tilde{\Sigma}_i \cap \tilde{\Sigma}_k$. Tubing $\tilde{\Sigma}_k$ along the part of $K_i$ between the two extremities of the intersection arc that is contained in the surgery picture transforms this arc of intersection into $\varepsilon f$ and removes the intersection of $K_i$ with $\tilde{\Sigma}_k$.
(-.2,.2)(4.4,3.3) (1.1,1.6)[1.1]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[95]{}[140]{} (1.1,1.6)[1.1]{}[140]{}[180]{} (1.1,1.6)[.7]{}[0]{}[95]{} (1.1,1.6)[.7]{}[95]{}[140]{} (1.1,1.6)[.7]{}[140]{}[180]{} (1.1,1.6)[.9]{}[95]{}[140]{} (1.1,1.6)[.9]{}[140]{}[240]{} (1.1,1.6)[.9]{}[240]{}[95]{} (2,4)(2.1,3.2)(2.4,2)(3.5,2.8) (2.3,1.6)[1.1]{}[0]{}[180]{} (2.3,1.6)[.7]{}[0]{}[180]{} (2.3,1.6)[.9]{}[-60]{}[180]{} (2.3,1.6)[.9]{}[180]{}[300]{} (2.1,3.2)(2.4,2)(3.5,2.8)(4.2,3.5) (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(0,.2)(-1,.2) (2.3,2.7)(3.4,1.6)(3.07,.83)(2.3,.4)(3.4,.2)(4.4,.2) (2.65,.9)[$\varepsilon m_k$]{} (.75,.9)[$f$]{} (0.1,.3)[$K_i$]{} (3.4,2.7)[$\varepsilon K_k$]{} (2.65,2.18)(2.85,2.52) (2.8,2.9)[$\tilde{\Sigma}_i \cap \tilde{\Sigma}_k$]{} (.4,1.55)[$\tilde{\Sigma}_i$]{} (2.75,2.9)(2.75,2.45) (1.7,.4)(1.1,.7) (1.7,.4)(2.3,.7) (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6)
(-.2,.2)(4.4,3.3) (1.1,1.6)[1.1]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[95]{}[140]{} (1.1,1.6)[1.1]{}[140]{}[180]{} (1.1,1.6)[.7]{}[0]{}[95]{} (1.1,1.6)[.7]{}[95]{}[140]{} (1.1,1.6)[.7]{}[140]{}[180]{} (1.1,1.6)[.8]{}[-6]{}[95]{} (1.1,1.6)[.8]{}[95]{}[140]{} (1.1,1.6)[.8]{}[140]{}[186]{} (1.1,1.6)[.6]{}[5]{}[95]{} (1.1,1.6)[.6]{}[95]{}[140]{} (1.1,1.6)[.6]{}[140]{}[173]{} (2,4)(2.1,3.2)(2.4,2)(3.5,2.8) (2.3,1.6)[1]{}[60]{}[185]{} (2.3,1.6)[.8]{}[60]{}[186]{} (2.3,1.6)[.6]{}[60]{}[173]{} (2.3,1.6)[1.2]{}[60]{}[176]{} (2.3,1.6)[1.1]{}[0]{}[180]{} (2.3,1.6)[.7]{}[0]{}[180]{} (2.1,3.2)(2.4,2)(3.5,2.8)(4.2,3.5) (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(0,.2)(-1,.2) (2.3,2.7)(3.4,1.6)(3.07,.83)(2.3,.4)(3.4,.2)(4.4,.2) (0.1,.3)[$K_i$]{} (3.4,2.7)[$\varepsilon K_k$]{} (2.3,2.46)(2.7,2.26)(2.8,2.5)(2.4,2.64) (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (.5,1.7)(1.1,1.7) (1.5,1.5)(1.9,1.5) (1.7,.4)[${\Sigma}_i$]{} (.3,1.5)(.8,1.5) (.1,1.4)[${\Sigma}_i \cap \varepsilon {\Sigma}_k$]{} (.8,1.5)(1.3,1.5)
If the third leaf is a meridian of $K_j$ for $i<j<k$ then perform the tubing along this leaf inside the tubing of $\tilde{\Sigma}_j$ along the meridional leaf of $m_k$. Let $\Sigma_k$ denote the surface obtained after all these tubings.
(-.2,.2)(4.4,3.3) (1.4,4)(1.3,3.2)(1,2)(-.1,2.8) (1.1,1.6)[1.1]{}[0]{}[180]{} (1.1,1.6)[.7]{}[0]{}[180]{} (1.1,1.6)[.9]{}[-14]{}[120]{} (1.1,1.6)[1]{}[-13]{}[120]{} (.7,2.26)(.6,2.5) (2,4)(2.1,3.2)(2.4,2)(3.5,2.8) (1.1,1.6)[.8]{}[-6]{}[186]{} (1.3,3.2)(1,2)(-.1,2.8)(-.8,3.5) (2.3,1.6)[1]{}[60]{}[185]{} (2.3,1.6)[.8]{}[60]{}[186]{} (2.3,1.6)[1.1]{}[0]{}[180]{} (2.3,1.6)[.9]{}[-14]{}[194]{} (2.3,1.6)[.7]{}[0]{}[180]{} (2.1,3.2)(2.4,2)(3.5,2.8)(4.2,3.5) (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(0,.2)(-1,.2) (2.3,2.7)(3.4,1.6)(3.07,.83)(2.3,.4)(3.4,.2)(4.4,.2) (0.1,.3)[$K_i$]{} (3.4,2.7)[$\varepsilon K_k$]{} (0,2.7)[$\varepsilon_j K_j$]{} (2.3,2.46)(2.7,2.26)(2.8,2.5)(2.4,2.64) (3.2,2.9)[${\Sigma}_i \cap {\Sigma}_j \cap {\Sigma}_k$]{} (1.7,.4)[${\Sigma}_i$]{} (3.2,2.9)(2.8,2.4) (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (1.4,1.4)(2,1.4) (2.1,1.4)(2.65,1.4) (3.2,1.3)[$ \varepsilon_j{\Sigma}_j \cap \Sigma_i$]{} (2.65,1.4)(3.2,1.4) (.3,1.5)(.8,1.5) (.1,1.4)[${\Sigma}_i \cap \varepsilon {\Sigma}_k$]{} (.8,1.5)(1.3,1.5) (1.5,1.5)(1.9,1.5)
It is left to the reader to check that the surfaces have the announced properties. [ ]{}
Say that a $Y$-link $G$ [*$\mu$-laces*]{} the trivial $r$-component link $U^{(r)}$ of a connected $3$-manifold if it laces it, and if for any triple $\{i,j,k\}$ of integers in $\{1,\dots,r\}$, there are exactly $|\mu(K_i,K_j,K_k)|$ components with one leaf that links $U_i$, one leaf that links $U_j$ and one leaf that links $U_k$.
It is known that any algebraically split link can be presented by a $Y$-link $G$ that laces the trivial link $U^{(r)}$ [@ggp Lemma 5.6], [@matv; @murnak]. We prove the following proposition that refines this result, and that, together with Lemma \[lemintclas\], proves Proposition \[proprealmil\].
\[propclaspmu\] For any algebraically split link $L=(K_1,\dots,K_r)$ in a connected $3$-manifold $M$ such that $H_2(M;{\mathbb{Z}})=0$, there exists a $Y$-link $G$ that $\mu$-laces the trivial link $U^{(r)}$ of $M$ such that $(G,U^{(r)})$ presents $L$.
This proposition will be a direct corollary from the slightly more general proposition \[propclaspmubased\] below, that may be used for the study of homology handlebodies.
Here, an [*$r$-component based link*]{} is an embedding $\Gamma_L$ of the following graph with $r$ loops, up to isotopy. Its underlying link is the restriction of the embedding to its $r$ loops.
(0,.6)(3.2,1.5) (.4,1.2)[$U_1$]{} (.7,1.2)[.2]{}[180]{}[0]{} (.7,1.2)[.2]{}[0]{}[180]{} (1.3,1.2)[.2]{}[180]{}[0]{} (1.3,1.2)[.2]{}[0]{}[180]{} (.7,1)(1.6,.7)(1.3,1) (1.6,.7)(2.5,1) (2.05,1.2)[$\dots$]{} (2.5,1.2)[.2]{}[180]{}[0]{} (2.5,1.2)[.2]{}[0]{}[180]{} (2.8,1.2)[$U_r$]{}
The [*trivial $r$-component based link*]{} $\Gamma_U^{(r)}$ is the above $r$-component based link seen in a $3$–ball.
\[propclaspmubased\] For any based $r$-component link $\Gamma_L$, whose underlying link $L=(K_1,\dots,K_r)$ is algebraically split, in a connected $3$-manifold $M$ such that $H_2(M;{\mathbb{Z}})=0$, there exists a $Y$-link $G$ in $M \setminus \Gamma_U^{(r)}$ that $\mu$-laces the trivial link $U^{(r)}$ of $M$ such that $(G,\Gamma_U^{(r)})$ presents $\Gamma_L$.
[[Proof: ]{}]{}For any sublink $L^{\prime}$ of $L$, there is a canonical subgraph $\Gamma_{L^{\prime}}$ of $\Gamma_L$ that is a based link with underlying link $L^{\prime}$. We prove Proposition \[propclaspmubased\] by proving the following statement by induction on the number $r$ of components of $L$.
[*Induction hypothesis*]{}\
Let $M$ be a connected $3$-manifold such that $H_2(M;{\mathbb{Z}})=0$. Let $\Gamma_{L \cup L^{\prime}}$ be a based algebraically split link in $M$ where $L$ has $r$ components. Let $\Gamma_{U^{(r)} \cup L^{\prime}}$ be the based link obtained from $\Gamma_{L \cup L^{\prime}}$ by replacing $\Gamma_L$ by $\Gamma_U^{(r)}$ so that each component of $U^{(r)}$ bounds a disk $D_i$ whose interior does not meet $\Gamma_{U^{(r)} \cup L^{\prime}}$. Then there exists a $Y$-link $G$ in of $M \setminus \Gamma_{L^{\prime}}$ such that the following set of properties $H(G,\Gamma_L,\Gamma_{L^{\prime}})$ is satisfied.
- $G \subset M \setminus \Gamma_{U^{(r)} \cup L^{\prime}}$,
- $G$ [*$\mu$-laces*]{} the trivial link $U^{(r)}$ of $M \setminus L^{\prime}$
- $(G,\Gamma_{U^{(r)} \cup L^{\prime}})$ presents $\Gamma_{L \cup L^{\prime}}$ in $M$,
- the only leaves of $G$ that link $L^{\prime}$ algebraically are meridional leaves of $L^{\prime}$,
- no component of $G$ contains more than one meridional leaf of a given component of $L^{\prime}$,
- For any triple $\{I,J,K\}$ of components of $L \cup L^{\prime}$ with at least one component in $L$, there are exactly $|\mu(I,J,K)|$ components of $G$ with one leaf that links $I$, one leaf that links $J$ and one leaf that links $K$.
This statement is obviously true for $0$-component links.
Assume that it is true for $(r-1)$-component links, we wish to prove it for $(L=(K_1,\dots,K_r), L^{\prime})$. Let $U^{(r-1)}=(U_1, \dots, U_{r-1})$ denote the trivial $(r-1)$-component link that bounds a disjoint union of disks $(D_i)_{i=1,\dots, r-1}$. By induction, there exists $(G_1 \subset M
\setminus \Gamma_{L^{\prime} \cup K_r})$ such that $H(G_1,\Gamma_{K_1,\dots,K_{r-1}}, \Gamma_{K_r \cup L^{\prime}})$ is satisfied.
Consider a two-dimensional disk $D$ that meets $K_r$ along an arc $\alpha$ of its boundary around which all the meridional leaves of $K_r$ are, and such that $D$ intersects all the meridional leaves, so that $$K^{\prime}_r = (K_r \setminus \stackrel{\circ}{\alpha}) \cup (-\partial D \setminus \alpha)$$ bounds a surface $\Sigma$ that meets neither $\Gamma_{L^{\prime} \cup U^{r-1}} \cup \cup_{i<r}D_i$, nor the path $\gamma_{r}$ from the vertex of $\Gamma_{L \cup L^{\prime}}$ to $K_r$, nor the leaves of $G_1$.
(0,-1.1)(5,1.6) (0,-.4)(.5,.1)(4.5,.1)(5,-.4) (2.5,-.35)(3.5,.05) (3,-.15)[$D$]{} (-.1,-.4)(.5,.2)(3,.2) (5.1,-.4)(4.5,.2)(3,.2) (5.3,-.6)(5.1,-.4)(3,-.4) (-.3,-.6)(-.1,-.4)(3,-.4) (3,-.5)[$K^{\prime}_r$]{} (1,.2)[.2]{}[-160]{}[160]{} (.7,1.2)[.2]{}[180]{}[0]{} (.7,1.2)[.2]{}[0]{}[180]{} (1.3,1.2)[.2]{}[180]{}[0]{} (1.3,1.2)[.2]{}[0]{}[180]{} (.7,1)(1,.7)(1.3,1) (1,.7)(1,.4) (2.2,.2)[.2]{}[-160]{}[160]{} (1.9,1.2)[.2]{}[180]{}[0]{} (1.9,1.2)[.2]{}[0]{}[180]{} (2.5,1.2)[.2]{}[180]{}[0]{} (2.5,1.2)[.2]{}[0]{}[180]{} (1.9,1)(2.2,.7)(2.5,1) (2.2,.7)(2.2,.4) (3.1,1.2)[$\dots$]{} (3,.4)[$\alpha$]{} (4,.2)[.2]{}[-160]{}[160]{} (3.7,1.2)[.2]{}[180]{}[0]{} (3.7,1.2)[.2]{}[0]{}[180]{} (4.3,1.2)[.2]{}[180]{}[0]{} (4.3,1.2)[.2]{}[0]{}[180]{} (3.7,1)(4,.7)(4.3,1) (4,.7)(4,.4)
The graph $G_1$ and the surface $\Sigma$ can be modified so that $\Sigma$ does not meet $G_1$ at all outside the meridional leaves of $L$, and the following set of assumptions\
$H_2(G_1,\Gamma_{K_1,\dots,K_{r-1}}, \Gamma_{K_r \cup L^{\prime}},\Sigma)$
- $\Sigma$ meets neither $\Gamma_{U^{(r-1)} \cup L^{\prime}} \cup \cup_{i<r}D_i$, nor $\gamma_{r}$, nor the leaves of $G_1$
- $H(G_1,\Gamma_{K_1,\dots,K_{r-1}}, \Gamma_{K_r \cup L^{\prime}})$ is satisfied except that components of $G_1$ are allowed to have no meridians of $L\setminus K_r$ provided that they have a meridian of $K_r$.
is still satisfied.
[[Proof: ]{}]{}We need to remove the intersections of $\Sigma$ with the edges of $G_1$. By isotopy, without loss, assume that no edge adjacent to a meridional leaf of $K_r$ intersects $\Sigma_r$ (push the intersection on the two other edges, if necessary). Similarly, assume that if a component contains only one meridian of $L$, the edge adjacent to this component does not meet $\Sigma_r$. Now, the intersections of the edges adjacent to non-meridional leaves can be removed by tubing $\Sigma_r$ around the part of the $Y$-graph that contains the corresponding leaf. Here [*tubing*]{} means replacing a small disk of $\Sigma_r$ in a neighborhood of an intersection point with an edge by the closure of its complement in the boundary of a regular neighborhood of the part of the $Y$-graph that contains the corresponding leaf, as below.
(-.5,0)(4,2) (0,0)(2,.5)(2,1.9)(0,1.4) (1,.95)(3,.95) (3.5,.95)(.5,.3)
(-.5,0)(4.5,2) (0,0)(2,.5)(2,1.9)(0,1.4) (1,.9)[.2]{} (1,.95)(3,.95) (3.5,.95)(.5,.3) (1,.75)(2.5,.7)(3.5,.45)(4.2,.95)(3.5,1.45)(2.5,1.1)(1,1.05) (3.3,.95)(3.5,1.05)(3.7,.95) (3.2,1.05)(3.3,.95)(3.5,.85)(3.7,.95)(3.8,1.05)
Thus, we are under the assumptions $H(E)$ that the only edge intersections occur on edges adjacent to a meridional leaf of some $K_j$, with $j<r$, of components with at least two meridional leaves of $L$. Define the complexity $c(\Sigma; G_1)$ of such a situation as follows. Define the complexity $c_e(Y)$ of a component $Y$ of $G_1$ as the number of intersection points of its edges with $\Sigma$. Define the complexity $c(\Sigma; G_1)$ as the pair (maximal complexity $c_e$ of the components, number of components with this complexity) ordered by the lexicographic order.
Now, to prove the lemma, it is enough to prove that there exists a pair $(\Sigma; G_1)$ with lower complexity such that $H_2(G_1,\Gamma_{K_1,\dots,K_{r-1}}, \Gamma_{K_r \cup L^{\prime}},\Sigma)$ and $H(E)$ are satisfied.
Consider a component $Y$ of $G_1$ with maximal complexity, and its edge $e$ with the maximal number of intersection points. By hypothesis, $e$ is adjacent with a meridional leaf $\ell$ of some component $K_i$ with $i<r$. Remove the point of $e \cap \Sigma$ that is closest to $\ell$ as follows. By our assumptions, $\Sigma$ intersects a neighborhood of $Y$ in the gray part of the following picture, where the intersection point that will be removed is at the top right corner. Perform the modification of Lemma \[lemY5\] so that the resulting three graphs are like in the following picture with respect to the positions of the possible intersections with $\Sigma$.
(-.1,-.6)(4,2.4) (0,0)(3.8,2) (.5,.5)[.15]{} (.5,1.5)[.15]{} (2.8,.5)[.15]{} (2.8,1.5)[.15]{} (2.8,1.5)[.25]{}[0]{}[180]{} (2.8,1.5)[.25]{}[180]{}[0]{} (.5,.5)[.3]{} (.5,1.5)[.3]{} (2.8,.5)[.3]{} (.5,.8)(.5,1.2) (.5,1)(1.6,1.6)(2.8,1.85)(3.2,1.6)(2.8,.8) (2.8,1.65)(2.8,2) (0,.5)(.35,.5) (0,1.5)(.35,1.5) (1.7,0)(1.7,2) (0,1.1)(.5,1.1)(.9,1.5)(.9,2) (0,.9)(.5,.9)(.9,.5)(.9,0) (2.5,1.5)[$m_r$]{} (3.25,1.5)[$e$]{} (3.15,.5)[$\ell$]{}
(-1.5,-.6)(4,2.4) (0,0)(2.8,2) (2.3,1.1)(2.5,1.1)(2.5,.6) (2.3,1.1)(1.2,.9) (.6,1)(1.45,1)(1.4,.8) (2.3,.5)[.4]{} (.5,1.5)[.15]{} (2.3,.5)[.15]{} (2.3,1.5)[.15]{} (.85,1.5)(1.7,1.4)(1.7,.5) (.5,.5)[.35]{} (.6,1)(.5,1.25) (.5,1.5)[.35]{}[180]{}[320]{} (.5,1.5)[.35]{}[-60]{}[180]{} (.6,1)(.5,.75) (.85,.5)(1,.9)(1.3,.4)(1.7,.5)(1.95,.5) (2.3,.5)[.4]{} (1.2,.7)[.2]{}[180]{}[5]{} (1.2,.7)[.2]{}[45]{}[105]{} (1.4,.7)[.1]{}[-65]{}[240]{} (2.3,1.5)[.25]{} (2.3,.5)[.25]{} (.5,.5)[.25]{} (.5,1.5)[.25]{} (.5,.5)[.15]{} (2.3,1.25)(2.3,1.1) (2.3,1.65)(2.3,2) (0,.5)(.35,.5) (0,1.5)(.35,1.5) (1.85,0)(1.85,2) (0,1.1)(.5,1.1)(.8,1.2)(.9,1.5)(.9,2) (0,.9)(.5,.9)(.8,.8)(.9,.5)(.9,0) (1.3,2.1)(1.3,1.55) (1.3,2.15)[$Y_1$]{} (-.1,1)(.5,1) (-.15,1)[$Y_2$]{} (2.9,1.5)(2.6,1.5) (2.95,1.5)[$Y_3$]{} (1.1,-.05)(1.1,.5) (1.1,-.1)[$\ell_1$]{}
Let $Y_1$ be the graph that replaces $Y$ with one edge intersection removed. Let $Y_3$ be the graph with a meridional leaf of $K_r$, a leaf parallel to $\ell$, and another trivial leaf $\ell_1$, and let $Y_2$ be the other one with one meridian of $\ell_1$. Remove all the intersection of $Y_3$ with $\Sigma$ outside its meridional leaf of $K_r$ by tubing. If $Y_2$ has only one meridional leaf of $L$, then remove its intersections as before, too. Otherwise, don’t change it, it has two meridional leaves, and its complexity $c_e$ is lower than $c_e(Y)$. Slide the meridional leaf of $K_r$ in $Y_3$ so that it is around the arc $\alpha$ of $K_r$. Thus, the obtained graph and the modified $\Sigma$ together satisfy $H_2(G_1,\Gamma_{K_1,\dots,K_{r-1}}, \Gamma_{K_r \cup L^{\prime}},\Sigma)$ and $H(E)$, and have lower complexity. The lemma is proved. [ ]{}
By Lemma \[lemysur\], $K_r \setminus \alpha$ is obtained from $\partial D \setminus \alpha$ by surgery on a $Y$–link $G_2$ in the neighborhood of $\Sigma \setminus D$ such that any component of $G_2$ contains exactly one meridian of $\partial D$. Let $U_r = \partial D$. Thus, $K_r$ is obtained from $U_r$ by surgery on $G_1 \cup G_2$, $G_1 \cup G_2$ $\mu$-laces the trivial link $U^{(r)}$ of $M \setminus L^{\prime}$, $(G_1 \cup G_2,\Gamma_{U^{(r)} \cup L^{\prime}})$ presents $\Gamma_{L \cup L^{\prime}}$ in $M$. Let us now modify $G=G_1 \cup G_2$ so that the last three conditions of $H(G,\Gamma_L,\Gamma_{L^{\prime}})$ are satisfied in addition to the previous ones.
$\bullet$ [*Cutting the leaves so that the only leaves that link $L^{\prime}$ algebraically are meridional leaves of $L^{\prime}$*]{}
Use Move $Y4$ of [@ggp] (Lemma \[lemY4\]) to cut the leaves of $G_2$ that are not meridians of $K_r$ so that they are either $0$-framed meridians of $L^{\prime}$ or they do not link $L^{\prime}$ at all. Indeed, this move allows us to cut the leaves into leaves that are homologically trivial in the complement of $L^{\prime}$, and meridians of the components of $L^{\prime}$ without creating further intersections of $G_2$ with the disk $D$. Define the complexity of a leaf as the minimal number of leaves in such a decomposition minus one. Define the complexity of a $Y$-graph as the sum of the complexities of its leaves. Finally define the complexity of a $Y$-link as the pair (maximal complexity of the components, number of components with this complexity) ordered by the lexicographic order. The leaves can be cut in order to make this complexity decrease without creating further intersections of $G$ with $D$.
$\bullet$ [*Sliding the handles so that no component of $G$ contains more than one meridional leaf of a given component of $L^{\prime}$.*]{}
Now, we wish to remove the $Y$-components with a meridional leaf of $K_r$ and two meridional leaves of the same component $J$ of $L^{\prime}$. By Lemma \[lemysur\], a surgery with respect to such a graph $G_3$ corresponds to a band sum with the boundary of a genus one Seifert surface as below.
(-.2,.2)(3.6,2.8) (1.1,1.6)[1.1]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[95]{}[140]{} (1.1,1.6)[1.1]{}[140]{}[180]{} (1.1,1.6)[.7]{}[0]{}[95]{} (1.1,1.6)[.7]{}[95]{}[140]{} (1.1,1.6)[.7]{}[140]{}[180]{} (1.1,1.6)[.9]{}[95]{}[140]{} (1.1,1.6)[.9]{}[140]{}[240]{} (1.1,1.6)[.9]{}[240]{}[95]{} (2.3,1.6)[1.1]{}[0]{}[40]{} (2.3,1.6)[1.1]{}[40]{}[85]{} (2.3,1.6)[1.1]{}[85]{}[180]{} (2.3,1.6)[.7]{}[0]{}[40]{} (2.3,1.6)[.7]{}[40]{}[85]{} (2.3,1.6)[.7]{}[85]{}[180]{} (2.3,1.6)[.9]{}[85]{}[180]{} (2.3,1.6)[.9]{}[180]{}[300]{} (2.3,1.6)[.9]{}[-60]{}[40]{} (2.3,1.6)[.9]{}[40]{}[85]{} (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(1.3,.4) (2.3,2.7)(3.4,1.6)(3.07,.83)(2.3,.4)(2.1,.4) (1.1,.4)(2.3,.4) (2.65,.9)[$\alpha$]{} (.75,.9)[$\beta$]{} (2.3,.4)(1.1,.7) (2.3,.4)(2.3,.7) (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6)
where $\alpha$ and $\beta$ are meridians of $J$. In this figure, a right-handed (resp.left-handed) Dehn twist of the surface along the simple curve $c(\alpha)$ freely homotopic to $\alpha$ transforms $\beta$ into $\alpha \beta$ (resp. $\alpha^{-1} \beta$) and does not change $\alpha$. Therefore, the $Y$-graph $G_3$ is equivalent to a $Y$-graph whose leaves are a meridian of $K_r$, the meridian $\alpha$, and the curve among $\alpha \beta$ and $\alpha^{-1} \beta$ that is null-homologous.
$\bullet$ [*Realizing the algebraic cancellations to the Milnor invariants $\mu(K_r,K_s,K_t)$ where $K_s$ and $K_t$ are components of $L^{\prime}$.*]{}
First recall from Lemma \[lemintclas\] that $\mu(K_r,K_s,K_t)$ is the sum of the contributions $\varepsilon \eta$ of the $Y$-graphs of type $(m_r,-\varepsilon m_s, \eta m_t)$ or $(m_r,\varepsilon m_t, \eta m_s)$ where $\varepsilon$ and $\eta$ belong to $\{-1,1\}$. Second, exchange the order of the $Y$-graphs that link $U_r$ so that all the graphs that contribute with a sign opposite to the Milnor invariant are followed by a graph that contributes with the sign of the Milnor invariant. In order to exchange two $Y$-graphs that link $U_r$, perform the following sequence of operations.
(0,-.4)(3.2,2.4) (0,.5)(3,.5) (1,.5)[.4]{}[-90]{}[165]{} (1,.5)[.4]{}[-160]{}[-90]{} (2,.5)[.4]{}[-90]{}[160]{} (2,.5)[.4]{}[-160]{}[-90]{} (1,.9)(1,1.8) (1,2)[.2]{}[135]{}[45]{} (1,2)[.2]{}[45]{}[135]{} (.3,2)[.2]{}[135]{}[45]{} (.3,2)[.2]{}[45]{}[135]{} (2,.9)(2,1.8) (2,2)[.2]{}[135]{}[45]{} (2,2)[.2]{}[45]{}[135]{} (2.7,2)[.2]{}[135]{}[45]{} (2.7,2)[.2]{}[45]{}[135]{} (.1,.4)[$U_r$]{} (2,1.3)(2.7,1.8) (1,1.3)(.3,1.8) (2,-.2)[$m$]{} (1,-.2)[$m^{\prime}$]{}
(0,-.4)(3.2,2.4) (0,.5)(3,.5) (1,.5)[.4]{}[-90]{}[165]{} (1,.5)[.4]{}[-160]{}[-90]{} (2,1.3)(2.5,.8)(2,.7)(1.33,.7) (2,.5)[.4]{}[-90]{}[140]{} (2,.5)[.4]{}[-160]{}[-90]{} (2,.5)[.4]{}[160]{}[165]{} (2,.9)(1,1.3)(1,1.8) (1,2)[.2]{}[135]{}[45]{} (1,2)[.2]{}[45]{}[135]{} (.3,2)[.2]{}[135]{}[45]{} (.3,2)[.2]{}[45]{}[135]{} (2,1.3)(2,1.8) (2,2)[.2]{}[135]{}[45]{} (2,2)[.2]{}[45]{}[135]{} (2.7,2)[.2]{}[135]{}[45]{} (2.7,2)[.2]{}[45]{}[135]{} (.1,.4)[$U_r$]{} (2,1.3)(2.7,1.8) (1,1.3)(.3,1.8) (1,-.2)[$m$]{} (2,-.2)[$m^{\prime}$]{}
(-1.1,-.4)(3.2,2.4) (.2,2)[.35]{}[135]{}[45]{} (1,1.5)(.2,1.8) (1,1.5)(1,1.8) (1,2)[.35]{}[135]{}[45]{} (-.8,.5)(3,.5) (.5,.5)[.4]{}[-90]{}[165]{} (.5,.5)[.4]{}[-160]{}[-90]{} (2,1.3)(1.35,1.05) (1.2,1.05)[.15]{} (.2,1.65)(1,.9)(1.2,.75)(2,1) (1.25,1.05)(.6,.86) (2,.5)[.3]{}[-90]{}[165]{} (2,.5)[.3]{}[-160]{}[-90]{} (2,.5)[.4]{}[160]{}[165]{} (1,2)[.2]{}[135]{}[45]{} (1,2)[.2]{}[45]{}[135]{} (.2,2)[.2]{}[135]{}[45]{} (.2,2)[.2]{}[45]{}[135]{} (1,2)[.35]{}[45]{}[135]{} (.2,2)[.35]{}[45]{}[135]{} (2,1.3)(2,1.8) (2,2)[.2]{}[135]{}[45]{} (2,2)[.2]{}[45]{}[135]{} (2.7,2)[.2]{}[135]{}[45]{} (2.7,2)[.2]{}[45]{}[135]{} (-.7,.4)[$U_r$]{} (2,1.3)(2.7,1.8) (2,1)(1.5,1.5) (1.5,1.5)(1.24,1.76) (2,1)(2,.8) (1.2,1.2)(1,1.5) (.5,-.2)[$m$]{} (2,-.2)[$m^{\prime}$]{}
First slide the meridian $m$ of one of them inside the other one $m^{\prime}$. Next use move $Y_4$ (Lemma \[lemY4\]) to cut $m^{\prime}$ into $m^{\prime}$ and a leaf that links the edge going to $m$. It is enough to slide inside components that contribute to $\mu(K_r,K_s,K_t)$. Thus, we do not lose properties of our graphs, (and otherwise we could just perform the surgery).
Last, transform a pair of $K_r$–adjacent $Y$-graphs with opposite contributions to $\mu(K_r,K_s,K_t)$ into a family of $Y$-graphs that do not individually contribute to $\mu(K_r,K_s,K_t)$. To do this, see the effect of the surgery along the two $K_r$–adjacent $Y$-graphs as a band sum with the boundary of a genus two surface $\Sigma$ whose $1$–handles are $\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$, and are meridians of $K_s$ and $K_t$.
(0,.2)(5.4,2.9) (1.1,1.6)[1.1]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[95]{}[140]{} (1.1,1.6)[1.1]{}[140]{}[180]{} (1.1,1.6)[.7]{}[0]{}[95]{} (1.1,1.6)[.7]{}[95]{}[140]{} (1.1,1.6)[.7]{}[140]{}[180]{} (1.1,1.6)[.9]{}[95]{}[140]{} (1.1,1.6)[.9]{}[140]{}[240]{} (1.1,1.6)[.9]{}[240]{}[95]{} (2.3,1.6)[1.1]{}[0]{}[40]{} (2.3,1.6)[1.1]{}[40]{}[85]{} (2.3,1.6)[1.1]{}[85]{}[180]{} (2.3,1.6)[.7]{}[0]{}[40]{} (2.3,1.6)[.7]{}[40]{}[85]{} (2.3,1.6)[.7]{}[85]{}[180]{} (2.3,1.6)[.9]{}[85]{}[180]{} (2.3,1.6)[.9]{}[180]{}[300]{} (2.3,1.6)[.9]{}[-60]{}[40]{} (2.3,1.6)[.9]{}[40]{}[85]{} (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(1.3,.4) (1.1,.4)(5.9,.4) (2.65,.9)[$\alpha_2$]{} (.75,.9)[$\beta_2$]{} (5.9,.4)(1.1,.7) (5.9,.4)(2.75,.82) (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (4.7,1.6)[1.1]{}[0]{}[95]{} (4.7,1.6)[1.1]{}[95]{}[140]{} (4.7,1.6)[1.1]{}[140]{}[180]{} (4.7,1.6)[.7]{}[0]{}[95]{} (4.7,1.6)[.7]{}[95]{}[140]{} (4.7,1.6)[.7]{}[140]{}[180]{} (4.7,1.6)[.9]{}[95]{}[140]{} (4.7,1.6)[.9]{}[140]{}[240]{} (4.7,1.6)[.9]{}[240]{}[95]{} (5.9,1.6)[1.1]{}[0]{}[40]{} (5.9,1.6)[1.1]{}[40]{}[85]{} (5.9,1.6)[1.1]{}[85]{}[180]{} (5.9,1.6)[.7]{}[0]{}[40]{} (5.9,1.6)[.7]{}[40]{}[85]{} (5.9,1.6)[.7]{}[85]{}[180]{} (5.9,1.6)[.9]{}[85]{}[180]{} (5.9,1.6)[.9]{}[180]{}[300]{} (5.9,1.6)[.9]{}[-60]{}[40]{} (5.9,1.6)[.9]{}[40]{}[85]{} (5.9,2.7)(7,1.6)(6.67,.83)(5.9,.4)(5.7,.4) (6.25,.9)[$\alpha_1$]{} (4.35,.9)[$\beta_1$]{} (5.9,.4)(4.7,.7) (5.9,.4)(5.9,.7) (3.4,1.6)(3.6,1.6) (4,1.6)(4.8,1.6) (5.2,1.6)(5.4,1.6) (5.8,1.6)(6.6,1.6)
We are in one of the following situations for the homology classes of the curves: Either $[\alpha_1]=[\alpha_2]$ and $[\beta_1]=-[\beta_2]$, or $[\alpha_1]=-[\alpha_2]$ and $[\beta_1]=[\beta_2]$, or $[\alpha_1]=[\beta_2]$ and $[\beta_1]=[\alpha_2]$, or $[\alpha_1]=-[\beta_2]$ and $[\beta_1]=-[\alpha_2]$.
Consider the following simple closed curves $c(\alpha_2)$, $c(\beta_1)$, $c(\beta_2)$, $c(\beta_1\alpha_2)$ and $c(\beta_1\beta_2)$ whose homology classes are $[\alpha_2]$, $[\beta_1]$, $[\beta_2]$, $[\beta_1\alpha_2]$ and $[\beta_1\beta_2]$, respectively.
(0,.2)(5.4,2.9) (1.1,1.6)[.95]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[0]{}[95]{} (1.1,1.6)[1.1]{}[95]{}[140]{} (1.1,1.6)[1.1]{}[140]{}[180]{} (1.1,1.6)[.7]{}[0]{}[95]{} (1.1,1.6)[.7]{}[95]{}[140]{} (1.1,1.6)[.7]{}[140]{}[180]{} (1.1,1.6)[.8]{}[95]{}[140]{} (1.1,1.6)[.8]{}[140]{}[240]{} (1.1,1.6)[.8]{}[240]{}[95]{} (2.3,1.6)[1.1]{}[0]{}[40]{} (2.3,1.6)[1.1]{}[40]{}[85]{} (2.3,1.6)[1.1]{}[85]{}[180]{} (2.3,1.6)[.7]{}[0]{}[40]{} (2.3,1.6)[.7]{}[40]{}[85]{} (2.3,1.6)[.7]{}[85]{}[180]{} (2.3,1.6)[.8]{}[85]{}[180]{} (2.3,1.6)[.8]{}[180]{}[300]{} (2.3,1.6)[.8]{}[-60]{}[40]{} (2.3,1.6)[.8]{}[40]{}[85]{} (1.1,2.7)(0,1.6)(.33,.83)(1.1,.4)(1.3,.4) (1.1,.4)(5.9,.4) (2.65,.95)[$c(\alpha_2)$]{} (.65,1)[$c(\beta_2)$]{} (.4,1.6)(1.2,1.6) (1.6,1.6)(1.8,1.6) (2.2,1.6)(3,1.6) (4.7,1.6)[1.2]{}[0]{}[95]{} (4.7,1.6)[1.2]{}[95]{}[140]{} (4.7,1.6)[1.2]{}[140]{}[180]{} (4.7,1.6)[.6]{}[0]{}[95]{} (4.7,1.6)[.6]{}[95]{}[140]{} (4.7,1.6)[.6]{}[140]{}[180]{} (4.7,1.6)[.7]{}[95]{}[140]{} (4.7,1.6)[.7]{}[140]{}[240]{} (4.7,1.6)[.7]{}[240]{}[95]{}
(5.62,1.1)[$c(\beta_1\beta_2)$]{} (4.7,1.6)[1]{}[95]{}[140]{} (4.7,1.6)[1]{}[140]{}[180]{} (4.7,1.6)[1]{}[-30]{}[95]{} (4.7,1.6)[1]{}[-90]{}[-30]{} (1.1,1.6)[.95]{}[95]{}[140]{} (1.1,1.6)[.95]{}[140]{}[270]{} (.15,1.6)(1.1,.65)(2.9,.6)(4.7,.6)(5.7,1.6) (1.1,2.55)(2.05,1.6)(3.55,1.2)(3.7,1.6)(4.7,2.6) (3.55,.95)[$c(\beta_1\alpha_2)$]{} (4.7,1.6)[.85]{}[95]{}[140]{} (4.7,1.6)[.85]{}[140]{}[180]{} (4.7,1.6)[.85]{}[270]{}[95]{} (2.3,1.6)[.95]{}[180]{}[270]{} (2.3,1.6)[.95]{}[0]{}[40]{} (2.3,1.6)[.95]{}[40]{}[85]{} (2.3,1.6)[.95]{}[85]{}[180]{} (1.35,1.6)(2.3,.65)(3.55,1)(4.7,.75) (2.3,.65)(3.55,1)(4.7,.75)(5.55,1.6) (2.3,2.55)(3.25,1.6)(3.55,1.4)(3.85,1.6)(4.7,2.45)
(5.9,1.6)[1.1]{}[0]{}[40]{} (5.9,1.6)[1.1]{}[40]{}[85]{} (5.9,1.6)[1.1]{}[85]{}[180]{} (5.9,1.6)[.7]{}[0]{}[40]{} (5.9,1.6)[.7]{}[40]{}[85]{} (5.9,1.6)[.7]{}[85]{}[180]{} (5.9,2.7)(7,1.6)(6.67,.83)(5.9,.4)(5.7,.4) (4.35,1.05)[$c(\beta_1)$]{} (3.4,1.6)(3.5,1.6) (4.1,1.6)(4.8,1.6) (5.2,1.6)(5.3,1.6) (5.9,1.6)(6.6,1.6)
For a curve $c$, let $\tau_c$ denote the right-handed Dehn twist around this curve. Recall the action of $\tau$ on homology classes $\tau_c(x) = x + \langle c , x\rangle_{\Sigma} c$. Then the homeomorphism $\tau_{c(\alpha_2)}^{-1}\tau_{c(\beta_1)}^{-1}\tau_{c(\beta_1\alpha_2)}$ of $\Sigma$ transforms $\alpha_2$ and $\beta_1$ to conjugate curves, where the conjugation paths are in the neighborhood of the genus $2$ surface and avoids the disks $D_i$, for $i \leq r$, and it transforms $\alpha_1$ and $\beta_2$ into curves homologous to $\alpha_1 \alpha_2^{-1}$ and $\beta_1\beta_2$. Therefore using this boundary-preserving homeomorphism in the first case allows us to transform the surgery on the initial pair of $Y$-graphs into a surgery on a pair of $Y$-graphs such that each of the graphs has a homologically trivial leaf and two meridional leaves. In the second case, use $\tau_{c(\alpha_2)}\tau_{c(\beta_1)}\tau_{c(\beta_1\alpha_2)}^{-1}$. Use $\tau_{c(\beta_2)}^{-1}\tau_{c(\beta_1)}^{-1}\tau_{c(\beta_1\beta_2)}$ and $\tau_{c(\beta_2)}\tau_{c(\beta_1)}\tau_{c(\beta_1\beta_2)}^{-1}$ in the third and fourth cases, respectively to achieve a similar reduction. [ ]{}
Proof of the formulae for algebraically split links
===================================================
\[secprooffas\]
We prove the surgery formulae of Theorem \[thmfas\] and Theorem \[thmfasmu\] following a strategy that was used in [@ggp] to compare the filtration of the space of ${\mathbb{Z}}$–spheres associated to algebraically split links to the filtration associated to $Y$-links.
According to Proposition \[propclaspmu\], it is enough to prove these theorems for links that are presented by pairs $(G,U^{(r)})$ where $G$ is a $Y$-link that $\mu$-laces the trivial link $U^{(r)}$ of $M$, that is for $U^{(r)} \subset M_G$, where $U^{(r)}$ is equipped with surgery coefficients $p_1/q_1$, $p_2/q_2$, …, $p_r/q_r$.
$$[M_G;U^{(r)}] = \sum_{H \subset G} (-1)^{\sharp H} [M;H \cup U^{(r)}]$$ where $$[M;H \cup U^{(r)}]= \sum_{J \subset H, I \subset \{1,2,\dots,r\}}(-1)^{\sharp J + \sharp I}M_{J,(U_i;p_i/q_i)_{i \in I}} \sharp \left(\sharp_{j \in \{1,2,\dots,r\} \setminus I} L(p_j,-q_j)\right).$$ Then $[M_G;U^{(r)}] \stackrel{n}{\equiv} \sum_{H \subset G ;\sharp H \leq 2n} (-1)^{\sharp H} [M;H \cup U^{(r)}]$.\
If there exists $i$, such that $U_i$ does not link any leaf of $H$, then $[M;H \cup U^{(r)}]=0$.\
If there exists $i$, such that $U_i$ links only one leaf of $H$, then let $Y_1$ be the component of this leaf. $$[M;H \cup U^{(r)}]=- [M_{Y_1}; H \setminus Y_1 \cup U^{(r)}].$$
Recall that the surgery on $Y_1$ is a surgery associated with a genus one surface bounded by some $K_i$ as in Subsection \[sublagboun\]. Then the inverse surgery of this subsection transforms $U_i$ into $-K_i$ and since it can be realized as a genus one cobordism, it can also be realized by a surgery on a $Y$-graph that laces $U_1$ and that sits in the complement of $G$. Let $Y_1^{-1}$ be such a graph. We can assume that its leaves are a meridian of $U_i$ and two leaves parallel to the two other leaves of $Y_1$. Compare with Lemma \[lemY6\].
Then $[M_{Y_1}; H \setminus Y_1 \cup U^{(r)}]= [M_{Y_1^{-1}}; H \setminus Y_1 \cup U^{(r)}]$ and $$[M;H \cup U^{(r)}] = \frac{1}{2} [M;H \cup Y_1^{-1} \cup U^{(r)}].$$
As long as there is a component $U_j$ that bounds a disk $D_j$ intersecting $H \cup Y_1^{-1} \cup \dots \cup Y_k^{-1}$ once (and necessarily) inside a meridional leaf of some component $Y_{k+1}$ of $H$, add $Y_{k+1}^{-1}$, and write $$[M;H \cup Y_1^{-1} \cup \dots \cup Y_k^{-1} \cup U^{(r)}]=
\frac{1}{2} [M;H \cup Y_1^{-1} \cup \dots \cup Y_k^{-1} \cup Y_{k+1}^{-1} \cup U^{(r)}].$$ $$[M;H \cup U^{(r)}]=\frac{1}{2^{k+1}}[M;H \cup Y_1^{-1} \cup \dots \cup Y_k^{-1} \cup Y_{k+1}^{-1} \cup U^{(r)}].$$ Finally, $[M_G;U^{(r)}]$ is a rational combination of terms of the form $[M; H^{\prime} \cup U^{(r)}]$ where each $U_i$ links at least two leaves of $H^{\prime}$. To be more specific, the considered $H^{\prime}$ are of the form $H \cup H_1^{-1}$, where $H$ is a sublink of $G$, and $H_1^{-1}$ is a link made of inverses of the components of a sublink $H_1$ of $H$. In particular, the leaves of a component of $H_1^{-1}$ have the same constraints as the leaves of a component of $G$. Since a leaf of $H^{\prime}$ links at most one $U_i$, such a $H^{\prime}$ has at least $2r$ leaves linking the $U_i$. In particular, if $2r>6n$, $[M_G;U^{(r)}] \stackrel{n}{\equiv} 0$.
- Under the hypotheses of Theorem \[thmfas\], assume $2r=6n$. Up to elements in ${{\cal F}}_{2n+1}$, $[M_G;U^{(r)}]$ is a rational combination of terms of the form $[M; H^{\prime} \cup U^{(r)}]$ where each $U_i$ links exactly two leaves of $H^{\prime}$, and each leaf of $H^{\prime}$ is a meridional leaf of some $U_i$. More precisely, let $G_3$ be the sublink of $G$ made of the components that have three meridional leaves, we have $$[M_G;U^{(r)}]\stackrel{n}{\equiv} \sum_{H}(-1)^{\sharp H}[M;H\cup U^{(r)}]$$ where the sum runs over the $H$ that read as the disjoint union of two $Y$-links $H_1$ and $H_2$ of $G_3$ such that for any component $U_i$ of $U^{(r)}$, either there is one meridional leaf of $U_i$ in $H_1$ and no meridional leaf of $U_i$ in $H_2$, or there is no meridional leaf of $U_i$ in $H_1$ and there are two meridional leaves of $U_i$ in $H_2$. Let ${{\cal H}}$ denote the set of the $(H_1,H_2)$ where $H_1 \cup H_2$ is a decomposition as above of such a graph.
- Under the hypotheses of Theorem \[thmfasmu\], at most two thirds of the leaves of the $H^{\prime}$ link the $U_i$ once, and the leaves of the other third do not link the $U_i$ at all. Therefore, if $2r>4n$, $[M_G;U^{(r)}]$ belongs to ${{\cal F}}_{2n+1}$. If $r=2n$, up to elements in ${{\cal F}}_{2n+1}$, $[M_G;U^{(r)}]$ is a rational combination of terms of the form $[M; H^{\prime} \cup U^{(r)}]$ where each $U_i$ links exactly two leaves of $H^{\prime}$, and in each component of $H^{\prime}$, there are two meridional leaves of $U^{(r)}$ and a null-homologous leaf. More precisely, let $G_2$ be the sublink of $G$ made of the components that have two meridional leaves, we have $$[M_G;U^{(r)}]\stackrel{n}{\equiv} \sum_{H}(-1)^{\sharp H}[M;H\cup U^{(r)}]$$ where the sum runs over the $H$ that read as the disjoint union of two $Y$-links $H_1$ and $H_2$ of $G_2$ such for any component $U_i$ of $U^{(r)}$, either there is one meridional leaf of $U_i$ in $H_1$ and no meridional leaf of $U_i$ in $H_2$, or there is no meridional leaf of $U_i$ in $H_1$ and there are two meridional leaves of $U_i$ in $H_2$. Let ${{\cal H}}$ denote the set of the $(H_1,H_2)$ where $H_1 \cup H_2$ is a decomposition as above of such a graph.
In both cases $$[M_G;U^{(r)}]\stackrel{n}{\equiv} \sum_{(H_1,H_2)\in {{\cal H}}}\left(\frac{-1}{2}\right)^{\sharp H_1} (-1)^{\sharp H_2}[M;H_1 \cup H_1^{-1} \cup H_2 \cup U^{(r)}]$$ where $H_1 \cup H_1^{-1} \cup H_2$ has $2n$ components (and therefore $(-1)^{\sharp H_2}=1$). Apply Theorem \[thmflag\] to compute it. The tripod associated to a surgery on an oriented $Y$-graph whose leaves are $\ell_1,\ell_2,\ell_3$ was computed in Lemma \[lemtriplag\] (thanks to Proposition \[propborlag\]). It is
(-.05,-.1)(.9,.7) (0.05,.3)(.45,.6) (0.05,.3)(.45,.3) (0.05,.3)(.45,0) (.55,0)[$\ell_3$]{} (.55,.3)[$\ell_2$]{} (.55,.6)[$\ell_1$]{}
while the tripod associated to an inverse of such a graph is
(-.05,-.1)(1,.7) (0.05,.3)(.45,.6) (0.05,.3)(.45,.3) (0.05,.3)(.45,0) (.55,0)[$\ell_{1 \parallel}$]{} (.55,.3)[$\ell_{2 \parallel}$]{} (.55,.6)[$\ell_{3 \parallel}$]{}
where the parallels are taken with respect of the parallelization of the leaves. Later, we shall consider twice the tripods of the components of $H_1$ and remove the $(-1)^{\sharp H_1}$. Recall the formula of Lemma \[lemvarlk\] $$lk_{M_{(U_i;p_i/q_i)}}(J,K)=lk_M(J,K)-\frac{q_i}{p_i}lk_M(U_i,J)lk_M(U_i,K).$$ If for some $i$, a contraction does not pair two curves linking $U_i$, then its contribution to $[M;H^{\prime} \cup (U \setminus U_i)]$ and its contribution to $[M_{U_i};H^{\prime} \cup (U \setminus U_i)]$ will be the same. Therefore, it won’t contribute to $[M;H^{\prime} \cup U ]$. Thus since there are exactly two leaves $m_i$ and $m^{\prime}_i$ linking $U_i$ in each $H^{\prime}$, the only pairings that will contribute will pair these pairs together, and the corresponding remaining linking number will be $\frac{q_i}{p_i}lk_M(U_i,m_i)lk_M(U_i,m^{\prime}_i)$.
- Under the hypotheses of Theorem \[thmfas\], there is one contributing pairing for every $(H_1,H_2)\in {{\cal H}}$. It can be seen as an edge-labelled Jacobi diagram $\Gamma(H_1,H_2)$ together with a bijection $b$ from the set of its vertices to the set of components of $H_1 \cup H_1^{-1}\cup H_2$ that maps a vertex $v$ with adjacent edges labelled by $i,j,k$ to a component $b(v)$ of $G$ of type $(\varepsilon_i m_i, \varepsilon_j m_j,\varepsilon_k m_k)$ where $\varepsilon_i$, $\varepsilon_j$, $\varepsilon_k$ are in $\{-1,1\}$, or to the inverse of such a component. Equip $\Gamma(H_1,H_2)$ with an orientation. Then if the orientation of a vertex $v$ as above is induced by the cyclic order $(i,j,k)$, assign it the sign $(-\varepsilon_i \varepsilon_j \varepsilon_k)$, and assign it $\varepsilon_i \varepsilon_j \varepsilon_k$ otherwise. Define $\mbox{sign}(\Gamma(H_1,H_2);b)$ as the products of the signs of the vertices. Then $$Z_n\left([M_G;U^{(r)}]\right)= \prod_{i=1}^{3n}\frac{q_i}{p_i}\sum_{(H_1,H_2)\in {{\cal H}}}\frac{1}{2^{\sharp H_1}}\mbox{sign}(\Gamma(H_1,H_2);b)[\Gamma(H_1,H_2)]$$
Now, let $f=f(b)$ be the map from $V(\Gamma(H_1,H_2))$ to the set of components of $G$ obtained from a bijection $b$ as above by setting $$\begin{array}{lll}
f(b)(v)&=b(v)& \mbox{if}\; b(v) \; \mbox{is a component of }\;
H_1 \cup H_2\\
&=Y_i& \mbox{if}\; b(v)=Y_i^{-1}.
\end{array}$$ There are $2^{\sharp H_1}$ $b$ such that $f(b)=f$, and, if $\sharp \mbox{Aut}_e(\Gamma)$ is the set of automorphisms of $\Gamma$ that induce the Identity on $E(\Gamma)$, there are $\sharp \mbox{Aut}_e(\Gamma)$ $b$ that define the same pairing. Since an automorphism that preserves the edges pointwise may only exchange vertices inside components ${ \begin{pspicture}[0.2](0,0)(.5,.4)
\pscircle(0.25,0.2){.25}
\psline{*-*}(0.05,.2)(.45,.2) \end{pspicture}}$, $\sharp \mbox{Aut}_e(\Gamma) = 2^{\theta(\Gamma)}$.
Orient $G$ arbitrarily. Let $\Gamma \in D_{e,n}$. Equip $\Gamma$ with an arbitrary orientation. Let $G(\Gamma)$ be the set of maps $g$ from $V(\Gamma)$ to the set of components of $G$ that map a vertex $v$ with adjacent edges labelled by $i,j,k$, with respect to the order induced by the orientation, to a component $g(v)$ of $G$ of type $(\varepsilon_i m_i, -\varepsilon_j m_j,\varepsilon_k m_k)$ or $(\varepsilon_i m_i, \varepsilon_k m_k, \varepsilon_j m_j)$. Define $\mbox{sign}(g,v)=\varepsilon_i \varepsilon_j \varepsilon_k$ for such a vertex. Define $\mbox{sign}(\Gamma;g)$ as the product of the signs associated to the vertices. Then $$Z_n\left([M_G;U^{(r)}]\right)= \prod_{i=1}^{3n}\frac{q_i}{p_i}
\sum_{\Gamma \in D_{e,n}, g \in G(\Gamma)}\frac{\mbox{sign}(\Gamma;g)}{2^{\theta(\Gamma)}}[\Gamma].$$ Now, Lemma \[lemintclas\] easily leads to the conclusion of the proof of Theorem \[thmfas\].
- Orient $G$ arbitrarily. Under the hypotheses of Theorem \[thmfasmu\], a contributing pairing for $(H_1,H_2)\in {{\cal H}}$ is a $2/3$–labelled Jacobi diagram $\Gamma$, equipped with a bijection from $V(\Gamma)$ to the set of components of $H_1 \cup H_1^{-1}\cup H_2$ that maps a vertex with two adjacent edges labelled by $i$ and $j$ to a component of type $(\varepsilon_i m_i, \varepsilon_j m_j,f)$ or $(\varepsilon_j m_j, -\varepsilon_i m_i,f)$. For a fixed $2/3$–labelled Jacobi diagram $\Gamma$, there are $\sharp \mbox{Aut}_{2/3}(\Gamma)$ bijections from $V(\Gamma)$ to the set of components of $H_1 \cup H_1^{-1}\cup H_2$ that will correspond to the same pairing.
Let $\Gamma \in D_{2/3,n}$. Equip $\Gamma$ with an orientation. Let $G(\Gamma)$ be the set of maps $g$ from $V(\Gamma)$ to the set of components of $G$ that map a vertex $v$ whose adjacent edges are labelled by $(i,j,\mbox{nothing})$ (with respect to the orientation of $\Gamma$) to a component of type $(\varepsilon_i m_i, \varepsilon_j m_j,f)$ or $(\varepsilon_j m_j, -\varepsilon_i m_i,f)$ of $G$. When $g \in G(\Gamma)$ is fixed, assign the framed oriented curve $\varepsilon_i\varepsilon_j f$ to the unlabelled edge of each $v\in V(\Gamma)$ as above. Then assign to each edge of $\Gamma$ the linking number of the two curves assigned to its half-edges (change a curve $f$ into its parallel $f_{\parallel}$, if the two curves coincide) and define $lk(\Gamma;g)$ as the product over the edges of $\Gamma$ of the associated linking numbers. $$Z_n\left([M_G;U^{(r)}]\right)= \prod_{i=1}^{2n}\frac{q_i}{p_i}
\sum_{\Gamma \in D_{2/3,n}, g \in G(\Gamma)}\frac{lk(\Gamma;g)}{\sharp \mbox{Aut}_{2/3}(\Gamma)}[\Gamma].$$ Now, Lemma \[lemintclas\] easily leads to the conclusion of the proof of Theorem \[thmfasmu\] when the Seifert surfaces are associated to a presentation of the link by a graph that $\mu$-laces the unlink as in Proposition \[propclaspmu\]. Fortunately, this is enough to conclude the proof of Theorem \[thmfasmu\] thanks to the following proposition \[propasmu\] that ensures that the right-hand side of the equality of Theorem \[thmfasmu\] does not depend on the choice of the Seifert surfaces.
[ ]{}
Let $n \in {\mathbb{N}}$. Let $D_{2/3,o,n}$ be the set of $2/3$-labelled unoriented Jacobi diagrams whose labelled edges are oriented. Forgetting the edge orientations transforms an element $\Gamma$ of $D_{2/3,o,n}$ into an element $f(\Gamma)$ of $D_{2/3,n}$, and an element of $D_{2/3,n}$ comes from $\frac{2^{2n}}{\sharp \mbox{Aut}_{2/3}(\Gamma)}$ elements of $D_{2/3,o,n}$.
Let $L=(K_i;p_i/q_i)_{i \in \{1,2,3,\dots, 2n\}}$ be a framed $2n$–component algebraically split link in a rational homology sphere $\;M$. Assume that for any $\{i,j,k\} \subset \{1,2,3,\dots, 2n\}$, $\mu(K_i,K_j,K_k)=0$. Let $(F_i^-)_{i \in \{1,2,3,\dots, 2n\}} \cup (F_i^+)_{i \in \{1,2,3,\dots, 2n\}}$ be a collection of transverse surfaces such that, for any $i$, $F_i^-$ and $F_i^+$ are two Seifert surfaces of $K_i$ that do not meet the $K_j$ for $j \neq i$.
Let $\Gamma \in D_{2/3,o,n}$. Orient $\Gamma$. In such a $\Gamma$ the half-edges of the labelled edges inherit a label from the edge orientation. Namely, Edge $i$ goes from $i^-$ to $i^+$.
For any vertex of $\Gamma$, whose half-edges are labelled by $(i^{\varepsilon},j^{\eta},$ nothing$)$ with respect to the cyclic order induced by the orientation, assign the intersection curve $F_i^{\varepsilon} \cap F_j^{\eta}$ to its unlabelled half-edge. To any unlabelled edge that is now equipped with two intersection curves associate the linking number of these curves. Then define $\ell_{\Gamma}((F_i^-,F_i^+)_{i=1,\dots 2n})$ as the product over all the unlabelled edges of $\Gamma$ of the corresponding linking numbers. Note that $\ell_{\Gamma}((F_i^-,F_i^+)_{i=1,\dots 2n})[\Gamma]$ does not depend on the orientation of $\Gamma$.
When $F_i^+$ is a parallel copy of $F_i^-$, then $$\sum_{\Gamma \in D_{2/3,n}} \frac{\ell((F_i^-)_{i=1, \dots 2n};\Gamma)}{\sharp \mbox{Aut}_{2/3}(\Gamma)}[\Gamma]=\sum_{\Gamma \in D_{2/3,o,n}} \ell_{\Gamma}((F_i^-,F_i^+)_{i=1,\dots 2n})\frac{1}{2^{2n}}[\Gamma]$$
\[propasmu\] With the notation and hypotheses above $$\sum_{\Gamma \in D_{2/3,o,n}} \ell_{\Gamma}((F_i^-,F_i^+)_{i=1,\dots 2n})\frac{1}{2^{2n}}[\Gamma]$$ is independent of the choice of the surfaces $(F_i^-,F_i^+)_{i=1,\dots 2n}$ in the complement of $\cup_{j\neq i}K_j$, it only depends on $L$.
[[Proof: ]{}]{}We study the effect of changing a surface $F_i^{\varepsilon}$ to another Seifert surface $F^{\prime}$ of $K_i$ disjoint from the $K_j$ for $i \neq j$, and transverse to the other ones. Obviously, for any $\Gamma$, the only modified ingredient is the linking number associated with the unlabelled edge $e$ that shares a vertex with $i^{\varepsilon}$ that reads $$\pm lk(F_i^{\varepsilon} \cap S_1, S_2 \cap S_3)$$ where $S_1$, $S_2$ and $S_3$ are the three other surfaces associated to the three other labelled half-edges containing the vertices of $e$.
Let us compute the variation of such a linking number. Recall that $H_2(M \setminus \cup_{j=1,2,\dots,2n}K_j)$ is generated by the homology classes of the boundaries $\partial N(K_j)$ of the tubular neighborhoods of the $K_j$, for $j\neq i$. Therefore the immersed oriented closed surface $(F_i^{\varepsilon} \cup - F^{\prime})$ cobounds a $3$-dimensional chain $C$ with some copies $\partial N(K_j)$. In particular, if $S_1$ is a Seifert surface for $K_{j(1)}$, the boundary of $C\cap S_1$ is the union of $(F^{\prime} \cap S_1 -F_i^{\varepsilon} \cap S_1)$ and some copies of $K_{j(1)}$. Since all the Milnor triple linking numbers vanish, $lk(K_{j(1)},S_2 \cap S_3)=0$, and $$lk(F^{\prime} \cap S_1-F_i^{\varepsilon} \cap S_1, S_2 \cap S_3)
=\pm \langle C\cap S_1, S_2 \cap S_3 \rangle=\pm \langle C, S_1 \cap S_2 \cap S_3 \rangle.$$ Now, consider the two elements of $D_{2/3,o,n}$ obtained from $\Gamma$ by changing the neighborhood of $e$ in $\Gamma$ as in the following figure.
(-1,-.2)(2,1.4) (0,1)(.2,.2) (.5,.5)(.5,1) (1,0)(.5,.5) (0,0)(.5,.5) (.4,.5)[$e$]{} (-.05,1)[$F_i^{\varepsilon}$]{} (-.05,0)[$S_1$]{} (1.05,0)[$S_2$]{} (.55,1)[$S_3$]{}
(-1,-.2)(2,1.4) (.5,.5)(.5,1) (1,0)(.5,.5) (0,0)(.5,.5) (.1,1)(.3,.3)(.8,.2) (-.05,0)[$S_1$]{} (1.05,0)[$S_2$]{} (-.05,1)[$F_i^{\varepsilon}$]{} (.55,1)[$S_3$]{}
(-1,-.2)(2,1.4) (.5,.2)(.5,1) (1,0)(.5,.2) (0,0)(.5,.2) (0,1)(.2,.6)(.7,.6)(.5,.8) (-.05,1)[$F_i^{\varepsilon}$]{} (-.05,0)[$S_1$]{} (1.05,0)[$S_2$]{} (.55,1)[$S_3$]{}
(Actually, since the current definition of Jacobi diagrams does not allow looped edges, some of the above graphs may not be Jacobi diagrams. In order to make this proof work, allow Jacobi diagrams with looped edges, and set them to be zero in ${{\cal A}}_n(\emptyset)$, so that the IHX relations involving such graphs are still valid and these graphs do not contribute to the sum of the statement.)
Assume without loss, that the orientations of the three graphs coincide outside the neighborhood of $e$ and are induced by the figure at the shown vertices. Then the coefficients of these three elements of $D_{2/3,o,n}$ are perturbed in the same way. (Note that we did not need to take care about the above signs, they are well-defined in each step, and the result only depends on the cyclic order of $S_1$, $S_2$, $S_3$.) Since the sum of the corresponding oriented graphs vanishes in ${{\cal A}}_n(\emptyset)$ and since all the graphs of $D_{2/3,o,n}$ can be grouped in three-element sets as above, the sum of the statement is independent of the surfaces. [ ]{}
Similarly, we can show the following proposition.
\[propmilfour\] Let $L=(K_0,K_1,K_2,K_3)$ be a rationally algebraically split link whose three-component sublinks have Milnor triple linking number $0$ in a rational homology sphere $M$. Let $a$, $b$ and $c$ be three real numbers such that $a+b+c=0$. Let $\Sigma_i$ be a Seifert surface for $K_i$ in the exterior of $L\setminus K_i$. Then $$\nu_{abc}(K_0,K_1,K_2,K_3) =$$ $$a lk(\Sigma_0 \cap \Sigma_1, \Sigma_2 \cap \Sigma_3) + b lk(\Sigma_0 \cap \Sigma_2, \Sigma_3 \cap \Sigma_1)
+c lk(\Sigma_0 \cap \Sigma_3, \Sigma_1 \cap \Sigma_2)$$ does not depend on the surfaces $\Sigma_i$ that satisfy the given assumption. The invariant $\nu_{abc}$ satisfies the following properties.
- It is invariant under self-crossing changes of the components of $L$.
- If $M=S^3$, $\nu_{abc}$ is the following combination of the Milnor invariants defined in [@mil], $$\nu_{abc}=b \mu(10,23) -c\mu(01,23).$$ $\mu(01,23)=\nu_{1,0,-1}$ and $\mu(10,23)=\nu_{-1,1,0}$.
[[Proof: ]{}]{}The proof of Proposition \[propasmu\] shows that $\nu_{abc}$ does not depend on the surfaces and that it is therefore well-defined. Let us prove that $\nu_{abc}$ does not vary under self-crossing changes and is therefore a homotopy invariant of these four-component links. To study the effect of a self-crossing change on $K_0$ inside a ball $B$, choose the surfaces $\Sigma_i$ for $i>0$ so that they intersect $B$ as parallel tubes around one strand of $K_0$. Then their intersections like $\Sigma_2 \cap \Sigma_3$ will not meet $B$, and will also bound a surface $\Sigma_{23}$ in the exterior of $K_0$ and $K_1$ that intersects $B$ as parallel tubes around the same strand of $K_0$. Now, $\Sigma_1 \cap \Sigma_{23}$ does not meet $B$, and then $$lk(\Sigma_0 \cap \Sigma_1, \Sigma_2 \cap \Sigma_3)=\pm lk(K_0,\Sigma_1 \cap \Sigma_{23})$$ does not vary under the considered crossing change of $K_0$.
According to [@mil], if the ambient $3$-manifold is $S^3$, there is a bijection from the set of homotopy classes of four-component algebraically split links $L$ whose three-component sublinks have Milnor triple linking number $0$ to ${\mathbb{Z}}\oplus {\mathbb{Z}}$ that maps $L$ to $(\mu(01,23)(L),\mu(10,23)(L))$.
Furthermore, if $(K_0,K_1,K_2)$ is the trivial three-component link with meridians $\alpha_0,\alpha_1,\alpha_2$, and if the homotopy class of $K_{01}$ in the exterior of $(K_0,K_1,K_2)$ reads (with the notation of [@mil]), $$\alpha_{2}^{k_0k_1}=\alpha_0\alpha_1 \alpha_2 (\alpha_0\alpha_1)^{-1}
(\alpha_0 \alpha_2^{-1} \alpha_0^{-1})
\alpha_2(\alpha_1 \alpha_2^{-1} \alpha_1^{-1})
=[\alpha_0,[\alpha_1,\alpha_2]]$$ then $\mu(01,23)(K_0,K_1,K_2,K_{01})=1$ and $\mu(10,23)(K_0,K_1,K_2,K_{01})=0$. More generally, if the homotopy class of $K_{3}$ reads $[\alpha_0,[\alpha_1,\alpha_2]]^{\mu_{01}}[\alpha_1,[\alpha_0,\alpha_2]]^{\mu_{10}}$, then $\mu(01,23)(K_0,K_1,K_2,K_3)=\mu_{01}$ and $\mu(10,23)(K_0,K_1,K_2,K_{3})=\mu_{10}$. The link presented by the following clasper
(-1.2,-.7)(2,1.6) (1,0)[.2]{}[0]{}[92]{} (1,.2)[.2]{}[90]{}[270]{} (1,.2)[.2]{}[-90]{}[90]{} (1,0)[.2]{}[90]{}[360]{} (.5,.7)[.2]{}[0]{}[180]{} (.3,.7)[.2]{}[180]{}[360]{} (.3,.7)[.2]{}[0]{}[180]{} (.5,.7)[.2]{}[180]{}[0]{} (1.5,1.2)[.2]{}[-2]{}[180]{} (1.3,1.2)[.2]{}[180]{}[0]{} (1.3,1.2)[.2]{}[0]{}[180]{} (1.5,1.2)[.2]{}[180]{}[360]{} (-.2,0)[.2]{}[180]{}[90]{} (-.2,.2)[.2]{}[90]{}[270]{} (-.2,.2)[.2]{}[-90]{}[90]{} (-.2,0)[.2]{}[90]{}[180]{} (-.2,0)[.2]{}[170]{}[190]{} (-.5,1.2)[.2]{}[0]{}[180]{} (-.7,1.2)[.2]{}[178]{}[0]{} (-.7,1.2)[.2]{}[0]{}[180]{} (-.5,1.2)[.2]{}[180]{}[360]{} (.7,.7)(1,.7)(1.3,1) (1,.7)(1,.4) (-.2,.7)(.1,.7) (-.2,.4)(-.2,.7)(-.5,1) (1.3,0)[$U_{01}$]{} (-.5,0)[$U_{1}$]{} (1.8,1.2)[$U_{0}$]{} (-1,1.2)[$U_{2}$]{}
has the same properties than $(K_0,K_1,K_2,K_{01})$ and, according to Lemma \[lemintclas\], $$\nu_{abc}((K_0,K_1,K_2,K_{01}))=-c.$$ More generally, if the homotopy class of $K_{3}$ reads $[\alpha_0,[\alpha_1,\alpha_2]]^{\mu_{01}}[\alpha_1,[\alpha_0,\alpha_2]]^{\mu_{10}}$, then $$\nu_{abc}(K_0,K_1,K_2,K_{3})=(b\mu_{10}- c\mu_{01})(K_0,K_1,K_2,K_{3}).$$ [ ]{}
On the polynomial form of the knot surgery formula: proofs and remarks
======================================================================
\[secproofpol\]
[Proof of Theorem \[thmpol\]:]{} Since the theorem easily follows from Theorem \[thmfboun\] for $n=1$, we assume $n \geq 2$. First assume $(p,q)=(1,0)$. A $\frac{1}{r}$-surgery on $K$ is equivalent to $|r|$ $\mbox{sign}(r)$-surgeries on parallel copies on $K$. These parallel copies form an $|r|$-component boundary link $L$ bounding parallel copies of $F$. We have $$M(K;\frac{1}{r})= \sum_{J \subset L} (-1)^{\sharp J} [M;J]$$ Up to elements of $\mbox{Ker}(Z_n)$, we only consider the sublinks $J$ of $L$ with at most $n$ components, according to Theorem \[thmfboun\]. There are $$\left(\begin{array}{c}|r|\\{j}\end{array}\right)=\frac{|r|(|r|-1)\dots(|r|-j+1)}{j!}$$ sublinks $J$ of $L$ with $j$ components and they are all isomorphic to the boundary link $L_j$ whose components are framed by $\mbox{sign}(r)$. This shows that $$\begin{array}{lll}
Z_n(M(K;\frac{1}{r}))-Z_n(M)&=\sum_{i=1}^n Y_{n,0}^{(i)}(K \subset M) r^i &\mbox{if}\; r \geq 0\\
&=\sum_{i=1}^n Y_{n,0}^{(i)-}(K \subset M) r^i &\mbox{if}\; r \leq 0
\end{array}$$ where $$Y_{n,0}^{(n)-}=Y_{n,0}^{(n)}=\frac{(-1)^n}{n!}Z_n\left([M;L_n]\right)$$ is given by Theorem \[thmfboun\]. Now, we prove that the two polynomial expressions, the one for $r>0$ and the one for $r<0$, coincide. Applying the above result to $M(K;\frac{1}{r_0})$ with $r_0 <-n$, implies that for any $r\geq r_0$, $$Z_n(M(K;\frac{1}{r}))-Z_n(M(K;\frac{1}{r_0}))=\sum_{i=1}^n Y_{n,0}^{(i)}(K \subset M(K;\frac{1}{r_0})) (r-r_0)^i.$$ The above result also implies that $Z_n(M(K;\frac{1}{r}))-Z_n(M(K;\frac{1}{r_0}))$ is $$\begin{array}{ll}\sum_{i=1}^n Y_{n,0}^{(i)}(K \subset M) r^i +Z_n(M)-Z_n(M(K;\frac{1}{r_0}))&\mbox{if}\; r \geq 0\\
\sum_{i=1}^n Y_{n,0}^{(i)-}(K \subset M) r^i +Z_n(M)-Z_n(M(K;\frac{1}{r_0}))&\mbox{if}\; r \leq 0. \end{array}$$ Therefore the coefficients of the two polynomials coincide. This proves the existence of the polynomial expression with its given leading term for $(p,q)=(1,0)$. Applying this result in $M(K;\frac{p}{q})$ and using the fact that a $\frac{p}{q+rp}$-surgery on $K$ is equivalent to a $p/q$-surgery on $K$ and a $1/r$ surgery on a parallel copy on $K$ gives a similar polynomial expression for $Z_n(M(K;\frac{p}{q+rp}))-Z_n(M(K;\frac{p}{q}))$ with the same leading coefficient since, according to Theorem \[thmfboun\], $Z_n([M;L_n])=Z_n([M(K;\frac{p}{q});L_n])$. Now, up to polynomials in $r$ of degree less than $(n-1)$, $$Z_n(M(K;\frac{p}{q+rp}))-Z_n(M)
=\frac{(-1)^n}{n!}Z_n\left([M;L_n]\right) r^n$$ $$+
\left(\frac{n(1-n)}{2}\frac{(-1)^n}{n!}Z_n\left([M;L_n]\right) + \frac{(-1)^{n-1}}{(n-1)!}Z_n\left([M(K;\frac{p}{q});L_{n-1}]\right)\right) r^{n-1}$$ $$=Y_{n,0}^{(n)}(K \subset M) (r^n + \frac{n q}{p} r^{n-1})
+ Y_{n,q/p}^{(n-1)}(K \subset M) r^{n-1}.$$ Thus $$Y_{n,q/p}^{(n-1)}(K \subset M) + n \frac{(-1)^nq}{n!p}Z_n\left([M;L_n]\right)$$ $$=\frac{(-1)^{n-1}}{(n-1)!}Z_n\left([M(K;\frac{p}{q});L_{n-1}]\right)
+\frac{1}{2}\frac{(-1)^{n-1}(n-1)}{(n-1)!}Z_n\left([M;L_n]\right).$$ Then $$Y_{n,q/p}^{(n-1)}(K \subset M) - Y_{n,0}^{(n-1)}(K \subset M)$$ $$= \frac{(-1)^{n-1}}{(n-1)!} \left(\frac{q}{p} Z_n\left([M;L_n]\right)
+Z_n\left([M(K;\frac{p}{q});L_{n-1}]-[M;L_{n-1}]\right)\right)$$ where $$Z_n\left([M(K;\frac{p}{q});L_{n-1}]-[M \sharp L(p,-q);L_{n-1}]\right)=-\frac{q}{p}Z_n([M;L_n])$$ by Theorem \[thmfboun\], and by additivity of $p^c(Z_n)=Z_n^c$ under connected sum, since $n\geq 2$, $$Z_n^c([M \sharp L(p,-q);L_{n-1}])=Z_n^c[M ;L_{n-1}].$$ Therefore $Y_{n,q/p}^{(n-1)^c}(K \subset M) = Y_{n,0}^{(n-1)^c}(K \subset M)$.\
The behaviour of $Y_{n,q/p}^{(i)}(K \subset M)$ under an orientation change of $M$ comes from the fact that $Z_n(-M)=(-1)^nZ_n(M)$, and the other assertions are easy to observe. [ ]{}
It is easy to see that $\langle \langle\; \bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\rangle \rangle$ is an invariant of the knot. First, it does not depend on the symplectic bases chosen for the Seifert surfaces because $H_1(F)$ may be identified to $H_1(F)^{\ast}$ via $(x \mapsto \langle x,. \rangle)$, and therefore the tensor $\left(\sum_i x_i \otimes y_i-\sum_i y_i \otimes x_i\right)$ may be identified with the intersection form of the surface that lives in $H_1(F)^{\ast} \otimes H_1(F)^{\ast}$. Now, $\langle \langle \bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\;\rangle \rangle$ is invariant under the addition of a hollow handle. (See [@go p. 27] or [@KeL] for a reference for the fact that for two Seifert surfaces of a knot $K$, there exists a third Seifert surface of $K$ that is obtained from the two former ones by adding hollow handles.) Indeed let $m$ be a meridian of a one-handle whose boundary is the union of the hollow handle and two disks, and let $\ell$ be a dual curve for it with respect to the intersection form of the stabilized surface $F$. Since the innermost copy of $m$ does not link any curve of the other copies of $F$, the pair $(m,\ell)$ does not contribute to the pairing. Now, the next innermost meridian does not link any other curve either... In such a way, it is easily seen that the pairs $(m,\ell)$ can be forgotten and this shows that $\langle \langle \;\bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\rangle \rangle$ is invariant under a stabilization of $F$ by addition of a hollow handle.
[Proof of Proposition \[propzsing\]:]{} Let $K_0=K_{\emptyset}$ be the positive desingularisation of $K^s$. Let $U^{(k)}$ be the trivial link that bounds a disjoint union of disks $D_i$ such that each $D_i$ meets $K^s$ exactly at one double point, and $\partial D_i$ does not algebraically link $K_0$, so that each desingularisation of $K^s$ is obtained from $K_0$ by surgery on a subset of $L=\{(U_i;-1)\}_{i \in \{1,\dots,k\}}$. Then $$\sum_{i=0}^n Y_{n,q/p}^{(i)}(K^s \subset M) (r+\frac{q}{p})^i=Z_n([M(K_0;\frac{p}{q+rp});L]).$$
Each $U_i$ bounds a genus one surface $\Sigma_i$ in $M \setminus K_0$ obtained from $D_i$ by tubing $K_0$, say in the $K^{\prime}_i$ part, where we fix the choice of the $K^{\prime}_i$ so that for any pair $\{i,j\}$, $K^{\prime}_i \cap K^{\prime}_j$ is connected.
Let us prove that such a choice is indeed possible for the $K^{\prime}_i$. Fixing the choice of $K^{\prime}_i$ amounts to choosing an interval of the circle between the two preimages of the double point $i$. If some of the two possible intervals for a double point $i$ does not contain a pair of preimages for another double point, pick such an interval. In the next steps, if some of the two intervals for a double point $i$ only contains pairs of preimages for another double point together with their associate already chosen intervals, then pick such an interval. It is easy to see that this process will stop when all the $K^{\prime}_i$ are chosen so that for any pair $\{i,j\}$, $K^{\prime}_i \cap K^{\prime}_j$ is connected.
Now, assume that the diameters of the tubes are all constant and different and that the tube for $U_j$ is thinner than the tube for $U_i$, if $K^{\prime}_j$ contains the two preimages of the double point $i$.
(-.4,-.4)(1.4,1.4) (.25,.05)(-.15,.25)(.25,.45)(.65,.25)(.25,.05) (-0.15,-0.15)(0.65,0.65) (-0.15,0.65)(0.65,-0.15) (0.25,.25) (.25,.45)(-.15,.25)(.25,.05)(.65,.25)(.25,.45) (.75,.25)[$\partial D_j$]{}
(-.9,-.4)(2,1.4) (.25,.05)(-.25,.25)(.25,.45)(.75,.25)(.25,.05) (0.05,0.25)(0.05,0.45) (0.05,-0.25)(0.05,0.45) (0.05,0.05)(0.05,0.45)(.25,.9)(0.45,0.45)(0.45,0.05) (0.05,0.05)(0.05,0.45)(.25,.9)(0.45,0.45)(0.45,0.05) (0.45,0.45)(0.45,0.25) (0.45,0.45)(0.45,-0.25) (.25,.45)(-.25,.25)(.25,.05)(.75,.25)(.25,.45) (.85,.25)[$\partial D_j=\partial \Sigma_j$]{} (.5,-.25)[$K_0$]{} (0.05,0.25)[2pt]{}[-180]{}[0]{} (0.45,0.25)[2pt]{}[-180]{}[0]{} (0.05,0.25)[4pt]{}[0]{}[180]{} (0.45,0.25)[4pt]{}[0]{}[180]{}
(-3,-.4)(1.5,1.4) (.45,-.1)(-.35,.25)(.45,.6)(1.15,.25)(.45,-.1) (0.05,0.65)[4pt]{}[0]{}[180]{} (.05,-0.25)(0.05,0.65) (.05,.25)(0.05,0.65) (0.05,-0.25)(0.05,0.65) (0.05,0.05)(0.05,0.65)(.45,1.3)(0.85,0.65)(0.85,0.05) (0.05,0.05)(0.05,0.65)(.45,1.3)(0.85,0.65)(0.85,0.05) (0.85,0.65)(0.85,0.25) (0.85,0.65)(0.85,-0.25) (.45,.6)(-.35,.25)(.45,-.1)(1.15,.25)(.45,.6) (1.25,.25)[$\partial \Sigma_j$]{} (.9,-.25)[$K_0$]{} (0.05,0.25)[2pt]{}[-180]{}[0]{} (0.05,0.25)[4pt]{}[-180]{}[0]{} (0.85,0.25)[2pt]{}[-180]{}[0]{} (0.05,0.25)[2pt]{}[0]{}[180]{} (0.05,0.25)[4pt]{}[0]{}[180]{} (0.85,0.25)[2pt]{}[0]{}[180]{} (0.05,-0.25)[4pt]{}[-180]{}[0]{} (0.05,0.65)[4pt]{}[-180]{}[0]{} (0.05,-0.25)[4pt]{}[0]{}[180]{} (-.15,-.25)[$\Sigma_i$]{} (-.4,.65)[$\Sigma_i \cap \Sigma_j$]{} (-.35,.5)(-.1,.2)
Then $\Sigma_i \cap \Sigma_j$ is empty if the pair $(D_i \cap K_0)$ does not link the pair $(D_j \cap K_0)$, and it is a meridian curve of $K_0$ in $D_j$ otherwise. Therefore, the $\mu$-invariants of the three-component sublinks of $L$ in $M(K_0;\frac{p}{q+rp})$ are zero and Theorem \[thmfasmu\] can be used to compute $Z_n([M(K_0;\frac{p}{q+rp});L])$.
In particular, if $k>2n$, $Z_n([M(K_0;\frac{p}{q+rp});L])=0$. Since the linking numbers between two intersection curves will be $\pm \frac{q+rp}{p}$ or zero, if $k=2n$, $Z_n([M(K_0;\frac{p}{q+rp});L])$ is a monomial in $\left(\frac{q+rp}{p}\right)^n$. [ ]{}
[Proof of Proposition \[propzsingun\]:]{} In this case, the link $U^{(k)}$ of the previous proof is a boundary link in $M(K_0;\frac{p}{q+rp})$ because the produced genus one surfaces are disjoint. Then Theorem \[thmfboun\] can be applied. It implies the first part of the proposition. Use bases $(m_i, \ell_i)$ for the Seifert surfaces where $m_i$ is a $0$-framed meridian of $K_0$, and $\ell_i$ is a curve along the tube of $\Sigma_i$ and $D_i$ that is homotopic to $K^{s\prime}_i$, and that does not link $K_0$. In $M(K_0;\frac{p}{q+rp})$, the linking number of two meridians is $\pm \frac{q+rp}{p}$, the linking number of a meridian and a longitude is $0$ or $\pm 1$ while the linking number of two longitudes is their linking number in $M$. Note that if one tube for $\Sigma_j$ goes inside another one for $\Sigma_i$ (if $K^{s,i}_j \neq K^{s\prime}_j$), and if $K^{\prime}_j$ is the positive desingularisation of $K^{s\prime}_j$ then $lk(\ell_i,\ell_j)=lk(\ell_i,K^{\prime}_j)=-lk(\ell_i,K_0-K^{\prime}_j)=-\ell_{ij}(K^s)$. There are at most $n$ pairings of meridians. Furthermore, since there is at least one innermost meridian that cannot be paired with a longitude, there is at least one pairing of meridians. Now, the number of pairs of meridians coincides with the number of pairs of longitudes in a pairing. [ ]{}
As an example, we compute $Y_{2}^{(i)c}(K^s)$ where $K^s$ is a singular knot with two unlinked double points.
\[propvarztwo\] Let $K^s$ be a singular knot with two unlinked double points. $$\sum_{I \subset \{1,2\}}(-1)^{\sharp I}Z^c_2(M(K_I;\frac{1}{r}))=$$ $$\frac{1}{4}\left(\left( 5 \ell_{12}(K^s)^2 + 2\ell_{11}(K^s)\ell_{22}(K^s) \right)r^2
-\ell_{12}(K^s)r\right){\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.1,0)(.2,.2)(.1,.4)
\pscurve(.5,0)(.4,.2)(.5,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}}.$$
[[Proof: ]{}]{}Use the strategy and the notation of the proofs of the two previous propositions. Choose Seifert surfaces of the two knots of the crossing changes with disjoint tubes whose longitudes $\ell_1$ and $\ell_2$ are homotopic to $K^{s,2}_1$ and $K^{s,1}_2$, respectively, so that $$lk(\ell_1,\ell_2)=lk(\ell_1,\ell_2^+)=lk(\ell_1^+,\ell_2)=\ell_{12}(K^s)$$ $$lk(m_1,m_2)=-r=lk(m_i,m_i^+)$$ $$lk(\ell_i,\ell_i^+)=-\ell_{ii}(K^s)$$ $$lk(m_i,\ell_i^+)=lk(m_1,\ell_2)=lk(m_2,\ell_1)=0$$ $$lk(m_i^+,\ell_i)=1.$$ Then, according to Theorem \[thmfboun\], $$\sum_{I \subset \{1,2\}}(-1)^{\sharp I}Z^c_2(M(K_I;\frac{1}{r}))=\frac{1}{4} p^c\left(\langle \langle \begin{pspicture}[0.5](-.2,-.5)(1.2,1.2)
\psline{-}(0.05,0)(.35,.15)
\psline{*-*}(0.05,0)(.05,.8)
\psline{-}(0.05,0)(.35,-.15)
\psline{-}(0.05,.8)(.35,.95)
\psline{-}(0.05,.8)(.35,.65)
\rput[lt](.5,-.15){$m_1$}
\rput[l](.5,.15){$\ell_1$}
\rput[l](.5,.65){$\ell_1^+$}
\rput[lb](.5,.95){$m_1^+$}
\end{pspicture}
\begin{pspicture}[0.5](-.2,-.5)(1.2,1.2)
\psline{-}(0.05,0)(.35,.15)
\psline{*-*}(0.05,0)(.05,.8)
\psline{-}(0.05,0)(.35,-.15)
\psline{-}(0.05,.8)(.35,.95)
\psline{-}(0.05,.8)(.35,.65)
\rput[lt](.5,-.15){$m_2$}
\rput[l](.5,.15){$\ell_2$}
\rput[l](.5,.65){$\ell_2^+$}
\rput[lb](.5,.95){$m_2^+$}
\end{pspicture} \rangle \rangle \right).$$
Note that $m_1$ and $m_2$ must be paired to another meridian. Then the right-hand side of the equality can be rewritten as $$\frac{r^2}{4} p^c\left(\langle \langle \begin{pspicture}[0.3](-.35,0)(2.1,1)
\pscurve(1.85,0)(2.1,.4)(1.85,.8)
\psline{*-*}(1.85,0)(1.85,.8)
\pscurve(0.05,0)(-.2,.4)(.05,.8)
\psline{*-*}(0.05,0)(.05,.8)
\psline{-}(0.05,0)(.35,.15)
\psline{-}(1.85,.8)(1.55,.65)
\psline{-}(1.85,0)(1.55,.15)
\psline{-}(0.05,.8)(.35,.65)
\rput[l](.5,.15){$\ell_1$}
\rput[l](.5,.65){$\ell_1^+$}
\rput[r](1.5,.15){$\ell_2$}
\rput[r](1.5,.65){$\ell_2^+$}
\end{pspicture}
+\begin{pspicture}[0.3](-.2,0)(2,1)
\pscurve(1.85,0)(.95,-.15)(.05,0)
\psline{*-*}(1.85,0)(1.85,.8)
\pscurve(0.05,.8)(.95,.95)(1.85,.8)
\psline{*-*}(0.05,0)(.05,.8)
\psline{-}(0.05,0)(.35,.15)
\psline{-}(1.85,.8)(1.55,.65)
\psline{-}(1.85,0)(1.55,.15)
\psline{-}(0.05,.8)(.35,.65)
\rput[l](.5,.15){$\ell_1$}
\rput[l](.5,.65){$\ell_1^+$}
\rput[r](1.5,.15){$\ell_2$}
\rput[r](1.5,.65){$\ell_2^+$}
\end{pspicture}
+ \begin{pspicture}[0.3](-.2,0)(2,1)
\pscurve(1.85,0)(.95,-.15)(.05,0)
\psline{*-*}(1.85,0)(1.85,.8)
\pscurve(0.05,.8)(.95,.95)(1.85,.8)
\psline{*-*}(0.05,0)(.05,.8)
\psline{-}(0.05,0)(.35,.15)
\psline{-}(1.85,.8)(1.55,.65)
\psline{-}(1.85,0)(1.55,.15)
\psline{-}(0.05,.8)(.35,.65)
\rput[l](.5,.15){$\ell_1$}
\rput[l](.5,.65){$\ell_1^+$}
\rput[r](1.5,.15){$\ell_2^+$}
\rput[r](1.5,.65){$\ell_2$}
\end{pspicture}
\rangle \rangle \right) -\frac{r}{4}\ell_{12}(K^s) {\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.1,0)(.2,.2)(.1,.4)
\pscurve(.5,0)(.4,.2)(.5,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}}.$$ Indeed, either two pairs of meridians are paired together. This leads to the quadratic contribution in $r$ above, or there is one pair of meridians, it is necessarily $(m_1,m_2)$ and in this case $m_1^+$ must be paired with $\ell_1$ and $m_2^+$ must be paired with $\ell_2$. Then $\ell_1^+$ and $\ell_2^+$ must be paired together, and this provides the linear contribution above. [ ]{}
Computation of the Casson-Walker knot invariant
===============================================
\[secproofcasone\]
Let $K$ be an order $O_K$ knot in a rational homology sphere $M$. Let $\widetilde{M\setminus K}$ be the infinite cyclic covering of $M\setminus K$. Denote the action of the homotopy class of the meridian of $K$ on $H_1(\widetilde{M\setminus K};{\mathbb{Q}})$ as the multiplication by $t$ so that a generator of $H_1(M\setminus K)/\mbox{Torsion}$ acts as the multiplication by $t^{1/O_K}$. As in [@pup Chapter 2], define the [*Alexander polynomial*]{} $\Delta(K)$ of $K$ as the order of the ${\mathbb{Q}}[t^{\pm 1/O_K}]$-module $H_1(\widetilde{M\setminus K};{\mathbb{Q}})$ normalized so that $$\Delta(K)(1)=|\mbox{Torsion}(H_1(M\setminus K))|=\frac{|H_1(M)|}{O_K}\; \mbox{and} \;\Delta(K)(t^{1/O_K})=\Delta(K)(t^{-1/O_K}).$$
Then the formula of [@pup p 12-13] implies the following lemma.
\[lemcaschirknot\] For any knot $K$ such that $lk(K,K) \in {\mathbb{Z}}$ in a rational homology sphere $M$, for any pair $(p,q)$ of coprime integers such that $q\neq 0$. $$\lambda(M(K;p/q))-\lambda(M)=\frac{q}{p}\left(\frac{O_K}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2} - \frac{1}{24O_K^2} + \frac{1}{24}\right) +\lambda(L(p,-q)).$$
[[Proof: ]{}]{}Recall that $\lambda(M)=\frac{\overline{\lambda}(M)}{|H_1(M)|}$ where $|H_1(M)|$ is the cardinality of $H_1(M;{\mathbb{Z}})$ and $\overline{\lambda}$ is the extension of $|H_1(M)|\lambda$ to oriented closed $3$-manifolds that is denoted by $\lambda$ in [@pup]. For any knot $K$ in a rational homology sphere $M$, according to [@pup 1.4.8,T2], for $q>0$, $$\lambda(M(K;p/q))-\lambda(M)=$$ $$\frac{q}{p} \left( \frac{O_K}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2} - \frac{1}{24O_K^2} -\frac{p^2+1}{24q^2}\right) +\frac{\mbox{sign}(pq)}{8} + \frac{s(p-qlk(K,K),q)}{2}$$ where the Dedekind sum $s(p-qlk(K,K),q)$ is defined in [@rg] (and in [@pup 1.4.5]). This formula makes clear that $$\lambda(M(K;p/q))-\lambda(M)=\frac{q}{p}\left(\frac{O_K}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2} - \frac{1}{24O_K^2} + \frac{1}{24}\right) +f(p,q)$$ for some $f(p,q)$ that depends neither on the knot $K$ with self-linking number $0$ nor on its ambient manifold $M$. Applying this formula to the trivial knot $U$ of $S^3$ concludes the proof of the lemma. [ ]{}
We now express $\Delta(K)$ from the Seifert form of a Seifert surface for $K$ in the following proposition.
\[propalexseif\] Let $K$ be a knot of order $d$, with self-linking number $(-a/b) \in {\mathbb{Q}}/{\mathbb{Z}}$, where $a$ and $b$ are coprime integers, in a rational homology sphere $M$. Let $N(K)$ be a tubular neighborhood of $K$. Let $\Sigma$ be a surface in $M$ whose boundary is made of $(d/b)$ parallel copies of a primitive curve of $\partial N(K)$. Let ${\cal B}_s$ be a symplectic basis for $H_1(\Sigma)/H_1(\partial \Sigma)$, and let $$\Delta_{\Sigma}(\tau)=\mbox{det} [lk(\tau^{1/2} b^{\prime +} - \tau^{-1/2}b^{\prime -},b)]_{(b, b^{\prime}) \in {\cal B}_s^2}$$ where $b^{\prime +}$ (resp. $b^{\prime -}$) is a representative of $b^{\prime}$ pushed away from $\Sigma$ in the direction of the positive (resp. negative) normal direction to $\Sigma$. Then $$\frac{d}{|H_1(M)|}\Delta(K)=\Delta_{\Sigma}(t^{1/d})\frac{b}{d}\frac{t^{\frac{1}{2b}}-t^{-\frac{1}{2b}}}{t^{\frac{1}{2d}}-t^{-\frac{1}{2d}}}.$$
[[Proof: ]{}]{}First assume that the self-linking number of $K$ is zero. Let $N(K)$ be a tubular neighborhood of $K$. There exists a genus $g$ surface $\Sigma$ in $M$ whose boundary is made of $d$ parallel copies of $K$. Consider a collar $\Sigma \times [-1,1]$ in $M$ such that $$(\Sigma \times [-1,1]) \cap N(K) =\partial \Sigma \times [-1,1].$$
Let $Y=M \setminus (N(K) \cup \Sigma \times ]-1,1[)$.
The infinite cyclic covering $\tilde{X}$ of $M \setminus N(K)$ can be seen as $$\left(\coprod_{k\in {\mathbb{Z}}}h^k(Y)
\coprod \coprod_{k\in {\mathbb{Z}}}
h^k(\Sigma \times [-1,1])\right)/\cong$$ where $h$ is a generator of the group of automorphisms of the covering $\tilde{X}$ and $\cong$ provides the following identifications. $$h^k\left((\sigma \in \Sigma, 1) \in Y\right) \cong h^k\left((\sigma \in \Sigma, 1) \in (\Sigma \times [-1,1])\right)$$ $$h^k\left((\sigma \in \Sigma, -1) \in Y \right) \cong h^{k+1}\left((\sigma \in \Sigma, -1) \in (\Sigma \times [-1,1])\right).$$
Then it is easy to see that, if the action of $h$ on $H_1(\tilde{X};{\mathbb{Q}})$ is denoted as a multiplication by $\tau$, $$H_1(\tilde{X};{\mathbb{Q}})=\frac{H_1(Y;{\mathbb{Q}}) \otimes {\mathbb{Q}}[\tau,\tau^{-1}]}{(\oplus_{b\in {\cal B}}(\tau b^+-b^-) {\mathbb{Q}})\otimes {\mathbb{Q}}[\tau,\tau^{-1}]},$$ as a ${\mathbb{Q}}[\tau,\tau^{-1}]$-module, where ${\cal B}$ is a basis of $H_1(\Sigma)$ and, for $b \in {\cal B}$, $b^+$ (resp. $b^-$) denotes the class of $b$ in $H_1(\Sigma \times \{1\})$ (resp. in $H_1(\Sigma \times \{-1\})$).
In particular, if ${\cal C}$ is a basis of $H^1(Y;{\mathbb{Q}})$, then $$\Delta(K)(\tau=t^{1/d})=\mbox{det}\left[\tau^{1/2} c(b^+) -\tau^{-1/2} c(b^-)\right]_{(c,b)\in {\cal C} \times {\cal B}}$$ up to a multiplication by a unit of ${\mathbb{Q}}[\tau,\tau^{-1}]$.\
[*Computation of $H^1(Y;{\mathbb{Q}})$.*]{}\
Let $Z=M \setminus (\Sigma \times ]-1,1[)$.
The collar $\Sigma \times [-1,1]$ is a genus $(2g+d-1)$-handlebody whose $H_1$ has a basis ${\cal B}$ made of the classes $\ell_1, \ell_2, \dots, \ell_{d-1}$ of $(d-1)$ boundary components of $\Sigma$, and a symplectic basis ${\cal B}_s$ for $H_1(\Sigma)/H_1(\partial \Sigma)$. Therefore, $Z$ has the rational homology of a genus $(2g+d-1)$-handlebody and $H^1(Z;{\mathbb{Q}})$ is freely generated by the linking numbers with the elements of ${\cal B}$.
Use the following exact sequence to compute $H^1(Y;{\mathbb{Q}})$
$$H^1(Z,Y;{\mathbb{Q}}) \hookrightarrow H^1(Z;{\mathbb{Q}}) \rightarrow H^1(Y;{\mathbb{Q}}) \rightarrow
H^2(Z,Y;{\mathbb{Q}}) \rightarrow 0.$$
The pair $(Z,Y)$ has the homology of the pair $(N(K), \partial N(K) \setminus (\partial \Sigma \times [-1,1]))$ where $\partial N(K) \setminus (\partial \Sigma \times [-1,1])$ is a disjoint union of $d$ annuli $A(\ell^{++}_i)$ whose cores are parallels $\ell^{++}_1, \ell^{++}_2, \dots, \ell^{++}_{d}$ of $K$, and such that $$\partial A(\ell^{++}_i) = \ell^+_{i} -\ell^-_{i+1}$$ (where $\ell^-_{d+1}=\ell^-_1$). In particular, $$\begin{array}{lll}H_j(Z,Y)&=0\;&\mbox{if}\; j \neq 1,2\\
&=\oplus_{i=2}^d {\mathbb{Z}}c_i\;&\mbox{if}\; j =1\\
&=\oplus_{i=2}^d {\mathbb{Z}}B_i\;&\mbox{if}\; j =2 \end{array}$$ where $c_i$ is the class of a path from $\ell^{++}_{1}$ to $\ell^{++}_{i}$ in $N(K)$, and $B_i$ is the class of an annulus whose boundary is $(\ell^{++}_{i}-\ell^{++}_{1})$.
The image of $H^1(Z,Y;{\mathbb{Q}}) \hookrightarrow H^1(Z;{\mathbb{Q}})$ is freely generated by the algebraic intersections $\langle .,-A(\ell^{++}_i)\rangle=lk(.,\ell^-_{i+1} -\ell^+_{i})$ for $i\in {2, \dots,d}$.
For $i\geq 2$, consider a curve $e_i$ that goes from $\ell_{i}$ to $\ell_{i+1}$ in $\Sigma$ and that avoids the chosen geometric symplectic basis of $H_1(\Sigma)/H_1(\partial \Sigma)$, and consider a closed loop $\mu_i$ in $N(K) \cup \Sigma \times [-1,1]$ that equals $e_i$ outside $N(K)$.
(-1,-.3)(4.2,3) (2.2,.7)(2.2,1.35) (2.2,1.35)(2.2,2) (2.2,2)(1.65,2.325) (1.65,2.325)(1.1,2.65) (1.1,2.65)(.55,2.325) (.55,2.325)(0,2) (0,2)(0,1.35) (0,1.35)(0,.7) (0,.7)(.55,.375) (.55,.375)(1.1,.05) (1.1,.05)(1.65,.375) (1.65,.375)(2.2,.7) (2.3,.7)[$\ell_1^-$]{} (2.3,2)[$\ell_1^+$]{} (1.7,2.4)[$\ell_1^{++}$]{} (-.1,1.35)[$\ell_2^{++}$]{} (1.7,.35)[$\ell_3^{++}$]{} (.2,2.4)[$\ell_2$]{} (.6,.25)[$\ell_3$]{} (-1.1,-1.088)(0,-.113)(.55,.375)(.7,1.35)(.55,2.325) (.55,.375)(.7,1.35)(.55,2.325)(0,2.813)(-1.1,3.788) (.7,1.4)[$\mu_2$]{} (4.1,1.35)(3,1.35)(2.2,1.35)(1.25,1.11)(.55,.375) (2.2,1.35)(1.25,1.11)(.55,.375)(0,-.113) (1.3,1.1)[$\mu_3$]{}
Then $lk(\partial B_i=\ell^{++}_{i}-\ell^{++}_{1},\mu_j)=\delta_{ij}$. Therefore the map $H^1(Y;{\mathbb{Q}}) \rightarrow
H^2(Z,Y;{\mathbb{Q}})$ admits a section whose image is $\oplus_{i=2}^d{\mathbb{Q}}lk(.,\mu_i)$.
Thus $$H^1(Y;{\mathbb{Q}})=\oplus_{b \in {\cal B}_s}{\mathbb{Q}}lk(.,b) \oplus \oplus_{i=2}^d {\mathbb{Q}}lk(.,\mu_i).$$ Since $lk(\ell_i^{\pm},b)=0$ for any $b \in {\cal B}_s$, up to units of ${\mathbb{Q}}[t^{\pm 1/O_K}]$, $$\Delta(K)=\Delta_{\Sigma}(\tau) \Delta(d)$$ with $$\Delta(d)=
\mbox{det} [lk(\tau^{1/2} \ell_i^+ - \tau^{-1/2}\ell_i^-,\mu_j)]_{(i,j) \in \{2,\dots,d\}^2}$$ where $\ell_i^+=\ell^{++}_{i}$ and $\ell_i^-= \ell^{++}_{i-1}$.
$$\Delta(d)=\frac{\tau^{d/2}-\tau^{-d/2}}{d(\tau^{1/2}-\tau^{-1/2})}.$$
[Proof of the sublemma:]{} By pushing $\mu_j$ along the negative normal of the Seifert surface of $\ell^{++}_{1}$ (or $K$) we see that $lk(\ell^{++}_{1},\mu_j)=-\frac{1}{d}$.
Set $z=\tau^{1/2}-\tau^{-1/2}$ and $\rho=\tau^{1/2}$, $\Delta(d)$ is the determinant of the following matrix $[\Delta_{ij}]_{(i,j) \in \{2,\dots,d\}^2}$ where $$\Delta_{2j}=lk(\rho \ell_2^{++} - \rho^{-1}\ell_1^{++},\mu_j)= \rho\delta_{2j}-\frac{z}{d},$$ and for $i>2$ $$\Delta_{ij}=lk(\rho (\ell_i^{++}-\ell_2^{++}) - \rho^{-1}(\ell^{++}_{i-1}-\ell_1^{++}),\mu_j)=\rho(\delta_{ij}-\delta_{2j}) -\rho^{-1}\delta_{(i-1)j},$$ that is for $d=5$, $$[\Delta_{ij}]=\left[\begin{array}{cccc}
\rho-\frac{z}{d}&-\frac{z}{d}&-\frac{z}{d}&-\frac{z}{d}\\
-\rho-\rho^{-1}&\rho&0&0\\
-\rho&-\rho^{-1}&\rho&0\\
-\rho&0&-\rho^{-1}&\rho\end{array}
\right],$$ $\Delta(2)=\frac{\rho+\rho^{-1}}{2}$ and $\Delta(3)=\frac{\tau+\tau^{-1}+1}{3}.$
In general the development with respect to the first column gives that $$\Delta(d)=(\rho-\frac{z}{d}) \rho^{(d-2)}
-\frac{z}{d}(\rho+\rho^{-1}) \left( \sum_{j=3}^d \rho^{(d-j)-(j-3)} \right)
-\frac{z}{d} \rho \sum_{i=4}^d \rho^{i-3} \sum_{j=i}^d \rho^{(d-j)-(j-i)}$$ where $$\rho^{-1}\left( \sum_{j=3}^d \rho^{(d-j)-(j-3)} \right)=
\sum_{j=3}^d \rho^{(d-2j+2)}=\rho^{(2-d)} + \rho^{(4-d)} + \dots + \rho^{(d-4)}
.$$ Thus, $$\Delta(d)=\rho^{(d-1)}- \frac{z}{d} \rho^{(d-2)} -\frac{ \rho^{(d-3)} - \rho^{(1-d)}}{d}
-\frac{z}{d} \rho^{-2} \sum_{i=3}^d \sum_{j=i}^d \rho^{(d+2i-2j)}.$$ $$\sum_{i=3}^d \sum_{j=i}^d \rho^{(d+2i-2j)}=(d-2)\rho^{d} + (d-3)\rho^{d-2}+ (d-4)\rho^{d-4}+ \dots
+ \rho^{(6-d)}$$ $$z \sum_{i=3}^d \sum_{j=i}^d \rho^{(d+2i-2j)}
=(d-2)\rho^{(d+1)} - \rho^{(d-1)} -\rho^{(d-3)} -\dots - \rho^{(5-d)}$$ $$d\Delta(d)=d\rho^{(d-1)}-\rho^{(d-1)} + \rho^{(1-d)}
-(d-2)\rho^{(d-1)} + \rho^{(d-3)} +\rho^{(d-5)} +\dots + \rho^{(3-d)}$$ $$=\rho^{(d-1)}+ \rho^{(d-3)} +\rho^{(d-5)} +\dots + \rho^{(3-d)} + \rho^{(1-d)}=\frac{\tau^{d/2}-\tau^{-d/2}}{(\tau^{1/2}-\tau^{-1/2})}.$$ [ ]{}
Back to the proof of Proposition \[propalexseif\], since $\Delta(K)(t=\tau^d)(1)=\frac{|H_1(M)|}{d}$, $$\Delta(K)(t)=\frac{|H_1(M)|}{d}\frac{t^{1/2}-t^{-1/2}}{d(t^{1/(2d)}-t^{-1/(2d)})}\Delta_{\Sigma}(\tau).$$ and Proposition \[propalexseif\] is proved in the self-linking number $0$ case. Let us now deduce the general case from this case. Let $K$ be a knot with order $d$ and with self-linking number $(-a/b)$ where $a$ and $b$ are coprime. Let $m$ be a meridian of $K$, there exist a parallel $L$ of $K$ and a surface $\Sigma$ in $M \setminus K$ whose boundary is made of $(d/b)$ parallel copies of $am+bL$. Then there exists a primitive curve $m_J$ such that $\langle m_J,am+bL \rangle=1$. Let $J$ be the knot with meridian $m_J$ and with complement $M\setminus K$. This knot has order $(d/b)$ and self-linking number $0$. Its Alexander polynomial is then given by the proposition. Furthermore, since it satisfies $\Delta(J)(1)=|\mbox{Torsion}(H_1(M \setminus K))|=\Delta(K)(1)$, $\Delta(J)(t_J=\tau^{d/b})=\Delta(K)(t_K=\tau^{d})$. Then $\Delta(K)(t_K)=\Delta(J)(t_J=t_K^{1/b})$ and we are done. [ ]{}
Proposition \[propalexseif\] implies the following lemma that together with Lemma \[lemcaschirknot\] proves Proposition \[propcasknot\] for $n=1$. We use the notation of Section \[seccasstate\].
\[lemalexorder\] Under the assumptions of Proposition \[propalexseif\], $$\frac{d}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2}=\frac{\langle \langle I(\Sigma) \rangle \rangle_{W_1}}{2d^2} + \frac{1}{24b^2} - \frac{1}{24d^2}.$$
[[Proof: ]{}]{}First note that when $K$ is null-homologous, $O_K=d=b=1$. Then since $\lambda= W_1 \circ Z_1$, Lemma \[lemcaschirknot\] together with Theorem \[thmfboun\] together imply that $$\frac{1}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2}=\frac{\langle \langle I(\Sigma) \rangle \rangle_{W_1}}{2}.$$ Therefore, according to Proposition \[propalexseif\] (that is well-known in this case), $$\frac{\Delta^{\prime\prime}_{\Sigma}(1)}{2}=\frac{\langle \langle I(\Sigma) \rangle \rangle_{W_1}}{2}.$$ Then since $\Delta_{\Sigma}(t)=\Delta_{\Sigma}(t^{-1})$, $$\Delta_{\Sigma}(\exp(u))=1 +\frac{\langle \langle I(\Sigma) \rangle \rangle_{W_1}}{2} u^2 +O(4)$$ where $O(4)$ stands for an element of $u^4{\mathbb{Q}}[[u]]$, and this formula remains true for any $\Sigma$ as in the statement of Proposition \[propalexseif\]. Since $$\exp(u)^{\frac{1}{2d}}-\exp(u)^{-\frac{1}{2d}}=\frac{u}{d}(1+\frac{u^2}{24d^2} +O(4)),$$ it is easy to conclude. [ ]{}
Now that Proposition \[propcasknot\] is shown for $n=1$, let us prove it for $n=2$. By the formula that is recalled in the beginning of the proof of Lemma \[lemcaschirknot\], $$\lambda(M(K;p/q))-\lambda(M)= \frac{q}{p}\frac{O_K}{|H_1(M)|}\frac{\Delta^{\prime\prime}(K)(1)}{2} +f(p,q,lk(K,K),O_K)$$ for some $f(p,q,lk(K,K),O_K)$ that only depends on $p$, $q$, $lk(K,K)$, $O_K$, and that therefore does not change under surgery on a knot $K_2$ that does not link $K$ algebraically, so that $$\sum_{I \subset \{1,2\}} (-1)^{\sharp I} \lambda \left(M_{(K_i;p_i/q_i)_{i \in I}}\sharp \sharp_{j \in \{1,2\} \setminus I} L(p_j,-q_j)\right)= \sum_{I \subset \{1,2\}} (-1)^{\sharp I} \lambda \left(M_{(K_i;p_i/q_i)_{i \in I}}\right)$$ $$=\frac{q_1}{p_1}\left(\frac{O_{K_1}\Delta^{\prime\prime}(K_1\subset M(K_2;p_2/q_2))(1)}{2|H_1(M(K_2;p_2/q_2))|}-\frac{O_{K_1}\Delta^{\prime\prime}(K_1\subset M)(1)}{2|H_1(M)|}\right)$$ $$=\frac{q_1}{2p_1O_{K_1}^2}\left(\langle \langle I(\Sigma_1) \subset M(K_2;p_2/q_2) \rangle \rangle_{W_1}
-\langle \langle I(\Sigma_1) \subset M \rangle \rangle_{W_1} \right)$$ according to Lemma \[lemalexorder\]. Therefore, Proposition \[propcasknot\] for $n=2$ follows from the following lemma.
\[lemvarlambdaprime\] Under the assumptions of Proposition \[propcasknot\], $$\langle \langle I(\Sigma_1) \subset M(K_2;p/q) \rangle \rangle_{W_1}
-\langle \langle I(\Sigma_1) \subset M \rangle \rangle_{W_1}
=-\frac{2q}{d_2^2p} lk\left(\Sigma_1 \cap \Sigma_2, (\Sigma_1 \cap \Sigma_2)_{\parallel}\right).$$
[[Proof: ]{}]{}Let $(x_i,y_i)_{i \in \{1,\dots,g\}}$ be a symplectic basis for $H_1(\Sigma_1)/H_1(\partial \Sigma_1)$. Because of the variation of linking numbers after surgery recalled in Lemma \[lemvarlk\], the variation of the expression of $\langle \langle I(\Sigma_1) \rangle \rangle_{W_1}$ given before Proposition \[propcasknot\] reads $$\langle \langle I(\Sigma_1) \subset M(K_2;p/q) \rangle \rangle_{W_1}
-\langle \langle I(\Sigma_1) \subset M \rangle \rangle_{W_1} =$$ $$2 \frac{q^2}{p^2} \sum_{(j,k) \in \{1,2,\dots, g\}^2} lk(x_j,K_2)lk(K_2,x_k^+)lk(y_j,K_2)lk(K_2,y_k^+)$$ $$-2 \frac{q^2}{p^2} \sum_{(j,k) \in \{1,2,\dots, g\}^2}lk(x_j,K_2)lk(K_2,y_k^+)lk(y_j,K_2)lk(K_2,x_k^+)$$ $$-2\frac{q}{p} \sum_{(j,k) \in \{1,2,\dots, g\}^2}\left(lk(x_j,K_2)lk(K_2,x_k^+)lk(y_j,y_k^+)-lk(x_j,K_2)lk(K_2,y_k^+)lk(y_j,x_k^+)\right)$$ $$-2\frac{q}{p}\sum_{(j,k) \in \{1,2,\dots, g\}^2}\left(lk(x_j,x_k^+)lk(y_j,K_2)lk(K_2,y_k^+)-lk(x_j,y_k^+)lk(y_j,K_2)lk(K_2,x_k^+)\right)$$ where the quadratic part in $q/p$ is obviously zero. On the other hand, when $c \in H_1(\Sigma_1)$, $$\langle c, \Sigma_1 \cap \Sigma_2 \rangle_{\Sigma_1} =d_2 lk(c,K_2).$$ Therefore in $H_1(\Sigma_1)$, $$\Sigma_1 \cap \Sigma_2=d_2\sum_{i=1}^g (lk(x_i,K_2)y_i -lk(y_i,K_2)x_i)$$ and $$lk(\Sigma_1 \cap \Sigma_2, (\Sigma_1 \cap \Sigma_2)^{+})$$ $$=d_2^2\sum_{(j,k) \in \{1,2,\dots, g\}^2}
lk\left(lk(x_j,K_2)y_j -lk(y_j,K_2)x_j, lk(x_k,K_2)y_k^+ -lk(y_k,K_2)x_k^+\right).$$ [ ]{}
Then Proposition \[propcasknot\] is proved for $n=2$. Since $$lk_{M(K_3;p_3/q_3)}(\Sigma_1 \cap \Sigma_2, (\Sigma_1 \cap \Sigma_2)_{\parallel})
-lk_M(\Sigma_1 \cap \Sigma_2, (\Sigma_1 \cap \Sigma_2)_{\parallel})
=-\frac{q_3}{p_3}lk_M(\Sigma_1 \cap \Sigma_2,K_3)^2$$ $$=-\frac{q_3}{d_3^2p_3}\langle \Sigma_1, \Sigma_2, \Sigma_3 \rangle^2$$ this in turn implies Proposition \[propcasknot\] for $n=3$. Now, since $lk(\Sigma_1 \cap \Sigma_2,K_3)$ does not vary under a surgery on a knot that does not link $K_1$, $K_2$ and $K_3$ algebraically, Proposition \[propcasknot\] is also true for $n\geq 4$ and hence for all $n$. [ ]{}
[Proof of Proposition \[propcasknottwo\]:]{} Use that $\lambda^{\prime}(K^s)=\lambda^{\prime}(U,K^-)$ where $U$ is a trivial knot that surrounds the crossing change. (See the proofs of Propositions \[propzsing\] and \[propvarztwo\] in Section \[secproofpol\].) [ ]{}
Proofs of the statements on $\lambda_2$ and $w_3$
=================================================
\[secprooflambdatwo\]
Theorem \[thmpol\] guarantees the existence of a polynomial surgery formula $$\lambda_2(M(K;p/q))-\lambda_2(M)=\lambda_2^{\prime \prime}(K) (q/p)^2 + w_3(K)(q/p) +C(K;q/p) + \lambda_2(L(p;-q))$$ where $C(K;q/p)$ only depends on $q/p$ mod ${\mathbb{Z}}$ and $C(U;q/p)=0$. Since $Z_2^c(-M)=Z_2^c(M)$, $w_3(K \subset M)=-w_3(K \subset (-M))$.
Furthermore, according to Proposition \[propvarztwo\], if $K^s$ is a singular link with two unlinked double points, then $w_3(K^s)=-\frac{\ell_{12}(K^s)}{2}$ and $C(K^s;q/p)=0$.
The only unproved assertion of Theorem \[thmw3\] is that the knot invariants $C(K;q/p)$ read $c(q/p)\lambda^{\prime}(K)$ for knots that bound a surface whose $H_1$ vanishes in $H_1(M)$. The proof of this assertion will be given in this section.
Also note that for any knot $K$ in a rational homology sphere $M$, $w_3(K \subset M)= w_3(K \subset M \sharp N)$ and $C(K \subset M;q/p)= C(K \subset M \sharp N;q/p)$.
Let $K^s$ be a singular knot with one double point in a rational homology sphere. Let $K^+$ and $K^-$ be its two desingularisations, and let $K^{\prime}$ and $K^{\prime \prime}$ be the two knots obtained from $K^s$ by smoothing the double point. Assume that $K^{\prime}$ and $K^{\prime \prime}$ are null-homologous, set $$f(K^s)=\frac{\lambda^{\prime}(K^{\prime}) + \lambda^{\prime}(K^{\prime \prime})}{2} -\frac{\lambda^{\prime}(K^+) + \lambda^{\prime}(K^-)+lk^2(K^{\prime},K^{\prime \prime})}{4}.$$
Note that $f(K^s \subset M)=f(K^s \subset M \sharp N)$.
In order to prove Proposition \[propvarwthree\], we shall successively prove the following lemmas. The two last ones Lemmas \[lemredcalcpar\] and \[lemcalcpar\] obviously imply Proposition \[propvarwthree\].
\[lemnotvarunderxing\]Let $K^s$ be a singular knot with one double point in a rational homology sphere. The invariants $C(K^s;q/p)$ and $(w_3-f)(K^s)$ do not vary under a surgery on a knot that is null-homologous in the complement of $K^s$.
\[lemsurpresgr\] Let $\Gamma$ be a non-necessarily connected graph in a rational homology sphere $M$, such that every loop of $\Gamma$ is null-homologous in $M$. Then there exist a graph $\Gamma_0$ in $S^3$, an algebraically split (rationally) framed link $L$ in $S^3$ whose components are null-homologous in $S^3 \setminus \Gamma_0$, and a rational homology sphere $N$, such that $(S^3(L),\Gamma_0)= (M,\Gamma) \sharp N$.
\[lemredcalcpar\] Let $K^s_n$ be the following singular knot
(-.2,-.1)(3,2.2) (1.2,.6)(1.2,.4)(1.2,.2)(1.5,0)(2,.2)(2,1.8)(1.6,2)(1.2,1.8) (.8,1)(.8,1.2)(.8,1.4)(.6,1.6)(.8,1.8)(1,2) (.6,1.6)(.8,1.8)(1,2)(1.2,1.8) (.8,1.8)(1,2)(1.2,1.8)(1.4,1.6)(1.2,1.4)(1.2,1.2)(1.2,1) (.6,.4)(1.4,1.2) (1,.8)[$2n$]{} (1,1.6)(.8,1.8)(.4,2)(.2,1.8) (.8,1.8)(.4,2)(0,1.8)(0,.2)(.4,0)(.8,.2)(.8,.4)(.8,.6) (2,1.8)(1.6,2)(1.2,1.8)(1,1.6)(.8,1.8)(.4,2) (2.2,1)[$K_n^s$]{}
(-.2,-.1)(2.2,2.2) (1,.6)(.8,.4)(1,.2)(1.5,0)(2,.2)(2,1.8)(1.6,2)(1.2,1.8) (1,1)(1.2,1.2)(1,1.4)(.6,1.6)(.8,1.8)(1,2) (1,.2)(1.2,.4)(1,.6)(.8,.8)(1,1) (.6,1.6)(.8,1.8)(1,2)(1.2,1.8) (.8,1.8)(1,2)(1.2,1.8)(1.4,1.6)(1,1.4)(.8,1.2)(1,1)(1.2,.8)(1,.6)(.8,.4)(1,.2) (1,1.6)(.8,1.8)(.4,2)(.2,1.8) (.8,1.8)(.4,2)(0,1.8)(0,.2)(.4,0)(1,.2)(1.2,.4)(1,.6) (1,.6)(.8,.8)(1,1)(1.2,1.2)(1,1.4) (2,1.8)(1.6,2)(1.2,1.8)(1,1.6)(.8,1.8)(.4,2) (2.2,1)[$K_2^s$]{}
where
(-.1,-.1)(.6,.6) (0,0)(.5,.5) (.25,.25)[$2n$]{}
represents $|n|$ vertical juxtapositions of the motive
(.65,.2)(1.35,.9) (1,.1)(1.15,.25)(1,.4)(.85,.55)(1,.7) (1,1)(.85,.85)(1,.7)(1.15,.55)(1,.4)(.85,.25)(1,.1) (1,.4)(.85,.55)(1,.7)(1.15,.85)(1,1)
if $n>0$ and $|n|$ vertical juxtapositions of the motive
(.65,.2)(1.35,.9) (1,.1)(.85,.25)(1,.4)(1.15,.55)(1,.7) (1,1)(1.15,.85)(1,.7)(.85,.55)(1,.4)(1.15,.25)(1,.1) (1,.4)(1.15,.55)(1,.7)(.85,.85)(1,1)
if $n<0$. Then for any singular knot $K^s$ with one double point $p$, such that the two knots $K^{\prime}$ and $K^{\prime \prime}$ obtained from $K^s$ by smoothing $p$ are null-homologous, $$(w_3-f)(K^s)=(w_3-f)(K^s_{-lk(K^{\prime},K^{\prime \prime})}).$$
\[lemcalcpar\] For all $n \in {\mathbb{Z}}$, $(w_3-f)(K_n^s)=0$.
We shall next prove the following proposition that generalizes a Casson lemma from integral to rational homology spheres.
\[propgmcar\] Let $C$ be a real-valued invariant of null-homologous knots in rational homology spheres such that
- $C(K \subset M)= C(K \subset M \sharp N)$,
- $C(U)=0$,
- $C(K)$ does not vary under a surgery on a knot $J$ such that $(J,K)$ is a boundary link,
- if $K^s$ is a singular knot with one double point, $C(K^s)$ does not vary under surgery on a knot that is null-homologous in the complement of $K^s$.
Then there exists $c \in {\mathbb{R}}$ such that
- if $K^s$ is singular knot with one double point $p$, such that the two knots $K^{\prime}$ and $K^{\prime \prime}$ obtained from $K^s$ by smoothing $p$ are null-homologous, then $C(K^s)=c lk(K^{\prime},K^{\prime \prime})$, and,
- if $K$ bounds a surface whose $H_1$ maps to zero in $H_1(M)$, $C(K)=c \lambda^{\prime}(K)$.
Since the $C(.;p/q)$ satisfy the hypotheses of the proposition above (thanks to Theorem \[thmfboun\] for the hypothesis on boundary links), this proposition will be sufficient to conclude the proof of Theorem \[thmw3\]. [ ]{}
Let us now prove all the lemmas and the proposition.
[Proof of Lemma \[lemnotvarunderxing\]:]{} Let $J$ be a null-homologous knot unlinked with $K^{\prime}$ and $K$. Let $F_J$ be a Seifert surface for $J$ that does not meet $K^s$, and let $(m, \ell)$ be the usual basis of the genus one surface obtained by tubing a trivial knot $V$ surrounding the double point of $K^s$, $m$ is a meridian of $K^-$, $\ell$ is homotopic to $K^{\prime}$ and $lk(\ell,K^-)=0$. By Theorem \[thmfboun\], $$Z_2^c(M(J;\frac{p_J}{q_J})(K^s;\frac{p}{q} ) )- Z_2^c(M(K^s;\frac{p}{q}) )$$ $$= \frac{q_J}{4p_J} p^c \left( \langle \langle \begin{pspicture}[0.4](-.5,-.3)(1.3,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(.8,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.15,.5){$\ell$}
\rput[b](.15,.5){$m$}
\rput[b](.65,.5){$m^+$}
\rput[lb](.95,.5){$\ell^+$}
\end{pspicture} I(F_J) \subset M(K^-;\frac{p}{q})\;\rangle \rangle \right).$$
Since, according to Lemma \[lemvarlk\], the pairing of $m$ and a curve $c$ in the contraction above will give rise to the coefficient $(-q/p)lk(K,c)=-rlk(K,c)$, $C(K^s;q/p)$ does not vary under a $(p_J/q_J)$-surgery on $J$.
$$w_3(K^s \subset M(J;\frac{p_J}{q_J})) - w_3(K^s \subset M) =$$ $$\left(\frac{\partial}{\partial r}\right)_{r=0} W_2\left( Z_2^c(K^s \subset M(J;\frac{p_J}{q_J})) - Z_2^c(K^s \subset M)\right)$$ where $m$ must be paired either with $m^+$ or with $I(F_J)$, and in the latter case $m^+$ must be paired with $\ell$ in order to lead to a linear contribution in $r$. $$w_3(K^s \subset M(J;\frac{p_J}{q_J})) - w_3(K^s \subset M) =$$ $$=- \frac{q_J}{4p_J} \langle \langle \begin{pspicture}[0.4](-.5,-.1)(1,.8)
\psline{*-*}(0,0.05)(.4,.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.4,0.05)(.55,.35)
\rput[b](-.15,.4){$\ell$}
\rput[b](.6,.4){$\ell^+$}
\end{pspicture} I(F_J) \subset M \rangle \rangle_{W_2} + \frac{q_j}{4p_J}
\langle \langle \begin{pspicture}[0.4](-.5,-.1)(1,.8)
\psline{*-*}(0,0.05)(.4,.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.4,0.05)(.55,.35)
\rput[b](-.15,.4){$K$}
\rput[b](.6,.5){$\ell^+$}
\end{pspicture} I(F_J) \subset M \rangle \rangle_{W_2}.$$
Since $K=K^{\prime \prime} + \ell$, as far as the connected pairing with $I(F_J)$ is concerned, $$\begin{pspicture}[0.15](-.6,-.15)(1.15,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K$}
\rput[lb](.65,-.05){$K^+$} \end{pspicture}
=\begin{pspicture}[0.15](-.5,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$\ell$}
\rput[lb](.65,-.05){$\ell^+$}
\end{pspicture} +
\begin{pspicture}[0.15](-.8,-.15)(1.3,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K^{\prime \prime}$}
\rput[lb](.65,-.05){$K^{\prime \prime +}$}
\end{pspicture}
+2
\begin{pspicture}[0.15](-.5,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$\ell$}
\rput[lb](.65,-.05){$K^{\prime \prime}$}
\end{pspicture}$$ and $$\begin{pspicture}[0.15](-.6,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K$}
\rput[lb](.65,-.05){$\ell^+$} \end{pspicture}
=
\begin{pspicture}[0.15](-.5,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$\ell$}
\rput[lb](.65,-.05){$\ell^+$}
\end{pspicture} +
\begin{pspicture}[0.15](-.8,-.15)(1.2,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K^{\prime \prime}$}
\rput[lb](.65,-.05){$\ell^{+}$}
\end{pspicture}.$$ Therefore, $$\begin{pspicture}[0.15](-.6,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K$}
\rput[lb](.65,-.05){$\ell^+$} \end{pspicture}
= \frac12
\begin{pspicture}[0.15](-.5,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$\ell$}
\rput[lb](.65,-.05){$\ell^+$}
\end{pspicture} - \frac12
\begin{pspicture}[0.15](-.8,-.15)(1.3,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K^{\prime \prime}$}
\rput[lb](.65,-.05){$K^{\prime \prime +}$}
\end{pspicture}
+ \frac12
\begin{pspicture}[0.15](-.6,-.15)(1.15,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K$}
\rput[lb](.65,-.05){$K^+$} \end{pspicture}.$$
$$w_3(K^s \subset M(J;\frac{p_J}{q_J})) - w_3(K^s \subset M) =$$ $$- \frac{q_J}{8p_J} \langle \langle \begin{pspicture}[0.4](-.5,-.2)(.8,1)
\psline{*-*}(0,0.05)(.4,.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.4,0.05)(.55,.35)
\rput[b](-.15,.5){$\ell$}
\rput[b](.55,.5){$\ell^+$}
\end{pspicture} I(F_J) \rangle \rangle_{W_2} - \frac{q_J}{8p_J}
\langle \langle \begin{pspicture}[0.4](-.5,-.2)(.8,1)
\psline{*-*}(0,0.05)(.4,.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.4,0.05)(.55,.35)
\rput[b](-.15,.5){$K^{\prime \prime}$}
\rput[b](.55,.5){$K^{\prime \prime +}$}
\end{pspicture} I(F_J) \rangle \rangle_{W_2}
+ \frac{q_J}{8p_J}
\langle \langle \begin{pspicture}[0.4](-.5,-.2)(1,1)
\psline{*-*}(0,0.05)(.4,.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.4,0.05)(.55,.35)
\rput[b](-.15,.5){$K$}
\rput[b](.55,.5){$K^+$}
\end{pspicture} I(F_J) \rangle \rangle_{W_2}.$$
Thus, according to Proposition \[propcasknotvar\], since $$\langle \langle \begin{pspicture}[0.3](-.7,-.3)(1.4,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$K$}
\rput[lb](.65,-.05){$K^+$} \end{pspicture} I(F_J) \;\rangle \rangle_{W_2}= \langle \langle \begin{pspicture}[0.3](-.6,-.3)(1.2,.6)
\psline{-}(.35,0.05)(-.05,.05)
\rput[rb](-.1,0){$K$}
\rput[lb](.4,0){$K^+$}
\end{pspicture} I(F_J) \;\rangle \rangle_{W_1},$$ $$w_3(K^s \subset M(J;\frac{p_J}{q_J})) - w_3(K^s \subset M)$$ $$=\frac{q_J}{2p_J} \left( \lambda^{\prime}(J,K^{\prime}) +
\lambda^{\prime}(J,K^{\prime \prime})- \lambda^{\prime}(J,K)\right)$$ $$=\frac{q_J}{p_J} \left( \frac{\lambda^{\prime}(J,K^{\prime})}{2} +
\frac{\lambda^{\prime}(J,K^{\prime \prime})}{2}- \frac{\lambda^{\prime}(J,K^+)}{4}
- \frac{\lambda^{\prime}(J,K^-)}{4}\right)$$ $$=f(K^s \subset M(J;\frac{p_J}{q_J})) - f(K^s \subset M).$$ [ ]{}
[Proof of Lemma \[lemsurpresgr\]:]{} After a possible connected sum with some lens spaces, the ${\mathbb{Q}}/{\mathbb{Z}}$–valued linking form of $M$ is diagonal [@wall], and the generators of $H_1(M;{\mathbb{Z}})$ can be represented by a link $L$ of algebraically unlinked curves $K_i$ that do not link $\Gamma$, algebraically. Then for each $K_i$, there exists a surface $\Sigma_i$ in the exterior $(M \setminus \mbox{Int}{N}(L))$ of $L$ whose boundary is a connected essential curve of $\partial N(K_i)$, and that does not meet $\Gamma$. Thus, $H^1(M \setminus \mbox{Int}{N}(L);{\mathbb{Z}})$ is freely generated by the algebraic intersections with the $\Sigma_i$, and there exists a surgery on $L$ that transforms $M$ into a homology sphere $H$. The manifold $H$ can in turn be transformed into $S^3$ by surgery on a boundary link of $H$ bounding a disjoint union $F_H$ of surfaces in $H$ that can be assumed to be disjoint from the first surgery link and from the image of $\Gamma$ in $H$. This proves the lemma. [ ]{}
[Proof of Lemma \[lemredcalcpar\]:]{} Apply Lemma \[lemsurpresgr\] to $\Gamma=K^s$, then $K^{s,0}=\Gamma_0$. Note that $lk(K^{\prime}_0,K^{\prime\prime}_0)=lk(K^{\prime},K^{\prime\prime})$. Recall that $(w_3-f)(K^s \subset M)=(w_3-f)(K^s \subset M \sharp N)$. Thanks to Lemma \[lemnotvarunderxing\], $(w_3-f)(K^s \subset M \sharp N)=(w_3-f)(K^{s,0} \subset S^3)$. Now that the proof has been reduced to the case where $M=S^3$, recall that a crossing change on $K^{\prime}$ or $K^{\prime \prime}$ may be realized by a surgery on a knot satisfying the hypotheses of Lemma \[lemnotvarunderxing\], that changes neither $lk(K^{\prime},K^{\prime\prime})$ nor $(w_3-f)(K^s)$. Unknotting $K^{\prime}$ first by crossing changes and next unknotting the parts of $K^{\prime \prime}$ between two consecutive intersection points with the disk bounded by $K^{\prime}$ transforms $K^s$ into $K^s_{-lk(K^{\prime},K^{\prime\prime})}$. [ ]{}
[Proof of Lemma \[lemcalcpar\]:]{} By the crossing change formula of Proposition \[propcasknottwo\], $\lambda^{\prime}(K^+_{n})-\lambda^{\prime}(K^+_{n-1})=-1$, and $\lambda^{\prime}(K^+_{n})=-n$. Since $K^-_n$, $K^{\prime}_n$ and $K^{\prime \prime}_n$ are trivial, $f(K^s_{n})=-\frac{n(n-1)}{4}$.
On the other hand, since $w_3(K^-_n)=0$, $w_3(K^s_n)=w_3(K^+_n)$. The unlinked double crossing change formula of Theorem \[thmw3\] implies that $$w_3(K^+_{n+2})-2w_3(K^+_{n+1}) +w_3(K^+_{n})=-\frac{1}{2}$$ Since $K^+_0$ is trivial, $w_3(K^+_{0})=0$, and since $K^+_1$ is the figure-eight knot that coincides with its mirror image, $w_3(K^+_{1})=0$, too. Then $w_3(K^s_n)=w_3(K^+_{n})=-\frac{n(n-1)}{4}$. [ ]{}
[Proof of Proposition \[propgmcar\]:]{} Let $K^s$ be as in the hypotheses of Proposition \[propvarwthree\]. The proof of Lemma \[lemredcalcpar\] shows that $C(K^s)=C(K^s_{-lk(K^{\prime},K^{\prime \prime})})$. Since $C(U)=0$, $C(K^s_n)=C(K^+_n)$. Since the hypotheses of the proposition imply that $C$ maps singular knots of $S^3$ with two unlinked double points to $0$, $C(K^+_{n+2})-2C(K^+_{n+1}) +C(K^+_{n})=0$, and $C(K^+_{n})$ is affine with respect to $n$. Since $C(K^+_{0})=0$, $C(K^+_{n})$ is linear. Then there exists $c$ such that $C(K^s)=c lk(K^{\prime},K^{\prime \prime})$.
Let $K$ be a knot that bounds a Seifert surface $\Sigma$ whose $H_1$ maps to zero in $H_1(M)$. Applying Lemma \[lemsurpresgr\] to the one-skeleton of $\Sigma$ allows us to reduce the proof that $C(K)=c \lambda^{\prime}(K)$ to the case of knots in $S^3$, thanks to the hypothesis on boundary links. Then this case is easily proved with the crossing change formula. [ ]{}
[Proof of Proposition \[propgenusone\]:]{} Consider the genus one surface $\Sigma$ in $H$ and its symplectic basis $(a,b)$ below.
(-.2,-1.5)(3.2,1.5) (2.9,.8)(2.9,1.4)(.1,1.4)(.1,.8) (.5,.8)(.5,1)(1.3,.7)(1.3,.5) (1.7,.5)(1.7,.7)(2.5,1)(2.5,.8) (2.7,1.4)(1.5,1.4) (.7,.925)(.9,.85) (1.9,.775)(2.1,.85) (1.2,-.5)(1.8,.5) (0,-.8)(.6,.8) (2.4,-.8)(3,.8) (.3,0)[$y$]{} (1.5,0)[$z$]{} (2.7,0)[$x$]{} (2.9,-.8)(2.9,-1.4)(.1,-1.4)(.1,-.8) (.5,-.8)(.5,-1)(1.3,-.7)(1.3,-.5) (1.7,-.5)(1.7,-.7)(2.5,-1)(2.5,-.8) (2.7,-1.4)(1.5,-1.4) (.7,-.925)(.9,-.85) (1.9,-.775)(2.1,-.85) (2.1,-1.1)[$a$]{} (1.4,-.5)(1.4,-1.2)(2.05,-1.2) (2.05,-1.2)(2.7,-1.2)(2.7,-.8) (2.1,1.1)[$a$]{} (2.7,.8)(2.7,1.25)(2.15,1) (2.15,1)(1.6,.75)(1.6,.5) (1,-1.05)[$b$]{} (.3,-.8)(.3,-1.3)(.95,-1.15) (.95,-1.15)(1.6,-1)(1.6,-.5) (.95,1.15)[$b$]{} (1.4,.5)(1.4,.95)(.85,1.1) (.85,1.1)(.3,1.25)(.3,.8)
$\langle a,b \rangle=1$, $lk(a,a^+)=\frac{x+z}{2}$, $lk(b,b^+)=\frac{y+z}{2}$, $lk(a,b^+)=\frac{-1-z}{2}$, $lk(a^+,b)=\frac{1-z}
{2}$, $$\lambda^{\prime}(K(x,y,z))= \frac{(x+z)(y+z) +1
-z^2}{4}=\frac{xy +yz +zx +1}{4}.$$ Note that $\lambda^{\prime}(\phi(X),\phi(Y)) =\lambda^{\prime}(\phi(Y),\phi(Z))
=\lambda^{\prime}(\phi(Z),\phi(X))$. In particular, both sides of the equality to be proved are symmetric under a cyclic permutation of $((X,x),(Y,y),(Z,z))$. Using this cyclic symmetry, the formula for the pretzel knot $K(x,y,z)$ follows from the crossing change formula starting with the trivial knot $K_{-1,1,1}$: $$4w_3(K(x+2,y,z))-4w_3(K(x,y,z))=\lambda^{\prime}(K(x+2,y,z)) +
\lambda^{\prime}(K(x,y,z))
+\left(\frac{y+z}{2}\right)^2.$$ $$16\left(w_3(K(x+2,y,z))-w_3(K(x,y,z))\right)=(2x+2)(y+z)+2 +4yz +y^2 +z^2.$$ $$32 w_3(K(x,y,z)) = 2x + 4xyz + xy^2 +xz^2 + x^2(y+z) +F(y,z).$$ Otherwise, the following lemma \[lemvargenone\] reduces the proof of Proposition \[propgenusone\] to the case where the knot $\phi(K(x,y,z))$ is in $S^3$, thanks to Lemma \[lemsurpresgr\], and next when the knot is a pretzel knot $K(x,y,z)$ by crossing changes on $X$ and $Y$.
\[lemvargenone\] Let $\phi$ be an embedding of $H$ in a rational homology sphere such that $\phi(X)$ and $\phi(Y)$ are null homologous in the exterior of $\phi(H)$. Let $J$ be a knot in the exterior of $\phi(H)$ that links neither $\phi(X)$ nor $\phi(Y)$, then $$w_3(\phi(K(x,y,z)) \subset M(J;p/q)) - w_3(\phi(K(x,y,z)) \subset M)$$ $$=\frac{q}{2p} \left(3 \lambda^{\prime}(\phi(X),\phi(Y),J) -x \lambda^{\prime}(\phi(X),J) - y \lambda^{\prime}(\phi(Y),J) -z \lambda^{\prime}(\phi(Z),J)\right).$$
[Proof of Lemma \[lemvargenone\]:]{} According to Theorem \[thmfboun\], if $F_J$ is a Seifert surface of $J$ in the complement of the genus one Seifert surface $\Sigma$ of $\phi(K(x,y,z))$ in $\phi(H)$, $$w_3(\phi(K(x,y,z)) \subset M(J;p/q)) - w_3(\phi(K(x,y,z)) \subset M)=
\frac{q}{4p} \langle \langle\; I(\Sigma) \; I(F_J)\; \rangle \rangle_{W_2}$$ where $$I(\Sigma)=\begin{pspicture}[0.4](-.5,-.3)(1.3,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(.8,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.15,.5){$a$}
\rput[b](.15,.5){$b$}
\rput[b](.65,.5){$b^+$}
\rput[lb](.95,.5){$a^+$}\end{pspicture}.$$
Write $$\frac{q}{4p} \langle \langle\; I(\Sigma) \; I(F_J)\; \rangle \rangle_{W_2}=C_A +C_B$$ where $C_A$ is the contribution of the pairings that pair two univalent vertices of $I(\Sigma)$, and $C_B$ is the contribution of the pairings that pair all the univalent vertices of $I(\Sigma)$ to univalent vertices of $I(F_J)$. $$C_A=\frac{q}{4p} \langle \langle\;
\left(\frac{x+z}{2} \begin{pspicture}[0.15](-.7,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$b$}
\rput[lb](.65,-.05){$b^+$}
\end{pspicture}\;
+\frac{y+z}{2}\begin{pspicture}[0.15](-.7,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$a$}
\rput[lb](.65,-.05){$a^+$}
\end{pspicture}
+z \begin{pspicture}[0.15](-.7,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$a$}
\rput[lb](.65,-.05){$b$}
\end{pspicture}
\right)I(F_J)\; \rangle \rangle_{W_2}.$$
From now on, we write $X$, $Y$ and $Z$ for $\phi(X)$, $\phi(Y)$ and $\phi(Z)$, respectively. $$C_A=\frac{q}{4p} \langle \langle\;
\left(\frac{x}{2} \begin{pspicture}[0.15](-.7,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$X$}
\rput[lb](.65,-.05){$X^+$}
\end{pspicture}\;
+\frac{y}{2}\begin{pspicture}[0.15](-.7,-.15)(1.1,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.05){$Y$}
\rput[lb](.65,-.05){$Y^+$}
\end{pspicture}
+\frac{z}{2} \begin{pspicture}[0.15](-1.8,-.15)(2.2,.6)
\pscurve{*-*}(0,0.05)(.2,-.1)(.4,0.05)
\pscurve(0,0.05)(.2,.2)(.4,0.05)
\psline{-}(0,0.05)(-.2,0.05)
\psline{-}(.4,0.05)(.6,0.05)
\rput[rb](-.25,-.1){$(X+Y)$}
\rput[lb](.65,-.1){$(X+Y)^+$}
\end{pspicture}
\right)I(F_J)\; \rangle \rangle_{W_2}.$$
Thus, according to Proposition \[propcasknotvar\], $$C_A=-\frac{q}{p}\left(\frac{x \lambda^{\prime}(X,J)}{2} + \frac{y \lambda^{\prime}(Y,J)}{2} +\frac{z \lambda^{\prime}(Z,J)}{2}\right).$$
Let us now compute the contribution of the pairings that are bijections from the set of univalent vertices of $I(\Sigma)$ to the set of univalent vertices of $I(F_j)$. For them, we may change $a$ to $Y$ and $b$ to $X$ and write $$I(\Sigma)=\begin{pspicture}[0.4](-.5,-.3)(1.3,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(.8,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.15,.5){$X$}
\rput[b](.15,.5){$Y$}
\rput[b](.65,.5){$Y^+$}
\rput[lb](.95,.5){$X^+$}
\end{pspicture}$$ where the superscripts $+$ distinguish two copies of $X$ (or $Y$) whose linking numbers with the curves of $F_J$ are the same.
Let us compute the contribution $C_B$ of the pairings that are bijections from the set of univalent vertices of $I(\Sigma)$ to the set of univalent vertices of some $$I(c,d,e,f)=\begin{pspicture}[0.4](-.5,-.3)(1.3,1)
\psline{-}(0,0.05)(.15,.35)
\psline{*-*}(0,0.05)(.8,.05)
\psline{-}(0,0.05)(-.15,.35)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.15,.5){$c$}
\rput[b](.15,.5){$d$}
\rput[b](.65,.5){$e$}
\rput[l](.95,.6){$f$}
\end{pspicture}$$ to $\langle \langle\; I(c,d,e,f)\; I(\Sigma) \;\rangle \rangle_{W_2}$.
Note the symmetry under the exchange of the pair $(X,Y)$ with the pair $(X^+,Y^+)$.
The contribution of the pairings that pair $c$ and $d$ to $X$ and $X^+$ is $$lk(c,X)lk(d,X)\langle \langle\; \begin{pspicture}[0.4](-.9,-.3)(1.3,1.2)
\psline{-*}(0,0.05)(.2,.45)
\psline{-}(.2,.45)(.35,.75)
\psline{*-*}(0,0.05)(.8,.05)
\psline(.2,.45)(-.2,.45)
\psline{-}(-.2,.45)(-.35,.75)
\psline{-*}(0,0.05)(-.2,.45)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.4,.75){$Y$}
\rput[bl](.4,.75){$Y^+$}
\rput[b](.65,.5){$e$}
\rput[l](.95,.6){$f$}
\end{pspicture} +
\begin{pspicture}[0.4](-.9,-.3)(1.3,1.2)
\pscurve{-*}(0,0.05)(.1,.2)(-.2,.45)
\pscurve[border=1pt]{-*}(0,0.05)(-.1,.2)(.2,.45)
\psline{-}(.2,.45)(.35,.75)
\psline{*-*}(0,0.05)(.8,.05)
\psline(.2,.45)(-.2,.45)
\psline{-}(-.2,.45)(-.35,.75)
\psline{-}(.8,0.05)(.95,.35)
\psline{-}(.8,0.05)(.65,.35)
\rput[br](-.4,.75){$Y$}
\rput[bl](.4,.75){$Y^+$}
\rput[b](.65,.5){$e$}
\rput[l](.95,.6){$f$}
\end{pspicture}\;\rangle \rangle_{W_2}$$ that is zero by the antisymmetry relation in the space of Jacobi diagrams. Similarly, the contribution of the pairings that pair $c$ and $d$ to $Y$ and $Y^+$ vanishes.
The contributions of the pairings that pair $d$ and $e$ to $X$ and $X^+$ is $$2lk(d,X)lk(e,X)lk(c,Y)lk(f,Y) W_2\left({\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.1,0)(.2,.2)(.1,.4)
\pscurve(.5,0)(.4,.2)(.5,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}}+ \begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline(.1,0)(.5,.4)
\psline[border=1pt](.5,0)(.1,.4)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}\right)$$ where $$W_2\left(\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline(.1,0)(.5,.4)
\psline[border=1pt](.5,0)(.1,.4)
\psline{*-*}(.1,0)(.5,0)
\psline{*-*}(.1,.4)(.5,.4)
\pscurve(.1,0)(0,.2)(.1,.4)
\pscurve(.5,0)(.6,.2)(.5,.4)
\end{pspicture}\right)=W_2\left({\begin{pspicture}[.2](-.2,-.1)(.8,.6)
\psline{*-}(0,0)(.6,0)(.3,.2)
\psline{*-*}(.6,0)(.3,.5)(.3,.2)
\psline{*-}(.3,.5)(0,0)(.3,.2)
\end{pspicture}}\right)=1.$$
Therefore, the contribution to $\langle \langle\; I(c,d,e,f)\; I(\Sigma) \;\rangle \rangle_{W_2}$ of the pairings that are bijections from the set of univalent vertices of $I(\Sigma)$ to the set of univalent vertices of $I(c,d,e,f)$ is $$\frac{3}{4} \langle \langle I(c,d,e,f) \begin{pspicture}[0.2](-.6,-.1)(1.2,.6)
\psline{-}(.35,0.1)(-.05,.1)
\rput[rb](-.1,0){$X$}
\rput[lb](.4,0){$X^+$}
\end{pspicture}\begin{pspicture}[0.2](-.6,-.1)(1.2,.6)
\psline{-}(.35,0.1)(-.05,.1)
\rput[rb](-.1,0){$Y$}
\rput[lb](.4,0){$Y^+$}
\end{pspicture}\rangle \rangle_{W_1}$$ Therefore, according to Proposition \[propcasknotvar\], $$C_B=\frac{q}{4p} \frac{3}{4} \langle \langle I(F_J) \begin{pspicture}[0.2](-.6,-.1)(1.2,.6)
\psline{-}(.35,0.1)(-.05,.1)
\rput[rb](-.1,0){$X$}
\rput[lb](.4,0){$X^+$}
\end{pspicture}\begin{pspicture}[0.2](-.6,-.1)(1.2,.6)
\psline{-}(.35,0.1)(-.05,.1)
\rput[rb](-.1,0){$Y$}
\rput[lb](.4,0){$Y^+$} \end{pspicture} \rangle \rangle_{W_1} = 3\frac{q}{2p} \lambda^{\prime}(J,X,Y)$$ $$= \frac{3}{2}\left(\lambda^{\prime}((X,Y) \subset M(J;q/p)) - \lambda^{\prime}((X,Y) \subset M))\right).$$ [ ]{}
More about surgeries on general knots in rational homology spheres
==================================================================
Theorem \[thmpol\] describes the polynomial behaviour of $Z_n$ under surgeries on null-homologous knots. It can easily be generalized to the case of non null-homologous knots $K$ a primitive satellite $\ell$ of which bounds a Seifert surface. Let $m_K$ be the meridian of such a knot $K$ such that $\langle m_K ,\ell\rangle_{\partial N(K)}=O_K$.
A surgery curve $\mu$ on $\partial N(K)$ is determined by its coordinates $(p_K,q_K)$ in the symplectic basis $(m_K, \frac{1}{O_K} \ell)$ of $H_1(\partial N(K);{\mathbb{Q}})$ where $p_K= \frac{1}{O_K} \langle \mu ,\ell \rangle$ is the linking number of $K$ and $\mu$, and $q_K=\langle m_K ,\mu \rangle$. The associate surgery coefficient is $p_K/q_K$.
\[thmpolprim\] Let $n \in {\mathbb{N}}$. Let $K$ be a knot of order $O_K$ in a rational homology sphere $M$ such that a primitive satellite $\ell$ of $K$ bounds a Seifert surface $F$. Let $F^1,\dots, F^n$ be parallel copies of $F$. Let $p_K/q_K \in {\mathbb{Q}}$ be a surgery coefficient for $K$. Then $$Z_n(M(K;\frac{p_K}{q_K}))-Z_n(M)=\sum_{i=0}^n Y_{n,q_K/(p_KO_K^2)}^{(i)}(K \subset M) (\frac{q_K}{p_K})^i$$ where $$Y_{n,q_K/(p_KO_K^2)}^{(n)}(K)= \frac{1}{n!2^n O_K^{2n}} \langle \langle \bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\;\rangle \rangle\;$$ $Y_{n,q_K/(p_KO_K^2)}^{(i)}$ only depends on $q_K/(p_KO_K^2)$ mod ${\mathbb{Z}}$, and, if $n\geq 2$, $p_c(Y_{n,q_K/(p_KO_K^2)}^{(n-1)})= Y_{n,q_K/(p_KO_K^2)}^{(n-1)c}$ does not depend on $p_K$ and $q_K$. Furthermore, if $m$ is a primitive satellite of $K$ such that $\langle m ,\ell\rangle_{\partial N(K)}=1$, and if $\hat{K} \subset \hat{M}$ is the knot with the same complement as $K$ whose meridian is $m$, then, if $n \geq 2$, $$Y_{n}^{(n-1)c}(K \subset M)=\frac{1}{O_K^{2n-2}}Y_{n}^{(n-1)c}(\hat{K} \subset \hat{M})
+n \langle m, m_K \rangle O_K Y_{n}^{(n)c}(K \subset M).$$
[[Proof: ]{}]{}Let $\mu=p_Km_K +(q_K/O_K)\ell$ be a surgery curve on $\partial N(K)$. Let $$\left(p=\langle \mu ,\ell \rangle = O_K p_K,q=\langle m ,\mu \rangle=p_K \langle m, m_K \rangle + q_K/O_K\right)$$ be the coordinates of $\mu$ in the symplectic basis $(m, \ell)$ of $H_1(N(K);{\mathbb{Z}})$. Note that changing $m$ to another curve such that $\langle m ,\ell\rangle_{\partial N(K)}=1$ leaves $p$ invariant and does not change the class of $\frac{q}{p}$ in ${\mathbb{Q}}/{\mathbb{Z}}$. When the other data are fixed, the mod ${\mathbb{Z}}$ congruence class of $$\frac{q}{p}=\frac{q_K}{p_K O_K^2} + \frac{\langle m, m_K \rangle}{O_K}$$ depends on the class of $\frac{q_K}{p_K O_K^2}$ in ${\mathbb{Q}}/{\mathbb{Z}}$. From the formula of Theorem \[thmpol\] $$Z_n(\hat{M}(\hat{K};\frac{p}{q+rp}))-Z_n(\hat{M})=\sum_{i=0}^n Y_{n,q/p}^{(i)}(\hat{K} \subset \hat{M}) (r+\frac{q}{p})^i,$$ we deduce $$Z_n(M(K;\frac{p_K}{q_K +r O_K^2 p_K}))-Z_n(M)$$ $$=\sum_{i=0}^n Y_{n,q/p}^{(i)}(\hat{K} \subset \hat{M}) (r+ \frac{q_K}{p_K O_K^2} + \frac{\langle m, m_K \rangle}{O_K})^i+Z_n(\hat{M}) -Z_n(M)$$ $$=\sum_{i=0}^n Y_{n,q_K/(p_K O_K^2)}^{(i)}(K \subset M) (rO_K^2+ \frac{q_K}{p_K })^i$$ where $$Y_{n,q_K/(p_K O_K^2)}^{(n)}(K \subset M)=\frac{1}{n!2^nO_K^{2n}}\langle \langle \bigsqcup_{i \in \{1,\dots,n\}} I(F^i) \;\;\rangle \rangle\;$$ and, if $n\geq 2$, $$Y_{n,q_K/(p_K O_K^2)}^{(n-1)}(K \subset M)=\frac{1}{O_K^{2n-2}}Y_{n,q/p}^{(n-1)}(\hat{K} \subset \hat{M})
+n \langle m, m_K \rangle O_K Y_{n}^{(n)}(K \subset M).$$ [ ]{}
A knot $K$ of order $O_K$ in a rational homology sphere has a primitive satellite that is null-homologous in its exterior if and only if the self-linking number of $K$ reads $d/O_K$ (mod ${\mathbb{Z}}$) where $d$ is coprime with $O_K$.
Like in the proof of Theorem \[thmpolprim\], the case of knots without null-homologous primitive satellites can be reduced to the case of knots of order $O_K >1$ with self-linking number $0$. This latter case is still unclear to me (except for the degree 1 case that can be treated with the methods of the article).
Relationships between surgery formulae for various $q/p$ can be found using some equivalences of surgeries. See [@go5].
Questions
=========
The statements of Theorems \[thmfas\] and \[thmfasmu\] make sense for rationally algebraically split links. Do they hold true in this case?
How do the properties of surgery formulae generalize for surgeries on non null-homologous knots?
What is the graded space associated to the filtration of the rational vector space generated by rational homology spheres, defined using Lagrangian-preserving surgeries?
The degree $n$ parts of the LMO invariant and the Kontsevich-Kuperberg-Thurston invariant coincide on the intersection of ${{\cal F}}_n$ with the vector space generated by homology spheres. The configuration space invariant for knots in $S^3$ is obtained from the Kontsevich integral by an isomorphism that inserts a (possibly trivial) specific two-leg box $\beta$ on each chord of a chord diagram. See [@les5] for a more specific statement. Do the LMO invariant and the Kontsevich-Kuperberg-Thurston invariant actually coincide? Is the Kontsevich-Kuperberg-Thurston invariant obtained from the LMO invariant by inserting the two-leg box $\beta$, $k$ (or $2k$ or $3k$) times on each degree $k$ component of a Jacobi diagram?
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---
abstract: 'A stationary wave pattern occurring in a flow of a two-component Bose-Einstein condensate past an obstacle is studied. We consider the general case of unequal velocities of two superfluid components. The Landau criterium applied to the two-component system determines a certain region in the velocity space in which superfluidity may take place. Stationary waves arise out of this region, but under the additional condition that the relative velocity of the components does not exceed some critical value. Under increase of the relative velocity the spectrum of the excitations becomes complex valued and the stationary wave pattern is broken. In case of equal velocities two sets of stationary waves that correspond to the lower and the upper Bogolyubov mode can arise. If one component flows and the other is at rest only one set of waves may emerge. Two or even three interfere sets of waves may arise if the velocities approximately of equal value and the angle between the velocities is close to $\pi/2$. In two latter cases the stationary waves correspond to the lower mode and the densities of the components oscillate out-of-phase. The ratio of amplitudes of the components in the stationary waves is computed. This quantity depends on the relative velocity, is different for different sets of waves, and varies along the crests of the waves. For the cases where two or three waves interfere the density images are obtained.'
author:
- 'L.Yu. Kravchenko'
- 'D.V. Fil'
date: 'Received: date / Accepted: date'
title: 'Stationary waves in a supersonic flow of a two-component Bose gas '
---
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction {#intro}
============
A unique feature of two-component Bose-Einstein condensates (BEC) is the possibility for two superfluids to flow with different velocities. The properties of such systems can be described by the three-velocity hydrodynamics (one normal and two superfluid velocities). This feature was already noticed by Khalatnikov [@hal]. The modern three-velocity superfluid hydrodynamic theory was formulated in the paper by Andreev and Bashkin [@ab]. As was shown in [@ab] the specifics of the three-velocity superfluid hydrodynamics is the presence of a non-dissipative drag between the components. The drag effect emerges at nonzero relative velocity of the components and consists in a dependence of the superfluid current of one component on the gradient of the phase of the order parameter of the other component. The microscopic theory of the non-dissipative drag effect was developed in [@fil1; @fil]. An important question that arises in the three-velocity superfluid hydrodynamics is the question on critical velocities. At equal velocities of the components the answer to this question can be obtained from the Galilean transformation. It yields that under neglecting of vortex excitations the critical velocity coincides with the minimal phase velocity of the lower Bogolyubov mode. In a wider sense one can introduce two critical velocities $c_-$ and $c_+$, one is for the lower mode and the other is for the upper mode [@nb; @bi2].
In case of unequal velocities of the components the situation becomes more complicated. The question was considered in Refs. [@yukalov; @6]. The authors of [@yukalov] have obtained the dispersion equation for the spectrum of elementary excitations in the presence of superfluid flows. However the analysis of critical velocities in [@yukalov] was based on the implication that the Landau criterium can be formulated as the condition on the relative velocity of the superfluid components. Such an implication can be put in question since there are two independent relative velocities in the three-velocity theory. In [@6] the Landau criterium was formulated as the condition of positiveness of energies of elementary excitations in a reference frame connected with a normal component. It yields a joint condition on absolute values of the superfluid velocities of the components and on the angle between their directions. If one component is at rest the superfluidity condition [@6] is reduced to [@yukalov].
The analysis carried out in [@6] shows that the Landau criterium may be fulfilled if one or even both superfluid velocities exceed the velocity of the lower mode $c_-$(in the latter case the velocities should have different directions). In view of unusual behaviour of critical velocities in such systems it is interesting to consider how this behavior can reveal itself in experiments. Two-component BECs have been realized experimentally in ultracold alkali metals gases confined in magnetic and magneto-optical traps. Two components may correspond to different hyperfine Zeeman states of the same isotope [@Rb2; @Rb-n], or to different isotopes [@K-Rb; @K-Rb1; @Rb3]. One of the methods to determine critical velocities for trapped ultracold gases consists in the observation of density excitations induced by some object moving through the condensate [@exp] (usually a laser beam is used as such an object). A motion of an object in a two-component gas with nonzero relative velocity of the superfluid components corresponds to the general case of the three-velocity hydrodynamics.
The Bogolyubov spectrum has a dispersion. Therefore, a motion of an object through a superfluid system (or a superfluid flow past an obstacle) can lead to an occurrence of stationary waves (the waves whose crests remain at rest relative to the obstacle). Such an effect called “ship waves” is well-known [@wi]. It was considered by Kelvin for the waves generated on a water surface by a ship moving in a deep water. Stationary waves in a one-component quasi-two-dimensional BEC were studied in [@ship]. It was shown that in a superfluid that flows past a point obstacle (the obstacle size is less than the healing length) the stationary wave pattern is similar to one for stationary capillary waves. The effect takes place if the superfluid velocity $s$ exceeds the minimal phase velocity of the Bogolyubov mode $c_0$. The stationary waves arise outside the Mach cone bounded by arms directed at the angles $\vartheta =\pm \arcsin(c_0/s)$ relative to the flow.
In recent papers the solitons [@bi2; @16] and stationary waves [@16] induced by an obstacle in a two-component superfluid system were studied (the paper [@16] was published as the electronic preprint when the present study was almost completed). But the authors of [@bi2; @16] considered only the case of equal superfluid velocities (a relative velocity of the components is equal to zero).
In the present paper we put emphasis on the general case of nonzero relative velocity. In Sec. \[sec2\] we obtain the equation for the spectrum and the eigenvectors of collective excitations and define two more critical velocities (in addition to $c_-$ and $c_+$). One of them is the maximum critical velocity $c_m$ for a given component ($c_m>c_-$). The superflow at the velocity $s\to c_m$ can be reached if the other component is at rest. The other is the relative critical velocity $c_{\rm sep}$ ($c_{\rm sep}>c_m$). If the relative velocity exceeds $c_{\rm
sep}$, the frequency of the lower mode becomes complex valued. The latter signals for an instability of the two-component system with respect to a spatial separation of the components. In Sec.\[sec3\] the equation that describes the stationary wave pattern is obtained. It is shown that the stationary waves emerge if the Landau criterium of superfluidity is violated (the energies of the excitations with certain wave vectors becomes negative), but the system remains stable with respect to the spatial separation (the critical relative velocity is not achieved). A number of stationary wave patterns are presented. It is shown that depending on the velocities of the components several qualitatively different situations are possible. If the velocities are the same in modulus and in direction, one set of stationary waves appears at $s>c_-$, and another set adds at $s>c_+$. The phase separation does not occur. If only one component moves with the velocity $s$, one set of stationary waves is formed at $s>c_m$ but if $s$ reaches $c_{\rm sep}$ the phase separation occurs. If the velocities are equal in modulus and the angle between their directions is $\pi/2$ or close to $\pi/2$ two or three sets of stationary waves occur at $s_1=s_2>c_m$. In Sec. \[sec4\] we investigate the structure of the stationary waves. The densities of the components always oscillate out-of-phase in the waves that correspond to the lower mode . The ratio of the amplitudes of the oscillations of the components depends on the relative velocity and varies along the crests of the waves. For complex density patterns where two or three waves interfere the density plots are presented. It is established that the stationary waves are visible in total density images as well as relative density images, but in most cases relative density images are more contrast. The only exception is the stationary waves that are exited at $s>c_+$ and correspond to the upper Bogolyubov mode.
The spectrum of collective modes {#sec2}
================================
To analyze the stationary waves in a two-component superfluid system one should obtain the collective modes spectrum in a moving condensate. It can be found from the matrix version of the Gross-Pitaevskii equation $$\begin{aligned}
\label{19}
i \hbar \frac{\partial \psi_1}{\partial t} = - \frac{\hbar^2}{2
m_1} \triangle \psi_1 + \gamma_1 |\psi_1|^2 \psi_1 + \gamma_{12}
|\psi_2|^2 \psi_1, \cr
i \hbar \frac{\partial \psi_2}{\partial t} =
- \frac{\hbar^2}{2 m_2} \triangle \psi_2 + \gamma_2 |\psi_2|^2
\psi_2 + \gamma_{12} |\psi_1|^2 \psi_2,\end{aligned}$$ where $\psi_i$ are the wave functions of the components, $m_i$ are the masses of the particles, $$\label{47}
\gamma_i = \frac{4 \pi \hbar^2 a_i}{m_i}, \; \gamma_{12} = \frac{2
\pi \hbar^2 (m_1 + m_2) a_{12}}{m_1 m_2}$$ are the interaction constants ($a_i$ and $a_{12}$ are scattering lengthes).
Here we restrict the consideration by the most convenient for the analysis symmetric case for which the components have equal masses of the particles $m_1 = m_2 = m$, equal densities $n_1=n_2=n_0$ of the components and equal interaction constants $\gamma_1 =
\gamma_2 = \gamma$. While this case is quite specific, it is possible to produce equal $\gamma_i$ using a Feshbach resonance. The symmetric case may also correspond to two quasi two-dimensional Bose clouds with a strong dipole interaction separated by a rather high (but thin) barrier that suppress the tunneling (see [@fil1]). On the qualitative level the results obtained for the symmetric case hold for the general case where such a symmetry between the components is broken. We assume the interaction between the particles of the same component is repulsive ($\gamma> 0$), and the stability condition with respect to a spatial separation of the components $(\gamma>
|\gamma_{12}|)$ is fulfilled. Going ahead we note that in case of different velocities that condition is necessary but not sufficient one.
If the temperature is much less than the temperature of the Bose-Einstein condensation the wave functions of the components can be presented as a sum of a large stationary part and a small fluctuating part $$\label{30}
\psi_i (\mathbf{r},t) = \psi_{0i}(\mathbf{r},t) + \delta \psi_i
(\mathbf{r},t).$$ The stationary part of the condensate wave function reads as $$\begin{aligned}
\label{31}
\psi_{0i}(\mathbf{r},t) = \sqrt{n_0} \,e^{i
\varphi_i(\mathbf{r})}e^{-\frac{i \mu_i t}{\hbar}} ,\end{aligned}$$ where $\mu_i = \displaystyle \frac{m_i \mathbf{s}_i^2}{2} +
(\gamma + \gamma_{12}) n_0$ are the chemical potentials of the components, and $\mathbf{s}_j = \displaystyle
\frac{\hbar}{m}\nabla \varphi_j$ are their superfluid velocities. We will search for the fluctuating part of Eq. (\[30\]) in the form: $$\begin{aligned}
\label{31a}
\delta \psi_i (\mathbf{r},t) = e^{i \varphi_i(\mathbf{r})} e^{-
\frac{i \mu_i t}{\hbar}} \left[\textrm{u}_i(\mathbf{r}) e^{-i
\omega t} + \textrm{v}_i^*(\mathbf{r}) e^{i\omega^* t}\right].\end{aligned}$$ Here the functions $\textrm{u}_i$ and $\textrm{v}_i$ are the plane waves $$\label{33}
\textrm{u}_i(\mathbf{r}) = A_i e^{i \mathbf{k} \mathbf{r}}, \;
\textrm{v}_i(\mathbf{r}) = B_i e^{i \mathbf{k} \mathbf{r}}.$$ Substituting Eqs. (\[30\]) – (\[33\]) into Eq. (\[19\]), in a linear in fluctuations approximation we obtain the following equation for the excitation energies $\hbar\omega$ and the eigenvectors: $$\label{40} \mathbf{M} \mathbf{V}= \omega
\mathbf{V},$$ where $$\label{41}
\mathbf{M} = \left(%
\begin{array}{cccc} \displaystyle
\left(\frac{k^2}{2} + 1 + \mathbf{s}_1 \mathbf{k}\right) & 1 & \gamma' & \gamma' \\
-1 &\displaystyle -\left(\frac{k^2}{2} + 1 - \mathbf{s}_1 \mathbf{k}\right) & -\gamma' & -\gamma' \\
\gamma' & \gamma' &\displaystyle \left(\frac{k^2}{2} + 1 + \mathbf{s}_2 \mathbf{k}\right) & 1 \\
-\gamma' & -\gamma' & -1 &\displaystyle -\left(\frac{k^2}{2} + 1 - \mathbf{s}_2 \mathbf{k}\right)\\
\end{array}%
\right)$$ and $$\label{ev}
\mathbf{V}=\left(%
\begin{array}{c}
A_1 \\
B_1 \\
A_2 \\
B_2 \\
\end{array}%
\right)$$ Here and below all physical quantities are expressed in terms of dimensionless length and time $$\begin{aligned}
\label{22}
\tilde{r_i} = \frac{r_i}{\sqrt{2} \xi},\; \tilde{t} =
\frac{c_0}{\sqrt{2} \xi} t,\end{aligned}$$ where $c_0 =\sqrt{\gamma n_0/m}$ is a sound velocity in a one-component condensate, and $\xi =\hbar/\sqrt{2 m \gamma n_0}$ is a healing length. We also define the dimensionless parameter of the interspecie interaction $\gamma' = \gamma_{12}/\gamma$.
The $u-v$ transformation procedure (\[31a\]) is equivalent to the diagonalization of a quadratic form on Bose operators [@pit; @bogol]. The matrix $\mathbf{M}$ has four eigenvalues. The components of the eigenvectors of Eq. (\[40\]) can be normalized as $|A_1|^2-|B_1|^2+|A_2|^2-|B_2|^2=\pm 1/V$ ($V$ is the volume of the system). For two physical modes the norm of the eigenvector should be positive [@bogol]. The spectra of the physical modes read as $$\label{42}
\omega_{\pm} = \frac{(\mathbf{s}_1+\mathbf{s}_2) \mathbf{k} %+
}{2} + \sqrt{k^2 \left(1 + \frac{k^2}{4}\right) +
\frac{(\mathbf{s}_- \mathbf{k} )^2}{4} \pm \sqrt{k^2\left(1+
\frac{k^2}{4}\right)(\mathbf{s}_- \mathbf{k} )^2 + k^4
\gamma'^2}},$$ where $\mathbf{s}_- = \mathbf{s}_1 - \mathbf{s}_2$ is the relative superfluid velocity. At $s_1=s_2 = 0$ the spectrums (\[42\]) has the standard Bogolyubov form $$\label{42-1}
\omega_{\pm}=k\sqrt{1\pm |\gamma'|+k^2/4}.$$ As is clear from Eq. (\[42-1\]), the stability condition with respect to a spatial separation $\gamma> |\gamma_{12}|$ is the requirement for the excitation spectrum be real valued. At small $\mathbf{k}$ the excitation spectrum is a sound one and the velocities of the modes are equal to $c_\pm = \sqrt{1\pm
|\gamma'|}$. The Landau criterium requires the spectrum (\[42\]) be positive valued at all wave vectors. One can see from Eq.(\[42\]) that at $\mathbf{s}_1 = \mathbf{s}_2
=\mathbf{s}$ the Landau criterium is reduced to the inequality $s
<c_-$. If only one component flows ($s_2=0$), the Landau criterium requires a fulfilment of the inequality $s_1 <c_m=c_-\sqrt{1 +
|\gamma'|} $. If the velocity of a given component exceeds $c_m$, the superfluidity condition is broken irrespective of a value and direction of the velocity of the other component [@6], i.e. $c_m$ can be called the maximum critical velocity.
At the velocities for which the energy (\[42\]) is negative valued in some range of ${\bf k}$ the Landau criterium is broken and stationary waves can occur in the system. In contrast to a one-component system, in a two-component system an increase of superfluid velocities may result in that the spectrum of the lower mode be complex valued. As follows from Eq. (\[42\]), the spectrum remains real valued if the relative velocity satisfies the condition $$\label{17}
|\mathbf{s}_1 - \mathbf{s}_2|< c_{\rm sep}= 2 c_- .$$ At complex valued frequencies (\[42\]) the amplitude of excitations grows with time that leads to a destruction of a homogeneous state and to a spatial separation of the components (or stratification). In contrast to Ref. [@yukalov], we consider the condition of stratification and the Landau criterium as different conditions. The Landau criterium yields a joint restriction on both superfluid velocities (and its mutual direction), while the condition of stratification is a restriction only on the modulus of the relative velocity. Note that $c_{\rm
sep}> c_m $, and under increase of the velocity of a given component, first, the Landau criterium is violated, and then, after further increase, the stability condition with respect to the stratification is broken. One can show, that the same situation takes place under simultaneous increase of two velocities (at nonzero relative velocity). Only at equal in value and oppositely directed superfluid velocities the Landau criterium and the stability condition are broken at the same point.
Since we are interested in stationary waves in a homogeneous (not stratified) system, we will consider only the velocities for which the condition (\[17\]) is satisfied.
The stationary wave pattern {#sec3}
===========================
Let us consider a two-component BEC that flows past an obstacle situated at the origin of coordinates. We assume the system is quasi-two-dimensional, i.e. it is thin enough to neglect the dependence of the condensate wave function on the transverse coordinate, and to consider all vector quantities as two-dimensional ones. If the size of an obstacle is much less than $ \xi $ it can be considered as a point one. Under violation of the Landau criterium the obstacle behaves as a point source of waves (below we are only interested in stationary waves). The waves propagate from an obstacle with the group velocities $\mathbf{v}_{g\pm} =
\partial \omega_{\pm} / \partial \mathbf{k}$, where $\omega_\pm =\omega_\pm (k_x, k_y)$ are given by Eq. (\[42\]) at $k_z=0$. Here and below the index $+(-)$ corresponds to the waves generated by the upper (lower) mode. The direction of the propagation for the wave with a given $\mathbf{k}$ is defined by the expression $$\label{9}
\tan\chi_{\pm} =\frac{\partial \omega_{\pm}/\partial k_y}{\partial
\omega_{\pm}/\partial k_x},$$ where $\chi_+ (\chi_-)$ is the angle between the group velocity direction for a given mode and the axis $x$.
For the stationary wave the frequency (\[42\]) is equal to zero, and the components of the wave vector $ \mathbf{k} = (k_x, k_y) $ are related by the equation $$\label{60}
\omega_\pm(k_x, k_y)=0.$$ It is convenient to use the angle $\eta$ between the wave vector and the opposite direction of the axis $x$ as an independent parameter, i.e. $$\label{12-0}
\mathbf{k} = (- k \cos\eta, k \sin\eta).$$ Let us denote the angle between the velocities by $\theta $ and select the axis $x$ along the bisectrix of this angle ($
\mathbf{s}_1 = (s_1 \cos \frac{\theta}{2}, - s_1 \sin
\frac{\theta}{2})$ and $\mathbf{s}_2 = (s_2 \cos \frac{\theta}{2},
s_2 \sin \frac{\theta}{2})$). For the stationary waves Eq. ($\ref{60}$) yields the following dependence of the wave number $k$ on the angle $\eta$ $$\label{12}
\begin{array}{c}
k_{\pm}(\eta) = \sqrt{2} \biggl[-2 + s_1^2 \cos^2\left(\eta -
\frac{\theta}{2}\right) + s_2^2 \cos^2\left(\eta+
\frac{\theta}{2}\right) \mp \\ \left.
\sqrt{4 \gamma'^2 + \left(s_1^2
\cos^2\left(\eta - \frac{\theta}{2}\right) - s_2^2
\cos^2\left(\eta
+ \frac{\theta}{2}\right)\right)^2}\right]^{1/2}. \\
\end{array}$$
A wave crest line is a line of a constant phase. For the stationary wave the phase can be obtained from the equation $$\label{13}
\phi(\mathbf{r}) = \int_0^{\mathbf{r}} \mathbf{k} d\mathbf{r}.$$ In Eq. (\[13\]) the integral is along a straight line going out from the origin of coordinates and directed parallel to the group velocity $\mathbf{v}_{g\pm}$. The angle $\mu$ between $\bf{k}$ and $ \bf{r}$ is defined by the expression $$\label{14-0}
\mu = \pi - \eta - \chi_\pm$$ (see Fig. \[scheme\]).
According to Eq.(\[13\]), the quantity $r$ for the points at the wave crest with a given phase $\phi$ satisfies the equation $$\label{15}
r=\frac{\phi}{k \cos\mu}.$$ Substituting Eq. (\[14-0\]) into Eq. (\[15\]), we get the equations for the stationary wave crest coordinates $x=r \cos\chi$ and $y=r \sin\chi$ in a parametric form $$\begin{aligned}
\label{16}
x_{\pm}= - \frac{\phi}{k_{\pm}(\eta)\cos\eta \left[1 -
\tan\chi_{\pm}\tan\eta\right]}, \cr y_{\pm}= -
\frac{\phi\tan\chi_{\pm}}{k_{\pm}(\eta)\cos\eta \left[1 -
\tan\chi_{\pm} \tan\eta\right]}.\end{aligned}$$ In Eq. (\[16\]) the values $\tan\chi_{\pm}$ are the functions of the parameter $\eta$. The explicit form of these functions is given by Eq. (\[9\]), in which after differentiation one should substitute Eqs. (\[12-0\]) and (\[12\]). The range of values of $ \eta $ is determined by the condition for the function $k_\pm
(\eta) $ be real valued. Eqs. (\[16\]) allow to draw the stationary wave pattern for arbitrary values of $s_1$, $s_2$ and $
\theta $.
Let us consider some special cases. For definiteness, we choose the parameter $\gamma'=0.5$. Such a choice corresponds to $c_- =
0.707$, $c_m=0.866$ and $c_+ = 1.22$ (in $c_0$ units).
1\. At equal superfluid velocities $\mathbf{s}_1 = \mathbf{s}_2
=\mathbf{s}$ Eqs. (\[16\]) yield the expected result. At $s>
c_-$ a set of stationary waves corresponding to the lower mode arises. The crests for these waves are outside the cone bounded by the arms $\chi_1 =\pm \arcsin(c_-/s) $. If the velocities $s> c_+$ are reached the second set of stationary waves appears. It corresponds to the upper mode and situated outside the arms $\chi
=\pm \arcsin (c_+/s) $. Since $s_- = 0$, the stratification does not occur at any $s$. As an example, the stationary wave pattern for $s=1.5$ is shown in Fig.\[waves10\].
2\. If only one component (say, the component 1) flows, the stationary waves arise at the velocities $s_1> c_m $. These waves correspond to the lower mode. The waves are outside the cone $
\chi_- =\pm \arcsin(c_m/s) $. In this case the frequency of the upper mode is always positive and the second set of waves cannot arise. At $s_1> c_{\rm sep}$ (for the parameters chosen $c_{\rm
sep} =1.41$) the system becomes unstable with respect to the stratification. The stationary wave pattern for the flow of one component with $s_1=1.0$ is shown in Fig.\[waves1\].
3\. It is interesting to analyze the situation when the angle between the velocities is $\theta=\pi/2$. In this case the obstacle emits waves only when the velocity of at least one component exceeds $c_m$. Let us consider a more specific case of the velocities equal in magnitude $s_1=s_2=s$. Then the velocity range in which the stationary waves occur is limited by the condition $c_m <s <\sqrt{2} c_- $ (for the parameters chosen $0.866 <s <1$). In Fig.\[waves4\] we present the stationary wave pattern for $s=0.9$. One can see that in such a situation two sets of stationary waves arise. It is important to emphasize that both sets correspond to the lower mode (the frequency of the upper mode remains positive for the velocities in the range $c_m <s <\sqrt{2}
c_-$).
In the limit $ \gamma'\to 0$ (that corresponds to the absence of the interaction between the components) each component should have its own set of stationary waves at any relative directions of the velocities. For rather large $\gamma'$ this feature survives only in a close vicinity to $\theta=\pi/2$. There should be a smooth transition from $\theta =\pi/2$ to $\theta=0$. The analysis of the wave patterns at different $\theta$ shows that at intermediate $\theta$ a quite complicated pattern emerges: the crests of a given set end with cusps, and bridges connect them with crests from the other set. Under decrease of $\theta$ the bridges and cusps disappear and the wave pattern becomes similar to one for $
\theta=0$. The stationary wave pattern with cusps and bridges is shown in Fig. \[waves11\].
In general, the stationary wave pattern is qualitatively similar to one of presented in Figs. \[waves10\]-\[waves11\].
The density pattern for the stationary waves {#sec4}
============================================
There is a number of methods of probing BECs to get their density profiles (see [@ket]). Advanced technics was developed for the study of density profiles of spinor BECs [@c0; @c1; @c2; @c3; @prl05] (that in certain sence can be considered as two-component ones). In particular, the density and the spin-density profiles of an atomic cloud were measured with a high resolution by the polarization-dependent phase-contrast imaging method [@prl05; @sk07].
In view of modern experimental possibilities it is important to find the ratio of the components in the stationary waves and to determine specific features of the total density and the relative density patterns for the stationary wave in two-component systems.
Since the stationary wave is the eigenmode, the ratio of the total density and relative density amplitudes can be found from the corresponding eigenvector (\[ev\]).
The density of a given component is the square modulus of the condensate wave function $ \rho_i = | \psi_i |^2$. This quantity can be presented as a sum of the unperturbed density $n_0$ and the perturbation $\delta n_{i}$ caused by the stationary wave. Using the equation for the condensate wave function (\[30\]) and Eqs. (\[31\]) – (\[33\]), we obtain $$\label{38} \delta n_{i}
= \psi_{0} \delta \psi_{i}^* + \delta \psi_{i} \psi_{0}^*=\delta
\rho_{i} \cos(\mathbf{k} \mathbf{r}),$$ where $$\begin{aligned}
\label{46}
\delta\rho_{i} = {\cal A} \sqrt{n_0} (A_{i} + B_{i}).\end{aligned}$$ here ${\cal A} $ is the amplitude of a given eigenmode. The amplitude ${\cal A}$ depends on the intensity of the source (obstacle) and on the distance from the source. As follows from Eqs. (\[40\],\[41\]), the components of the eigenvectors are real valued quantities. That is why Eq. (\[46\]) may correspond to in-phase oscillations of the densities or to the oscillation with the phase shift equal to $\pi$ (out-of-phase). Here we consider the case $\gamma'>0$ for which the lower Bogolyubov mode corresponds to out-of-phase oscillations and the higher mode - to in-phase oscillations.
The total density and the relative density oscillation amplitudes in the stationary wave are $\delta\rho_\pm=\rho_1\pm \rho_2$. The specific of the symmetric case (components with equal masses, densities and interaction constants) is that in the absence of the flow the oscillations of the total density vanish for the lower Bogolyubov mode ($\delta\rho_+=0$), and the oscillations of the relative density are absent in the higher mode ($\delta\rho_-=0$). As was shown in the previous section, in most cases the stationary waves are caused by the lower mode. Therefore it is important to clarify whether is the total density disturbed in the stationary waves.
Using the analytical expressions for the eigenvectors of the matrix (\[41\]) one finds that the exact relations $\delta\rho_1
=-\delta\rho_2$ for the lower mode and $\delta\rho_1 =
\delta\rho_2$ for the upper mode are hold in case of equal velocities (${\bf s}_1={\bf s}_2$). In means that in the stationary wave pattern shown in Fig. \[waves10\] the lower mode set is visible in the relative density image, while the upper mode set - in the total density image.
In the general case ${\bf s}_1\ne {\bf s}_2$ we find that $\delta\rho_1 \ne -\delta\rho_2$ and the total density oscillations are nonzero for the stationary waves that correspond to the lower mode, in particular, for the pattern shown in Figs. \[waves1\] - \[waves11\]. For these cases the ratios $|\delta\rho_1|/|\delta\rho_2|$ along the crests are shown in Fig. \[density1\]. Since this ratio differs from unity, both, $\delta\rho_+$ and $\delta \rho_-$ are nonzero and stationary wave patterns should be visible in the total density image as well as in the relative density image. We also would like to point out the feature that follows from Fig. \[density1\]. If only one component flows past the obstacle the stationary waves contain admixture of both components, but the flowing component has larger amplitude.
In cases shown in Figs. \[waves1\],\[waves4\] two or three sets of stationary waves interfere. Thus the density pattern depends not only on the eigenvectors but on the amplitudes of the eigenmodes, and another approach should be used to analyze the density profiles. Here we use the approach developed in [@16].
The interaction of the Bose gas with the obstacle is described by the Hamiltonian $$H_{int}=\int d {\bf r} \sum_i V_i (\mathbf{r}) |
\psi_i |^2,$$ where $V_i (\mathbf{r})$ is the potential of the interaction between the $i$-th component and the obstacle. Considering the obstacle as a point source and assuming that the obstacle interacts identically with both components we set $V_1
(\mathbf{r}) =V_2 (\mathbf{r}) =V_0 \delta({\bf r})$. Respectively, the terms $V_0 \delta({\bf r}) \psi_i$ should be added to the right hand parts of the Gross-Pitaevskii equations (\[19\]). The Gross-Pitaevskii equations can be rewritten in terms of density and velocity fields (the hydrodynamic form). Linearizing the hydrodynamic equations with respect to the density and velocity fluctuations (see, for instance, [@6]), and excluding the velocity fluctuations we arrive at the equations for the Fourier components of $\delta n_i$: $$\label{111}
\left[- \left(\mathbf{s}_i \mathbf{k}\right)^2 + k^2 \left(1 +
\frac{k^2}{4}\right)\right] \delta n_{i}({\bf k}) + k^2 \gamma'
\delta {n}_{3-i}({\bf k}) = - k^2 \tilde{V}_0,$$ where $\tilde{V}_0=V_0/\gamma $. In deriving Eq. (\[111\]) we take into account that all time derivatives are zero for the stationary waves.
Solving Eq.(\[111\]) and taking the inverse Fourier transformation we get $$\label{111a}
\delta n_i = - \frac{4 V_0}{\pi^2\gamma\xi^2} {\rm Re}
\int_{-\pi/2}^{\pi/2} d\eta \int_0^{\infty} k d k \frac{ 1 - \gamma'
+ \frac{k^2}{4} - \frac{\left(\mathbf{s}_i \mathbf{k}\right)^2}{k^2}
}{(k^2 - k_+^2)(k^2 - k_-^2)} \, e^{i k r \cos \mu},$$ where the quantities $k_\pm$ are given by Eq. (\[12\]). The angle $\mu(\eta)$ is determined by Eqs. (\[14-0\]) and (\[9\]). The integral over $k$ in Eq.(\[111a\]) can be evaluated analytically with the use of the residue theorem. The poles $k=k_-$ and $k=k_+$ (for real $k_\pm$) yield the contribution of the lower and the upper Bogolyubov modes, respectively. The integral over $\eta$ is evaluated by the stationary phase method. The number of stationary phase points coincides with the number interfered waves in the stationary wave pattern (1 point for Figs. \[waves10\] and \[waves1\], 2 points for Fig. \[waves4\] and 3 points for Fig. \[waves11\]).
The answer can be presented in the following form $$\label{112}
\delta n_i\approx \frac{V_0}{\gamma\xi^2}\sum_{i,\lambda}
C_{i,\lambda}(\chi)\frac{\cos\left [\kappa_{i,\lambda}(\chi)
r-\pi/4\right]}{\sqrt{r}},$$ where the index $\lambda$ numbers the sets that contribute to the stationary wave pattern in a given sector of $\chi$. In Eq. (\[112\]) short range terms $\propto 1/r^2$ are omitted. The coefficients $C_{i,\lambda}(\chi)$ and $\kappa_{i,\lambda}(\chi)$ are quite complicate and we do not present the explicit expressions here.
The case ${\bf s}_1={\bf s}_2={\bf s}$ can be analyzed directly from Eq. (\[111a\]). At such velocities $$\label{113}
\delta n_1= \delta n_2= \frac{ V_0}{\pi^2\gamma\xi^2} {\rm Re}
\int_{-\pi/2}^{\pi/2} d\eta \int_0^{\infty} k d k \frac{ e^{i
{\bf k r}}}{k^2 - k_+^2} ,$$ where $k_+=2\sqrt{s^2 \cos^2 \eta -c_+^2}$. One can see from Eq. (\[113\]) that, first, $\delta n_1=\delta n_2$ and the relative density remains unperturbed, and, second, the stationary waves patterns are caused by the upper mode only and emerges at $s>c_+$. Such features have clear explanation: the obstacle with $V_1=V_2$ cannot excite the mode with $\delta n_1=-\delta n_2$. We emphasize that these features are specific for the symmetric two-component condensate. If the bare modes of the components differ from each other the lower mode is not pure relative density oscillations ($\delta n_1\ne-\delta n_2$) and the obstacle with $V_1=V_2$ may excite the lower as well as the upper mode.
Using Eq. (\[112\]) and explicit expressions for $C_{i,\lambda}$ and $\kappa_{i,\lambda}$ we obtain the density plots for the interfered stationary waves given in Figs. \[waves4\], \[waves11\]. The results for the total density and relative density patterns are shown in Figs. \[f9\], \[f10\]. One can see that the images presented have clear interference structure and both the total density and relative density measurements can be used for the visualization of the stationary waves. Note that in the case considered the relative density pattern in much more contrast than the total density pattern.
Ending this section we would note the following. The excitation of stationary and propagating waves by the obstacle can be considered as a kind of Cherenkov radiation. Cherenkov radiation arises when the velocity of a radiating object in the medium exceeds the phase velocity of radiated waves in this medium. In this respect it was unclear why the critical velocity may be larger than the velocity of the lower mode $c_-$. The answer is the following. At relative motion of the components the eigenvector for the lower mode differs from one for the condensate at rest or at $s_-=0$. In other words, in the flow with $s_- \neq 0$ the structure of the modes is modified, and the mode radiated at nonzero relative velocity is not the $c_-$ lower Bogolyubov mode.
Conclusions {#sec5}
===========
In conclusion, the properties of stationary waves arising in a flow of a two-component BEC past an obstacle have been studied. The problem was considered for a special case of the symmetric two-component system (with the components of equal masses of the particles, equal densities and equal interaction constants). Nevertheless, the majority of conclusions are applicable also to the cases in which such a symmetry between the components is absent. Let us recite these conclusions.
In a two-component flowing superfluid system the energies of the excitations can take on not only negative, but complex values. At reaching of negative values the Landau criterium is broken, and at reaching of complex values the system becomes unstable with respect to a spatial separation. The Landau criterium is a joint condition of both superfluid velocities (and the angle between them) in the lab reference frame. The stability condition with respect to a spatial separation is a condition solely on an absolute value of the relative velocity (which does not depend on the reference frame). In case of equal (in modulus and in direction) velocities of the components a spatial separation does not arise at any velocities. Under increase of the velocities the Landau criterion is broken first, and then the stability condition is broken. Stationary waves arise when the Landau criterion is already broken, but the system remains stable with respect to a spatial separation. At equal velocities the stationary waves generated by the lower and the upper modes can arise. If only one component flows, the one set of stationary waves (corresponding to the lower mode) may emerge. If the angle between the velocities is close to $ \pi/2$, two sets of stationary waves can arise, and both of them correspond to the lower mode. In general, the stationary waves are visible at the total density and relative density images.
We did not consider here the ways of creation of relative flow of the components in two-component atomic vapors, but we would like to mention some other possibilities. It is quite simple to realize such a flow for two components separated with a thin barrier. In this case the interaction between the components should contain a long-range (for example, dipole) part. Strictly speaking, in such systems the spectrum of excitations may differ from (\[42\]) (due to a long-range interaction). Nevertheless one can expect that the stationary waves will behave qualitatively the same as in the case considered in this paper. Similar phenomena may emerge in some other systems, for example, multilayer electronic systems with superfluid indirect excitons [@kf]. Another perspective object is superfluid polaritons in semiconductor microcavities, where superflow can be controlled by the laser beam. In a one-component polariton system the observation of stationary waves was reported recently [@amo]. Relative motion of the components may also arise if one component is electrically charged and the system is subjected by an electromagnetic field. For instance, such a situation takes place in neutron stars [@bab].
![The scheme of a wavefront crest[]{data-label="scheme"}](figure1.eps){width="6cm"}
![The wave pattern forming at the flow of components with equal velocities $(s=1.5)$. The waves caused by the lower and the upper modes are shown with solid and dashed lines, respectively.[]{data-label="waves10"}](figure2.eps){width="6cm"}
![The wave pattern forming at the flow of one component with the velocity $s_1 = 1.0 \, (s_2 = 0)$. The axis $x$ is along the flow direction.[]{data-label="waves1"}](figure3.eps){width="6cm"}
![The wave pattern forming by two orthogonally directed flows with the velocities $s_1 = s_2 = 0.9$. The directions of the velocities are shown by the arrows.[]{data-label="waves4"}](figure4.eps){width="6cm"}
![The wave pattern at $s_1=s_2 =1$ and $ \theta = 0.45 \pi
$[]{data-label="waves11"}](figure5.eps){width="6cm"}
![The ratio of densities of the components in the stationary waves along the chests for: a) one component flow $(s_1
= 1, s_2 = 0)$; b) two component flow in mutually orthogonal directions $(\theta = \pi/2, s_1 = s_2 = 0.9)$; c) the case of $\theta = 0.45\pi, s_1 =
s_2 = 1$.[]{data-label="density1"}](figure6.eps){width="8cm"}
![The total density (a) and the relative density (b) patterns (in relative units) for the stationary waves at $ \theta = \pi/2$ and $s_1 = s_2 = 0.9$.[]{data-label="f9"}](figure7.eps){width="15cm"}
![The same as in Fig. \[f9\] at $ \theta = 0.45\pi$ and $s_1 = s_2 = 1$.[]{data-label="f10"}](figure8.eps){width="15cm"}
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---
abstract: 'The vacuum persistence probability, $P_{vac}(t)$, for a system of charged fermions in a fixed, external, and spatially homogeneous electric field, was derived long ago by Schwinger; $w \equiv - \log[P_{vac}(t)]/ V t$ has often been identified as the rate at which fermion-antifermion pairs are produced per unit volume due to the electric field. In this paper, we separately compute exact expressions for both $w$ and for the rate of fermion-antifermion pair production per unit volume, $\Gamma$, and show that they differ. While $w$ is given by the standard Schwinger mechanism result of $w = \frac{(q E)^2}{4 \pi^3 } \, \sum_{n=1}^\infty \frac {1}{n^2} \, \exp \left( -\frac{n \pi m^2 }{q E } \right )$, the pair production rate, $\Gamma$, is just the first term of that series. Our calculation is done for a system with periodic boundary conditions in the $A_0=0$ gauge but the result should hold for any consistent set of boundary conditions. We discuss, the physical reason why the rates $w$ and $\Gamma$ differ.'
author:
- 'Thomas D. Cohen'
- 'David A. McGady'
title: The Schwinger mechanism revisited
---
Introduction
============
Fermion-antifermion pair production from a static classical electric field, known as the Schwinger mechanism, has given rise to a vast literature, since its formulation in $1951$[@Schwinger]. Invoked to gain insights on topics as diverse as the string breaking rate in QCD[@CNN; @Neuberger] and on black hole physics[@GR], this mechanism has become a textbook topic in quantum field theory[@IZ]. Topics such as back reaction[@BR] and finite size effects[@Wang_Wong] have been addressed. In a classic paper, Schwinger exactly calculated the rate at which the vacuum decays due to pair production in the external field. If the electric field is treated classically—*i.e.* the formal limit of $q\rightarrow 0$, $E \rightarrow \infty$ with $q E$ fixed—the vacuum persistence probability is[@Schwinger]: $$\begin{aligned}
P_{\rm vac}(t)& \equiv& |\langle {\rm vac}| U(t) |{\rm vac} \rangle |^2 = \exp(-w V t) \label{SF1}\\
{\rm with} \; \; w &=& \frac{(q E)^2}{4 \pi^3 } \, \sum_{n=1}^\infty \frac {1}{n^2} \,
\exp \left( -\frac{n \pi m^2 }{q E } \right ), \label{SF2}\end{aligned}$$ where $V$ is the spatial volume of the system and $w$ is the rate of vacuum decay per unit volume. While the Schwinger formula of Eq. (\[SF2\]) is very well known, the Schwinger mechanism has often not been fully appreciated in an essential way. The quantity $w$ in Eq. (\[SF2\]) appears to have a very natural interpretation as the rate of production of pairs per unit volume. Schwinger suggested this interpretation in his original paper. There is a plausible, if heuristic, argument in its favor[@CNN; @IZ]. Start by considering a more general situation in which the pair production rate can vary in space and time. The vacuum persistence probability can then be written as $$P_{\rm vac}=e^{ - \int {\rm d}^4x \ w(x) } \; .$$ Next approximate the integral as a discrete sum over space-time cells of volume $\Delta v_i$ points centered at points $x_i$. The vacuum persistence probability then becomes the limit of $$P_{\rm vac}= \lim_{\Delta v \rightarrow 0} \prod_i e^{-w(x_i) \Delta v_i}=\prod_i (1- w(x_i) \Delta v_i) \; .$$ Itzykson and Zuber[@IZ] note that this is the form expected from an independent contribution to the vacuum persistence coming from each cell if $w(x_i)$ is the local rate of pair production. This is taken to confirm Schwinger’s interpretation of $w$. This argument [*is*]{} plausible; it is not surprising that a considerable body of literature has adopted Schwinger’s interpretation and used $w$ to compute particle production rates.
However, as it happens, the interpretation is not correct: in general $w$ does [*not*]{} give the rate of pair creation. As a logical matter the rate characterizing the vacuum’s decay (by the production of the first charged pair), is not necessarily the same as the continuous rate of pair production. While the argument given above is highly plausible, it is also heuristic. A more compelling approach is via a comparison of a direct computation of the pair production rate with $w$, the rate in Eq. (\[SF2\]). Nikishov directly computed the pair production rate per unit volume long ago[@Niki] and found it to be $$\Gamma = \frac{(q E)^2}{4 \pi^3 } \,
\exp \left( -\frac{\pi m^2 }{q E } \right ) \; ; \label{SF3}$$ Remarkably it does [*not*]{} agree with $w$: the entire rate is given by the first term in the series for $w$.
Since $w$ is still commonly confused with the pair production rate, it is useful to rederive Nikishov’s result in a physically transparent context which clearly illustrates why $\Gamma$ rather than $w$ is the rate of pair creation per unit volume. This will first be done in the context of (1+1) dimensions where the issues are particularly clear; the generalization to (3+1) dimensions is straightforward.
Before embarking on the calculation, it is useful to discuss briefly why one might expect $\Gamma$ (the pair production rate per unit volume) to differ from $w$ (the rate characterizing the rate of vacuum decay per unit vacuum). We begin with the trivial observation that pairs are created in distinct modes. If the probability for any given mode to create a pair is constant in time (and thus uncorrelated with each other), one expects that total rate of pair creation to equal $w$. However, if there are temporal correlations between likelihood for decay at various times for the various modes, then this need not be the case.
The precise time that a pair is created is ambiguous; however, once a pair is created and the fermion and antifermion become well separated, they each accelerate in the electric field. Thus particles with large (mechanical) momentum are likely to have been created earlier than particles with small momentum. To the extent that the natural way to characterize modes is in momentum space, there are very large temporal correlations in pair creation. As will be seen in detail in the explicit calculations, momentum space [*is*]{} the natural way to characterize modes for this problem and it is natural that $\Gamma$ differs from $w$. This makes clear why the argument of Ref. [@IZ] that $w$ is the pair creation rate breaks down: there it is assumed that the rate of pair production at distinct space-time points are independent. Implicit in this is the notion that the pairs are created at well-defined points in space-time. However, if the pairs are created at well-defined (or nearly well-defined) momenta they are delocalized in position.
Pair creation in (1+1) dimensions
=================================
Before embarking on the original problem, it is useful to consider the analogous problem in (1+1) dimensions. The (1+1) dimensional case is somewhat simpler and the issues are more straightforward. Moreover, the (1+1) dimension result serves as an important input into the (3+1) dimensional calculation. One can adapt Schwinger’s calculation[@Schwinger] for smaller dimensions[@1plus1; @2plus1]; in (1+1) dimensions $P_{\rm vac} (t)$ is $$\begin{aligned}
P_{\rm vac}^{1+1}(t)& =& \exp(-w^{1+1} L t) \nonumber \\
{\rm with} \; \; w^{1+1} &=& \frac{(q E)}{2 \pi } \, \sum_{n=1}^\infty \frac {1}{n}\,
\exp \left( -\frac{n \pi m^2 }{q E } \right )\nonumber \\
& =& - \frac{(q E)}{2 \pi }\log \left( 1 - \exp \left( -\frac{ \pi m^2 }{q E } \right ) \right )\label{SF1p1}\end{aligned}$$ where $L$ is the length of the system.
The imposition of appropriate boundary conditions in both space and time is crucial in the treatment of this problem. Accordingly, it is critical to compute both $w$ and $\Gamma$, [*with the same boundary conditions*]{} so they can be directly compared with each-other. We adopt the following strategy in choosing boundary conditions in space and time.
The most natural way to identify the number of pairs created is to formulate the problem in terms of an electric field which is switched on for a fixed period and then switched off. One begins with the system in its vacuum state and the electric field turned off. The electric field is turned on at $t=0$ and left on until $t=T$ and then switched off. With the field now off, one can unambiguously use a free particle basis to count the number of pairs. The act of turning the system on and off can yield transient effects; thus one needs to consider the limit of large $T$ ($T \gg m^{-1}$ , $T \gg qE^{-1/2}$) so that the steady state Schwinger mechanism dominates.
To simplify the computation it is useful to keep the electric field a constant over the entire system while making the system of finite length. In this case the modes become discrete and are easily counted. The large $L$ limit can be taken at the end. Continuous acceleration of particles over extended time plays a key role in the analysis. Thus it is important to choose spatial boundary conditions which allow a charged particle to continue accelerating as it reaches the end of the system. Dirichlet boundary conditions are thus inappropriate as a particle striking the end of the system will bounce back and, instead of continuing to accelerate, will then decelerate. Continuous acceleration over long times can be achieved simply via the implementation of periodic boundary conditions. However, periodic boundary conditions are sensible only when the Hamiltonian is invariant under translations by $L$. Because the gauge choice $A_0=0$ ensures translational invariance, we work in this gauge. Of course, our results must be gauge invariant and we can work in any gauge. However, if we choose to adopt a different gauge the boundary conditions needed to ensure continuous acceleration become time dependent. Ultimately we can check whether this choice of boundary condition (and its associated gauge choice) is sensible by comparing the derived vacuum persistence probability with the Schwinger result of Eq. (\[SF1p1\]). The Dirac equation becomes $$\begin{aligned}
&\left ( \alpha (-i \partial_x -q A_x(t)) + \beta m \right ) \psi(x,t) = i \partial_t \, \psi(x,t) \nonumber\\
&A_x(t)= \begin{cases} 0 & {\rm for} \; t<0 \\ -E t & {\rm for} \; 0 \le t<T\\ -ET & {\rm for} \; T \le t \end{cases} \nonumber \\
&\psi(x,t) = \psi(x+L,t)\end{aligned}$$ where $\psi$ is a two-dimensional spinor and $\alpha$ and $\beta$ are the appropriate two-dimensional Dirac matrices. There exists a complete set of a solutions of the form $$\begin{aligned}
\psi(x,t) & = e^{i p_k x} \chi_k(t) \nonumber \\
p_k & = k \, \frac{2 \pi}{L} \; \; \;({\rm integer} \; k) \nonumber\\
h_k(t)& = \alpha(p_k -q A_x(t)) + \beta m \nonumber \\
h_k(t) \chi_k(t) & = i \partial_t \, \chi_k(t)
\label{mode}\end{aligned}$$ where the restriction to integer $k$ comes from the periodic boundary conditions. Finally, it is convenient to introduce a unitary transformation, $U_k(t)$, into a basis in which $h_k$ is diagonal $$\begin{aligned}
\chi_k(t) & = U_k(t) \Phi_k (t) \nonumber \\ \epsilon_k(t) & = \sqrt{(p_k + q A(t))^2+m^2} \nonumber \\
U_k^\dagger(t) h_k(t) U_k(t) &= \begin{pmatrix} \epsilon_k(t) & 0 \\ 0 & -\epsilon_k(t) \end{pmatrix}
\label{trans}\end{aligned}$$
For convenience we choose $T$ such that $T= j \tau$ with $\tau \equiv \frac{2 \pi}{qE L}$ and j a positive integer. It is easy to see that $\epsilon_{k}(T)=\epsilon_{k+j}(0)$: the spectrum of the system after $E$ is turned off is identical to the original spectrum, although the mode labels have changed. This is hardly surprising, as the system for $t \ge T$ is simply that of a free particle in the absence of an electric field—exactly as it is for $t < 0$. The restriction to integer $j$ ensures that the boundary conditions are the same. At $t=0$ the system is in its vacuum state with all of its negative energy levels filled. Thus the appropriate boundary condition for the fermionic modes is $\Phi_k^T(0)= (0, 1)$.
The equation of motion for $\Phi_k^T(t) \equiv (\phi_k^+(t), \phi_k^-(t))$ is obtained from Eqs. (\[mode\]) and (\[trans\]) and for $0 \le t \le T$ is given by $$\begin{aligned}
\begin{pmatrix} \epsilon_k(t) & \frac{q E m}{\epsilon_k^2(t)} \\ \frac{q E m}{\epsilon_k^2(t)} & -\epsilon_k(t) \end{pmatrix} \begin{pmatrix} \phi_k^+(t) \\ \phi_k^-(t) \end{pmatrix}& = i \partial_t \begin{pmatrix} \phi_k^+(t) \\ \phi_k^-(t) \end{pmatrix} \nonumber \\
{\rm with} \; \; \; \begin{pmatrix} \phi_k^+(0) \\ \phi_k^-(0) \end{pmatrix} &= \begin{pmatrix} 0 \\ 1 \end{pmatrix} \; . \label{EOM}\end{aligned}$$ By construction, $|\phi_k^+(T)|^2$ is the probability for the positive energy state of the $k^{\rm th}$ mode to be occupied after the field has been switched off; it represents the probability of pair creation for this mode. The expectation value for the total number of pairs produced, $N$, and the vacuum persistence probability are thus given by $$\begin{aligned}
\langle N(T)\rangle &= \sum_k |\phi_k^+(T)|^2 \label{number}\\
P_{\rm vac}^{1+1}(T) &= \prod_k \left (1 - |\phi_k^+(T)|^2 \right )\nonumber \\
& =\exp \left (\sum_k \log\left (1 - |\phi_k^+(T)|^2 \right ) \right ) \; . \label{pvac}\end{aligned}$$
To proceed further we need solutions of Eq. (\[EOM\]). An exact solution can be obtained in terms of parabolic cylinder functions with complex arguments. The form is cumbersome and will not be given here. The important point here is the asymptotic behavior valid when $|p_k| \gg m$, and $|p_k + qE T| \gg m$, in which case the solution reduces to $$\begin{split}
|\phi_k^+(T)|^2 & = \theta (-p_k)\theta (p_k + qE T ) \exp \left( -\frac{ \pi m^2 }{q E } \right ) \\ & \times \left( 1 + {\cal O}\left(\frac{m}{|p_k|} ,\frac{m}{|p_k + qE T|}\right ) \right ) \; . \label{lz}
\end{split}$$
The form of Eq. (\[lz\]) is not surprising. The equation of motion for the two level system in Eq. (\[mode\]) is precisely of the form considered long ago by Landau and Zener[@LZ]; $\exp \left( -\frac{ \pi m^2 }{q E } \right )$ is simply the Landau-Zener transition probability which is valid when the initial and final levels are well separated on the scale of $m$. Provided that $qE T \gg m$ the correction to Eq. (\[lz\]) is small except for a small number of modes. Inserting this form into Eqs. (\[number\]) and (\[pvac\]) and using the identity $\log(1 - \theta(y) \theta(z) x) = \theta(y) \theta(z) \log(1-x)$ yields $$\begin{aligned}
\langle N(T)\rangle &= \sum_k \theta(-p_k) \theta (p_k + qE T ) e^{-\frac{ \pi m^2 }{q E }} \nonumber \\
&\times \left ( 1 + {\cal O} \left ( \frac{m}{qE T} \right ) \right ) \label{number2}\\
\log \left (P_{\rm vac}^{1+1}(T) \right )
& =\sum_k \theta(-p_k) \theta (p_k + qE T ) \log\left (1 - e^{ -\frac{ \pi m^2 }{q E } } \right )\nonumber \\
& \times \left (1 + {\cal O} \left ( \frac{m}{qE T}\right ) \right ) \, \; . \label{pvac2}\end{aligned}$$ where the scale of the correction term reflects the fraction of modes for which the correction to the leading term in Eq. (\[lz\]) is non-negligible. From the definition of $p_k$ in Eq. (\[mode\]) one sees that $\sum_k \theta(-p_k) \theta (p_k + qE T )= \frac{qE T L}{2 \pi}$. Using this fact, and taking the long time limit to remove transient effects, we see that $P_{\rm vac}^{1+1}(t)$ in Eq. (\[pvac2\]) exactly reproduces the standard result for the vacuum persistence probability in Eq. (\[SF1p1\]), indicating that our boundary conditions were chosen consistently.
Further, the rate of pairs produced by the electric field in the long time limit is, $$\Gamma^{1+1} \equiv \frac{\langle N \rangle}{L T} = \frac{q E}{2 \pi} \exp \left( -\frac {\pi m^2}{q E} \right ). \label{number3}$$ Critically, we see that $\Gamma^{1+1}$, the rate of pairs produced per unit length per unit time, is [*not*]{} $w^{1+1}$ but rather the first term in the series as expected from Ref. [@Niki].
The massless limit
------------------
The $m \rightarrow 0$ limit in (1+1) dimensions illuminates the essential issues quite clearly. From Eq. (\[SF1p1\]) it is clear that $w^{1+1}$ diverges as $m \rightarrow 0$. Thus, the vacuum persistence probability becomes zero. If the interpretation of $w$ as the pair production rate were correct this would mean that the rate of pair production per unit length would, perversely, be infinite for massless particles. However, the particle production rate per unit length is given by $\Gamma^{1+1}_{m \rightarrow 0} = \frac {q E}{2 \pi}$ and is finite.
It is easy to visualize this physically. With the boundary conditions used here, the energies of the modes for $m=0$ are given by $\pm (p_k + qE t)$ as shown in Fig. \[zeromass\]. Moreover the occupation numbers are preserved by the equation of motion; only the mass term induces transitions. There is a subtlety when $p_k+ qE t=0$; at which point the positive and negative energies cross. Since $\phi^+$ indicates the amplitude for the positive energy solution, the labels $\phi^+$ and $\phi^-$ switch at $p_k+ qE t=0$. (Note that when any finite mass is put in, the level crossings are avoided. However, as $m \rightarrow 0$ the Landau-Zener probability which the occupation number switches from the negative to positive level goes to unity and one reproduces the zero mass result.)
Initially, the system is in the vacuum state with all negative energy levels filled. Thus except for the special case of $k=0$, $|\phi^+(t)|^2$ is given by $\theta(-p_k) \theta(qEt + p_k)$, exactly as one expects from Eq. (\[lz\]). The $k=0$ modes are special in that they have exactly zero energy at $t=0$; we take them to be half occupied. Consider the occupation of positive energy levels as time increases. As seen in Fig. \[zeromass\] when $t$ increases by $\tau \equiv \frac{2 \pi}{qE L}$ exactly one new level crosses from negative to positive and this corresponds to the creation of a pair. At $t=j \tau$ one has created on average $j-1/2$ pairs (the $1/2$ coming form the $1/2$ filled $k=0$ mode): the number of pairs increases linearly with time at a rate of $1/\tau = \frac{qE L}{2 \pi}$, precisely the rate in Eq. (\[number3\]). However, for $j \ge 2$ the vacuum persistence probability is strictly zero; at $j=2$ we [*know*]{} with unit probability that a pair has been created in the $k=-1$ mode. This cleanly illustrates both the distinction between $\Gamma$ and $w$ and the critical role played by temporal correlations between modes. It is the fact that pairs are created in different modes at different times which accounts for the difference.
![Energy levels for massless (1+1) charged fermions in a constant electric field as a function of time in units of $\tau \equiv \frac{2 \pi}{qE L}$. The dotted lines correspond to empty–initially positive energy–levels and the solid lines to filled–initially negative energy–levels; the dashed lines are the half-filled levels of the zero-mode. The solid circles correspond to particles and the open circles to antiparticles; the half-filled circles correspond to the creation of a particle (antiparticle) with 50% probability []{data-label="zeromass"}](zeromass.eps){width="3.0in"}
A natural way to visualize the massless case in (1+1) dimensions is to think of the (1+1) dimensional “universe” as a circular loop of radius $R=L/(2 \pi)$. Suppose further that this loop is imbedded in a (3+1) dimensional world with a time-varying magnetic flux perpendicular to the loop and threading through its center. This flux is turned on at $t = 0$, after which it linearly increases in time, for a time $t = j \tau$ after which it is held fixed; $\tau$ is defined as the amount of time for one flux quantum to thread through. Because $\frac {\partial B}{\partial t} = - \nabla \times E$, this is identical to an $E$ field pointing tangent to the loop at all points, for a total time $\Delta t = j \tau$. The ground state of this system at $t=0$ is simply a filled Dirac sea of massless, charged particles—in just the same manner as the ground state of a normal wire represents a filled [*Fermi*]{} sea of massive, charged, particles (electrons). This system clearly has periodic boundary conditions, and thus gives another physical picture for understanding pair creation of charged particles in (1+1) dimensions. While the $E-$field is on, magnetic flux-quanta thread through the ring at a rate of $1/\tau$. Since the system is circular, the natural variable labeling modes is the angular momentum, $L_k$ (where $L_k$ is related to the momentum in the previous formulation via $L_k = R p_k$). After $j$ flux quanta have thread through the loop, then $L_{k} \rightarrow L_{k+j}$. This both drives $j$ empty levels into the Dirac sea and $j$ filled levels out of the Dirac sea, creating $j$ fermion-antifermion pairs. The rate is $1/\tau $ and is precisely that given in Eq. (\[number3\]).
A card game
-----------
The distinction between the rate per length at which pairs over time, $\Gamma^{1+1}$, and the rate per length of vacuum decay, $w^{1+1}$, is essentially one of classical probabilities; for this (1+1)-dimensional system this can be simply illustrated in the context of a card game.
One might imagine that a casino has introduced a new game called “vacuum.” The rules of the game are simple: A very large number of decks of cards are shuffled together. One class of card is specified—this class could be rather general ([*e.g., *]{} all black cards or all diamonds), somewhat more specific ([*e.g., *]{} all sevens), or quite specific ([*e.g., *]{} aces of spades). The pile of cards is said to be in the vacuum. Cards are then turned over at a fixed rate of $1/\tau$. The pile remains in the vacuum until the first card of the specified class is turned over. Wagers are placed on how long the pile is in the vacuum and on how many cards of the specified class are turned over in a specified time.
It is easy to see that if one continues to turn over cards at the rate of $1/\tau$, the average number of cards in our specified class which are turned over after $t= j \tau$ (where $j$ is a positive integer) is $\langle N_{\rm class} \rangle = f t/\tau$ where $f$ is the fraction of the cards in the class ([*i.e.,*]{} $f=1/2$ if the class is black cards; $f=1/52$ if the class consists of aces of spades). Thus, $\gamma$, the average rate of production for cards in the class, is given by $$\gamma = \frac{1}{\tau} f \; . \label{gamma}$$ The probability of the pile remaining in the vacuum after $t=j \tau$ is obviously given by $P_{\rm vac}= (1-f)^{j}=(1-f)^{t/\tau}$. This can be rewritten as $$\begin{aligned}
P_{\rm vac}& = \exp (-\omega t) \nonumber \\
\omega & = - \frac{1}{\tau} \log (1 - f) = \frac{1}{\tau} \sum_n \frac{f^n}{n} \label{omega} \, .\end{aligned}$$ It is obvious that the structure of Eqs. (\[gamma\]) and (\[omega\]) are analogous to Eqs. (\[number3\]) and (\[SF1p1\]) with $f$ playing the role of $e^{-\frac{\pi m^2}{q E}}$.
In the context of the game it is obvious why $\gamma$ and $\omega$ differ. Suppose as an extreme case one considers as the class [*all*]{} cards. The rate at which cards in the class are produced is obviously $1/\tau$; after the first card has been turned over the probability that the pile is in the vacuum is clearly zero. This is the analog of the $m \rightarrow 0$ limit.
Pair creation in (3+1) dimensions
=================================
Having illustrated the essential distinction between the vacuum decay rate, $w$, and the rate of particle production, $\Gamma$, in the (1+1) dimensional system, we turn to the case of (3+1) dimensions. For simplicity we specify $\overrightarrow{E} = |E|\hat{z}$, and require solutions of the Dirac equation to satisfy periodic boundary conditions along the transverse direction (making the spectrum discrete and counting explicit). In (3+1) dimensions there are two spin states for the fermions. However, in the energy basis analogous to Eq. (\[trans\]), the system is diagonal in spin; thus its sole effect is an overall factor of $2$ in $w$ and $\Gamma$. In direct analogy to Eq. (\[EOM\]), the equation of motion for the mode specified by the spin state, $s$, and $\overrightarrow{k} = (k_x, k_y, k_z) = (n_x \frac {2\pi}{L_x}, n_y \frac {2\pi}{L_y}, n_z \frac {2\pi}{L_z})$ is $$\begin{aligned}
\begin{pmatrix} \epsilon_{\overrightarrow{k}}(t) & \frac{q E m}{\epsilon_{\overrightarrow{k}}^2(t)} \\ \frac{q E m}{\epsilon_{\overrightarrow{k}}^2(t)} & -\epsilon_{\overrightarrow{k}}(t) \end{pmatrix} \begin{pmatrix} \phi_{\overrightarrow{k}, s}^+(t) \\ \phi_{\overrightarrow{k}, s}^-(t) \end{pmatrix}& = i \partial_t \begin{pmatrix} \phi_{\overrightarrow{k}, s}^+(t) \\ \phi_{\overrightarrow{k}, s}^-(t) \end{pmatrix}\ \label{EOM3plus1}\end{aligned}$$ with $\epsilon_{\overrightarrow{k}}(t) = \sqrt{(k_z + qA_z(t))^2+(p_x^2+p_y^2+m^2)}$ the instantaneous energy of the mode $\overrightarrow{k}$ at time $t$. As in (1+1) dimensions, the initial conditions are $(\phi_{\overrightarrow{k}, s}^+(0), \phi_{\overrightarrow{k}, s}^-(0)) = ( 0 , 1 )$. Note that since $A_{\mu}$ (in this gauge) does not change the transverse momentum, it acts exactly like an additional mass-term in the Landau-Zener tunneling rate. Thus, for convenience, we will define $m_T^2 \equiv m^2 + \overrightarrow{k}_T^2$.
All of the previous arguments leading up to Eq. (\[lz\]) go through, after the substitution $m^2 \rightarrow m_T^2$. Thus, we derive $$\begin{aligned}
\frac {\langle N(T)\rangle}{VT} &=& 2 \sum_{k_T, k_z} \frac {\theta(-k_z) \theta (k_z + qE T )}{L_x L_y L_z T} e^{\left [ -\frac {\pi (m^2+ k_T^2)}{q E} \right ]} \label{number4}\nonumber\\
&=& 2 \frac{qE}{2 \pi} \frac{L_z T}{L_x L_y L_z T} \sum_{k_x,k_y} e^{ -\frac {\pi (m^2+k_x^2+k_y^2)}{qE}} \nonumber\\
&=& \frac {qE}{4 \pi^3} e^{ -\frac {\pi m^2}{qE}} \frac{L_x L_y}{L_x L_y} \int dk_x dk_y e^{ -\frac {\pi (k_x^2+k_y^2)}{qE} } \ .\end{aligned}$$ Evaluation of this double integral exactly reproduces the rate of pair production given in Eq. (\[SF3\]). A similar calculation of log$[P_{vac}(T)] \equiv -w VT$ yields, $$\begin{aligned}
&\log [P_{vac}(T)] \equiv -w VT ,\nonumber \\
&= 2 \, \sum_{k_z,k_T} \theta(-k_z) \theta (k_z + q E T ) \log [1 - e^{-\frac {\pi (m^2+k_T^2)}{q E} }] \nonumber \\
&= \frac {q E L_z T}{\pi} \frac{L_x}{2 \pi}\frac{L_y}{2 \pi} \int dk_x dk_y \log [1- e^{ -\frac {\pi (m^2+k_T^2)}{q E} } ] \nonumber\\
&= - VT \frac {(qE)^2}{4 \pi^3} \sum_{n =1}^{\infty} \frac {1}{n} \int dk_x dk_y e^{ -\frac {n \pi (m^2+k_T^2)}{q E} } \ .\end{aligned}$$ Evaluation of the double integrals, and identification of this quantity with $-w VT$, yields the expression for $w$ in Eq. (\[SF2\]). Exponentiating this rate, we derive the standard vacuum persistence probability in Eq. (\[SF1\]).
Discussion
==========
In summary, we have explicitly shown that the rate associated with vacuum decay, $w$, and the rate of pair creation, $\Gamma$, differ: $\Gamma$ is given by the first term in the series for $w$. This result is in a very real sense quite well known: it was first derived by Nikishov [@Niki] quite long ago. It can be derived in various other elegant formulations[@Rafelski; @NewRefs]. However, it is not as widely appreciated in the community as it should be: much of the literature in the field is still based on Schwinger’s initial assumption that $w$ gives the rate of pair production per unit volume. It is hoped that the physically transparent way that this paper shows that $\Gamma$ differs from $w$ will help clarify the issue. The massless case in (1+1) dimensions is particularly illuminating. It is clear from Fig. \[zeromass\] why the pair production rate remains finite even while the vacuum persistence probability goes to zero at finite times indicating an infinite value for $w$. While this calculation was with periodic boundary conditions in the $A_0=0$ gauge, we expect the result to hold generically for any consistent set of boundary conditions (*i.e.,* those that reproduce the Schwinger result for $w$). This will be investigated in future work.
In practice, the distinction between $w$ and $\Gamma$ is exponentially small for weak fields. However, for strong fields, $w$ exceeds $\Gamma$ by a factor as large as $\zeta(2) = \frac {\pi^2}{6} \sim 1.64$. More generally, in $d$ space-time dimensions $w^d$ will exceed $\Gamma^d$ by a factor of $\zeta(d/2)$ in the strong field limit.
T.D.C. was supported by the United States D.O.E. through grant number DE-FGO2-93ER-40762. We thank S. Nussinov and I. Shovkovy for useful discussions, and C. Goebel for a careful reading of the manuscript. J. Rafelski was very helpful in pointing out the relevant literature.
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abstract: 'Lower boundedness, global minimality, and uniqueness are established for the solutions of a physically-motivated class of inverse electromagnetic-radiation problems in (meta)material backgrounds. The radiating source is reconstructed by minimizing its $L^{2}$-norm subject to a prescribed radiated field and a vanishing reactive power. The minimization of the $L^{2}$-norm constitutes a useful criterion for the minimization of the physical resources of the source. The reactive power is the power cycling through the inductive and capacitive parts of the source and its vanishing corresponds to the maximization of the power transmitted in the far-field.'
author:
- 'M. R. Khodja[^1]'
title: 'On the Boundedness, Minimality, and Uniqueness of Constrained Optimal Solutions for an Inverse Radiation Problem in Metamaterial Media'
---
[***Keywords—*** inverse radiation problem, Maxwell’s equations, optimization, complex media, metamaterials]{}
Introduction {#Introduction}
============
We revisit the following problem [@MKB_SIAM_2008] $$\min\limits_{\mathbf{J}\in X}\left(\mathbf{J},\mathbf{J}\right),
\label{optim-problem-1}$$ $$X\equiv\left\{\mathbf{J}\in L^{2}\left(V;\mathbb{C}^{3}\right):\left(\mathcal{B}_{l,m}^{(j)},\mathbf{J}\right)=a_{l,m}^{(j)},\;\Im\left[\mathcal{P}\right]=0\right\},
\label{optim-problem-2}$$ where $ (\mathbf{J},\mathbf{J})\equiv\int \mathbf{J}^{\ast}(\mathbf{r})\cdot\mathbf{J}(\mathbf{r})\,d\mathbf{r}\equiv\mathcal{E}$. This corresponds to the problem of reconstructing a radiating electromagnetic source by minimizing its $L^{2}$-norm (so-called “source energy”) subject to a prescribed radiated field (the prescription constraint being $(\mathcal{B}_{l,m}^{(j)},\mathbf{J})=a_{l,m}^{(j)}$) and a vanishing reactive power (the tuning constraint being $\Im\left[\mathcal{P}\right]=0$; $\Im$ stands for the imaginary part). The electromagnetic source to be reconstructed is assumed to be embedded in a lossless (meta)material substrate with volume $V\equiv\{\mathbf{r}\in\mathbb{R}^3:r\equiv\|\mathbf{r}\|\leq a\}$. The electric permittivity and magnetic permeability distributions are of the form $f(\mathbf{r}) =f\theta(a-r)+f_{0}\theta(r-a)$, wherein $f=\epsilon,\mu$, stands for the permittivity and the permeability, respectively; subscript $0$ refers to the the vacuum; and $\theta$ denotes the Heaviside unit step. Since the source’s substrate is lossless, its propagation constant $k \equiv \omega \sqrt{\epsilon} \sqrt{\mu} \in \mathbb{R}$. One identifies the following cases: (1) $k>k_{0} \equiv \omega\sqrt{\epsilon_{0}\mu_{0}}$ for ordinary materials ($ k_0$ is the propagation constant in vacuum), (2) $0<k<k_{0}$ for double-positive (DPS) metamaterials (*i.e.*, media with $\epsilon > 0 \wedge \mu > 0$), (3) $k<0$ for double-negative (DNG) metamaterials (*i.e.*, media with $\epsilon < 0 \, \wedge \, \mu < 0$), and (4) $k=0$ for nihility metamaterials. The source radiates a prescribed field outside $V$. The field is considered to be time-harmonic and the generally frequency-dependent constitutive parameters are taken at a given central frequency. Propagation is goverened by the propagator equation $$\mbox{\boldmath${\nabla}$}\times\left( \frac{\mbox{\boldmath${\nabla}$}\times
\mathbf{\underline{G}}(\mathbf{r},\mathbf{r^{\prime}})}{\mu(\mathbf{r})}\right)
-\omega^{2}\epsilon(\mathbf{r})\mathbf{\underline{G}}(\mathbf{r},\mathbf{r^{\prime}})=i\omega\delta(\mathbf{r}-\mathbf{r^{\prime}})\underline{\mathbf{I}},$$ wherein $\mathbf{\underline{G}}(\mathbf{r},\mathbf{r^{\prime}})$ is the dyadic propagator, $\underline{\mathbf{I}}$ is the identity dyadic, and $\delta$ is the Dirac delta. The minimization of the “source energy” constitutes a useful criterion for the minimization of the actual physical resources of the radiating source and the vanishing of the reactive power $\Im[\mathcal{P}] \equiv \Im[-\frac{1}{2}(\mathbf{J}(\mathbf{r}),\int d\mathbf{r^{\prime}\underline{\mathbf{G}}}(\mathbf{r},\mathbf{r^{\prime}})\cdot\mathbf{J}(\mathbf{r^{\prime}}))]$ is typically desirable in radiating systems as it corresponds to the vanishing of the “useless” power. The source is assumed to be a finite spherical system with generalized (*i.e.*, possibly metamaterial) constitutive parameters.
Here, we establish the lower boundedness, global minimality, and uniqueness of the tuned minimal sources. Their existence and explicit forms have been presented in [@MKB_SIAM_2008].
Electromagnetic and elastic inverse problems in complex media continue to arouse a great deal of interest in the mathematics [@LiChenLi2017; @Valdivia2012; @Li2011; @DengLiuUhlmann2017; @KenigSaloUhlmann2011; @CaroZhou2014; @Mederski2015; @LassasZhou2016; @Uhlmann2014], physics [@ReinkeEtAl2011; @MulkeyDilliesDurach2017; @LiuGabrielliLipsonEtAl2013; @RezaDehbashiaBialkowskiAbbosh2017; @RaghunathanBudko2010], and engineering [@DenisovaRezvov2012; @OkhmatovskiAronssonShafai2012; @ElKahloutKiziltas2011] communities. The continued interest is due in part to the extraordinary propagation properties made possible by the advent of metamaterials and previously unknown to be possible. This has provided researchers with a much appreciated flexibility in the design of propagation systems with desired properties and fascinating potential applications in fields as diverse as medicine, nanotechnology, photovoltaics, seismic protection, and exploration geophysics. From a mathematical point of view, metamaterials have also introduced a richness to the old problems due to their generalized nature.
Boundedness {#Sec_Boundedness}
===========
The untuned minimum source energy $\mathcal{E}_{\mathcal{E}}$ is a lower bound on the tuned minimum source energy $\mathcal{E}_{\mathcal{E},\mathcal{P}}$ for source substrates with a propagation constant $k \in \mathbb{R}^{*}$, i.e., with generalized constitutive parameters $\epsilon_{r}$ and $\mu_{r}$ that satisfy: $\epsilon_{r} \, \mu_{r} > 0$. \[Theorem\_Boundedness\]
The necessary and sufficient condition for $\mathcal{E}_{\mathcal{E}}$ to be a unique lower bound on $\mathcal{E}_{\mathcal{E},\mathcal{P}}$ may be expressed as $$\mathcal{E}_{\mathcal{E}} \leq \mathcal{E}_{\mathcal{E},\mathcal{P}}.
\label{Lower_Bound_Condition-1}$$ Substituting the explicit expressions of $\mathcal{E}_{\mathcal{E}}$ and $\mathcal{E}_{\mathcal{E},\mathcal{P}} $ yields $$\sum_{j,l,m}\left(\left.R_{l,m}^{(j)}\right|_{\chi}-\left.R_{l,m}^{(j)}\right|_{0}\right)\left|a_{l,m}^{(j)}\right|^{2} \geq 0,
\label{Lower_Bound_Condition-2}$$ Our best hope is to have the sufficient (but not necessary) condition satisfied $$\left.R_{l,m}^{(j)}\right|_{\chi}-\left.R_{l,m}^{(j)}\right|_{0} \geq 0, \quad \forall j,l,m.
\label{Lower_Bound_Condition-3}$$ Substituting the explicit expression for $R_{l,m}^{(j)}$ yields the following conditions $$\begin{aligned}
&\left[\int_{0}^{a} \left( \left| j_{l}(kr) \right|^{2}+\frac{\left| k \, r \, u_{l}(kr)\right|^{2}}{l\left( l+1 \right)} \right)dr \right] \left[\int_{0}^{a} \left( \left| j_{l}(Kr) \right|^{2}+\frac{\left| K \, r \, u_{l}(Kr)\right|^{2}}{l\left( l+1 \right)} \right)dr\right]\nonumber
\\
& \qquad \qquad \qquad \qquad \geq \left| \int_{0}^{a} \left( j_{l}(kr) j_{l}(Kr) +\frac{k \, K \,r^2\,u_{l}(kr) u_{l}(Kr) }{l\left( l+1 \right)} \right)dr \right|^2;\quad j=1
\label{Lower_Bound_Condition-4}
\end{aligned}$$ and $$\left[\int_{0}^{a} \left| r j_{l}(k r) \right|^2 dr \right] \left[ \int_{0}^{a} \left| r j_{l}(K r) \right|^2 dr\right] \geq \left| \int_{0}^{a} j_{l}(k r) j_{l}(K r) r^2 dr\right|^2;\quad j=2,
\label{Lower_Bound_Condition-5}$$ The parameter $K \equiv \sqrt{k^2-\chi\mu\omega}$ is the propagation constant that appears in the wave equation governing the spatiotempral variation of the source’s optimized current distribution, that is, $\mbox{\boldmath${\nabla}$}\times\mbox{\boldmath${\nabla}$}\times \mathbf{J}_{\mathcal{E},\mathcal{P}}(\mathbf{r})-K^{2}\mathbf{J}_{\mathcal{E},\mathcal{P}}(\mathbf{r})=\mathbf{0}$. To have a time-harmonic current distribution transmitting a time-harmonic electromagnetic field through the source’s substrate, we need to have $K^2>0$. Consequently, we shall focus on double-positive (DPS) and double-negative (DNG) substrates.
Inequality (\[Lower\_Bound\_Condition-5\]) is satisfied by virtue of the Cauchy-Schwarz-Bunyakovsky theorem since for $a<\infty$, $$\begin{aligned}
\int_{0}^{a}\left|r j_{l}(\alpha r)\right|^2 dr &=\frac{a^3}{2}\left[j_{l}^{2}(\alpha a)-j_{l+1}(\alpha a)j_{l-1}(\alpha a)\right]; \quad \alpha \in \mathbb{R} \nonumber
\\
&< \infty.
\label{Lower_Bound_Condition-6}
\end{aligned}$$ The integral is Lommel’s first integral (see, for instance, [@GradshteynRyzhik]).
For $j=1$, standard relations involving spherical harmonics $Y_{l,m}$ and vector spherical harmonics $\mathbf{Y}_{l,m}$, in particular [@Hill] $$\begin{aligned}
\int_\Omega Y_{l,m}\,\overline{Y}_{l^{\prime},m^{\prime}}\,d\hat{\mathbf{r}}&=\delta_{ll^{\prime}}\delta_{mm^{\prime}},
\\
\int_\Omega\mathbf{Y}_{l,m}\cdot\overline{\mathbf{Y}}_{l^{\prime},m^{\prime}}\,d\hat{\mathbf{r}}&=l\left(l+1\right)\delta_{ll^{\prime}}\delta_{mm^{\prime}},
\label{SphericalHarmonicsIdentity2}
\end{aligned}$$ and [@marengo2000b] $$\mbox{\boldmath${\nabla}$} \times \left[ f_{l}(r)\mathbf{Y}_{l,m}\right]=\frac{i\,l(l+1)}{r} f_{l}(r){Y}_{l,m}\,\hat{\mathbf{r}}+\frac{1}{r}\frac{d}{dr}[r f_{l}(r)]\,\hat{\mathbf{r}}
\times\mathbf{Y}_{l,m},
\label{SphericalHarmonicsIdentity1}$$ where $f_{l}$ is a radial function, allow us to recast inequality (\[Lower\_Bound\_Condition-4\]) as $$\begin{aligned}
&\left[\int_{\Omega}d\hat{\mathbf{r}}\int_{0}^{a} \left|\mbox{\boldmath${\nabla}$} \times \left[ j_{l}(k r)\mathbf{Y}_{l,m}\right] \right|^2dr \right] \left[\int_{\Omega}d\hat{\mathbf{r}}\int_{0}^{a} \left|\mbox{\boldmath${\nabla}$} \times \left[ j_{l}(K r)\mathbf{Y}_{l,m}\right] \right|^2 dr \right]\nonumber
\\
& \qquad \qquad \qquad \geq \left| \int_{\Omega}d\hat{\mathbf{r}}\int_{0}^{a} \overline{\mbox{\boldmath${\nabla}$} \times \left[ j_{l}(k r)\mathbf{Y}_{l,m}\right]} \cdot \mbox{\boldmath${\nabla}$} \times \left[ j_{l}(K r)\mathbf{Y}_{l,m}\right]dr \right|^2;\quad j=1.
\label{Lower_Bound_Condition-7}
\end{aligned}$$ This clearly shows that we have another inequality of the Cauchy-Schwarz-Bunyakovsky type provided that $$\int_{\Omega}d\hat{\mathbf{r}} \int_{0}^{a} \left|\mbox{\boldmath${\nabla}$} \times \left[ j_{l}(\alpha r)\mathbf{Y}_{l,m}\right]\right|^{2}<\infty.
\label{Lower_Bound_Condition-8}$$ For this (square-integrability criterion) to be satisfied, it is sufficient to show that the spherical Bessel functions $j_{l}:r\rightarrow j_{l}(\alpha r)$ and their combinations $r \, u_{l}:r\rightarrow r \, u_{l}(\alpha r)$ are square-integrable. In this case, it is easier to forgo the analytical expressions of the finite-domain integrals and use the (sufficient) extended square-integrability conditions $$\begin{aligned}
\int_{0}^{\infty} \left| j_{l}(\alpha r) \right|^{2} dr =\frac{\pi}{2(2l+1)}\frac{1}{\alpha}<\infty;\quad \alpha \in \mathbb{R}^{*}
\label{Lower_Bound_Condition-10-1}
\end{aligned}$$ and $$\int_{0}^{\infty} \left| r \, u_{l}(\alpha r)\right|^{2} dr=0<\infty; \quad \alpha \in \mathbb{R}^{*}
\label{Lower_Bound_Condition-10-2}$$ (The integrals were computed with Mathematica.)
Note that the finite size of the source allows one to write the following extended versions of the integrals (*e.g.*, those which appear in inequalities (\[Lower\_Bound\_Condition-4\]) and (\[Lower\_Bound\_Condition-5\])) $$\int_{0}^{a} \rightarrow \int_{0}^{\infty} \theta(a-r)$$ where $\theta$ is the Heaviside unit step.
The values of the substrate’s propagation constant that would satisfy the square-integrability conditions and be consistent with the physical assumptions of the problem are $k \in \mathbb{R}^{*}$ (or, $\epsilon_{r} \, \mu_{r} > 0$) which corresponds to DPS (meta)materials and DNG metamaterials.
The equality in (\[Lower\_Bound\_Condition-4\]), (\[Lower\_Bound\_Condition-5\]), and (\[Lower\_Bound\_Condition-7\]) holds if and only if $j_{l}(k r)=j_{l}(K r)$, *i.e.*, when $0 \in \Xi$. In that case $\mathcal{E}_{\mathcal{E}}$ would be *the greatest lower bound* and *the unique global minimum* (see theorem (\[Theorem\_Global\_Minimality\_1\])). This completes the proof of the lower-boundedness theorem.
Global Minimality {#Sec_Global_Minimality}
=================
Let $\Xi$ be the set of all the Lagrange multipliers that satisfy the tuning constraint, that is $$\Xi\equiv\left\{\chi\in\mathbb{R}:\Im\left[\mathcal{P}\right]=0\right\},$$ and let $\chi_0 \in \Xi$ be the element that satisfies $$\left|\chi_{0}\right| =\inf\limits_{\chi\in\Xi}\left\{ \left|\chi\right| \right\}.
\label{Chi_Nought_Def}$$
When the unique lower bound $\mathcal{E}_{\mathcal{E}}$ belongs to the set of tuned minimum source energies $\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi}$, it is also the unique global minimum. \[Theorem\_Global\_Minimality\_1\]
This corollary follows directly from the lower-boundedness of $\mathcal{E}_{\mathcal{E}}$ (theorem \[Theorem\_Boundedness\]).
The tuned minimum source energy $\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi_{0}}$ where $\left|\chi_{0}\right| =\inf\limits_{\chi\in\Xi}\left\{ \left|\chi\right|\right\}$ and $\Xi\equiv\left\{\chi\in\mathbb{R}:\Im\left[\mathcal{P}\right]=0\right\}$ is the unique global minimum on the tuned minimum energy $\mathcal{E}_{\mathcal{E},\mathcal{P}}$ for source substrates with a propagation constant $k \in \mathbb{R}^{*}$, i.e., with generalized constitutive parameters $\epsilon_{r}$ and $\mu_{r}$ that satisfy: $\epsilon_{r} \, \mu_{r} > 0$. \[Theorem\_Global\_Minimality\_2\]
The necessary and sufficient condition for $\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi_{0}}$ to be a unique global minimum on $\mathcal{E}_{\mathcal{E},\mathcal{P}}$ may be expressed as $$\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi_{0}} < \left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi},
\label{Global_Min_Condition-1}$$ Substituting the explicit expressions of $\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi_{0}}$ and $\left.\mathcal{E}_{\mathcal{E},\mathcal{P}}\right|_{\chi}$ yields $$\sum_{j,l,m}\left(\left.R_{l,m}^{(j)}\right|_{\chi}-\left.R_{l,m}^{(j)}\right|_{\chi_{0}}\right)\left|a_{l,m}^{(j)}\right|^{2} > 0,
\label{Global_Min_Condition-2}$$ Again, our best hope is to have $$\left.R_{l,m}^{(j)}\right|_{\chi}-\left.R_{l,m}^{(j)}\right|_{\chi_{0}} > 0, \quad \forall j,l,m.
\label{Global_Min_Condition-3}$$ When $0 \notin \Xi$, condition (\[Global\_Min\_Condition-3\]) leads to the following conditions $$\begin{aligned}
&\frac{\int_{0}^{a} \left( \left| j_{l}(Kr) \right|^{2}+\frac{\left| K \, r \, u_{l}(Kr)\right|^{2}}{l\left( l+1 \right)} \right)dr}{\left| \int_{0}^{a} \left( j_{l}(kr) j_{l}(Kr) +\frac{k \, K \,r^2\,u_{l}(kr) u_{l}(Kr) }{l\left( l+1 \right)} \right)dr \right|^2} \nonumber
\\
&\qquad \qquad \geq \frac{\int_{0}^{a} \left( \left| j_{l}(K_{0}r) \right|^{2}+\frac{\left| K_{0} \, r \, u_{l}(K_{0}r)\right|^{2}}{l\left( l+1 \right)} \right)dr}{\left| \int_{0}^{a} \left( j_{l}(kr) j_{l}(K_{0}r) +\frac{k \, K_{0} \,r^2\,u_{l}(kr) u_{l}(K_{0}r) }{l\left( l+1 \right)} \right)dr \right|^2} ;\quad j=1
\label{Global_Min_Condition-4}
\end{aligned}$$ and $$\frac{\int_{0}^{a} \left|r j_{l}(Kr) \right|^{2} dr}{\left| \int_{0}^{a} j_{l}(kr) j_{l}(Kr) r^2 dr \right|^2} \geq \frac{\int_{0}^{a} \left|r j_{l}(K_{0}r) \right|^{2} dr}{\left| \int_{0}^{a} j_{l}(kr) j_{l}(K_{0}r) r^2 dr \right|^2};\quad j=2.
\label{Global_Min_Condition-5}$$ Next, the idea is to expand the functions around $\chi=0$. Let us start with the case $j=2$. $$\frac{\int_{0}^{a} \left|r j_{l}(Kr) \right|^{2} dr}{\left| \int_{0}^{a} j_{l}(kr) j_{l}(Kr) r^2 dr \right|^2} = f_{0}^{(2)}+f_{2}^{(2)} \, \chi^2+O[\chi^3]
\label{Global_Min_Condition-7}$$ where $$\begin{aligned}
f_{0}^{(2)} \equiv \frac{2}{a^3 \left[j_l^2(k a)-j_{l-1}(k a) j_{l+1}(k a)\right]}
\label{Global_Min_Condition-8}
\end{aligned}$$ and $$\begin{aligned}
f_{2}^{(2)} &\equiv \bigg\{\mu ^2 \omega ^2 \Big[a^2 k^2 j_{l-1}^4(k a) \left(a^2 k^2-l^2+1\right)\Big.\bigg. \nonumber
\\
&+4 a k (l-1) j_l(k a) j_{l-1}^3(k a) \left(-a^2 k^2+l^2+l\right) \nonumber
\\
&+4 a k j_l^3(k a) j_{l-1}(k a) \left(l (l (l+2)-2)-a^2 k^2 (l-1)\right) \nonumber
\\
&+a^2 k^2 j_l^4(k a) \left(a^2 k^2-l (l+4)\right) \nonumber
\\
&+\bigg.\Big.2 j_l^2(k a) j_{l-1}^2(k a) \left(a^4 k^4+a^2 k^2 ((l-6) l+2)-2 l^4+2 l^2\right)\Big]\bigg\} \nonumber
\\
&\times \bigg\{6 a^5 k^6 \left[j_l^2(k a)-j_{l-1}(k a) j_{l+1}(k a)\right]^3\bigg\}^{-1}
\label{Global_Min_Condition-9}
\end{aligned}$$ In other words, to lowest order in $\chi$ condition (\[Global\_Min\_Condition-5\]) is equivalent to $$\chi^{2} > \chi_{0}^{2},
\label{Global_Min_Condition-10}$$ which is true by assumption.
For $j=1$, the explicit expressions contain rather involved combinations of Bessel’s functions and hypergeometric functions. To somewhat simplify the calculations, we adopt a slightly different approach. We write that $$\begin{aligned}
&\frac{\int_{0}^{a} \left( \left| j_{l}(Kr) \right|^{2}+\frac{\left| K \, r \, u_{l}(Kr)\right|^{2}}{l\left( l+1 \right)} \right)dr}{\left| \int_{0}^{a} \left( j_{l}(kr) j_{l}(Kr) +\frac{k \, K \,r^2\,u_{l}(kr) u_{l}(Kr) }{l\left( l+1 \right)} \right)dr \right|^2} \nonumber
\\
& \qquad \qquad \qquad \equiv \frac{\int_0^a\left(c_{0}+c_{1} \, \chi+c_{2} \, \chi^2+O[\chi^3]\right)dr}{\left\{\int_0^a \left(d_{0}+d_{1} \, \chi+d_{2} \, \chi^2+O[\chi^3]\right)dr\right\}^2} \nonumber
\\
& \qquad \qquad \qquad \equiv f_{0}^{(1)}+f_{1}^{(1)} \, \chi+f_{2}^{(1)} \, \chi^2+O[\chi^3],
\label{Global_Min_Condition-11}
\end{aligned}$$ where $c_0, \, c_1, \, c_2, \, d_0, \, d_1,$ and $d_2$ are functions of $r$ while $f_{0}^{(1)}, \, f_{1}^{(1)},$ and $f_{2}^{(1)}$ are independent of $r$. $$\begin{aligned}
f_{0}^{(1)} & \equiv \frac{c_{0}}{d_{0}^2}
\\
f_{1}^{(1)} & \equiv \frac{c_{1} d_{0}-2 c_{0} d_{1}}{d_{0}^3}
\\
f_{2}^{(1)} & \equiv \frac{3c_{0}d_{1}^{2}+c_{2}d_{0}^{2}-2c_{0}d_{0}d_{2}-2c_{1}d_{0}d_{1}}{d_{0}^4}.
\label{Global_Min_Condition-12}
\end{aligned}$$ The idea here is to show that $$f_{1}^{(1)} = 0, \, \forall l \in \mathbb{N}^{*}.
\label{Global_Min_Condition-13}$$ With the aid of Mathematica, it was possible to show that eq.(\[Global\_Min\_Condition-13\]) does indeed hold. (The explicit expressions of $c_0,\,c_1,\,d_0$ and $d_1$ are provided in \[Appendix\_vanishing\_condition\_J1\].) Hence, to lowest order in $\chi$ condition (\[Global\_Min\_Condition-4\]) too is equivalent to eq.(\[Global\_Min\_Condition-10\]). This completes the proof of theorem \[Theorem\_Global\_Minimality\_2\].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by King Fahd University of Petroleum and Minerals.
Appendix A {#appendix-a .unnumbered}
==========
Explicit Expressions of $c_0,\,c_1,\,d_0$ and $d_1$ which Appear in Eq.(\[Global\_Min\_Condition-12\]) {#Appendix_vanishing_condition_J1 .unnumbered}
======================================================================================================
$$\begin{aligned}
c_0 &= \int_0^a \frac{1}{l (l+1) (2 l+1)^2}\bigg[k^2 (l+1)^2 r^2 j_{l-1}(r \left| k\right| )^2-2 k^2 l (l+1) r^2 j_{l-1}(r \left| k\right| ) j_{l+1}(r \left| k\right| ) \bigg. \nonumber
\\
&+l \bigg.\left(k^2 l r^2 j_{l+1}(r \left| k\right| ){}^2+(l+1) (2 l+1)^2 j_l(r \left| k\right| )^2\right)\bigg] \, dr
\\
c_1 &= \int_0^a \frac{-1}{l (l+1) \left| k\right| ^2}\bigg[\mu \omega j_l(r \left| k\right| ) \bigg((l+1) \left(-k^2 r^2+2 l^2+l\right) j_l(r \left| k\right| ) \bigg.\bigg. \nonumber
\\
&+\bigg.\bigg.r \left| k\right| \left(k^2 r^2-2 l^2-2 l\right) j_{l+1}(r \left| k\right| )\bigg) \bigg] \, dr
\\
d_0 &= \int_0^a \frac{r^2 k^{2-l} \left| k\right| ^l \left[(l+1) j_{l-1}(k r)-l j_{l+1}(k r)\right]^2}{l (l+1) (2 l+1)^2}+j_l(k r) j_l(r \left| k\right| ) \, dr
\\
d_1 &= \int_0^a \frac{-1}{2 l (l+1)} \bigg[\mu \omega k^{-l-2} \left| k\right| ^l j_l(k r) \bigg((l+1) \left(-k^2 r^2+2 l^2+l\right) j_l(k r) \bigg. \bigg.\nonumber
\\
&+\bigg. \bigg.k r \left(k^2 r^2-2 l^2-2 l\right) j_{l+1}(k r)\bigg)\bigg] \, dr
\end{aligned}$$
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[^1]: King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia. (Email: mrkhodja@kfupm.edu.sa)
|
---
abstract: 'Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. In this paper, we prove that the number of $z$-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least $6$, and finite otherwise.'
address: 'IISER Mohali, Knowledge city, Sector 81, SAS Nagar, P.O. Manauli, Punjab 140306, INDIA'
author:
- Sushil Bhunia
title: Conjugacy classes of centralizers in the group of upper triangular matrices
---
[^1]
Introduction
============
Let $G$ be a group. Two elements $x$ and $y$ in $G$ are said to be $z$-equivalent, denoted by $x\sim_{z} y$, if their centralizers in $G$ are conjugate, i.e., $\mathcal Z_G(y)=g\mathcal Z_G(x)g^{-1}$ for some $g\in G$, where $\mathcal Z_G(x):=\{y\in G \mid xy=yx \}$ denotes centralizer of $x$ in $G$. Clearly, $\sim_{z}$ is an equivalence relation on $G$. The equivalence classes with respect to this relation are called **$z$-classes**. It is easy to see that if two elements of a group $G$ are conjugate then their centralizers are conjugate thus they are also $z$-equivalent. However, in general, the converse is not true. In geometry, $z$-classes describe the behaviour of dynamical types (see for example [@Ku1], [@Ci], and [@Go]). That is, if a group $G$ is acting on a manifold $M$ then understanding (dynamical types of) orbits is related to understanding (conjugacy classes of) centralizers.
Robert Steinberg [@St] (Section 3.6 Corollary 1 to Theorem 2) proved that for a reductive algebraic group $G$ defined over an algebraically closed field, of good characteristic, the number of $z$-classes is finite. A natural question that followed: [*Is the number of $z$-classes finite for algebraic group $G$ defined over an arbitrary field $k$?*]{} In [@Si], A. Singh studied $z$-classes for real compact groups of type $\mathrm{G}_2$. Ravi S. Kulkarni, in [@Ku], proved that the number of $z$-classes in $\mathrm{GL}_n(k)$ is finite if the field $k$ has only finitely many field extensions of any fixed finite degree. Unless otherwise specified, we will always assume that $k$ is a field of $\mathrm{char}\; \neq 2$. Let $V$ be an $n$-dimensional vector space over a field $k$ equipped with a non-degenerate symmetric or skew-symmetric bilinear form $B$. Then, in [@GK], it is proved that there are only finitely many $z$-classes in orthogonal groups $\mathrm{O}_n(k)$ and symplectic groups $\mathrm{Sp}_n(k)$ if $k$ has only finitely many field extensions of any fixed finite degree. Let $k$ be a perfect field with a non-trivial Galois automorphism of order $2$. Let $V$ be an $n$-dimensional vector space over $k$ equipped with a non-degenerate Hermitian form $H$. Suppose that the fixed field $k_0$ has only finitely many field extensions of any fixed finite degree. Then, in [@BS], we proved that the number of $z$-classes in the unitary group $\mathrm{U}_n(k_0)$ is finite. All the groups mentioned so far are special types of reductive algebraic groups. A natural problem to study would be to consider $z$-classes in non-reductive ones. In particular, one may consider the following problem:
\[prob2\] Is the number of $z$-classes finite for a solvable algebraic group?
Let $G$ be a connected solvable linear algebraic group over an algebraically closed field $k$, then by Lie-Kolchin theorem (see Theorem 17.6 [@Hu1]) $G$ is a subgroup of the group of upper triangular matrices in $\mathrm{GL}_n(k)$ for some $n$.
Let $\mathrm{B}_n(k)$ denote the group of upper triangular matrices in $\mathrm{GL}_n(k)$. In this paper, we solve the Problem \[prob2\] for this special classes of groups. In a sequel, we will do this for general nilpotent and solvable groups. The main result of this paper is the following theorem, which solves Problem \[prob2\] for the group of upper triangular matrices:
\[maintheorem\]
1. For $2\leq n \leq 5$, the number of $z$-classes in $\mathrm{B}_n(k)$ is finite.
2. For $n\geq 6$, the number of $z$-classes in $\mathrm{B}_n(k)$ is infinite.
In Section 2, we explore the semisimple $z$-classes for the group of upper triangular matrices. In Section 3, we study unipotent conjugacy classes and unipotent $z$-classes in $\mathrm{B}_n(k)$. In Section 4, we prove our main theorem of this paper. Throughout the paper, we assume that $k$ is an infinite field of $\mathrm{char}\; k\neq 2$. Here we include an appendix, which contains an explicit computation of unipotent conjugacy classes and their representatives in the group of upper triangular matrices $\B_n(k)$ for $n=2, 3, 4, 5$ using Belitskii’s algorithm.
Semisimple -classes in
=======================
Let $n$ be a positive integer with a partition $\lambda=(1^{k_{1}} 2^{k_{2}}\ldots n^{k_{n}})$, denoted by $\lambda \vdash n$, i.e., $n=\sum_{i}ik_{i}$.
\[semisimplez\] The number of semisimple $z$-classes in $\B_n(k)$ is $$\displaystyle \sum_{(1^{k_1} 2^{k_2}\ldots n^{k_n})\vdash n}\frac{n!}{\prod_{j=1}^{n}(j!)^{k_j}(k_j!)}.$$ So, in particular, the number of semisimple $z$-classes in $\B_n(k)$ is finite.
Semisimple elements in $\B_n(k)$ are nothing but diagonals up to conjugacy (for details see the first-page second paragraph in [@Ro]). So the number of semisimple $z$-classes in $\B_n(k)$ is equal to the number of $z$-classes of diagonals in $\B_n(k)$. Now we give a combinatorial argument to count the $z$-classes of diagonals in $\B_n(k)$, which is as follows: Let $\lambda=(1^{k_{1}} 2^{k_{2}}\ldots n^{k_{n}})$ be a partition of $n$. Let us consider the following multiset $$M:=\{\underbrace{a_{11}, a_{12},\ldots, a_{1k_1}}_{k_1}; \underbrace{a_{21}, a_{21}, \ldots, a_{2k_2}, a_{2k_2}}_{2k_2}; \ldots, \underbrace{a_{i1},\ldots, a_{i1}, \ldots, a_{ik_i}, \ldots, a_{ik_i}}_{ik_i}; \ldots\}.$$ So the number of ways to order the above multiset is $$\frac{n!}{(1!)^{k_1}(2!)^{k_2}\cdots (n!)^{k_n}}.$$ Now look at the action of $S_{k_1}\times S_{k_2}\times \cdots \times S_{k_n}$ on the ordered tuples via $$(\sigma_1\times \sigma_2\times \cdots \times \sigma_n) (\cdots a_{ij} \cdots)=(\cdots a_{i\sigma_i(j)}\cdots),$$ where $\sigma_i \in S_{k_i}$ and $S_{k_i}$’s are the symmetric groups on $k_i$ symbols. Therefore the size of each orbit under the above action is $\prod_{j=1}^{n}(k_j!)$ as the stabilizer is the identity group. So the number of orbits is $$\frac{n!}{\prod_{j=1}^{n}(j!)^{k_j}\prod_{j=1}^{n}(k_j!)}.$$ Hence the result.
A numerical example should make the above argument transparent.
Let $n=5$, then the number of partitions of $n$ is equal to $7$ and are given by $(5^1), (1^14^1), (2^13^1), (1^23^1), (1^12^2), (1^32^1), (1^5)$. Now
1. For $\lambda=(5^1)\vdash 5$, the number of semisimple $z$- classes is $\frac{5!}{5!}=1$ and representative is the following: $$\mathrm{diag}\;(\alpha, \alpha, \alpha, \alpha, \alpha).$$
2. For $\lambda=(1^14^1)\vdash 5$, the number of semisimple $z$- classes is $\frac{5!}{4!}=5$ and representatives are given by the following: $$\begin{aligned}
\mathrm{diag}\;(\alpha, \alpha, \alpha, \alpha, \beta);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \alpha, \alpha, \beta, \alpha);\\
\mathrm{diag}\;(\alpha, \alpha, \beta, \alpha, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \beta, \alpha, \alpha, \alpha);
\end{aligned}$$ $$\mathrm{diag}\;(\beta, \alpha, \alpha, \alpha, \alpha).$$
3. For $\lambda=(2^13^1)\vdash 5$, the number of semisimple $z$- classes is $\frac{5!}{3!2!}=10$ and representatives are given by the following: $$\begin{aligned}
\mathrm{diag}\;(\alpha, \alpha, \alpha, \beta, \beta);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \alpha, \beta, \alpha, \beta);\\
\mathrm{diag}\;(\alpha, \beta, \alpha, \alpha, \beta);&\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \alpha, \alpha, \beta);\\
\mathrm{diag}\;(\beta, \alpha, \alpha, \beta, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \beta, \alpha, \alpha);\\
\mathrm{diag}\;(\beta, \beta, \alpha, \alpha, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \alpha, \beta, \beta, \alpha);\\
\mathrm{diag}\;(\alpha, \beta, \beta, \alpha, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \beta, \alpha, \beta, \alpha).
\end{aligned}$$
4. For $\lambda=(1^23^1)\vdash 5$, the number of semisimple $z$-classes is $\frac{5!}{(1!)^22!3!}=10$ and representatives are given by the following: $$\begin{aligned}
\mathrm{diag}\;(\alpha, \alpha, \alpha, \beta, \gamma); &\hspace{15mm}
\mathrm{diag}\;(\alpha, \alpha, \beta, \alpha, \gamma); \\
\mathrm{diag}\;(\alpha, \beta, \alpha, \alpha, \gamma); &\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \alpha, \alpha, \gamma); \\
\mathrm{diag}\;(\beta, \alpha, \alpha, \gamma, \alpha); &\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \gamma, \alpha, \alpha); \\
\mathrm{diag}\;(\alpha, \beta, \gamma, \alpha, \alpha); &\hspace{15mm}
\mathrm{diag}\;(\alpha, \alpha, \beta, \gamma, \alpha); \\
\mathrm{diag}\;(\alpha, \beta, \alpha, \gamma, \alpha); &\hspace{15mm}
\mathrm{diag}\;(\beta, \gamma, \alpha, \alpha, \alpha).
\end{aligned}$$
5. For $\lambda=(1^12^2)\vdash 5$, the number of semisimple $z$-classes is $\frac{5!}{(2!)^2 2!}=15$ and representatives are given as follows: $$\begin{aligned}
\mathrm{diag}\;(\alpha, \alpha, \beta, \beta, \gamma); &\hspace{10mm}
\mathrm{diag}\;(\alpha, \beta, \alpha, \beta, \gamma); &
\mathrm{diag}\;(\beta, \alpha, \alpha, \beta, \gamma);\\
\mathrm{diag}\;(\beta, \alpha, \beta, \gamma, \alpha); &\hspace{10mm}
\mathrm{diag}\;(\alpha, \beta, \beta, \gamma, \alpha); &
\mathrm{diag}\;(\gamma, \alpha, \alpha, \beta, \beta); \\
\mathrm{diag}\;(\alpha, \alpha, \beta, \gamma, \beta); &\hspace{10mm}
\mathrm{diag}\;(\alpha, \alpha, \gamma, \beta, \beta); &
\mathrm{diag}\;(\alpha, \beta, \gamma, \alpha, \beta); \\
\mathrm{diag}\;(\alpha, \beta, \gamma, \beta, \alpha); &\hspace{10mm}
\mathrm{diag}\;(\alpha, \gamma, \alpha, \beta, \beta); &
\mathrm{diag}\;(\alpha, \gamma, \beta, \alpha, \beta); \\
\mathrm{diag}\;(\alpha, \gamma, \beta, \beta, \alpha); &\hspace{10mm}
\mathrm{diag}\;(\gamma, \alpha, \beta, \alpha, \beta); &\;
\mathrm{diag}\;(\gamma, \beta, \alpha, \alpha, \beta).
\end{aligned}$$
6. For $\lambda=(1^32^1)\vdash 5$, the number of semisimple $z$-classes is $\frac{5!}{(1!)^33!2!}=10$ and representatives are given by the following: $$\begin{aligned}
\mathrm{diag}\;(\alpha, \alpha, \beta, \gamma, \delta); &\hspace{15mm}
\mathrm{diag}\;(\beta, \gamma, \delta, \alpha, \alpha);\\
\mathrm{diag}\;(\beta, \gamma, \alpha, \delta, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\beta, \gamma, \alpha, \alpha, \delta);\\
\mathrm{diag}\;(\beta, \alpha, \gamma, \alpha, \delta);&\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \alpha, \gamma, \delta);\\
\mathrm{diag}\;(\alpha, \beta, \alpha, \gamma, \delta);&\hspace{15mm}
\mathrm{diag}\;(\alpha, \beta, \gamma, \alpha, \delta);\\
\mathrm{diag}\;(\alpha, \beta, \gamma, \delta, \alpha);&\hspace{15mm}
\mathrm{diag}\;(\beta, \alpha, \gamma, \delta, \alpha).
\end{aligned}$$
7. For $\lambda=(1^5)\vdash 5$, the number of semisimple $z$-classes is $\frac{5!}{(1!)^55!}=1$ and the representative is given as follows: $$\mathrm{diag}\;(\alpha, \beta, \gamma, \delta, \eta).$$
Therefore the total number of semisimple $z$-classes in $\B_5(k)$ is $52$.
Unipotent -classes in
======================
Let $\alpha\in k$, define $$x_{\alpha}=\begin{pmatrix}
0&1&\alpha&0&0&0\\&0&0&0&1&0\\&&0&1&1&0\\
&&&0&0&0\\&&&&0&1\\&&&&&0\end{pmatrix}.$$ Let $u_{\alpha}=I_6+x_{\alpha}\in \B_6(k)$ be a unipotent element. The following result is already known. We record this result as we are going to use it.
\[unicong\]
1. For $2\leq n \leq 5$, the number of unipotent conjugacy classes in $\B_n(k)$ is finite. In particular, the numbers are $2, 5, 16, 60$ for $n=2, 3, 4, 5$ respectively.
2. For $n\geq 6$, the number of unipotent conjugacy classes in $\B_n(k)$ is infinite.
<!-- -->
1. Use Belitskii’s algorithm, for details see [@CXLF], [@Ko] and [@Th]. Also, see the appendix for explicit calculations of the number of unipotent conjugacy classes. We also give representatives of the unipotent conjugacy classes in $\B_n(k)$ for $2\leq n\leq 5$.
2. The proof was originally given by M. Roitman in [@Ro] for $n=12$. Then latter Djokovic and Malzan, in [@DM], proved this for $n=6$ and in fact, it is the minimum value for which this happens to be true by part (1). For completeness, we will give this prove again. It is enough to prove this for $n=6$. Let $u_{\alpha}$ and $u_{\beta}$ are conjugate in $\B_6(k)$, i.e., $Pu_{\alpha}P^{-1}=u_{\beta}$ for some $P\in \B_6(k)$. Then $Px_{\alpha}=x_{\beta}P$. Let $P=(p_{ij})$, then we get the following: $$\begin{aligned}
p_{23}&= p_{45}=0\\
p_{11}&= p_{22}\\
p_{22}&= p_{55}\\
p_{55}&= p_{33}\\
\alpha p_{11}&= p_{23}+\beta p_{33}.
\end{aligned}$$ Therefore from the above equation, we get $\alpha=\beta$. So the number of unipotent conjugacy classes in $\B_6(k)$ is infinite as $k$ is an infinite field. Hence it is true for $\B_n(k)$ provided $n\geq 6$.
\[unipotentz1\] For $2\leq n \leq 5$, the number of unipotent $z$-classes in $\B_n(k)$ is finite.
If two elements are conjugate, then they are also $z$-conjugate. So this follows from the first part of Proposition \[unicong\].
Now the centralizer of $u_{\alpha}$, $\mathcal{Z}_{\B_6(k)}(u_{\alpha})$ is the following: $$\left\{ \begin{pmatrix}
a&b_1&b_2&b_3&b_4&b_5\\&a&0&b_2-\alpha b_6&b_1+b_2-\alpha b_7&b_4-\alpha b_8\\
&&a&b_6&b_7&b_8\\&&&a&0&(\alpha+1)b_7-b_1-b_2\\
&&&&a&b_1+b_2-\alpha b_7\\
&&&&&a\end{pmatrix}\mid a\in k^{\times}, b_i\in k \right\}.$$
\[unipotentz2\] For $n\geq 6$, the number of unipotent $z$-classes in $\B_n(k)$ is infinite.
It is enough to prove this lemma for $n=6$. Now assume that $n=6$. Suppose that $u_{\alpha}$ and $u_{\beta}$ are $z$-conjugate, then $P\mathcal{Z}_{\B_6(k)}(u_{\alpha})P^{-1}=
\mathcal{Z}_{\B_6(k)}(u_{\beta})$ for some $P\in \B_{6}(k)$.
$\alpha=\beta$.
Now $PAP^{-1}\in \mathcal{Z}_{\B_6(k)}(u_{\beta})$ for all $A\in \mathcal{Z}_{\B_6(k)}(u_{\alpha})$. So $PAP^{-1}=A'$ for some $A' \in \mathcal{Z}_{\B_6(k)}(u_{\beta})$. Observe that two upper triangular matrices are conjugate via a upper triangular matrix implies that they have the same diagonal entries. Let $P=(p_{ij})\in \B_6(k)$ and $A, A'$ have the form described as above. Then we get $$\begin{aligned}
a'&= a\\
b_1'&= b_1p_{11}p_{22}^{-1}\\
b_2'&= (b_2p_{11}-b_1p_{11}p_{23}p_{22}^{-1})p_{33}^{-1}\\
b_6'&= b_6p_{33}p_{44}^{-1}\\
b_7'&= (b_7p_{33}-b_6p_{33}p_{45}p_{44}^{-1})p_{55}^{-1}.
\end{aligned}$$ $$\begin{aligned}
\label{eq6}
(b_2-\alpha b_6)p_{22}+b_6p_{23}&= (b_2'-\beta b_6')p_{44}.
\end{aligned}$$ $$\begin{aligned}
\label{eq7}
((\alpha+1)b_7-b_1-b_2)p_{44}+(b_1+b_2-\alpha b_7)p_{45}&=
((\beta+1)b_7'-b_1'-b_2')p_{66}.
\end{aligned}$$ From Equation (\[eq6\]), we get, $p_{23}=0$ and $\alpha p_{22}=\beta p_{33}$.
(If $b_6=b_2=0$ and $b_1\neq 0,$ then $b_6'=0$ and $b_2'=-b_1p_{11}p_{23}p_{22}^{-1}p_{33}^{-1}$. So $$-b_1p_{11}p_{23}p_{22}^{-1}p_{33}^{-1}p_{44}=0,$$ hence $p_{23}=0$, since $b_1\neq 0$ and $p_{ii}\neq 0$. And if $b_1=b_2=0$ and $b_6\neq 0$, then $b_1'=0=b_2'$ and $ b_6'= b_6p_{33}p_{44}^{-1}$. So $$-\alpha b_6p_{22}+b_6p_{23}=-\beta b_6p_{44}p_{33}p_{44}^{-1}.$$ Now, since $p_{23}=0$, we get $\alpha b_6p_{22}=\beta b_6a_{33}$, which implies $\alpha p_{22}=\beta p_{33}$, as $b_6\neq 0$).
From Equation (\[eq7\]), we get, $p_{22}=p_{33}$.
(If $b_1=b_6=b_7=0$ and $ b_2\neq 0,$ then $b_1'=b_6'=b_7'=0$ and $ b_2'=b_2p_{11}p_{33}^{-1}$. So from Equation (\[eq6\]) we get $$p_{44}-p_{45}=p_{11}p_{66}p_{33}^{-1},$$ since $b_2\neq 0$. Again if $b_2=b_6=b_7=0$ and $ b_1\neq 0$, then $b_6'=b_7'=0$ and $b_1'= b_1p_{11}p_{22}^{-1}; b_2'=-b_1p_{11}p_{23}p_{22}^{-1}p_{33}^{-1}=0$, as $p_{23}=0$. So again from Equation (\[eq6\]) we get, $$p_{44}-p_{45}=p_{11}p_{66}p_{22}^{-1},$$ since $b_1\neq 0$. Therefore from the above two we get $p_{22}=p_{33}$). Hence $\alpha=\beta$. Therefore the result is true for $n=6$. Now for $n >6$. Let $$U_{\alpha}:=\left(\begin{array}{c|c}
u_{\alpha}&0\\
\hline
0&I_{n-6}
\end{array}\right),\, U_{\beta}:=\left(\begin{array}{c|c}
u_{\beta}&0\\
\hline
0&I_{n-6}
\end{array}\right)\in \B_n(k)$$ be two unipotent elements which are $z$-conjugate in $\B_n(k)$. Then $$Q\mathcal{Z}_{\B_n(k)}(U_{\alpha})Q^{-1}=\mathcal{Z}_{\B_n(k)}(U_{\beta})$$ for some $Q\in \B_n(k)$. Therefore $QCQ^{-1}=C'$ for some $C\in \mathcal{Z}_{\B_n(k)}(U_{\alpha})$ and $C' \in \mathcal{Z}_{\B_n(k)}(U_{\beta})$. Now write $Q, C$ and $C'$ in block form, we get $$\left(\begin{array}{c|c}
P&*\\
\hline
0&*
\end{array}\right)
\left(\begin{array}{c|c}
A&*\\
\hline
0&*
\end{array}\right)=\left(\begin{array}{c|c}
A'&*\\
\hline
0&*
\end{array}\right)
\left(\begin{array}{c|c}
P&*\\
\hline
0&*
\end{array}\right)$$ for some $P, A, A'\in \B_6(k)$. Hence $PA=A'P$, which reduces to the case of $n=6$. Therefore the number of unipotent $z$-classes in $\B_n(k)$ ($n\geq 6$) is infinite.
The unipotent $z$-classes for $\B_6(k)$ is parametrized by elements of the field $k$.
Let $u$ be a unipotent element of $\B_6(k)$. Then using Belitskii’s algorithm (see appendix and [@Ko] for details) we get $bub^{-1}=u_{\alpha}$ for some $b\in \B_6(k)$ and for some $\alpha \in k$, where $u_{\alpha}$ is defined at the beginning of this section. Again from Lemma \[unipotentz2\] we have $u_{\alpha}\sim_{z} u_{\beta}$ if and only if $\alpha=\beta$, where $\alpha, \beta \in k$. Therefore unipotent $z$-classes in $\B_6(k)$ are completely determined by the elements of $k$ via the map $\alpha \mapsto u_{\alpha}$.
-classes in
============
\[techlemma\] Let $g\in G$ and $g=g_sg_u$ be the Jordan decomposition of $g$, and $\alpha \in G$. Then we have $$\alpha \mathcal{Z}_{\mathcal{Z}_G(g_s)}(g_u)\alpha^{-1}
=\mathcal{Z}_{\alpha \mathcal{Z}_G(g_s)\alpha^{-1}}(\alpha g_u\alpha^{-1}).$$
Let $x\in \mathcal{Z}_{\mathcal{Z}_G(g_s)}(g_u)$ then $xg_s=g_sx$ and $xg_u=g_ux$. Therefore $\alpha xg_u\alpha^{-1}=\alpha g_u x\alpha^{-1}$ implies that $(\alpha x\alpha^{-1})(\alpha g_u \alpha^{-1})=(\alpha g_u \alpha^{-1})(\alpha x\alpha^{-1})$. Therefore $\alpha x\alpha^{-1}\in \mathcal{Z}_{\alpha \mathcal{Z}_G(g_s)\alpha^{-1}}(\alpha g_u\alpha^{-1})$.
On the other hand let $y\in \mathcal{Z}_{\alpha \mathcal{Z}_G(g_s)\alpha^{-1}}(\alpha g_u\alpha^{-1})$, then $(\alpha^{-1}y \alpha)g_s=g_s(\alpha^{-1}y\alpha)$ and $y(\alpha g_u\alpha^{-1})=(\alpha g_u\alpha^{-1})y$. Now the last equation is same as $(\alpha^{-1}y\alpha)g_u=g_u(\alpha^{-1}y\alpha)$. So $\mathcal{Z}_{\alpha \mathcal{Z}_G(g_s)\alpha^{-1}}(\alpha g_u\alpha^{-1})\subseteq \alpha \mathcal{Z}_{\mathcal{Z}_G(g_s)}(g_u)\alpha^{-1}$. Hence the result.
\[remark\] Let us assume that the number of semisimple $z$-classes in $G$ is $n$ and representatives are given by $s_1, s_2, \ldots, s_n$. Let $g\in G$ then $g=g_sg_u$ is the Jordan decomposition of $g$. By the above assumption, $g_s$ will be $z$-conjugate to $s_i$ for some $i=1, 2, \ldots, n$. Without loss of generality, say $g_s\sim_z s_1$, i.e., $\alpha \mathcal{Z}_G(g_s)\alpha^{-1}=\mathcal{Z}_G(s_1)$ for some $\alpha \in G$. Then $$\alpha \mathcal{Z}_G(g)\alpha^{-1}=\alpha \mathcal{Z}_{\mathcal{Z}_G(g_s)}(g_u)\alpha^{-1}
=\mathcal{Z}_{\mathcal{Z}_G(s_1)}(\alpha g_u\alpha^{-1}).$$ The first equality follows from the uniqueness of the Jordan decomposition, i.e., $$\mathcal{Z}_G(g)=\mathcal{Z}_G(g_s)\cap \mathcal{Z}_G(g_u)=\mathcal{Z}_{\mathcal{Z}_G(g_s)}(g_u),$$ and the second equality follows from Lemma \[techlemma\]. Now if the number of unipotent $z$-classes in $\mathcal{Z}_G(s_i)$ is finite for all $i=1,2,\ldots, n$. Then the number of $z$-classes in $G$ is finite. So the upshot is the following:
If we know that the number of semisimple $z$-classes in $G$ is finite, and the number of unipotent $z$-classes in centralizer of semisimple elements is finite, then the number of $z$-classes in $G$ is finite.
Proof of the Theorem \[maintheorem\]:
-------------------------------------
1. The number of semisimple $z$-classes in $\B_n(k)$ is finite follows from Proposition \[semisimplez\]. The number of unipotent $z$-classes in $\B_n(k)$ is finite for $2\leq n\leq 5$ follows from Corollary \[unipotentz1\]. So the number of $z$-classes in $\B_n(k)$, for $2\leq n \leq 5$, is finite follows from Lemma \[techlemma\] (see also Remark \[remark\]).
2. The number of unipotent $z$-classes in $\B_n(k)$ is infinite for $n\geq 6$ follows from Lemma \[unipotentz2\]. Therefore the number of $z$-classes in $\B_n(k)$ is infinite provided $n\geq 6$.
Appendix
========
Two matrices $A, B \in \B_n(k)$ are said to be **conjugate** if $B=PAP^{-1}$ for some $P\in \B_n(k)$. Here we are following [@Ko].
Let $A=(a_{ij})\in \B_n(k)$. Elements of the matrix $A$ are ordered by $$a_{nn};a_{n-1 n-1}, a_{n-1 n};\ldots; a_{11}, a_{12}, \ldots, a_{1n},$$ i.e., a sequence from bottom to top and in each row from left to right.
The aim of this algorithm is to simplify the first entry in the above sequence, then the second entry and so on. By “simplifying" we mean replacing the entry by $0$ or $1$ (conjugating the matrix $A$ by upper triangular matrices) if possible. If not, then we continue with the next entry in the above sequence. At each step, we take care not to disturb any of the reductions obtained so far.
Let $e_{ij}(\alpha)$ be an elementary matrix, with $(i,j)^{\mathrm{th}}$ element equal to $\alpha$ and $0$ everywhere else. Two matrices $A$ and $ B$ are conjugate if and only if one reduces to the other by a sequence of the following two elementary transformations:
1. Multiply row $i$ by $\alpha \neq 0$, then multiply column $i$ by $\alpha^{-1}$; the elementary transformations can be obtained as $A\mapsto PAP^{-1}$, $P=I+e_{ii}(\alpha -1)$.
2. For $i<j$, multiply $I+e_{ij}(\alpha)$ from the left; then multiply $I+e_{ij}(-\alpha)$ from the right; the elementary transformations can be obtained as $A\mapsto PAP^{-1}$, $P=I+e_{ij}(\alpha)$.
**Step 1:** Let $a_{pq}$ be the first unreduced entry of $A$. If the column of $a_{pq}$ contains an entry $a_{iq}=1$ located under $a_{pq}$ (i.e., $i>p$) and $a_{iq}$ is the first nonzero entry in the row, then $a_{pq}=0$ by transformation of the type (2) with $P=I+e_{pi}(-a_{pq})$.
**Step 2:** Suppose that the column of $a_{pq}$ does not contain such an entry $a_{iq}$. If $a_{pq}$ is the first nonzero entry of that row, then $a_{pq}=1$ by transformation of the type (1) with $P=I+e_{pp}(a_{pq}^{-1}-1)$.
**Step 3:** Suppose $a_{pr}=1$ is the first nonzero entry in the row of $a_{pq}$, then $a_{pq}=0$ by the transformation of the type (2) with $P=I+e_{rq}(a_{pq})$. But this might disturb the row $a_{r*}$, which was reduced before. However, this does not happen if the row $a_{q*}$ is zero.
**Step 4:** If $a_{ri}=1=a_{qj}$ are the first nonzero entries of corresponding rows and $i<j$, then the above transformation by $P=I+e_{rq}(a_{pq})$ disturbs the row of $a_{ri}$, which can be restored by the transformation of the type (2) with $P=I+e_{ij}(a_{pq})$ (this transformation does not disturb the reduced entries since the matrix is $5\times 5$ this need not be true for $6\times 6$ matrices). In this case, we get $a_{pq}=0$.
**Step 5:** If $a_{qj}$ is the first nonzero entry of the row and $a_{ri}=0$ for all $i<j$, then $a_{pq}=1$ by transformation of the type (1) with $P=I+e_{qq}(a_{pq}-1)$. But this transformation disturbs row of $a_{qj}$ by changing $a_{qj}=1$ into $a_{qj}:=a_{pq}$. This can be restored by a transformation of type (1) with $P=I+e_{jj}(a_{pq}-1)$. The latter transformation does not disturb already reduced entries if $j^{\text{th}}$ row is zero. If $j^{\text{th}}$ row is not zero, then we have $j=4$ since the dimension is $5$ and the element $a_{45}=1$ was changed to $a_{45}:=a_{pq}$. This can be restored by a transformation of the type (1) with $P=I+e_{55}(a_{pq}-1)$.
Here we use Belitskii’s algorithm (for details see [@Ko]) for unipotent elements. Number of unipotent conjugacy classes in $\B_5(k)$ is $60$ and representatives are the following: $$\begin{pmatrix}1&1&&&\\&1&&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&1\\&1&&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&&\\&1&1&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&&1&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&&&1\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&&\\&1&&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix};
\begin{pmatrix}1&&&&\\&1&&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&1&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&&\\&1&&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&&\\&1&&&1\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&1&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&1&\\&1&&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&1\\&1&&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&&1&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&1&&\\&1&&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&1&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&1\\&1&1&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&1&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&&&\\&1&1&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&1&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&&&1\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&1&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&1&&\\&1&&&1\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&&1\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&1\\&1&&1&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix};
\begin{pmatrix}1&&&&\\&1&1&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&&&\\&1&&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&\\&1&1&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&1&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&&&\\&1&1&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&1&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&1&1&\\&1&&&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&&&1\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&1&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&1&&\\&1&&&1\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&1&&\\&1&&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&1&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&&1\\&1&1&&\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&&&\\&1&1&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&1&&\\&1&&1&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&1&&\\&1&&&\\&&1&&1\\
&&&1&\\&&&&1\end{pmatrix};
\begin{pmatrix}1&1&&&\\&1&1&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&1&&\\&1&&&\\&&1&1&\\
&&&1&1\\&&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&&&\\&1&1&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&&&1&\\&1&1&1&\\&&1&&\\
&&&1&1\\&&&&1\end{pmatrix}
\begin{pmatrix}1&1&1&&\\&1&&&1\\&&1&1&\\
&&&1&\\&&&&1\end{pmatrix};
\begin{pmatrix}1&&&&\\&1&&&\\&&1&&\\
&&&1&\\&&&&1\end{pmatrix}.$$ Therefore we have obtained also all unipotent conjugacy classes for $\B_n(k)$ for $n=2,3,4$. In particular, all we have to do is to look for the bottom right $2\times 2$, $3\times 3$ and $4\times 4$ corners of the above $5\times 5$ case.
For $n=2$, representatives are the following: $\begin{pmatrix}1&1\\&1\end{pmatrix} ; \begin{pmatrix}1&\\&1\end{pmatrix}$.
For $n=3$, representatives are the following: $$\begin{pmatrix}1&1&\\&1&\\&&1\end{pmatrix}
\begin{pmatrix}1&&1\\&1&\\&&1\end{pmatrix}
\begin{pmatrix}1&&\\&1&1\\&&1\end{pmatrix};
\begin{pmatrix}1&1&\\&1&1\\&&1\end{pmatrix} ;
\begin{pmatrix}1&&\\&1&\\&&1\end{pmatrix}.$$ For $n=4$, representatives are the following: $$\begin{pmatrix}1&1&&\\&1&&\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&&1&\\&1&&\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&&&1\\&1&&\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&&&\\&1&1&\\&&1&\\&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&\\&1&&1\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&&&\\&1&&\\&&1&1\\&&&1\end{pmatrix};
\begin{pmatrix}1&1&&\\&1&1&\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&1&&\\&1&&1\\&&1&\\&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&1&&\\&1&&\\&&1&1\\&&&1\end{pmatrix}
\begin{pmatrix}1&&1&\\&1&&1\\&&1&\\&&&1\end{pmatrix}
\begin{pmatrix}1&&1&\\&1&&\\&&1&1\\&&&1\end{pmatrix}
\begin{pmatrix}1&&&\\&1&1&\\&&1&1\\&&&1\end{pmatrix}$$ $$\begin{pmatrix}1&&&1\\&1&1&\\&&1&\\&&&1\end{pmatrix};
\begin{pmatrix}1&1&&\\&1&1&\\&&1&1\\&&&1\end{pmatrix}
\begin{pmatrix}1&1&1&\\&1&&\\&&1&1\\&&&1\end{pmatrix};
\begin{pmatrix}1&&&\\&1&&\\&&1&\\&&&1\end{pmatrix}.$$
**Acknowledgement:** The author would like to acknowledge Dr. Anupam Singh and Dr. Rohit Joshi of IISER Pune for many helpful discussions. The author thanks Dr. Pranab Sardar and Dr. Krishnendu Gongopadhyay of IISER Mohali for encouragement.
[99]{} Sushil Bhunia; Anupam Singh, *“Conjugacy classes of centralizers in unitary groups"*, to appear in the Journal of Group Theory. Joana Cirici, *“Classification of isometries of spaces of constant curvature and invariant subspaces"*, Linear Algebra and its Applications 450: 250-279, (2014). Yuan Chen; Yunge Xu; Huanhuan Li; Wenhao Fu, *“Belitskii’s canonical forms of upper triangular nilpotent matrices under upper triangular similarity"*, Linear Algebra and its Applications 506: 139-153, (2016). D. Z. Djokovic; J. Malzan, *“Orbits of nilpotent matrices"*, Linear Algebra and its Applications 32: 157-158, (1980). K. Gongopadhyay, *“The z-classes of quaternionic hyperbolic isometries"*, J. Group Theory, 16 (6), 941-964, (2013). K. Gongopadhyay and R. S. Kulkarni, *“The $z$-classes of isometries"*, J. Indian Math. Soc. (N.S.) 81, no. 3-4,245-258, (2014). James E. Humphreys, *“Linear algebraic groups"*, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, (1975). Damjan Kobal, *“Belitskii’s canonical form for $5\times 5$ upper triangular matrices under upper triangular similarity"*, Linear Algebra and its Applications 403: 178-182, (2005). R.S. Kulkarni, *“Dynamical types and conjugacy classes of centralizers in groups"*, J. Ramanujan Math. Soc. 22 (1), 35-56, (2007). R. S. Kulkarni, *“Dynamics of linear and affine maps"*, Asian J. Math. 12, no.3, 321-344, (2008). Moshe Roitman, *“A Problem on Conjugacy of Matrices"*, Linear Algebra and its Applications 19: 87-89, (1978). A. Singh, *“Conjugacy Classes of Centralizers in $\mathrm{G}_{2}$"*, J. Ramanujan Math. Soc. 23, no. 4, 327-336, (2008). R. Steinberg, *“Conjugacy Classes in Algebraic Groups"*, notes by V. Deodhar, Lecture Notes in Mathematics 366, Springer-Verlag, (1974). P. Thijsse, *“Upper triangular similarity of upper triangular matrices"*, Linear Algebra and its Applications 260: 119-149, (1997).
[^1]: The author is supported by the SERB, India (No. PDF/2017/001049) and DST-RFBR joint Indo-Russian project (No. INT/RUS/RFBR/P-288).
|
---
author:
- 'Ben Hoare,'
- 'Timothy J. Hollowood'
- 'and J. Luis Miramontes'
title: 'A Relativistic Relative of the Magnon S-Matrix'
---
Imperial/TP/11/BH/02
Introduction {#s1}
============
There have been many remarkable implications of the fact that the Green-Schwarz superstring world-sheet theory defined on a $\text{AdS}_5\times S^5$ background is an integrable theory.[^1] Integrable field theories have some characteristic features: at the classical level, for instance, they exhibit the remarkable property of admitting more than one Hamiltonian formulation. For example, the mKdV integrable system, which is the simplest analogue of the string theory system, admits two distinct Poisson brackets, one of which is relativistically invariant. In fact, the latter is the formulation of the integrable system as the sine-Gordon theory. The two Poisson brackets are compatible, or “coordinated", so that one can actually write down an interpolating family of Poisson brackets. The string world sheet theory for $\text{AdS}_5\times S^5$ mirrors this structure precisely. The string theory Poisson bracket structure is coordinated with the Poisson bracket of a relativistic system which appears as the Pohlmeyer reduction of the world-sheet theory [@Mikhailov:2005qv; @Mikhailov:2005sy; @Schmidtt:2010bi; @Schmidtt:2011nr]. This Pohlmeyer reduced form of the $\text{AdS}_5 \times S^5$ superstring [@Grigoriev:2007bu] has received recent attention due to its classical equivalence to the original Green-Schwarz superstring [@Pohlmeyer:1975nb; @Hofman:2006xt], while possessing a relativistic two-dimensional Lorentz symmetry, preserving the integrability of the superstring world sheet theory and also the UV-finiteness [@Roiban:2009vh]. The reduction procedure is a fermionic generalisation [@Grigoriev:2007bu; @Mikhailov:2007xr; @Grigoriev:2008jq] of the original relation between the classical $O(3)$ sigma model and the sine-Gordon model [@Pohlmeyer:1975nb] (for a review of the reduction of more general bosonic models see [@Miramontes:2008wt] and references therein). For the case of the superstring on $\text{AdS}_5 \times S^5$, the relativistic theory is in a class known as the semi-symmetric space sine-Gordon (SSSSG) theories that are related to a superspace generalization of a symmetric space known as a semi-symmetric space [@Serg; @Zarembo:2010sg]. The particular semi-symmetric space in question is the coset [$$\begin{split}
{\raisebox{.5ex}{$PSU(2,2|4)$}\Big/\raisebox{-.5ex}{$Sp(2,2)\times Sp(4)$}}\ .
\label{sss1}
\end{split}$$]{} The bosonic part of this coset is $\text{AdS}_5\times S^5$ itself. In more detail, the SSSSG theory is a gauged WZW model for the coset [$$\begin{split}
{\raisebox{.5ex}{$Sp(2,2)\times Sp(4)$}\Big/\raisebox{-.5ex}{$SU(2)^{\times4}$}}\ ,
\label{doo}
\end{split}$$]{} coupled in a particular way to a set of fermions, deformed by the addition of a potential which breaks conformal invariance.
As is seemingly ubiquitous, the integrability of the theory is controlled by a twisted affine loop algebra. Such algebras are defined by a Lie algebra, say $\mathfrak f$, and a finite order automorphism $\sigma$, $\sigma^N=1$. The algebra is defined over a set of generators [$$\begin{split}
{\cal L}(\mathfrak f,\sigma)=\bigoplus_{n\in\mathbb Z}z^n\otimes\mathfrak f_{n\ \text{mod}\ N}\ ,
\label{hee}
\end{split}$$]{} where $\mathfrak f_n\subset\mathfrak f$ are the eigenspaces $\sigma(\mathfrak f_n)=e^{2\pi in/N}\mathfrak f_n$. The algebra takes the form [$$\begin{split}
[z^m\otimes u,z^n\otimes v]=z^{m+n}\otimes[u,v]\ .
\end{split}$$]{} In the present case, $\mathfrak f=\mathfrak{psl}(2,2|4)$ and $\sigma$ is an automorphism of order 4 [@Grigoriev:2008jq; @Hollowood:2011fq; @Schmidtt:2010bi]. The potential term in the action has the effect of breaking the large affine symmetry to a smaller subalgebra which, remarkably, contains [$$\begin{split}
{\cal L}\big(\mathfrak p\big(\msl(2|2)\oplus\msl(2|2)\big),\sigma\big)\subset{\cal L}\big(\mathfrak{psl}(2,2|4),\sigma)\ .
\label{zxy}
\end{split}$$]{} Elements of the this algebra are associated to conserved charges whose Lorentz spin equals one half the grade. The zero-graded bosonic subalgebra of this is the Lie algebra of the group $SU(2)^{\times4}$ in the denominator of the WZW coset . This is the group that is gauged in the WZW model but its global part remains as a symmetry of the spectrum. The algebra has single elements of grade $\pm2$ whose associated conserved charges are the lightcone components of the 2-momentum. These elements are centres of the algebra and this implies that the affine loop superalgebra , contains as a subalgebra the elements of grade between $-2$ and $2$, that is Lorentz spins $(0,\pm\frac12,\pm1)$, which generate the finite centrally-extended Lie algebra[^2] [$$\begin{split}
\big(\mathfrak{psl}(2|2) \oplus \mathfrak{psl}(2|2))\ltimes \mathbb{R}^2 \ .
\label{symalg}
\end{split}$$]{} The central elements here, are components of the 2-momentum graded $\pm2$. This algebra is a non-trivial $\mathcal N = (8,8)$ supersymmetry algebra of the theory which acts in a (mildly) non-local way on the Lagrangian fields of the theory [@Schmidtt:2011nr; @Hollowood:2011fq; @Goykhman:2011mq]. This non-locality motivates the idea that the algebra may become $q$ deformed in the quantum theory with $q\to1$ in the classical limit. Lorentz transformations can naturally be included by extending the algebra to include the derivation (the operator that grades the elements). The supersymmetry algebra includes the bosonic symmetry $SU(2)^{\times4}$, the global part of the gauge group of the gauged WZW model, which plays the rôle of a non-abelian R-symmetry.
Generalizing methods from the analysis of bosonic theories [@Hollowood:2011fm; @Hollowood:2010dt], the classical solitons of the theory were constructed and quantized in [@Hollowood:2011fq] and were found to transform in short (or atypical) representations of the symmetry algebra of dimension $4a\times 4a$ with a mass spectrum of the form [$$\begin{split}
m_a=\mu\sin\left(\frac{\pi a}{2k}\right)\ ,\qquad a=1,2,\ldots,k\ .
\label{mass2}
\end{split}$$]{} Here, $k$ is the level of the WZW model which is assumed to be a positive integer. Notice that the spectrum of soliton states is naturally truncated and also the states of lowest mass $a=1$ are identified with the perturbative states of the theory. This latter point deserves to be highlighted because it may seem surprising at first that perturbative excitations are actually solitons, a point described in detail in [@Hollowood:2010dt].
The appearance of the doubly extended superalgebra $\mpsl(2|2)\ltimes\mathbb R^2$ is intriguing because the triply extended superalgebra $\mpsl(2|2)\ltimes\mathbb R^3$ plays a central rôle in the integrability and the magnon S-matrix on the string theory side. The additional central term can simply be understood as arising from a conventional central extension of the affine loop superalgebra ${\cal L}(\msl(2|2),\sigma)$ which vanishes in the SSSSG theory. All this evidence suggests that the over-arching algebraic structure that organizes both the integrability of the string and the SSSSG theory is a quantum group, or $q$, deformation of a centrally extended loop algebra that we will denote $\msl(2|2)^{(\sigma)}$, based on an affinization of $\msl(2|2)$ with twisting by the outer automorphism $\sigma$ of order 4.[^3] This algebra includes the triply extended superalgebra $\mpsl(2|2)\ltimes\mathbb R^3$ as a finite subalgebra.[^4] Roughly speaking, the string and SSSSG theories involve a quantization of the classical loop algebra ${\cal L}(\msl(2|2),\sigma)$ of two different kinds, the former by generating a non-trivial central charge and the latter by $q$ deformation. This implies that there should be an interpolating non-relativistic theory, with both a central charge and $q$ deformation, from which the string theory is obtained by taking a limit $q\to1$ and the relativistic SSSSG theory by taking the affine central charge to 0. In fact, given that we identify the deformation parameter as [$$\begin{split}
q=e^{i\pi/k}\ ,
\label{rrd}
\end{split}$$]{} where $k$ is the level of the WZW model, the former is obtained in the limit $k\to\infty$, the classical limit of the SSSSG theory, and the latter in classical large tension limit of the string world-sheet theory. The fact that the SSSSG theory involves a quantum group is expected because it has already been shown that S-matrix of the (bosonic) symmetric space sine-Gordon theory associated to $\mathbb CP^{n+1}$ involves the affine (loop) quantum group $U_q(\msl(n)^{(1)})$ [@Hollowood:2010rv].
The relativistic S-matrix theory that we construct turns out to be fit neatly into a previously known class of S-matrices associated to affine quantum groups $U_q(\hat\mg)$ [@deVega:1990av; @Bernard:1990ys; @Hollowood:1992sy; @Hollowood:1993fj; @Delius:1995tc; @Gandenberger:1997pk]. It is interesting that there are also existing examples involving affine superalgebras $\mathfrak{osp}(2|2)^{(1)}$ [@Bassi:1999ua]. The particles transform in representations of the affine quantum loop group and the S-matrix elements are proportional to the R-matrix for the affine quantum loop group. In the present case, the appropriate affine superalgebra is above. In particular, the particles transform in representations of the finite supersymmetry algebra .
The construction of the S-matrix begins with the R-matrix associated to the quantum group $U_q(\mathfrak{psl}(2|2) \ltimes \mathbb R^3)$ which was constructed in [@Beisert:2008tw]. This R-matrix is the $q$-deformation of the R-matrix that lies behind the magnon S-matrix. The first hint that this is the correct R-matrix is that a particular classical relativistic limit of this R-matrix identified in [@Beisert:2010kk] bears a remarkable resemblance to the tree-level S-matrix of the reduced $\text{AdS}_5 \times S^5$, or SSSSG, theory [@Hoare:2009fs]. In [@Hoare:2011fj] this relativistic limit was extended to all orders in the coupling and the resulting S-matrix proposed as a candidate for the S-matrix of the Lagrangian field excitations of the theory. In particular, the minimal CCD factor fixed by unitarity and crossing symmetry agrees with a perturbative computation. However, the origin of the quantum deformation in the perturbative computation is still an open question but not surprising given that the quantum groups play an important rôle in the quantization of WZW models [@Bernard:1990ys; @Alekseev:1990vr; @Caneschi:1996sr; @Gawedzki:1990jc; @Alekseev:1992wn].
In [@Hoare:2011fj] the one-loop perturbative S-matrix for the reduced $\text{AdS}_3 \times S^3$ and the reduced $\text{AdS}_5 \times S^5$ theories were computed.[^5] For the reduced $\text{AdS}_3 \times S^3$ theory it was found that (with the addition of a suitable local counterterm) the perturbative S-matrix satisfies the Yang-Baxter equation and is invariant under a $\mathcal N = (4,4)$ quantum-deformed supersymmetry. Due to the non-abelian nature of the $SU(2)^{\times4}$ gauge group for the reduced $\text{AdS}_5 \times S^5$ theory, the one-loop result did not satisfy the Yang-Baxter equation. However, motivated by the reduced $\text{AdS}_3 \times S^3$ example it was conjectured that the physical symmetry of the theory should be given by a quantum-deformation of . A similar quantum-deformation is conjectured to occur in related bosonic models, examples of which are discussed in [@Hollowood:2010rv; @Hollowood:2009sc; @Hollowood:2009tw].
This paper is organized as follows. In section \[s2\], we review some aspects of the S-matrix theories associated to affine quantum groups that will be useful in generalising to the superalgebra case of interest in the present paper. In particular, we highlight some important issues concerning the bootstrap/fusion procedure, whereby certain simple poles of the S-matrix on the physical strip in rapidity space correspond to bound states in the direct or crossed channel. These bound states are then added to the spectrum and their S-matrix elements can then be deduced by the fusion equations. In section \[s3\], we draw heavily on [@Beisert:2008tw] and review the construction of the quantum group $U_q(\mathfrak{psl}(2|2) \ltimes \mathbb R^3)$ and discuss the relativistic limit identified in [@Beisert:2010kk; @Hoare:2009fs]. In this section, the interpretation of the triply extended algebra as a subalgebra of a central extension of the affine loop superalgebra ${\cal L}(\msl(2|2),\sigma)$ is discussed and the magnon representation and relativistic soliton representations are then compared in some detail. In section \[s4\], we turn to the representation theory of the quantum-deformed superalgebra which is key to solving the bootstrap programme. In particular, the representation theory must mesh precisely with the analytic structure of the S-matrix in order to have a consistent S-matrix theory. Unfortunately the representation theory of the quantum-deformed algebra has not been investigated in detail. Experience with low dimensional representations suggests that, just as for a $q$ deformation of an ordinary Lie algebra, the representations are simple deformations of the representations of the undeformed algebra — at least when $q$ is not a root of unity. Pending a more detailed investigation, we will assume that it is true and so we will review the representation theory of the undeformed superalgebra in some detail based, in particular, on [@Zhang:2004qx]. A novel feature of the representation theory, is that tensor products are reducible but indecomposible and this feature will require careful treatment when we turn to the S-matrix. This we do in section \[s5\] where we write down an S-matrix for the basic excitations transforming in the four-dimensional evaluation representation of $U_q(\mathfrak{sl}(2|2)^{(\sigma)})$ based on the R-matrix of [@Beisert:2008tw] appended with a suitable scalar, or CDD, factor to ensure unitarity and crossing. In section \[s6\], we turn to the question of bound states and the bootstrap programme. We show that the behaviour of tensor products of the quantum supergroup, including all the complications of indecomposable representations, meshes perfectly with the bootstrap/fusion procedure of S-matrix theory. The resulting bootstrap procedure gives a mass spectrum that precisely matches the semi-classical mass spectrum of the solitons of the SSSSG theory in providing strong evidence that the tensor product of two copies of our relativistic S-matrix with an appropriate scalar factor is the S-matrix for the soliton excitations — including the perturbative modes — of the reduced theory.
We conclude the paper with a discussion of the closure of the bootstrap procedure and other open questions. The closure is key to defining a consistent quantum S-matrix and we suggest a number of ways it which it could happen. The most cogent possibility is that the S-matrix theory requires that $k$, which is the level of the WZW model, is a positive integer and so the deformation parameter $q$ in is a root of unity. Although, the representation theory of $U_q(\mathfrak{psl}(2|2)\ltimes\mathbb R^3)$ with $q$ a root of unity has not been investigated in any detail, experience with ordinary quantum groups suggests that the effect is to restrict the set of allowed representations and this would provide a mechanism for truncating the spectrum of states as indicated in .
Quantum Group S-Matrices {#s2}
========================
S-matrix theories with symmetries that are associated to affine quantum groups arising as deformations of affine Lie algebras have been studied in the past [@deVega:1990av; @Hollowood:1992sy; @Hollowood:1993fj; @Delius:1995tc; @Gandenberger:1997pk]. The extension to the case of the affine superalgebra $\mathfrak{osp}(2|2)^{(1)}$ appears in [@Bassi:1999ua]. In this section, we review, following loosely the approach described in [@Delius:1995tc], some of the features of this body of work that will assist in applying the construction to our Lie superalgebra case.
The general setting involves the quantum group deformation $U_q(\hat\mg)$ of the universal enveloping algebra of an affine Lie algebra $\hat\mg$. The quantum group is defined by the Chevalley generators $\mH_j$, $\mE_j$, $\mF_j$, $j=0,1\ldots,r$, which have non-vanishing commutators [$$\begin{split}
[\mH_j,\mE_k]=A_{jk}\mE_k\ ,\qquad[\mH_j,\mF_k]=A_{jk}\mF_k\ ,\qquad
[\mE_j,\mF_k]=\delta_{jk}[\mH_j]_{q_j}\ .
\label{ola}
\end{split}$$]{} They also obey quantum Serre relations that we will not write. In the above, $A_{jk}$ is the Cartan matrix of $\hat\mg$, and $q_j=q^{d_j}$, where $d_j$ are coprime intergers such that $d_jA_{jk}$ is symmetric. In the above, [$$\begin{split}
[x]_q=\frac{q^x-q^{-x}}{q-q^{-1}}\ .
\end{split}$$]{}
The QFT has particle multiplets of masses $m_a$ whose Hilbert spaces $V_a(\theta)$ are modules for certain finite dimensional unitary representations $\pi_a$ of $U_q(\hat\mg)$ with vanishing central charge.[^6] The representations and associated modules are labelled by the rapidity $\theta$, which is associated algebraically to a gradation of $\hat\mg$ defined by a set of real numbers $\{s_j\}$. The representation with rapidity is then defined by [$$\begin{split}
\pi_a^\theta(\mE_j)=e^{s_j\theta}\pi_a(\mE_j)\ ,\qquad\pi_a^\theta(\mF_j)= e^{-s_j\theta}\pi_a(\mF_j)\ .
\end{split}$$]{} The quantum group has an associated co-product $\Delta$ which describes how the generators act on a tensor product: [$$\begin{split}
\Delta(\mathfrak\mH_j)&=\mH_j\otimes1+1\otimes\mH_j\ ,\\
\Delta(\mathfrak E_j)&=\mathfrak E_j\otimes 1+q^{-\mathfrak H_j}\otimes \mathfrak E_j\ ,\\
\Delta(\mathfrak F_j)&=\mathfrak F_j\otimes q^{\mathfrak H_j}+1\otimes\mathfrak F_j\ .
\label{cop}
\end{split}$$]{} This can be used to define the representation on a multi-particle state; for example for two particles [$$\begin{split}
\pi^{\theta_1\theta_2}_{ab}(u)=(\pi^{\theta_1}_a\otimes \pi^{\theta_2}_b)\,\Delta(u)\ ,\qquad u\in U_q(\hat\mg)\ .
\end{split}$$]{}
The S-matrix of a relativistic integrable theory are determined by the 2-body S-matrix elements $S_{ab}(\theta_{12})$, which act as intertwiners between the incoming and outgoing Hilbert spaces: [$$\begin{split}
S_{ab}(\theta_{12}):\quad V_a(\theta_1)\otimes V_b(\theta_2)\longrightarrow V_b(\theta_2)\otimes V_a(\theta_1)\ ,
\end{split}$$]{} where $\theta_{12}=\theta_1-\theta_2$. This illustrated in figure \[f1\].
\[line width=1.5pt,inner sep=2mm, place/.style=[circle,draw=blue!50,fill=blue!20,thick]{}\]
[foreground layer]{} at (1.5,1.5) \[place\] (sm) [$\theta_{12}$]{};
at (0,0) (i1) [$V_a(\theta_1)$]{}; at (3,0) (i2) [$V_b(\theta_2)$]{}; at (0,3) (i3) [$V_b(\theta_2)$]{}; at (3,3) (i4) [$V_a(\theta_1)$]{}; (i1) – (i4); (i2) – (i3);
For consistency the 2-body S-matrix element must satisfy the Yang-Baxter Equation [$$\begin{split}
\big(S_{bc}(\theta_{23})\otimes1)&\big(1\otimes S_{ac}(\theta_{13})\big)\big(S_{ab}(\theta_{12})\otimes1)\\ &
=\big(1\otimes S_{ab}(\theta_{12})\big)\big(S_{ac}(\theta_{13})\otimes1)\big(1\otimes S_{bc}(\theta_{23})\big)\ ,
\end{split}$$]{} which is illustrated in figure \[f2\].
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The theory is invariant under the affine quantum symmetry in the sense that [$$\begin{split}
\pi^{\theta_2\theta_1}_{ba}(u)S_{ab}(\theta_{12})=
S_{ab}(\theta_{12})\pi^{\theta_1\theta_2}_{ab}(u)\ , \qquad u\in U_q(\hat\mg)\ .
\end{split}$$]{} So the S-matrix elements lie in the commutant of $U_q(\hat\mg)$ acting on a tensor product representation. It is often useful to write [$$\begin{split}
S_{ab}(\theta)=X_{ab}(\theta)\check R_{ab}(x=e^{\lambda\theta})\ ,
\end{split}$$]{} where $X_{ab}(\theta)$ is a scalar factor which carries the important analytic structure of the S-matrix, in particular all the bound state poles, and $\check R_{ab}(x)$ is the quantum group $R$-matrix that carries all the tensorial structure.
The S-matrix of a relativistic QFT has to satisfy the two important conditions of unitarity and crossing symmetry. Unitarity requires [$$\begin{split}
S_{ab}(\theta)S_{ba}(-\theta)=1\otimes1\ .
\label{unit}
\end{split}$$]{} Note that this is not the same as unitarity of the underlying QFT. This latter form of unitarity is intimately related to the behaviour of the S-matrix at bound state singularities as we explain below. Crossing symmetry relies on the fact that each particle multiplet $V_a$ has a degnerate anti-particle multiplet $V_{\bar a}$, which transforms in a conjugate representation of $U_q(\hat\mg)$. For real representations, it is possible to have $\bar a=a$. Charge conjugation is then an invertible map [$$\begin{split}
{\cal C}:\quad V_a\longrightarrow V_{\bar a}
\end{split}$$]{} and then crossing symmetry requires [$$\begin{split}
S_{ab}(\theta)=({\cal C}^{-1}\otimes1)\big(\sigma\cdot S_{\bar ba}(i\pi-\theta)\big)^{t_1}\cdot\sigma\cdot(1\otimes{\cal C})\ ,
\label{cs}
\end{split}$$]{} where $\sigma$ is the permutation on the tensor product $\sigma(V_a\otimes V_b)=V_b\otimes V_a$ and $t_1$ means transpose on first space in the tensor product which is well defined because $\sigma\cdot S_{ba}(\theta)\in\text{End}(V_b\otimes V_a)$. Charge conjugate relies in an algebraic sense on the antipode operation of the quantum group.
Let $\mg_0$ be the zero graded Lie subalgebra of $\hat\mg$. In many cases, the modules $V_a$ are just finite dimensional irreducible representations of $\mg_0$, but there are situations in which a finite dimensional representation of $\mg_0$ cannot be lifted to $\hat\mg$ — it is not [*affinizable*]{} — as we shall highlight later. In these cases, $V_a$ is a reducible representation of $\mg_0$.
The most non-trivial aspect of S-matrix theory is the analytic structure and its explanation in terms of bound states and anomalous thresholds. Bound states give rise to simple poles of the S-matrix on the physical strip, the region $0<{\operatorname{Im}}\,\theta<\pi$. For integrable field theories these poles occur at purely imaginary values: if a bound state corresponding to particle $V_c$ is exchanged in the direct channel then $S_{ab}(\theta)$, $\theta=\theta_{12}$, has a simple pole at the imaginary value $\theta=iu_{ab}^c$, with $0<u_{ab}^c<\pi$ where [$$\begin{split}
m_c^2=m_a^2+m_b^2+2m_am_b\cos u_{ab}^c\ .
\end{split}$$]{} Note that if $c$ is a bound state of $a$ and $b$, then $a$ is a bound state of $c$ and $\bar b$, the anti-particle of $b$, and $b$ is a bound state of $c$ and $\bar a$, [$$\begin{split}
u_{ab}^c+u_{b\bar c}^{\bar a}+u_{a\bar c}^{\bar b}=2\pi\ ,
\end{split}$$]{} as illustrated in figure \[f3\]
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The position of the bound-state poles must mesh with the representation theory of the quantum affine algebra. For generic values of the rapidities, the representation $\pi^{\theta_1\theta_2}_{ab}$ is irreducible. So, for consistency, at the bound-state pole $\theta_{12}=iu_{ab}^c$ the representation must become reducible and contain $V_c$ as a component. At this special point, if we write for $U_q(\mg_0)$ representations [$$\begin{split}
V_a\otimes V_b=V_c\oplus V_c^\perp\ ,
\end{split}$$]{} then we require that $V^\perp_c$ lies in the kernel of $\text{Res}\,S_{ab}(iu_{ab}^c)$: [$$\begin{split}
\text{Res}\,S_{ab}(iu_{ab}^c):\; V_c^\perp\longrightarrow0\ .
\label{ipm}
\end{split}$$]{} In general, the affinizable representation $V_c$ is reducible under $U_q(\mg_0)$. Suppose we write the decomposition as [$$\begin{split}
V_c=\oplus_jV_c^{(j)}\ ,
\end{split}$$]{} then near the pole we have [$$\begin{split}
S_{ab}(\theta)\thicksim \frac i{\theta-iu_{ab}^c}\sum_j\rho_j\,\Pr_{ab}^{c,j}\ ,
\label{fgf}
\end{split}$$]{} where $\Pr_{ab}^{c,j}$ is the $U_q(\mg_0)$ invariant intertwiner which is only non-vanishing on $V_c^{(j)}\subset V_a\otimes V_b$. It can be expressed as [$$\begin{split}
\Pr_{ab}^{c,j}=\EuScript P^{ba}_{c,j}\,\EuScript P_{ab}^{c,j}\ ,
\end{split}$$]{} where $\EuScript P_{ab}^{c,j}:\ V_a\otimes V_b\rightarrow V_c^{(j)}$ and $\EuScript P^{ba}_{c,j}:\ V^{(j)}_c\rightarrow V_b\otimes V_a$. We remark that when $a=b$, $\Pr_{aa}^{c,j}$ is a projection operator. In , the numbers $\rho_j$ are required to be real, and unitarity of the underlying QFT dictates the sign. In simple cases, the sign is related to the parity of the bound state as found by Karowski [@Karowski:1978ps]. For our S-matrix, the issue of unitarity and the signs of the residues is left for future analysis. The coupling of asymptotic states to the bound state is illustrated in figure \[f4\].
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(i1) – (p1); (i2) – (p1); (i3) – (p2); (i4) – (p2); (p2) – (p1);
The fact that $c$ can appear as a bound state of $a$ and $b$ means that the S-matrix elements of $c$ with other states, say $d$, can be written in terms of those of $a$ and $b$. This is the essence of the bootstrap, or fusion, programme. The relation between the S-matrix elements can be written concretely as [$$\begin{split}
S_{dc}(\theta)&=\left(\sum_j\sqrt{|\rho_j|}\,\EuScript P_{ab}^{c,j}\otimes1\right) \Big(1\otimes S_{db}(\theta+i\bar u_{b\bar c}^{\bar a})\Big) \\ &\qquad\times\left(S_{da}(\theta-i\bar u_{a\bar c}^{\bar b})\otimes1\right) \left(1\otimes\sum_l\frac1{\sqrt{|\rho_l|}}\,\EuScript P_{c,l}^{ab}\right)\ ,
\label{bfs}
\end{split}$$]{} where $\bar u_{ab}^c=\pi-u_{ab}^c$. This expression follows in an obvious way from the equality illustrated in figure \[f5\].
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The most difficult aspect of building a consistent QFT is finding closure of the bootstrap programme; that is being able to account for all the poles of the S-matrix on the physical strip either in terms of direct- or cross-channel bound states, which lead to simple poles, or anomalous thresholds, which in $1+1$-dimensions manifest as poles of arbitrary order.
Examples
--------
The first example involves the affine algebra $\msl(n)^{(1)}$ with the homogeneous gradation where only $s_0\neq0$ and so $\mg_0=\msl(n)$. We will denote $x=e^{s_0\theta/2}$. The $R$-matrix for the tensor product $V_1\otimes V_1$, where $V_1$ is the vector representation of $\msl(n)$, can be written [@Jimbo] [$$\begin{split}
\check R(x)=(xq-x^{-1}q^{-1})\Pr_++(x^{-1}q-xq^{-1})\Pr_-\ ,
\end{split}$$]{} where $\Pr_\pm$ are $U_q(\msl(n))$ invariant projectors onto the rank-2 symmetric and anti-symmetric representations. These projectors are related to a $q$ deformation of the symmetric group known as the Hecke algebra. The form of $\check R(x)$ above shows that there are special points when $x^2=q^{\pm2}$ where $\check R(x)$ becomes a projector onto the symmetric and anti-symmetric representations. Both these representations are affinizable. QFTs can be constructed which include either the symmetric or anti-symmetric representations in the spectrum by appropriate choices of $s_0$ and the scalar factor.
We now turn to an affine algebra $\mathfrak{so}(2n)^{(1)}$, again with the homogeneous gradation. In this case the $\check R(x)$ matrix for a tensor product $V_1\otimes V_1$, where $V_1$ is the vector representation takes the form [@Jimbo] [$$\begin{split}
\check R(x)&=(xq-x^{-1}q^{-1})(xq^{n/4}-x^{-1}q^{-n/4})\Pr_+\\ &\qquad+(x^{-1}q-xq^{-1})(xq^{n/4}-x^{-1}q^{-n/4})\Pr_-\\ &\qquad\qquad+(x^{-1}q-xq^{-1})(x^{-1}q^{n/4}-xq^{-n/4})\Pr_\bullet\ ,
\end{split}$$]{} where $\bullet$ is the singlet. In this case, at the special points $x=\pm q^{-1}$ we have [$$\begin{split}
\check R(\pm q^{-1})&=(q^2-q^{-2})(q^{n/4-1}-q^{-n/4+1})\Pr_-\\ &\qquad+(q^2-q^{-2})(q^{n/4+1}-q^{-n/4-1})\Pr_\bullet\ ,
\end{split}$$]{} which is not a projector onto a single irreducible representation of $U_q(\mathfrak{so}(2n))$; rather the representation is reducible and the $\check R$-matrix is a weighted sum of projection operators. This is symptom of the fact that anti-symmetric representation by itself is not “affinizable", and in order to find an irreducible representation of the affine algebra one must take the reducible module $V_2=V_-\oplus V_\bullet$. In contrast, notice that when $x=\pm q$ the $\check R$-matrix becomes a projector onto the symmetric representation.
Centrally Extended $\mathfrak{psl}(2|2)$ and its Quantum Group {#s3}
==============================================================
In this section we describe the theory of the Lie superalgebra $\mathfrak{psl}(2|2)$, its central extensions and its $q$, or quantum group, deformation. We pay particular attention to the defining representation and the differences between the magnon and soliton representations. In the following section, we turn to its rich representation theory. Since this algebra has been extensively studied, our discussion will not be comprehensive and, in particular, we shall draw extensively on the discussion by Beisert and Koroteev [@Beisert:2008tw] and use their notation throughout.
The Lie superalgebra $\mathfrak{su}(2|2)$ is generated by $4\times 4$ anti-Hermitian matrices which, in $2\times2$ block notation, are of the form [$$\begin{split}
M=-M^\dagger=\left(\begin{array}{c|c}m & \theta \\\hline \eta & n\end{array}\right)\ .
\label{vxx}
\end{split}$$]{} The matrices $m$ and $n$ are Grassmann even, with $m^\dagger=-m$ and $n^\dagger=-n$, and $\theta$ and $\eta$ are Grassmann odd, with $\eta=-\theta^\dagger$.[^7] These matrices are required to have vanishing supertrace $\text{str}\,M=-{\operatorname{tr}}\,m+{\operatorname{tr}}\,n=0$. In most of the following, we will work with the complex form of the algebra $\msl(2|2)$. Notice that the algebra includes the element ${\bf 1}$ as a centre, and this allows one to define $\mathfrak{psl}(2|2)$ as the quotient $\mathfrak{sl}(2|2)/{\bf1}$.
The Lie superalgebra $\mathfrak{psl}(2|2)$ is unique in that it can be extended with 3 independent centres. Using the notation of [@Beisert:2008tw], the centrally extended algebra can be written as follows. The generators of $\mathfrak{psl}(2|2)$ consist of the $\mg_0=\mathfrak{sl}(2)\oplus\msl(2)$ even generators $\mathfrak R^a{}_b$ and $\mathfrak L^\alpha{}_\beta$, with traceless conditions $\mathfrak R^1{}_1=-\mathfrak R^2{}_2$ and $\mathfrak L^1{}_1=-\mathfrak L^2{}_2$, and the odd generators $\mathfrak Q^\alpha{}_b$ and $\mathfrak S^a{}_\beta$, the latter to be multiplied by Grassmann numbers to give an algebra element. The brackets of the algebra involving elements of $\mg_0$ are $$\begin{aligned}
{2}
\notag [\mR^a{}_b,\mR^c{}_d]&=\delta^c_b\mR^a{}_d-\delta^a_d\mR^c{}_b\ ,\qquad&
[\mL^\alpha{}_\beta,\mL^\gamma{}_\delta]&=\delta^\gamma_\beta
\mL^\alpha{}_\delta-\delta^\alpha_\delta
\mL^\gamma{}_\beta
\ ,\\
[\mR^a{}_b,\mQ^\gamma{}_d]&=-\delta^a_d\mQ^\gamma{}_b+\tfrac12
\delta^a_b\mQ^\gamma{}_d\ ,\qquad&
[\mL^\alpha{}_\beta,\mQ^\gamma{}_d]&=\delta^\gamma_\beta\mQ^\alpha{}_d-\tfrac12\delta^\alpha_\beta\mQ^\gamma{}_d\ ,\label{com1}\\
\notag[\mR^a{}_b,\mS^c{}_\delta]&=\delta^c_b\mS^a{}_\delta-\tfrac12\delta^a_b\mS^c{}_\delta\ ,&
[\mL^\alpha{}_\beta,\mS^c{}_\delta]&=-\delta^\alpha_\delta\mS^c{}_\beta+\tfrac12\delta^\alpha_\beta\mS^c{}_\delta\ ,\end{aligned}$$ while the odd generators satisfy the anti-commutation algebra [$$\begin{split}
\{\mQ^\alpha{}_b,\mS^c{}_\delta\}&=\delta^c_b\mL^\alpha{}_\delta+\delta^\alpha_\delta\mR^c{}_b+\delta^c_b\delta^\alpha_\delta\mC\ ,\\
\{\mQ^\alpha{}_b,\mQ^\gamma{}_d\}&=\varepsilon^{\alpha\gamma}\varepsilon_{bd}\mP\ ,
\qquad\{\mS^a{}_\beta,\mS^c{}_\delta\}=\varepsilon^{ac}\varepsilon_{\beta\delta}\mK\ .\label{com2}
\end{split}$$]{} In the above, $\mC$, $\mP$ and $\mK$ are the 3 central extensions which commute with all other generators.
The question before us is what is the relation to the defining 4-dimensional representation of the real form $\msu(2|2)$ described above. Introducing the basis ${\boldsymbol{e}}_{ij}$, $i,j\in\{1,2,3,4\}$, as the matrix with a 1 in position $(i,j)$ and 0 elsewhere, the $\mg_0$ generators are simply $$\begin{aligned}
{3}
\notag \mR^1{}_1&=-\mR^2{}_2=
\tfrac12\big({\boldsymbol{e}}_{11}-{\boldsymbol{e}}_{22}\big)\ ,\quad&\mR^1{}_2&={\boldsymbol{e}}_{12}\ ,\quad&\mR^2{}_1&={\boldsymbol{e}}_{21}\ ,\\
\mL^1{}_1&=-\mL^2{}_2=
\tfrac12\big({\boldsymbol{e}}_{33}-{\boldsymbol{e}}_{44}\big)\ ,\quad&\mL^1{}_2&={\boldsymbol{e}}_{34}\ ,\quad&\mL^2{}_1&={\boldsymbol{e}}_{43}\ .\end{aligned}$$ For the odd generators there is some freedom: $$\begin{aligned}
{2}
\notag\mQ^1{}_1&=a {\boldsymbol{e}}_{31}+b{\boldsymbol{e}}_{24}\ ,\qquad&\mQ^1{}_2&=a{\boldsymbol{e}}_{32}-b{\boldsymbol{e}}_{14}\ ,\\
\mQ^2{}_1&=a{\boldsymbol{e}}_{41}-b{\boldsymbol{e}}_{23}\ ,\qquad&\mQ^2{}_2&=a{\boldsymbol{e}}_{42}+b{\boldsymbol{e}}_{13}
\label{genA}\end{aligned}$$ and $$\begin{aligned}
{2}
\notag\mS^1{}_1&=d{\boldsymbol{e}}_{13}+c{\boldsymbol{e}}_{42}\ ,\qquad&\mS^1{}_2&=d{\boldsymbol{e}}_{14}-c{\boldsymbol{e}}_{32}\ ,\\
\mS^2{}_1&=d{\boldsymbol{e}}_{23}-c{\boldsymbol{e}}_{41}\ ,\qquad&\mS^2{}_2&=d{\boldsymbol{e}}_{24}+c{\boldsymbol{e}}_{31}\ ,
\label{genB}\end{aligned}$$ where the parameters satisfy [$$\begin{split}
ad-bc=1\ .
\label{con}
\end{split}$$]{} In this 4-dimensional representation, the three centres are all proportional to the identity matrix which is the centre of $\msl(2|2)$: $\mathfrak C=C\cdot{\bf 1}$, $\mP=P\cdot{\bf1}$ and $\mK=K\cdot{\bf1}$, where [$$\begin{split}
C=\tfrac12(ad+bc)\ ,\qquad P=ab\ ,\qquad K=cd
\end{split}$$]{} and due to they are subject to the constraint [$$\begin{split}
C^2-PK=\tfrac14\ .
\label{conL2}
\end{split}$$]{} Different choices for $\{a,b,c,d\}$ give rise to different representations of the algebra, for example later we will focus on two particular representations associated to the magnons of string theory and the solitons of the SSSSG theory.
The $\mg=\mpsl(2|2)\ltimes\mathbb R^3$ algebra admits a ${\mathbb Z}$-gradation [$$\begin{aligned}
{3}
\notag&s(\mR^a{}_b)=s(\mL^\alpha{}_\beta)=0\ ,\qquad& s(\mQ^\alpha{}_b)&=1\ ,\qquad& s(\mS^a{}_\beta)&=-1\ ,\\
&s(\mC)=0\ ,\qquad& s(\mP)&=2\ ,\qquad& s(\mK)&=-2\ ,
\label{zgr}\end{aligned}$$]{} which can be associated to an additional element that can be added to the algebra known as the [*derivation*]{} $\mD$ which, for the basis elements $u\in\{\mR^a{}_b,\mL^\alpha{}_\beta,\mQ^\alpha{}_b,\mS^a{}_\beta,\mP,\mK,\mC\}$ of the algebra, acts as [$$\begin{split}
[\mD,u]=s(u)u\ .
\label{soo}
\end{split}$$]{} In the relativistic soliton theory, the grade of an element will be identified with minus twice the Lorentz spin. In this interpretation, $\mQ^\alpha{}_b$ and $\mS^a{}_\beta$ have spins $\mp\tfrac12$ and so are interpreted as supersymmetry generators and $\mathfrak P$ and $\mathfrak K$ will be identified, up to an overall constant, with the lightcone components of the 2-dimensional momentum, of spin $\mp1$. Notice that, in comparison with the case in section \[s2\], the momentum generators are part of the symmetry algebra as one expects in a supersymmetric theory. The derivation is then the generator of Lorentz transformations.
It should be apparent that the algebra $\mg=\mpsl(2|2)\ltimes\mathbb R^3$ has the whiff of an affine algebra about it even though it is finitely generated. The finite set of elements have grades restricted to the interval $[-2,+2]$. The algebra $\mg$ can be thought of as a finite-dimensional subalgebra of the centrally extended loop superalgebra $\msl(2|2)^{(\sigma)}$ defined below, associated to a $\mathbb Z_4$ automorphism $\sigma$. We do not have a complete understanding of the rôle of such an infinite algebra, but we can make the following observations. The appearance of a $\mathbb Z_4$ automorphism is not a surprise since the semi-symmetric space is defined by such an automorphism of the superalgebra $\mpsl(2,2|4)$. If we take the generators of $\msl(2|2)$ as $\mL^\alpha{}_\beta$, $\mR^a{}_b$, $\mQ^\alpha{}_b$, $\mS^a{}_\beta$ along with the unit matrix $\bf1$, then the action of the automorphism $\sigma$ in this basis is simply related to the $\mathbb Z$ grade by [$$\begin{split}
\sigma(u)=e^{i\pi s(u)/2}u\ ,
\label{dwe}
\end{split}$$]{} along with $\sigma({\bf 1})=-{\bf1}$. Denoting the eigenspaces of the algebra under $\sigma$, $\sigma(\mg_j)=e^{i\pi j/2}\mg_j$, we have [$$\begin{aligned}
{2}
\notag \msl(2|2)_0&=\{\mR^a{}_b,\mL^\alpha{}_\beta\}\ ,\qquad&
\msl(2|2)_1&=\{\mQ^\alpha{}_b\}\ ,\\
\msl(2|2)_2&=\{{\bf1}\}\ ,\qquad&
\msl(2|2)_3&=\{\mS^a{}_\beta\}\,.
\label{identify}\end{aligned}$$]{} The algebra $\msl(2|2)^{(\sigma)}$ is obtained as a central extension of the loop algebra ${\cal L}(\msl(2|2),\sigma)$, completed with a derivation: [$$\begin{split}
\msl(2|2)^{(\sigma)}={\cal L}(\msl(2|2),\sigma)\oplus\mathbb C\,\mC\oplus\mathbb C\,\mD\ ,
\end{split}$$]{} which takes the form [$$\begin{split}
&[z^m\otimes u,z^n\otimes v]=z^{m+n}\otimes[u,v]+m\,\text{str}(uv)\delta_{m+n,0}\,\mC\ ,\\
&[\mC,z^m\otimes u]=0\ ,\qquad[\mD,z^m\otimes u]=mz^m\otimes u\ ,\qquad[\mC,\mD]=0\ .
\end{split}$$]{} Notice that $\mC$, the third central term of $\mg$, is identified with the conventional central charge of the infinite algebra. Although it would be interesting to further investigate the structure of this infinite algebra, for our purposes it will be enough to deal with the much more manageable finite algebra $\mg=\mathfrak{psl}(2|2)\ltimes\mathbb R^3$ which is a finite subalgebra of $\msl(2|2)^{(\sigma)}$ consisting of all the elements of grades $[-2,+2]$ with $\mD$ identified with the derivation in . In particular, the unique elements of grade $\pm2$ are identified with the two centres $\mK$ and $\mP$, respectively: [$$\begin{split}
z^2\otimes{\bf1}=\mP\ ,\qquad z^{-2}\otimes{\bf1}=\mK\ .
\end{split}$$]{} The fact that these elements are the only elements of grade $\pm2$ and they are in the centre of the algebra is the reason why $\mg$ is a closed finite subalgebra of the full infinite dimensional affine algebra. Before proceeding, we point out a connection with the affine algebra $\msl(2|2)^{(2)}$ discussed in [@Gould1]. The outer automorphism used to define this twisted affinization differs from ours by an inner automorphism. Consequently the affine algebras $\msl(2|2)^{(\sigma)}$ and $\msl(2|2)^{(2)}$ are isomorphic. However, the difference by an inner automorphism means that the algebras have different $\mathbb Z$ gradations. The difference in gradations has physical consequences, for instance, the zero graded algebra is $\msl(2)\oplus\msl(2)$ in our case, but $\mathfrak{osp}(2|2)\simeq\msl(2|1)$ for the gradation implicit in [@Gould1].
The centrally extended Lie superalgebra $\mathfrak{psl}(2|2)\ltimes\mathbb R^3$ admits a group of outer automorphisms $SL(2, \mathbb C)$ [@Beisert:2006qh; @Arutyunov:2009ga] which acts on the Grassman odd generators $\mathfrak Q^\alpha{}_b$ and $\mathfrak S^a{}_\beta$ and, thus, on the central elements, leaving the combination [$$\begin{split}
\vec{\mC}^2=\mC^2 - \mP\mK
\end{split}$$]{} invariant. It was used in [@Beisert:2006qh] to construct representations of $\mathfrak{psl}(2|2)\ltimes \mathbb R^3$ for generic values of the three central elements in terms of the representations of $\mathfrak{sl}(2|2)$. In eqs. and , these automorphisms relate different choices of the parameters $\{a,b,c,d\}$ which lead to inequivalent realizations of the basis of generators. In particular, the action of the outer automorphism does not act in a way that is consistent with the $\mathbb Z$ grades of the generators .
Before we describe the quantum group $U_q(\mg)$, it is helpful to introduce a Chevalley basis for the complex algebra consisting of generators $\{\mathfrak E_i,\mathfrak F_i,\mathfrak H_i\}$. Following [@Beisert:2008tw], we choose $$\begin{aligned}
{3}
\notag\mE_1&=\mR^2{}_1\ ,\qquad&\mE_2&=\mQ^2{}_2\ ,\qquad&\mE_3&=\mL^1{}_2\ ,\\
\mF_1&=\mR^1{}_2\ ,\qquad&\mF_2&=\mS^2{}_2\ ,\qquad&\mF_3&=\mL^2{}_1\ ,
\label{pqo}\end{aligned}$$ in which case [$$\begin{split}
\mH_1=-2\mR^1{}_1\ ,\qquad\mH_2=-\mC-\tfrac12\mH_1-\tfrac12\mH_3\ ,\qquad\mH_3=-2\mL^1{}_1\ .
\label{wsw}
\end{split}$$]{} The Chevalley generators satisfy the algebra [$$\begin{split}
[\mH_i,\mE_j]=A_{ij}\mE_j\ ,\qquad[\mH_i,\mF_j]=-A_{ij}\mF_j
\label{bvv}
\end{split}$$]{} and [$$\begin{split}
[\mE_1,\mF_1]=\mH_1\ ,\qquad\{\mE_2,\mF_2\}=-\mH_2\ ,\qquad[\mE_3,\mF_3]=-\mH_3\ .
\label{buu}
\end{split}$$]{} In the above, [$$\begin{split}
A_{ij}=\left(\begin{array}{rrr}2 & \ -1 & 0\\ -1 & 0 & 1\\ 0 & 1 & \ -2\end{array}\right)
\end{split}$$]{} is the (degenerate) Cartan matrix of $\mg$. The remaining (anti-)commutators are written down in [@Beisert:2008tw].
The quantum deformation {#s3.1}
-----------------------
In order to proceed, we need to describe the quantum group deformation $U_q(\mg)$. For the Chevalley generators, it corresponds to the deformation of to [$$\begin{split}
[\mE_1,\mF_1]=[\mH_1]_q\ ,\qquad\{\mE_2,\mF_2\}=-[\mH_2]_q\ ,\qquad[\mE_3,\mF_3]=-[\mH_3]_q\ .
\label{buu2}
\end{split}$$]{} In contrast, the commutators are not modified, but it is convenient to write them in the exponentiated form [$$\begin{split}
q^{\mH_i}\mE_j=q^{A_{ij}}\mE_jq^{\mH_i}\ ,\qquad
q^{\mH_i}\mF_j=q^{-A_{ij}}\mF_jq^{\mH_i}\ .
\end{split}$$]{} The Serre relations are also deformed although we shall not need the explicit expressions here. In the following, we take the deformation parameter[^8] [$$\begin{split}
q=e^{i\pi/k}\ ,
\end{split}$$]{} where $k$ is a positive real number which we assume is not an integer. The case with $k$ an integer is considered briefly in section \[s6.1\].
At the level of the 4-dimensional representation we can achieve the $q$ deformation by modifying the Chevalley generators with appropriate factors of $q$: $$\begin{aligned}
{3}
\notag\mE_1&=q^{1/2}\mR^2{}_1\ ,\qquad&\mE_2&=\mQ^2{}_2\ ,\qquad&\mE_3&=q^{-1/2}\mL^1{}_2\ ,\\
\mF_1&=q^{-1/2}\mR^1{}_2\ ,\qquad&\mF_2&=\mS^2{}_2\ ,\qquad&\mF_3&=q^{1/2}\mL^2{}_1\ .
\label{onn}\end{aligned}$$ As before, in the 4-dimensional representation, the centres are $P=ab$ and $K=cd$, but now the other centre is defined implicitly by [$$\begin{split}
ad=[C+\tfrac12]_q\ ,\qquad bc=[C-\tfrac12]_q\ .
\label{ll1}
\end{split}$$]{} Then, the constraint is modified to [$$\begin{split}
(ad-qbc)(ad-q^{-1}bc)=1\ ,
\label{ll2}
\end{split}$$]{} which is equivalent to [$$\begin{split}
[C]_q^2-PK=[\tfrac12]_q^2\ .
\label{con2}
\end{split}$$]{}
For later use we can write the action of the Chevalley generators on the 4-dimensional representation by introducing a basis $\{\ket{\phi^1},\ket{\phi^2},\ket{\psi^1},\ket{\psi^2}\}$ for the action of the basis matrices ${\boldsymbol{e}}_{ij}$: [$$\begin{aligned}
{3}
\notag &\mathfrak H_1\ket{\phi^1}=-\ket{\phi^1},\qquad &&\mathfrak E_1
\ket{\phi^1}=q^{1/2}\ket{\phi^2},\qquad&&\mathfrak F_2\ket{\phi^1}=c\ket{\psi^1},\\
\notag&\mathfrak H_1\ket{\phi^2}=\ket{\phi^2},\ &&\mathfrak E_2
\ket{\phi^2}=a\ket{\psi^2},\ &&\mathfrak F_1\ket{\phi^2}=q^{-1/2}\ket{\phi^1},\\
&\mathfrak H_3\ket{\psi^2}=\ket{\psi^2},\ &&\mathfrak E_3
\ket{\psi^2}=q^{-1/2}\ket{\psi^1},\ &&\mathfrak F_2\ket{\psi^2}=d\ket{\phi^2},\\
\notag&\mathfrak H_3\ket{\psi^1}=-\ket{\psi^1},\ &&\mathfrak E_2
\ket{\psi^1}=b\ket{\phi^1},\ &&\mathfrak F_3\ket{\psi^1}=q^{1/2}\ket{\psi^2}\ ,
\label{fppq}\end{aligned}$$]{} where the action of $\mH_2$ is not written since it is determined by . The Chevalley generators $\mE_j$ and $\mF_j$ have $\mathbb Z$ grades $\pm s_j$, respectively, with $s_j=(0,1,0)$.
A useful parameterization of $\{a,b,c,d\}$ for the quantum group was introduced in [@Beisert:2008tw]: [$$\begin{aligned}
{2}
\notag a&=\sqrt g\gamma\ ,\qquad&
b&=\frac{\sqrt g\alpha}{\gamma}\left(1-q^{2C-1}\frac{x^+}{x^-}\right)\ ,\\
c&=\frac{i\sqrt g\gamma}\alpha\frac{q^{-C+1/2}}{x^+}\ ,\qquad &
d&=\frac{i\sqrt g}\gamma q^{C+1/2}\left(x^--q^{-2C-1}x^+\right)\ ,\end{aligned}$$]{} subject to a constraint [$$\begin{split}
\frac{x^+}q+\frac q{x^+}-qx^--\frac1{qx^-}+ig(q-q^{-1})\left(\frac{x^+}{qx^-}-\frac{qx^-}{x^+}\right)=\frac ig\ ,
\label{xll}
\end{split}$$]{} which follows from and . The three centres are given by [$$\begin{split}
q^{2C}&=q\frac{(q-q^{-1})/x^+-ig^{-1}}{(q-q^{-1})/x^--ig^{-1}}
=q^{-1}\frac{(q-q^{-1})x^++ig^{-1}}{(q-q^{-1})x^- + ig^{-1}}\ ,\\
P&=g\alpha\left(1-q^{2C}\frac{x^+}{qx^-}\right)\ ,\qquad
K=g\alpha^{-1}\left(q^{-2C}-\frac{qx^-}{x^+}\right)\ ,
\end{split}$$]{} which satisfy the constraint . It is important to understand the nature of the parameters above. The parameters $g$, $\alpha$, and of course $q$, are constants whereas $x^\pm$ and $\gamma$, of which two are independent due to the constraint , are kinematic variables that can vary on each one particle state. In [@Beisert:2008tw] Beisert and Koroteev identify a choice of $\gamma$ that has nice analytic properties. Introducing an arbitrary non-kinematic phase ${\varphi_1}$ that will be useful for discussing reality conditions in the later parts of this section, this choice of $\gamma$ is given by[^9] [$$\begin{split}
\gamma=e^{i{\varphi_1}}\frac{\sqrt{-i\alpha q^{C+1/2}U(x^+-x^-)}}{(1-(q-q^{-1})^2g^2)^{1/4}}\ ,
\label{rev}
\end{split}$$]{} where $U=(x^+/qx^-)^{1/2}$ so that there is only a single kinematic variable which will be identified with the momentum of a one particle state.
At this point, we focus on two particular representations of $U_q(\mg)$ that are associated to the magnons and the solitons that are obtained as particular limits of the parameterization above. The magnon representation has been very well studied in the context of the string theory on $\text{\text{AdS}}_5\times S^5$ or $\mathcal{N}=4$ super Yang-Mills [@Beisert:2005tm; @Beisert:2006qh; @Klose:2006zd; @Arutyunov:2006yd] and as such we just discuss the limit briefly. The soliton representation, however, is new and in the rest of this paper we will investigate this representation and its associated R-matrix.
[**Magnons:**]{} This representation is constructed by taking the limit of vanishing $q$-deformation, that is $q\to1$, or $k\to\infty$. The combination [$$\begin{split}
\frac{x^+}{x^-}=e^{ip}\ ,
\label{poy}
\end{split}$$]{} where $p$, the kinematic variable, is the world sheet momentum of the string. The constant $g$ is a coupling which in the $\text{\text{AdS}}_5\times S^5$ setting is related to the ’t Hooft coupling by $g^2=\lambda/8\pi$. In the limit, the variable $\gamma$ is determined via with ${\varphi_1}=0$ to be [$$\begin{split}
\gamma=\sqrt{-i\alpha e^{ip/2}(x^+-x^-)}\ .
\end{split}$$]{}
[**Solitons:**]{} This representation is obtained by keeping $q$ fixed and taking the limit $g\to\infty$, which is the limit of large string tension (or ’t Hooft coupling). In this case, in contrast to , [$$\begin{split}
\frac{x^+}{x^-}=q\ .
\end{split}$$]{} First of all, taking the limit $g\to\infty$ of gives [$$\begin{split}
\gamma=e^{i{\varphi_1}}\sqrt{\frac{\alpha x^+[\tfrac12]_q}{g}}\ ,\qquad [\tfrac12]_q=\frac1{q^{1/2}+q^{-1/2}}=\frac1{2\cos\tfrac\pi{2k}}
\end{split}$$]{} and then to leading order in $g^{-1}$ [$$\begin{split}
a=e^{i{\varphi_1}}\sqrt{\alpha x^+[\tfrac12]_q}\ ,\quad b=ie^{-2i{\varphi_1}}q^{-1/2}a\ ,\quad c=ie^{2i{\varphi_1}}q^{1/2}d\ ,\quad d=e^{-i{\varphi_1}}\sqrt{\frac{[\tfrac12]_q}{\alpha x^+}}\ .
\label{ma1}
\end{split}$$]{} This implies that the central term $C=0$ and so the condition becomes [$$\begin{split}
-PK=[\tfrac12]_q^2\ .
\label{con3}
\end{split}$$]{} This will be identified as a relativistic mass shell condition with $P$ and $K$ proportional to the lightcone components of the 2-momentum. We can define the kinematic variable $\theta$, to be identified with the rapidity, and take [$$\begin{split}
x^\pm=\frac{q^{\pm1/2}}{\alpha}e^{-\theta +i{\varphi_2}}\ .
\label{ma2}
\end{split}$$]{} In the above we have introduced a second arbitrary phase $e^{i\varphi_2}$.[^10] With this parameterization [$$\begin{split}
P=ab=i[\tfrac12]_q\, e^{-\theta + i{\varphi_2}}\ ,\qquad K=cd=i[\tfrac12]_q\, e^{\theta-i{\varphi_2}}\ .
\end{split}$$]{} In our interpretation, $\theta$ is the rapidity of the state, and $P$ and $K$ are proportional to the lightcone components [$$\begin{split}
p_\pm=\mu\sin\left(\frac{\pi}{2k}\right)e^{\pm\theta}
\end{split}$$]{} of the 2-momentum via [$$\begin{split}
P=\frac{ip_-e^{i{\varphi_2}}}{\mu\sin\frac{\pi}k} ,\qquad K=\frac{ip_+e^{-i{\varphi_2}}}{\mu\sin\frac\pi k}\ .
\label{haa2}
\end{split}$$]{} The constraint , is then interpreted as the mass-shell condition as promised: [$$\begin{split}
p_+p_-=\mu^2\sin^2\left(\frac{\pi}{2k}\right)\ .
\end{split}$$]{} Notice that just as described in section \[s2\] the rapidity appears in precisely the way dictated by the $\mathbb Z$ gradation with Lorentz spin equal to minus half the $\mathbb Z$ grade. At the moment, it still is not obvious that we can associate $P$ and $K$ with the relativistic 2-momentum. The consistency of this identification will come when we consider the action of $\mP$ and $\mK$ on multi-particle states: the action will have the required additive property.
In the soliton representation, using , the $\mathbb Z_4$ automorphism can be written in the following way acting on the generators of the 4-dimensional representation of the quantum group [$$\begin{split}
\sigma(M) = -{\cal K} M^{st} {\cal K}^{-1}\,,\label{z4aut}
\end{split}$$]{} where the super-transpose (with $M$ as in ) and ${\cal K}$ are defined as [$$\begin{split}
M^{st}=\left(\begin{array}{c|c}m^t & -\eta^t \\\hline \theta^t & n^t\end{array}\right)\,,\quad
{\cal K}=\left(\begin{array}{c|c} q^{-1/4}J & 0 \\\hline 0&-e^{2i{\varphi_1}}q^{1/4}J\end{array}\right)\ ,\quad J=\left(\begin{array}{cc} 0 & 1\\ -1 & 0\end{array}\right)\ .
\end{split}$$]{} In the limit $q\to1$, with ${\varphi_1}= \frac\pi2$, the $\mathbb Z_4$ automorphism here becomes exactly the one used to define the semi-symmetric space . We remark that, in this limit, $\sigma$ is an outer automorphism of order 4 in the group of all automorphisms but, since $\sigma^2$ is inner, it has order 2 in the group of outer automorphisms [@Arutyunov:2009ga; @Serg3].
The soliton representation has the reality properties $a^*=-i\,e^{-i{\varphi_2}}\,b$, $d^*=-i\,e^{i{\varphi_2}}\,c$ so that [$$\begin{aligned}
{2}
\notag({\mR}^a{}_b)^\dagger&={\mR}^b{}_a\ ,\qquad&({\mL}^\alpha{}_\beta)^\dagger&={\mL}^\beta{}_\alpha\ ,\\\label{rcpr}(\mQ^\alpha{}_b)^\dagger&=ie^{-i{\varphi_2}}\,\epsilon_{\alpha\beta}\epsilon^{ba}\,\mQ^\beta{}_a\ ,\qquad&
(\mS^a{}_\beta)^\dagger&=ie^{i{\varphi_2}}\,\epsilon_{ab}\epsilon^{\beta\alpha}\,\mS^b{}_\alpha\ ,\\
\notag{\mP}^\dagger&=-e^{-2i{\varphi_2}}\,{\mP}\ ,\qquad&{\mK}^\dagger&=-e^{2i{\varphi_2}}\,{\mK}\ .\end{aligned}$$]{} These are different to the usual reality conditions taken for the magnon representation, for which $a=d^*$ and $b=c^*$, implying [@Arutyunov:2006yd] [$$\begin{aligned}
{2}
\notag( {\mR}^a{}_b)^\dagger&= {\mR}^b{}_a\ ,\qquad&( {\mL}^\alpha{}_\beta)^\dagger&= {\mL}^\beta{}_\alpha\ ,\\ \label{rcgs}
( {\mQ}^\alpha{}_b)^\dagger&= {\mS}^b{}_\alpha\ ,\qquad&
( {\mS}^a{}_\beta)^\dagger&= {\mQ}^\beta{}_a\ , \\
\notag {\mC}^\dagger&= {\mC}\ ,\qquad& {\mP}^\dagger&= {\mK}\ .\end{aligned}$$]{}
Of course, there will always exist an $SL(2,\mathbb C)$ automorphism that relates the magnon and soliton representations that amounts to a change in the basis of the generators of the algebra. One may think that this allows a Lorentz symmetry to be defined on the magnon representation. This is discussed further in the appendix, however, it appears that this is not consistent as the representations are not equivalent when one considers the way that they act on tensor product representations, a subject to which we now turn. At this point and for the rest of the paper we choose the arbitrary phases ${\varphi_1}$ and ${\varphi_2}$ to vanish.
The co-product
--------------
The action of the quantum group on a tensor product in the Lie superalgebra case involves the co-product which generalizes [$$\begin{aligned}
{2}
\notag\Delta(\mathfrak E_j)&=\mathfrak E_j\otimes 1+q^{-\mathfrak H_j}\mathfrak U^{s_j}\otimes \mathfrak E_j\ ,\qquad&
\Delta(\mathfrak F_j)&=\mathfrak F_j\otimes q^{\mathfrak H_j}+\mathfrak U^{-s_j}\otimes\mathfrak F_j\ ,\\
\label{coprod}
\Delta(\mC)&=\mC\otimes 1+1\otimes\mC\ ,\qquad &
\Delta(\mP)&=\mP\otimes 1+q^{2\mC}\mU^2\otimes\mP\ ,\\
\notag\Delta(\mK)&=\mK\otimes q^{-2\mC}+\mU^{-2}\otimes\mK\ ,\qquad &
\Delta(\mU)&=\mU\otimes\mU\ .
\label{coprod} \end{aligned}$$]{} It involves a new abelian generator $\mU$ introduced in the magnon example to describe non-localities in the action of the supersymmetry generators on two-particle states.
For consistency the coproduct for the central extensions should equal themselves under the action of the permutation operator on the tensor product. This imposes the following constraints [@Beisert:2008tw] [$$\begin{split}
\mP=g\alpha(1-q^{2\mC}\mU^2)\ ,\qquad\mK=g\alpha^{-1}(q^{-2\mC}-\mU^{-2})\ .
\end{split}$$]{} The braiding factor satisfies $\mU^2=x^+/qx^-\cdot{\bf1}$ with [$$\begin{split}
\mU\ket{\phi^a}=\sqrt{\frac{x^+}{qx^-}}\ket{\phi^a}\ ,\qquad\mU\ket{\psi^\alpha}=
-\sqrt{\frac{x^+}{q x^-}}\ket{\psi^\alpha}\ .
\end{split}$$]{} In the magnon representation, from , $\mU$ acts as $e^{\mp ip/2}$ on states, while in the soliton representation the braiding factor simplifies to [$$\begin{split}
\mU\ket{\phi^a}=\ket{\phi^a}\ ,\qquad\mU\ket{\psi^\alpha}=
-\ket{\psi^\alpha}\ ,
\end{split}$$]{} which is just the fermion number. This is a significant result and required for the interpretation of the symmetry structure as the supersymmetry algebra of a relativistic QFT since, on physical grounds, one requires a factor $-1$ when moving a supersymmetry, including $\mathfrak E_2$ and $\mathfrak F_2$, past a fermionic state.[^11] Of crucial significance also is that in the soliton representation the non-trivial centres have a trivial co-product: [$$\begin{split}
\Delta(\mP)=\mP\otimes 1+1\otimes\mP\ ,\qquad\Delta(\mK)=\mK\otimes 1+1\otimes\mK\ ,
\end{split}$$]{} which is required if we are to interpret them as the lightcone components of the 2-momentum.
The relation between the coproducts of the magnon and soliton representations is discussed more fully in the appendix.
Representation Theory {#s4}
=====================
The representation theory of Lie superalgebras is much more convoluted than conventional Lie algebras. For a Lie algebra, arbitrary irreducible representations can be built up by taking tensor products of a small set of basic representations and decomposing. On the other hand, for Lie superalgebras, such tensor products are generally reducible but indecomposable. This feature, in particular, will play a prominent rôle in our story because physically the basic particles transform in the 4-dimensional representation of $\mg=\mpsl(2|2)\ltimes\mathbb R^3$ (or rather a tensor product of two copies thereof) and bound states are in representations that lie in tensor products of this representation.
A further complication is that our algebra is $q$ deformed and this modifies the representation theory. Since we lack a comprehensive analysis of the representation theory of $U_q(\mg)$ we will make certain assumptions (although see [@Ky:1994cr; @Ky:1994we]). We will take $q$ to be a generic deformation, in particular we will assume that $q$ is not a root of unity (a situation we shall analyse later). We will also assume that, as in the case of the $q$ deformation of an ordinary Lie algebra, the representations of $U_q(\mg)$ are simply deformations of the representations of $\mg$. This is supported by the explicit constructions of low-dimensional representations.
To start with we consider the undeformed algebra $\mg=\mathfrak{psl}(2|2)\ltimes{\mathbb R}^3$. As explained in section \[s3\], we can construct representations of this algebra by considering the analogous problem in the Lie superalgebra $\mathfrak{sl}(2|2)$ which has a single centre. The three centres can then be generated by the outer-automorphism group.
The long and the short representations
--------------------------------------
Arbitrary representations of the related algebra $\mathfrak{gl}(2|2)$ were constructed in [@Zhang:2004qx] (see also [@Beisert:2006qh; @Beisert:2008tw]). The algebra $\mathfrak{gl}(2|2)$ consists of the algebra $\msl(2|2)$ plus the additional generator ${\cal F}={\boldsymbol{e}}_{11}+{\boldsymbol{e}}_{22}-{\boldsymbol{e}}_{33}-{\boldsymbol{e}}_{44}$ which plays the rôle of the fermion number. The simplest set of representations are of dimension $16(2J_1+1)(2J_2+1)$ and are labelled $(J_1,J_2,\mq,\mpf)$, where $J_i$ are $\msl(2)$ spins. Here, $\mq$ (not a deformation parameter) is identified with the eigenvalue of the central element, which in the defining representation is the identity matrix ${\bf1}$, and $\mpf$ is the fermion number label; $\mq$ and $\mpf$ are complex numbers. If we ignore the fermion label $\mpf$ then these representations give representations of $\mathfrak{sl}(2|2)$. These are the [*long*]{}, or [*typical*]{}, representations denoted $\{m,n\}$, with $m=2J_1$ and $n=2J_2$, in [@Beisert:2008tw]. These representations exist for generic values of the single centre $C=\mq$ and by making use of the $\msl(2,\C)$ automorphism we can use them to construct representation of the case with general values for all 3 centres with $C^2-PK=\mq^2$.
When the centre $\mq$ takes special values the long representations become reducible but indecomposable. This is called a [*shortening*]{} condition and it is very similar to a BPS condition in a supersymmetric QFT. What happens is that the corresponding module $V_{\{m,n\}}$ splits as [$$\begin{split}
V_{\{m,n\}}=V_\text{sub-rep}\oplus V^\perp\ ,
\label{kjj}
\end{split}$$]{} where $V_\text{sub-rep}$ is an invariant subspace under the action of $\mg$. This is therefore a representation of $\mg$, the [*sub-representation*]{}. What makes $\{m,n\}$ indecomposable is that $V^\perp$ is [*not*]{} an invariant subspace. However, the quotient [$$\begin{split}
V_\text{factor}={\raisebox{.5ex}{$V_{\{m,n\}}$}\Big/\raisebox{-.5ex}{$V_\text{sub-rep}$}}
\end{split}$$]{} defines another representation of $\mg$, the [*factor representation*]{}. In a basis for the module ${\begin{pmatrix} u\\ v\end{pmatrix}}$, $u\in V_\text{sub-rep}$ and $v\in V^\perp$, when the shortening condition holds, the generators of the algebra take the form [$$\begin{split}
{\begin{pmatrix} * & *\\ 0 & *\end{pmatrix}}\ .
\end{split}$$]{}
The sub- and factor representations are known as short, or atypical, representations. There are four possibilities that we consider below [@Zhang:2004qx]:
\(i) $\mq=\tfrac12(m+n+2)$. In this case, the sub-representation has dimension [$$\begin{split}
4(2mn+3m+n+2)
\end{split}$$]{} and we denote it $\langle m,n+1\rangle$ to agree with the notation of [@Beisert:2008tw]. The corresponding factor representation is then $\langle m+1,n\rangle$ and has dimension [$$\begin{split}
4(2mn+m+3n+2)\ .
\end{split}$$]{}
\(ii) $\mq=-\tfrac12(m+n+2)$. In this case the situation is the same as (i) except that the rôles of the sub and factor representations are interchanged.
\(iii) $\mq=\tfrac12(m-n)$, $m\neq n$. In this case, the sub-representation has dimension [$$\begin{split}
4(2mn+m+n)
\end{split}$$]{} and we denote it $\langle m,n\rangle_2$. The corresponding factor representation has dimension [$$\begin{split}
4(2mn+3m+3n+4)\ .
\end{split}$$]{} and we denote it as $\langle m,n\rangle_3$.
\(iv) $\mq=-\tfrac12(m-n)$, $m \neq n$. In this case the situation is the same as (i) except that the rôles of the sub and factor representations are interchanged.
\(v) $\mq = 0$, $m = n \neq 0$. In this case, the sub-representation has dimension [$$\begin{split}
2(2m^2+4m+1)
\end{split}$$]{} and we denote it $\langle m,m \rangle_4$. The corresponding factor representation has dimension [$$\begin{split}
2(6m^2+12m+7)\,.
\end{split}$$]{}
\(vi) $\mq = 0$, $m = n = 0$. For this special case we have that the subrepresentation is a singlet, denoted by $\bullet$, and [$$\begin{split}
\{0,0\}\longrightarrow\bullet\oplus\text{\bf adj}\oplus\bullet\ .
\end{split}$$]{}
For our purposes, we will be mostly interested in the atypical representations $\langle m,n\rangle$ whose dimension is $4(m+1)(n+1)+4mn$. For later use, the representations $\langle m,0\rangle$ have dimension $4(m+1)$, $\mq=\frac{1}{2}(m+1)$, and $\mg_0$ content[^12] [$$\begin{split}
\langle m,0\rangle=(m+1,0)\oplus(m,1)\oplus(m-1,0)\ .
\label{juu}
\end{split}$$]{} The 4-dimensional defining representation corresponds to $\langle0,0\rangle$.
For the algebra with 3 non-vanishing centres, the shortening conditions (i) and (ii) can be written [$$\begin{split}
\mq^2=C^2-PK=\tfrac14(m+n+2)^2:\qquad\{m,n\}\longrightarrow\langle m,n+1\rangle+\langle m+1,n\rangle\ ,
\label{er2}
\end{split}$$]{} where one of the representations on the right-hand side is the sub-representation and one the factor representation. In the similar way, for $m \neq n$ the shortening conditions (iii), (iv) are [$$\begin{split}
\mq^2=C^2-PK=\tfrac14(m-n)^2:\qquad\{m,n\}\longrightarrow\langle m,n\rangle_2+\langle m,n\rangle_3\ .
\label{er1}
\end{split}$$]{}
When we move to the quantum algebra $U_q(\mg)$ the shortening conditions on $\{m,n\}$ in the above must become suitably deformed. Following, and generalizing [@Beisert:2008tw], we propose that the conditions and become [$$\begin{split}
[C]_q^2-PK=[\tfrac12(m+n+2)]_q^2\ ,\qquad
[C]_q^2-PK=[\tfrac12(m-n)]_q^2\ ,\label{sh2}
\end{split}$$]{} respectively. This can be checked for small values of $m$ and $n$ explicitly and we shall assume that it is true generally. In the deformed theory, the atypical representations $\langle m,n\rangle$ require [$$\begin{split}
[C]_q^2-PK=[\tfrac12(m+n+1)]_q^2\ .
\label{bps}
\end{split}$$]{}
As we reported in section \[s2\], an S-matrix theory constructed from the quantum group requires a perfect meshing of the representation theory with the analytic structure and in this regard the shortening conditions will play a key rôle. For the construction of the S-matrix, we will be particularly interested in the representations $\{m,0\}$ of $U_q(\mg)$. For these representations, the first shortening condition in corresponds to [$$\begin{split}
\{m,0\}\longrightarrow\langle m+1,0\rangle\oplus\langle m,1\rangle\ ,
\end{split}$$]{} while for $m>0$ the second condition in corresponds to [$$\begin{split}
\{m,0\}\longrightarrow\langle m,0\rangle_2\oplus\langle m,0\rangle_3\ .
\end{split}$$]{} Note that $\langle m,0\rangle_2\equiv\langle m-1,0\rangle$ for $m>0$. The special case where $m=0$ and the shortening condition $\eqref{sh2}$ is satisfied is given by case (vi) above.
The other important information we need, is the decomposition of a tensor product of the particular short representations $\langle m,0\rangle$, $m\geq0$. These take the form [$$\begin{split}
\langle m,0\rangle\otimes \langle n,0\rangle= \sum_{k=0}^{\text{min}(m,n)} \{m+n-2k,0\}\ .
\label{xxy}
\end{split}$$]{}
The Basic S-Matrix {#s5}
==================
For our relativistic QFT, we will be identifying the particle states with the short representations $\pi^\theta_a=\langle a-1,0\rangle$ with associated modules $V_a(\theta)$, $a\in\mathbb N$. The masses of the states follow from the shortening, or BPS, condition with $C=0$ and $P$ and $K$ related to the lightcone components of the momentum as in : [$$\begin{split}
m_a=\mu\sin\left(\frac{\pi a}{2k}\right)\ ,
\label{mass}
\end{split}$$]{} which is the mass formula . Obviously, this formula suggests that $a$ should somehow be cut-off appropriately, an issue we will return to later. In particular, the basic states transform in the 4-dimensional representation $\pi_1^\theta$. The 2-body S-matrix elements of the basic particles involves the tensor product [$$\begin{split}
\pi^{\theta_1\theta_2}_{11}=\langle 0,0\rangle\otimes\langle 0,0\rangle\ .
\end{split}$$]{}
The $\check R$-matrix for the tensor product $V_1\otimes V_1$ can be extracted from the general solution in [@Beisert:2008tw] by taking the limit $g\to\infty$ [@Beisert:2010kk; @Hoare:2011fj] and matching the parameters as in and . The explicit expression for the $\check R$-matrix in the basis $\{\ket{\phi^a},\ket{\psi^\alpha}\}$, with $x=e^{\theta_{12}}$, is [$$\begin{split}
\check R(x)\ket{\phi^a\phi^a}&=A\ket{\phi^a\phi^a}\ ,\qquad \check R(x)\ket{\psi^\alpha\psi^\alpha}=D\ket{\psi^\alpha\psi^\alpha}\ ,\\
\check R(x)\ket{\phi^1\phi^2}&=\frac{q(A-B)}{q^2+1}\ket{\phi^2\phi^1}+
\frac{q^2 A+B}{q^2+1}\ket{\phi^1\phi^2}+\frac{C}{1+q^2}\ket{\psi^1\psi^2}-\frac{qC}{1+q^2}\ket{\psi^2\psi^1}\ ,\\
\check R(x)\ket{\phi^2\phi^1}&=\frac{q(A-B)}{q^2+1}\ket{\phi^1\phi^2}
+\frac{q^2 B+A}{q^2+1}\ket{\phi^2\phi^1}-\frac{qC}{1+q^2}\ket{\psi^1\psi^2}+\frac{q^2C}{1+q^2}\ket{\psi^2\psi^1}\ ,\\
\check R(x)\ket{\psi^1\psi^2}&=\frac{q(D-E)}{q^2+1}\ket{\psi^2\psi^1}
+\frac{q^2 D+E}{q^2+1}\ket{\psi^1\psi^2}+\frac{F}{1+q^2}\ket{\phi^1\phi^2}-\frac{qF}{1+q^2}\ket{\phi^2\phi^1}\ ,\\
\check R(x)\ket{\psi^2\psi^1}&=\frac{q(D-E)}{q^2+1}\ket{\psi^1\psi^2}
+\frac{q^2 E+D}{q^2+1}\ket{\psi^2\psi^1}-\frac{qF}{1+q^2}\ket{\phi^1\phi^2}+\frac{q^2F}{1+q^2}\ket{\phi^2\phi^1}\ ,\\
\check R(x)\ket{\phi^a\psi^\alpha}&=G\ket{\psi^\alpha\phi^a}+H\ket{\phi^a\psi^\alpha}\ ,\qquad
\check R(x)\ket{\psi^\alpha\phi^a}=K\ket{\psi^\alpha\phi^a}+L\ket{\phi^a\psi^\alpha}\ ,
\end{split}$$]{} with[^13] [$$\begin{aligned}
{2}
\notag A&=\frac{(q x-1)(x+1)}{q^{1/2}x}\ ,\qquad& D&=\frac{(q-x)(x+1)}{q^{1/2}x}\ ,\\
\notag B&=\frac{q^3-(q^3-2q^2+2q-1)x-x^2}{q^{3/2}x} \ ,\qquad& E&=\frac{q^3x^2-(q^3-2q^2+2q-1)x-1}{q^{3/2}x}\ , \\
\notag C&=F=\frac{i(q-1)(q^2+1)(x-1)}{q^{3/2} x^{1/2}}\ ,\qquad& G&=L=x-x^{-1}\ ,\\ H&=K=\frac{(q-1)(x+1)}{q^{1/2}x^{1/2}}\ .\end{aligned}$$]{} Since the tensor product has $U_q(\mg_0)$ content [$$\begin{split}
\langle0,0\rangle\otimes\langle0,0\rangle=(2,0)\oplus(0,2)\oplus2(1,1)\oplus2(0,0)\ ,
\label{dcp}
\end{split}$$]{} another way to express the $\check R$-matrix is in terms of $U_q(\mg_0)$ invariant projectors [$$\begin{split}
\check R(x)&=\frac{(x+1)(q-x)}{x\sqrt q}\Pr_{(0,2)}
+\frac{(x+1)(qx-1)}{x\sqrt q}\Pr_{(2,0)}\\ &+
\frac{(x+1)(\sqrt{qx}-1)(\sqrt q+\sqrt x)}{x\sqrt q}\Pr_{(1,1)}^{(+)}+
\frac{(x+1)(\sqrt{qx}+1)(\sqrt q-\sqrt x)}{x\sqrt q}\Pr_{(1,1)}^{(-)}\\ &+
f_+(x)\Pr_{(0,0)}^{(+)}+f_-(x)\Pr_{(0,0)}^{(-)}\ ,
\label{srm}
\end{split}$$]{} where [$$\begin{split}
f_\pm(x)&=\frac1{2q^{3/2}x^2}\Big(
q^3(x-1)^2+4qx(q-1)
\pm(x-1)\Big(1-x+\big((x-1)^2+q^6(x-1)^2\\ &\qquad\qquad\qquad
+4xq(2q^4-3q^3-3q+2)+2q^3(1+10x+x^2)\big)^{1/2}\Big)\Big)\ .
\end{split}$$]{}
In order to construct an S-matrix we will need the unitarity condition [$$\begin{split}
\check R(x^{-1})\check R(x)=\frac{(q-x)(qx-1)(x+1)^2}{qx^2}1\otimes 1
\end{split}$$]{} and the crossing symmetry relation [$$\begin{split}
\check R(x)=({\cal C}^{-1}\otimes1)(\sigma\cdot
\check R(-x^{-1}))^{st_1}\cdot\sigma\cdot(1\otimes{\cal C})\ ,
\end{split}$$]{} where $\sigma$ is the graded permutation operator and $st$ indicates the super-transpose on first space in the tensor product.[^14] The charge conjugation matrix on the basic states takes the form [$$\begin{aligned}
{2}
\notag{\cal C}\ket{\phi^1}&=q^{-1/2}\ket{\phi^2}\ ,\qquad&{\cal C}\ket{\phi^2}&=-q^{1/2}\ket{\phi^1}\ ,\\
\quad{\cal C}\ket{\psi^1}&=q^{-1/2}\ket{\psi^2}\ ,\qquad&{\cal C}\ket{\psi^2}&=-q^{1/2}\ket{\psi^1}\ . \end{aligned}$$]{} In order that the S-matrix satisfies the unitarity and crossing symmetry constraints, and , we multiply the $\check R$-matrix by a scalar function [$$\begin{split}
\widetilde S_{11}(\theta)=Y(\theta)Y(i\pi-\theta)\check R(e^{\theta})\ .
\end{split}$$]{} This form guarantees that crossing symmetry is satisfied and unitarity requires [$$\begin{split}
Y(\theta)Y(i\pi-\theta)Y(-\theta)Y(i\pi+\theta)=\frac{1}{16\sinh(\frac\theta{2}+\frac{i\pi}{2k})
\sinh(-\frac\theta{2}+\frac{i\pi}{2k})\cosh^2(\frac\theta{2})}\ .
\end{split}$$]{} The solution to this is not unique, however, there is a minimal solution with the smallest number of poles and zeros and, in particular, with no poles on the physical strip; namely, [$$\begin{split}
Y(\theta)=\frac1{\sqrt2\pi}
\prod_{l=0}^\infty\frac{\Gamma(\tfrac{\theta}{2i\pi}+l+\tfrac1{2k})\Gamma(\tfrac{\theta}{2i\pi}+l-\tfrac1{2k}+1)}{\Gamma(\tfrac{\theta}{2i\pi}+l+\tfrac1{2k}+\tfrac12)\Gamma(\tfrac{\theta}{2i\pi}+l-\tfrac1{2k}+\tfrac32)}\cdot\frac{\Gamma(\tfrac{\theta}{2i\pi}+l+\tfrac1{2})^2}{\Gamma(\tfrac{\theta}{2i\pi}+l+1)^2}\ .
\end{split}$$]{} We can also write the integral representation [$$\begin{split}
&Y(\theta)Y(i\pi-\theta)
=\frac{{\cal F}(\theta)}{2(q-q^{-1})}\ ;\\
&{\cal F}(\theta)=
\exp\left[-2\int_0^\infty\frac{dt}t
\,\frac{\cosh^2(t(1-\tfrac1k))\sinh(t(1-\tfrac\theta{i\pi}))
\sinh(\tfrac{t\theta}{i\pi})}{\sinh t\cosh^2t}\right]\ .
\label{gqq}
\end{split}$$]{} Notice that ${\cal F}(\theta)$ is real and positive when $\theta$ is purely imaginary.
The Bootstrap Programme {#s6}
=======================
As in the non-graded affine Lie algebra case described in section \[s2\], the representation $\pi^{\theta_1\theta_2}_{11}$ is irreducible for generic values of $\theta_1$ and $\theta_2$. In fact it is identified with the 16-dimensional long representation $\{0,0\}$. However, drawing on the results reported earlier in section \[s4\], for special values of the rapidity difference $\theta_{12}$ the representation becomes reducible. Setting $C_1=C_2=0$, the tensor product has $P=P_1+P_2$ and $K=K_1+K_2$ and so the first shortening condition in becomes [$$\begin{split}
-(P_1+P_2)(K_1+K_2)=[\tfrac32]_q^2\qquad\implies\qquad
\theta_{12}=\pm\frac{i\pi}k\ .
\label{ewe}
\end{split}$$]{} At these special points, the representation becomes reducible [$$\begin{split}
\{0,0\}\longrightarrow\langle1,0\rangle\oplus\langle0,1\rangle
\end{split}$$]{} and for the upper sign $\langle0,1\rangle$ and $\langle1,0\rangle$ are the sub- and factor representation, respectively. For the lower sign these designations swap over.
At these points the $\check R$-matrix, and hence the S-matrix gains a non-trivial kernel. The fact that $\widetilde S_{11}(\theta)$ lies in the commutant of $U_q(\mg)$ means that, for consistency, the kernel must be the invariant subspace corresponding to the sub-representation. The bound state is consequently associated to the factor representation in the tensor product. At this point we have a choice to make. By picking the sign of $k$, we can choose either special point to be on the physical strip. Here, we will take $k$ to be positive, in which case the special point $\theta_{12}=\frac{i\pi}k$ lies on the physical strip and the potential bound state corresponds to the representation $V_2=\langle1,0\rangle$. In this case, the kernel of $\widetilde S_{11}(\tfrac{i\pi}k)$ corresponds to the sub-reprentation $\langle0,1\rangle$: [$$\begin{split}
\widetilde S_{11}(\tfrac{i\pi}k):\quad V_{\langle0,1\rangle}\longrightarrow0\ .
\end{split}$$]{} The bound state transforms in the factor representation and we write [$$\begin{split}
\pi^\theta_2\subset\pi^{\theta+\tfrac{i\pi}{2k},\theta-\tfrac{i\pi}{2k}}_{11}\Big|_\text{factor}\ .
\end{split}$$]{} On the other hand, because $\widetilde S_{11}(\frac{i\pi}k)$ permutes the rapidities it swops over the special points and so maps the factor representation $\langle1,0\rangle\subset\{0,0\}$ to the sub-representation $\langle1,0\rangle\subset\{0,0\}$, and we write [$$\begin{split}
\pi^\theta_2\subset\pi^{\theta-\tfrac{i\pi}{2k},\theta+\tfrac{i\pi}{2k}}_{11}\Big|_\text{sub}\ .
\end{split}$$]{}
Since the whole issue of the shortening of the tensor product representation is key to the construction of the S-matrix, we will pause to discuss it in explicit detail. The tensor product module $V_1\otimes V_1$ can be decomposed in terms of $U_q(\mg_0)$ modules, following the decomposition , as: [$$\begin{split}
V^{{{\color{white}()}}}_{\{0,0\}}=V^{{{\color{white}()}}}_{(2,0)}\oplus V_{(1,1)}^{(+)}\oplus V_{(1,1)}^{(-)}\oplus V^{{{\color{white}()}}}_{(0,2)}\oplus V_{(0,0)}^{(+)}\oplus V_{(0,0)}^{(-)}\ .
\end{split}$$]{} Explicitly, we have the bases[^15] [$$\begin{split}
V^{{{\color{white}()}}}_{(2,0)}:\quad&\ket{\phi^1\phi^1}\ ,\quad q^{1/2}\ket{\phi^1\phi^2}+q^{-1/2}\ket{\phi^2\phi^1}\ ,\quad\ket{\phi^2\phi^2}\ ,\\
V^{(\pm)}_{(1,1)}:\quad&\ket{\phi^1\psi^1}\pm\ket{\psi^1\phi^1}\ ,\quad\ket{\phi^2\psi^1}\pm\ket{\psi^1\phi^2}\ ,\\ &\ket{\phi^1\psi^2}\pm\ket{\psi^2\phi^1}\ ,\quad
\ket{\phi^2\psi^2}\pm\ket{\psi^2\phi^2}\ ,\\
V^{{{\color{white}()}}}_{(0,2)}: \quad &\ket{\psi^2\psi^2}\ ,\quad q^{1/2}\ket{\psi^1\psi^2}+q^{-1/2}\ket{\psi^2\psi^1}\ ,\quad\ket{\psi^1\psi^1}\ ,\\
V^{(+)}_{(0,0)}: \quad& \frac{i(q^{1/2}-q^{-1/2})}{q+q^{-1}}\big(q^{-1/2}\ket{\phi^1\phi^2}- q^{1/2}\ket{\phi^2\phi^1}\big)
+q^{-1/2}\ket{\psi^1\psi^2}-q^{1/2}\ket{\psi^2\psi^1}\ ,\\
V^{(-)}_{(0,0)}: \quad& q^{-1/2}\ket{\phi^1\phi^2}- q^{1/2}\ket{\phi^2\phi^1}
-\frac{i(q^{1/2}-q^{-1/2})}{q+q^{-1}}\big(q^{-1/2}\ket{\psi^1\psi^2}-q^{1/2}\ket{\psi^2\psi^1}\big)\ .
\end{split}$$]{}
Generically, $\mE_2$ and $\mF_2$ act between the $U_q(\mg_0)$ modules according to figure \[f6\]. When $\theta_{12}=\frac{ i\pi}k$ the situation is shown in figure \[f7\] where the action along the dotted lines is in one direction only as indicated by the arrows. The subspace $V_{(0,2)}\oplus V_{(1,1)}^{(-)}\oplus V_{(0,0)}^{(-)}$ becomes an invariant subspace and forms a module for the sub-representation $\langle0,1\rangle$. When $\theta_{12}=-\frac{i\pi}k$ the arrows are reversed
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V\_[(2,0)]{}& & & V\_[(0,2)]{}\
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]{}; (m-2-1) edge (m-1-2); (m-1-2) edge (m-1-3); (m-3-2) edge (m-3-3); (m-3-3) edge (m-2-4); (m-1-2) edge (m-3-2); (m-1-3) edge (m-3-3); (m-2-1) edge (m-3-3); (m-1-2) edge (m-2-4);
The S-matrix of the bound-state $V_2$ can then be found by using the bootstrap/fusion equations . At the special rapidity difference, using , we have [$$\begin{split}
\widetilde S_{11}(\tfrac{i\pi}k)&=\frac{{\cal F}(i\pi/k)}{2(q-q^{-1})}\check R(q)\\ &=
{\cal F}\left({i\pi}/k\right)\left[
\frac{q+1}{2q^{1/2}}{\mathbb P}_{(2,0)}+{\mathbb P}_{(1,1)}^{(+)}+
\frac{q^4-q^3+4q^2-q+1}{2q^{3/2}(q+1)}{\mathbb P}_{(0,0)}^{(+)}\right]\ .
\label{res}
\end{split}$$]{} which is non-vanishing on the factor representation $\langle1,0\rangle$ and vanishing on the sub-representation $\langle0,1\rangle$, as required for the consistency of the bootstrap . Notice, that the S-matrix residue is not a projector rather it is a weighted sum of $U_q(\mg_0)$ projectors; precisely the situation described in section \[s2\]. In the above, ${\cal F}(\frac{i\pi}k)$ is a postive real number given by the exponential in evaluated at $\theta=\tfrac{i\pi}k$ and for $k\in{\mathbb R}>0$, the $\rho_j$ above are all real numbers — in fact positive — which is a necessary condition for the unitarity of the underlying QFT.[^16]
The bootstrap/fusion equations can be used to write down the S-matrix for the scattering of $V_1$ with $V_2$ [$$\begin{split}
\widetilde S_{12}(\theta)=\left(1\otimes\widetilde S_{11}(\theta+\tfrac{i\pi}{2k})\right)\left(\widetilde S_{11}(\theta-\tfrac{i\pi}{2k})\otimes1\right)\Big|_{V_1\otimes V_2}\ ,
\label{fus1}
\end{split}$$]{} Note, that in the above, we have not shown the explicit projection factors present in since in this case the decomposition of $V_2$ into $U_q(\mg_0)$ representations is non-degenerate and the S-matrix acts diagonally: $V_1\otimes V_2^{(j)}\to V_2^{(j)}\otimes V_1$. In that case, is only non-vanishing when $j=l$ and so the $\rho_j$ factors are not needed. The projection onto $V_2$ is then only indicated implicitly.
The bootstrap programme now continues. The blueprint for the resulting theory is as follows. Particles are associated to the representations $\pi^\theta_a=\langle a-1,0\rangle$ with masses as given in . Each particle is self conjugate $a=\bar a$. The appearance of bound states is governed by the three-point couplings at rapidity angles [$$\begin{split}
u_{ab}^{a+b}=\frac{\pi(a+b)}{2k}\ ,\qquad u_{ab}^{|a-b|}=\pi-\frac{\pi|a-b|}{2k} \label{ded}
\end{split}$$]{} illustrated in figure \[f8\]. Note, here, that the second is implied by the first and the fact that the particles are self conjugate. The scalar factor $X_{ab}(\theta)$ provides simple poles on the physical strip at these rapidity differences, as well as poles corresponding to bound states in the crossed channel at $\theta=i(\pi-u_{ab}^{a+b})$ and $\theta=i(\pi-u_{ab}^{|a-b|})$.
\[line width=1.5pt,inner sep=2mm, place/.style=[circle,draw=blue!50,fill=blue!20,thick]{},proj/.style=[circle,draw=red!50,fill=red!20,thick]{}\] at (2,2) \[proj\] (p1) ; at (0.4,0.4) (i1) [$a$]{}; at (3.6,0.4) (i2) [$b$]{}; at (2,4) (i3) [$a+b$]{}; (i1) – (p1); (i2) – (p1); (i3) – (p1); at (2,0.3) [$\frac{\pi(a+b)}{2k}$]{}; at (8,2) \[proj\] (p1) ; at (6.4,0.4) (i1) [$a$]{}; at (9.6,0.4) (i2) [$b$]{}; at (8,4) (i3) [$|a-b|$]{}; (i1) – (p1); (i2) – (p1); (i3) – (p1); at (8,0.3) [$\pi-\frac{\pi|a-b|}{2k}$]{};
For instance the S-matrix element $S_{12}(\theta)$ has four simple poles. The poles at $\theta=\frac{3i\pi}{2k}$ and $\frac{i\pi}{2k}$ should correspond to particles $V_3$ and $V_1$, respectively, in the direct channel. The questions is does this mesh with the quantum group representation theory? The tensor product representation in question is [$$\begin{split}
\pi^{\theta_1\theta_2}_{12}=\langle0,0\rangle\otimes\langle1,0\rangle\ .
\end{split}$$]{} According to the representation theory of the undeformed algebra $\mg$, we expect this to be the irreducible representation $\{1,0\}$ for generic $\theta_{12}$. Now the simple pole at $\theta_{12}=\frac{3i\pi}{2k}$ occurs precisely at the first shortening condition in of the representation $\{1,0\}$ corresponding to the factor representation $\langle2,0\rangle$. At this point we can verify directly that the S-matrix $\widetilde S_{12}$ that is constructed from the fusion equations indeed is only non-vanishing on $\langle2,0\rangle$ which has $U_q(\mg_0)$ content $(3,0)\oplus(2,1)\oplus(1,0)$: [$$\begin{split}
\widetilde S_{12}(\tfrac{3i\pi}{2k})&={\cal F}\left({2i\pi}/k\right){\cal F}\left({i\pi}/k\right)
\left[
\frac{(q^2+1)(q^2+q+1)}{4q^2}\Bbb P_{(3,0)}\right.\\ &\left.+\frac{(q^2+1)(q+\sqrt q+1)}{4q^{3/2}}\Bbb P_{(2,1)}+\frac{q^3+q^{5/2}+q^{2}+q^{1/2}+1}{4q^{3/2}}
\Bbb P_{(1,0)}\right]\ .\label{wts12a}
\end{split}$$]{} This matches the fact that, from , particle 3 can be formed as a bound state of 1 and 2. The coefficients of the intertwiners in the above are all real and positive.
However, the representation $\{1,0\}$ admits another shortening condition, the second in , occurring on the physical strip at $\theta_{12}=i\pi-\frac{i\pi}{2k}$ corresponding to the representation $\langle0,0\rangle$. Once again we can verify directly that the S-matrix $\widetilde S_{12}$-matrix that is constructed from the fusion equations is only non-vanishing on $\langle0,0\rangle$ which has $U_q(\mg_0)$ content $(1,0)\oplus(0,1)$: [$$\begin{split}
\widetilde S_{12}(i\pi-\tfrac{i\pi}{2k})\propto i(q^{1/2}-q^{-1/2})\Bbb P_{(1,0)}+\big(q+q^{-1}\big)\Bbb P_{(0,1)}\label{wts12b}
\end{split}$$]{} and this matches the other three-point coupling in .
The picture for general $S_{ab}(\theta)$ is now clear. The tensor product representation [$$\begin{split}
\pi^{\theta_1\theta_2}_{ab}=\langle a-1,0\rangle\otimes\langle b-1,0\rangle
\end{split}$$]{} is, according to the decomposition for the undeformed algebra, the reducible representation [$$\begin{split}
\{a+b-2,0\}\oplus\{a+b-4\}\oplus\cdots\oplus\{|a-b|,0\}
\end{split}$$]{} the simple pole at $\theta_{12}=\frac{i\pi(a+b)}{2k}$ occurs precisely at the special point where $\{a+b-2,0\}$ becomes reducible with a factor representation $\langle a+b-1,0\rangle$. At this point, $\widetilde S_{ab}(\tfrac{i\pi(a+b)}{2k})$ is only non-vanishing on this subspace. Correspondingly, at the simple pole $\theta_{12}=i\pi-\frac{i\pi|a-b|}{2k}$ the representation $\{|a-b|,0\}$ becomes reducible with a factor representation $\langle |a-b|-1,0\rangle$. At this point, $\widetilde S_{ab}(i\pi-\tfrac{i\pi|a-b|}{2k})$ is only non-vanishing on that subspace.
Although we have not proved the above picture for arbitrary $a$ and $b$, we have checked that the S-matrix has the required projection properties for the case $a=1$ and $b=3$.
A magnon-like relativistic S-matrix {#s6.1}
-----------------------------------
The S-matrix building blocks $\widetilde S_{ab}(\theta)$ can then be put together with a scalar factor which supplies necessary poles on the physical strip. A magnon-like S-matrix is obtained by putting together two such blocks in a graded tensor product so that the symmetry algebra is a product $U_q(\mg)\times U_q(\mg)$. In addition, the central elements are identified since each factor has the same momentum or rapidity. Particles transform in product representations $\langle a-1,0\rangle\otimes_\text{gr}\langle a-1,0\rangle$. The S-matrix elements take the form[^17] [$$\begin{split}
S_{ab}(\theta)=X_{ab}(\theta)\,\widetilde S_{ab}(\theta)\otimes_\text{gr}\widetilde S_{ab}(\theta)\ ,\label{smfull}
\end{split}$$]{} where the tensor product is graded meaning that it respects boson/fermion statistics. The scalar factor $X_{ab}$ satisfies the bootstrap equations by itself and so is defined by specifying it on the basic particle $a=b=1$:[^18] [$$\begin{split}
X_{11}(\theta)=\frac{\sinh(\tfrac{\theta}{2}+\tfrac{i\pi}{2k})}{\sinh(\tfrac{\theta}{2}-\tfrac{i\pi}{2k})}\cdot
\frac{\cosh(\tfrac{\theta}{2}-\tfrac{i\pi}{2k})}{\cosh(\tfrac{\theta}{2}+\tfrac{i\pi}{2k})}\label{x11}\,.
\end{split}$$]{} Note that this supplies simple poles at $iu_{11}^2=\frac{i\pi}k$ corresponding to the direct channel bound state $\langle 1,0\rangle$, and at $i(\pi-u_{11}^2)=i\pi-\frac{i\pi}k$, corresponding to the cross channel bound state $\langle 1,0\rangle$. We remark that it is very common in the construction of integrable S-matrix theories to take a tensor product structure of the form [@Hollowood:1993fj]. The resulting S-matrix are trigonometric generalizations of the S-matrices of the principal chiral models which are obtained in the rational limit $q\to1$.
By applying the bootstrap equations it follows that $X_{ab}(\theta)$ has four simple poles at $iu_{ab}^{a+b}$, $iu_{ab}^{|a-b|}$ and their crossed positions $i(\pi-u_{ab}^{a+b})$ and $i(\pi-u_{ab}^{|a-b|})$. If we define the standard S-matrix building blocks [$$\begin{split}
[x]=\{x\}\{2k-x\}\ ,\qquad\{x\}=(x-1)(x+1)\ ,\qquad
(x)=\frac{\sinh(\frac\theta2+\frac{i\pi x}{4k})}
{\sinh(\frac\theta2-\frac{i\pi x}{4k})}\ ,
\end{split}$$]{} then [$$\begin{split}
X_{ab}(\theta)=[a+b-1][a+b-3]\cdots[|a-b|+1]\ .
\label{smb}
\end{split}$$]{}
The closure of the bootstrap
----------------------------
It is clear from the mass formula that the particle states can only exist for $a<2k$. Indeed, the bound state pole in $S_{ab}(\theta)$ at $\theta=\frac{i\pi(a+b)}{2k}$ moves off the physical strip for $a+b>2k$. So the spectrum of states must be bounded. The situation is very similar to the breather states in the sine-Gordon theory [@Zamolodchikov:1978xm]. Their masses are also given by a formula similar to with [$$\begin{split}
k=\frac{8\pi}{\beta^2}\left(1-\frac{\beta^2}{8\pi}\right)\ ,
\end{split}$$]{} where $\beta$ is the sine-Gordon coupling. In the sine-Gordon case, the breather spectrum is actually cut off at $a\sim k$. The potential bound state pole in $S_{ab}(\theta)$ for $a+b>k$, but $<2k$, so that it is still on the physical strip, is actually an anomalous threshold arising from a graph involving the soliton states of the theory and is therefore not a bound state pole.[^19]
Since the mechanism above is not available for the present theories, there are two other ways that the additional poles may be removed. The first is inspired by the S-matrix of the non-simply-laced Toda theories [@Delius:1991kt; @Corrigan:1993xh]. The theories associated to the pair of affine Lie algebras $(c_n^{(1)},d_{n+1}^{(2)})$ have a mass spectrum of the form with $H=2k$ and $a=1,2,\ldots,n$, where $n$ is the largest integer $\leq k$. The 2-point couplings are precisely those of figure \[f8\] but with $a,b,a+b$ all restricted to $\leq n$. The S-matrix elements can be written as in above with a modified building block [$$\begin{split}
\{x\}=\frac{(x-1)(x+1)}{(x+B-1)(x-B+1)}\ ,\qquad B=2(k-n)\ ,
\end{split}$$]{} where $0\leq B\leq 2$. In the Toda theory, $B$ is determined by the Toda coupling as [$$\begin{split}
B=\frac1{2\pi}\cdot\frac{\beta^2}{1+\beta^2/4\pi}\ .
\end{split}$$]{} In the modified S-matrix factor $X_{ab}(\theta)$, the simple pole of the original S-matrix at $\theta=\frac{i\pi(a+b)}{2k}$, for $a+b>k$, is now absent. There are new simple poles on the physical strip whose origin is not due to bound states but in a generalized Coleman-Thun mechanism. Notice, however, that the resulting S-matrix is not an analytic function of $k$.
Another way in which the spectrum can truncate with the additional poles on the physical strip for $a+b>k$ being removed, happens when $q$ is a root of unity. Although the representation theory of the quantum group $U_q(\mg)$ has not been developed in this case, we can infer what will happen by considering the behaviour of the bosonic subalgebra $U_q(\msl(2))\times U_q(\msl(2))$. In this case, it is well known that when $q=e^{i\pi/k}$ with $k$ a positive integer, then the series of representations of spin $\tfrac m2$ are truncated to the finite set $m=0,1,\ldots,k-2$. Since the representations $\langle a-1,0\rangle$ of $U_q(\mg)$ contain the $U_q(\mg_0)$ representations in , i.e. $(a,0)\oplus(a-1,1)\oplus(a-2,0)$, this implies that the spectrum of states only includes the finite set $a=1,2,\ldots,k$. Representations near the top of the tower with $a=k$ and $k-1$ have a modified $U_q(\mg_0)$ content. For example, in the truncated representation theory, the representation $a=k$, that is $\langle k-1,0\rangle$, consists of the $U_q(\mg_0)$ representation $(k-2,0)$ only, while for $a=k-1$ we have $\langle k-2,0\rangle=(k-2,1)\oplus(k-3,0)$. The truncated spectrum of states matches the semi-classical spectrum of soliton states in the SSSSG theories exactly [@Hollowood:2011fm; @Hollowood:2011fq]. Notice that the S-matrix theory actually only makes sense for $k>2$ in order that the fundamental representation $\langle0,0\rangle$ exists. This suggests that $k$ in the S-matrix may be shifted relative to the $k$ in the action of the WZW model by a finite amount, namely $k\to k+2$, which is a common feature of the quantization of a WZW model, where 2 is the dual Coxeter number of $\msl(2)$. However, in the perturbative computation [@Hoare:2011fj] and in the related $\mathcal{N}=(2,2)$ supersymmetric sine-Gordon theory there is no evidence of such a shift. These issues will require further analysis to reconcile.
Discussion
==========
In this paper, we have shown how a conventional relativistic factorizable S-matrix can be constructed in an algebraic setting that is continuously connected to the non-relativistic S-matrix that describes the magnons on the string world sheet in $\text{AdS}_5\times S^5$. It would be interesting to see whether there exists a consistent, but necessarily non-relativistic, S-matrix theory that interpolates between our S-matrix and the magnon S-matrix. This would provide a quantum version of the classical picture of a Hamiltonian structure that interpolates the string world-sheet theory and the relativistic semi-symmetric space sine-Gordon theory.
We have not completed the analysis of the full S-matrix and shown that all the singularities on the physical strip can be accounted for as bound-state poles or anomalous thresholds and that the bootstrap closes: this remains for future work. However, the picture we have arrived at is very compelling and involves a delicate meshing of the representation theory of the affine quantum supergroup and the bootstrap/fusion procedure of S-matrix theory.
The algebraic setting lying behind the family of S-matrix theories is rich and fascinating and at its heart seems to be a quantum deformation of a centrally extended loop superalgebra $\msl(2|2)^{(\sigma)}$. The R-matrices are also associated to other integrable systems, namely the one-dimensional Hubbard model and its deformations [@Beisert:2008tw]. In a recent paper [@Beisert:2011wq] the full R-matrix was shown to be related to what appears to be a different affine extension than our $\msl(2|2)^{(\sigma)}$, and it would be useful to gain an overview of how all the different facets of $\mpsl(2|2)\ltimes\mathbb R^3$ and its affinizations and $q$-deformations are related.
In the context of the magnon S-matrix, one notable feature has been the apparent inability to use the bootstrap procedure [@Chen:2006gq; @Roiban:2006gs; @Chen:2006gp; @Dorey:2007xn; @Dorey:2007an; @Arutyunov:2008zt]. It would clearly be interesting to re-visit this issue given that we have found that for the relativistic S-matrix the bootstrap equations seem to mesh perfectly with the peculiar representation theory of a $q$-deformed Lie superalgebra.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Arkady Tseytlin for helpful comments on a draft of this paper.
BH is supported by EPSRC.
JLM thanks LPTHE (UPMC Paris–CNRS) for the kind hospitality while this work was in progress. He also acknowledges the support of MICINN (FPA2008-01838 and FPA2008-01177), Xunta de Galicia (Consejería de Educación and INCITE09.296.035PR), the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042), and FEDER.
[**Appendix : The Co-Product**]{}
\[A\]
For the relativistic Pohlmeyer-reduced theory the interpretation of the triply extended algebra as a finite subalgebra of an affine superalgebra $\mathfrak{sl}(2|2)^{(\s)}$ (discussed in section \[s3.1\]) naturally gives the reality conditions given by . These differ from the ones that are usually used in the context of the $\text{AdS}_5 \times S^5$ string theory/${\cal N }= 4$ SYM magnon discussion .
The reality conditions are related to those in with ${\varphi_2}=\frac{\pi}{2}$ by a change in the basis of the generators. This is explicitly given by [$$\begin{aligned}
{2}
\notag &\tilde{\mR}^a{}_b = \mR^a{}_b\ , \qquad & &\tilde{\mL}^\alpha{}_\beta = \mL^\alpha{}_\beta\ ,
\\\notag&\tilde{\mQ}^\alpha{}_b = \frac{1}{\sqrt{2}}(\mQ^\alpha{}_b -
\varepsilon^{\alpha\beta}\varepsilon_{ba}\mS^a{}_\beta)\ , \qquad&
&\tilde{\mS}^a{}_\beta = \frac{1}{\sqrt{2}}(\mS^a{}_\beta +
\varepsilon^{ab}\varepsilon_{\beta\alpha}\mQ^\alpha{}_b)\ ,
\\&\tilde{\mP} = \frac{1}{2}(\mP + \mK) - \mC \ , &
&\tilde{\mK} = \frac{1}{2}(\mP + \mK) + \mC \ ,\label{cob}
\\\notag&&&\hspace{-2.5cm}\tilde{\mC} = \frac{1}{2}(\mP - \mK)\ .\end{aligned}$$]{}
Moreover, this change of basis preserves the commutation relations , , i.e. it is an element of the outer-automorphism group $SL(2,\C)$. In particular it is in the $SL(2,\R)$ subgroup that contains those automorphisms that amount to a change of the basis for the generators of the real algebra.
For convenience, in this appendix we denote the eigenvalues of the central charges as $(P,\,K,\,C)$ and $(\tilde P,\,\tilde K,\,\tilde C)$ for the two different bases. Further we will refer to the basis satisfying the hermiticity relations with ${\varphi_2}=\frac{\pi}{2}$ as basis (i) and to that satisfying as basis (ii).
For the soliton representation the central charges acting on the one-particle state have the eigenvalues [$$\begin{aligned}
{3}
\notag P=&-[\tfrac12]_q e^{-\o} \ , \qquad &
K=& [\tfrac12]_q e^\o \ , \qquad &
C=&0 \ ,
\\
\tilde P = &[\tfrac12]_q \sinh \o \ , \qquad &
\tilde K = &[\tfrac12]_q \sinh \o \ , \qquad &
\tilde C = &[\tfrac12]_q \cosh \o \ .\label{prevtilde}\end{aligned}$$]{} Of particular interest is the final equation where we see that the eigenvalue for the central charge $\tilde \mC$ is proportional to the two-dimensional energy. This is reminiscent of the magnon representation for which this third central extension is identified with the Hamiltonian.
It is interesting to note that for the relativistic representation in either basis the Lorentz symmetry can be identified with the subgroup of the real outer-automorphism group $SL(2,\R)$ that preserves the hermiticity relations. (The full subgroup of the complex outer-automorphism group $SL(2,\C)$ that preserves the hermiticity relations is $SU(1,1)$ [@Arutyunov:2009ga].)
From it is apparent that the Lorentz symmetry grading is clearer in basis (i) with the eigenvalue for the central charge $\mC$ vanishing. One may wonder if this change of basis could allow one to identify a hidden Lorentz symmetry of the magnon representation. To proceed we quote the central charge eigenvalues for the magnon representation [@Beisert:2005tm; @Beisert:2006qh; @Arutyunov:2006yd] [$$\begin{aligned}
{3}
\tilde P &= -ge^{i(\frac{p}{2}+2\xi)}\sin\frac{p}{2} \ , \qquad &
\tilde K &= -ge^{-i(\frac{p}{2}+2\xi)}\sin\frac{p}{2} \ , \qquad &
\tilde C^2 &= \frac{1}{4}+g^2\sin^2\frac{p}{2}\ ,\end{aligned}$$]{} where $p$ is the world sheet momentum, and identifying $\tilde C$ with the Hamiltonian the final equation gives the dispersion relation. $\xi$ is an arbitrary phase. The usual choice that is made is $\xi = 0$ for which the coproduct is given by with $q=1$ and $\mU = e^{i \bP}$ where $\bP$ is the world sheet momentum operator. One can choose different values for $\xi$, however the coproduct needs to be modified accordingly [@Arutyunov:2009ga].
Transforming into the alternative basis we find [$$\begin{split}
P &= -g \sin\frac{p}{2}\cos\Big(\frac{p}{2}+2\xi\Big) +
\tilde C(p) \ , \\
K &=-g \sin\frac{p}{2}\cos\Big(\frac{p}{2}+2\xi\Big)
- \tilde C(p) \ , \\
C &= ig \sin\frac{p}{2}\sin\Big(\frac{p}{2}+2\xi\Big)\ .
\end{split}$$]{} Choosing $\xi=-\frac{p}{4}$ (recall that one needs to modify the coproduct accordingly) the central charge $C$ vanishes as in the relativistic Pohlmeyer reduced theory.
In this basis, and with this choice for $\xi$, one may expect to be able to associate a Lorentz symmetry grading to the generators. Indeed for the generators acting on the single-particle states this is just [$$\begin{aligned}
{1}
P & \rightarrow e^{-\lambda} P \ ,\qquad\qquad K \rightarrow e^{\lambda} K \ , \notag
\\\sin \frac{p}{2} &\rightarrow -\frac{e^{-\lambda} P +e^{\lambda}K}{2g} = \sin \frac{p}{2}\cosh \lambda +\tilde C(p)\sinh \lambda \ .\label{gslorentz}\end{aligned}$$]{} While this transformation preserves the reality of $P$ and $K$ it clearly does not preserve the reality of the world sheet momentum $p$.
Even so we may ask if it describes some formal hidden Lorentz symmetry of the Green-Schwarz theory. However, this appears not to be case as the symmetry does not extend to the action of the symmetry on tensor product representations. In particular this is a consequence of the coproduct being constructed with the generators satisfying the reality conditions . By constructing the coproduct for the alternative basis using one can see explicitly that the symmetry identified in does not extend to higher representations.
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[^1]: There is a large literature on this subject, see for example the series of review articles [@Beisert:2010jr] and references therein.
[^2]: Here, and in the following, we use the notation $\mathbb R^2=\mathbb R\oplus\mathbb R$.
[^3]: Note that $\sigma^2$ is an inner automorphism [@Serg3; @Arutyunov:2009ga] and so in the notation of [@Kac] this algebra would be denoted $\msl(2|2)^{(2)}$. However, it differs by an inner automorphism from the twisting described in [@Gould1]. Another issue is that for the quotient algebra $\mpsl(2|2)$ the automorphism lies inside a continuous group of $SL(2,\mathbb C)$ automorphisms that are connected to the identity. However, for $\msl(2|2)$ this is no longer the case meaning that it is genuinely different from the untwisted algebra $\msl(2|2)^{(1)}$ [@Serg2]. We refer to $\msl(2|2)^{(\sigma)}$ as a centrally extended loop algebra because, since $\msl(2|2)$ is not simple (it is reductive) it is not clear whether it fits into the usual class of Kac Moody superalgebras: for instance it has an infinite number of centres that span a Heisenberg subalgebra.
[^4]: More precisely we need two copies of this algebra as in with central terms identified.
[^5]: Similar computations for bosonic models were investigated in [@Hoare:2010fb].
[^6]: Note that the representations in question are [*not*]{} unitary highest weight representations of the affine algebra, since such representations only exist when the centre of the affine algebra is a positive integer. Rather they are [*evaluation representations*]{} that lift from the finite Lie algebra $\mg$ to the loop algebra realization of $\hat\mg$.
[^7]: Here, $\dagger$ is the usual hermitian conjugation, $M^\dagger=(M^*)^t$, but with the definition that complex conjugation is anti-linear on products of Grassmann odd elements [$$\begin{split}
(\theta_1\theta_2)^*=\theta_2^*\theta_1^*\ ,
\end{split}$$]{} which guarantees that $(M_1M_2)^\dagger=M_2^\dagger M_1^\dagger$.
[^8]: In [@Hoare:2011fj] the quantum deformation parameter was taken to be related to $k$ as $q=e^{-i\pi/k}$. This amounts to choosing the bosonic states of the factorized S-matrix to originate from the $\text{AdS}_5$ sector and the bound states to transform in the short representations $\langle 0, a \rangle$ (see section \[s4\]). Here, to mirror the construction of the bound states in the superstring theory, we take the bosonic states of the factorized S-matrix to originate from the $S^5$ sector. Then, the bound states transform in the short representations $\langle a, 0 \rangle$, and correspondingly $q=e^{i\pi/k}$.\[footnote1\]
[^9]: As $\gamma$ can be understood as parametrising the normalisation of the fermionic states relative to the bosonic states the phase $e^{i\varphi_1}$ is not be physical as it can always be incorporated into the definition of the states. In the latter sections of this paper, we will take ${\varphi_1}=0$, whereas in [@Hoare:2011fj] it was taken to be equal to $\frac{\pi}{4}$.
[^10]: In the later sections of this paper we take $\varphi_2 = 0$, whereas in [@Hoare:2011fj] it was taken to be $-\frac{\pi}{2}$.
[^11]: Note that in previous disucssions [@Beisert:2006qh; @Arutyunov:2006yd; @Beisert:2008tw] this minus sign was put in by hand and $\mU$ had the same eigenvalue on all states. It seems particularly nice that the minus sign can be incorporated into the definition of $\mU$.
[^12]: That is, this is the decomposition into representations of the zero graded part of the algebra $\mg_0=\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$.
[^13]: Our functions are those of [@Beisert:2008tw] multiplied by $(x-q)(x+1)/(q^{1/2}x)$ in order to ensure that $\check R(x)$ has no poles. Similarly compared to those in [@Hoare:2011fj] we have multiplied the functions by $x-x^{-1}$ and rescaled the fermionic states by a factor of $e^{-i\pi/4}$. This is related to the choice of $\gamma$ (which controls the normalisation of fermions relative to bosons) which differs from that of [@Hoare:2011fj] by precisely this factor. Also recall that in [@Hoare:2011fj] $q$ was taken to be related to $k$ as $q=e^{-i\pi/k}$, see footnote \[footnote1\].
[^14]: Note that $\sigma\cdot\check R(-x^{-1})\in\text{End}\,(V_1\otimes V_1)$ so the transpose is well defined. In terms of explicit indices $(A^{st})_{ab}=A_{ba}$, $(A^{st})_{a\alpha}=-A_{\alpha a}$, $(A^{st})_{\alpha a}=A_{a\alpha}$ and $(A^{st})_{\alpha\beta}=A_{\beta\alpha}$.
[^15]: The choice of the singlets is made so that the invariant subspaces that we define in due course are simpler.
[^16]: A more in-depth analysis will be needed to constrain the signs the residues.
[^17]: One can check that the graded tensor product structure does not upset the Yang-Baxter Equation that is satisfied by each factor separately.
[^18]: This choice for the scalar factor along with the phase factor implies the full S-matrix for the scattering of two basic particles agrees with that constructed in [@Hoare:2011fj] up to the sign in $k$. The sign in $k$ amounts to the opposite choice of which states are bosonic and fermionic for the factorised S-matrix.
[^19]: A careful explanation of this appears in [@Dorey:1996gd].
|
---
abstract: |
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the *Steiner distance* $d_G(S)$ among the vertices of $S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n,k$ be two integers with $2\leq k\leq n$. Then the *Steiner $k$-eccentricity $e_k(v)$* of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S
\}$. Furthermore, the *Steiner $k$-diameter* of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in V(G)\}$. In 2011, Chartrand, Okamoto and Zhang showed that $k-1\leq sdiam_k(G)\leq n-1$. In this paper, graphs with $sdiam_4(G)=3,4,n-1$ are characterized, respectively.\
[**Keywords:**]{} Diameter, Steiner tree, Steiner $k$-diameter\
[**AMS subject classification 2010:**]{} 05C05; 05C12; 05C75.
author:
- |
Zhao Wang$^{1,2}$, Yaping Mao$^{2,4}$[^1], Hengzhe Li$^{3}$, Chengfu Ye$^{2,4}$\
$^1$School of Mathematical Sciences, Beijing Normal\
University, Beijing 100875, China\
$^2$Department of Mathematics, Qinghai Normal\
University, Xining, Qinghai 810008, China\
$^3$School of Mathematical Sciences, Henan Normal\
University, Xinxiang 453007, China\
$^4$Center for Mathematics and Interdisciplinary Sciences\
of Qinghai Province, Xining, Qinghai 810008, China\
E-mails: wangzhao@mail.bnu.edu.cn; maoyaping@ymail.com;\
hengzhe\_li@126.com; yechf@qhnu.edu.cn
title: '**The Steiner $4$-diameter of a graph** [^2]'
---
Introduction
============
All graphs in this paper are undirected, finite and simple. We refer to [@Bondy] for graph theoretical notation and terminology not described here. For a graph $G$, let $V(G)$, $E(G)$, $e(G)$, $\delta(G)$, and $\overline{G}$ denote the set of vertices, the set of edges, the size, minimum degree, and the complement of $G$, respectively. In this paper, we let $K_{n}$, $P_n$, $K_{1,n-1}$ and $C_n$ be the complete graph of order $n$, the path of order $n$, the star of order $n$, and the cycle of order $n$, respectively. For any subset $X$ of $V(G)$, let $G[X]$ denote the subgraph induced by $X$; similarly, for any subset $F$ of $E(G)$, let $G[F]$ denote the subgraph induced by $F$. We use $G\setminus X$ to denote the subgraph of $G$ obtained by removing all the vertices of $X$ together with the edges incident with them from $G$; similarly, we use $G\setminus F$ to denote the subgraph of $G$ obtained by removing all the edges of $F$ from $G$. If $X=\{v\}$ and $F=\{e\}$, we simply write $G-v$ and $G\setminus e$ for $G-\{v\}$ and $G\setminus \{e\}$, respectively. For two subsets $X$ and $Y$ of $V(G)$ we denote by $E_G[X,Y]$ the set of edges of $G$ with one end in $X$ and the other end in $Y$. If $X=\{x\}$, we simply write $E_G[x,Y]$ for $E_G[\{x\},Y]$. We divide our introduction into the following four subsections to state the motivations and our results of this paper.
Distance and its generalizations
--------------------------------
Distance is one of the most basic concepts of graph-theoretic subjects. For a graph $G$, let $V(G)$, $E(G)$, and $e(G)$ denote the set of vertices, the set of edges, and the size of $G$, respectively. If $G$ is a connected graph and $u,v\in
V(G)$, then the *distance* $d_G(u,v)$ between $u$ and $v$ is the length of a shortest path connecting $u$ and $v$. If $v$ is a vertex of a connected graph $G$, then the *eccentricity* $e(v)$ of $v$ is defined by $e(v)=\max\{d_G(u,v)\,|\,u\in V(G)\}$. Furthermore, the *radius* $rad(G)$ and *diameter* $diam(G)$ of $G$ are defined by $rad(G)=\min\{e(v)\,|\,v\in V(G)\}$ and $diam(G)=\max
\{e(v)\,|\,v\in V(G)\}$. These last two concepts are related by the inequalities $rad(G)\leq diam(G) \leq 2 rad(G)$. The *center* $C(G)$ of a connected graph $G$ is the subgraph induced by the vertices $u$ of $G$ with $e(u)=rad(G)$. Recently, Goddard and Oellermann gave a survey paper on this subject, see [@Goddard].
The distance between two vertices $u$ and $v$ in a connected graph $G$ also equals the minimum size of a connected subgraph of $G$ containing both $u$ and $v$. This observation suggests a generalization of distance. The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian and Zou in 1989, is a natural and nice generalization of the concept of classical graph distance. For a graph $G(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, *an $S$-Steiner tree* or *a Steiner tree connecting $S$* (or simply, *an $S$-tree*) is a subgraph $T(V',E')$ of $G$ that is a tree with $S\subseteq V'$. Let $G$ be a connected graph of order at least $2$ and let $S$ be a nonempty set of vertices of $G$. Then the *Steiner distance* $d_G(S)$ among the vertices of $S$ (or simply the distance of $S$) is the minimum size among all connected subgraphs whose vertex sets contain $S$. Note that if $H$ is a connected subgraph of $G$ such that $S\subseteq V(H)$ and $|E(H)|=d_G(S)$, then $H$ is a tree. Observe that $d_G(S)=\min\{e(T)\,|\,S\subseteq V(T)\}$, where $T$ is subtree of $G$. Furthermore, if $S=\{u,v\}$, then $d_G(S)=d(u,v)$ is the classical distance between $u$ and $v$. Set $d_G(S)=\infty$ when there is no $S$-Steiner tree in $G$.
Let $n$ and $k$ be two integers with $2\leq k\leq n$. The *Steiner $k$-eccentricity $e_k(v)$* of a vertex $v$ of $G$ is defined by $e_k(v)=\max \{d(S)\,|\,S\subseteq V(G), |S|=k,~and~v\in
S \}$. The *Steiner $k$-radius* of $G$ is $srad_k(G)=\min \{
e_k(v)\,|\,v\in V(G)\}$, while the *Steiner $k$-diameter* of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\,v\in V(G)\}$. Note for every connected graph $G$ that $e_2(v)=e(v)$ for all vertices $v$ of $G$ and that $srad_2(G)=rad(G)$ and $sdiam_2(G)=diam(G)$. Each vertex of the graph $G$ of Figure 1 $(c)$ is labeled with its Steiner $3$-eccentricity, so that $srad_3(G)=4$ and $sdiam_3(G)=6$.
\[obs2\] Let $k,n$ be two integers with $2\leq k\leq n$.
$(1)$ If $H$ is a spanning subgraph of $G$, then $sdiam_k(G)\leq
sdiam_k(H)$.
$(2)$ For a connected graph $G$, $sdiam_k(G)\leq sdiam_{k+1}(G)$.
In [@ChartrandOZ], Chartrand, Okamoto, Zhang obtained the following result.
[[@ChartrandOZ]]{}\[th1\] Let $k,n$ be two integers with $2\leq k\leq n$, and let $G$ be a connected graph of order $n$. Then $k-1\leq sdiam_k(G)\leq n-1$. Moreover, the upper and lower bounds are sharp.
In [@DankelmannSO2], Dankelmann, Swart and Oellermann obtained a bound on $sdiam_k(G)$ for a graph $G$ in terms of the order of $G$ and the minimum degree $\delta$ of $G$, that is, $sdiam_k(G)\leq
\frac{3n}{\delta+1}+3k$. Later, Ali, Dankelmann, Mukwembi [@AliDM] improved the bound of $sdiam_k(G)$ and showed that $sdiam_k(G)\leq \frac{3n}{\delta+1}+2k-5$ for all connected graphs $G$. Moreover, they constructed graphs to show that the bounds are asymptotically best possible.
As a generalization of the center of a graph, the *Steiner $k$-center* $C_k(G)\ (k\geq 2)$ of a connected graph $G$ is the subgraph induced by the vertices $v$ of $G$ with $e_k(v)=srad_k(G)$. Oellermann and Tian [@OellermannT] showed that every graph is the $k$-center of some graph. In particular, they showed that the $k$-center of a tree is a tree and those trees that are $k$-centers of trees are characterized. The *Steiner $k$-median* of $G$ is the subgraph of $G$ induced by the vertices of $G$ of minimum Steiner $k$-distance. For Steiner centers and Steiner medians, we refer to [@Oellermann; @Oellermann2; @OellermannT].
The *average Steiner distance* $\mu_k(G)$ of a graph $G$, introduced by Dankelmann, Oellermann and Swart in [@DankelmannOS], is defined as the average of the Steiner distances of all $k$-subsets of $V(G)$, i.e. $$\mu_k(G)={n\choose k}^{-1}\sum_{S\subseteq V(G),|S|=k}d_G(S).$$ For more details on average Steiner distance, we refer to [@DankelmannOS; @DankelmannSO].
Let $G$ be a $k$-connected graph and $u$, $v$ be any pair of vertices of $G$. Let $P_k(u,v)$ be a family of $k$ inner vertex-disjoint paths between $u$ and $v$, i.e., $P_k(u,v)=\{P_1,P_2,\cdots,P_k\}$, where $p_1\leq p_2\leq \cdots \leq p_k$ and $p_i$ denotes the number of edges of path $P_i$. The *$k$-distance* $d_k(u,v)$ between vertices $u$ and $v$ is the minimum $p_k$ among all $P_k(u,v)$ and the *$k$-diameter* $d_k(G)$ of $G$ is defined as the maximum $k$-distance $d_k(u,v)$ over all pairs $u,v$ of vertices of $G$. The concept of $k$-diameter emerges rather naturally when one looks at the performance of routing algorithms. Its applications to network routing in distributed and parallel processing are studied and discussed by various authors including Chung [@Chung], Du, Lyuu and Hsu [@Du], Hsu [@Hsu; @Hsu2], Meyer and Pradhan [@Meyer].
Application background of Steiner distance
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Let $G$ be a $k$-connected graph and $u$, $v$ be any pair of vertices of $G$. Let $P_k(u,v)$ be a family of $k$ internally vertex-disjoint paths between $u$ and $v$, i.e. $P_k(u,v)=\{P_{p_1},P_{p_2},\cdots,P_{p_k}\}$, where $p_1\leq p_2\leq \cdots \leq p_k$ and $p_i$ denotes the number of edges of path $P_{p_i}$. The *$k$-distance* $d_k(u,v)$ between vertices $u$ and $v$ is the minimum $|p_k|$ among all $P_k(u,v)$ and the *$k$-diameter* $d_k(G)$ of $G$ is defined as the maximum $k$-distance $d_k(u,v)$ over all pairs $u,v$ of vertices of $G$. The concept of $k$-diameter emerges rather naturally when one looks at the performance of routing algorithms. Its applications to network routing in distributed and parallel processing are studied and discussed by various authors including Chung [@Chung], Du, Lyuu and Hsu [@Du], Hsu [@Hsu; @Hsu2], Meyer and Pradhan [@Meyer].
The Wiener index $W(G)$ of the graph $G$ is defined as $W(G)=\sum_{\{u,v\} \subseteq V(G)} d_G(u,v)$. Details on this oldest distance–based topological index can be found in numerous surveys, e.g., in [@Rouv1; @Rouv2; @Dobrynin; @Xu]. Li et al. [@LMG] put forward a Steiner–distance–based generalization of the Wiener index concept. According to [@LMG], the [*$k$-center Steiner Wiener index*]{} $SW_k(G)$ of the graph $G$ is defined by $$\label{sw}
SW_k(G)=\sum_{\overset{S\subseteq V(G)}{|S|=k}} d(S)\,.$$ For $k=2$, the above defined Steiner Wiener index coincides with the ordinary Wiener index. It is usual to consider $SW_k$ for $2 \leq k \leq n-1$, but the above definition would be applicable also in the cases $k=1$ and $k=n$, implying $SW_1(G)=0$ and $SW_n(G)=n-1$. A chemical application of $SW_k$ was recently reported in [@GFL]. Gutman [@GutmanSDD] offered an analogous generalization of the concept of degree distance. Later, Furtula, Gutman, and Katanić [@FurtulaGK] introduced the concept of Steiner Harary index and gave its chemical applications. For more details on Steiner distance indices, we refer to [@FurtulaGK; @GFL; @GutmanSDD; @LMG; @LMG2; @MWG; @MWGK; @MWGL].
Our results
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From Theorem \[th1\], we have $k-1\leq sdiam_k(G)\leq n-1$. In [@Mao], Mao characterized the graphs with $sdiam_3(G)=2,3,n-1$, respectively, and studied the Nordhaus-Gaddum-type problem of the parameter $sdiam_k(G)$.
In this paper, graphs with $sdiam_4(G)=3,4,n-1$ are characterized, respectively.
\[th2\] Let $G$ be a connected graph of order $n \ (n\geq 4)$.
$(i)$ If $n=4$, then $sdiam_4(G)=3$;
$(ii)$ If $n\geq 5$, then $sdiam_4(G)=3$ if and only if $n-3\leq\delta(G)\leq n-1$ and $C_4$ is not a subgraph of $\overline{G}$.
A graph $H_1$ is defined as a connected graph of order $n \ (n\geq 5)$ obtained from a $K_4$ with vertex set $\{u_1,u_2,u_3,u_4\}$ and four stars $K_{1,a}, K_{1,b}, K_{1,c}, K_{1,d}$ by identifying the center of one star and one vertex in $\{u_1,u_2,u_3,u_4\}$, where $0\leq a\leq
b\leq c\leq d$, $d\geq 1$, and $a+b+c+d=n-4$; see Figure \[fig1\].
Figure 1: Graphs for Theorem \[th3\].\[fig1\]
A graph $H_2$ is defined as a connected graph of order $n \ (n\geq 5)$ obtained from $K_4-e$ with vertex set $\{u_1,u_2,u_3,u_4\}$, $e=u_1u_4$ and two stars $K_{1,a}$, $K_{1,b}$ by identifying the center of a star and one vertex in $\{u_2,u_3\}$, and then adding the paths $u_1z_iu_4 \ (1\leq i\leq c)$, where $0\leq a\leq
b$, $b\geq 0$, $c\geq 0$ and $a+b+c=n-4$; see Figure \[fig2\].
A graph $H_3$ is defined as a connected graph of order $n \ (n\geq 5)$ obtained from a cycle $C_4=u_1u_2u_3u_4u_1$ by adding the paths $u_1x_iu_2 \ (1\leq i\leq a)$ and the paths $u_3y_ju_4 \ (1\leq j\leq b)$, where $0\leq a\leq
b$, $b\geq 1$ and $a+b=n-4$; see Figure \[fig2\].
A graph $H_4$ is defined as a connected graph of order $n \ (n\geq 5)$ obtained from a star $K_{1,3}$ with vertex set $\{u_1,u_2,u_3,u_4\}$ and a star $K_{1,a}$ by identifying $u_3$ and the center of $K_{1,a}$, where $u_3$ is the center of $K_{1,3}$, and then adding the vertices $y_i$ and the edges $y_iu_j \ (1\leq i\leq b, \ j=1,2,4)$, where $0\leq a\leq
b$, $b\geq 1$ and $a+b=n-4$; see Figure \[fig2\].
\[th3\] Let $G$ be a connected graph of order $n \ (n\geq 5)$. Then $sdiam_4(G)=4$ if and only if $G$ satisfies one of the following conditions.
$(i)$ $\delta(G)=n-3$ and $C_4$ is a subgraph of $\overline{G}$;
$(ii)$ $\delta(G)\leq n-4$ and each $H_i\ (1\leq i\leq 4)$ is not a spanning subgraph of $\overline{G}$ (see Figure \[fig1\]).
We now define some graph classes.
- Let $T_{a,b,c,d} \ (0\leq a,b,c,d\leq n-1, a+b+c+d\leq n-1)$ be a tree of order $n \ (n\geq 5)$ obtained from three paths $P_1,P_2,P_3$ of length $n-b-c-1,b,c$ respectively by identifying the $(a+1)$-th vertex of $P_1$ and one endvertex of $P_2$, and then identifying the $(n-b-c-d)$-th vertex of $P_1$ and one endvertex of $P_3$ (Note that $u$ and $v$ can be the same vertex);
- Let $\triangle_{a,b,c,d} \ (0\leq a,b,c,d\leq n-2, a+b+c+d\leq n-2)$ be an unicyclic graph of order $n \ (n\geq 5)$ obtained from three paths $P_1,P_2,P_3$ of length $n-b-c-1,b+1,c$ respectively by identifying the $(a+1)$-th vertex of $P_1$ and one endvertex of $P_2$, and then identifying the $(n-b-c-d)$-th vertex of $P_1$ and one endvertex of $P_3$, and then adding an edge $u_{b+1}v_{a+2}$ (Note that $v_{a+2}$ and $v$ can be the same vertex).
- Let $\triangle_{a,b,c,d}' \ (0\leq a,b,c,d\leq n-3, \ a+b+c+d\leq n-3)$ be an bicyclic graph of order $n \ (n\geq 5)$ obtained from three paths $P_1,P_2,P_3$ of length $n-b-c-1,b+1,c+1$ respectively by identifying the $(a+1)$-th vertex of $P_1$ and one endvertex of $P_2$, and then identifying the $(n-b-c-d)$-th vertex of $P_1$ and one endvertex of $P_3$, and then adding two edges $u_{b+1}v_{a+2}$ and $w_{c+1}x_{d+2}$ (Note that $v_{a+2}$ and $v$ can be the same vertex).
- Let $G_2$ be a graph of order $n \ (n\geq 5)$ obtained from a cycle of order $4$ and four paths $P_1,P_2,P_3,P_4$ of length $a,b,c,d \ (0\leq a,b,c,d\leq n-4, \ a+b+c+d=n-4)$ respectively by identifying each vertex of this cycle with an endvertex of one of the four paths.
![Graphs for Theorem \[th4\].](2.eps "fig:")\
\[fig2\]
- Let $G_3$ be a graph of order $n \ (n\geq 5)$ obtained from $K_4^-$ and four paths $P_1,P_2,P_3,P_4$ of length $a,b,c,d \ (0\leq a,b,c,d\leq n-4, \ a+b+c+d=n-4)$ respectively by identifying each vertex of $K_4^-$ with an endvertex of one of the four paths, where $K_4^-$ denotes the graph obtained from a clique of order $4$ by deleting one edge.
\[th4\] Let $G$ be a connected graph of order $n \ (n\geq 5)$. Then $sdiam_4(G)=n-1$ if and only if $G=T_{a,b,c,d}$ or $G=\triangle_{a,b,c,d}$ or $G=\triangle^\prime_{a,b,c,d}$ or $G=G_1$ or $G=G_2$ or $G=G_3$.
Proofs of Theorem $2$ and $3$
=============================
In this section, we characterize graphs with $sdiam_4(G)=3,4$ and give the proofs of Theorems $2$ and $3$.
\[lem1\] Let $G$ be a connected graph of order $n$, and let $k$ be an integer with $3\leq k\leq n-1$. Then $sdiam_k(G)=n-1$ if and only if the number of non-cut vertices in $G$ is at most $k$.
Let $r$ be the number of non-cut vertices in $G$. Suppose $sdiam_k(G)=n-1$. We claim that $r\leq k$. Assume, to the contrary, that $r\geq k+1$. For any $S\subseteq V(G)$ with $|S|=k$, there exists a non-cut vertex in $G$, say $u$, such that $u\in
V(G)\setminus S$. Then $G\setminus u$ is connected, and hence $G\setminus u$ contains a spanning tree of size $n-2$. From the arbitrariness of $S$, we have $sdiam_{k}(G)\leq d_{T}(S)\leq n-2$, a contradiction. So $r\leq k$, as desired.
Conversely, we suppose $r\leq k$. Let $v_1,v_2,\cdots,v_r$ be all the non-cut vertices in $G$. Then the remaining vertices are all cut vertices of $G$. Choose $v_{i_1},v_{i_2},\cdots,v_{i_{k-r}}\in V(G)\setminus
\{v_1,v_2,\cdots,v_r\}$. Set $S=\{v_1,v_2,\cdots,v_r,v_{i_1},v_{i_2},\cdots,v_{i_{k-r}}\}$. Note that each vertex in $V(G)\setminus S$ is a cut vertex of $G$. Therefore, any $S$-Steiner tree $T$ occupies all the vertices of $G$, and hence $sdiam_k(G)\geq d_G(S)\geq n-1$. From Theorem \[th1\], we have $sdiam_k(G)=n-1$, as desired.
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The following corollary is immediate from the above lemma.
\[cor1\] Let $G$ be a connected graph of order $n$, and let $k$ be an integer with $3\leq k\leq n-2$. Then $sdiam_k(G)\leq n-2$ if and only if the number of non-cut vertices in $G$ is at least $k+1$.
Mao [@Mao] obtained the following result, which will be used later.
[[@Mao]]{}\[lem2\] Let $n,k$ be two integers with $2\leq k\leq n$, and let $G$ be a connected graph of order $n$. If $sdiam_k(G)=k-1$, then $0\leq
\Delta(\overline{G})\leq k-2$, namely, $n-k+1\leq \delta(G)\leq
n-1$.
**Proof of Theorem \[th2\]:** If $n=4$, then $sdiam_4(G)=3$. So we assume that $n\geq 5$. Suppose $sdiam_4(G)=3$. For Lemma \[lem2\], if $sdiam_4(G)=3$, then $n-3\leq\delta(G)\leq n-1$. We claim that $C_4$ is not a subgraph of $\overline{G}$. Assume, to the contrary, that $C_4$ is a subgraph of $\overline{G}$. Choose $S=V(C_4)$. Since $G[S]$ is not connected, it follows that any $S$-Steiner tree must contain one vertex in $V(G)\setminus S$, and hence $sdiam_4(G)\geq d_G(S)\geq 4$, a contradiction. So $C_4$ is not a subgraph of $\overline{G}$.
Conversely, we suppose that $n-3\leq\delta(G)\leq n-1$ and $C_4$ is not a subgraph of $\overline{G}$. Since $n-3\leq\delta(G)\leq n-1$, it follows that $G$ is a graph obtained from the complete graph of order $n$ by deleting some independent paths and cycles. For any $S\subseteq V(G)$, since $C_4$ is not a subgraph of $\overline{G}$, it follows that $\overline{G}[S]=4K_1$ or $\overline{G}[S]=K_2\cup 2K_1$ or $\overline{G}[S]=2K_2$ or $\overline{G}[S]=P_4$ or $\overline{G}[S]=K_3\cup K_1$ or $\overline{G}[S]=P_3\cup K_1$. Then $G[S]=K_4$ or $G[S]=K_4\setminus e$ or $G[S]=C_4$ or $G[S]=P_4$ or $G[S]=K_{1,3}$ or $G[S]=K_{1,3}^{+}$, where $K_{1,3}^{+}$ is the graph obtained from a star $K_{1,3}$ by adding an edge. Since $G[S]$ is a connected graph, it follows that $d_G(S)\leq 3$. From the arbitrariness of $S$, we have $sdiam_4(G)\leq 3$ and hence $sdiam_4(G)=3$ by Theorem \[th1\]. The proof is complete.
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**Proof of Theorem \[th3\]:** Suppose that $G$ is a graph with $sdiam_4(G)=4$. From Theorem \[th2\], we have $\delta(G)=n-3$ and $C_4$ is a subgraph of $\overline{G}$, or $\delta(G)\leq n-4$. For the former, we have $\delta(G)=n-3$ and $C_4$ is a subgraph of $\overline{G}$, as desired. Suppose $\delta(G)\leq n-4$. It suffices to prove that each $H_i\ (1\leq i\leq 4)$ is not a spanning subgraph of $\overline{G}$, and we have the following claims.
**Claim 1.** $H_1$ is not a spanning subgraph of $\overline{G}$.
**Proof of Claim 1.** Assume, to the contrary, that $H_1$ is a spanning subgraph of $\overline{G}$. Choose $S=\{u_1,u_2,u_3,u_4\}\subseteq V(H_1)=V(\overline{G})$. Then the subgraph in $\overline{G}$ induced by the vertices in $S$ is a complete graph of order $4$, and hence $G[S]=4K_1$ is not connected. Therefore, any $S$-Steiner tree $T$ must occupy a vertex in $V(G)\setminus S$, say $x$. Because $H_1$ is a spanning subgraph of $\overline{G}$, we have $xu_1\notin
E(G)$ or $xu_2\notin E(G)$ or $xu_3\notin E(G)$ or $xu_4\notin
E(G)$. Thus, the $S$-Steiner tree $T$ must occupy another vertex in $V(G)\setminus S$, and hence the tree $T$ must occupy at least two vertices in $V(G)\setminus S$. Then $d_G(S)\geq 5$, and hence $sdiam_4(G)\geq 5$, a contradiction. So $H_1$ is not a spanning subgraph of $\overline{G}$, as desired.
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**Claim 2.** $H_2$ is not a spanning subgraph of $\overline{G}$.
**Proof of Claim 2.** Assume, to the contrary, that $H_2$ is a spanning subgraph of $\overline{G}$. Choose $S=\{u_1,u_2,u_3,u_4\}\subseteq V(H_2)=V(\overline{G})$. Since $G[S]$ is not connected, it follows that any $S$-Steiner tree $T$ must occupy a vertex in $V(G)\setminus
S$, say $x$. From the structure of $H_2$, since $H_2$ is a spanning subgraph of $\overline{G}$, we have $xu_1,xu_4\in
E(\overline{G})$ or $xu_2\in E(\overline{G})$ or $xu_3\in
E(\overline{G})$. If $xu_1,xu_4\in E(\overline{G})$, then there are at most three edges in $\{xu_2,xu_3,u_1u_4\}$ belonging to $G[S\cup
\{x\}]$. In order to connect to $u_1$ or $u_4$, the $S$-Steiner tree $T$ uses at least two vertex of $V(G)\setminus S$. If $xu_2\in
E(\overline{G})$, then there are at most three edges in $\{xu_1,xu_3,xu_4,u_1u_4\}$ belonging to $G[S\cup \{x\}]$. In order to connect to $u_2$, the $S$-Steiner tree $T$ must use at least two vertex of $V(G)\setminus S$. The same is true for $xu_3\in
E(\overline{G})$. Therefore, $e(T)\geq 5$ and $d_G(S)\geq 5$, which results in $sdiam_4(G)\geq 5$, a contradiction. So $H_2$ is not a spanning subgraph of $\overline{G}$.
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**Claim 3.** $H_3$ is not a spanning subgraph of $\overline{G}$.
**Proof of Claim 3.** Assume, to the contrary, that $H_3$ is a spanning subgraph of $\overline{G}$. Choose $S=\{u_1,u_2,u_3,u_4\}\subseteq V(H_3)=V(\overline{G})$. Since $G[S]=2K_2$ is not connected, it follows that any $S$-Steiner tree $T$ must occupy a vertex in $V(G)\setminus S$, say $x$. From the structure of $H_3$, since $H_3$ is a spanning subgraph of $\overline{G}$, we have $xu_1,xu_2\in
E(\overline{G})$ or $xu_3,xu_4\in E(\overline{G})$. If $xu_1,xu_2\in
E(\overline{G})$, then there are at most four edges in $\{xu_3,xu_4,u_1u_2,u_3u_4\}$ belonging to $G[S\cup \{x\}]$. In order to connect to $u_1$ or $u_2$, the $S$-Steiner tree $T$ uses at least two vertex of $V(G)\setminus S$. If $xu_3,xu_4\in
E(\overline{G})$, then there are at most four edges in $\{xu_1,xu_2,u_1u_2,u_3u_4\}$ belonging to $G[S\cup \{x\}]$. In order to connect to $u_3$ or $u_4$, the $S$-Steiner tree $T$ must use at least two vertex of $V(G)\setminus S$. Therefore, $e(T)\geq 5$ and $d_G(S)\geq 5$, which results in $sdiam_4(G)\geq 5$, a contradiction. So $H_3$ is not a spanning subgraph of $\overline{G}$.
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**Claim 4.** $H_4$ is not a spanning subgraph of $\overline{G}$.
**Proof of Claim 4.** Assume, to the contrary, that $H_4$ is a spanning subgraph of $\overline{G}$. Choose $S=\{u_1,u_2,u_3,u_4\}\subseteq V(H_4)=V(\overline{G})$. Since $G[S]=K_3\cup K_1$ is not connected, it follows that any $S$-Steiner tree $T$ must occupy a vertex in $V(G)\setminus S$, say $x$. From the structure of $H_4$, since $H_4$ is a spanning subgraph of $\overline{G}$, we have $xu_3\in
E(\overline{G})$ or $xu_1,xu_2,xu_4\in E(\overline{G})$. If $xu_3\in
E(\overline{G})$, then there are at most six edges in $\{xu_1,xu_2,xu_4,u_1u_2,u_1u_4,u_2u_4\}$ belonging to $G[S\cup
\{x\}]$. In order to connect to $u_3$, the $S$-Steiner tree $T$ uses at least two vertex of $V(G)\setminus S$. If $xu_1,xu_2,xu_4\in
E(\overline{G})$, then there are at most four edges in $\{xu_3,u_1u_2,u_1u_4,u_2u_4\}$ belonging to $G[S\cup \{x\}]$. In order to connect to $u_1$ or $u_2$ or $u_4$, the $S$-Steiner tree $T$ uses at least two vertex of $V(G)\setminus S$. Therefore, $e(T)\geq 5$ and $d_G(S)\geq 5$, which results in $sdiam_4(G)\geq
5$, a contradiction. So $H_4$ is not a spanning subgraph of $\overline{G}$.
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From the above argument, we know that the result holds.
Conversely, suppose that $G$ is a connected graph satisfying one of the following conditions.
$\bullet$ $\delta(G)=n-3$ and $C_4$ is a subgraph of $\overline{G}$;
$\bullet$ $\delta(G)\leq n-4$ and $H_i \ (1\leq
i\leq 4)$ is not a spanning subgraph of $\overline{G}$.
Suppose that $\delta(G)=n-3$ and $C_4$ is a subgraph of $\overline{G}$. Since $\delta(G)=n-3$, it follows that $G$ is a graph obtained from the complete graph of order $n$ by deleting some pairwise independent paths and cycles. Then $\overline{G}$ is a union of pairwise independent paths, cycles, and isolated vertices. For any $S=\{u,v,w,z\}\subseteq V(G)$, since $\overline{G}$ contains $C_4$ as its subgraph, it follows that $\overline{G}[S]=C_4$ or $\overline{G}[S]=4K_1$ or $\overline{G}[S]=K_2\cup 2K_1$ or $\overline{G}[S]=2K_2$ or $\overline{G}[S]=P_4$ or $\overline{G}[S]=K_3\cup K_1$ or $\overline{G}[S]=P_3\cup K_1$. Then $G[S]=2K_2$ or $G[S]=K_4$ or $G[S]=K_4\setminus e$ or $G[S]=C_4$ or $G[S]=P_4$ or $G[S]=K_{1,3}$ or $G[S]=K_{1,3}^{+}$, where $K_{1,3}^{+}$ is the graph obtained from $K_{1,3}$ by adding an edge. If $G[S]=2K_2$, then $|E_G[x,S]|=4$ for any $x$ in $V(G)\setminus S$, since $\delta(G)=n-3$. Thus, we have $d_G(S)=4$. For the other cases, $G[S]$ is connected, and so $d_G(S)=3$. From the arbitrariness of $S$, we have $sdiam_4(G)=4$, as desired.
Suppose $\delta(G)\leq n-4$ and each $H_i \ (1\leq
i\leq 4)$ is not a spanning subgraph of $\overline{G}$. For any $S\subseteq V(G)$ and $|S|=4$, if there exists a vertex $x\in V(G)\setminus S$ such that $|E(\overline{G}[x,S])|=0$, then $|E(G[x,S])|=4$, and hence the tree $T$ induced by the four edges in $E(G[x,S])$ is an $S$-Steiner tree in $G$, and hence $d_G(S)\leq 4$, as desired. From now on, we assume for any $S\subseteq V(G)$ and $|S|=4$, and any $x\in V(G)\setminus S$, $|E(\overline{G}[x,S])|\geq
1$.
From the definition of $sdiam_4(G)$ and Theorem \[th2\], it suffices to show that $d_G(S)\leq 4$ for any set $S\subseteq V(G)$ and $|S|=4$. It is clear that $0\leq
|E(G[S])|\leq 6$. If $4\leq |E(G[S])|\leq 6$, then $G[S]$ is connected, and hence $G[S]$ contains a spanning tree, which is an $S$-Steiner tree in $G$. So $d_G(S)=3<4$, as desired. From now on, we assume $0\leq |E(G[S])|\leq 3$.
If $|E(G[S])|=0$, then $\overline{G}[S]=K_4$. Since $|E(\overline{G}[x,S])|\geq 1$ for any $x\in V(G)\setminus S$, it follows that $H_1$ is a spanning subgraph of $\overline{G}$, a contradiction.
Suppose $|E(G[S])|=1$. Set $S=\{u_1,u_2,u_3,u_4\}$. Without loss of generality, let $u_1u_4\in E(G)$ and $u_1u_2,u_1u_3,u_2u_3,u_2u_4,u_3u_4\notin E(G)$. Then $u_1u_2,u_1u_3,u_2u_3,u_2u_4,u_3u_4\in E(\overline{G})$, and hence $\overline{G}[S]$ is a graph obtained from $K_4$ by deleting one edge. Since $H_2$ is not a spanning subgraph of $\overline{G}$, it follows that there exists a vertex $x\in
V(G)-S$ such that $xu_1\in E(\overline{G})$ but $xu_2,xu_3,xu_4\notin E(\overline{G})$ or $xu_4\in E(\overline{G})$ but $xu_2,xu_3,xu_1\notin E(\overline{G})$. By symmetry, we only to consider the former case. Clearly, $xu_2,xu_3,xu_4\in E(G)$. Combining this with $u_1u_4\in E(G)$, the tree $T$ induced by the edges in $\{u_1u_4,xu_2,xu_3,xu_4\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$, as desired.
Suppose $|E(G[S])|=2$. Without loss of generality, we can assume that $u_1u_2,u_3u_4\in E(G)$ or $u_1u_2,u_1u_4\in E(G)$. First, we consider the case $u_1u_2,u_3u_4\in E(G)$. Clearly, $u_1u_3,u_1u_4,u_2u_3,u_2u_4\notin E(G)$, and hence $u_1u_3,u_1u_4,u_2u_3,u_2u_4\in E(\overline{G})$. Note that for any $S\subseteq V(G)$ and $|S|=4$, and any $x\in V(G)\setminus S$, $|E(\overline{G}[x,S])|\geq
1$. Since $H_3$ is not a spanning subgraph of $\overline{G}$, it follows that there exists a vertex $x\in V(G)\setminus S$ satisfying one of the following.
- \(1) $xu_1\in E(\overline{G})$ but $xu_2,xu_3,xu_4\notin
E(\overline{G})$;
- \(2) $xu_1,xu_4\in E(\overline{G})$ but $xu_2,xu_3\notin E(\overline{G})$;
- \(3) $xu_1,xu_3\in E(\overline{G})$ but $xu_2,xu_4\notin
E(\overline{G})$;
- \(4) $xu_2\in E(\overline{G})$ but $xu_1,xu_3,xu_4\notin E(\overline{G})$;
- \(5) $xu_2,xu_4\in E(\overline{G})$ but $xu_1,xu_3\notin E(\overline{G})$;
- \(6) $xu_2,xu_3\in
E(\overline{G})$ but $xu_1,xu_4\notin E(\overline{G})$;
- \(7) $xu_3\in E(\overline{G})$ but $xu_1,xu_2,xu_4\notin E(\overline{G})$;
- \(8) $xu_3,xu_1\in
E(\overline{G})$ but $xu_2,xu_4\notin E(\overline{G})$;
- \(9) $xu_2,xu_3\in E(\overline{G})$ but $xu_1,xu_4\notin
E(\overline{G})$;
- \(10) $xu_4\in E(\overline{G})$ but $xu_1,xu_2,xu_3\notin E(\overline{G})$;
- \(11) $xu_1,xu_4\in E(\overline{G})$ but $xu_2,xu_3\notin
E(\overline{G})$;
- \(12) $xu_2,xu_4\in E(\overline{G})$ but $xu_1,xu_3\notin E(\overline{G})$.
By symmetry, we only consider the first three cases, and other cases can be similarly proved. If $xu_1\notin E(G)$ but $xu_2,xu_3,xu_4\in E(G)$, then the tree $T$ induced by the edges in $\{xu_2,xu_4,u_1u_2,u_3u_4\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_4\notin
E(G)$ but $xu_2,xu_3\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_3u_4,xu_2,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_3\notin E(G)$ but $xu_2,xu_4\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_3u_4,xu_2,xu_4\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$, as desired.
Next, we consider the case $u_1u_2,u_1u_4\in E(G)$. Clearly, $u_1u_3,u_2u_3,u_2u_4,u_3u_4\notin E(G)$, and hence $u_1u_3,u_2u_3,u_2u_4,u_3u_4\in E(\overline{G})$. Note that for any $S\subseteq V(G)$ and $|S|=4$, and any $x\in V(G)\setminus S$, $|E(\overline{G}[x,S])|\geq
1$. Since $H_4$ is not a spanning subgraph of $\overline{G}$, it follows that there exists a vertex $x\in
V(G)\setminus S$ satisfying one of the following.
- \(1) $xu_1\in E(\overline{G})$ but $xu_2,xu_3,xu_4\notin
E(\overline{G})$;
- \(2) $xu_1,xu_4\in E(\overline{G})$ but $xu_2,xu_3\notin E(\overline{G})$;
- \(3) $xu_1,xu_2\in E(\overline{G})$ but $xu_4,xu_3\notin
E(\overline{G})$;
- \(4) $xu_2\in E(\overline{G})$ but $xu_1,xu_3,xu_4\notin E(\overline{G})$;
- \(5) $xu_2,xu_4\in E(\overline{G})$ but $xu_1,xu_3\notin
E(\overline{G})$;
- \(6) $xu_1,xu_2\in E(\overline{G})$ but $xu_4,xu_3\notin E(\overline{G})$;
- \(7) $xu_4\in E(\overline{G})$ but $xu_1,xu_2,xu_3\notin E(\overline{G})$;
- \(8) $xu_1,xu_4\in
E(\overline{G})$ but $xu_2,xu_3\notin E(\overline{G})$;
- \(9) $xu_2,xu_4\in E(\overline{G})$ but $xu_1,xu_3\notin E(\overline{G})$.
By symmetry, we only consider the first three cases. Note that $u_1u_2,u_1u_4\in E(G)$. If $xu_1\notin E(G)$ but $xu_2,xu_3,xu_4\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_2,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_4\notin
E(G)$ but $xu_2,xu_3\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_2,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_2\notin E(G)$ but $xu_4,xu_3\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_4,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$, as desired.
Suppose $|E(G[S])|=3$. Without loss of generality, let $u_1u_2,u_1u_4,u_2u_3\in E(G)$ or $u_1u_2,u_1u_4,u_2u_4\in E(G)$. If $u_1u_2,u_1u_4,u_2u_3\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,u_2u_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)=3\leq 4$, as desired. If $u_1u_2,u_1u_4,u_2u_4\in
E(G)$, then $u_1u_3,u_2u_4,u_2u_3\notin E(G)$, and hence $u_1u_3,u_2u_4,u_2u_3\in E(\overline{G})$. Since $H_4$ is not a spanning subgraph of $\overline{G}$, it follows that there exists a vertex $x\in
V(G)\setminus S$ satisfying one of the following.
- \(1) $xu_1\in E(\overline{G})$ but $xu_2,xu_3,xu_4\notin
E(\overline{G})$;
- \(2) $xu_1,xu_4\in E(\overline{G})$ but $xu_2,xu_3\notin E(\overline{G})$;
- \(3) $xu_1,xu_2\in E(\overline{G})$ but $xu_3,xu_4\notin
E(\overline{G})$;
- \(4) $xu_2\in E(\overline{G})$ but $xu_1,xu_3,xu_4\notin E(\overline{G})$;
- \(5) $xu_2,xu_4\in E(\overline{G})$ but $xu_1,xu_3\notin
E(\overline{G})$;
- \(6) $xu_1,xu_2\in E(\overline{G})$ but $xu_3,xu_4\notin E(\overline{G})$;
- \(7) $xu_4\in E(\overline{G})$ but $xu_2,xu_3,xu_1\notin E(\overline{G})$;
- \(8) $xu_1,xu_4\in
E(\overline{G})$ but $xu_2,xu_3\notin E(\overline{G})$;
- \(9) $xu_4,xu_2\in E(\overline{G})$ but $xu_3,xu_1\notin E(\overline{G})$.
By symmetry, we only consider the first three cases. Recall that $u_1u_2,u_1u_4,u_2u_4\in
E(G)$. If $xu_1\notin E(G)$ but $xu_2,xu_3,xu_4\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_2,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_4\notin
E(G)$ but $xu_2,xu_3\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_2,xu_3\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$. If $xu_1,xu_2\notin E(G)$ but $xu_3,xu_4\in E(G)$, then the tree $T$ induced by the edges in $\{u_1u_2,u_1u_4,xu_3,xu_4\}$ is an $S$-Steiner tree in $G$ and hence $d_G(S)\leq 4$.
From the arbitrariness of $S$, we have $sdiam_4(G)\leq 4$. Since $\delta(G)=n-3$ and $C_4\in\overline{G}$, or $\delta(G)\leq n-4$, it follows from Theorem \[th2\] that $sdiam_4(G)=4$. The proof is now complete.
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Proof of Theorem $4$
====================
The following lemma is a preparation of our main result.
\[pro1\] Let $G$ be a connected graph, and $H$ be a connected subgraph of $G$. Then the number of non-cut vertices of $G$ is not less than the number of non-cut vertices of $H$.
It suffices to show that there exists an injective mapping $f$ from the set of non-cut vertices of $H$ to the set of non-cut vertices of $G$. We define such a mapping $f$ as follows. Let $v$ be a non-cut vertex of $H$. If $v$ is a non-cut vertex of $G$, then let $f(v) = v$. If $v$ is a cut-vertex of $G$, then let $G_1$ be a component of $G\setminus v$ not containing any vertex of $H$. Let $T_1$ be a spanning tree of $G_1$, and let $w$ be an end-vertex of $T_1$ distinct from $v$. Then $w$ is a non-cut vertex of $G$, and we define $f(v)=w$. Now $f$ maps non-cut vertices of $H$ to non-cut vertices of $G$, and $f$ is injective since either $f(v)=v$ or $f(v$) is in a component of $G\setminus V(H)$ which is (in $G$) attached only to $v$, and to no other vertex in $V(H)$.
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0.3cm
From Proposition \[pro1\], the following corollaries are immediate.
\[cor2\] Let $G$ be a connected graph of order $n \ (n\geq 3)$, and let $c(G)$ be the circumference of the graph $G$. If $c(G)\leq n$, then there are at least $c(G)$ non-cut vertices in $G$.
\[cor3\] Let $G$ be a connected graph of order $n \ (n\geq 3)$. Let $C_1,C_2,\cdots,C_r \ (r\geq 2)$ are cycles of the graph $G$ with $|V(C_i)|=n_i \ (1\leq i\leq r)$. If $|V(C_i)\cap V(C_j)|\leq 1$ for any $i,j \ (1\leq i,j\leq r, \
i\neq j)$, then the graph $G$ has at least $n_1+n_2+\cdots+n_r-2(r-1)$ are non-cut vertices in $G$.
We are now in a position to give the proof of Theorem \[th4\].
**Proof of Theorem \[th4\]:** Suppose $G=T_{a,b,c,d}$ or $G=\triangle_{a,b,c,d}$ or $G=\triangle^\prime_{a,b,c,d}$ or $G=G_1$ or $G=G_2$ or $G=G_3$. Since there are at most four non-cut vertices in $G$, it follows from Lemma \[lem1\] that $sdiam_4(G)=n-1$.
Conversely, suppose $sdiam_4(G)=n-1$. If $G$ is a tree, then it follows from Lemma \[lem1\] that $G$ contains at most non-cut four vertices, and hence $G=T_{a,b,c,d}$. Now, we assume that $G$ contains cycles. Recall that $c(G)$ is the circumference of the graph $G$. Obviously, $3\leq c(G)\leq n$. If $5\leq c(G)\leq n$, then it follow from Corollaries \[cor1\] and \[cor2\] that $sdiam_4(G)\leq
n-2$, a contradiction. Therefore, $c(G)=3$ or $c(G)=4$. If $c(G)=4$, then it follows from Lemma \[lem1\] and Corollaries \[cor1\] and \[cor3\] that $G$ contains four non-cut vertices, and from Corollaries \[cor3\] that $G$ contains no two cycles $C_1,C_2$ with $|V(C_1)|=4$ or $|V(C_2)|=4$ such that $|V(C_1)\cap V(C_2)|\leq
1$. From Proposition \[pro1\], we have the following facts.
$\bullet$ $G\setminus V(C_i) \ (i=1,2)$ is a union of pairwise independent paths;
$\bullet$ The number of these paths are at most four;
$\bullet$ The endvertices of each pair of these paths share the different neighbors in $C_i$.
From these facts, we have $G=G_1$ or $G=G_2$ or $G=G_3$. If $c(G)=3$, then it follows from Lemma \[lem1\] and Corollaries \[cor1\] and \[cor3\] that $G$ contains at most four non-cut vertices, and $G$ contains exactly one triangle or at most two cycles $C_1$ and $C_2$ with $|V(C_1)|=3$ and $|V(C_2)|=3$ such that $|V(C_1)\cap V(C_2)|\leq 1$. Suppose $G$ contains only one triangle. Let $K_{1,3}^*$ be the subdivision of star $K_{1,3}$ of order $t$. Then we have the following facts.
$\bullet$ The graph obtained from $G$ by deleting this triangle is $P_r\cup P_s\cup K_{1,3}^* \ (r+s+t=n-3)$ or $P_r\cup K_{1,3}^* \ (r+t=n-3)$ or $P_r\cup P_s \ (r+s=n-3)$ or $P_{n-3}$ or $K_{1,3}^* \ (t=n-3)$ or $P_r\cup P_s\cup P_p\ (r+s+p+q=n-3)$ or $P_r\cup P_s\cup P_p\cup P_q\ (r+s+p+q=n-3)$;
$\bullet$ If the graph obtained from $G$ by deleting this triangle is not $P_r\cup P_s\cup P_p\cup P_q\ (r+s+p+q=n-3)$, then the endvertices of each pair of these paths share the different neighbors in the triangle.
$\bullet$ If the graph obtained from $G$ by deleting this triangle is $P_r\cup P_s\cup P_p\ (r+s+p+q=n-3)$ or $P_r\cup P_s\cup P_p\cup P_q\ (r+s+p+q=n-3)$, then each vertex of this triangle share at least one common neighbor of each path.
Clearly, we have $G=\triangle_{a,b,c,d}$. If $G$ contains at most two cycles $C_1$ and $C_2$ with $|V(C_1)|=3$ and $|V(C_2)|=3$ such that $|V(C_1)\cap V(C_2)|\leq 1$, then $G\setminus (V(C_1)\cup V(C_2)$ is a union of pairwise independent paths, and the number of these path is at most five. So $G=\triangle^\prime_{a,b,c,d}$. The proof is complete.
------------------------------------------------------------------------
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[^1]: Corresponding author
[^2]: Supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11371205, 11161037, 11101232, 11461054) and the Science Found of Qinghai Province (Nos. 2016-ZJ-948Q, and 2014-ZJ-907).
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abstract: 'We outline the general construction of three-player games with incomplete information which fulfill the following conditions: (*i*) symmetry with respect to the permutations of players; (*ii*) the existence of an upper bound for total payoff resulting from Bell inequalities; (*iii*) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (*i*)$\div$(*iii*) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.'
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<span style="font-variant:small-caps;"></span>
[^1]\
**\
Introduction
============
One of the most important features of quantum theory is the nonlocality - the existence of correlations that cannot be explained in the framework of any local realistic theory. In particular, the correlations admitted by the latter must satisfy certain set of inequalities which can be violated on the quantum level [@Bell].
Violation of Bell inequalities has been confirmed experimentally [@Aspect]. Nonlocality inherent to quantum physics appears to be useful in practice, in particular for information processing (see [@Ekert], [@Acin], [@Buhrman] and numerous other references). Bell inequalities can be also discussed in the context of game theory. One can pose the question how the properties of the game are modified due to the existence of nonlocal correlations between the players. The first attempts to construct the games based on quantum mechanical correlations concerned those of complete information [@Meyer]$\div$[@Flitney]. It appeared that the quantum versions of classical games offer additional strategies which allow to resolve dilemmas that occur in classical games (for example, the Prisoner’s Dilemma). It has been also shown that quantum games can be realized experimentally [@Du], [@Prevedel].
On the other hand the conclusion that the advantages of quantum counterparts of classical games with complete information result from the specific properties of quantum correlations has been much debated and criticized [@Benjamin], [@Enk]. In order to make the relation between quantum nonlocality and the advantages of quantum strategies more transparent the quantum versions of games with incomplete information [@Harsanyi] have been proposed [@Cheon1]. In this way the connection has been established between the Bell theorem and the Bayesian games. In order to understand it let us note that, as it is nicely explained by Fine [@Fine], [@Fine1] (see also [@Halliwell], [@Halliwell1]), the violation of Bell inequalities is directly related to the existence of noncommuting observables. Now, the unknown elements of the game with incomplete information are represented by the concept of the player type. On the quantum level the player types are, in turn, represented by different, in general noncommuting, observables. This leads to the violation of Bell inequalities. If the payoff functions of the players are related to Bell expressions, the players sharing nonlocal resources can outperform the ones having access to classical resources only.
In the particular example proposed by Cheon and Iqbal [@Cheon1], which is a mixture of Battle of Sexes and Chicken games, the bound on classical payoffs is related to Cereceda inequalities [@Cereceda1]. The ideas of Cheon and Iqbal were further developed in the papers [@Iqbal], [@Flitney1], [@Iqbal1], [@Hill]. Quite recently Brunner and Linden [@Brunner] considered the more general situation where nonlocal resources provide an advantage over any classical strategy because the bounds on some combinations of payoff functions follow from Bell inequalities.
The examples of games presented in [@Brunner] are the games of common interest. Pappa et al. [@Pappa] gave an example of conflicting interest game where quantum mechanics also offers an advantage over its classical counterpart. The game they consider is a two-player one obtained as a combination of the Battle of Sexes and CHSH games. It is symmetric with respect to the permutation of players although this property is somewhat hidden due to the specific numbering of strategies and types used by the authors. An important point is that, in the classical version of the game, the total payoff (the sum of payoffs of both players) is bounded from above due to the Bell inequality. This implies that it is a conflicting interest game provided there exist unfair equilibria saturating the bound resulting from Bell inequality. On the other hand, on the quantum level all equilibria are the fair ones because the payoff functions become equal. Moreover, there exist fair quantum equilibria where the parties have strictly higher payoffs than for any classical fair equilibrium.
Further examples of conflicting interest games where quantum mechanics offers an advantage were given by Situ [@Situ]. Moreover, by a slight modification of the payoff functions proposed in [@Pappa] Roy et.al [@Roy] gave the examples of games where quantum strategies can outperform even the unfair classical equilibrium strategies. Depending on the values of additional parameters entering the payoff functions their games have only fair equilibria, both fair and unfair equilibria or only unfair ones.
In final instance, all physics behind any example of quantum game with incomplete information is related to quantum nonlocality manifesting in violation of some Bell inequalities. The advantages of quantum strategies are the consequence of quantum entanglement built into the game.
In the multipartite case the structure of nonlocal correlations is richer and less understood [@Brunner1]. There exist different definitions of nonlocality which refine the bipartite definition. Therefore, it is advantegous to consider the three- or multiparty generalizations of quantum Bayesian games. One example of three-party (three-player) game has been provided by Situ et al. [@Situ1]. It is based on Svetlichny inequality [@Svetlichny] and allows to analyse the advantages of the game based on fully quantum correlations over the one where the correlations can be reduced to the mixtures of two-player ones related locally to the third player.
In the present paper we consider the three-player counterpart of the game considered in Ref. [@Pappa]. In Sec.II we outline the general construction of the three-player games with incomplete information possessing the upper bound for total payoff following from Bell inequalities and both fair and unfair equilibria saturating this bound. As we have mentioned above such games are automatically conflicting interest games. In Sec.III we provide an explicit example of such a game. As a next step we consider in Sec.IV a quantum counterpart of our game, i.e. we keep the utilities intact but replace the classical advisor by a quantum one. It appears that the quantum game possesses only fair equilibria and the corresponding payoffs are strictly higher than the classical ones. We show that the nonlocality inherent in quantum mechanics plays the twofold role: it raises, due to violation of Bell inequalities, the payoffs corresponding to fair equilibria and excludes the unfair ones.
To conclude this section let us note that the game-theoretic language has much wider range of applications and is a very convenient tool for describing the peculiar properties of quantum correlations. It can be used, for example, to study the entanglement in spin systems [@Miszczak], [@Ozaydin] or decoherence phenomena [@Gawron], [@Gawron1], [@Dajka].
Three-player games
==================
We define a three-player Bayesian game following the analogous discussion of two-player case by Pappa et al. [@Pappa] (see also [@Brunner]). There are three players, Alice (A), Bob (B) and Charlie (C); each player acquires a type $x_i$, $i\in{\left\{A,B,C\right\}}$, $x_i\in{\left\{0,1\right\}}$, according to the probability distribution $P(\underline{x})\equiv P{\left(x_A,x_B,x_C\right)}$. They decide on their actions $y_i$, $y_i\in{\left\{0,1\right\}}$, according to a chosen strategy. The average payoff of each player reads $$F_i=\sum_{{\left(x,y\right)}}P{\left(\underline{x}\right)}p{\left(\underline{y}|\underline{x}\right)}u_i{\left(\underline{x},\underline{y}\right)}$$ where $p{\left(\underline{y}|\underline{x}\right)}\equiv p{\left(y_A,y_B,y_C|x_A,x_B,x_C\right)}$ is the probability the players choose actions $\underline{y}\equiv{\left(y_A,y_B,y_C\right)}$ given their types were $\underline{x}\equiv{\left(x_A,x_B,x_C\right)}$; $u_i{\left(\underline{x},\underline{y}\right)}$ are the utility functions determining the gains of players depending on their types and actions. The properties of the game are determined by the form of utility functions and the restrictions imposed on the probabilities $p{\left(\underline{y}|\underline{x}\right)}$.
In order to set the question of fair and unfair equilibria in the proper framework we consider the games symmetric with respect to the permutations of players. This implies the relations (which hold up to a possible renumbering of types and/or strategies) $$\begin{split}
& u_A{\left(x_A,x_B,x_C,y_A,y_B,y_C\right)}=u_B{\left(x_B,x_A,x_C,y_B,y_A,y_C\right)}\\
& u_C{\left(x_A,x_B,x_C,y_A,y_B,y_C\right)}=u_C{\left(x_B,x_A,x_C,y_B,y_A,y_C\right)}
\end{split}\label{a}$$ together with the similar relations obtained by choosing the remaining two pairs of players. As far as the probabilities $p{\left(\underline{y}|\underline{x}\right)}$ are concerned we assume they obey the no-signalling conditions [@Brunner1] $$\sum_{y_C}p{\left(y_A,y_B,y_C|x_A,x_B,x_C\right)}=\sum_{y_C}p{\left(y_A,y_B,y_C|x_A,x_B,x_C'\label{aa}\right)}$$ and similar conditions for the remaining two players. Apart from eq. (\[aa\]) we have the normalization conditions $$\sum_{\underline{y}}p{\left(\underline{y}|\underline{x}\right)}=1 \quad \text{for all} \quad \underline{x}.$$ Given no-signalling condition we consider two types of probability distributions:
- : one assumes further constraints on $p{\left(\underline{y}|\underline{x}\right)}$ in form of Bell inequalities [@Hardy], [@Cereceda2], [@Brunner1]. According to Fine [@Fine], [@Fine1], [@Halliwell], [@Halliwell1] this leads to the hidden variables representation of the relevant probabilities (actually, Fine’s theorem concerns two-parties case but we assume it holds for three-parties as well): $$p{\left(y_A,y_B,y_C|x_A,x_B,x_C\right)}=\int \text{d}\lambda \rho(\lambda)p_A{\left(y_A|x_A,\lambda\right)}\cdot p_B{\left(y_B|x_B,\lambda\right)}\cdot p_C{\left(y_C|x_C,\lambda\right)}\label{a4}$$ $\lambda$ being a set of hidden variables distributed with probability density $\rho{\left(\lambda\right)}$. Note that since there are only two possible actions per player it is sufficient to consider only hidden variables providing three bits so that $\lambda\equiv{\left(\lambda_A,\lambda_B,\lambda_C\right)}$, $p_A{\left(y_A|x_A,\lambda\right)}=p_A{\left(y_A|x_A,\lambda_A\right)}$ etc.\
In game-theoretic language one says that the players receive advice from a classical source that is independent of the inputs $\underline{x}$; $\rho{\left(\lambda\right)}$ can be viewed as characterizing an advisor. In particular, if the strategies of the players are uniquely determined by their types and advices they received (deterministic hidden variable model), $$y_A=c_A{\left(x_A,\lambda\right)},\quad \text{etc.}$$ one finds $$F_i=\sum_{\underline{x}}P{\left(\underline{x}\right)}\int\text{d}\lambda \rho(\lambda)u_i{\left(x_A,x_B,x_C,c_A{\left(x_A,\lambda\right)},c_B{\left(x_B,\lambda\right)},c_C{\left(x_C,\lambda\right)}\right)}$$ On the other hand if the players are insensitive to the advisor suggestions, $p_A{\left(y_A|x_A,\lambda\right)}=p_A{\left(y_A|x_A\right)}$, etc., the probability factorizes $$p{\left(y_A,y_B,y_C|x_A,x_B,x_C\right)}=p_A{\left(y_A|x_A\right)}p_B{\left(y_B|x_B\right)}p_C{\left(y_C|x_C\right)}$$
- : quantum probabilities (a quantum source/advisor) are defined by the choice of tripartite density matrix $\rho$ (which characterizes an advisor) and the choice of three pairs of observables $A_x$, $B_x$ and $C_x$, $x=0,1$, acting in twodimensional Hilbert spaces of individual players and admitting the spectral decompositions $$A_x=1\cdot A_x^1+{\left(-1\right)}\cdot A_x^0, \quad \mathbbm{1}=A_x^1+A_x^0,\quad \text{etc.}$$ with $A_x^y$, etc., being the corresponding projectors. The resulting payoffs read $$F_i=\sum_{\underline{x},\underline{y}}P{\left(\underline{x}\right)}\text{Tr}{\left(\rho{\left(A_{x_A}^{y_A}\otimes B_{x_B}^{y_B}\otimes C_{x_C}^{y_C}\right)}\right)}u_i{\left(\underline{x},\underline{y}\right)}.\label{b2}$$
Note that the general form of our quantum variables reads $$A_x=\vec{n}_x^{(A)}\cdot\vec{\sigma}$$ where $\vec{\sigma}$ are Pauli matrices while $\vec{n}_{0,1}^A$- the unit vectors,\
$\vec{n}_x^{(A)}={\left(\sin\theta_x^A\cos\varphi_x^A,\sin\theta_x^A\sin\varphi_x^A,\cos\theta_x^A\right)}$; similar formulae are valid for $B$ and $C$.\
In principle, we could also consider superquantum no-signalling distributions [@Popescu]; however, we shall not dwell on this question.
In what follows we assume that the distribution of the player types is uniform, $$P{\left(\underline{x}\right)}=\frac{1}{8}\quad \text{for all}\quad \underline{x}={\left(x_A,x_B,x_C\right)}. \label{ca}$$ In order to construct the examples of games with conflicting interests which possess fair quantum equilibria with higher payoffs than those corresponding to classical equilibria we start with the utility functions $u_i{\left(\underline{x},\underline{y}\right)}$, $i\in{\left\{A,B,C\right\}}$. We demand they obey the symmetry conditions (\[a\]). Moreover, we assume that the sum of payoffs $F_A+F_B+F_C$ is expressible in terms of the expression(s) entering the Bell inequality(ies). The relevant Bell inequality reads [@Hardy], [@Cereceda2], [@Brunner1] $${\left|{\left<A_0B_1C_1\right>}+{\left<A_1B_0C_1\right>}+{\left<A_1B_1C_0\right>}-{\left<A_0B_0C_0\right>}\right|}\leq 2\label{a2}$$ where $A_x$, $B_x$ and $C_x$ acquire the values $\pm 1$. Rewritting the above inequality in terms of relevant probabilities and comparying the resulting expression with $F_A+F_B+F_C$ one finds the conditions on utility functions. However, there is an important difference between two- and three-players cases. In the latter one the resulting equations are more stringent and imply that the utility functions lead to a trivial game. This can be cured as follows. Note that the properties of a game (i.e. the structure of its Nash equilibria) are invariant under the transformations $$u_i{\left(\underline{x},\underline{y}\right)}\rightarrow \alpha u_i{\left(\underline{x},\underline{y}\right)}+\beta\label{a1}$$ with arbitrary $\alpha$ and $\beta$. Therefore, if the constraints on $u_i's$ are not invariant under (\[a1\]) their solutions must be so special that they lead to a trivial game.
However, note that we have eight Bell inequalities at our disposal. In fact, the remaining ones are obtained from (\[a2\]) by making the replacement $0\leftrightarrow 1$ for one, two and three players. In particular, in the latter case we arrive at the inequality $${\left|{\left<A_1B_0C_0\right>}+{\left<A_0B_1C_0\right>}+{\left<A_0B_0C_1\right>}-{\left<A_1B_1C_1\right>}\right|}\leq 2.\label{a3}$$ By demanding that $F_A+F_B+F_C$ is expressible in terms of the linear combination (actually, the difference) of expressions entering (\[a2\]) and (\[a3\]) one finds much more reasonable conditions on utility functions (in particular, they are invariant under the transformations (\[a1\])).
The symmetry conditions (\[a\]) and the one imposed on $F_A+F_B+F_C$ allow us to express the utilities $u_i{\left(\underline{x},\underline{y}\right)}$ in terms of a number of independent parameters.\
As a next step we select some set of strategies as the candidates for nonfair classical equilibria. Additionally, we demand that, for these equilibria, the sum $F_A+F_B+F_C$ saturates the uper bound following from Bell inequalities. If this is the case we can take for granted that, for any fair equilibrium, at least one player will gain smaller payoff than for the unfair one. The resulting general conditions (derived with the help of MATHEMATICA) are too complicated to present them here explicitly. Instead, we give an example of a game sharing the properties discussed above.
The example of three-player game
================================
The utilities in our example are presented in Table \[t1\].
$u_A{\left(\underline{x},\underline{y}\right)}$:
--------- --------- ------------------------------------------------------------- ------------------------------------------------------------------------ ----------------------------------------------------------------------- ---------------------------------------------------------------------------------------
$y_B=0$ $y_B=1$ $y_B=0$ $y_B=1$
$x_C=0$ $x_B=0$ $\left[\begin{array}{cc} 2 & 0\\ 2 & 1\end{array}\right]$ $\left[\begin{array}{cc} \frac{3}{2} & 1\\ 0 & 2\end{array}\right] $ $\left[\begin{array}{cc} \frac{3}{2} & 1\\ 0 & 2\end{array}\right]$ $\left[\begin{array}{cc} 4 & 1\\ 4 & \frac{19}{3}\end{array}\right] $
$x_B=1$ $\left[\begin{array}{cc} 0 & -1\\ -1 & 1\end{array}\right]$ $ \left[\begin{array}{cc} -\frac{1}{2} & 2\\ 1 & 0\end{array}\right] $ $\left[\begin{array}{cc} 1 & -1\\ \frac{1}{2} & 0\end{array}\right] $ $ \left[\begin{array}{cc} -2 & -\frac{19}{6}\\ -1 & -\frac{1}{2}\end{array}\right] $
$x_C=1$ $x_B=0$ $\left[\begin{array}{cc} 0 & -1\\ -1 & 1\end{array}\right]$ $\left[\begin{array}{cc} 1& -1\\ \frac{1}{2} & 0\end{array}\right] $ $\left[\begin{array}{cc} -\frac{1}{2} & 2\\ 1 & 0\end{array}\right] $ $\left[\begin{array}{cc} -2 & -\frac{19}{6}\\ -1 & -\frac{1}{2}\end{array}\right] $
$x_B=1$ $\left[\begin{array}{cc} 2 & 2\\ 0 & -2\end{array}\right] $ $\left[\begin{array}{cc} 1 & 1\\ 2 & \frac{1}{2}\end{array}\right]$ $ \left[\begin{array}{cc} 1 & 1\\ 2 & \frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc} 0 & 4\\ -1 & \frac{2}{3} \end{array}\right]$
--------- --------- ------------------------------------------------------------- ------------------------------------------------------------------------ ----------------------------------------------------------------------- ---------------------------------------------------------------------------------------
: The utilities of players
\
$ u_B{\left(\underline{x},\underline{y}\right)}$:
--------- --------- -------------------------------------------------------------------------------- ---------------------------------------------------------------------- ----------------------------------------------------------------------- ------------------------------------------------------------------------------------------
$y_B=0$ $y_B=1$ $y_B=0$ $y_B=1$
$x_C=0$ $x_B=0$ $\left[\begin{array}{cc} 2 & \frac{3}{2}\\ 0 & -\frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc} 0 & 1\\ -1 & 2\end{array}\right]$ $\left[\begin{array}{cc} \frac{3}{2} &4\\ 1 & -2\end{array}\right]$ $\left[\begin{array}{cc} 1 & 1\\ -1 & -\frac{19}{6}\end{array}\right]$
$x_B=1$ $\left[\begin{array}{cc} 2 & 0\\ -1 & 1\end{array}\right]$ $\left[\begin{array}{cc} 1 & 2\\ 1 & 0\end{array}\right]$ $\left[\begin{array}{cc} 0 & 4\\ \frac{1}{2} & -1\end{array}\right]$ $\left[\begin{array}{cc} 2 & \frac{19}{3}\\ 0 & -\frac{1}{2}\end{array}\right]$
$x_C=1$ $x_B=0$ $\left[\begin{array}{cc} 0 & 1\\ 2 & 1\end{array}\right]$ $\left[\begin{array}{cc} -1 & -1\\ 2 & 1\end{array}\right]$ $\left[\begin{array}{cc} -\frac{1}{2} & -2\\ 1 & 0\end{array}\right]$ $\left[\begin{array}{cc} 2 & -\frac{19}{6}\\ 1 & 4\end{array}\right]$
$x_B=1$ $\left[\begin{array}{cc}-1 & \frac{1}{2}\\ 0 & 2\end{array}\right]$ $\left[\begin{array}{cc} 1 & 0\\ -2 & \frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc} 1 & -1\\ 2 & -1\end{array}\right]$ $\left[\begin{array}{cc} 0 & -\frac{1}{2}\\ \frac{1}{2} & \frac{2}{3}\end{array}\right]$
--------- --------- -------------------------------------------------------------------------------- ---------------------------------------------------------------------- ----------------------------------------------------------------------- ------------------------------------------------------------------------------------------
: The utilities of players
\
$ u_C{\left(\underline{x},\underline{y}\right)}$:
--------- --------- -------------------------------------------------------------------------------- ----------------------------------------------------------------------- --------------------------------------------------------------------- -----------------------------------------------------------------------------------------
$y_B=0$ $y_B=1$ $y_B=0$ $y_B=1$
$x_C=0$ $x_B=0$ $\left[\begin{array}{cc} 2 & \frac{3}{2}\\ 0 & -\frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc} \frac{3}{2} & 4\\ 1 & -2\end{array}\right]$ $\left[\begin{array}{cc} 0 & 1\\ -1 & 2\end{array}\right]$ $\left[\begin{array}{cc} 1 & 1\\ -1 & -\frac{19}{6}\end{array}\right]$
$x_B=1$ $\left[\begin{array}{cc} 0 & 1\\ 2 & 1\end{array}\right]$ $\left[\begin{array}{cc} -\frac{1}{2} & -2\\ 1 & 0\end{array}\right]$ $\left[\begin{array}{cc} -1 & -1\\ 2 & 1\end{array}\right]$ $\left[\begin{array}{cc} 2 & -\frac{19}{6}\\ 1 & 4\end{array}\right]$
$x_C=1$ $x_B=0$ $\left[\begin{array}{cc} 2 & 0\\ -1 & 1\end{array}\right]$ $\left[\begin{array}{cc} 0 & 4\\ \frac{1}{2} & -1\end{array}\right]$ $\left[\begin{array}{cc} 1 & 2\\ 1 & 0\end{array}\right]$ $\left[\begin{array}{cc} 2 & \frac{19}{3}\\ 0 & -\frac{1}{2}\end{array}\right]$
$x_B=1$ $\left[\begin{array}{cc}-1 & \frac{1}{2}\\ 0 & 2\end{array}\right]$ $\left[\begin{array}{cc} 1 & -1\\ 2 & -1\end{array}\right]$ $\left[\begin{array}{cc} 1 & 0\\ -2 &\frac{1}{2}\end{array}\right]$ $\left[\begin{array}{cc} 0 & -\frac{1}{2}\\ \frac{1}{2} &\frac{2}{3}\end{array}\right]$
--------- --------- -------------------------------------------------------------------------------- ----------------------------------------------------------------------- --------------------------------------------------------------------- -----------------------------------------------------------------------------------------
: The utilities of players
\[t1\]
The elements of the matrices entering the Table \[t1\] are indexed by $x_A$ (rows) and $y_A$ (columns). Some elements of the utility functions above are negative (loss instead of gain) but this can be easily cured, if necessary, using the symmetry transformations (\[a1\]). The resulting game may seem slightly complicated but the underlying principles are very simple: (*i*) symmetry with respect to the permutations of players, (*ii*) expressibility of the total payoff $F_A+F_B+F_C$ in terms of Bell operators and (*iii*) saturation of the bound for total payoff following from Bell inequalities. The latter reads in our example $$F_A+F_B+F_C\leq \frac{9}{4}.\label{a5}$$ Actually, in order to obtain the utilities presented in Table \[t1\], we have used still one constraint to be discussed below.
The game defined by the utilities given in Table \[t1\] possesses the correlated Nash equilibria described in Table \[t\]. The rows in first three columns present the values of $y$’s for $x=0$ and $x=1$.
$y_A$ $y_B$ $y_C$ $F_A$ $F_B$ $F_C$
---------------------- ----------------------- ---------------------- ----------------- ----------------- -----------------
${\left(0,1\right)}$ ${\left(0,0\right)} $ ${\left(0,0\right)}$ $\frac{5}{8}$ $\frac{13}{16}$ $\frac{13}{16}$
${\left(0,0\right)}$ ${\left(0,1\right)} $ ${\left(0,0\right)}$ $\frac{13}{16}$ $\frac{5}{8}$ $\frac{13}{16}$
${\left(0,0\right)}$ ${\left(0,0\right)} $ ${\left(0,1\right)}$ $\frac{13}{16}$ $\frac{13}{16}$ $\frac{5}{8}$
${\left(1,0\right)}$ ${\left(0,1\right)} $ ${\left(0,1\right)}$ $\frac{11}{8}$ $\frac{7}{16}$ $\frac{7}{16}$
${\left(0,1\right)}$ ${\left(1,0\right)} $ ${\left(0,1\right)}$ $\frac{7}{16}$ $\frac{11}{8}$ $\frac{7}{16}$
${\left(0,1\right)}$ ${\left(0,1\right)} $ ${\left(1,0\right)}$ $\frac{7}{16}$ $\frac{7}{16}$ $\frac{11}{8}$
${\left(0,1\right)}$ ${\left(1,1\right)} $ ${\left(1,1\right)}$ $\frac{3}{4}$ $\frac{3}{4}$ $\frac{3}{4}$
${\left(1,1\right)}$ ${\left(0,1\right)} $ ${\left(1,1\right)}$ $\frac{3}{4}$ $\frac{3}{4}$ $\frac{3}{4}$
${\left(1,1\right)}$ ${\left(1,1\right)} $ ${\left(0,1\right)}$ $\frac{3}{4}$ $\frac{3}{4}$ $\frac{3}{4}$
: “Pure” Nash equilibria
\[t\]
In order to show that the configurations presented in Table \[t\] provide the Nash equilibria we note first that the relevant probabilities are of the form (\[a4\]). Consider, for example, the first row in Table \[t\]. The probabilities corresponding to the strategies entering it read $$\begin{split}
& p_A{\left(y_A|x_A,\lambda\right)}=\delta_{y_A,x_A}\\
& p_B{\left(y_B|x_B,\lambda\right)}=\delta_{y_B,0}\\
& p_C{\left(y_C|x_C,\lambda\right)}=\delta_{y_C,0}.
\end{split}\label{a6}$$
Eqs. (\[a6\]) define an equilibrium. To see this consider the Alice payoff. Eqs. (\[a4\]) and (\[a6\]) yield $$p{\left(\underline{y}|\underline{x}\right)}=\delta_{y_B,0}\delta_{y_C,0}\int\text{d}\lambda\rho{\left(\lambda\right)}p_A{\left(y_A|x_A,\lambda\right)}\equiv p_A{\left(y_A|x_A\right)}\delta_{y_B,0}\delta_{y_C,0}$$ and, consequently, $$\begin{split}
& F_A=\frac{1}{8}\sum_{{\left(x_A,y_A\right)}}p_A{\left(y_A|x_A\right)}\sum_{x_B,x_C}u_A{\left(x_A,x_B,x_C,y_A,0,0\right)}\equiv\\
& \quad\equiv \frac{1}{8}\sum_{x_A,y_A}p_A{\left(y_A|x_A\right)}u_A{\left(x_A,y_A\right)}.
\end{split}$$ $F_A$ should be maximized on the convex set $\sum\limits_{y_A}p_A{\left(y_A|x_A\right)}=1$, $x_A=0,1$; $F_A$ acquires maximum on some of extremal points of this set. The same reasoning applies to Bob and Charlie. So it remains to check the equilibrium condition on $2^6$ strategies $y_i{\left(x_i\right)}$, $i\in{\left\{A,B,C\right\}}$.
By inspecting the Table \[t\] we see that we have 3 groups, each containing 3 equilibria; each set is invariant under the permutation of players. Two sets represent unfair equilibria; the remaining one contains fair ones. The game is a conflicting interest one as it is clearly seen from Table \[t\]: there is no common equilibrium preferred by all players. In fact, even if some mixed (i.e. the one with some $0<p{\left(\underline{y}|\underline{x}\right)}<1$) fair equilibrium existed, the payoff of each player could not exceed $\frac{3}{4}$ due to the bound on the total payoff following from Bell inequalities.
The quantum counterpart of three-player game
============================================
Let us now pass to the quantum case. The density matrix $\rho$ entering eq. (\[b2\]) is chosen as $$\rho={\left|\Psi\right>}{\left<\Psi\right|}$$ where ${\left|\Psi\right>}$ is the GHZ state $${\left|\Psi\right>}=\frac{1}{\sqrt{2}}{\left({\left|111\right>}+i{\left|000\right>}\right)}.$$ The choice of $\rho$ determines the properties of advisor while the players strategies are described by the probabilities $p{\left(\underline{y}|\underline{x}\right)}$ which, in turn, are determined by choosing the unit vectors $\vec{n}_x^{(A)}$, $\vec{n}_x^{(B)}$ and $\vec{n}_x^{(C)}$; one needs twelve angles to characterize them. This makes the problem complicated. Therefore, we restrict ourselves to the special case $\theta_x^A=\theta_x^B=\theta_x^C=\frac{\pi}{2}$. Let us denote by ${\left(\varphi_1,\varphi_2\right)}$, ${\left(\varphi_3,\varphi_4\right)}$ and ${\left(\varphi_5,\varphi_6\right)}$ the angles characterizing the observables $A_x$, $B_x$ and $C_x$, respectively. It is then not difficult to find the relevant payoffs $$\begin{split}
& F\equiv F_{A,B,C}=\frac{1}{48}\left( 26+3\sin{\left(\varphi_1+\varphi_3+\varphi_5\right)}+2\sin{\left(\varphi_2+\varphi_3+\varphi_5\right)}+\right.\\
& \quad +2\sin{\left(\varphi_1+\varphi_4+\varphi_5\right)}-3\sin{\left(\varphi_2+\varphi_4+\varphi_5\right)}+2\sin{\left(\varphi_1+\varphi_3+\varphi_6\right)}+\\
& \quad \left.-3\sin{\left(\varphi_2+\varphi_3+\varphi_6\right)}-3\sin{\left(\varphi_1+\varphi_4+\varphi_6\right)}-2\sin{\left(\varphi_2+\varphi_4+\varphi_6\right)}\right).
\end{split}$$ We have fixed the values of utility functions in such a way that all payoff functions are equal; this is the additional condition we have mantioned before. Due to this property all Nash equilibria must be fair.
$F$ is invariant under the transformations $$\begin{split}
& \varphi_1\rightarrow\varphi_1+\chi_1,\quad \varphi_2\rightarrow\varphi_2+\chi_1,\quad \varphi_3\rightarrow\varphi_3+\chi_2,\quad\varphi_4\rightarrow\varphi_4+\chi_2\\
& \varphi_5\rightarrow\varphi_5+\chi_3,\quad \varphi_6\rightarrow\varphi_6+\chi_3
\end{split}\label{b3}$$ provided $\chi_1+\chi_2+\chi_3=2n\pi$. This follows from the relation $$e^{i\frac{\chi_1}{2}\sigma_3}\otimes e^{i\frac{\chi_2}{2}\sigma_3}\otimes e^{i\frac{\chi_3}{2}\sigma_3}{\left|GHZ\right>}={\left(-1\right)}^n{\left|GHZ\right>}.$$ Maximizing $F$ one obtains the Nash equilibrium. Due to the symmetry (\[b3\]) we get two parameter family of equilibria. To fix one we put $\varphi_1=\varphi_3=0$. Then the remaining angles (obtained numerically) read $\varphi_2=-\frac{\pi}{2}$, $\varphi_4=-\frac{\pi}{2}$, $\varphi_5=2.1588$, $\varphi_6=0.5880$ (up to the multiples of $2\pi$). The corresponding gain of each player is $$F_A=F_B=F_C=0.842$$ We conclude that the quantum version of the game possesses only fair equilibria and the corresponding payoffs are higher than in any classical fair equilibrium which, due to the inequality (\[a5\]), cannot exceed 0,75. Let us note that our game is genuinely a quantum one (in spite of the restriction $\theta_x^i=\frac{\pi}{2}$ imposed) since the strategies are represented by, in general, noncommuting observables. However, the result obtained (the existence of only fair equilibria) might occur as a consequence of artificial constraint imposed on the $\theta$ angles. To get some feeling what is going on consider the general quantum game with no constraints on $\theta's$. Let us take into account the unfair equilibrium corresponding to the first row of Table \[t\]. It is defined by the probabilities $p{\left(\underline{y}|\underline{x}\right)}$ which cannot appear on quantum level. In fact, assume we have six pairs of onedimensional projectors $A_x^y$, $B_x^y$ and $C_x^y$ obeying $${\left<\Psi\right|}{\left(A_{x_A}^{y_A}\otimes B_{x_B}^{y_B}\otimes C_{x_C}^{y_C}\right)}{\left|\Psi\right>}=\delta_{y_A,x_A}\delta_{y_B,0}\delta_{y_C,0}.$$ Summing over $y_A$ and $y_B$ yields $${\left<\Psi\right|}{\left(\mathbbm{1}\otimes\mathbbm{1}\otimes C_{x_C}^{y_C}\right)}{\left|\Psi\right>}=\delta_{y_C,0}.$$ Now, $\mathbbm{1}\otimes\mathbbm{1}\otimes C_{x_C}^{y_C}$ is a projector so that $${\left(\mathbbm{1}\otimes\mathbbm{1}\otimes C_{x_C}^{y_C}\right)}{\left|\Psi\right>}={\left|\Psi\right>}$$ which is impossible (${\left|\Psi\right>}\equiv{\left|GHZ\right>}$).
We conclude that not all classical strategies can be reproduced on quantum level. In particular, this concerns strategies leading to unfair equilibria. One can say that quantum entanglement plays here twofold role: it excludes at least some (unfair) equilibria and raises the payoffs corresponding to fair equilibria.
Finally, let us note that our results concern the case of uniform distribution of the player types (cf. eq. (\[ca\])). If this condition is relaxed new interesting possibilities arise. In the nice recent paper [@Auletta] Auletta et al. presented some examples of three-party GHZ games with nonuniform distributions of types; in particular, they constructed a game with the following feature: no no-signalling (superquantum) distribution can help to achieve a better fair equilibrium than that achieved by a quantum strategy. However, it should be stressed that the assumption concerning the nonuniform distribution of types is here crucial.
Conclusions
===========
We have outlined the construction of general three-player game with incomplete information such that: ($i$) it is symmetric under the permutation of players, ($ii$) the upper bound on the total payoff results from Bell inequalities, ($iii$) there exist both fair and unfair Nasha equilibria saturating this bound. Such games are necessarily conflicting interes ones. Although the general formulae are rather involved, the basic assumptions and the algorithm for constructing the game are clearly described which allows to produce easily numerous examples. One example is presented in detail. Contrary to the case of two-player game [@Pappa], [@Situ], [@Roy] one has to combine at least two Bell inequalities to obtain a nontrivial game.
A quantum counterpart of the game is obtained by keeping the same utility functions but replacing the classical advisor by a quantum one. As it has been already shown by Pappa et al [@Pappa] the quantum strategies can outperform the classical ones due to the quantum phenomenon of entanglement which leads to the violation of Bell inequalities. The description of entanglement in the three-partite (and multi-partite) case is more complicated than in two-partite one (see, for example, Ref. [@Brunner1]). One can consider, for example, the three-partite correlations which are the mixtures of quantum and classical ones [@Svetlichny] and construct a three-players game based on Svetlichny inequalities [@Situ1]. It is desirable to construct also the three-player games based on Bell inequalities. In such a case one has to use more than one Bell inequality. Another important point which should be mentioned is that in order to ensure the violation of Bell inequalities (which allows the quantum strategies to outperform the classical ones) one has to choose a particular form of quantum advisor. It appears that it can be chosen in such a way that the payoff functions of the players coincide. This happens to be the case in the example considered in Ref. [@Pappa] as well as in the one described in the present paper. The quantum game possesses then only fair equilibria.
Acknowledgement {#acknowledgement .unnumbered}
---------------
I would like to thank Prof. Piotr Kosiński for helpful discussion and useful remarks. The research was supported by the NCN Grant no. DEC-2012/05/D/ST2/00754.
[99]{}
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[^1]: kbolonek@uni.lodz.pl
|
---
abstract: 'A method for calculating coupling impedances and power losses for off-axis beams is developed. It is applied to calculate impedances of small localized discontinuities like holes and slots, as well as the impedance due to a finite resistivity of chamber walls, in homogeneous chambers with an arbitrary shape of the chamber cross section. The approach requires to solve a two-dimensional electrostatic problem, which can be easily done numerically in the general case, while for some particular cases analytical solutions are obtained.'
author:
- |
Sergey S. Kurennoy\
Physics Department, University of Maryland,\
College Park, MD 20742, USA
title: |
IMPEDANCES AND POWER LOSSES\
FOR AN OFF-AXIS BEAM[^1]
---
Introduction
============
The beam-chamber coupling impedances, as well as power losses due to a finite conductivity of the chamber wall, may depend essentially on the beam position inside the chamber. While for the power loss in a circular pipe this dependence is well-known [@AChao], developing an approach working for other chamber cross sections seems to be worthwhile.
In the present note, we consider the problem for the vacuum chamber with an arbitrary but constant cross section, and calculate, for an off-axis beam, the coupling impedance due to either resistive wall or a small localized discontinuity, like a hole. Analytical results are presented for circular and rectangular cross sections.
Longitudinal Impedance
======================
Let us consider an infinite cylindrical chamber with an arbitrary cross section $S$. The $z$ axis is directed along the chamber axis, an ultrarelativistic point charge $q$ moves parallel to the axis with the transverse offset $\vec{a}$ from it. A small discontinuity (e.g., a hole) located on the chamber wall at the point ($\vec{b},z=0$), contributes as an inductance to the longitudinal coupling impedance $$Z(k;\vec{a}\,) = - ikZ_0e^2_\nu(\vec{a}\,) \left (\psi -
\chi \right )/2 \ , \label{Z}$$ where $Z_0 = 120 \pi$ Ohms is the impedance of free space, $k=\omega/c$, and $\psi$ and $\chi$ are magnetic and electric polarizabilities of the discontinuity . The dependence on the beam position, as well as on the hole position in the cross section, is via $$e_\nu(\vec{a}\,) = - \sum_s k^{-2}_s
e_s(\vec{a}\,) \nabla _{\nu}e_s(\vec{b}\,) \label{enorm}$$ which is merely the normalized electrostatic field produced at the hole location by a filament charge displaced from the chamber axis by distance $\vec{a}$. It is expressed in terms of eigenvalues $k^2_{nm}$ and orthonormalized eigenfunctions (EFs) $e_{nm}(\vec{r})$ of the 2D boundary problem in $S$: $$\left (\nabla ^2+ k^2_{nm}\right ) e_{nm} =
0 \ ; \qquad e_{nm}\big\vert_{\partial S} = 0 \ . \label{boundpr}$$ We denote $\hat{\nu}$ and $\hat{\tau}$ the outward normal and tangent unit vectors to the boundary $\partial S$ of the chamber cross section $S$, so that $\{ \hat{\nu},\hat{\tau},\hat{z}\}$ form a right-handed basis. One should note the normalization condition $$\oint_{\partial S}\! dl \ e_\nu = 1 \ , \label{norma}$$ where integration goes along the boundary ${\partial S}$, which reflects the Gauss law. It follows from the fact that Eq. (\[enorm\]) gives the boundary value of $\vec{e}_\nu(\vec{a}\,) \equiv -\vec{\nabla}
\Phi(\vec{r}\,-\vec{a}\,)$, where $\Phi(\vec{r}\,-\vec{a}\,)$ is the Green function of boundary problem (\[boundpr\]): $\nabla^2 \, \Phi(\vec{r}\,-\vec{a}\,) = -
\delta(\vec{r}\,-\vec{a}\,)$. For the symmetric case of an on-axis beam in a circular pipe of radius $b$ from Eq. (\[norma\]) immediately follows $e_\nu(0)=1/(2\pi b)$.
Likewise, a finite resistivity of the chamber wall leads to the resistive impedance per unit length of the chamber, e.g. [@RLGetc92], $$Z_L(k;\vec{a}\,)/L = Z_s(k) \oint_{\partial S}\! dl \
e^2_\nu(\vec{a}\,) \ , \label{Zres}$$ where the surface impedance $Z_s(k)$ is equal to $Z_0k\delta/2$ when skin-depth $\delta$ is smaller than the wall thickness.
Therefore, the problem of the impedance dependence on the beam position is reduced to evaluating $e_\nu(\vec{a}\,)$, cf. Eqs. (\[Z\]) and (\[Zres\]). It can be performed analytically for simple cross sections when the EFs are known, or numerically in a general case, applying any 2D electrostatic code and imposing (\[norma\]) for normalization of a numerical solution.
Beam-Position Dependence
========================
Circular Chamber
----------------
Using known eigenfunctions (e.g., [@Collin] or see [@KGS]) for a circular cross section of radius $b$, we sum up in Eq. (\[enorm\]) to get $$e_\nu(\vec{a}\,)= \frac{1}{2\pi b} \ \frac{b^2-a^2}
{b^2-2ab \cos (\varphi_a -\varphi_h) +a^2} \, . \label{ecirc}$$ Here $a$ is the beam offset, $\varphi_a, \, \varphi_h$ are azimuth positions of the beam and hole. Result (\[ecirc\]) coincides with the known distribution of the wall current, e.g. [@NasSach]. Figure 1 shows the beam-position dependence of the hole impedance (\[Z\]). Integrating in (\[Zres\]) yields the well-known beam-position dependence for the power loss, e.g. [@AChao], $$\oint_{\partial S}\! dl \ e^2_\nu(\vec{a}\,) =
\frac{1}{2\pi b} \, \frac{b^2 + a^2}{b^2 - a^2} \, . \label{e2circ}$$
Rectangular Chamber
-------------------
The eigenvalues and EFs for a rectangular chamber of width $w$ and height $h$ are well known, see in [@Collin] or [@KGS]. Let a hole be located in the side wall at $x=w, \ y=y_h$. Then from Eq. (\[enorm\]) for the beam offset $\vec{a}=(x,y)$; ($|x| \le w/2,
\, |y| \le h/2$) from the axis at $(w/2,h/2)$ follows $$\begin{aligned}
e_\nu(\vec{a}\,) = \frac{2}{h} \left [ \, \sum_{n=0}^\infty
(-1)^n \sin \frac{(2n+1)\pi y_h}{h} \times \right.
\qquad \quad \nonumber\\
\cos \frac{(2n+1)\pi y}{h} \; \frac{\sinh [(n+1/2)\pi (w+2x)/h]}
{\sinh [(2n+1)\pi w/h]} \label{erect} \\
+ \ \sum_{n=1}^\infty (-1)^n \sin \frac{2n\pi y_h}{h}
\sin \frac{2n\pi y}{h} \, \times \qquad \nonumber \\
\left. \frac{\sinh [n\pi (w+2x)/h]}
{\sinh [2n\pi w/h]} \right ]. \nonumber\end{aligned}$$ Despite a rather long expression, this series is fast convergent and convenient for evaluations, and it looks much simpler for a centered beam, with $x=y=0$, cf. [@SK92]. Figure 2 shows that the impedance increases significantly as the beam is displaced closer to the hole.
For integrated $e^2_\nu$ we obtain $$\begin{aligned}
\oint_{\partial S}\! dl \ e^2_\nu(\vec{a}\,) \ = \
\frac{4}{w} \left [ \ \sum_{n=0}^\infty
\cos^2 \frac{(2n+1)\pi x}{w} \times \right. \nonumber \\
\ \frac{\sinh^2 [(n+1/2)\pi (h+2y)/w]}
{\sinh^2 [(2n+1)\pi h/w]} \, + \label{e2rect}\\
\left. + \ \sum_{n=1}^\infty \sin^2 \frac{2n\pi x}{w}
\frac{\sinh^2 [n\pi (h+2y)/w]} {\sinh^2 [2n\pi h/w]} \ \right ]
\nonumber \\
+ \ \left \{ x \leftrightarrow y; \ w \leftrightarrow h
\right \} \ . \nonumber\end{aligned}$$ An example of a square pipe is illustrated in Fig. 3.
For a centered beam, i.e. $x=y=0$, it is reduced to $$\oint_{\partial S}\! dl \ e^2_\nu(0) =
\frac{1}{w} \, \sum_{n=0}^\infty
\cosh^{-2} \frac{(2n+1)\pi h}{2w}
+ \left \{w \leftrightarrow h \right \} , \label{e2r0}$$ the result obtained in [@RLGetc92], which was also expressed in a closed form in terms of elliptic integrals [@RGO].
On Transverse Impedance
=======================
The longitudinal and transverse wake functions are related by Panofsky-Wenzel theorem $$\vec{\nabla\,} W(z,\vec{a}\,) = \frac{\partial}
{\partial z} \vec{W}_\bot (z,\vec{a}\,) \ . \label{PW}$$ The longitudinal wake function corresponding to the inductive impedance (\[Z\]) of the hole is $W(z,\vec{a}\,)=\delta'(z)F(\vec{a}\,)$, where $F(\vec{a}\,)=Z_0 e^2_\nu(\vec{a}\,) (\psi - \chi)/2$. Together with Eq. (\[PW\]), it implies $\vec{W}_\bot (z,\vec{a}\,) = \delta(z) \vec{\nabla\,} F(\vec{a}\,)$, and the monopole transverse impedance defined as the Fourier transform of $\vec{W}_\bot (z,\vec{a}\,)$ in $\tau=z/c$, is $$\vec{Z}^{mon}_\bot (k,\vec{a}\,) = \frac{1}{c} \,
\vec{\nabla\,} F(\vec{a}\,) = Z_0 \frac{\psi - \chi}{2}
\vec{\nabla\,} e^2_\nu(\vec{a}\,) \ . \label{Zt0}$$ Defined in such a way $\vec{Z}^{mon}_\bot$ has dimension of Ohms, and can be easily calculated when $e_\nu(\vec{a}\,)$ is found, e.g.Eqs. (\[ecirc\]) or (\[erect\]). In an axisymmetric pipe, $\vec{Z}^{mon}_\bot = 0$, e.g. [@AChao], which formally follows from the fact that $Z_{long}$ is independent of the beam position in such a case. However, presence of a hole breaks this symmetry, so that $\vec{Z}^{mon}_\bot$ does not vanish even on the axis. For example, for a circular chamber with a hole $$\vec{Z}^{mon}_\bot(k,0) = Z_0 \frac{\psi - \chi}{4\pi^2 b^3}
\vec{h\,} \ , \label{Zt0c0}$$ where $\vec{h\,}$ is a unit vector from the axis toward the hole. The presence of a second, symmetric hole (or a few of them) restores the symmetry, and this effect disappears.
The transverse kick obtained by a test charge $q_t$ which follows, at distance $z \ge 0$, the leading charge $q_s$, is $$\vec{p}_\bot (z,\vec{a}\,) = \frac{q_t q_s}{c}
\vec{W}_\bot (z,\vec{a}\,) = \frac{q_t q_s}{c}
\delta(z) \vec{\nabla\,} F(\vec{a}\,) \ . \label{tkick}$$ As an example, Fig. 4 shows the direction and magnitude of the monopole impedance and corresponding transverse kick in a circular pipe. For a rectangular chamber, the picture is similar. The result (\[tkick\]) looks suspicious due to $\delta(z)$, which means there is no influence on any test charge with $z>0$, while self-influence of the source charge diverges. One should attribute this unphysical behavior to the approximations used: (i) point-like discontinuity, (ii) ultrarelativistic charge, and (iii) instant induction of effective dipoles on the hole. A rigorous approach, taking into account $\beta <1$ and a finite hole size, would lead to a more appropriate longitudinal dependence, although calculations will be certainly complicated. An involved direct calculation (using the method of the second paper of Ref. , again with $\beta =1$) of the integrated transverse force acting on an on-axis charge passing a hole in a circular pipe leads to divergent sums which, however, would be natural to put equal to zero[^2]. Anyway, this question remains open.
The more usual dipole transverse coupling impedance in the chamber with a hole, e.g. [@SK92; @SKrev; @KGS], reflects the influence of a couple of opposite-charged particles with transverse offsets $(\vec{s},-\vec{s})$ on a test charge with offset $\vec{t}$: $$\vec{Z}^{dip}_\bot (k,\vec{s},\vec{t}\,) = -i Z_0
\frac{\psi - \chi}{2} \; \frac{e_\nu(\vec{s}\,)-e_\nu(-\vec{s}\,)}
{2s} \, \vec{\nabla}e_\nu(\vec{t}\,) \ , \label{Zt1}$$ where the limit $\vec{s} \to \vec{t} \to 0$ is usually assumed. If instead one considers $\vec{t} \to 0$ while keeping $\vec{s}=\vec{a}$ finite, we get corrections to the transverse dipole impedance. For example, in a circular pipe ($\varepsilon=a/b<1$) $$\begin{aligned}
\vec{Z}^{dip}_\bot (k,\vec{a}) & = & -i Z_0 \frac{\psi - \chi}
{2\pi^2 b^4}\; \vec{h}\, \cos (\varphi_a -\varphi_h) \times \label{Zt1c}
\\ & & \ \frac{1-\varepsilon^2}{(1+\varepsilon^2\,)^2
-4\varepsilon^2 \cos^2 (\varphi_a -\varphi_h) } \ . \nonumber \end{aligned}$$ In the limit of $a \to 0$ it reproduces the known result for the transverse dipole impedance of the hole, the first line in (\[Zt1c\]), cf. . It corresponds to the deflecting force directed toward (or opposite to) the hole with its magnitude proportional to the beam offset and depending on beam azimuth position $\varphi_a$ as $\cos (\varphi_a -\varphi_h)$. Expanding in powers of $\varepsilon$ yields sextupole term and higher-order corrections: $$\cos{(\varphi_a -\varphi_h)} + \varepsilon^2
\cos{3(\varphi_a -\varphi_h)} + O\left(\varepsilon^4\right) \ .$$ Results for rectangular pipes are obtained in a similar way from Eq. (\[erect\]) in terms of series.
[9]{} A.W. Chao, [*Physics of Collective Beam Instabilities in High Energy Accelerators*]{}. New York: John Wiley, 1993. S.S. Kurennoy, [*Part. Acc.*]{}, [**39**]{}, 1 (1992);\
R.L. Gluckstern, [*Phys. Rev. A*]{}, [**46**]{}, 1106 (1992). S.S. Kurennoy, in [*Proceed. EPAC*]{} (Berlin, 1992), p. 871; more details in IHEP 92-84, Protvino, 1992. R.L. Gluckstern, J. van Zeijts, and B. Zotter, [*Phys. Rev. E*]{}, [**47**]{}, 656 (1993). R.E. Collin, [*Field Theory of Guided Waves.*]{} NY: McGraw-Hill, 1960. S.S. Kurennoy, R.L. Gluckstern and G.V. Stupakov, [*Phys. Rev. E*]{}, [**52**]{}, 4354 (1995). G. Nassibian and B. Sacherer, [*NIM*]{}, [**159**]{}, 21 (1979). F. Ruggiero, [*Phys. Rev. E*]{}, [**53**]{}, 2802 (1996). S.S. Kurennoy, Phys. Part. Nucl. [**24**]{}, 380 (1993).
[^1]: Presented at the 5th European Particle Accelerator Conference, Sitges (Barcelona), Spain, June 10-14, 1996
[^2]: Remark due to R.L. Gluckstern
|
---
author:
- |
[ Jacques Giacomoni and Guillaume Warnault]{}\
[LMAP (UMR CNRS 5142) Bat. IPRA,]{}\
[Avenue de l’Université ]{}\
[F-64013 Pau, France]{}\
[e-mail: jacques.giacomoni@univ-pau.fr]{}\
- |
[S. Prashanth]{}\
[TIFR-Centre For Applicable Mathematics]{}\
[Post Bag No. 6503, Sharada Nagar,]{}\
[GKVK Post Office,]{}\
[Bangalore 560065, India]{}\
[e-mail: pras@math.tifrbng.res.in]{}\
title: 'Biharmonic equation with Singular nonlinearity in ${{\mathbb R}}^N$ [^1] '
---
Introduction
============
Let $\Omega$ be a bounded smooth domain in ${{\mathbb R}}^N$, $N\geq 2$ and denote $\rho(x):=d(x,\partial\Omega),\ x \in \Omega$. Denote by $\phi_1$ and $\lambda_1$ respectively the first (positive) eigenfunction and the first eigenvalue of $-\Delta$ in the space $H^1_0(\Omega)$. Also, let $G$ denote the positive Green’s function for $-\Delta$ in $\Omega$. Assume that $K\in C^\nu_{loc}(\Omega)$ ($\nu\in(0,1)$) is such that $$\inf_{\Omega} K >0, \;\;\text{ and }\;\; K =O(\rho^{-\beta}) \;\;\text{ near }\;\; \partial \Omega, \;\;\text{ for some }\;\; \beta \geq 0.$$ Given $\alpha>0$, we consider the following fourth order singular elliptic problem: $$\begin{aligned}
( P)\qquad \displaystyle\left\{\begin{array}
{ll}
& \Delta^2 u
= K(x)u^{-\alpha}
\quad \mbox{ in }\,\Omega , \\
&u> 0\quad \mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega}=0, \,\Delta u\vert_{\partial\Omega} = 0.
\end{array}\right.
\end{aligned}$$
There is a large literature concerning such singular problems (as well as the corresponding systems) for second order elliptic operators wherein questions of existence, uniqueness and multiplicity, regularity, asymptotic behaviour, symmetry, etc. have been investigated (see for instance [@AdGi], [@BaGi], [@CaGi], [@CrRaTa], [@DhGiPrSa], [@Gh1], [@Ha], [@HeMaVe], [@HiSaSh]). Similar results for the quasilinear case have been obtained in [@GiScTa] and [@GiScTa1]. We refer the reader to the two excellent surveys [@GhRa] and [@HeMa] for more details.
There are very few results available which concern fourth order singular problems similar to $(P)$. In [@Gh], the author studies the problem $ \Delta^2 u = u^{-\alpha}$, $\alpha<1,$ but with Dirichlet boundary condition. Furthermore, the author assumes that the domain is a perturbation of the ball to ensure positivity of the associated Green’s function. Using the Schauder fixed point theorem to a suitable integral formulation of the problem in an appropriate cone of positive continuous functions, the existence and the uniqueness of a solution in $C^2(\Omega)\cap C^1_0(\overline{\Omega})$ that behaves like $\rho^2$ near the boundary is shown in this work. Since such a boundary behaviour is expected, the restriction $\alpha<1$ is necessary.
In contrast with [@Gh], we consider the problem $(P)$ for a general smooth bounded domain $\Omega$ with Navier boundary conditions. We first clarify the notion of a solution to $(P)$:
\[first-defi\] A function $u\in C^2(\overline{\Omega})$ is a solution to $(P)$ if $u>0$ in $\Omega$, $u=\Delta u=0$ on $\partial\Omega$ and satisfies the following integral identity for any $\psi\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$: $$\label{varia-form}
\int_{\Omega}\Delta u\Delta\psi{\rm d}x=\int_{\Omega}K(x) u^{-\alpha}\psi{\rm d}x.$$
\[remark1\]
1. We require $C^2(\overline{\Omega})$ regularity to be able to define $\Delta u=0$ on $\partial\Omega$.
2. A consequence of the above definition is that a solution $u$ to $(P)$ necessarily satisfies $$\int_{\Omega} K(x) u^{-\alpha}(x)\rho(x){\rm d}x<\infty.$$ To see this, plug in the test function $\psi=\phi_1$ in , where $\phi_1$ is the first (normalized) positive eigenfunction of $-\Delta$ on $H^1_0(\Omega)$.
3. Definition \[first-defi\] is similar to the concept of very weak solution given in [@BrMaPo] (see definition 0.2 there) for solving second order elliptic problems with $L^1$- ( or measure) data. We adapt this notion here for fourth order elliptic equations.
The solution to $(P)$ can be defined equivalently using the Green’s representation formula (see proposition \[defi-equiv\] in section \[section3\]). It is easy to see that the equation in $(P)$ is equivalent to the following second order elliptic system: $$\begin{aligned}
( PS)\qquad \displaystyle\left\{\begin{array}
{ll}
& -\Delta u
= v
\quad \mbox{ in }\,\Omega , \, u> 0\quad \mbox{ in }\,\Omega,\\
&-\Delta v=K(x) u^{-\alpha} \quad \mbox{ in }\,\Omega ,\\
& \;\;u\vert_{\partial\Omega}=0, \,v\vert_{\partial\Omega} = 0.
\end{array}\right.
\end{aligned}$$ Nevertheless, $(PS)$ is not a cooperative system and hence monotone methods can not be used to prove existence of solutions to $(PS)$, as is done in [@CrRaTa] for the single equation. Furthermore, for $\alpha\in (0,1)$, the problem $(P)$ has a variational structure and the energy functional $J$ associated to $(P)$ is defined as follows: $$\begin{aligned}
\label{china}
\qquad J(w){\stackrel{{\rm{def}}}{=}}\frac{1}{2}\int_{\Omega}(\Delta w)^2 {\rm d}x- \frac{1}{1-\alpha}\int_{\Omega} K(x)w^{1-\alpha}{\rm d}x \;\;\text{ for }\;\; w\in X{\stackrel{{\rm{def}}}{=}}H^2(\Omega)\cap H^1_0(\Omega).\end{aligned}$$ Clearly, $J$ is well defined in the cone of nonnegative functions in $X$ provided $K$ has moderate singularity near $\partial\Omega$. But the main difficulty is that truncation techniques (which work in case of second order elliptic equations) can not be used directly since we are in the $H^2$-framework. This makes it difficult to employ variational methods for studying $(P)$. Another difficulty is that the Schauder fixed point theorem (used in [@Gh]) works only in the case $\alpha<1$ where the invariance of the solution operator with respect to a cone of positive solutions can be ensured.
For these reasons, our approach in this paper is slightly different. We first approximate the singular problem $(P)$ by a family of problems $(P_\epsilon)$ with regular terms as given below and use apriori estimates to show the existence of solution. We now state the results that we prove:
\[main\] Assume that $\alpha+\beta<2$. Then there exists a unique solution $u$ to $(P)$. Furthermore, there exist $c_1, c_2>0$ such that $$\begin{aligned}
\label{behaviour-bound}
c_1 \rho(x)\leq u(x)\leq c_2 \rho(x).\end{aligned}$$
The idea behind the proof (see section \[section2\]) is to approximate the problem in the following way: $$\begin{aligned}
( P_\epsilon)\qquad \displaystyle\left\{\begin{array}
{ll}
& \Delta^2 u
= K_\epsilon(x) (u+\epsilon)^{-\alpha}
\quad \mbox{ in }\,\Omega , \;\; K_\epsilon:=\min(\frac{1}{\epsilon}, K),\\
&u> 0\quad \mbox{ in }\,\Omega, \;\;u\vert_{\partial\Omega}=0,\, \Delta u\vert_{\partial\Omega} = 0.
\end{array}\right.
\end{aligned}$$ The existence of the solution $u_\epsilon$ to $(P_\epsilon)$ can be obtained by the Schauder fix point theorem. We then prove a priori estimates on $\{u_\epsilon\}_{\epsilon>0}$ using crucially the restrictions on $\alpha,\beta$ and pass to the limit as $\epsilon\to 0^+$.
The following nonexistence result proves that the restriction $\alpha+\beta<2$ is sharp in the above results:
\[nonexistence\] Assume that $\alpha+\beta\geq 2$. Then, there is no solution to $(P)$.
Next, we use Theorem \[main\] to obtain the existence of a path-connected branch of solutions to the following bifurcation problem: $$\begin{aligned}
\label{bifurcation-problem}
( P_\lambda)\qquad \displaystyle\left\{\begin{array}
{ll}
& \Delta^2 u
= K(x) u^{-\alpha}+\lambda f(u)
\quad \mbox{ in }\,\Omega ,\\
&u> 0\quad \mbox{ in }\,\Omega, \;\; u\vert_{\partial\Omega}=0, \,\Delta u\vert_{\partial\Omega} = 0
\end{array} \right.\end{aligned}$$ where $\lambda$ is the bifurcation parameter and $f$ a function satisfying the following assumptions:
- $f:[0,\infty) \to [0,\infty)$ is a twice continuously differentiable map with $f(0)=0$.
- $f(t)$ is a finite product of functions of the form $g(t^{p}), p > 0,$ where $g$ is a real entire function on ${{\mathbb R}}$.
- $f' \geq 0$ and $\displaystyle \liminf_{t \to \infty} \frac{f(t)}{t} >0.$
Given a positive continuous function $\phi$ on $\Omega$, denote by $$\mathcal{C}_{\phi}(\Omega):= \Big \{ u\in C(\overline{\Omega}): \displaystyle \sup_{\Omega} \Big|\frac{u}{\phi} \Big| < +\infty \Big \},$$ with the norm $$\| u\|_{\mathcal{C}_{\phi}(\Omega)}:= \displaystyle \sup_{\Omega} \Big|\frac{u}{\phi} \Big|$$ and the “positive cone" $$\mathcal{C}_{\phi}^+(\Omega):= \Big \{ u\in \mathcal{C}_{\phi}(\Omega): \displaystyle \inf_{\Omega} \frac{u}{\phi} >0 \Big \}.$$ We define the inverse of the biharmonic operator denoted as $(\Delta^2)^{-1}$ as follows: $$(\Delta^2)^{-1}h = u$$ where for $h$ in an appropriate space, $u$ solves the inhomogeneous problem: $$\begin{aligned}
\label{biharmonic-f}
\displaystyle\left\{\begin{array}
{ll}
& \Delta^2 u
= h
\quad \mbox{ in }\,\Omega ,\\
&u\vert_{\partial\Omega}= \Delta u\vert_{\partial\Omega} = 0.
\end{array}\right.\end{aligned}$$
The bifurcation analysis is done in the space ${{\mathbb R}}\times \mathcal{C}_{\phi_1}(\Omega)$. Therefore, we consider the following set of all solutions (in the sense of definition \[first-defi\]) $$\label{soln set}
\mathcal{S}= \{u\in C^4(\Omega)\cap C^2(\overline{\Omega}), u>0 \text{ solves } (P_\lambda)\} \subset {{\mathbb R}}\times \mathcal{C}_{\phi_1}(\Omega).$$ Consider the following solution operator associated to $(P_\lambda)$ : $$F(\lambda ,u)=u- (\Delta^2 )^{-1}(K(x)u^{-\alpha }+\lambda f(u)),\;
(\lambda ,u)\in \mathbb{R} \times {\mathcal C}_{\phi_1 }^+(\Omega ), \;0<\alpha+\beta<2. \label{3}$$ Using the framework of analytic bifurcation theory as developed in the works [@BuDaTo] and [@BuDaTo2] (see also [@BoGiPr] and [@BuTo]), we obtain an analytic global unbounded path of solutions to $(P_\lambda)$:
\[th1\] Let $f$ satisfy conditions $(f_0)-(f_2)$ and $\alpha+\beta<2$. Then, $F : \mathbb{R} \times {\mathcal C}_{\phi_1 }^+(\Omega ) \to {\mathcal C}_{\phi_1 }(\Omega )$ is an analytic map (see definition \[anal map\]). Furthermore, there exists $\Lambda \in(0,\infty)$ and an unbounded set ${\mathcal A} \subset (-\infty,\Lambda]\times {\mathcal C}^+_{\phi_1}(\Omega) \subset \mathcal{S}$ of solutions to $(P_\lambda)$ which is globally parametrised by a continuous map :
$$(-\infty,\infty) \ni s \to (\lambda(s),u(s)) \in \mathcal{A} .$$
Moreover, the following properties hold along this path $\mathcal{A}$:
- $(\lambda(s),u(s))\to (0,u_0)$ in ${{\mathbb R}}\times {\mathcal C}_{\phi_1}(\Omega)$ as $s \to 0$, where $u_0$ is the unique solution to $(P)$.
- $\Vert u(s)\Vert_{{\mathcal C}_{\phi_1}(\Omega)}\to \infty$ as $s\to\infty$ .
- $\mathcal{A}$ has at least one asymptotic bifurcation point $\Lambda_a\in [0,\Lambda]$. That is, there exist sequences $\{s_n\}_{n\in{{\mathbb N}}}\subset (0,\infty)$, $\{(\lambda(s_n), u(s_n))\} \subset {\mathcal A}$ such that $s_n\to\infty$, $\lambda(s_n)\to\Lambda_a$ and $\Vert u(s_n)\Vert_{{\mathcal C}_{\phi_1}(\Omega)} \to \infty$.
- $\left\{s\geq 0\,:\,\partial_u F(\lambda(s),u(s)) \text{ is not invertible }\right\}$ is a discrete set.
- ($\mathcal{A}$ is an “analytic" path) At each of its points ${\mathcal A}$ has a local analytic re-parameterization in the following sense: For each $s^*\in {{\mathbb R}}$ there exists a continuous, injective map $\rho^*\,:\, (-1,1)\to {{\mathbb R}}$ such that $\rho^*(0)=s^*$ and the re-parametrisation $$\begin{aligned}
(-1,1) \ni t\to (\lambda(\rho^*(t)),u(\rho^*(t))) \in \mathcal{A} \mbox{ is analytic}.\end{aligned}$$ Furthermore, the map $s \mapsto \lambda(s)$ is injective in a neighborhood of $s=0$ and for each $s^* >0$ there exists $\epsilon^*>0$ such that $\lambda$ is injective on $[s^*,s^*+\epsilon^*]$ and on $[s^*-\epsilon^*,s^*]$.
- For any $\lambda\leq 0$, there exists atmost one solution to $(P_\lambda)$ and ${\mathcal A} \cap (-\infty,0) \times {\mathcal C}_{\phi_1}(\Omega )$ is a single analytic curve which is a graph from the $\lambda$ axis consisting of non-degenerate solutions $u_\lambda$. In particular, we can take $\lambda(s)=s$ for $s<0$.
The paper is organized as follows: In Section \[section2\], we prove Theorem \[main\] using a version of Hopf principle recalled in proposition \[prop-Hopf\]. In Section \[section3\], we study the equivalence between the two definitions of a solution and prove Theorem \[nonexistence\]. Finally, in Section \[section4\] we prove Theorem \[th1\].
Some preliminary results for Theorem \[main\] {#prel}
=============================================
We first prove a version of Hopf principle.
\[prop-Hopf\] Let $h\in L^\infty(\Omega)$ be a nonnegative function. Let $u$ be the classical solution to . Then there exists a constant $C>0$ (independent of $h$) such that the following inequality holds: $$\label{1}
u(x)\geq C \rho(x)\int_{\Omega}h(y)\rho(y){\rm d}y.$$
Since $h\in L^\infty(\Omega)$, $u$ solves the following system: $$\begin{aligned}
\label{haf}
\displaystyle\left\{\begin{array}
{ll}
& -\Delta u
= v
\quad \mbox{ in }\,\Omega ,\\
&-\Delta v=h\quad \mbox{ in }\,\Omega ,\\
& \;\;u\vert_{\partial\Omega}= v\vert_{\partial\Omega} = 0.
\end{array}\right.
\end{aligned}$$ Recall from lemma 3.2 in [@BrCa] that for any nonnegative function $h\in L^\infty(\Omega)$, the unique solution $w$ to the problem $$\begin{aligned}
\displaystyle\left\{\begin{array}{ll}
& -\Delta w = h \quad \mbox{ in }\,\Omega ,\\
&w\vert_{\partial\Omega}=0
\end{array}\right.\end{aligned}$$ satisfies the estimate: $$w(x)\geq C\rho(x)\int_{\Omega}h(y)\rho (y){\rm d}y \quad x \in \Omega,$$ where the constant $C$ does not depend to $h$.
We apply the previous inequality to $u$ and $v$ to get $$\begin{aligned}
u(x)\geq C \rho(x)\int_{\Omega}v(y)\rho(y){\rm d}y\geq C^2\rho(x)\int_{\Omega}\rho(y)^2{\rm d}y\int_{\Omega}h(z)\rho(z){\rm d}z\end{aligned}$$ which completes the proof.\
By a simple approximation argument and the maximum principle, we have the
\[dee\] Let $h \rho \in L^1(\Omega)$ and nonnegative. Then any $u$ solving (in the sense of definition \[first-defi\]) satisfies the inequality .
We next have the following regularity and uniform estimate result :
\[jee\] Let $h\in C^\nu_{loc}(\Omega)$ be a nonnegative function such that $h \rho^{\delta} \in L^{\infty}(\Omega)$ for some $0<\delta<2$. Let $u\in C^2(\overline{\Omega})$ be the solution (in the sense of definition \[first-defi\]) to . Then there exist constants $C>0$ (dependent on $\|h\rho^{\delta}\|_{L^{\infty}(\Omega)}$, $\nu$ and $\delta$) and $0<\theta<1$ (depending on $\nu$ and $\delta$) such that the following inequality holds: $$\label{1b}
\|u\|_{C^{2,\theta}(\overline{\Omega})} \leq C.$$
Since $h \rho \in L^1(\Omega)$, from the above corollary we obtain that $u \geq c \rho$ for some $c>0$. Since $u \in C^2(\overline{\Omega}) \cap C_0(\overline{\Omega})$, we obtain that $u \sim \rho$ near $\partial\Omega$. We note that $v:=-\Delta u\in C^{2,\nu}_{loc}(\Omega)$ by elliptic regularity and is a nonnegative function by the maximum principle. Consider the equivalent system for $u,v$ as in . Then we have $$\label{cam}
\vert\Delta v\vert\leq C_0\rho^{-\delta}, \quad \text{ where }\;\; C_0:= \|h\rho^{\delta}\|_{L^{\infty}(\Omega)}.$$ Let $w:=w(\delta)$ denote the unique positive solution to $$\begin{aligned}
\left\{\begin{array}{ll}
&-\Delta w = w^{-\delta}\mbox{ in }\Omega,\\
&w= 0\mbox{ on }\partial\Omega.
\end{array}\right.\end{aligned}$$ From [@CrRaTa], there exist positive constants $c_1< c_2$ such that the following estimates hold: $$\begin{aligned}
c_1\rho \leq& w &\leq c_2\rho \;\;\text{ if }\;\;0<\delta <1, \notag \\
c_1\rho\ln(\frac{D}{\rho})\leq& w &\leq c_2\rho\ln(\frac{D}{\rho}) \;\;\text{ if }\;\; \delta=1 \;\; (D:= diam(\Omega)) \;\;\text{ and }\;\; \notag\\
c_1\rho^{\frac{2}{\delta+1}}\leq& w &\leq c_2\rho^{\frac{2}{\delta+1}} \;\;\text{ if }\;\;1<\delta<2. \notag\end{aligned}$$ Choosing the constant $M>0$ large enough (depending on $C_0,\ c_1$ and $\delta$) and using the weak comparison principle, we can conclude $v-Mw \leq 0$. Thus, we have $$\label{mil}
0\leq v \leq M w \leq Mc_2 \rho^{\mu} \;\;\text{ for some} \;\; \mu>0.$$ By noting and , appealing to Proposition 3.4 in [@GuLi] we obtain that $v\in C^{0,\theta}(\overline{\Omega})$ for some $\theta:=\theta(\delta, \nu)\in (0,1)$ and $\Vert v\Vert_{C^{0,\theta}(\overline{\Omega})}\leq C=C(C_0, \delta)$. We then apply the classical elliptic theory to get $u\in C^{2,\theta}(\overline{\Omega})$ and $\Vert u\Vert_{C^{2,\theta}(\overline{\Omega})}\leq \tilde C=\tilde C(C_0, \delta,\nu)$.
We can now show the following result on existence of $C^2(\overline{\Omega})$ solution (as in definition \[first-defi\]) by means of a simple approximation argument:
\[gul\] Let $h$ be a nonnegative function such that $h \rho^{\delta} \in L^{\infty}(\Omega)$ for some $0<\delta<2$. Then there exists a unique solution $u \in C^2(\overline{\Omega})$ solving .
Define $h_n := \min\{h, n\}$. Let $u_n \in C^2(\overline{\Omega})$ be the unique solution to with $h=h_n$. We note that given $\psi\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$ there exist $p>1$ such that $$\Big\{h_n \psi\Big\} \quad \text{ is a bounded sequence in } \;\; L^p(\Omega).$$ Then, by Vitali’s convergence theorem $$\int_{\Omega} h_n \psi \to \int_{\Omega} h \psi \;\;\text{ as }\;\; n \to \infty.$$ By appealing to lemma \[jee\] we obtain as well that for some $\theta \in (0,1)$, $$\Big\{u_n\Big\} \quad \text{ is a bounded sequence in } \;\; C^{2,\theta}(\overline{\Omega}).$$ Thus, upto a subsequence $u_n \to u$ in $C^{2}(\overline{\Omega}).$ It is then easy to see that $u$ solves .
We recall the Hardy Inequality for $H^s$ spaces:
\[Hardy\] Let $s\in [0,2]$ such that $s\neq \frac{1}{2}$ and $s\neq \frac{3}{2}$. Then the following generalisation of Hardy’s inequality holds: $$\|\rho^{-s} g\|_{L^{2}(\Omega)}\leq C \|g\|_{H^{s}(\Omega)}\quad \mbox{ for all }g\in H_0^s(\Omega). \label{eq4b}$$
Finally, we state the following regularity result.
\[hum\] Assume that $h$ is a nonnegative function such that $h \rho^{\delta} \in L^{\infty}(\Omega)$ for some $0<\delta<2$. Let $u \in C^2(\overline{\Omega}) \cap C_0(\overline{\Omega})$ be the solution of . Then $u \in W^{4,p}_{loc}(\Omega) \cap H^{4-s}(\Omega)$ for any $1 \leq p< \infty$ and $s \in (\delta -\frac{1}{2},2] \setminus \{\frac{1}{2}, \frac{3}{2} \}$.
That $u \in W^{4,p}_{loc}(\Omega)$ for $1\leq p <\infty$ follows from standard elliptic regularity result. Fix any $s \in (\delta -\frac{1}{2},2] \setminus \{\frac{1}{2}, \frac{3}{2} \}$. We claim that $ h \in H^{-s}(\Omega)$. Indeed, using lemma \[Hardy\], for any $\xi \in H^{s}_0(\Omega)$, $$\begin{aligned}
\Big|\int_{\Omega }h \xi \Big | & = & \left\vert \int_{\Omega }(h \rho^{s})(\xi \rho ^{-s})\right\vert \notag \\
& \leq & C\int_{\Omega }(\rho^{s-\delta })(\rho^{-s}|\xi |) \notag \\
& \leq & C\Vert \rho^{s-\delta}\Vert
_{L^{2}(\Omega )}\Vert \rho^{-s}\xi \Vert _{L^{2}(\Omega )}
\notag \\
& \leq & C\Vert \xi \Vert _{H_{0}^{s}(\Omega )} \notag.
\label{17}\end{aligned}$$ Hence by elliptic regularity used successively to $v$ and $u$ we obtain that $u \in H^{4-s}(\Omega)$.
Proof of Theorem \[main\] {#section2}
=========================
We first show that the solution is unique. Let $u_1$ and $u_2$ be two solutions to $(P)$. Then, $$\begin{aligned}
\label{arg-uniqueness}
\int_{\Omega}(\Delta(u_1-u_2))^2{\rm d}x=\int_{\Omega}K(x) (u_1^{-\alpha}-u_2^{-\alpha})(u_1-u_2){\rm d}x\leq 0.\end{aligned}$$ Therefore, since $u_1, u_2\in H^1_0(\Omega)$, we obtain $u_1\equiv u_2$.
Fix $\epsilon>0$. We next prove the existence of a unique solution to $(P_\epsilon)$. Let ${\mathcal W}$ be the positive cone of $C_0(\overline{\Omega})$, i.e. $$\begin{aligned}
{\mathcal W}{\stackrel{{\rm{def}}}{=}}\left\{u\in C_0(\overline{\Omega})\,\vert\, u\geq 0 \mbox{ in }\Omega\right\}. \end{aligned}$$ We define the functional $\Phi\, :\, {\mathcal W}\to {\mathcal W}$ as the solution to the following problem: $$\begin{aligned}
\displaystyle\left\{\begin{array}
{ll}
& \Delta^2 \Phi(u)
= K_\epsilon(x)(u+\epsilon)^{-\alpha}
\quad \mbox{ in }\,\Omega ,\\
&\Phi(u)\vert_{\partial\Omega}=0, \,\Delta \Phi(u)\vert_{\partial\Omega} = 0.
\end{array}\right.\end{aligned}$$ By the elliptic regularity theory, $\Phi$ is a compact linear operator on $C_0(\overline{\Omega})$, and by the weak comparison principle, leaves the closed convex set ${\mathcal W}$ invariant. Hence, by the Schauder fixed point theorem, there exists $u_\epsilon\in {\mathcal W}$ solution to $(P_\epsilon)$. Using a similar argument as in , $u_\epsilon$ is the unique solution to $(P_\epsilon)$. By elliptic regularity, we also have $u_\epsilon \in C^2(\overline{\Omega})$.
Multiplying the equation satisfied by $u_\epsilon$ by $\phi_1$, we obtain that $$\label{ji}
\lambda_1^2\int_{\Omega}u_\epsilon\phi_1{\rm d}x=\int_{\Omega}\Delta^2u_\epsilon\phi_1{\rm d}x=\int_{\Omega}K_\epsilon(x)\phi_1 (u_\epsilon(x)+\epsilon)^{-\alpha}{\rm d}x.$$
First we show a uniform lower bound:
\[jyo\] There exists a constant $C>0$ independent of $\epsilon$ such that $u_\epsilon \geq C \rho$ in $ \Omega$.
We first show the following fact: $$\label{cha}
\displaystyle \inf_{\epsilon>0}\int_{\Omega} K_\epsilon \phi_1 (u_\epsilon+\epsilon)^{-\alpha}{\rm d}x>0.$$ We argue by contradiction. Suppose, up to a subsequence, $$\int_{\Omega}K_\epsilon \phi_1 (u_\epsilon+\epsilon)^{-\alpha}{\rm d}x\to 0 \;\;\text{ as }\;\; \epsilon\to 0^+.$$
Using this implies that $\int_{\Omega}u_\epsilon \phi_1{\rm d}x\to 0$ and hence $
u_\epsilon\to 0$ in $L^1_{\rm loc}(\Omega)$ as $\epsilon\to 0^+$.
Again up to a subsequence, we deduce that $$\begin{aligned}
\label{conv-pp}
u_\epsilon\to 0 \;\; \text{ and }\;\; K_\epsilon (u_\epsilon +\epsilon)^{-\alpha}\to 0\;\mbox{ a.e. in } \Omega\mbox{ as }\epsilon\to 0^+\end{aligned}$$ which is a contradiction. This proves above.
By elliptic regularity theory, $u_\epsilon\in C^{2,\gamma}(\overline{\Omega})$ for any $\gamma\in (0,1)$ and from Proposition \[prop-Hopf\] the estimate $$\begin{aligned}
\label{Hopf-epsilon}
u_\epsilon(x)\geq C\rho(x)\int_{\Omega}K_\epsilon (u_\epsilon+\epsilon)^{-\alpha}\rho \;{\rm d}y\end{aligned}$$ holds. The conclusion follows from .
\[chr\] There exists $\theta \in (0,1)$ independent of $\epsilon>0$ such that $$\sup_{\epsilon>0} \|u_\epsilon\|_{C^{2,\theta}(\Omega)} < +\infty.$$
From the last proposition, it follows that $$\label{ahm}
K_\epsilon (u_\epsilon + \epsilon)^{-\alpha} \rho^{\alpha+\beta} \in L^{\infty}(\Omega).$$ Noting that $0<\alpha+\beta<2$ and invoking lemma \[jee\], the conclusion follows.
Let $ u_\epsilon \to u$ in $C^2(\overline{\Omega})$ as $\epsilon \to 0$. From , we note that given $\psi\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$ there exists $p>1$ such that $$\Big\{K_\epsilon (u_\epsilon + \epsilon)^{-\alpha} \psi \Big\}_{\epsilon>0} \quad \text{is a bounded family in } L^{p}(\Omega).$$ We can now use Vitali’s convergence theorem to directly pass to the limit as $\epsilon \to 0$ in $(P_\epsilon)$ to conclude that $u$ solves $(P)$.
Proof of Theorem \[nonexistence\] {#section3}
==================================
We first prove the following equivalent way of defining a solution to $(P)$:
\[defi-equiv\] $u\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$ is a solution to $(P)$ (in the sense of definition \[first-defi\]) if $u>0$ in $\Omega$ and verifies $$\begin{aligned}
\label{green-funct}
u(x)=\int_{\Omega}G(x,y)\left(\int_{\Omega}G(y,z)K(z) u^{-\alpha}(z) {\rm d}z\right){\rm d}y.\end{aligned}$$
Assume first that $u$ satisfies Definition \[first-defi\]. From the estimates in Proposition 4.13 in [@GaGrSw] and noting $\int_{\Omega}K(z)\rho(z)u^{-\alpha}(z){\rm d}z<\infty$ (see Remark \[remark1\]), we obtain by Fubini’s theorem that for any $x\in \Omega$, $$\begin{aligned}
\int_\Omega G(x,y)\left(\int_\Omega G(y,z)K(z) u^{-\alpha}(z) {\rm d}z\right){\rm d}y=\int_{\Omega}K(z)u^{-\alpha}(z) {\rm d}z \int_{\Omega}G(x,y)G(y,z){\rm d}y<\infty.\end{aligned}$$ Therefore, by classical arguments, $u$ satisfies .
Now assume that $u\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$, $u>0$ in $\Omega$ and verifies . Let us show that $u$ satisfies Definition \[first-defi\]. For that, observe that for $\eta>0$ small enough, $$\begin{aligned}
\label{phi1-control}
u(x)-\frac{\eta}{\lambda_1^2}\phi_1(x)=\int_{\Omega}G(x,y)\left(\int_{\Omega}G(y,z)(K(z) u^{-\alpha}(z)-\eta\phi_1(z)){\rm d}z\right)\geq 0.\end{aligned}$$ Thus $u \geq c \rho$ for some $c>0$. From the $C^2$-regularity of $u$, we also have that $u \leq C\rho$ for $C>0$. Therefore, by the assumptions on $K$, for any $\psi\in C^2(\overline{\Omega})\cap C_0(\overline{\Omega})$, $$\begin{aligned}
\int_{\Omega} K(x) u^{-\alpha}(x) \psi(x){\rm d}x<\infty\end{aligned}$$ and hence $u$ satisfies .
Now we prove Theorem \[nonexistence\]. Let $\alpha+\beta\geq 2$ and $u$ be a solution to $(P)$. From Proposition \[defi-equiv\], the inequality holds and noting that $u\in C^1(\overline{\Omega})$, we obtain that $u \sim \rho$ in $\Omega$. Since $\alpha+\beta\geq 2$, from Theorem 2.4 in [@Gh1], we get the required contradiction.
Bifurcation results {#section4}
===================
In this section we prove Theorem \[th1\]. We consider the following bifurcation framework (see Chapter 9 in [@BuTo] or Theorem 1.13 in [@BoGiPr] for more details):
Let $\mathcal{X}$, $\mathcal{Y}$ be real Banach spaces, $\mathcal{U}\subset{{\mathbb R}}^+\times \mathcal{X}$ an open set. Let $\Psi : \mathcal{U} \to \mathcal{Y}$ be a map.
\[anal map\] $\Psi$ is said to be real analytic on $\mathcal{U}$ if for each $x \in \mathcal{U}$ there is an $\varepsilon > 0$ and continuous $k$-homogeneous polynomials $P_k : \mathcal{U} \to \mathcal{Y}$ such that $\Psi(x + h) = \sum_{k=0}^{\infty} P_k (h)$ if $\|h\| < \varepsilon$.
Define the solution set $${\mathcal S}=\displaystyle\left\{(\lambda,x)\in \mathcal{U}\,:\, \Psi(\lambda,x)=0\right\}$$ and the non-singular solution set $$\mathcal{N}= \displaystyle\left\{(\lambda,x)\in \mathcal{S}\,:\,Ker( \partial_x \Psi(\lambda,x))=\{0\}\right\}.$$
\[def dist arc\] A distinguished arc is a maximal connected subset of $\mathcal{N}$.
Suppose that
- Bounded closed subsets of ${\mathcal S}$ are compact in ${{\mathbb R}}\times \mathcal{X}$.
- $\partial_x\Psi(\lambda, x)$ is a Fredholm operator of index zero for all $(\lambda,x) \in \mathcal{S}$.
- There exists an analytic function $(\lambda,u) \, : (-\epsilon,\epsilon) \to \mathcal{S}$ such that $\partial_x \Psi(\lambda(s),u(s))$ is invertible for all $s\in (-\epsilon,\epsilon)$ and $\displaystyle\lim_{s\to 0^+}(\lambda(s), u(s))=(0,u_0)$ where $u_0\in {\mathcal X}$ is the unique solution to $\Psi(0,u_0)=0$.
Let $${\mathcal A_0}=\left\{(\lambda(s),u(s))\,:\, s\in (-\epsilon,\epsilon)\right\}.$$ Obviously, ${\mathcal A_0}\subset {\mathcal S}$. The following result gives a global extension of the function $(\lambda, u)$ from $(-\epsilon,\epsilon)$ to $(-\infty,\infty)$ in the real analytic case.
\[theo9.1.1\] Suppose (G1)-(G3) hold. Then, $(\lambda,u)$ can be extended as a continuous map (still called) $(\lambda,u) : (-\infty,\infty) \to \mathcal{S}$ with the following properties:
- Let ${\mathcal A} {\stackrel{{\rm{def}}}{=}}\{(\lambda(s),u(s)): s \in{{\mathbb R}}\}.$ Then, ${\mathcal A} \cap {\mathcal N}$ is an atmost countable union of distinct distinguished arcs $\bigcup_{i=1}^n {\mathcal A_i},\; n \leq \infty$.
- ${\mathcal A_0}\subset {\mathcal A_1}$.
- $\{s\in{{\mathbb R}}\,:\,ker(\partial_x \Psi(\lambda(s),u(s))) \neq \{0\}\}$ is a discrete set.
- At each of its points ${\mathcal A}$ has a local analytic re-parameterization in the following sense: For each $s^*\in {{\mathbb R}}$ there exists a continuous, injective map $\rho^*\,:\, (-1,1)\to {{\mathbb R}}$ such that $\rho^*(0)=s^*$ and the re-parametrisation $$\begin{aligned}
(-1,1) \ni t\to (\lambda(\rho^*(t)),u(\rho^*(t))) \in \mathcal{A} \mbox{ is analytic}.\end{aligned}$$ Furthermore, the map $s \mapsto \lambda(s)$ is injective in a neighborhood of $s=0$ and for each $s^*\neq 0$ there exists $\epsilon^*>0$ such that $\lambda$ is injective on $[s^*,s^*+\epsilon^*]$ and on $[s^*-\epsilon^*,s^*]$.
- Only one of the following alternatives occurs:
- $\Vert(\lambda(s),u(s))\Vert_{{{\mathbb R}}\times \mathcal{X}}\to\infty$ as $s\to +\infty$ (resp. $s \to -\infty$).
- a subsequence $\{(\lambda(s_n),u(s_n))\}$ approaches the boundary of $\mathcal{U}$ as $s_n \to +\infty$ (resp. $s_n \to -\infty$).
- ${\mathcal A}$ is the closed loop : $${\mathcal A}=\left\{(\lambda(s),u(s))\,:\, -T\leq s\leq T, (\lambda(T),u(T))=(\lambda(-T),u(-T)) \text{ for some } T>0 \right\}.$$ In this case, choosing the smallest such $T>0$ we have $$\begin{aligned}
(\lambda(s+2T),u(s+2T))=(\lambda(s),u(s)) \mbox{ for all } s \in {{\mathbb R}}.\end{aligned}$$
- Suppose $\partial_x \Psi(\lambda(s_1),u(s_1))$ is invertible for some $s_1\in{{\mathbb R}}$. If for some $s_2\neq s_1$, we have $(\lambda(s_1),u(s_1))=(\lambda(s_2),u(s_2))$ then (e)(iii) occcurs and $\vert s_1-s_2\vert$ is an integer multiple of $2T$. In particular, the map $s \mapsto (\lambda(s), u(s))$ is injective on $[-T,T)$.
\[bif from u0\] We remark that theorem 9.1.1 in [@BuTo] deals with “bifurcation from the first eigenvalue" type of situation whereas Theorem 1.13 in [@BoGiPr] concerns the bifurcation from origin. The conditions $(G1)-(G3)$ assumed there are required only to ensure that the starting analytic path corresponding to $\mathcal{A}_0$ is available for global extension. In our case, we make this as an assumption $(G3)$ above. Hence the proof given in [@BuTo] and in [@BoGiPr] holds good in our case as well.
We recall the following result from [@BoGiPr] (proposition 2.1).
\[g anal\] Let $g : {{\mathbb R}}\to {{\mathbb R}}$ be an entire function with $g(0)=0$. Define $M_k (a)= max_{[-a,a]}g^{(k)}, \; k=1,2,3,..$. Assume that for any $a \geq 0$, there exists $\mu>0$ such that the series $\sum_{k=0}^{\infty} \frac{M_k(a)}{k!}\mu^k $ converges. Then, for any $\phi \in C_0(\overline{\Omega}), \phi>0$ in $\Omega$, we have ${\mathcal C}_\phi(\Omega) \ni u \mapsto g(u) \in {\mathcal C}_\phi(\Omega)$ is an analytic map. Furthermore, if $\inf_{[0,\infty)}g' >0$, then $g$ maps $
{\mathcal C}_\phi^+(\Omega)$ into itself.
Consider now the solution operator $F$ associated to $(P_\lambda)$ defined in .
\[analyticity\] The map $F$ takes $\mathbb{R}\times {\mathcal C}_{\phi_1 }^+(
\Omega )$ into ${\mathcal C}_{\phi_1}(\Omega)$ and is analytic. Furthermore, if $\lambda\geq 0$, then $F(\lambda, \cdot)$ maps $ {\mathcal C}_{\phi_1 }^+(
\Omega )$ into ${\mathcal C}_{\phi_1}^+(\Omega)$.
[*Step 1: The map ${\mathcal C}_{\phi_1 }^+(\Omega )\ni u\longmapsto
K(x)u^{-\alpha } + \lambda f(u) \in {\mathcal C}_{\phi_1 ^{-\alpha -\beta }}(\Omega )$ is analytic.*]{}
Given $u\in {\mathcal C}_{\phi_1 }^+(\Omega )$, it follows that $K(x)u^{-\alpha} \in {\mathcal C}_{\phi_1^{-\alpha-\beta }}(\Omega )$. Then following the arguments in Step 1 of prop.2.3 in [@BoGiPr], we obtain the analyticity of the map.
[*Step 2: The map ${\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega )\ni u\longmapsto (\Delta^2)
^{-1}u\in {\mathcal C}_{\phi_1}(\Omega)$ is a linear continuous map (and hence analytic). Furthermore, this map takes ${\mathcal C}^+_{\phi_1 ^{-\alpha-\beta }}(\Omega )$ into ${\mathcal C}^+_{\phi_1}(\Omega)$.*]{}
We observe that $(\Delta^2)^{-1}$ is well defined on ${\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega )$. Indeed, since $\alpha+\beta<2$, from lemma \[jee\], there exists a unique solution $w\in C^{2, \theta}(\overline{\Omega }), 0<\theta<1,$ solving $$\left\{
\begin{array}{ll}
\Delta^2 w=u\ \ in\ \ \Omega ,\ \ u\in {\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega ), & \\
w=\Delta w=0 \mbox{ on} \ \partial \Omega. &
\end{array}\right. \label{19}$$Clearly, $w := (\Delta^2)
^{-1}u \in {\mathcal C}_{\phi_1 }(\Omega )$. If additionally $u \in {\mathcal C}^+_{\phi_1 }(\Omega )$, from the Hopf principle in corollary \[dee\], we also have that $w\in {\mathcal C}_{\phi_1}^+(\Omega)$. The proof of the proposition follows by combining steps 1 and 2.We now prove the existence of ${\mathcal A_0}$:
\[A+\] Let $0<\alpha+\beta<2$. There exists a $\lambda_0>0$ such that for all $\lambda\in (-\lambda_0,\lambda_0)$, there exists a non degenerate solution $u_\lambda\in {\mathcal C}_{\phi_1}^+(\Omega)$ to $(P_\lambda)$. Furthermore, the map $$(-\lambda_0,\lambda_0)\ni\lambda\to u_\lambda\in {\mathcal C}_{\phi_1}^+(\Omega)$$ is analytic and $\Vert u_\lambda-u_0\Vert_{{\mathcal C}_{\phi_1}(\Omega)}\to 0$ as $\lambda\to 0$ where $u_0$ is the unique solution to $(P)$.
We would like to apply the analytic version of the implicit function theorem. Given $u \in {\mathcal C}^+_{\phi_1 }$, we can check that the linearised operator $\partial_u F(\lambda,u): {\mathcal C}_{\phi_1 }(
\Omega ) \to {\mathcal C}_{\phi_1 }(
\Omega )$ is given by $$\label{kab}
\partial_u F(\lambda,u )\phi=\phi+ (\Delta^2)^{-1}\Big( [\alpha K(x)u^{-\alpha-1} -\lambda f^{\prime}(u)]\phi \Big).$$
Note that $K(x)u^{-\alpha-1}\phi \in {\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega )$. Indeed, for some $C>0$, $$\| K(x)u^{-\alpha-1}\phi\|_{{\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega )} \leq C \|\phi\|_{{\mathcal C}_{\phi_1 }(\Omega )}.$$ Therefore from lemma \[jee\], we obtain a constant $\theta \in (0,1)$ (depending only on $\nu,\alpha$ and $\beta$) such that $$\Big\{ K(x)u^{-\alpha-1}\phi \Big\} \;\text{ bounded in }\; {\mathcal C}_{\phi_1 ^{-\alpha-\beta }}(\Omega ) \Longrightarrow \Big\{(\Delta^2)^{-1}( K(x)u^{-\alpha-1}\phi)\Big\} \;\text{ bounded in }\; C^{2,\theta}(\overline{\Omega}).$$ We infer then that $\partial_u F(\lambda,u)$ is a compact perturbation of the identity and hence is a Fredholm operator of $0$-index. Next, we show that $\partial_u F(\lambda,u)$ is invertible for $\lambda \leq 0$. If $\phi$ belongs to the kernel of $\partial_uF(\lambda,u)$, denoted by $N(\partial_uF(\lambda,u))$, we will have $$\begin{aligned}
\int_{\Omega}(\Delta\phi)^2{\rm d}x+ \alpha \int_{\Omega}K(x) u^{-\alpha-1} \phi^2 {\rm d}x - \lambda \int_{\Omega} f^{\prime}(u)\phi^2 {\rm d}x=0.\end{aligned}$$ Using $(f_2)$ and non positivity of $\lambda$ we get $\phi\equiv\, 0$ from the above identity. Therefore, if $\lambda \leq 0$, we have $N(\partial_u F(\lambda,u))=\{0\}$ which implies that $\partial_u F(\lambda,u)$ is invertible.
Appealing to the real analytic version of implicit function theorem (see [@BuTo]), we obtain a $\lambda_0>0$ and an analytic branch of solutions, $\lambda\to (\lambda ,u_\lambda)$ to $F(\lambda, u)=0$ for $\lambda\in (-\lambda_0,\lambda_0)$. By taking $\lambda_0$ smaller if required, from the smoothness of the map $F$ we obtain that $\partial_u F(\lambda,u_\lambda)$ is invertible for all $-\lambda_0<\lambda < \lambda_0$. That is, the solution $u_\lambda$ is non degenerate for all such $\lambda$.
\[nar\] There exists $\Lambda>0$ such that $(P_\lambda)$ admits no solution for $\lambda>\Lambda$.
Using the assumption on $K$ and $(f_2)$ we note that for some positive constants $c_1,c_2$ $$K(x)t^{-\alpha}+\lambda f(t) \geq c_1+ c_2 \lambda t \;\; \text{ for all }\;\; x\in \Omega, t >0.$$ Let $\lambda>0$. We multiply the equation in $(P_\lambda)$ by $\phi_1$ and use the above inequality to get for any solution $u$ to $(P_\lambda)$: $$\begin{aligned}
\lambda_1^2\int_{\Omega}u\phi_1{\rm d}x=\int_{\Omega}K(x)\phi_1u^{-\alpha}{\rm d}x+\lambda\int_{\Omega}f(u)\phi_1{\rm d}x\geq c_1 \int_{\Omega} \phi_1{\rm d}x + c_2 \lambda\int_{\Omega}u\phi_1{\rm d}x.\end{aligned}$$ Hence, necessarily, $\lambda\leq \frac{\lambda_1^2}{c_2}$.
We finally prove Theorem \[th1\]. Let ${\mathcal U}{\stackrel{{\rm{def}}}{=}}{{\mathbb R}}\times {\mathcal C}_{\phi_1 }^+(\Omega )$. We first check that $F$ satisfies the conditions $(G1)-(G3)$ in order to apply Theorem \[theo9.1.1\]. From the regularity estimate in lemma \[jee\], we deduce that any bounded subset of ${\mathcal S}$ is relatively compact in ${{\mathbb R}}\times {\mathcal C}_{\phi_1}$, i.e. $(G1)$ holds. $(G2)-(G3)$ follow from the above proposition. Hence theorem \[theo9.1.1\] asserts the existence of ${\mathcal A} \subset {\mathcal S}$ satisfying $(a)-(e)$.
\(i) follows from proposition \[A+\].\
(ii) We first prove that assertion in the alternative $(e)(i)$ of theorem \[theo9.1.1\] occurs. We do this by ruling out the possibilities $(e)(ii)$ and $(e)(iii)$.
The case $(e)(ii)$ can be ruled out as follows. Suppose there exists a sequence $\{(\lambda(s_n),u(s_n))\} \subset {\mathcal A}$ such that $(\lambda(s_n),u(s_n)) \to (\tilde\lambda, \tilde u) \in \partial {\mathcal U}$ in ${{\mathbb R}}\times {\mathcal C}_{\phi_1 }(\Omega )$ as $s_n\to\infty$. In particular, $\tilde u\not\in {\mathcal C}_{\phi_1}^+(\Omega)$. Applying corollary \[dee\], we get for some $C>0$ independent of $n$, $$\begin{aligned}
\label{hopf-n}
u(s_n)(x)\geq C\rho(x)\int_{\Omega}(K(y) u(s_n)^{-\alpha} +\lambda(s_n)f(u(s_n)) \rho(y) {\rm d}y \;\; \forall x \in \Omega.\end{aligned}$$ Passing to the limit as $n\to\infty$ and using Fatou’s Lemma, we get $$\begin{aligned}
\label{hopf-limit}
\tilde u(x)\geq C\rho(x)\int_{\Omega}(K(y) \tilde u^{-\alpha}+\tilde\lambda f(\tilde u ))\rho(y) {\rm d}y\end{aligned}$$ which contradicts the assumption $\tilde u\not\in {\mathcal C}_{\phi_1}^+(\Omega)$ if $\tilde \lambda \geq 0$.
Next we rule out alternative $(e)(iii)$. For that, we observe that $u_0$ is the unique solution to $(P_0)$ and from the implicit function theorem, ${\mathcal A}_0$ is the unique branch of solutions emanating from $(0,u_0)$. Therefore, ${\mathcal A}$ can not bend back to join the point $(0,u_0)$.
Hence alternative $(e)(i)$ of theorem \[theo9.1.1\] holds. From proposition \[nar\], the conclusion (ii) of theorem follows.
\(iii) follows in view of (ii) and the fact that there is no solution for all large $\lambda$ (prop. \[nar\]).
\(iv) and (v) of theorem \[th1\] follow directly from $(c)$ and $(d)$ of theorem \[theo9.1.1\].
\(vi) We also note that (see Proposition \[A+\]) since $\partial_u F(\lambda,u_\lambda)$ is an invertible operator for $\lambda<0$, the negative portion of ${\mathcal A}$ i.e., ${\mathcal A} \cap (-\infty,0) \times {\mathcal C}_{\phi_1}(\Omega )$, is a single analytic curve (indeed a graph from the $\lambda$ axis) consisting of non-degenerate solutions $u_\lambda$. In particular, this curve does not undergo any bifurcations.
This completes the proof of the theorem.
[**Acknowledgements:**]{} the authors were funded by IFCAM (Indo-French Centre for Applied Mathematics) UMI CNRS under the project “Singular phenomena in reaction diffusion equations and in conservation laws”.
[9]{}
[^1]: All the authors of this work were supported by IFCAM.
|
---
abstract: 'Quantum theory puts forward phenomena unexplainable by classical physics—or information, for that matter. A prominent example is *non-locality*. Non-local correlations cannot be explained, in classical terms, by shared information but only by communication. On the other hand, the phenomenon does not *allow* for (potentially faster-than-light) message transmission. The fact that some non-local and non-signaling correlations are predicted by quantum theory, whereas others fail to be, asks for a criterion, as simple as possible, that characterizes which joint input-output behaviors are “quantum” and which are not. In the context of the derivation of such criteria, it is of central importance to understand when non-local correlations can be [*amplified*]{} by a non-interactive protocol, [*i.e.*]{}, whether some types of weak non-locality can be distilled into stronger by local operations. Since it has been recognized that the searched-for criteria must inherently be [*multi-partite*]{}, the question of distillation, extensively studied and understood [*two*]{}-party scenarios, should be adressed in the multi-user setting, where much less is known. Considering the space of intrinsically $n$-partite correlations, we show the possibility of distilling weak non-local boxes to the algebraically maximal ones without any communication. Our protocols improve on previously known methods which still required partial communication. The price we have to pay for dropping the need for communication entirely is the assumption of [*permutation invariance*]{}: Any correlation that can be realized between *some* set of players is possible between [*any*]{} such set. This assumption is very natural since the laws of physics are invariant under spacial translation.'
author:
-
bibliography:
- 'refs.bib'
title: 'Trading Permutation Invariance for Communication in Multi-Party Non-Locality Distillation'
---
Motivation and Outline
======================
Einstein, Podolsky, and Rosen [@EPR35] raised the question “Can quantum-mechanical description of physical reality be considered complete?” In direct response to that question, Bell [@B64] showed that quantum mechanics is incompatible with a *local hidden variable theory*: The theory predicts correlations that are, in classical terms, not explainable by shared information but only by communication. It is important to note, *however*, that, on the other hand, the arising correlations do not allow for message transmission.
With the goal of a systematic and generalized understanding of non-local correlations, quantum and beyond, Popescu and Rohrlich described an input-output behavior that maximally violates the *Bell-inequality*, but that still fulfills the non-signaling condition [@PR94]. Their bipartite input-output behavior or [*box*]{} can classically be realized with 75% only, by quantum states with 85% [@C80], whereas even the perfect approximation would still be compatible with the non-signaling principle. This means that, strangely enough, quantum physics is not maximally non-local and cannot be singled out by the non-signaling principle. It is a fascinating and conceptually important question whether there is an (information-theoretical) principle that is able to describe exactly the quantum correlations. One such attempt has been to generalize the non-signaling principle to parties that are allowed to use limited communication, to the so-called *information-causality principle* [@P09]. Other authors have looked for principles characterizing quantum correlations as the ones that do not improve the efficiency of *non-local computation* [@LPSW07] or do not collapse *communication complexity* [@BBLMTU06]. Two more physically motivated principles are *macroscopic locality* [@NW10] and *local orthogonality* [@FSA12]. In each case, it has been shown that quantum physics respects the corresponding principle, whereas some “super-quantum” correlations violate it. For none of the principles, however, it was possible to show that [*every*]{} non-quantum behavior is in violation.
In the search of a principle [*exactly*]{} singling out quantum theory, the possibility of making (weak) non-local correlations stronger by local wirings is paramount since it offers the possibility of generating systems violating some principle from correlations which respect it. Therefore, a systematic understanding of the power and limitations of distillation of non-locality potentially leads to deep insights into the mysterious nature of quantum theory.
The question of non-locality distillation was mainly studied in the bipartite scenario: There exist weak non-local correlations that can be distilled to an almost perfect Popescu-Rohrlich box by an adaptive protocol [@FWW09; @BS09; @HR10]. On the other hand, [*isotropic*]{} correlations seem to be undistillable [@DW08].
In the context both of information principles able to single out quantum theory [@mult11] as well as for information-processing tasks such as [*randomness generation*]{} [@rand12], it has turned out that [ *multi-party*]{}, as opposed to only bipartite correlations, play a crucial role. Nevertheless, much less is known for that case. One effort was to generalize the XOR protocol [@HW10], but it fails to distill maximal non-local boxes. It was shown in [@EW13a; @EW13b] that the large class of *full-correlation boxes* can be distilled by a multipartite version of Brunner-Skrzypczyk’s protocol [@BS09] under the (strong) assumption that [*partial communication*]{} is allowed. This latter assumption, unfortunately, puts into question the relevance of the protocol in the context described above, namely of finding information-based criteria singling out quantum theory.
We introduce a new kind of multi-party distillation protocols by showing that the need for communication can be dropped entirely. The price for this is the need for the assumption that any correlation which can be realized between *some* set of players is also possible between any *other* set. We believe this assumption to be quite natural since we imagine the correlations to arise from the interaction with a concrete physical system. In this sense, the assumption is true in every world in which the laws of physics are invariant under spacial translation.
This is an outline of the present article. We first characterize full-correlation boxes and give for them a criterion for being maximally non-local (Section \[sec:full\]). Second, we present a new distillation protocol that does not use partial communication, but that is still able to distill every non-isotropic faulty version of a full-correlation box to a close-to-perfect one. This new protocol requires the different parties to be able to arbitrarily distribute the input-and output-interfaces of the weak boxes (Section \[sec:dist\]). In Section \[sec:ex\], we illustrate our result with an example.
Definitions
===========
Systems and Boxes
-----------------
In an *$n$-partite input-output system*, each of the $n$ parties inputs an element $x_i$ and receives immediately an output $a_i$. The behavior of this system is defined by a conditional distribution $$P_{A_1A_2\cdots A_n\vert X_1X_2\cdots X_n}\ ,$$ where $X_i$ is the input and $A_i$ the output variable of the $i$th party.
An $n$-partite system with conditional probability distribution $P\left( a_1 a_2 \cdots a_n \vert x_1 x_2 \cdots x_n\right)$ is said *non-signaling* if the marginal distribution for each subset of parties $\lbrace a_{k_1} , a_{k_2},..., a_{k_m}\rbrace$ only depends on its their own inputs $$P\left( a_{k_1} \cdots a_{k_m}\vert x_1 \cdots x_n\right) = P\left( a_{k_1} \cdots a_{k_m}\vert x_{k_1} \cdots x_{k_m}\right)\ .$$
If the $n$-partite system fulfills the *non-signaling condition*, *i.e.,* the system cannot be used to transmit instantaneously information from one party to another one, then this system is called *n-partite box*.
Multipartite Locality
---------------------
Of central interest for us are $n$-partite boxes with the property that the parties cannot simulate the behavior of the box without communication, but by shared randomness only. This property is called *non-locality*.
An $n$-partite box with input variables $X_1$, $X_2$, ..., $X_n$ and output variables $A_1$, $A_2$, ..., $A_n$ is *local* if $$P_{A_1 A_2 \cdots A_n|X_1 X_2 \cdots X_n}=\sum_{r\in{\cal R}}{P_R(r)\cdot P_{A_1|X_1}^r\cdots P_{A_n|X_n}^r}\$$ for some random variable $R$.
Full-Correlation Boxes {#sec:full}
======================
Definition and Related Boxes
----------------------------
We focus our attention to one of the most general type of boxes: the *$n$-partite full-correlation boxes* that were introduced by Barrett and Pironio [@BP05]. This kind of boxes has the property that it displays correlation only with respect to the *full* set of parties. An *$n$-partite full-correlation box* takes as inputs $\textbf{x}=(x_1,x_2, \dots x_n)$ and as outputs $\textbf{a}=(a_1,a_2, \dots a_n)$, where all $x_i, a_i \in \{0, 1\}$. The input-output behavior is characterized by the following conditional distribution: $$P(\textit{\textbf{a}}\vert \textit{\textbf{x}}) = \begin{cases} \frac{1}{2^{n-1}}&\text{$\sum\limits_i a_i \equiv f(\textit{\textbf{x}})$ (mod 2)}\\0&\text{otherwise,}\end{cases}$$ where $f(\textit{\textbf{x}})$ is a Boolean function of the inputs.
A special case of the full-correlation boxes is the $n$-party generalization of the *Popescu-Rohrlich box* [@PR94] ($n$-PR box) that is defined by the conditional distribution: $$P^{\text{PR}}_n(\textit{\textbf{a}}\vert \textit{\textbf{x}}) = \begin{cases} \frac{1}{2^{n-1}}&\bigoplus\limits_{i=1}^n a_i = \prod\limits_{i=1}^n x_i\\0&\text{otherwise.}\end{cases}$$
If the output does not depend on all inputs, then we call the box an $(n,k)$-PR box ($k\leq n$): $$P^{\text{PR}}_{(n,k)}(\textit{\textbf{a}}\vert \textit{\textbf{x}}) = \begin{cases} \frac{1}{2^{n-1}}&\bigoplus\limits_{i=1}^n a_i = \prod\limits_{i=1}^k x_i\\0&\text{otherwise.}\end{cases}$$
Note that an $n$-PR box and an $(n,n)$-PR box are identical.
Construction of $n$-Partite PR Boxes {#sec:constr}
------------------------------------
In [@BLMPPR05] is shown how a $3$-PR box can be constructed from three PR boxes. This construction can be used to construct an $n$-PR recursively: Assume that the first $n-1$ parties share a $n-1$-PR box and every of this parties input their input $x_i$ to this box. The outputs (say $a_i'$) of the box fulfills $${i=1}^{n-1} a_i' =\bigwedge\limits_{i=1}^{n-1} x_i\ .$$ Every of these $n-1$ parties inputs the output bit in the PR box shared with the $n$th party and the $n$th party inputs in every PR box his input bit. The output bit of the $n$th party is the XOR of all his outputs from the PR boxes (see Fig. \[fig:nPR\]). In the end, the outputs fulfill $$\begin{aligned}
\bigoplus\limits_{i=1}^{n} a_i & = &\bigoplus\limits_{i=1}^{n-1} a_i\oplus \bigoplus\limits_{i=1}^{n-1} a_i \nonumber \\
& = & \bigoplus\limits_{i=1}^{n-1} (a_i\oplus b_i) \nonumber \\
& = & \bigoplus\limits_{i=1}^{n-1} (x_n\wedge a_i') \nonumber \\
& = & \bigwedge\limits_{i=1}^{n} x_i \ .\end{aligned}$$
\(A) at (0,0) ; (B) at (3,1.6) [PR Box]{}; (C) at (3,0.6) [PR Box]{}; (D) at (3,-1.6) [PR Box]{}; (0,-1) to \[\] (0,1); () at (-1.3,-0.2) [$\vdots$]{}; () at (3,-0.2) [$\vdots$]{};
(0.5,1.8) to\[\] (2,1.8); (0.5,0.8) to\[\] (2,0.8); (0.5,-1.4) to\[\] (2,-1.4); (1.2,1.4) to\[\] (2,1.4); (1.2,0.4) to\[\] (2,0.4); (1.2,-1.8) to\[\] (2,-1.8); () at (0.85,1.4) [$x_n$]{}; () at (0.85,0.4) [$x_n$]{}; () at (0.85,-1.8) [$x_n$]{};
(-1,1.8) to\[\] (-0.5,1.8); (-1,0.8) to\[\] (-0.5,0.8); (-1,-1.4) to\[\] (-0.5,-1.4); () at (-1.3,1.8) [$x_1$]{}; () at (-1.3,0.8) [$x_2$]{}; () at (-1.3,-1.4) [$x_{n-1}$]{};
(4,1.8) to\[\] (4.5,1.8); (4,0.8) to\[\] (4.5,0.8); (4,-1.4) to\[\] (4.5,-1.4); (4,1.4) to\[\] (4.5,1.4); (4,0.4) to\[\] (4.5,0.4); (4,-1.8) to\[\] (4.5,-1.8); () at (4.8,1.8) [$a_1$]{}; () at (4.8,0.8) [$a_2$]{}; () at (4.85,-1.4) [$a_{n-1}$]{}; () at (4.8,1.4) [$b_1$]{}; () at (4.8,0.4) [$b_2$]{}; () at (4.85,-1.8) [$b_{n-1}$]{};
() at (1.75,-2.5) [$a_n = b_1 \oplus b_2 \oplus \dots \oplus b_{n-1}$]{};
Construction of Full-Correlation Boxes
--------------------------------------
In the same way as in [@EW13a; @EW13b], we look how full-correlation boxes can be constructed by generalized PR-boxes and how they can be characterized.
\[lem:boolf\] If $f$ is a Boolean function of the input elements $x_1,x_2,...,x_n$, then it can be written as $$f(x_1,...,x_n) = \bigoplus\limits_{I \in \mathcal{I}} \left( a_I\cdot \bigwedge\limits_{i \in I}x_i \right)\ ,$$ where $\mathcal{I} = \mathcal{P}\left( \lbrace 1,2,...,n \rbrace\right) $ and $a_I \in \lbrace 0, 1\rbrace$ for all $I \in \mathcal{I}$.
Hence, it is obvious that the full-correlation box associated to the Boolean function $f$ can be constructed by $\sum_{I \in \mathcal{I}}a_I$ $n$ - PR boxes. For an example, see Fig. \[fig:equivalenz\]. Note that the $n$-PR boxes belonging to an $a_I$ where $\vert I \vert \leq 1$ are local and can be simulated by local operations and shared randomness.
\(A) at (0,0) ; (B) at (0,0) [3-PR Box]{}; (C) at (1.575,0) [3-PR Box]{}; (D) at (-1.575,0) [3-PR Box]{};
\(E) at (-5.5,0) [$1\oplus xy\oplus xz$]{}; (-3,0) to\[\] (-4,0); (-4,0) to\[\] (-3,0);
(-5.5,0.75) to\[\] (-5.5,0.5); (-6.1,0.75) to\[\] (-6.1,0.5); (-4.9,0.75) to\[\] (-4.9,0.5); () at (-5.5,1) [$y$]{}; () at (-6.1,1) [$x$]{}; () at (-4.9,1) [$z$]{};
(-5.5,-0.5) to\[\] (-5.5,-0.75); (-6.1,-0.5) to\[\] (-6.1,-0.75); (-4.9,-0.5) to\[\] (-4.9,-0.75); () at (-5.5,-1) [$b$]{}; () at (-6.1,-1) [$a$]{}; () at (-4.9,-1) [$c$]{};
(0,1) to\[\] (0,0.75); (1.575,1) to\[\] (1.575,0.75); (-1.575,1) to\[\] (-1.575,0.75); () at (0,1.25) [$y$]{}; () at (1.575,1.25) [$z$]{}; () at (-1.575,1.25) [$x$]{};
(0,-0.75) to\[\] (0,-1.3); (1.575,-0.75) to\[\] (1.575,-1); (-1.575,-0.75) to\[\] (-1.575,-1); () at (0,-1.7) [$b = b_1\oplus b_2\oplus b_3$]{}; () at (1.575,-1.25) [$c = c_1\oplus c_2\oplus c_3$]{}; () at (-1.575,-1.25) [$a = a_1\oplus a_2\oplus a_3$]{};
(0,0.4) to\[\] (0,0.2); (1.575,0.4) to\[\] (1.575,0.2); (-1.575,0.4) to\[\] (-1.575,0.2); () at (0,0.55) [$y$]{}; () at (1.575,0.55) [$1$]{}; () at (-1.575,0.55) [$1$]{};
(0,-0.2) to\[\] (0,-0.4); (1.575,-0.2) to\[\] (1.575,-0.4); (-1.575,-0.2) to\[\] (-1.575,-0.4); () at (0,-0.55) [$b_2$]{}; () at (1.575,-0.55) [$b_3$]{}; () at (-1.575,-0.55) [$b_1$]{};
(0.5,0.4) to\[\] (0.5,0.2); (2.075,0.4) to\[\] (2.075,0.2); (-1.075,0.4) to\[\] (-1.075,0.2); () at (0.5,0.55) [$1$]{}; () at (2.075,0.55) [$z$]{}; () at (-1.075,0.55) [$1$]{};
(0.5,-0.2) to\[\] (0.5,-0.4); (2.075,-0.2) to\[\] (2.075,-0.4); (-1.075,-0.2) to\[\] (-1.075,-0.4); () at (0.5,-0.55) [$c_2$]{}; () at (2.075,-0.55) [$c_3$]{}; () at (-1.075,-0.55) [$c_1$]{};
(-0.5,0.4) to\[\] (-0.5,0.2); (1.075,0.4) to\[\] (1.075,0.2); (-2.075,0.4) to\[\] (-2.075,0.2); () at (-0.5,0.55) [$x$]{}; () at (1.075,0.55) [$x$]{}; () at (-2.075,0.55) [$1$]{};
(-0.5,-0.2) to\[\] (-0.5,-0.4); (1.075,-0.2) to\[\] (1.075,-0.4); (-2.075,-0.2) to\[\] (-2.075,-0.4); () at (-0.5,-0.55) [$a_2$]{}; () at (1.075,-0.55) [$a_3$]{}; () at (-2.075,-0.55) [$a_1$]{};
We define the set of all non-local $n$-PR boxes that are needed to simulate the full-correlation box: Let $$\label{equ:j}
\mathcal{J} := \lbrace I \in \mathcal{I}\, \vert\, a_I = 1 \text{ and } \vert I\vert \geq 2\rbrace\ .$$ This set can be partitioned into pairwisely disjoint subsets $\lbrace J_1, J_2, ..., J_{n_\mathcal{J}}\rbrace$ such that all $A \in J_i$ and $B\in J_j$ fulfill $A\cap B = \emptyset$ for all $i \neq j$. We define the maximal number of such subsets as $n_{\mathcal{J}}$ and denote this partition as the *empty-overlap partition of $\mathcal{J}$*.
Non-Locality of Full-Correlation Boxes
--------------------------------------
We write the Boolean function that characterizes a full-correlation box as in Lemma \[lem:boolf\], so it is easy to determine if this box is local or not.
It is obvious that a full-correlation box is local if the associated Boolean function can be written as the XOR of single inputs of the box and a constant. Assume that the function consists of at least one AND-term, then this box can be reduced to a PR box by distributing all input-and output- interfaces only to two parties such that both of them get at least one input that belongs to the AND-term. Therefore, a full-correlation box is local if and only if the Boolean function can be written as the XOR of a constant and single inputs of the box
$$f(x_1,...,x_n) = \bigoplus\limits_{I \in \mathcal{L}} \left( a_I\cdot \bigwedge\limits_{i \in I}x_i \right)\ ,$$
where $\mathcal{L} = \lbrace \emptyset, \lbrace x_1\rbrace, \lbrace x_2\rbrace, ..., \lbrace x_n\rbrace \rbrace$ and $a_I \in \lbrace 0, 1\rbrace$ for all $I \subseteq \mathcal{L}$.
We show that for every non-local full-correlation box, there exists a closest local box (measured in the $L^1$-norm) that is also a full-correlation box. Let $P$ be the joint probability distribution of a non-local full-correlation box. Then the joint probability distribution of the *closest local full-correlation box $P^*$* is defined by $$\Vert P - P^* \Vert_1 = \underset{P' \textnormal{ loc. full-corr. box}}{\textnormal{min}}\left( \Vert P - P'\Vert_1\right)\ .$$
Let P be the joint probability distribution of an $n$-partite full-correlation box. Then the closest local full-correlation box is one of the closest local boxes. That means
$$\Vert P - P^* \Vert_1 = \textnormal{min}\left( \left\Vert P - \sum_{r\in{\cal R}}{P_R(r)\cdot P_{A_1|X_1}^r\cdots P_{A_n|X_n}^r} \right\Vert_1 \right) \ ,$$
for some random variable $\mathcal{R}$.
It is obvious that every deterministic local strategy $r \in \mathcal{R}$ can achieve at most the same number of input-output behaviors (XOR of the outputs equal to a Boolean function of the inputs) as the closest local full-correlation box. So every local box (that is a convex combination of these deterministic local strategies) has at least the same distance from the given full-correlation box as the closest local full-correlation box.
Extremal Boxes of the Non-Signaling Polytope
--------------------------------------------
It is a well-known fact that all full-correlation boxes are non-signaling, since they can be simulated by PR boxes [@BP05]. In [@PBS11], all tripartite extremal boxes of the non-signaling polytope have been characterized, but for more parties it is not known which of the (full-correlation) boxes are extremal.
\[thm\_fullcorr\] Let $P$ be an $n$-partite full-correlation box associated to the Boolean function $f$ that depends on $k$ input variables. Then $P$ is an extremal box of the non-signaling polytope if and only if $n_\mathcal{J} = 1$ and $k=n$ hold.
Theorem \[thm\_fullcorr\] follows from Lemmas \[lemma2\], \[lemma3\], \[lemma5\], and \[lemma6\].
\[lemma2\] Let $P$ be an $n$-partite full-correlation box with associated function $f$ that depends on $k$ input variables. If $k \neq n$, then $P$ is not an extremal box.
$P$ can be written as a convex combination of the following two non-signaling boxes: $$P^1(\textbf{a}\vert \textbf{x}) = \begin{cases} \frac{1}{2^{k-1}}&\bigoplus\limits_{i=1}^{k}a_i = f(x_1,x_2,...,x_k) \\ & \text{and $\bigoplus\limits_{i=k+1}^{n}a_i =0$}\\0&\text{otherwise,}\end{cases}$$ and $$P^2(\textbf{a}\vert \textbf{x}) = \begin{cases} \frac{1}{2^{k-1}}&\bigoplus\limits_{i=1}^{k}a_i = 1\oplus f(x_1,x_2,...,x_k) \\ & \text{and $\bigoplus\limits_{i=k+1}^{n}a_i =1$}\\0&\text{otherwise.}\end{cases}$$ So $P = \frac{1}{2}P^1 + \frac{1}{2}P^2$. Therefore, $P$ is not an extremal box of the non-signaling polytope.
\[lemma3\] Let $P$ be an $n$-partite full-correlation box with associated function $f$ that depends on $k$ input variables. Let $k =n$. If $n_\mathcal{J} \geq 2$, then $P$ is not extremal.
Since $n_\mathcal{J}$ is at least $2$, we are able to split the Boolean function $f$ in two other Boolean functions, $f_1$ and $f_2$, such that they do not depend on the same input variables. Without loss of generality, we assume that $f_1$ depends on the input variables $x_1, x_2, ..., x_m$ and $f_2$ depends on $x_{m+1}, ..., x_n$ ($m<n$). Therefore, $f$ can be written as $f(x_1,..., x_n) = f_1(x_1, ..., x_m) \oplus f_2(x_{m+1}, ..., x_n)$. So the box $P$ can be written as a convex combination of the following two boxes: $$P^1(\textbf{a}\vert \textbf{x}) = \begin{cases} \frac{1}{2^{n-2}}&\bigoplus\limits_{i=1}^{m}a_i = f_1(x_1,x_2,...,x_m) \\ & \text{and $\bigoplus\limits_{i=m+1}^{n}a_i = f_2(x_{m+1},..., x_n)$ }\\0&\text{otherwise,}\end{cases}$$ and $$P^2(\textbf{a}\vert \textbf{x}) = \begin{cases} \frac{1}{2^{n-2}}&\bigoplus\limits_{i=1}^{m}a_i = \neg f_1(x_1,x_2,...,x_m) \\ & \text{and $\bigoplus\limits_{i=m+1}^{n}a_i = \neg f_2(x_{m+1},..., x_n)$ }\\0&\text{otherwise.}\end{cases}$$ So $P = \frac{1}{2}P^1 + \frac{1}{2}P^2$. Therefore, $P$ is not an extremal box of the non-signaling polytope.
\[lemma5\] Every $n$-PR box is extremal.
The proof is based on the same argument as in [@HRW10] for showing that *any non-locality implies some secrecy*. Assume that the $n$-PR box $P$ can be written as a convex combination of two other non-signaling boxes $P^1$ and $P^2$ $$P = \epsilon P^1 + (1 - \epsilon) P^2,$$ where $0 < \epsilon <1$. It is obvious that both of the boxes must fulfill that the XOR of their output elements is equal to the AND of their input elements, *i.e.*, $$\label{equ:prob}
\text{Prob}\left[ \bigoplus\limits_{i=1}^{n} A_i = \prod\limits_{i=1}^{n} X_i\mid X_i = x_i ~\forall ~1\leq i\leq n \right] = 1$$ for all input elements $x_i \in \{0, 1\}$. We will show that all possible biases, $p_i := \text{Prob}\left[ A_i=0 \vert X_k=0 \text{ for all } k\right] $ for all $1\leq i \leq n-1$ such that the box is non-signaling, must be $p_i = 1/2$. Therefore, $P$ cannot be written as a convex combination of other non-signaling boxes.
Assume without loss of generality that all $p_i \geq 1/2$ for all $1\leq i \leq n-1$. Because of Equation (\[equ:prob\]), the bias $p_n$ can be computed from the biases $p_i$ for $i \in \lbrace 1, 2, ..., n-1\rbrace$.
Since our box is non-signaling, all biases are independent of the other parties’ inputs. We determine step by step the biases $p'_i := \text{Prob}\left[ A_i=0 \vert X_i=1\right]$ for all $i$ and get that $p'_i = p_i$. If not all biases are $1/2$, then this is a contradiction to Equation (\[equ:prob\]) for the input $(1, 1, ..., 1)$.
\[lemma6\] Let $P^1$ and $P^2$ be extremal $m$ and $k$-partite full-correlation boxes with associated functions $f_1$ and $f_2$, where $f_1$ depends on the input variables $x_1, x_2, ..., x_m$ and $f_2$ depends on $x_l, x_{l+1}, ..., x_{l+k-1}$ ($l\leq m$). Then the box $P$ with associated function $$f(x_1, ..., x_{l+k-1}) = f_1(x_1, ..., x_m)\oplus f_2(x_l, ..., x_{l+k-1})$$ is also extremal.
We assume that the box with associated function $f$ can be written as a convex combination of two other non-signaling boxes $P_1$ and $P_2$ $$P = \epsilon P_1 + (1 - \epsilon) P_2,$$ where $0 < \epsilon <1$. As before, it is obvious that both of the boxes must fulfill that the XOR of their output elements is equal to the XOR of the Boolean functions $f_1$ and $f_2$. Therefore, we define $f(X_1, ..., X_{l+k+1}) = f_1(X_1,...,X_m) \oplus f_2(X_l, ..., X_{l+k+1})$. We have $$\text{Prob}\left[ \bigoplus\limits_{i=1}^{n} A_i =f(X_1, ..., X_{l+k+1})\mid X_i = x_i ~\forall ~i \right] = 1$$ for all input elements $x_i \in \{0, 1\}$.
Assume that all parties $i \in \{x_m, ..., x_{l+k-1}\}$ input 0 to the box. Therefore, the box acts like the box $P^1$ (assume that the parties $m$ to $k+l-1$ are the same or are able to communicate to each other), and we have found a convex combination of this box. This is in contradiction to the assumption that $P^1$ is extremal. Therefore, the new box is also extremal.
Note that the $n$-partite full-correlation box associated to the function $f(x_1, ..., x_n) =\prod_{i=1}^{n}x_i \oplus x_1$ is also an extremal box since it can be constructed with an $n$-PR and an $(n-1)$-PR box by flipping the input bit $x_1$.
Distillation of Full-Correlation Boxes {#sec:dist}
======================================
We introduce a new noncommunicative protocol for distillation which requires the parties to arbitrarily distribute the input-and output-interfaces of the weak boxes between the parties. Therefore, the parties have no longer a fixed access to the box, it is even possible that one party has no access to a box, but another one has multiple ones.
Distilling $n$-Partite PR Boxes
-------------------------------
Using the generalization of the Brunner-Skrzypczyk protocol [@BS09] that were presented in [@EW13a; @EW13b] we are able to distill imperfect $n$-partite PR boxes $P^{\text{PR}}_{n,\varepsilon}$, where $$P^{\text{PR}}_{n,\varepsilon} = \varepsilon P^{\text{PR}}_{n} + (1 - \varepsilon) P^{\text{c}}_{n}\ ,$$ and $$P^{\text{c}}_n(\textit{\textbf{a}}\vert \textit{\textbf{x}}) = \begin{cases} \frac{1}{2^{n-1}}&\bigoplus\limits_{i}a_i = 0\\0&\text{otherwise.}\end{cases}$$
All n parties share two boxes, where we denote by $x_i$ the value that the $i$th party inputs to the first box and by $y_i$ the value that the $i$th party inputs to the second box. The output bit of the first box for the $i$th party is $a_i$, and the output bit of the second box is $b_i$. The n parties proceed as follows: $y_i = x_i\bar{a}_i$ and they output, finally, $c_i = a_i \oplus b_i$ (see also Fig. \[fig:dbsprot\]). \[prot\]
\(A) at (0,0) ; (B) at (0,0.75) ; (C) at (0,-0.75) ; (-1.4,1.8) to\[\] (-1.4,1.5); (-0.6,1.8) to\[\] (-0.6,1.5); (1.4,1.8) to\[\] (1.4,1.5); () at (-1.4,2) [$x_1$]{}; () at (-0.6,2) [$x_2$]{}; () at (1.4,2) [$x_n$]{};
(-1.4,1.15) to\[\] (-1.4,0.95); (-0.6,1.15) to\[\] (-0.6,0.95); (1.4,1.15) to\[\] (1.4,0.95); () at (-1.4,1.25) [$x_1$]{}; () at (-0.6,1.25) [$x_2$]{}; () at (1.4,1.25) [$x_n$]{};
(-1.4,0.55) to\[\] (-1.4,0.35); (-0.6,0.55) to\[\] (-0.6,0.35); (1.4,0.55) to\[\] (1.4,0.35); () at (-1.4,0.2) [$a_1$]{}; () at (-0.6,0.2) [$a_2$]{}; () at (1.4,0.2) [$a_n$]{};
(-1.4,-0.35) to\[\] (-1.4,-0.55); (-0.6,-0.35) to\[\] (-0.6,-0.55); (1.4,-0.35) to\[\] (1.4,-0.55); () at (-1.4,-0.25) [$x_1\bar{a}_1$]{}; () at (-0.6,-0.25) [$x_2\bar{a}_2$]{}; () at (1.4,-0.25) [$x_n\bar{a}_n$]{};
(-1.4,-0.95) to\[\] (-1.4,-1.15); (-0.6,-0.95) to\[\] (-0.6,-1.15); (1.4,-0.95) to\[\] (1.4,-1.15); () at (-1.4,-1.3) [$b_1$]{}; () at (-0.6,-1.3) [$b_2$]{}; () at (1.4,-1.3) [$b_n$]{};
(-1.4,-1.5) to\[\] (-1.4,-1.8); (-0.6,-1.5) to\[\] (-0.6,-1.8); (1.4,-1.5) to\[\] (1.4,-1.8); () at (-1.4,-2) [$c_1$]{}; () at (-0.6,-2) [$c_2$]{}; () at (1.4,-2) [$c_n$]{}; () at (0,-2.5) [$c_i = a_i \oplus b_i$]{};
\[thm:bsprot\] The generalized BS protocol takes two copies of an arbitrary box $P^{PR}_{n,\varepsilon}$ with $0<\varepsilon<1$ to an $n$-partite correlated non-local box $P^{PR}_{n,\varepsilon'}$ with $\varepsilon'>\varepsilon$, i.e., is distilling non-locality. In the asymptotic case of many copies, any $P^{PR}_{n,\varepsilon}$ with $0<\varepsilon$ is distilled arbitrarily closely to the n-PR box.
Equivalence Between $n$-and $(n,k)$-PR Boxes
--------------------------------------------
We show that $k$-PR boxes and $(n,k)$-PR boxes are equivalent in the sense that using one of the boxes and shared randomness, the other box can be simulated and *vice versa*. This property is transitive.
Assume $k$ parties share a $k$-PR box, where each of the parties has an input and an output. All of these $k$ parties share a random variable with $n-k$ additional parties. This random variable helps to create a local distribution between the parties that has the property that the XOR of all $n$ outputs is zero. Combining these two distributions with an XOR, the new distribution correspond the distribution of the $(n,k)$-PR box.
To see the opposite implication, let $k$ parties share a $(n,k)$ - PR box, where every party has an input interface that has influence on the distribution in the end, and the corresponding output. The left inputs (and the corresponding outputs) can be distributed to arbitrary parties. In the end, these parties have to take the XOR of their output to get their final output. Therefore, the $k$ parties simulate a $k$-PR box.
Distillation of $(n,k)$-PR Boxes
--------------------------------
Using the generalized BS Protocol we are able to distill imperfect $(n,k)$-PR boxes $P^{\text{PR}}_{(n,k),\varepsilon}$, where $$P^{\text{PR}}_{(n,k),\varepsilon} = \varepsilon P^{\text{PR}}_{(n,k)} + (1 - \varepsilon) P^{\text{c}}_{n}\ .$$ With the same construction as we showed the equivalence between $n$-and $(n,k)$-PR boxes, we are able to show equivalence between $P^{\text{PR}}_{(n,k),\varepsilon}$ and $P^{\text{PR}}_{k,\varepsilon}$. Because of Theorem \[thm:bsprot\], we know that $P^{\text{PR}}_{k,\varepsilon}$ can be distilled arbitrarily closely to $P^{\text{PR}}_{k}$. Again, $P^{\text{PR}}_{k}$ is equivalent to $P^{\text{PR}}_{(n,k)}$. Therefore, $P^{\text{PR}}_{(n,k),\varepsilon}$ can be distilled arbitrarily closely to $P^{\text{PR}}_{(n,k)}$.
Distillation of Full-Correlation Boxes {#distillation-of-full-correlation-boxes}
--------------------------------------
As shown in Lemma \[lem:boolf\], every (imperfect) $n$-partite full-correlation box can be simulated by (imperfect) $n$-PR boxes, where some parties input the constant one. These boxes are exactly the $(n,k)$-PR boxes. If we could isolate each of these (imperfect) $(n,k)$-PR boxes, then we could distill each of them to almost perfect, and so, the whole full-correlation box can be distilled to almost perfect.
Let us assume that $P$ is the full-correlation box, $P^*$ the closest local full-correlation box, and $\varepsilon$ a parameter between 0 and 1. We show that every convex combination $$P_\varepsilon = \varepsilon P + (1-\varepsilon ) P^*$$ can be distilled arbitrarily closely to the full-correlation box $P$. Without loss of generality, assume that $P$ has no local part, since we change only the local part of the box that can be reached by a local strategy. That implies that $P^* = P_n^{\text{c}}$.
An $(n,k)$-PR box belonging to $a_I$, $I \in \mathcal{J}$, can be isolated from the full-correlation box, if there exist no $J \in \mathcal{J}$ such that $J \subseteq I$. Then, the box can be isolated when every party $i \notin \mathcal{J}$ inputs 0 to the full-correlation box. Otherwise, the box cannot be isolated in this way, but there exist two other possibilities: Assume there is an $(n,k)$-PR box belonging to $a_I$ and an $(n,l)$-PR box ($k>l$) belonging to $a_J$, $J\subset I$.
1. If there exists an $(n,m)$-PR box with $m\geq k$ that can be isolated, then we replace our box with it.
2. Otherwise, we take the $(n,m)$-PR box with the biggest $m$, isolate or distill a PR-box, and use the recursive construction for $k$-PR boxes \[sec:constr\].
Therefore, every imperfect full-correlation box can be distilled in this way and we are also in the multipartite case able to distill boxes that are close to the local bound to maximal non-local ones.
Example {#sec:ex}
=======
In this example we distill the following full-correlation box: $$P^1(\textit{\textbf{a}}\vert \textit{\textbf{x}}) = \begin{cases} \frac{1}{8}&\sum\limits_{i=1}^4 a_i = x_1x_2x_3x_4\oplus x_1x_2x_3\oplus x_3x_4\\0&\text{otherwise.}\end{cases}$$ The closest local box is $P_4^{\text{c}}$. Let us assume that we have imperfect boxes that are close to the local bound $$P^1_{\varepsilon} = \varepsilon P^1 + (1-\varepsilon) P_n^{\text{c}} \ .$$ First, we isolate the tripartite PR box between the first three parties. Therefore, the first two parties take the first two inputs and the corresponding outputs of the box, and the third party takes the third and fourth inputs and outputs of the box. Into the fourth input, the third party inputs 0 (see Fig. \[fig:isolation\]). We apply the same method to isolate the bipartite PR box. As soon as the boxes are isolated, they can be distilled to almost perfect.
\(A) at (-1.5,0) [$x_1x_2x_3$]{}; (E) at (-5.8,0) [$P^1$]{}; (-3,0) to\[\] (-4,0); (-4,0) to\[\] (-3,0);
(-5.2,0) ellipse (0.6cm and 1.6cm); () at (-6.1,1.5) [3*rd* party]{};
(-5.5,0.75) to\[\] (-5.5,0.5); (-6.1,0.75) to\[\] (-6.1,0.5); (-4.9,0.75) to\[\] (-4.9,0.5); (-6.7,0.75) to\[\] (-6.7,0.5); () at (-6.7,1) [$x_1$]{}; () at (-5.5,1) [$x_3$]{}; () at (-6.1,1) [$x_2$]{}; () at (-4.9,1) [$0$]{};
(-5.5,-0.5) to\[\] (-5.5,-0.75); (-6.1,-0.5) to\[\] (-6.1,-0.75); (-4.9,-0.5) to\[\] (-4.9,-0.75); (-6.7,-0.5) to\[\] (-6.7,-0.75); () at (-5.5,-1) [$a_3$]{}; () at (-6.1,-1) [$a_2$]{}; () at (-4.9,-1) [$a_4$]{}; () at (-6.7,-1) [$a_1$]{};
(-2.3,0.75) to\[\] (-2.3,0.5); (-1.5,0.75) to\[\] (-1.5,0.5); (-0.7,0.75) to\[\] (-0.7,0.5); () at (-2.3,1) [$x_1$]{}; () at (-1.5,1) [$x_2$]{}; () at (-0.7,1) [$x_3$]{};
(-2.3,-0.5) to\[\] (-2.3,-0.75); (-1.5,-0.5) to\[\] (-1.5,-0.75); (-0.7,-0.5) to\[\] (-0.7,-0.75); () at (-2.3,-1) [$a_1$]{}; () at (-1.5,-1) [$a_2$]{}; () at (-0.7,-1) [$a_3\oplus a_4$]{};
Since the 4-PR box cannot be isolated and there does also not exist an $n$-PR box with $n\geq 4$, we have to construct it by smaller ones. Therefore, we use the almost-perfect 3-PR and 2-PR box and use the construction given in Section \[sec:constr\]. We can now put the three boxes together and receive the almost perfect full-correlation box $P^1$.
Conclusion
==========
We have studied the problem of non-locality distillation in the multipartite setting, where the parties are not allowed to communicate to each other. First, we characterized maximally non-local full-correlation boxes and showed that for every full-correlation box, there exists a closest local box which is also a full-correlation box. Second, based on the generalized Brunner/Skrzypzyk protocol, we showed that every full-correlation box can be distilled to almost perfect without communication if the interfaces of the box can be arbitrarily distributed by the parties. This implies that it is possible in the multipartite case to distill boxes that are arbitrarily close to the local bound to boxes that are maximally non-local. It remains an open question to classify and find distillation protocols for multipartite non-local boxes that are not full-correlation boxes.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors thank Jibran Rashid, Benno Salwey, and Marcel Pfaffhauser for helpful discussions. This work was supported by the Swiss National Science Foundation (SNF), the NCCR “Quantum Science and Technology" (QSIT), and the COST action on “Fundamental Problems in Quantum Physics."
|
---
author:
- 'A. Lamberts'
- 'G. Dubus'
- 'G. Lesur'
- 'S. Fromang'
bibliography:
- 'KHrot.bib'
title: Impact of orbital motion on the structure and stability of adiabatic shocks in colliding wind binaries
---
[The collision of winds from massive stars in binaries results in the formation of a double-shock structure with observed signatures from radio to X-rays.]{} [We study the structure and stability of the colliding wind region as it turns into a spiral due to orbital motion. We focus on adiabatic winds, where mixing between the two winds is expected to be restricted to the Kelvin-Helmholtz instability. Mixing of the Wolf-Rayet wind with hydrogen-rich material is important for dust formation in pinwheel nebulae such as WR 104, where the spiral structure has been resolved in infrared.]{} [We use the hydrodynamical code RAMSES to solve the equations of hydrodynamics on an adaptive grid. A wide range of binary systems with different wind velocities and mass loss rates are studied with 2D simulations. A specific 3D simulation is performed to model . ]{} [Orbital motion leads to the formation of two distinct spiral arms where the Kelvin-Helmholtz instability develops differently. We find that the spiral structure is destroyed when there is a large velocity gradient between the winds, unless the collimated wind is much faster. We argue that the Kelvin-Helmholtz instability plays a major role in maintaining or not the structure. We discuss the consequences for various colliding wind binaries. When stable, there is no straightforward relationship between the spatial step of the spiral, the wind velocities, and the orbital period. Our 3D simulation of WR 104 indicates that the colder, well-mixed trailing arm has more favourable conditions for dust formation than the leading arm. The single-arm infrared spiral follows more closely the mixing map than the density map, suggesting the dust-to-gas ratio may vary between the leading and trailing density spirals. However, the density is much lower than what dust formation models require. Including radiative cooling would lead to higher densities, and also to thin shell instabilities whose impact on the large structure remains unknown.]{}
Introduction
============
Massive stars possess highly supersonic winds due to radiation pressure on atomic lines. Wind mass-loss rates range from $\dot{M}\simeq 10^{-8} M_{\odot}$ yr$^{-1}$ for O or B type stars to $\dot{M}\simeq 10^{-4} M_{\odot}$ yr$^{-1}$ for Wolf-Rayet stars (WR) . Most massive stars lie in binary systems, where the interaction of the two supersonic stellar winds creates two strong shocks separated by a contact discontinuity. The geometry of the colliding wind region depends on the momentum flux ratio of the winds [@1990FlDy...25..629L] $$\label{eq:eta}
\eta\equiv \frac{\dot{M}_2v_2}{\dot{M}_1v_1}$$ where $v$ is the wind velocity. The subscript 1 usually stands for the stronger wind, the subscript 2 for the weaker one so that $\eta\leq 1$. There are several important observational signatures of the colliding wind region. The shock-heated gas generates observable thermal X-ray emission [e.g. @1976PAZh....2..356C; @Luo:1990mp; @Usov:1992re; @Stevens:1992on]. The presence of intra-binary structures causes variations of the emission line profiles with orbital phase [e.g. @1988ApJ...334.1021S; @1993ApJ...407..252W]. The high densities reached in the colliding wind region are thought to enable the formation of dust, explaining the large infrared emission from binary systems with WR stars , and the formation of spiral structures extending to distances up to 300 times the binary separation (“pinwheel nebulae", @1999Natur.398..487T). In some systems, diffusive shock acceleration of particles leads to non-thermal radio emission . The radio emission has been resolved by long baseline interferometry and shown to have a morphology changing with orbital phase [e.g. @2005ApJ...623..447D]. A new exotic class of colliding wind binaries is gamma-ray binaries, where the non-thermal emission is thought to arise from the interaction of a pulsar relativistic wind with the wind of its massive stellar companion . Interpreting all this observational data requires increasingly detailed knowledge of the physics of colliding winds, hence numerical simulations, notably of the large scale regions that can be resolved in radio or infrared.
On large scales, orbital motion is expected to turn the shock structure into a spiral, although we will show that this is not always true. Orbital motion breaks the symmetry with respect to the binary axis and no analytic solution predicts the detailed structure of the colliding wind region. Material in the spiral is generally thought to behave ballistically, so that the step of the spiral is the wind velocity $v$ times the orbital period $P_{\rm orb}$. The wind velocity to use is unclear. @2008ApJ...675..698T took the speed of the dominant wind $v_1$ (dominant in the sense that $\dot{M}_1 v_1 \geq \dot{M}_2 v_2$), whereas @2008MNRAS.388.1047P assume that it is the slower wind that determines the step of the spiral but focus their study on binaries with equal wind velocities. Simple dynamical models of the shocked layer have been developed for use with radiative transfer codes, assuming that the double shock structure is infinitely thin (thin shell hypothesis) and that the material is ballistic [@Harries2004; @2008MNRAS.388.1047P]. The spiral structure is then reproduced at small computational cost but this neglects the impact of the pressure, which creates a distinction between both arms of the spiral , the influence of the reconfinement of the weaker wind for small $\eta$ [@PaperI] and the large-scale evolution of instabilities in the colliding wind region (see below). Up to now no 2D or 3D hydrodynamical simulation has modelled a complete step of the spiral. We achieve this by using the hydrodynamical code RAMSES with Adaptive Mesh Refinement (AMR). AMR allows large scale simulations to be performed while keeping a high enough resolution close to the binary in order to form the shocks properly (§2).
Small scale simulations without orbital motion (see @Stevens:1992on and references in @PaperI, hereafter Paper I) have shown that several instabilities are at work in colliding wind binaries. Thin shell instabilities occur when cooling is important so that the shocked zone narrows to a thin layer, which is easily perturbed [@1994ApJ...428..186V]. They provoke strong distortions of the whole colliding region . However, these instabilities are unlikely to be dominant in wide binary systems where cooling is inefficient, the shocks adiabatic, and the colliding wind region wider [@Stevens:1992on]. In this case, the velocity difference between both winds triggers the Kelvin-Helmholtz instability (KHI) at the contact discontinuity (Paper I). Including orbital motion has led to contradictory results. @Lemaster:2007sl found that eddies develop even when the winds are completely identical, because orbital motion introduces a velocity difference. @2009MNRAS.396.1743P found no eddies in a simulation with a similar setup. also found no eddies, although their simulation has an initial non zero velocity difference $\beta=v_1/v_2=3/4$, and argued that orbital motion stabilises the KHI. Larger velocity differences between the winds have not been investigated. We performed a set of 2D simulations to study how orbital motion changes the shock structure and the development of the KHI, focusing exclusively on binaries with adiabatic shocks (§3). The size of the spiral step and the stability of the spiral on large scales are addressed in §4.
is an example of a long orbital period binary system where the collision between the winds of the WR and its early-type companion is expected to be close to adiabatic (see §5). The infrared emission is very well matched by an Archimedean spiral although its brightest point is shifted by 13 milli-arcseconds from its centre, possibly because dust formation is inhibited closer in (@2008ApJ...675..698T, hereafter T2008). The WR wind is hostile to dust formation due to its high temperature, low density, and absence of hydrogen . The wind collision region is more favourable, providing high densities, shielding from the UV radiation of the WR star, and the possibility of mixing with hydrogen from the companion star [@2007ASPC..367..213M]. We carried out 2D and 3D hydrodynamical simulations using the parameters of WR 104 to investigate these questions (§5). We then relate all our results to observations (§6).
Numerical Simulations {#numerics}
=====================
Equations
---------
We use the hydrodynamical code RAMSES for our simulations . This code uses a second order Godunov method to solve the equations of hydrodynamics $$\begin{aligned}
\frac{\partial\rho}{\partial t}+\nabla \cdot(\rho \mathbf{v}) &=& 0\\
\frac{\partial(\rho \mathbf{v})}{\partial t}+\nabla \cdot (\rho \mathbf{v}\mathbf{v})+\nabla P &=& 0 \\
\frac{\partial E }{\partial t}+\nabla \cdot[\mathbf{v}(E+P)] &=& 0\end{aligned}$$ where $\rho$ is the density, $\mathbf{v}$ the velocity, and $P$ the pressure of the gas. The total energy density $E$ is given by $$\nonumber
E= \frac{1}{2}\rho v^2+\frac{P}{(\gamma -1)}$$ $\gamma$ is the adiabatic index, set to 5/3 to model adiabatic flows.
Numerical parameters
--------------------
We use the MinMod slope limiter together with the exact Riemann solver (§3-4) or the HLLC (§5) Riemann solver to avoid numerical quenching of instabilities. We perform 2D and 3D simulations on a Cartesian grid with outflow boundary conditions. We use AMR which enables to locally increase the spatial resolution according to the properties of the flow. We base the refinement criterion on velocity gradients. In section §3 we perform small scale simulations where the size of the computational domain is $l_{box}=40a$, with $a$ the binary separation. We have $n_x=64$ for the resolution of the coarse (unrefined) grid and use 7 levels of refinement. This gives an equivalent resolution which is at least two times better than in former studies . In section §4 we perform larger scale simulations where $l_{box}=400a$ , the coarse grid also has $n_x=64$ but we use up to 9 levels of refinement. In some cases we adapt the size of our grid to larger or smaller values to model a complete step of the spiral.
Generation of the winds
-----------------------
To simulate the winds, we keep the same method as used in Paper I, which was largely inspired by @Lemaster:2007sl. Around each star, we create a wind by imposing a given density, pressure, and velocity profile in a spherical zone called mask. The masks are reset to their initial values at each time step to create steady winds. We add two passive scalars $s_1$ and $s_2$ to distinguish both winds and to quantify mixing. We initialise the passive scalars in the masks; their evolution is determined by $$\label{eq:passive_scalar}
\frac{\partial{\rho s_i}}{\partial t}+\nabla\cdot(\rho s_i\mathbf{v})=0 \qquad i=1,2$$ In the free wind of the first star $s_1=1$ and $s_2=0$, in the second wind it is the other way round. In the shocked zone both scalars have an intermediate value which accounts for the mixing of the winds. The rotation of the stars is clockwise in the figures, their positions are updated using a leapfrog method. For each simulation, the input parameters are the mass $M$, mass loss rate $\dot{M}$, wind velocity $v$ (which we suppose to be constant), and Mach number $\mathcal{M}$ at $r=a$ for each star. The exact value of the Mach number does not matter for the colliding wind region, as long as it is high enough that pressure terms can be neglected (Paper I), which is the case for massive star winds. Here, the Mach numbers of both winds are set to 30. In all our simulations, the star with the highest momentum flux is considered as the first star. We will refer to its wind as the stronger wind. The values of the parameters of the winds in the simulations are given in the table in Appendix B. Both stars have a mass of 15 $M_{\odot}$ and the binary separation $a$ is 1 AU. The corresponding orbital period $P_{\rm orb}$ is 0.18 yr (67 days). The orbital velocity of the stars is $v_{orb}=81$ km s$^{-1}$. We only study circular orbits.
2D and 3D simulations
---------------------
We perform our 2D simulations in the orbital plane of the binary. We thus model the cylindrical $(r,\theta)$ plane instead of the $(r,z)$ plane as classically done (e.g. ). This implies the density evolves $\propto r^{-1}$ instead of $\propto r^{-2}$ in a spherical geometry. For a given $\eta$ the structure of the colliding wind binary is thus different in 2D and 3D. However, as discussed in Paper I, the mapping $\sqrt{\eta_{\rm 3D}}\rightarrow \eta_{\rm 2D}$ captures most of the 3D structure in the 2D simulations. This point is re-discussed in §5 where we compare the results of a 2D simulation of WR 104 with a full 3D simulation including orbital motion. A major advantage of our 2D setup is the possibility to implement orbital motion for a modest computational cost, enabling the study of the flow structure up to scales currently inaccessible to full 3D calculations.
Impact of orbital motion on the shock arms {#small_scale}
==========================================
We carried out simulations of adiabatic colliding winds identical to those carried out in paper I except that they now include orbital motion to study its impact on the shock structure and development of the KHI. The simulations explore $\eta=1$ and $\eta=0.0625$ for different velocity ratios $\beta\equiv v_1/v_2=1,2,20$, all in a box of size $8a$. Briefly, the results without orbital motion were that (1) no instability is seen when $\beta$=1; (2) for $\beta \geqslant 2$, the instabilities affect the position of the contact discontinuity, for $\beta=20$ the KHI also affects the shocks positions; (3) for $\eta=0.0625$ the instabilities remain confined to the weaker wind. We present first the results of the simulations for $\beta$=1, where the KHI instability may be triggered (or not) by orbital shear (§3.1). We then discuss the simulations with $\beta\neq 1$. In these cases, the dominant wind is slower and much denser than the weaker wind. For $\eta=1$, there is no difference between simulations where $\beta=B$ and $\beta=1/B$.
A view of the overall colliding wind structure is given in Fig. \[fig:geometry\]. We define the leading arm as the arm preceding the second star, with respect to orbital motion (clockwise motion). The trailing arm is the second part of the spiral. Note that there is no dominant wind when $\eta=1$ so that the definition of leading/trailing is degenerate in this case. (The definition also has no link with the definition commonly used in galactic dynamics.) In each arm there is a shock in the wind from the first star and a shock in the wind from the second star, separated by a contact discontinuity. In 2D simulations, when $\eta <0.25$ the second wind is confined by the intersection of the shocks. In 3D simulations this occurs for $\eta \simeq 0.06$ (Paper I).
Close to the binary, relative motion of the stars creates an “aberration” @2008MNRAS.388.1047P of the shocked zone.@2008MNRAS.388.1047P introduce the skew angle $\mu$ which measures the offset between the line of centers of the stars and the symmetry axis of the shocked region. It is given by
$$\label{eq:skew}
\tan \mu =\frac{v_{orb}}{v}$$
where $v$ is taken as the speed of the slowest wind. This angle remains small unless the velocities of the winds are strongly reduced or orbital motion becomes important We measured $\mu$ in our simulations by finding the best fit of the analytic position of the contact discontinuity derived by @Canto:1996jj. An example is given on Fig. \[fig:skew\]. We measured $\mu\simeq 22^{ \circ}$ in the simulation $\{\eta=0.0625,\beta=0.05\}$ while the theory predicts $\mu=21^{\circ}$. The simulation of WR 104 (see §5) gives $\mu\simeq 9^{\circ}$ while the theory predicts $\mu=8^{\circ}$. $\mu$ is too small in our other simulations to allow correct measurements.
Simulations with $\beta=1$
--------------------------
Fig. \[fig:identical\] shows the density, velocity, and mixing map for a simulation with identical winds $\{\eta=1,\beta=1\}$. We determine the mixing by the product of the passive scalars $s_1\times s_2$. The free (unshocked) winds correspond to the low density parts at the top and bottom. The denser parts are the shocked winds. The radial inhomogeneities visible in the unshocked wind region of the velocity map is a numerical artefact: it corresponds to minute anisotropies in the stellar wind due to the finite size of the masks. As shown in Appendix A the orbital period is much longer than the local shear timescale. The Coriolis force does not impact on the development of the KHI. However, the velocity map shows that a $\simeq 20 \%$ velocity difference develops in each arm at a distance $\simeq 20a$ from the binary. Orbital motion makes the shocked material leading the contact discontinuity (red region on the velocity map) accelerate in the lower density free wind region while the shocked wind trailing the contact discontinuity (in green) moves into the denser, shocked material of the other wind. The resulting velocity difference is sufficient to trigger the KHI even though the original wind speeds are equal. The instability is clearly present in the mixing map. We therefore confirm the results of @Lemaster:2007sl that orbital motion triggers the KHI even when the winds are identical. We also observe, like they do, an artificial enhancement of the instabilities when the shocks align with the grid. Our simulations contradict the results from who find no KHI. Their simulations are performed with a Lax Friedrich Riemann solver. We ran a test simulation with the Lax Friedrich Riemann solver and observed no development of the KHI either, due to the important numerical diffusivity.
Fig. \[fig:small\_scale\_0625\] shows the density, velocity, and mixing for $\{\eta=0.0625,\beta=1\}$. Because of the low value of $\eta$, there is a reconfinement shock (Paper I) behind the second star. The various discontinuities are indicated in the velocity map, to be compared with the simpler geometry shown previously in Fig. \[fig:geometry\]. Our simulations without orbital motion showed no KHI because the initial velocities are identical. Here, as in the $\eta=1$ case, orbital motion leads to velocity shear and mixing at the contact discontinuity. The KHI is confined to narrow regions close to the discontinuity because of the low $\eta$ and not because of orbital motion: we had found the same behaviour in the models explored in paper I. We also see that complex velocity structures arise in the colliding wind region even in this [*a priori*]{} simple case where both winds have the same velocity, highlighting the possible difficulties in interpreting spectral line features arising from this region without guidance from numerical simulations.
. \[fig:small\_scale\_0625\]
Simulations with $\beta \ne 1$
------------------------------
Fig. \[fig:small\_scale\] shows the density and mixing for simulations with $\{\eta=1,\beta=2,20\}$ and $\{\eta=0.0625,\beta=0.05,0.5,2,20\}$. These maps show the impact of rotation on arm geometry and the development of the instabilities.
The leading and trailing arms become markedly different when the velocity difference increases, even when $\eta$=1. The shocked zone preceded by the unshocked wind with the higher velocity and lower density is larger than the zone preceded by the lower velocity, higher density wind. The latter shocked zone is compressed by the high velocity wind into a high density region . We verified that, as expected if this explanation is correct, for $\beta>1$ the compressed arm is the trailing arm while for $\beta<1$ the compressed arm is the leading arm (see Fig. \[fig:small\_scale\]). For $\beta=20$, compression results in a rim of puffed up matter where the density increases by two orders of magnitude with respect to the simulation with $\beta=1$. The differentiation of both arms is independent of the instabilities in the winds but plays a role in their development.
The KHI starts similarly in both arms close to the binary major axis as the velocity difference and density jump across the contact discontinuity are the same in both arms. The symmetry between both arms is broken as the flow moves outwards. The compression of the shocked zone in the narrower arm results in a thin mixed zone with small scale structures, whereas the eddies are stretched out in the wider arm. Mixing covers a larger area in the wider zone. This is not just a geometrical effect. We show in Appendix A that when media have different densities ($\alpha\ne 0$, see Eq. A.10) mixing by the KHI occurs preferentially in the least dense medium. Both velocity and density profiles thus play a role in the development of the KHI in colliding wind binaries. They both impact the large scale outcome of the spiral structure as will be shown in §\[large\_scale\].
We checked that the mixing is physical and not numerical in the narrow arm. If numerical, the physical size of the mixed zone increases with decreasing resolution. If mixing follows from instabilities, the physical size of the mixed zone is constant with resolution. We measure the width of the narrow mixed zone across the contact discontinuity at the upper edge of the simulation box for the case $\{\eta=0.0625,\beta=2\}$. The limit of the zone is determined by $s_1\times s_2=0.01$. For the simulations with the highest resolution (7 levels of refinement), we measure a width of 0.87$a$ while we measure a width of 0.93$a$ in a simulation with 6 levels of refinement. We conclude that numerical diffusion has a limited impact and that the mixing in our high resolution simulations is robust.
Formation of a spiral structure {#large_scale}
===============================
We now consider the large-scale evolution of the previous simulations (§\[small\_scale\]) in a box of size $l_{box}=400a$. Density and mixing maps are shown for $\eta=1$ in Fig. \[fig:large\_scale\_1\] and for $\eta=0.0625$ in Fig. \[fig:large\_scale\_00625\], with $\beta$ increasing from top to bottom in both figures. The spatial scale is the same in all plots except for the top two panels of Fig. \[fig:large\_scale\_00625\], where we reduced the size of the domain to avoid unnecessary computational costs. The different behaviour of mixing in both arms discussed in §3 persists on the larger scales, eventually causing both contact discontinuities to merge into one single spiral in simulations with $\eta=0.0625$ (Fig. \[fig:large\_scale\_00625\]). We cannot exclude this merger results from numerical artefacts due to the use of a cartesian grid to describe an inherently spherical phenomenon. Merger should still occur naturally from inhomogeneities in the winds. Figures \[fig:large\_scale\_1\]-\[fig:large\_scale\_00625\] also show that the colliding wind region does not always turn into a stable spiral and that, for given $\eta$ the appearance depends strongly on the velocity ratio $\beta$. We discuss below the step size that we measure when a steady spiral forms before addressing the issue of the stability of the pattern.
The step of the spiral
----------------------
We found that, when a stable spiral structure is formed, an Archimedean spiral with a step size $S$ provides a good fit to the results of our simulations. As in @2009MNRAS.396.1743P, we find that the fit with an Archimedean spiral is not perfect at the apex. However, the deviation is small and limited to a region $\simeq 10a$. Mixing follows closely the contact discontinuities in the arms so we used the mixing maps to trace the spiral. In fact, the spiral is not always clearly apparent in the density maps ([*e.g.*]{} $\{\eta=0.0625,\beta=1\}$ in Fig. \[fig:large\_scale\_00625\]), especially when a complex flow is established by the presence of a reconfinement shock behind the weaker star (Fig. 1). The fitted $S$ for various values of $\eta$ and $\beta$ is compared to the theoretical estimate $S_1$ in Fig. \[fig:pas\_spirales\]. $S_1$ assumes the velocity of the stronger wind controls the structure scale so that $S_1=P_{\rm orb}v_1$ (e.g. T2008). When $v_1=v_2$, there is no ambiguity in the velocity that sets the step size and we verify that, in this case, $S=S_1$ for all $\eta$. This also rules out any significant numerical issue with the way the spiral develops. There are significant deviations from $S_1$ in all the other cases, except when $\eta\ll 1$ [*i.e.*]{} when the first wind largely dominates momentum balance. For more balanced ratios $\eta$, the spiral step is smaller than expected when the weaker wind is slower than the stronger wind, and vice-versa when the weaker wind is the fastest. Using the slowest wind speed instead of $v_1$ [e.g. @2008MNRAS.388.1047P] does not work better. The results do not suggest a straightforward analytical correction using $\eta$ and $\beta$ that could be used to interpret observations of pinwheel nebulae without requiring hydrodynamical simulations.
![Step of the spiral $S/S_1$ as a function of $\beta$ for $\eta$ =1 (diamonds), 0.5 (diagonal crosses), and 0.0625 (crosses). The symbols are respectively joined by a solid, dashed, and dash-dotted line for easier identification.[]{data-label="fig:pas_spirales"}](pas_spirales){width="0.75\linewidth"}
Stability of the spiral structure \[spiral\]
--------------------------------------------
The complete disruption of the spiral structure for large velocity ratios (Figs. \[fig:large\_scale\_1\]-\[fig:large\_scale\_00625\]) was unexpected. The breakdown can be traced to the KHI: we found that the structure was stabilised when we tested a setup with a Lax-Friedrich Riemann solver and a smaller resolution, in which case the KHI is artificially suppressed (Paper I). The amplitude of the KHI appears to destroy the spiral when strong velocity gradients are present, resulting in widespread turbulence and important mixing throughout the domain. Curiously, for $\eta=0.625$, the structure is unstable when $v_1=20v_2$ while it is stable for the opposite velocity gradient $v_1=0.05v_2$: the strong density gradient ($\dot{M_1}/\dot{M_2}=320$) is also likely to play a role in the stability. Indeed, we measure $\alpha\simeq 0.95$ in the mixed region at $r\simeq 50 a$, which implies a strong reduction of the KHI growth rate.
Table 1 summarises the presence or absence of spiral structures, for all the simulations we performed. We compared these results against the KHI growth rate, normalised by the rate at which eddies propagate (see Appendix A for details). Although the growth rate gives no indication on the saturation in the non-linear regime, we suspect that the spiral is destabilized most easily when the eddies grow quickly before propagating further away.
Figure \[fig:growth\] shows the normalised growth rate (see Eq. A.12) as a function of $\beta$ for $\eta=1, 0.5$, and 0.0625. For $\eta=1$ the curve is symmetric. The KHI does not develop when there is no velocity difference, peaks at $\beta_{{\rm max}}\simeq 2$ and drops for higher values of $\beta$ because the density gradient dampens the growth rate. The symmetry with respect to $\beta=1$ is broken when $\eta< 1$: the normalised growth rate is weaker for $\beta<1$ and stronger for $\beta>1$. The lower the value of $\eta$, the stronger this asymmetry. Hence, stable structures are expected near $\beta=1$, for $\beta\gg 1$, and for $\beta \ll 1$. In addition, when $\eta\neq 1$, structures with $\beta<1$ should be more stable than with $\beta>1$.
The results of the simulations are in qualitative agreement with these expectations. When $\eta=1$, there is a spiral for $\beta=1,2,4$ but not for $\beta=8,20$ (Tab. 1), which is consistent with the faster growth of the KHI when $\beta$ increases from 1. However, the transition from stable to unstable spirals occurs further away than expected from Fig \[fig:growth\] ($\beta_{\rm max}\simeq 2$). Also, we were not able to recover a stable final structure for very high $\beta$ (or, equivalently in the case $\eta=1$, very low $\beta$), up to $\beta=200$ (Fig. \[fig:growth\]). Tests at higher $\beta$ are computationally too expensive (and may not have much astrophysical relevance). For $\eta=0.5$, we find that the spiral is maintained for values close to $\beta=1$ but is quickly destroyed for higher/lower values of $\beta$. In this case, a stable spiral is recovered when $\beta \leq 0.01$, consistent with the lower growth rate, while the spiral remains destroyed for the symmetric value of $\beta=20$ (higher growth rate). We observe a similar behaviour for $\eta=0.0625$. Stabilisation is possible for a higher $\beta=0.05$, which is consistent with the lower growth rate of the instability for $\beta<1$ as $\eta$ decreases. We conclude the presence of a spiral depends on $\eta$ and $\beta$ in a way that is consistent with having stable structures for near-equal velocity winds ($v_{1}\simeq v_{2}$) or when the weaker wind is much faster ($\dot{M}_{1}v_{1}\geq \dot{M}_{2}v_{2}$ and $v_{2}\gg v_{1}$).
[c c c c c c c c c c c]{} $\eta \backslash \beta$ & .01 & .05& .1 & .5 & 1& 2 & 4 &8 & 20 & 200\
1 & & X & & S & S &S& S & X & X & X\
0.5 & S & X & X & S & S &S& & X & X & X\
0.0625 & & S & X & S & S &S& S/X & X & X &\
![Theoretical 2D growth rate of the KHI in colliding wind binaries as a function of the velocity ratio $\beta=v_{1}/v_{2}$ of the winds. The solid, dashed and dash-dotted lines correspond to $\eta=1, 0.5 , 0.0625$ respectively.[]{data-label="fig:growth"}](taux_croissance){width="0.85\linewidth"}
The pinwheel nebula WR 104 {#wr104}
==========================
WR 104 is a binary composed of an early type star and a WR star. The system shows an excess of IR emission related to dust production. The IR emission has been resolved into a spiral structure with several steps imaged (T2008). The high temperatures and low densities in WR winds are difficult to reconcile with dust formation, which requires a temperature around $1000$ K and a number density range between $10^{6}$ cm$^{-3}$ and $10^{13}$ cm$^{-3}$ [@1995IAUS..163..346C]. An additional constraint for dust formation arises from the absence of hydrogen in the WR wind, leading to uncommon chemical processes . Dust production appears closely-related to binarity and the presence of dense colliding wind structures: in eccentric systems, such as WR 48 or WR 112, dust production is limited to orbital phases close to periastron while it is continuous in systems with circular orbits. Systems viewed pole-on show an extended spiral structure in infrared. WR 104 is the prototype system of these pinwheel nebulae. Our aim is to determine whether a hydrodynamical model with adiabatic winds reproduces the observed large-scale structure of WR 104, study mixing and identify regions where dust production may be possible. Detailed modelling of dust formation and growth in colliding wind binaries is beyond the scope of this study.
Simulation parameters
---------------------
\[tab:param\_wr104\]
[c c c]{} & WR & OB\
$v$ (km s$^{-1}$) & 1200 (a) & 2000 (b)\
$\dot{M}$ ($M_{\odot}$ yr$^{-1}$) & $0.8\times 10^{-5}-3 \times 10^{-5}$ (c)& $6\times10^{-8}$ (d)\
\(a) , (b) estimate according to spectral type [@Harries2004], (c) @1997MNRAS.290L..59C, (d) using the mass-loss luminosity relation by @1989ApJS...69..527H.
Table 2 has the wind parameters for the binary system WR 104. The characteristics of the companion to the WR star are not well constrained [@2001NewAR..45..135V] and, like T2008, we will refer to the companion star as the “OB" star. The orbital period (241.5$\pm 0.5$ days), eccentricity $e<0.06$, inclination ($i<16\degr$), and angular outflow velocity of the spiral 0.28 mas day$^{-1}$ in WR 104 were found by fitting an Archimedean spiral to the IR maps (T2008). The orbital separation $a$ is about 2.1-2.8 [au]{} for a total mass of 20-50 M$_\odot$. We took $e=0$ and $a=2.34$ [au]{}. Given the uncertainties on mass loss rate and velocities $\eta$ varies between $0.0125=1/80$ and $0.0033=1/300$. Assuming a constant velocity for the OB wind and $R_{\rm OB}=10$ R$_\odot$ [@Harries2004], the second shock forms at $2.7 R_\mathrm{OB}<r<5.1 R_\mathrm{OB}$ depending on $\eta$.
The shock position can be influenced by additional physical processes. The OB wind is accelerated on distances of $\simeq 2-3$ stellar radii and has not necessarily reached its final velocity at the shock, which modifies the effective momentum flux ratio of the collision. The shock position moves to $2.2R_\mathrm{OB} <r<4.7 R_\mathrm{OB}$ if acceleration is taken into account by using the velocity law $v=v_{\infty}(1-R_\mathrm{OB}/r)$. Radiative braking of the WR wind by the OB radiation field [@1997ApJ...475..786G] can also play a role in WR 104 (T2008). A slower WR wind moves the shock away from the OB star (up to 12 $R_{\rm OB}$ if radiative braking is able to stop the WR wind completely, which is only marginally possible in WR 104, see T2008). The magnitude of both effects, their compensating influence, and the uncertainties in the wind parameters did not justify including these processes. We adopted constant velocity winds and $\eta=0.0033$ to ease comparison with T2008.
Radiative cooling can significantly change the shock structure. The ratio $\chi$ of the cooling timescale $t_{cool}$ over the dynamical timescale $t_{esc}$ provides an estimate of its importance [@Stevens:1992on] $$\label{eq:chi}
\chi=\frac{t_{cool}}{t_{esc}}=\frac{k_BT_s}{4 n_w\Lambda(T_s)}\frac{c_s}{a}$$ where $\Lambda\approx 2\times 10^{-23}$ ergcm$^{3}$ s$^{-1}$ is the emission rate, $n_{w}$ the number density of the unshocked wind, $k_{B} T_{s}=(3/16) \mu m_{p} v_{w}^{2} $ the shock temperature and $c_{s}$ the associated sound speed. The system is adiabatic if $\chi >1$ and isothermal if $\chi \ll 1$. We find $3<\chi_\mathrm{OB}<15$ for the OB star and $0.3<\chi_\mathrm{WR}<1.2$ for the WR star. The system is at the transition between the two regimes. The escape timescale is assumed to be $\simeq a/c_s$ in Eq. \[eq:chi\] but could be as short as $2.7-5.1 R_\mathrm{OB}/c_s$ (increasing $\chi$ by a factor 10-20) if one takes the distance from the OB star ($\sim$ shock curvature radius, @Stevens:1992on). In the following, we neglected radiative cooling in the energy equation and assumed an adiabatic shock.
The low value of $\eta$ is challenging for numerical simulations (see discussion in paper I). The mask of the star needs to be as small as possible so that the shocks can form properly. A minimum length of 8 computational cells per direction is needed to obtain spherical symmetry of the winds. Numerical resolution on scales much smaller than a stellar radius (0.05 [au]{}) is thus required close to the binary. Further away, we need to maintain a high resolution in order to properly study the instabilities, while following a spiral step requires a box size $\geq 200$ [au]{}. We carried out two complementary simulations: a 3D simulation covering scales up to $12a$ and a 2D simulation to model a whole step of the spiral structure.
We use the large scale 2D simulation to determine the step of the spiral and the impact of mixing. As explained in §2.4, we use the mapping $\sqrt{\eta_{\rm 3D}}\rightarrow \eta_{\rm 2D}$ to obtain comparable 2D and 3D results. We took $\eta_{\rm 2D}=0.0625$ to help comparisons with the results in §3-4, which is close enough to $\eta_{\rm 2D WR104}\simeq 0.057$ derived from a straight application of the mapping. It is important to have the right velocity difference for the Kelvin-Helmholtz instability. We thus adapted $\dot{M}_{WR}$ in order to have $\eta_{\rm 2D}=0.0625$ for the 2D simulation. We use a 200$a\approx 500$ [au]{} simulation box with $n_x=128$ and 12 levels of refinement. This gives an equivalent resolution equal to $2^{19}\simeq 5\times 10^5$ cells. We use nested grids to slowly decrease the maximum allowed resolution away from the binary.
We use the smaller scale 3D simulations for quantitative results on the density and temperature in the winds. The 3D simulation follows $1/8$th of an orbit of WR 104 in a 12$a\approx 30$ [au]{} simulation box, large enough to see the impact of orbital motion. The orbital plane is the mid-plane of the box and the centre of mass of the binary is placed in a corner of the box to maximise the use of the simulated volume. We use adaptive mesh refinement with a maximal equivalent resolution of 4096$^3$. We limit the high resolution to a narrow zone of $3a$ close to the binary where the instabilities develop. It corresponds to the same equivalent resolution as in our 2D model. We model only $\simeq 20$ layers at this high resolution in the $z$ direction and we gradually reduce the resolution when going away from the orbital plane.
Global structure
----------------
Figure \[fig:wr104\_3D\] shows the density, velocity, mixing, and temperature in the binary orbital plane of the 3D simulation (top row). The bottom row has the corresponding 2D map on the same scale. The comparison confirms the mapping in $\eta$ captures adequately the 3D structure in the 2D simulation. The positions of the shocks and contact discontinuity along the line of centres are similar in both 2D and 3D simulation and match the analytic solutions (Paper I). The opening angle defined by the contact discontinuities, well traced in the mixing map, is $15 \pm 1^{\circ}$. This angle is consistent with the analytical estimates that have been used (@Harries2004, T2008). However, the opening angle defined by the location of the shocks is wider in 2D than in 3D, which may have some influence on the density structure at larger scales (see §5.3). Because of the low $\eta$, there is a reconfinement shock behind the OB wind at a distance $\simeq 0.75a$ in the 3D simulation (1.5$a$ in the 2D simulation). All of the OB wind is involved in the collision and no fraction escapes freely to infinity.
Material piles up in both arms of the spiral. T2008 suggest different strengths of the shock can change the conditions for dust formation in each arm. In the 3D simulation, the Mach number of the trailing arm at $r\simeq 12a$ is 13$\%$ higher than in the leading arm, in agreement with the results in @Lemaster:2007sl. The small temperature difference is unlikely to affect dust formation. A more significant effect is that compression keeps a hotter temperature in the leading arm than in the trailing arm. Material in the mixing zone of the trailing arm experiences a temperature an order-of-magnitude cooler than in the mixing zone of the leading arm. Dust formation may be favoured in this arm, seeding the spiral structure when the contact discontinuities merge farther out (see below).
The amount of mixing increases with the distance to the binary. Integrating in spheres of increasing radii, the ratio of mixed material to the total amount of material within $r$ increases from $\simeq 0.01\%$ at $r=a$ to $0.4\%$ at $r=10a$. These values are constant during the last stages of the simulation (lasting $\simeq 0.1 P_{\rm orb}$), indicating the development of the instabilities has reached a steady state. As can be seen on Fig. \[fig:wr104\_3D\] and as expected from theory (Appendix A), mixing occurs mostly in the lower density regions of the colliding wind zone. The velocity map shows that the velocity is mostly radial and that matter is accelerated on a distance of a few times the binary separation. After substraction of the radial component, we find the velocity of the flow along the spiral in the 3D simulation reaches a maximal value of $\simeq 800$ km s$^{-1}$. This corresponds to the low density region in the center of the spiral. In the outer regions of the spiral, the velocity along the spiral reaches $\simeq$ 500 km s$^{-1}$.
Figure \[fig:wr104\_2D\] shows the 2D simulation on the largest scale (200$a$ or about 470 [au]{}). A stable spiral structure forms as expected for $\beta=0.6$ and $\eta=0.0625$ (Tab. 1). The collimated OB wind generates a low density spiral bounded on each side by walls of material where the density is $\sim$100 times larger. The initially different mixing in both arms blurs at a distance of $\simeq 50 a$. The mixing zones more or less merge and follow the leading arm, overlapping slightly with the density enhancement of the arm. The step of the spiral is 1.05$S_{\rm WR}$ where $S_{\rm WR}=v_{\rm WR} P_{\rm orb}=170$ [au]{}$=77a$. T2008 assumed $S_{\rm WR}$ to determine a distance of 2.6 kpc from the observed step size. The 5% correction to this distance due to the intrinsically larger spiral step is smaller than the uncertainty on the measured WR velocity and observed angular step size.
The single-armed spiral observed in infrared is more reminiscent of the mixing region than the double spiral in the density map. A double-armed spiral structure, separated by a very under-dense region of angular size $\simeq 27$ mas (at 2.6 kpc), would have been resolved if the IR emission correlated with density ([*i.e.*]{} for a constant gas-to-dust ratio). However, we caution that the width of the low density zone may be overestimated in this 2D simulation since it is likely to be related to the opening angle of the shocks, which Fig. \[fig:wr104\_3D\] shows to be wider in 2D than in 3D. Including dust radiative transfer is required for a closer comparison of the observations with the hydrodynamical simulation.
Conditions for dust formation
-----------------------------
One criterion for dust formation is a high enough density. @1995IAUS..163..346C indicate different paths towards the formation of amorphous carbon for number densities $n$ ranging from 10$^6$ to 10$^{13}$ cm$^{-3}$ and give a detailed study for $n=10^{10}$cm$^{-3}$. This gives $\rho=1.4 \times 10^{-14}$ g cm$^{-3}$ assuming a mean molecular weight $\mu=1.4$, typical for a ionized WC wind [@Stevens:1992on]. Such a density is only present in our 3D simulation at the edge of the spiral, up to a distance $\simeq$ 2$a$ from the WR star. In the 2D simulation, along the walls we find that the density drops as $\rho \propto r^{-1}$. Using this as guidance, we expect $\rho \propto r^{-2}$ in 3D on large scales. The minimum value $n$=10$^{6}$ cm$^{-3}$ considered by @1995IAUS..163..346C is reached at [$r\simeq 25 $ $a$ at the inner wall of the spiral. This is equivalent to 1/3 of a turn]{} along the spiral. The density is too low for dust formation beyond this distance so that any dust present far away has been advected out.
Temperature is another criterion. Models show that dust condensation is possible for 1000K $<T<6000$K and does not vary much within that temperature range . A strong temperature gradient remains between leading and trailing regions of the shocked zone, with the more compressed material maintaining a very high temperature. Still, the temperature falls from $\simeq 10^7$ K close to the stars to $\simeq 10^5$ K in parts of the mixing zone at the outer edge of the 3D simulation box ($\simeq 10a$). The temperature is expected to fall below 6000 K at half a turn of the spiral by extrapolating using $T \propto r^{-4/3}$ (from $P\propto \rho^{5/3}\propto \rho T$ and $\rho\propto r^{-2}$ in the arms). This can be taken as an upper limit to the dust condensation distance as this is calculated in the adiabatic approximation. It is consistent with the infrared observations of a quarter-orbit shift between the maximum infrared emission and the binary centre. Radiative cooling and photoionization heating of the wind would have to be included to have an accurate determination of the temperature and of the impact of shielding from the stellar radiation fields.
Discussion
==========
Asymmetries due to orbital motion
---------------------------------
Orbital motion breaks the symmetry around the binary axis and introduces significant differences from the stationary case with adiabatic winds. It causes a velocity difference that triggers the KHI even when both winds are strictly identical. It results in differentiation of the two arms flanking the weaker star. The arm moving into the densest unshocked wind is compressed, dampening the KHI while the other arm expands and sees Kelvin-Helmholtz eddies of larger size. The density difference between the inner cavity and the bracketing walls can reach two orders of magnitude. According to @2011ApJ...726..105P radiation pressure can have a similar effect, either enhancing or reducing the initial difference. @2004MNRAS.351.1307V modelled the variations in emission line profiles of by a rotating cone with dense edges, allowing them to constrain the opening angle of the colliding wind region. They found a wider opening angle than expected using the analytic formula for the opening angle of the contact discontinuity [@Canto:1996jj] with the standard value of $\eta$ for this system. Our simulations also show that matter accumulates at the shock rather than at the contact discontinuity. The observed opening angle thus corresponds to the opening angle of the shocks, which is wider than the opening angle of the contact discontinuity. This also increases the fraction of the WR wind involved in the collision compared to estimates using the contact discontinuity. Some of the spectral line features that are not explained by models where both arms have equal emission (absorption) could be due to differences between leading and trailing arm [@1999MNRAS.302..549S]. The skew angle that we measured in the simulations matches the theoretical value given by Eq. \[eq:skew\]. However, as we found that the step of the spiral is mostly determined by the speed of the stronger wind (and not the speed of the slower wind), we wonder whether this could also be the case for the skew angle. This is implicitely assumed by @2007ApJ...662.1231K. Our simulations do not allow us to answer this question.
To spiral or not to spiral
--------------------------
The simulations presented here are the first including at least one step of the spiral. We have shown that a structure is maintained on these scales when the two winds have nearly equal velocities ($\beta\simeq 1$). This is consistent with the observations of pinwheel nebulae in several WR + O star binaries , since their winds do have comparable velocities. The spiral is destabilised when the stronger wind has a velocity between 10-50% of the weaker wind (Tab. 1). For example, the episodic ejection of large amounts of (initially) slow-moving material could have temporarily destabilised any spiral structure in the luminous blue variable (LBV) / WR binary [@2007ApJ...658L..25N; @2011AJ....142..191G]. may be a case where any large-scale structure generated near apastron (when the system is closer to being adiabatic) is destroyed because of the destabilising velocity ratio $\beta\simeq 1/6$ — although this would have to be assessed against the effects of the high orbital eccentricity [@2011ApJ...726..105P]. We expect our results to hold for eccentric orbits if the system stays adiabatic. If the system moves from adiabatic to radiative along its orbit then thin shell instabilities develop with unknown consequences on the large scale structure.
The spiral is stabilised again when the velocity ratio $\beta\ll 1$. Such a situation may occur in gamma-ray binaries composed of a young non-accreting pulsar and an early-type star . In this case, the stellar wind interacts with the tenuous, relativistic pulsar wind. have argued that the KHI would destroy any large-scale structure. Assuming our results hold in the relativistic regime, we find that a stable spiral can form on large scales if the stellar wind dominates because $\beta \ll 1$ in this situation. The structure is unstable if the pulsar wind dominates, pointing to the intriguing possibility that the interaction may switch from one regime to another in gamma-ray binaries with Be companions such as . The highly eccentric orbit takes the pulsar close to the equatorial disc where the slow-moving stellar outflow dominates momentum balance [@1997ApJ...477..439T], leading to a stable colliding wind region. However, at apastron, the pulsar wind may dominate over the radiatively-driven stellar wind and be unable to form a stable structure. Strong mixing of the two winds leads to rapid Coulomb or bremsstrahlung losses for the high energy particles, which has an impact on the gamma-ray emission [@2010MNRAS.403.1873Z]. Extended radio emission was detected around PSR B1259-63 near periastron [@2011ApJ...732L..10M]. Regular changes in radio morphology with orbital phase have been observed in other gamma-ray binaries that are compatible with non-thermal synchrotron emission from a stable collimated pulsar wind structure on scales $\leq v_w P_{\rm orb}$ . The luminosity and frequency of the radiation are probably too low to be able to detect the spiral structure on larger scales.
An example of colliding winds with $\beta \gg 1$ also involves pulsars, this time with a low-mass companion [@2011AIPC.1357..127R]. The weak stellar wind is overwhelmed by the relativistic wind of the recycled millisecond pulsar. No stable spiral is expected in this case. Another possible case is eruptive symbiotics like , or . These systems are composed of a red giant, with a very slow wind ($\simeq 20$ km s$^{-1}$) and a hot companion, a white dwarf in nova outburst at the origin of a fast outflow of several 1000 km s$^{-1}$ . We expect the spiral structure to be destroyed if the hot companion dominates. The radio maps of AG Peg have been interpreted assuming a stable spiral structure and a reversal in $\eta$ with time [@2007ApJ...662.1231K]. In many cases there is probably no time to form a spiral, because of the long orbital period compared to the outburst timescale. The radio maps of HM Sge (possible 90 year orbit) show a more fragmented emission region than expected from colliding wind models [@2005ApJ...619..527K], possibly because of thin shell instabilities triggered by radiative losses.
Finally, we note that the KHI also intervenes in the bow shock structure created by the adiabatic interaction of the wind of a fast-moving star with the interstellar medium (e.g. ). The controlling parameter is the ratio of wind speed to star velocity. Much like with spirals, fast growth of the KHI may strongly disturb the cometary structure at large distances, as seen in some hydrodynamical simulations [@2006MNRAS.372L..63W; @2007ApJ...660L.129W].
Dust formation in pinwheel nebulae
----------------------------------
Dust formation in WR 104 and other pinwheel nebulae should be helped by the mixing with hydrogen-rich material from the early-type star that we observe in the 2D and 3D simulations. @2009MNRAS.395.1749W argued that stronger dust emission in the trailing arm would explain better the IR high-resolution images of WR 140, and attributed this to density variations. The winds have nearly identical velocities but the WR has a mass-loss rate 10 times higher than its O companion. The O wind is therefore bracketed by two high-density regions. Our simulations do not suggest very different densities. However, larger amplitude mixing is expected in the trailing arm because it propagates in the more tenuous O wind, possibly enhancing dust formation in this arm. The lower temperature in the trailing arm also helps. Hence a different dust-to-gas ratio between both arms could be an alternative explanation. High density eddies triggered by the KHI in the arms could also be responsible for the observation of IR obscuration events by dust clouds in other WR+O star systems .
The offset of the peak IR emission in WR 104 is consistent with the distance at which we estimated the temperature to fall below the dust sublimation temperature. However, the densities in the colliding wind region are on the low side compared to what dust formation models require. Adiabatic shocks only enhance the density by a factor 4 so radiative cooling is required. Close to the binary, the WR wind is likely to present some cooling, resulting in a thinner and denser shocked layer. @2009MNRAS.396.1743P show the post-shock density is about 100 times higher in their model cwb1, which has strong cooling, than in their adiabatic model cwb3. Radiative cooling also decreases the temperature, bringing the region where dust condensation is possible closer to the binary. Thin shell instabilities can develop when cooling is strong, enhancing mixing of the winds. Given the impact of the (weaker) KHI in adiabatic colliding winds, thin shell instabilities can also be expected to significantly influence the large scale structure. @2009MNRAS.396.1743P have shown that the differenciation of the arms remains when thin shell instabilities are present but the large scale outcome has not been studied yet.
Strong cooling is not necessarily present in all WR binaries. @2011arXiv1111.5194W present evidence from long-term IR observations of WR 48 for dust production throughout the orbit. The stellar winds in this system have similar characteristics than in WR 104 but the (tentative) orbital period is much longer, 32 years. @2011arXiv1111.5194W estimate the system to be adiabatic with an average $\chi\simeq 11$. The value will be even higher at apastron in the eccentric orbit ($e=0.6$), yet dust formation is present. High densities will be much more difficult to reach than in WR 104, requiring dust formation at hitherto lower densities than have been considered possible.
Conclusion
==========
We have studied the large scale impact of orbital motion and the Kelvin-Helmholtz instability on adiabatic shocks in colliding wind binaries. We used hydrodynamical simulations with adaptive mesh refinement to perform the first simulations of complete spiral steps. Orbital motion induces differentiation between both arms of the spiral. The arm propagating in the higher density wind gets compressed while the arm propagating in the lower density wind expands. We explain that this is due to a stronger growth of the KHI in the wider arm and discuss possible observational signatures in spectral lines. We confirm that the KHI arises even when both winds have identical speeds. We compute the step of the spiral and caution that there can be large differences with the standard estimates. We discover that the large-scale spiral structure is destroyed when the velocity gradient between the winds is important. Strong density gradients have a stabilizing effect. According to our simulations we can predict whether certain types of binaries present an extended spiral or not. Systems with stable spirals are those with near-equal velocity winds and those where the weaker wind is much faster. Performing high resolution simulations of pinwheel nebula WR 104, we show that in an adiabatic model significant mixing of the WR wind occurs with the hydrogen-rich wind of the companion. The temperature drop allows the formation of dust at roughly half a step of the spiral, consistent with the spatial offset in peak IR emission. However, the density in those regions falls short of the critical density for dust condensation. Including radiative cooling would lead to higher densities, and also to thin shell instabilities. The impact of these instabilities on the differentiation of the two arms and on the spiral structure is unknown: resolving the thin shock layer in a large scale simulation remains a very challenging numerical problem.
AL and GD are supported by the European Community via contract ERC-StG-200911. Calculations have been performed at CEA on the DAPHPC cluster and using HPC resources from GENCI- \[CINES\] (Grant 2011046391)
The Kelvin Helmholtz Instability in stratified flows
====================================================
Linear theory
-------------
![Configuration of the stratified flow[]{data-label="config"}](configuration){width="0.5\linewidth"}
We work in the incompressible approximation, assuming we have a system with a mean profile $\mathbf{U}=\pm U\mathbf{e_x}$. Above $y=0$, the flow has a density $\rho^+$ and $\rho^-$ for $y<0$ (see Fig. \[config\]). We neglect the Coriolis force since the local shear timescale $\tau_S=\Delta x/\Delta U\sim 10^{-6}\,\mathrm{yr}$ is much sorter than the orbital period $\tau_\Omega\sim 10^{-1}\,\mathrm{yr}$. In this approximation, the linearised equation of motion reads: $$\begin{aligned}
\rho \frac{\partial\mathbf {v}}{ \partial t}+\rho U\frac{\partial \mathbf{v}}{\partial x} +\nabla P&=& 0 \\
\nabla \cdot \mathbf{v} &=& 0\end{aligned}$$ In the following, each quantity is Fourier transformed in $x$ and $t$ thanks to homogeneity: $Q=Q\exp[i(\omega t - k x)$\]. Rewriting the equation of motions and combining them leads to $$\label{eqorr}\partial_y^2v_y-k^2v_y=0,$$ which is solved with two decaying solutions $$\begin{aligned}
v_y^+ = A^+\exp(-ky)&\mathrm{for}&y>0,\\
v_y^- = A^-\exp(ky)&\mathrm{for}&y<0,\end{aligned}$$ $A^+$ and $A^-$ being two arbitrarily chosen constants which are adjusted by jump conditions at the interface $y=0$ : pressure should be continuous and fluid particles should stick to the interface on both sides. The pressure condition reads: $$\label{cont1}
\rho^+\frac{\sigma^+}{k}\partial_y v_y^+=\rho^-\frac{\sigma^-}{k}\partial_y v_y^+,$$ where $\sigma^\pm=\omega\pm U$. The second condition is obtained defining a displacement vector $\xi(x)$ which follows the interface. By definition, a fluid particle located at $(x,\xi(x)-\epsilon)$ satisfies $$v_y=\frac{D\xi}{Dt}=\partial_t\xi+U\partial_x\xi=i\sigma\xi.$$ Applying this to both side of interface ($\pm\epsilon$) leads to the jump condition $$\label{cont2} \frac{v_y^+}{\sigma^+}=\frac{v_y^-}{\sigma^-}.$$ Combining (\[cont1\]) and (\[cont2\]) and looking for non trivial solutions gives $$\omega^2+2\alpha\omega kU+(kU)^2=0,
\label{omega}$$ where $\alpha=(\rho^+-\rho^-)/(\rho^++\rho^-)$. An instability arises whenever $$\Delta=(1-\alpha^2)(-k^2U^2)<0$$ which is always true since $-1\leq\alpha\leq1$. The growth rate is $1/\tau_{\rm KHI}=\sqrt{-\Delta}=|kU|\sqrt{1-\alpha^2}$. Hence, a density contrast $|\alpha|$ close to 1 strongly dampens the growth rate of the KHI.
In colliding wind binaries, the density and velocity of both winds are related through the momentum flux ratio $\eta$. Using Eq. \[eq:eta\] and mass conservation for both winds then, far enough from the binary so that $r_1\simeq r_2$, the density ratio is roughly $$\label{eq:mass_cons}
\frac{\rho_2}{\rho_{1}}\simeq \eta \beta^{2}$$
$$\label{eq:growth_khi}
\frac{\tau_{\rm adv}}{\tau_{\rm KHI}}=\frac{|\beta-1|\sqrt{1-\alpha^2}}{-\alpha(\beta-1)+(\beta+1)}=\frac{\eta^{1/2}\beta |\beta-1|}{1+\eta \beta^3}$$
Nonlinear evolution
-------------------
### 2D Evolution
In order to investigate the evolution of the KHI in the nonlinear regime, we have performed numerical simulations for increasing $\alpha$. The 2D setup is as follows: box size $(lx=8, ly=4)$, resolution $(1024\times 256)$, code: PLUTO [@2007ApJS..170..228M], adiabatic equation of state $P\propto \rho^{5/3}$, background pressure $P=1$ in the initial state (using units dimensioned to the box length, density, and velocity shear). Reflective boundary conditions are enforced in $y$ to confine the instability in the simulation box. We always have $\rho^+>\rho^-$ [*i.e.*]{} the densest medium is found where $y>0$.
In addition to that, we follow a passive scalar $s$, initialised with $s=2\Theta(y)-1$ where $\Theta$ represents the Heaviside function. We performed simulations for $\{\alpha=0,0.5,0.9,0.99\}$. Kelvin-Helmholtz eddies are clearly present in the density snapshot shown in Fig. \[fig:snapshot\_d\] for model $\alpha=0.5$. In order to show the diffusion of the passive scalar as a function of time, we plot the evolution of $\overline{s}(y,t)=\int s\,dx$ as a function of $y$ and $t$ in Fig. \[d2D\]. These results demonstrate that when $\alpha\ne 0$, the scalar diffusion propagates much less in the denser medium ($y>0$) and that diffusion looks less efficient when $|\alpha|$ increases, in the sense that the region with intermediate values of the scalar $s$ becomes smaller when $\alpha$ increases.
![Snapshot of the density at t=21 (in dimensionless units) for $\alpha=0.5$.[]{data-label="fig:snapshot_d"}](d_KH_21){width="0.85\linewidth"}
![Diffusion of a passive scalar by the KHI. From left to right, top to bottom: $\alpha=0,0.5,0.9,0.99$.[]{data-label="d2D"}](mixing_0_0 "fig:"){width="0.45\linewidth"} ![Diffusion of a passive scalar by the KHI. From left to right, top to bottom: $\alpha=0,0.5,0.9,0.99$.[]{data-label="d2D"}](mixing_0_5 "fig:"){width="0.45\linewidth"}\
![Diffusion of a passive scalar by the KHI. From left to right, top to bottom: $\alpha=0,0.5,0.9,0.99$.[]{data-label="d2D"}](mixing_0_9 "fig:"){width="0.45\linewidth"} ![Diffusion of a passive scalar by the KHI. From left to right, top to bottom: $\alpha=0,0.5,0.9,0.99$.[]{data-label="d2D"}](mixing_0_99 "fig:"){width="0.45\linewidth"}
![ Diffusion of the passive scalar in the 3D simulations with $\alpha=0$ (left) and $\alpha=0.9$ (right).[]{data-label="d3D"}](mixing_0_0_3D "fig:"){width="0.45\linewidth"} ![ Diffusion of the passive scalar in the 3D simulations with $\alpha=0$ (left) and $\alpha=0.9$ (right).[]{data-label="d3D"}](mixing_0_9_3D "fig:"){width="0.45\linewidth"}\
3D evolution
------------
We have performed simulations for $\alpha=0$ and 0.9 in 3D to compare them to the 2D ones. They are very similar to the 2D configuration, except for the resolution which is reduced to $500\times 100 \times 100$ in order to reduce computational costs. We set $l_z=l_x=4.0$. $\overline{s}(y,t)$ is shown on Fig. \[d3D\]. The direct comparison with the 2D cases indicates that faster diffusion into the more tenuous region is still verified in 3D; diffusion is also slightly less efficient in 3D.
Parameters of the simulations
=============================
\[tab:param\_simu\]
[c c c c c c c ]{} $\{\eta,\beta\}$ &$v_1$ (km s$^{-1}$) &$v_2$ (km s$^{-1}$) &$\dot{M}_1$ ($10^{-7}M_{\odot}$ yr$^{-1}$) & $\dot{M}_2$ ($10^{-7}M_{\odot}$ yr$^{-1}$)& spiral?& $S/S_1$\
$\{1,1\}$ &2000 &2000 & 1 & 1 &S & 1\
$\{1,2\}$ &4000 &2000 & 0.5 & 1 &S & 0.67\
$\{1,4\}$ &2000 &500 & 0.25 & 1 &S & 0.35\
$\{1,8\}$ &4000 &500 & 0.25 & 2 &X &\
$\{1,20\}$ &40000&2000 & 0.05 & 1 &X &\
$\{1,200\}$ &8000 &40 & 0.05 & 10 &X &\
$\{0.5,0.01\}$ &40 &4000 & 100 & 0.5 &S & 2.5\
$\{0.5,0.05\}$ &200 &4000 & 40 & 1 &X &\
$\{0.5,0.1\}$ &400 &4000 & 20 & 1 &X &\
$\{0.5,0.5\}$ &1000 &2000 & 4 & 1 &S & 1.3\
$\{0.5,1\}$ &2000 &2000 & 2 & 1 &S & 1\
$\{0.5,2\}$ &4000 &2000 & 1 & 1 &S & 0.8\
$\{0.5,8\}$ &4000 &500 & 1 & 4 &X &\
$\{0.5,20\}$ &8000 &400 & .5 & 5 &X &\
$\{0.5,200\}$ &8000 &40 & .05 & 5 &X &\
$\{0.0625,0.05\}$&100 &2000 & 320 & 1 &S & 1.1\
$\{0.0625,0.1\}$ &200 &2000 & 160 & 1 &X &\
$\{0.0625,0.5\}$ &1000 &2000 & 32 & 1 &S & 1.04\
$\{0.0625,1\}$ &2000 &2000 & 16 & 1 &S & 1\
$\{0.0625,2\}$ &4000 &2000 & 8 & 1 &S & 0.9\
$\{0.0625,4\}$ &4000 &1000 & 4 & 1 &S/X &\
$\{0.0625,8\}$ &4000 &500 & 4 & 2 &S & 0.8\
$\{0.0625,20\}$ &40000&2000 & .8 & 1 &S &\
|
---
abstract: 'Analysis of seven optimization techniques grouped under three categories (hardware, back-end, and front-end) is done to study the reduction in average user response time for Modular Object Oriented Dynamic Learning Environment (Moodle), a Learning Management System which is scripted in PHP5, runs on Apache web server and utilizes MySQL database software. Before the implementation of these techniques, performance analysis of Moodle is performed for varying number of concurrent users. The results obtained for each optimization technique are then reported in a tabular format. The maximum reduction in end user response time was achieved for hardware optimization which requires Moodle server and database to be installed on solid state disk.'
author:
- Priyanka Manchanda
title: Analysis of Optimization Techniques to Improve User Response Time of Web Applications and Their Implementation for MOODLE
---
Manchanda, P. (2013). Analysis of Optimization Techniques to Improve User Response Time of Web Applications and Their Implementation for MOODLE.\
\
In Papasratorn, B.; Charoenkitkarn, N.; Vanijja, V.; Chongsuphajaisiddhi, V. (Eds.), Proceedings of the 6th International Conference, IAIT 2013, Bangkok, Thailand, December 12-13, 2013.\
\
To appear in volume 0409 of Springer CCIS series.\
The original final publication will be available at [www.springerlink.com](www.springerlink.com)
Introduction
============
The Internet has seen a significant growth of web based applications over the last few years. These have now become an inseparable part of numerous industries like airline, banking, business, computer, education, financial services, healthcare, publishing and telecommunications. They are preferred because of their zero installation time (as they run on a browser), availability of centralised data, their global reach, and their availability (24 hours a day, 7 days a week). According to [@blog], in June 2011 an average US user spent 74 minutes a day using web applications as compared to 64 minutes a day in June 2010.
In current scenario, improvement in the user response time is the most important issue for enhancing the performance of web applications. With reference to [@report], a delay of one second in the performance of web applications can impact customer satisfaction by up to 16%.
Web applications make use of a wide range of technologies including JavaScript, Apache, CSS, HTML, MySQL, PHP and protocols like HTTP headers. Optimizing the way they use these technologies can significantly improve user response time. Furthermore, the browser and hardware capabilities can be employed to reduce the user response time.
Many research groups and authors have addressed this problem and reported their solutions. These include teams such as Yahoo Exceptional Performance Team [@yahoo], book authors [@book_steve] and research papers [@paper].
In this contribution, seven optimization techniques grouped under three categories are analysed. Further, implementation of these seven techniques is done for the Modular Object Oriented Distance Learning Environment (Moodle) [@moodle]. The efficiency of these techniques is studied by comparing the original and the improved average user response time.
Performance Analysis of Moodle
==============================
Performance Analysis for Varying Number of Concurrent Users {#subsec:performance .unnumbered}
-----------------------------------------------------------
Moodle is a free source Learning Management System (LMS) which is used by thousands of educational institutions around the world to provide an organized interface for e-learning. As of June 2013, it has 83059 currently active sites that have been registered from 236 countries [@moodle_stats]. Moodle LMS is written in PHP and uses XHTML 1.0 Strict, CSS level 2 and JavaScript for its web user interface[@paper].
With reference to [@moodle_install], it has been reported that Moodle can support 50 concurrent users for every 1GB RAM. An experiment was performed to verify this result.\
**Experimental Setup**\
The experiment was perfomed on a machine with the following specifications:\
[*Hardware*]{}: IntelCorei5-2310 CPU @2.90GHz x 4 processor, 8GB Hard disk and 1GB RAM.\
[*Operating system*]{}: Ubuntu 12.10\
[*Web server*]{}: Apache v2.2.22 and PHP v5.4.6 for Moodle v2.5 for Ubuntu 12.10\
[*Database software*]{}: MySQL v5.5.31 for Ubuntu 12.10
- The experiment was performed using Apache JMeter 2.9, an open source load testing tool by the Apache Software Foundation [@jmeter].
- The test script was generated by using the JMeter Script Generator plugin for Moodle by James Brisland [@jmeter_plugin].
- The bandwidth of the network was set to 1024 kbps (1 Mbps) using JMeter.
- The load testing of Moodle was done for a chat activity.
- The sequence of pages visited on Moodle was :\
Login to site -$>$ View Course -$>$ View Chat page -$>$ View Chat window -$>$ Initialize Chat -$>$ Initialize Initial Update
- After initializing chat the following tasks were performed five times for each concurrent user : Post Chat Message -$>$ Initialize Update
- To test the performance of Moodle in the worst case scenario the ramp-up period, that is the amount of time for creating the total number of threads, was set to zero so as to ensure immediate creation of all the threads by JMeter.
\[table:per\]
***Number of Concurrent Users*** ***Average Response Time(s)*** ***Throughput (per min)***
---------------------------------- -------------------------------- ----------------------------
10 3.671 147.6
20 8.874 129
30 15.303 99.6
40 129.786 16.8
49 243.469 11.4
50 364.480 7.8
51 Database Overload Database Overload
: Average Response Time and Throughput for load testing Moodle\
on 1GB RAM and 8GB HARD DISK
![Average Respose Time(in s) for Varying Number of Concurrent Users[]{data-label="fig:fig_perf1"}](figure1.png){height="4cm"}
![Throughput(per minute) for Varying Number of Concurrent Users[]{data-label="fig:fig_perf2"}](figure2.png){height="4cm"}
While load testing Moodle for 51 concurrent users, it was observed that the connection to the database was aborted due to database overload and the testing process was killed by JMeter.
Hardware Optimization {#subsec:ssd}
=====================
Employing Solid State Disk {#employing-solid-state-disk .unnumbered}
--------------------------
The performance of the web applications can be highly enhanced by using a solid state disk drive to reduce the latency of the input and output operations carried out by the server.
A Solid State Disk, or SSD is a high performance plug and play data storage device which uses integrated circuit assemblies as memory to store data persistently [@wiki_ssd]. An SSD incorporates solid state flash memory and emulates a hard disk drive to store data [@paper_ssd]. However, unlike the traditional electromechanical disks like hard disk and floppy disks, an SSD is a flash-based and DRAM-based storage device which does not contain any moving parts [@article_ssd].
An experiment was performed by replacing the Hard Disk Drive(HDD) of the Moodle Server with a 128GB Kingston Solid State Disk Drive.\
**Experimental Setup**\
To conform to the experiment performed in section \[subsec:performance\] and to compare the performance of Moodle on HDD vs. SSD, the space allocated to Moodle server and database collectively was 8GB of 128GB SSD and the RAM size was limited to 1GB. The experiment was performed on a machine with following specifications:\
***Hardware:* IntelCorei5-2310 CPU @2.90GHz x 4 processor,\
8GB Solid State disk and 1GB RAM.**\
[*Operating system*]{}: Ubuntu 12.10\
[*Web server*]{}: Apache v2.2.22 and PHP v5.4.6 for Moodle v2.5 for Ubuntu 12.10\
[*Database software*]{}: MySQL v5.5.31 for Ubuntu 12.10\
[*Bandwidth*]{}: 1024 Kbps (1 Mbps)\
The experiment was performed for the chat activity mentioned in section \[subsec:performance\] using Apache JMeter 2.9.
![Average user response time (in s) for Moodle on HDD vs. SSD[]{data-label="fig:fig_ssd"}](figure5.png){height="3.6cm"}
[|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|]{} **No. of concurrent users** & **Average Response Time on HDD(s)** & **Average Response Time on SSD (s)** & **Reduction in Response time %**\
10 & 3.671 & 0.349 & 90.49\
20 & 8.874 & 1.048 & 88.19\
30 & 15.303 & 1.938 & 87.34\
40 & 129.786 & 3.438 & 97.35\
50 & 364.480 & 5.274 & 97.83\
60 & Database Overload & 5.97& -\
70 & Database Overload & 6.492 & -\
80 & Database Overload & 8.009 & -\
90 & Database Overload & 8.085 & -\
100 & Database Overload & 9.797 &-\
110 & Database Overload & 13.759 &-\
120 & Database Overload & 16.828 & -\
130 & Database Overload & 22.991 & -\
140 & Database Overload & 30.187& -\
150 & Database Overload & 36.119 & -\
151 & Database Overload & 39.141 & -\
152 & Database Overload & Database Overload & -\
From Table 2, it is concluded that the number of concurrent users supported by Moodle installed on SSD for 1 GB RAM is increased to **151** as compared to **50** concurrent users for Moodle installed on HDD with 1 GB RAM. Also, there is a reduction of **87% to 98%** in average user response time after installing Moodle server and database on SSD.
Back-End Optimization
=====================
Switching to LNMP Stack from LAMP Stack {#sec:back .unnumbered}
---------------------------------------
The Moodle web application runs on the LAMP stack which is a software bundle comprising of Linux based operating system, Apache HTTP server, MySQL database software and PHP object oriented scripting language. LNMP stack is almost similar to LAMP, except the change of web server from Apache to Nginx.
Apache is a process-based server, while nginx is an event-based asynchronous web server and is more scalable than Apache. In Apache, each simultaneous connection requires a thread which incurs significant overhead whereas nginx is event-driven and handles requests in a single (or at least, very few) threads [@wiki_apache_nginx].
The performance of Moodle or any web application that runs on Apache and frequently encounters heavy load, can be boosted by replacing Apache by Nginx. An experiment was performed to compare the performance of Moodle installed on Apache vs. Nginx web server.\
**Experimental Setup**\
Since it was observed in Section \[subsec:ssd\] that the performance of Moodle is highly enhanced by installing it on SSD, the experiment was performed on a machine with Moodle installed on 128 GB Solid State Disk and 4GB RAM.
All the other specifications (Operating system, Database software, Web server and Bandwidth) of the machine were kept same as in section \[subsec:ssd\]. The experiment was performed using Apache BenchMark 2.4 [@ab] for Moodle’s login page.
From Table 3, it is observed that there is a reduction of **24% to 34%** in average user response time after installing Moodle on Nginx v1.4.1 web server.
[|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|]{} **No. of concurrent users** & **Average Response Time on Apache(s)** & **Average Response Time on Nginx (s)** & **Reduction in Response time %**\
50 & 2.209 & 1.652 & 25.22\
100 & 4.505 & 3.359 & 25.43\
150 & 6.098 & 4.630 & 24.07\
200 & 8.192 & 5.408 & 33.98\
250 & 10.729 & 7.156 & 33.30\
![Average user response time (in s) for Moodle on Apache vs. Nginx Web Server[]{data-label="fig:fig_nginx"}](figure6.png){height="3cm"}
Front-End Optimization
======================
For any web application, only 10% to 20% of the end user response time is spent downloading the HTML document from the web server to the client’s browser. The other 80% to 90% is spent in performing the front end operations, i.e., in downloading the other components of web page [@book_steve].
A set of specific rules for speeding up the front – end operations carried by a web application is presented in Ref. [@book_steve]. Five of the most efficient techniques which showed significant reduction in user response time for Moodle Learning Management System are described in this section.
Browser Caching by Using Far Future Expires Header {#subsec:caching}
--------------------------------------------------
Browsers and proxies use cache to reduce the number and size of the HTTP requests thereby speeding up the web applications. A first-time visitor may have to make several HTTP requests, but by using a Far Future Expires header the developer can significantly improve the performance of web applications for returning visitors. A server uses the Expires header in HTTP response to inform the client that it can use the current copy of a component until the specified time [@book_steve].
Moodle sends requests with an Expires Header which is set in past **(20th Aug 1969 09:23 GMT)**. An experiment was performed by changing it to future date of **16th Apr 2015 20:00 GMT**. Also max-age directive was used in Cache control header so as to set the cache expiration window to 10 years in future and the pragma header was unset to enable caching.\
\
Given below are the lines which were added to the headers.conf file of Apache2 Web Server:
<FilesMatch ".(ico|pdf|flv|jpg|jpeg|png|gif|js|css|swf|php|html)$">
Header set Expires "Thu, 16 Apr 2015 20:00:00 GMT"
Header set Cache-Control " max-age=315360000"
Header unset Pragma
</FilesMatch>
The experimental setup is the same as section \[subsec:ssd\] and the experiment was performed using Apache Jmeter 2.9. From Table 4, it is observed that there is a reduction of **70% to 80%** in average user response time after implementing Far Future Expires Header Optimization Technique.
[|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|>p[3cm]{}|]{} **No. of concurrent users** & **Average Response Time Without Expires Header(s) (no caching)** & **Average Response Time with Expires Header(s) (caching)** & **Reduction in Response time %**\
10 & 0.625 & 0.144 & 76.96\
20 & 1.839 & 0.408 & 77.81\
40 & 5.061 & 1.210 & 76.09\
60 & 7.086 & 1.778 & 74.91\
80 & 8.124 & 2.426 & 70.14\
100 & 9.882 & 3.071 & 68.92\
Reduce DNS Lookups {#subsec:dns}
------------------
The Internet uses IP addresses to find webservers. Before establishing a network connection to a web server, the browser must resolve the hostname of the web server to an IP address by using Domain Name Systems (DNS). The latency introduced due to DNS lookups can be minimized if the DNS resolutions are cached by client’s browser [@book_steve]. The response time for Moodle’s login page of Institutional Moodle websites of 13 universities situated in six continents of the world was recorded for two cases: With DNS Cache and Without DNS Cache.\
The experiment was performed for a client located in IIT Bombay, India with 128GB SSD, 4GB RAM, IntelCorei5-2310 CPU @2.90GHz x 4 processor and 2 Mbps average download speed.
[|>p[1.5cm]{}|>p[1.5cm]{}|>p[4.3cm]{}|>p[1.5cm]{}|>p[1.5cm]{}|>p[1.6cm]{}|]{} **Continent** & **Country** & **University** & **Response time With DNS Cache(s)** & **Response time Without DNS Cache(s)** & **Reduction in Response time(%)**\
Asia & India & IIT, Bombay [@iitb] & 2.357 & 1.426 & 39.50\
Asia & India & IIT, Madras [@iitm] & 2.516 & 1.612 & 35.93\
Asia & Singapore & SIM University [@sim] & 1.381 & 1.055 & 23.61\
Asia & Japan & Sojo University, Kumamoto [@sojo] & 6.223 & 3.116 & 49.93\
Europe & Spain & Graduate School of Management, Barcelona [@spain] & 3.138 & 1.813 & 42.22\
Europe & UK & University of Nottingham [@uk] & 4.174 & 2.041 & 51.10\
North America & US & UCLA, California [@ucla] & 4.600 & 3.657 & 20.50\
South America & Argentina & Pontifical Catholic University of Argentina, Buenos Aires [@arg]& 2.534 & 1.710 & 32.52\
South America & Colombia & University of Grand Colombia, Bogotá, D.C. [@colombia] & 2.341 & 1.438 & 38.57\
Africa & Egypt & Oriflame University [@egypt] & 5.497 & 4.288 & 21.99\
Africa & South Africa & Virtual Academy of South Africa [@sa]& 4.936 & 2.588 & 47.57\
Australia & Australia & Australian National University [@aun]& 4.525 & 3.559 & 21.35\
Australia & Australia & Monash University [@monash] & 4.947 & 4.141 & 16.29\
From Table 5 it is concluded that there is a reduction of **16% to 51% depending on the geographical location** of Moodle server, if the resolved hostname for a web page is found in DNS cache.
Another experiment was carried out on the same client to compare the performance of Moodle by changing the number of DNS cache entries, DNS cache expiration period and HTTP keep alive timeout for Mozilla Firefox 21.0 browser. The following three scenarios were tested for 100 iterations of Moodle’s login page of Moodle websites of six universities situated in six continents of the world using iMacros 9.0 Firefox extension [@imacros] and HttpFox addon for Firefox [@httpfox].
Scenario 1 (S1):
DNS Cache Entries = 20
DNS Cache Expiration Period = 60 seconds
HTTP Keep Alive Timeout = 115 seconds
Scenario 2 (S2):
DNS Cache Entries = 512
DNS Cache Expiration Period = 3600 seconds
HTTP Keep Alive Timeout = 115 seconds
Scenario 3 (S3):
DNS Cache Entries = 512
DNS Cache Expiration Period = 3600 seconds
HTTP Keep Alive Timeout = 0 second
[|>p[1.7cm]{}|>p[2.5cm]{}|>p[1.5cm]{}|>p[1.5cm]{}|>p[2cm]{}|>p[1.5cm]{}|>p[2cm]{}|]{} **Continent** & **University** & **Response time for S1** & **Response time for S2** & **Difference (s) between S1 & S2** & **Response time for S3** & **Difference(s) between S2 & S3**\
North America & UCLA, USA [@ucla] & 173.984 & **169.284** & 4.7 & 178.69 & 9.406\
Asia & IIT, Madras, India [@iitm] & 108.93 & **105.677** & 3.253 & 110.548 & 4.871\
Australia & Australian National University [@aun] & 347.361 & **344.961** & 2.400 & 354.336 & 9.375\
Africa & Oriflame University, Egypt [@egypt] & 244.035 & **240.246** & 3.789 & 256.08 & 15.834\
Europe & University of Nottingham, UK [@uk] & 153.71 & **150.213** & 3.497 & 156.76 & 6.547\
South America & University of Grand Colombia, Colombia [@arg] & 142.241 & **135.908** & 6.333 & 146.763 & 10.855\
From Table 6, it is observed that the end user response time is minimum under Scenario 2. Hence, it can be concluded that the performance of a web application can be enhanced by reducing DNS Lookups, which was achieved by:
- Increasing the number of DNS cache entries,
- Increasing DNS expiration period, and
- Using a Network that supports HTTP keep-alive mechanism
Gzip Components {#subsec:gzip}
---------------
Gzip compression of web pages can significantly minimize the latency introduced due to transfer of the web page files from web server to client’s browser. Starting with HTTP/1.1, web clients indicate support for compression with the Accept-Encoding header in the HTTP request [@book_steve].
Accept-Encoding: gzip, deflate
After the web server sees this header, it compresses the response using one of the methods listed by the client. The web server uses the Content-Encoding header in the response to inform the client about the compressed response [@book_steve].
Content-Encoding: gzip
An experiment was performed on the client mentioned in section \[subsec:dns\] for Moodle installed on the machine with specifications as mentioned in section \[sec:back\] using Web Developer Extension for Mozilla Firefox 21.0 [@web_dev].
[|>p[2.8cm]{}|>p[2cm]{}|>p[3cm]{}|>p[2.5cm]{}|>p[2cm]{}|]{} **Moodle Page** & **No. of Files Requested** & **Response Size without Compression(KB)** & **Response size with Compression(KB)** & **Reduction in Response Size (%)**\
Index & 42 & 926 & 215 & 76.78\
Login & 13 & 597 & 138 & 76.88\
View Course & 42 & 804 & 187 & 76.74\
& 41 & 802 & 187 & 76.68\
& 35 & 889 & 218 & 75.48\
View Calendar & 42 & 861 & 207 & 75.96\
& 40 & 806 & 188 & 76.67\
1 page quiz with 5 questions & 49 & 847 & 198 & 76.62\
View Assignments & 43 & 804 & 187 & 76.74\
It is observed that Gzip compression reduces the response size by **75%77%**.
Deactivate ETags {#subsec:etag}
----------------
Entity tags (ETags) are used by web servers and browsers to determine whether the component in the browser’s cache An experiment was performed on the client mentioned in section \[subsec:dns\] for Moodle installed on the machine with specifications as mentioned in section \[sec:back\] using Firebug Extension 1.11.4 for Mozilla Firefox 21.0 [@firebug]. Moodle uses Etags for its style sheets, scripts and images. It was observed that deactivating Etags reduces the response There is a reduction of **an average of 1737 bytes** in response header size after deactivating the ETags.
[|>p[2.3cm]{}|>p[3cm]{}|>p[2.5cm]{}|>p[2.5cm]{}|>p[1.8cm]{}|]{} **Moodle Page** & **Total No. of CSS style sheets, JS Files and Images Requested** & **Response Header size with activated Etags(Bytes)** & **Response Header Size with deactivated Etags(Bytes)** & **Reduction in Response Size (Bytes)**\
Index & 40 & 20357 & 18454 & 1921\
Login & 11 & 5695 & 5172 & 523\
View Course & 42 & 21364 & 19345 & 2019\
1 page quiz with 5 questions & 49 & 19850 & 17978 & 1872\
View Assignments & 39 & 24978 & 22628 & 2350\
**Average** & & & & **1737**\
Optimize AJAX
-------------
AJAX (Asynchronous JavaScript and XML) is a collection of technologies, primarily JavaScript, CSS, DOM and asynchronous data retreival which is used to exchange data with a server, and update parts of a web page without reloading the whole page. Though AJAX allows the server to provide instantaneous feedback to the user, it does not guarantee that the user won’t have to wait for the asynchronous JavaScript and XML responses. The performance of the web application can be improved by optimizing the AJAX requests. The techniques mentioned in section \[subsec:caching\], \[subsec:dns\] and \[subsec:gzip\] are collectively used to optimize the ajax components of Moodle.
The AJAX components are made cacheable by modifying the expires header which is defined in **OutputRenderers.php** file located in **lib** directory of main Moodle directory. An experiment was performed on the client mentioned in section \[subsec:dns\] for Moodle installed on the machine with specifications as mentioned in section \[sec:back\] using Firebug Extension 1.11.4 for Mozilla Firefox 21.0 [@firebug].\
*Modifications made to expires header in OutputRenderers.php file*
Default:
@header('Expires: Sun, 28 Dec 1997 09:32:45 GMT'); Line 3345
Modified:
@header('Expires: Sun, 28 Dec 2020 09:32:45 GMT'); Line 3345
[|>p[2.3cm]{}|>p[3cm]{}|>p[2.5cm]{}|>p[2.5cm]{}|>p[1.8cm]{}|]{} **Activity** & **Response Time before optimizing AJAX(s)** & **Response Time after optimizing AJAX(s)** & **Reduction in Response Time(%)**\
Drag and Drop Sections & 309 & 227 & 26.54\
Drag and Drop Activities & 2.17 & 1.62 & 25.35\
Drag and Drop Files & 201 & 175 & 12.94\
Drag and Drop Blocks & 440 & 354 & 19.55\
AJAX Chat Box & 36 & 24 & 33.33\
**Average** & & & **23.54**\
There is a reduction of an average of 23.54% in user response time after optimizing the AJAX components.
Conclusion
==========
In this presented paper, seven methods to optimize web applications have been analysed and tested for Moodle LMS. These methods can be further used to optimize other essential web applications including webmail, online retail sales, online auctions, wikis and e-learning. It is observed that Moodle shows faster response time under heavy traffic, if it is loaded on a solid state disk. This technique can be used to scale high traffic web applications.
The caching mechanism can be used by the client’s browser to optimize the front-end operations that can reduce the end-user response time by up to 80%. This mechanism can be used for content that changes infrequently, that is, application’s static assets like graphics, style sheets and scripts. In addition to application’s static assets, DNS resolutions can be cached by client’s browser and can reduce the end-user response time by up to 50%.
High traffic web applications can employ more than one server to share the load. In such a scenario, Etags become invalid and deactivating them can further enhance the performance of the web application.
The seven web optimization techniques discussed in this paper were successfully tested for the Moodle LMS which showed a maximum reduction of 98% in average user response time by using the hardware optimization technique used in (Section \[subsec:ssd\]). These best practices can be further applied to a novel or existing web application to improve its performance by reducing end user response time and thereby increasing the number of concurrent users and throughput.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank the members of Department of Computer Science, Indian Institute of Technology, Bombay, India for their kind support and encouragement.
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<http://developer.yahoo.com/performance> Souders, S.(2007) High Performance Web Sites, O’Reilly Media
Horat D. and Arencibia A. Q. (2009) : Web Applications: A Proposal to Improve Response Time and Its Application to MOODLE. In: Moreno Díaz, R. ; Pichler, F. ; Arencibia, A.Q. (eds.) Computer Aided Systems Theory - EURO-CAST 2009. LNCS, vol 5717, pp. 218–225, Springer, Heidelberg (2009).
Project Moodle,<http://moodle.org> Moodle Statistiscs, <https://moodle.org/stats/> Joint Information Systems Committee, Regional Support Centre, West Midlands Moodle Wiki, <http://wiki.rscwmsystems.org.uk/index.php/Moodle> Apache JMeter, [ http://jmeter.apache.org/]( http://jmeter.apache.org/) moodle-jmeter-script-generator,\
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<http://en.wikipedia.org/wiki/Solid-state_drive> G.Wong : SSD Market Overview. In: Micheloni, R. ; Marelli, A. ; Eshghi, K. (eds.) Inside Solid State Drives (SSDs), Springer Series in Advanced Microelectronics, Volume 37, pp. 1-18, Springer Science+Business Media Dordrecht(2013).
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Apache vs nginx, <http://www.wikivs.com/wiki/Apache_vs_nginx> ab - Apache HTTP server benchmarking tool,\
<http://httpd.apache.org/docs/current/programs/ab.html> Indian Institute of Technology, Bombay : Moodle,\
<https://moodle.iitb.ac.in/login/index.php> Indian Institute of Technology, Madras : Moodle,\
<http://www.cse.iitm.ac.in/moodle/> Singapore Institute of Management University, Singapore : Moodle,\
<http://cp.unisim.edu.sg/moodle/> Sojo University, Nishi-ku, Kumamoto, Japan : Moodle,\
<http://md.ed.sojo-u.ac.jp/> Graduate School of Management, Barcelona, Spain : Moodle,\
<http://moodle.gsmbarcelona.eu/> University of Nottingham, Nottingham, UK : Moodle,\
<https://moodle.nottingham.ac.uk/login/index.php> University of California, Los Angeles : Physics and Astronomy Dept. Moodle,\
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<https://addons.mozilla.org/en-us/firefox/addon/httpfox/> Web Developer Extension for Mozilla Firefox 21.0,\
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|
---
abstract: 'We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale transform which is a duality between periodic difference equations of different orders. We describe periodic rational maps generalizing the classical Gauss map.'
address:
- 'Sophie Morier-Genoud, Institut de Mathématiques de Jussieu UMR 7586 Université Pierre et Marie Curie 4, place Jussieu, case 247 75252 Paris Cedex 05 '
- ' Valentin Ovsienko, CNRS, Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France'
- ' Richard Evan Schwartz, Department of Mathematics, Brown University, Providence, RI 02912, USA '
- ' Serge Tabachnikov, Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA, and ICERM, Brown University, Box1995, Providence, RI 02912, USA '
author:
- 'Sophie Morier-Genoud'
- Valentin Ovsienko
- Richard Evan Schwartz
- Serge Tabachnikov
title: ' Linear difference equations, frieze patterns and combinatorial Gale transform'
---
Introduction
============
Linear difference equations appear in many fields of mathematics, and relate fundamental objects of geometry, algebra and analysis. In this paper, we study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. (The solutions of the equation are periodic if the order of the equation is odd, and antiperiodic if the order is even.) The space of such equations is a very interesting algebraic variety. Despite the fact that this subject is very old and classical, this space has not been studied in detail.
We prove that the space ${\mathcal E\/}_{k+1,n}$ of $n$-periodic linear difference equations of order $k+1$ is equivalent to the space ${\mathcal F\/}_{k+1,n}$ of tame ${{\mathrm {SL}}}_{k+1}$-frieze patterns of width $w=n-k-2$. (Tameness is the non-vanishing condition on certain determinants; See Definition \[tamedef\].) We also show that these spaces are closely related to a certain moduli space ${\mathcal C\/}_{k+1,n}$ of $n$-gons in $k$-dimensional projective space. While ${\mathcal C\/}_{k+1,n}$ can be viewed as the quotient of the Grassmannian ${{\mathrm {Gr}}}_{k+1,n}$ by a torus action, see [@GM], the space ${\mathcal E\/}_{k+1,n}$ is a subvariety of ${{\mathrm {Gr}}}_{k+1,n}$. We show that, in the case where $k+1$ and $n$ (and thus $w+1$ and $n$) are coprime, the two spaces are isomorphic.
A frieze pattern is just another, more combinatorial, way to represent a linear difference equation with (anti)periodic solutions. However, the theory of frieze patterns was created [@Cox; @CoCo] and developed independently, see [@Pro; @ARS; @BR; @MGOT; @MG]. The current interest to this subject is due to the relation with the theory of cluster algebras [@CaCh]. We show that the isomorphism between ${\mathcal E\/}_{k+1,n}$ and ${\mathcal F\/}_{k+1,n}$ immediately implies the periodicity statement which is an important part of the theory. Let us mention that a more general notion of ${{\mathrm {SL}}}_{k+1}$-tiling was studied in [@BR]; a version of ${{\mathrm {SL}}}_{3}$-frieze patterns, called $2$-frieze patterns, was studied in [@Pro; @MGOT].
Let us also mention that an ${{\mathrm {SL}}}_{k+1}$-frieze pattern can be included in a larger pattern sometimes called a $T$-system (see [@DK] and references therein), and better known under the name of [*discrete Hirota equation*]{} (see, for a survey, [@Zab]). Although an ${{\mathrm {SL}}}_{k+1}$-frieze pattern is a small part of a solution of a $T$-system, it contains all the information about the solution. We will not use this viewpoint in the present paper.
The main result of this paper is a description of the duality between the spaces ${\mathcal E\/}_{k+1,n}$ and ${\mathcal E\/}_{w+1,n}$ when $(k+1)+(w+1)=n$. We call this duality the [*combinatorial Gale transform*]{}. This is an analog of the classical Gale transform which is a well-known duality on the moduli spaces of point configurations, see [@Gal; @Cob; @Cob1; @EP]. We think that the most interesting feature of the combinatorial Gale transform is that it allows one to change the order of an equation keeping all the information about it.
Let us give here the simplest example, which is related to Gauss’ [*pentagramma mirificum*]{} [@Gau]. Consider a third-order difference equation $$V_{i}=a_{i}V_{i-1}-b_iV_{i-2}+V_{i-3},
\qquad i\in{{\mathbb Z}}.$$ Assume that the coefficients $(a_i)$ and $(b_i)$ are $5$-periodic: $a_{i+5}=a_{i}$ and $b_{i+5}=b_{i}$, and that all the solutions $(V_i)$ are also $5$-periodic, i.e., $V_{i+5}=V_i$. The combinatorial Gale transform in this case consists in “forgetting the coefficients $b_i$”; to this equation it associates the difference equation of order $2$: $$W_{i}=a_{i}W_{i-1}-W_{i-2}.$$ It turns out that the solutions of the latter equation are $5$-antiperiodic: $W_{i+5}=-W_{i}$. Conversely, as we will explain later, one can reconstruct the initial third order equation from the second order one. Geometrically speaking, this transform sends (projective equivalence classes of) pentagons in ${{\mathbb {P}}}^2$ to those in ${{\mathbb {P}}}^1$. In terms of the frieze patterns, this corresponds to a duality between $5$-periodic Coxeter friezes and $5$-periodic ${{\mathrm {SL}}}_3$-friezes.
Our study is motivated by recent works [@OST; @Sol; @OST1; @GSTV; @Sch2; @MB; @MB1; @KS; @KS1] on a certain class of discrete integrable systems arising in projective differential geometry and cluster algebra. The best known among these maps is the pentagram map [@Sch; @Sch1] acting on the moduli space of $n$-gons in the projective plane.
This paper is organized as follows.
In Section \[OblectS\], we introduce the main objects, namely the spaces ${\mathcal E\/}_{k+1,n}$, ${\mathcal F\/}_{k+1,n}$, and ${\mathcal C\/}_{k+1,n}$.
In Section \[GeoS\], we construct an embedding of ${\mathcal E\/}_{k+1,n}$ into the Grassmannian ${{\mathrm {Gr}}}_{k+1,n}$. We then formulate the result about the isomorphism between ${\mathcal E\/}_{k+1,n}$ and ${\mathcal F\/}_{k+1,n}$ and give an explicit construction of this isomorphism. We also define a natural map from the space of equations to that of configurations of points in the projective space.
In Section \[TheGaleS\], we introduce the Gale transform which is the main notion of this paper. We show that a difference equation corresponds not to just one, but to two different frieze patterns. This is what we call the Gale duality. We also introduce a more elementary notion of projective duality that commutes with the Gale transform.
In Section \[DeTS\], we calculate explicitly the entries of the frieze pattern associated with a difference equation. We give explicit formulas for the Gale transform. These formulas are similar to the classical and well-known expressions often called the André determinants, see [@And; @Jor].
Relation of the Gale transform to representation theory is described in Section \[RePSS\]. We represent an ${{\mathrm {SL}}}_{k+1}$-frieze pattern (and thus a difference equation) in a form of a unitriangular matrix. We prove that the Gale transform coincides with the restriction of the involution of the nilpotent group of unitriangular matrices introduced and studied by Berenstein, Fomin, and Zelevinsky [@BFZ].
In Section \[PerSec\], we present an application of the isomorphism between difference equations and frieze patterns. We construct rational periodic maps generalizing the Gauss map. These maps are obtained by calculating consecutive coefficients of (anti)periodic second and third order difference equations and using the periodicity property of ${{\mathrm {SL}}}_{2}$- and ${{\mathrm {SL}}}_{3}$-frieze patterns. We also explain how these rational maps can be derived, in an alternate way, from the projective geometry of polygons.
In Section \[TriThmS\], we explain the details about the relations between the spaces we consider. We also outline relations to the Teichmüller theory.
Difference equations, ${{\mathrm {SL}}}_{k+1}$-frieze patterns and polygons in ${{\mathbb {RP}}}^k$ {#OblectS}
===================================================================================================
In this chapter we will define the three closely related spaces ${\mathcal E\/}_{k+1,n}$, ${\mathcal F\/}_{k+1,n}$, and ${\mathcal C\/}_{k+1,n}$ discussed in the introduction. All three spaces will be equipped with the structure of algebraic variety; we choose ${{\mathbb R}}$ as the ground field.
Difference equations
--------------------
Let ${\mathcal E}_{k+1,n}$ be the space of order $k+1$ difference equations $$\label{REq}
V_{i}=a_{i}^1V_{i-1}-a_{i}^{2}V_{i-2}+ \cdots+(-1)^{k-1}a_{i}^{k}V_{i-k}+(-1)^{k}V_{i-k-1},$$ where $a_{i}^{j}\in {{\mathbb R}}$, with $i\in{{\mathbb Z}}$ and $1\leq j\leq k$, are coefficients and $V_i$ are unknowns. (Note that the superscript $j$ is an index, not a power.) Throughout the paper, a solution $(V_i)$ will consist of either real numbers, $V_i\in {{\mathbb R}}$, or of real vectors $V_i\in{{\mathbb R}}^{k+1}$. We always assume that the coefficients are [*periodic*]{} with some period $n\geq{}k+2$: $$a_{i+n}^{j}=a_{i}^{j}$$ for all $i,j$, and that all the solutions are [*$n$-(anti)periodic*]{}: $$\label{APeriod}
V_{i+n}=(-1)^{k}\,V_{i}.$$
The algebraic variety structure on ${\mathcal E}_{k+1,n}$ will be introduced in Section \[AlgVarS\].
The simplest example of a difference equation (\[REq\]) is the well-known discrete [*Hill*]{} (or [*Sturm-Liouville*]{}) equation $$V_{i}=a_{i}V_{i-1}-V_{i-2}$$ with $n$-periodic coefficients and $n$-antiperiodic solutions.
${{\mathrm {SL}}}_{k+1}$-frieze patterns
----------------------------------------
An ${{\mathrm {SL}}}_{k+1}$-[*frieze pattern*]{} (see [@BR]) is an infinite array of numbers such that every $(k+1)\times (k+1)$-subarray with adjacent rows and columns forms an element of ${{\mathrm {SL}}}_{k+1}$.
More precisely, the entries of the frieze are organized in an infinite strip. The entries are denoted by $(d_{i,j})$, where $i\in{{\mathbb Z}}$, and $$i-k-1\leq{}j\leq{}i+w+k,$$ for a fixed $i$, and satisfy the following “boundary conditions” $$\left\{
\begin{array}{rccccl}
d_{i,i-1}&=&d_{i,i+w}&=&1& \hbox{for all}\; i,\\[4pt]
d_{i,j}&=&0&&&\hbox{for}\;j<i-1\;\hbox{or}\;j>i+w,
\end{array}
\right.$$ and the “${{\mathrm {SL}}}_{k+1}$-conditions” $$\label{DEq}
D_{i,j}:=
\left\vert
\begin{array}{llll}
d_{i,j}&d_{i,j+1}&\ldots&d_{i,j+k}\\[4pt]
d_{i+1,j}&d_{i+1,j+1}&\ldots&d_{i+1,j+k}\\[4pt]
\ldots& \ldots&& \ldots\\
d_{i+k,j}&d_{i+k,j+1}&\ldots&d_{i+k,j+k}
\end{array}
\right\vert=1,$$ for all $(i,j)$ in the index set.
An ${{\mathrm {SL}}}_{k+1}$-frieze pattern is represented as follows $$\label{FREq}
\begin{array}{ccccccccccccc}
&&&& \vdots&&&& \vdots&&&\\
&0&&0&&0&&0&&0&&\ldots\\[6pt]
\ldots&&1&&1&&1&&1&&1&\\[6pt]
&\ldots&&\;d_{0,w-1}&&\;d_{1,w}&&\;d_{2,w+1}&&\ldots&&\ldots\\
&&&\! \iddots&& \iddots&& \iddots&&&&\\
\ldots&& d_{0,1}&&d_{1,2}&&d_{2,3}&&d_{3,4}&&d_{4,5}&\\[6pt]
& d_{0,0}&&d_{1,1}&&d_{2,2}&&d_{3,3}&&d_{4,4}&&\ldots\\[6pt]
\ldots&&1&&1&&1&&1&&1&\\[6pt]
&0&&0&&0&&0&&0&&\ldots\\
&&&& \vdots&&&& \vdots&&&\\
\end{array}$$ where the strip is bounded by $k$ rows of 0’s at the top, and at the bottom. To simplify the pictures we often omit the bounding rows of 0’s.
The parameter $w$ is called the [*width*]{} of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern. In other words, the width is the number of non-trivial rows between the rows of $1$’s.
\[tamedef\] An ${{\mathrm {SL}}}_{k+1}$-frieze pattern is called [*tame*]{} if every $(k+2)\times (k+2)$-determinant equals $0$.
The notion of tame friezes was introduced in [@BR]. Let ${\mathcal F}_{k+1,n}$ denote the space of tame ${{\mathrm {SL}}}_{k+1}$-frieze patterns of width $w=n-k-2$.
\(a) The most classical example of friezes is that of [*Coxeter-Conway frieze patterns*]{} [@Cox; @CoCo] corresponding to $k=1$. For instance, a generic Coxeter-Conway frieze pattern of width $2$ looks like this: $$\begin{array}{ccccccccccc}
\cdots&&1&& 1&&1&&\cdots
\\[4pt]
&x_1&&\frac{x_2+1}{x_1}&&\frac{x_1+1}{x_2}&&x_2&&
\\[4pt]
\cdots&&x_2&&\frac{x_1+x_2+1}{x_1x_2}&&x_1&&\cdots
\\[4pt]
&1&&1&&1&&1&&
\end{array}$$ for some $x_1,x_2$. (Note that we omitted the first and the last rows of $0$’s.) This example is related to so-called Gauss [*pentagramma mirificum*]{} [@Gau], see also [@Sche].
\(b) The following width $3$ Coxeter pattern is not tame: $$\begin{array}{cccccccccccc}
\cdots&&1&& 1&&1&&1&&\cdots
\\[4pt]
&1&&0&&2&&0&&3&
\\[4pt]
\cdots&&-1&&-1&&-1&&-1&&\cdots
\\[4pt]
&0&&1&&0&&2&&0&
\\[4pt]
\cdots&&1&&1&&1&&1&&\cdots
\end{array}$$
Generic ${{\mathrm {SL}}}_{k+1}$-frieze patterns are tame. We understand the genericity of an ${{\mathrm {SL}}}_{k+1}$-frieze pattern as the condition that every $k\times k$-determinant is different from $0$. Then the vanishing of the $(k+2)\times (k+2)$-determinants follows from the Dodgson condensation formula, involving minors of order $k+2$, $k+1$ an $k$ obtained by erasing the first and/or last row/column. The formula can be pictured as follows $$\begin{vmatrix}
*&*&*&*\\
*&*&*&*\\
*&*&*&*\\
*&*&*&*\\
\end{vmatrix}
\begin{vmatrix}
&&&\\
\;&*&*&\,\\
\;&*&*&\,\\
\;&&&\,\\
\end{vmatrix}
=
\begin{vmatrix}
\;*&*&*&\;\; \\
\;*&*&*&\;\;\\
\;*&*&*&\;\;\\
&&&\;\;\\
\end{vmatrix}
\begin{vmatrix}
\;&\;&&\\
\;&\;*&*&*\;\;\\
\;&\;*&*&*\;\;\\
\;&\;*&*&*\;\;\\
\end{vmatrix}
-
\begin{vmatrix}
&\;*&*&*\;\;\\
&\;*&*&*\;\;\\
&\;*&*&*\;\;\\
&\;&&\\
\end{vmatrix}
\begin{vmatrix}
&&&\;\\
\;*&*&*&\;\;\\
\;*&*&*&\;\;\\
\;*&*&*&\;\;\\
\end{vmatrix}.$$ where the deleted columns/rows are left blank.
\[NotNot\] Given an ${{\mathrm {SL}}}_{k+1}$-frieze pattern $F$ as in , it will be useful to define the following $(k+1)\times{}n$-matrices[^1] $$\label{TheEmbM}
M^{(i)}_F:=\left(
\begin{array}{cccccccccccccc}
1 & d_{i,i}& \ldots &\ldots&d_{i,w+i-1}&1 \\[4pt]
& \ddots & & &&&\ddots\\[12pt]
& & 1 & d_{k+i,k+i} &\ldots& \ldots&d_{k+i,k+w+i-1} &1\\[4pt]
\end{array}
\right).$$ For a subset $I$ of $k+1$ consecutive elements of ${{\mathbb Z}}/n{{\mathbb Z}}$, denote by $\Delta_I(M)$ the minor of a matrix $M$ based on columns with indices in $I$. There are $n$ such intervals $I$. By definition of ${{\mathrm {SL}}}_{k+1}$-frieze pattern, $\Delta_I(M^{(i)}_F)=1$.
The matrix $M^{(i)}_F$ determines a unique tame ${{\mathrm {SL}}}_{k+1}$-frieze pattern. Indeed, from $M^{(i)}_F$ one can compute all the entries $d_{i,j}$, one after another, using the fact that $(k+2)\times (k+2)$-minors of the frieze pattern vanish.
We will also denote the North-East (resp. South-East) diagonals of an ${{\mathrm {SL}}}_{k+1}$-frieze pattern by $\mu_i$ (resp. $\eta_j$):
\#1\#2\#3\#4\#5[ @font ]{}
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Moduli space of polygons
------------------------
A [*non-degenerate*]{} $n$-gon is a map $$v:{{\mathbb Z}}\to{{\mathbb {RP}}}^{k}$$ such that $v_{i+n}=v_i$, for all $i$, and no $k+1$ consecutive vertices belong to the same hyperplane.
Let ${\mathcal C}_{k+1,n}$ be the moduli space of projective equivalence classes of non-degenerate $n$-gons in ${{\mathbb {RP}}}^{k}$.
The space ${\mathcal C}_{k+1,n}$ has been extensively studied; see, e.g., [@Gal; @GM; @Kap; @EP]. Our interest in this space is motivated by the study of the pentagram map, a completely integrable map on the space ${\mathcal C}_{3,n}$; see [@Sch; @Sch1; @OST; @Sol; @OST1].
Geometric description of the spaces ${\mathcal E}_{k+1,n}$, ${\mathcal F}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$ {#GeoS}
=============================================================================================================
In this section, we describe the structures of algebraic varieties on the spaces ${\mathcal E}_{k+1,n}$, ${\mathcal F}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$. We also prove that the spaces ${\mathcal E}_{k+1,n}$ and ${\mathcal F}_{k+1,n}$ are isomorphic. The spaces ${\mathcal E}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$ are also closely related. We will define a map from ${\mathcal E}_{k+1,n}$ to ${\mathcal C}_{k+1,n},$ which turns out to be an isomorphism, provided $k+1$ and $n$ are coprime. If this is not the case, then these two spaces are different.
The structure of an algebraic variety on ${\mathcal E}_{k+1,n}$ {#AlgVarS}
---------------------------------------------------------------
The space of difference equations ${\mathcal E}_{k+1,n}$ is an affine algebraic subvariety of ${{\mathbb R}}^{nk}$. This structure is defined by the condition (\[APeriod\]).
The space of solutions of the equation is $(k+1)$-dimensional. Consider $k+1$ linearly independent solutions forming a sequence $(V_i)_{i\in {{\mathbb Z}}}$ of vectors in ${{\mathbb R}}^{k+1}$ satisfying . Since the coefficients are $n$-periodic, there exists a linear map $T$ on the space of solutions called the [*monodromy*]{} satisfying: $$V_{i+n}=TV_i,$$ One can view the monodromy as an element of ${{\mathrm {GL}}}_{k+1}$ defined up to a conjugation.
\[monoid\] The space ${\mathcal E}_{k+1,n}$ has codimension $k(k+2)$ in ${{\mathbb R}}^{nk}$ with coordinates $a_{i}^{j}$.
Since the last coefficient in (\[REq\]) is $(-1)^{k}$, one has[^2] $$\left|V_i,V_{i+1},\ldots,V_{i+k}\right| =
\left|V_{i+1},V_{i+2},\ldots,V_{i+k+1}\right|.$$ Hence the monodromy is volume-preserving and thus belongs to the group ${{\mathrm {SL}}}_{k+1}({{\mathbb R}})$.
If furthermore all the solutions of (\[REq\]) are $n$-(anti)periodic, then the monodromy is $(-1)^{k} \mathrm{Id}$. Since $\dim {{\mathrm {SL}}}_{k+1}=k(k+2)$, this gives $k(k+2)$ polynomial relations on the coefficients.
Embedding of ${\mathcal F}_{k+1,n}$ into the Grassmannian {#SubSGr}
---------------------------------------------------------
Consider the Grassmannian ${{\mathrm {Gr}}}_{k+1,n}$ of $(k+1)$-dimensional subspaces in ${{\mathbb R}}^n$. Let us show that the space of ${{\mathrm {SL}}}_{k+1}$-frieze patterns ${\mathcal F}_{k+1,n}$ can be viewed as an $(n-1)$-codimensional algebraic subvariety of ${{\mathrm {Gr}}}_{k+1,n}$.
Recall that the the Grassmannian can be described as the quotient $$\label{Grass}
{{\mathrm {Gr}}}_{k+1,n}\simeq
{{\mathrm {GL}}}_{k+1}\backslash {{\mathrm {Mat}}}^*_{k+1,n}({{\mathbb R}}),$$ where ${{\mathrm {Mat}}}^*_{k+1,n}({{\mathbb R}})$ is the set of real $(k+1)\times{}n$-matrices of rank $k+1$. Every point of ${{\mathrm {Gr}}}_{k+1,n}$ can be represented by a $(k+1)\times{}n$-matrix $M$. The Plücker coordinates on ${{\mathrm {Gr}}}_{k+1,n}$ are all the $(k+1)\times{}(k+1)$-minors of $M$, see, e.g., [@Ful].
Note that the $(k+1)\times{}(k+1)$-minors depend on the choice of the matrix but they are defined up to a common factor. Therefore, the Plücker coordinates are homogeneous coordinates independent of the choice of the representing matrix. The minors $\Delta_I(M)$ with consecutive columns, see Notation \[NotNot\], play a special role.
\[EmbProp\] The subvariety of $Gr_{k+1,n}$ consisting in the elements that can be represented by the matrices such that all the minors $\Delta_I(M)$ are equal to each other: $$\label{EEqu}
\Delta_I(M)=\Delta_{I'}(M)
\qquad\hbox{for all}\quad
I,I'$$ is in one-to-one correspondence with the space ${{\mathcal F}}_{k+1,n}$ of ${{\mathrm {SL}}}_{k+1}$-frieze patterns.
If $M\in{{\mathrm {Gr}}}_{k+1,n}$ satisfies $\Delta_I(M)=\Delta_{I'}(M)$ for all intervals $I,I'$, then it is easy to see that there is a unique representative of $M$ of the form (\[TheEmbM\]), and therefore a unique corresponding ${{\mathrm {SL}}}_{k+1}$-frieze pattern.
Conversely, given an ${{\mathrm {SL}}}_{k+1}$-frieze pattern $F$, the corresponding matrices $M_F^{(i)}$ satisfy . Fixing $i$, for instance, taking $i=1$, we obtain a well-defined embedding $${\mathcal F}_{k+1,n}\subset{{\mathrm {Gr}}}_{k+1,n}.$$ Hence the result.
Note that the constructed embedding of the space ${\mathcal F}_{k+1,n}$ into the Grassmannian depends on the choice of index $i$. A different choice leads to a different embedding.
The space ${\mathcal C}_{k+1,n}$ as a quotient of the Grassmannian {#QuotGr}
------------------------------------------------------------------
A classical way to describe the space ${\mathcal C}_{k+1,n}$ as the quotient of ${{\mathrm {Gr}}}_{k+1,n}$ by the torus action: $$\label{TheQuot}
{\mathcal C}_{k+1,n}\simeq{{\mathrm {Gr}}}_{k+1,n}\slash{{\mathbb T}}^{n-1},$$ is due to Gelfand and MacPherson [@GM].
Let us comment on this realization of ${\mathcal C}_{k+1,n}$. Given an $n$-gon $v:{{\mathbb Z}}\to{{\mathbb {RP}}}^k$, consider an arbitrary lift $V:{{\mathbb Z}}\to{{\mathbb R}}^{k+1}$. The result of such a lift is a full rank $(k+1)\times{}n$-matrix, and thus an element of ${{\mathrm {Gr}}}_{k+1,n}$. Recall that the action of ${{\mathbb T}}^{n-1}$, in terms of the matrix realization (\[Grass\]), consists in multiplying $(k+1)\times{}n$-matrices by diagonal $n\times{}n$-matrices with determinant $1$. The projection of $V$ to the quotient ${{\mathrm {Gr}}}_{k+1,n}\slash{{\mathbb T}}^{n-1}$ is independent of the lift of $v$ to ${{\mathbb R}}^{k+1}$.
Triality {#1stIso}
--------
Let us briefly explain the relations between the spaces ${\mathcal E}_{k+1,n}$, ${\mathcal F}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$. We will give more details in Section \[TriThmS\].
The spaces of difference equations ${\mathcal E}_{k+1,n}$ and that of ${{\mathrm {SL}}}_{k+1}$-frieze patterns ${\mathcal F}_{k+1,n}$ are always isomorphic. These spaces are, in general, different from the moduli space of $n$-gons, but isomorphic to it if $k+1$ and $n$ are coprime.
\[TriThm\] (i) The spaces ${\mathcal E}_{k+1,n}$ and ${\mathcal F}_{k+1,n}$ are isomorphic algebraic varieties.
\(ii) If $k+1$ and $n$ are coprime, then the spaces ${\mathcal E}_{k+1,n}$, ${\mathcal F}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$ are isomorphic algebraic varieties.
The complete proof of this theorem will be given in Section \[TriThmS\]. Here we just construct the isomorphisms.
[*Part (i)*]{}. Let us define a map $${\mathcal E}_{k+1,n}
\buildrel{\simeq}\over\longrightarrow
{\mathcal F}_{k+1,n}$$ which identifies the two spaces. Roughly speaking, we generate the solutions of the recurrence equation (\[REq\]), starting with $k$ zeros, followed by one, and put them on the North-East diagonals of the frieze pattern. Let us give a more detailed construction.
Given a difference equation satisfying the (anti)periodicity assumption , we define the corresponding ${{\mathrm {SL}}}_{k+1}$-frieze pattern (\[FREq\]) by constructing its North-East diagonals $\mu_i$. This diagonal is given by a sequence of real numbers $V=(V_s)_{s\in {{\mathbb Z}}}$ that are the solution of the equation with the initial condition $$(V_{i-k-1},V_{i-k},\ldots,V_{i-1})=(0,0,\ldots,0,1);$$ this defines the numbers $d_{i,j}$ via $$\label{ConstEq}
d_{i,j}:=V_j.$$ Since the solution is $n$-(anti)periodic, we have $$d_{i,i+w+1}=\ldots=d_{i,i+n-2}=0,
\qquad
d_{i,i+n-1}=(-1)^k.$$ Furthermore, from the equation (\[REq\]) one has $$d_{i,i+w}=1.$$ The sequence of (infinite) vectors $(\eta_i)$, i.e., of South-East diagonals, satisfies Equation (\[REq\]).
\[HillEx\] [For an arbitrary difference equation, the first coefficients of the $i$-th diagonal are $$\begin{array}{rcl}
d_{i,i}&=&a_{i}^1, \\[4pt]
d_{i,i+1}&=&a_{i}^1a_{i+1}^1-a_{i+1}^2, \\[4pt]
d_{i,i+2}&=&a_{i}^1a_{i+1}^1a_{i+2}^1-a_{i+1}^2a_{i+2}^1- a_{i}^1a_{i+2}^2+a_{i+2}^3.
\end{array}$$ We will give more general determinant formulas in Section \[DeTS\]. ]{}
The idea that the diagonals of a frieze pattern satisfy a difference equation goes back to Conway and Coxeter [@CoCo]. The isomorphism between the spaces of difference equations and frieze patterns was used in [@OST; @MGOT] in the cases $k=1,2$.
[*Parts (ii)*]{}. The construction of the map ${\mathcal E}_{k+1,n}\to{\mathcal C}_{k+1,n}$ consists in the following two steps:
1. the spaces ${\mathcal E}_{k+1,n}$ and ${\mathcal F}_{k+1,n}$ are isomorphic;
2. there is an embedding ${\mathcal F}_{k+1,n}\subset{{\mathrm {Gr}}}_{k+1,n}$, and a projection ${{\mathrm {Gr}}}_{k+1,n}\to{\mathcal C}_{k+1,n}$, see (\[TheEmbM\]) and (\[TheQuot\]).
More directly, given a difference equation (\[REq\]) with (anti)periodic solutions, the space of solutions being $k+1$-dimensional, we choose any linearly independent solutions $(V_i^{(1)}),\ldots,(V_i^{(k+1)})$. For every $i$, we obtain a point, $V_i\in {{\mathbb R}}^{k+1}$, which we project to ${{\mathbb {RP}}}^k$; the (anti)periodicity assumption implies that we obtain an $n$-gon. Furthermore, the constructed $n$-gon is non-degenerate since the $(k+1)\times (k+1)$-determinant $$\label{detconst}
\left|
V_i, V_{i+1},\ldots,V_{i+k}
\right|=\mathrm{Const}\not=0.$$ A different choice of solutions leads to a projectively equivalent $n$-gon. We have constructed a map $$\label{MaPP}
{\mathcal E}_{k+1,n} \longrightarrow {\mathcal C}_{k+1,n}.$$
We will see in Section \[CoPSeC\], that this map is an isomorphism if and only if $k+1$ and $n$ are coprime.
\[NewProp\] Suppose that $\gcd(n,k+1)=q \neq 1$, then the image of the constructed map has codimension $q-1$.
We will give the proof in Section \[ProProSec\].
It will be convenient to write the ${{\mathrm {SL}}}_{k+1}$-frieze pattern associated with a difference equation (\[REq\]) in the form: $$\label{friezeGG}
\begin{array}{ccccccccccccccccccccccccccc}
&\ldots&1&&1&&1&&1&&1&&1\\[4pt]
&&&\ldots&&\alpha^{1}_{n}&&\alpha^{1}_1&&\alpha^{1}_2&& \ldots&&\alpha^{1}_n&\\[4pt]
&&&&\alpha^{2}_n&&\alpha^{2}_1&& \alpha^{2}_2&&&&\alpha^{2}_n&\\
&\ldots&& \iddots && \iddots&& \iddots&&&& \iddots&&\ldots\\
&&\alpha^{w}_{n}&& \alpha^{w}_{1}&&\alpha^{w}_{2}&& \ldots&& \alpha^{w}_{n}&&\ldots\\[4pt]
&1&&1&&1&&1&&1&&1&&\ldots\\
\end{array}$$ The relation between the old and new notation for the entries of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern of width $w$ is: $$\label{Corresp}
d_{i,j}=\alpha_{i-1}^{w-j+i},\qquad
\alpha_{i}^{j}=d_{i+1,w+i-j+1},$$ where $d_{i,j}$ is the general notation for the entries of an ${{\mathrm {SL}}}_{k+1}$-frieze pattern. See formula (\[FREq\]).
The combinatorial Gale transform {#TheGaleS}
================================
In this section, we present another isomorphism between the introduced spaces. This is a combinatorial analog of the classical Gale transform, and it results from the natural isomorphism of Grassmannians: $${{\mathrm {Gr}}}_{k+1,n}\simeq{{\mathrm {Gr}}}_{w+1,n},$$ for $n=k+w+2$.
On the spaces ${{\mathcal E}}_{k+1,n}$ and ${{\mathcal F}}_{k+1,n}$ we also define an involution, which is a combinatorial version of the projective duality. The Gale transform commutes with the projective duality so that both maps define an action of the Klein group $\left({{\mathbb Z}}/2{{\mathbb Z}}\right)^2$.
Statement of the result {#StatS}
-----------------------
We say that a difference equation (\[REq\]) with $n$-(anti)periodic solutions is [*Gale dual*]{} of the following difference equation of order $w+1$: $$\label{TheDualEq}
W_i=\alpha_{i}^1W_{i-1}-\alpha_{i}^2W_{i-2}+
\cdots+(-1)^{w-1}\alpha_{i}^wW_{i-w}+(-1)^{w}W_{i-w-1},$$ where $\alpha_r^s$ are the entries of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern (\[friezeGG\]) corresponding to (\[REq\]).
The following statement is the main result of the paper.
\[TheGailThm\] (i) All solutions of the equation (\[TheDualEq\]) are $n$-(anti)periodic, i.e., $$W_{i+n}=(-1)^wW_i.$$
\(ii) The defined map ${\mathcal E}_{k+1,n}\to{\mathcal E}_{w+1,n}$ is an involution.
We obtain an isomorphism $$\label{TheGale}
{{\mathcal G}}:{\mathcal E}_{k+1,n}
\buildrel{\simeq}\over\longrightarrow{\mathcal E}_{w+1,n}$$ between the spaces of $n$-(anti)periodic difference equations of orders $k+1$ and $w+1$, provided $$n=k+w+2.$$ We call this isomorphism the *combinatorial Gale duality* (or the *combinatorial Gale transform*).
An equivalent way to formulate the above theorem is to say that there is a duality between ${{\mathrm {SL}}}_{k+1}$-frieze patterns of width $w$ and ${{\mathrm {SL}}}_{w+1}$-frieze patterns of width $k$: $${{\mathcal G}}:{\mathcal F}_{k+1,n}
\buildrel{\simeq}\over\longrightarrow{\mathcal F}_{w+1,n}.$$
\[GDFprop\] The ${{\mathrm {SL}}}_{w+1}$-frieze pattern associated to the equation (\[TheDualEq\]) is the following $$\label{friezeG}
\begin{array}{ccccccccccccccccccccccccccc}
&\ldots&1&&1&&1&&1&&1&&1\\[4pt]
&&&\ldots&& a^1_n&&a^1_1&&a^1_2&& \ldots&& a^1_n&\\[4pt]
&&&& a^2_n&&a^2_1&& a^2_2&&&&a^2_n&\\
&\ldots&& \iddots && \iddots&& \iddots&&&& \iddots&&\ldots\\
&&a^k_n&& a^k_1&&a^k_2&& \ldots&& a^k_n&&\ldots\\[4pt]
&1&&1&&1&&1&&1&&1&&\ldots\\
\end{array}$$
We say that the ${{\mathrm {SL}}}_{w+1}$-frieze pattern (\[friezeG\]) is [*Gale dual*]{} to the ${{\mathrm {SL}}}_{k+1}$-frieze pattern (\[friezeGG\]).
The combinatorial Gale transform is illustrated by Figure \[Illustre\].
Proof of Theorem \[TheGailThm\] and Propositon \[GDFprop\] {#PruS}
----------------------------------------------------------
$\,$
Let us consider the following $n$-(anti)periodic sequence $(W_i)$. On the interval $(W_{-w},\ldots,W_{n-w-1})$ of length $n$ we set: $$\begin{array}{lccccccccccccccccccccccc}
(W_{-w},&\ldots,&W_{-1},&W_{0},&W_{1},&W_{2},&\ldots,
&W_{n-w-2},&W_{n-w-1}):=\\[6pt]
(0,&\ldots,&0,&1,&a_n^{k},&a_n^{k-1},&\ldots,&a_n^{1},&1)\\ \end{array}$$ and then continue by (anti)periodicity: $W_{i+n}=(-1)^wW_i$.
\[SoLem\] The constructed sequence $(W_i)$ satisfies the equation (\[TheDualEq\]).
[*Proof of the lemma*]{}. By construction of the frieze pattern (\[friezeGG\]), its South-East diagonals $\eta_i$ satisfy .
Consider the following selection of $k+2$ diagonals in the frieze pattern (\[friezeGG\]), and form the following $(k+2)\times{}n$-matrix: $$\begin{blockarray}{cccccccc}
\eta_{n-k-1} & \eta_{n-k} & & & \eta_{n} \\[10pt]
\begin{block}{(ccccccc)c}
\alpha^{1}_1 & 1 &0 & \cdots & 0 & \\[6pt]
\alpha^{2}_2 & \alpha^{1}_2 & 1 & 0 & \vdots& \\[6pt]
\vdots & & \ddots & \ddots & 0 &\\[6pt]
\vdots & & & \alpha^{1}_{k+1} & 1& \\[10pt]
\alpha^{w}_w & & & & \alpha^{1}_{k+2}&\\[6pt]
1 & \alpha^{w}_{w+1}& & & \vdots& \\[6pt]
0& \ddots & & & \vdots &\\[6pt]
\vdots&& 1& \alpha^{w}_{n-2}& \alpha^{w-1}_{n-2} &\\[6pt]
0& & 0 & 1 & \alpha^{w}_{n-1} &\\[6pt]
(-1)^k & 0& \cdots& 0 & 1 & \\[6pt]
\end{block}
\end{blockarray}$$ The above diagonals satisfy the equation: $$\eta_n=a_n^1\eta_{n-1}-\ldots-(-1)^ka_n^k\eta_{n-k}+(-1)^k\eta_{n-k-1}.$$ Let us express this equation for each component (i.e., each row of the above matrix).
The first row gives $a_n^k=\alpha^{1}_1$, which can be rewritten as $W_1=\alpha^{1}_1W_0$. Since, by construction, $W_{-1}=\cdots=W_{-w}=0$, we can rewrite this relation as $$W_1= \alpha^{1}_1W_0-\alpha^{2}_1W_{-1}+\cdots+(-1)^wW_{-w},$$ which is precisely (\[TheDualEq\]) for $i=1$. Then considering the second component of the vectors $\eta$, we obtain the relation $
0=a_n^{k-1}-a_n^k \alpha^{1}_2+ \alpha^{2}_2,
$ which reads as $$\begin{array}{cclllllllllllllllllllllllllllll}
W_2&=& \alpha^{1}_2W_1&-& \alpha^{2}_2W_0&&\Longleftrightarrow\\[6pt]
W_2&=& \alpha^{1}_2W_1&- &\alpha^{2}_2W_0&+
&\alpha^{3}_2W_{-1}&+\cdots+&(-1)^wW_{1-w}.
\end{array}$$ Continuing this process, we obtain $n$ relations which correspond precisely to the equation (\[TheDualEq\]), for $i=1,\ldots,n$. Hence the lemma. [$\Box$]{}
Shifting the indices, in a similar way we obtain $w$ other solutions of the equation (\[TheDualEq\]) that, on the period $(W_{-w},\ldots,W_{n-w-1})$, are given by: $$\begin{array}{cccccccccccccccccccccccc}
(0,&\ldots,&0,&1,&a_{n-1}^{k},&a_{n-1}^{k-1},&\ldots,&a_{n-1}^{1},&1,&0)\\[6pt]
(0,&\ldots,&1,&a_{n-2}^{k},&a_{n-2}^{k-1},&\ldots,&a_{n-2}^{1},&1,&0,&0)\\
\vdots\\
(1,&a_{n-w}^{k},&a_{n-w}^{k-1},&\ldots,&a_{n-w}^{1},&1,&0,&\dots,&0,&0)
\end{array}$$ Together with the solution from Lemma \[SoLem\], these solutions are linearly independent and therefore form a basis of $n$-(anti)periodic solutions of equation (\[TheDualEq\]). We proved that this equation indeed belongs to ${{\mathcal E}}_{w+1,n}$, so that the map ${{\mathcal G}}$ is well-defined.
The relation between the equations (\[REq\]) and (\[TheDualEq\]) can be described as follows: the solutions of the former one are the coefficients of the latter, and vice-versa. Therefore, the map ${{\mathcal G}}$ is an involution. This finishes the proof of Theorem \[TheGailThm\]. [$\Box$]{}
To prove Propositon \[GDFprop\], it suffice to notice that the diagonals of the pattern are exactly the solutions of the equation (\[TheDualEq\]) with initial conditions $(0,\ldots,0,1,a_i^{k})$. This is exactly the way we associate a frieze pattern to a differential equation, see Section \[1stIso\]. [$\Box$]{}
Let us give the most elementary, but perhaps the most striking, example of the combinatorial Gale transform. Suppose that a difference equation (\[REq\]) of order $k+1$ is such that all its solutions $(V_i)$ are $(k+3)$-(anti)periodic: $$V_{i+k+3}=(-1)^k\,V_i.$$ Consider the Hill equation $$W_{i}=a_{i}^1W_{i-1}-W_{i-2},$$ obtained by “forgetting” the coefficients $a_{i}^{j}$ with $j\geq2$. Theorem \[TheGailThm\] then implies that all the solutions of this equation are antiperiodic with the same period: $W_{i+k+3}=-W_i$.
Conversely, any difference equation (\[REq\]) of order $k+1$ with $(k+3)$-(anti)periodic solutions can be constructed out of a Hill equation.
At first glance, it appears paradoxical that forgetting almost all the information about the coefficients of a difference equation, we still keep the information about the (anti)periodicity of solutions.
Comparison with the classical Gale transform {#ClasS}
--------------------------------------------
Recall that the classical Gale transform of configurations of points in projective spaces is a map: $${{\mathcal G}}_{\mathrm{class}}:{\mathcal C}_{k+1,n}
\buildrel{\simeq}\over\longrightarrow{\mathcal C}_{w+1,n}\;,$$ where $n=k+w+2$, see [@Gal; @Cob; @Cob1; @EP].
The classical Gale transform is defined as follows. Let $A$ be a $(k+1)\times{}n$-matrix representing an element of ${\mathcal C}_{k+1,n}$, and $A'$ a $(w+1)\times{}n$-matrix representing an element of ${\mathcal C}_{w+1,n}$, see (\[Grass\]) and (\[TheQuot\]). These elements are in Gale duality if there exists a non-degenerate diagonal $n\times{}n$-matrix $D$ such that $$AD{A'}^T=0,$$ where ${A'}^T$ is the transposed matrix. This is precisely the duality of the corresponding Grassmannians combined with the quotient (\[TheQuot\]).
To understand the difference between the combinatorial Gale transform and the classical Gale transform, recall that the space ${\mathcal E}_{k+1,n}\simeq{\mathcal F}_{k+1,n}$ is a subvariety of the Grassmannian ${{\mathrm {Gr}}}_{k+1,n}$. Given an ${{\mathrm {SL}}}_{k+1}$-frieze pattern $F$, the $(k+1)\times{}n$-matrix $M^{(i)}_F$ representing $F$, as in (\[TheEmbM\]), satisfies the condition that every $(k+1)\times(k+1)$-minor $$\Delta_I(M^{(i)}_F)=1.$$ This implies that the diagonal matrix $D$ also has to be of a particular form.
\[FrGale\] Let $F$ be an ${{\mathrm {SL}}}_{k+1}$-frieze pattern and ${{\mathcal G}}(F)$ its Gale dual ${{\mathrm {SL}}}_{w+1}$-frieze pattern. Then the corresponding matrices satisfy $$\label{TheGaleM}
M^{(i)}_F\,D\,{M^{(j)}_{{{\mathcal G}}(F)}}^T=0,$$ where $i-j=w+1\mod n$, and where the diagonal matrix $$\label{DDiagM}
D=\left(
\begin{array}{cccc}
1&0&\ldots&0\\[4pt]
0&-1&\ldots&0\\
\vdots&\vdots&\ddots&\vdots\\[4pt]
0&0&\ldots&(-1)^{n-1}
\end{array}
\right).$$
The matrices are explicitly given by $$M^{(i)}_F=
\left(
\begin{array}{cccccccccccccc}
1 & \alpha^{w}_{i-1}& & & &&\alpha^{1}_{i-1}&1 \\[4pt]
& \ddots & & & &&&&\ddots\\[4pt]
& & 1 & \alpha^{w}_{i+k-1} &&&&&\alpha^{1}_{i+k-1} &1\\[6pt]
\end{array}\right)\;,$$ of size $(k+1)\times n$ and $$M^{(j)}_{{{\mathcal G}}(F)}=
\left(
\begin{array}{cccccccccccccc}
1 & a^{k}_{j-1}& & &&a^{1}_{j-1}&1 \\[4pt]
& 1& a^{k}_{j}& &&&a^{1}_{j}&1\\[2pt]
&& \ddots & & & &&&&\ddots\\[4pt]
& & & 1 & a^{k}_{j+w-1} &&&&&a^{1}_{j+w-1} &1\\[6pt]
\end{array}\right)\;,$$ of size $(w+1)\times n$. The columns of the matrix $M^{(i)}_F$ correspond to the diagonals $\eta_{i-1}, \cdots, \eta_{i+k+w}$ in $F$, which are solutions of . This gives immediately the relation $$M^{(i)}_F\,D\,{M^{(j)}_{{{\mathcal G}}(F)}}^T=0,
\quad \text{ where } \quad
D=\mathrm{diag}(1,-1,1,-1,\ldots ).$$
Recall that in Section \[1stIso\], we defined a projection from the space of equations to the moduli space of $n$-gons. The Gale transform agrees with this projection.
The following diagram commutes $$\xymatrix{
{{\mathcal E}}_{k+1,n} \ar[r]^{{{\mathcal G}}} \ar[d]& {{\mathcal E}}_{w+1,n} \ar[d]\\
{{\mathcal C}}_{k+1,n} \ar[r]_{{{\mathcal G}}_{\mathrm{class}}} &{{\mathcal C}}_{w+1,n}
}$$
The projection ${{\mathcal E}}_{k+1,n}\to{{\mathcal C}}_{k+1,n}$, written in the matrix form, associates to a matrix representing an element of ${{\mathcal E}}_{k+1,n}$ a coset in the quotient ${{\mathrm {Gr}}}_{k+1,n}\slash{{\mathbb T}}^{n-1}$ defined by left multiplication by diagonal $n\times{}n$-matrices. If two representatives of the coset satisfy (\[TheGaleM\]), then any two other representatives also do.
$\,$
Choose $k=2, w=3, n=7$. The two friezes of Figure \[DualFriezes\] are Gale dual to each other.
One immediately checks in this example that the corresponding matrices satisfy $$ADA'^T=0,\quad \text{ for } \quad
D=\mathrm{diag}(1,-1,1,-1,1,-1,1).$$
The projective duality {#DualSect}
----------------------
Recall that the dual projective space $({{\mathbb {RP}}}^k)^*$ (which is of course itself isomorphic to ${{\mathbb {RP}}}^k$) is the space of hyperplanes in ${{\mathbb {RP}}}^k$. The notion of [*projective duality*]{} is central in projective geometry.
Projective duality is usually defined for generic $n$-gons as follows. Given an $n$-gon $(v_i)$ in ${{\mathbb {RP}}}^k$, the [*projectively dual $n$-gon*]{} $(v_i^*)$ in $({{\mathbb {RP}}}^k)^*$ is the $n$-gon such that each vertex $v_i^*$ is the hyperplane $(v_i , v_{i+1},\ldots,v_{i+k-1})\subset{{\mathbb {RP}}}^k$. This procedure obviously commutes with the action of ${{\mathrm {SL}}}_{k+1}$, so that one obtains a map $$*:{\mathcal C}_{k+1,n}\to{\mathcal C}_{k+1,n},$$ which squares to a shift: $*\circ*:(v_i)\mapsto(v_{i+k-1})$.
In this section, we introduce an analog of the projective duality on the space of difference equations and that of frieze patterns: $$*:{\mathcal E}_{k+1,n}\to{\mathcal E}_{k+1,n},
\qquad
*:{\mathcal F}_{k+1,n}\to{\mathcal F}_{k+1,n}.$$ The square of $*$ also shifts the indices, but this shift is “invisible” on equations and friezes so that it is an [*involution*]{}: $*\circ*=\mathrm{id}$.
The difference equation $$\label{DualREq}
V^*_{i}=a_{i+k-1}^kV^*_{i-1}-a_{i+k-2}^{k-1}V^*_{i-2}+
\cdots+(-1)^{k-1}a_{i}^1V^*_{i-k}+(-1)^{k}V^*_{i-k-1}$$ is called the [*projective dual of equation*]{} (\[REq\]).
Here $(V_i^*)$ is just a notation for the unknown.
\[DeEx\] The projective dual of the equation $V_{i}=a_{i}V_{i-1}-b_iV_{i-2}+V_{i-3}$ is: $$V^*_{i}=b_{i+1}V^*_{i-1}-a_iV^*_{i-2}+V^*_{i-3}.$$
The above definition is justified by the following statement.
\[DualProp\] The map ${\mathcal E}_{k+1,n}\to{\mathcal C}_{k+1,n}$ from Section \[1stIso\] commutes with projective duality.
Recall that an $n$-gon $(v_i)$ is in the image of the map ${\mathcal E}_{k+1,n}\to{\mathcal C}_{k+1,n}$ if and only if it is a projection of an $n$-gon $(V_i)$ in ${{\mathbb R}}^{k+1}$ satisfying the determinant condition (\[detconst\]).
Let us first show that the dual $n$-gon $(v_i^*)$ is also in the image of the map ${\mathcal E}_{k+1,n}\to{\mathcal C}_{k+1,n}$. Indeed, by definition of projective duality, the affine coordinates of a vertex $v_i^*\in({{\mathbb {RP}}}^k)^*$ of the dual $n$-gon can be calculated as the $k\times{}k$-minors of the $k\times(k+1)$-matrix $$\left(
V_i \, V_{i+1}\,\ldots\,V_{i+k-1}
\right)$$ where $V_j$ are understood as $(k+1)$-vectors (i.e., the columns of the matrix). Denote by $V^*_i$ the vector in $({{\mathbb R}}^{k+1})^*$ with coordinates given by the $k\times{}k$-minors. In other words, the vector $V^*_i$ is defined by the equation $$\left|
V_i , V_{i+1},\ldots,V_{i+k-1},V_i^*
\right|=1.$$
A direct verification then shows that $(V_i^*)$ satisfy the equation (\[DualREq\]).
The isomorphism ${\mathcal E}_{k+1,n}\simeq{\mathcal F}_{k+1,n}$ allows us to define the notion of projective duality on ${{\mathrm {SL}}}_{k+1}$-frieze patterns.
\[Turn\] The projective duality of ${{\mathrm {SL}}}_{k+1}$-frieze patterns is just the symmetry with respect to the median horizontal axis.
We will prove this statement in Section \[DeTSTwo\]. The proof uses the explicit computations. See Proposition \[DualDiag\].
The projective duality commutes with the Gale transform: $$*\circ{{\mathcal G}}={{\mathcal G}}\circ*.$$
The self-dual case {#SDS}
------------------
An interesting class of difference equations and equivalently, of ${{\mathrm {SL}}}_{k+1}$-frieze patterns, is the class of [*self dual*]{} equations. In the case of frieze patterns, self-duality means invariance with respect to the horizontal axis of symmetry.
a\) Every ${{\mathrm {SL}}}_{2}$-frieze pattern is self-dual.
b\) Consider the following ${{\mathrm {SL}}}_{3}$-frieze patterns of width $2$: $$\begin{array}{rrrrrrrr}
1&&1&&1&&1&\\
&2&&2&&2&&2\\
2&&2&&2&&2&\\
&1&&1&&1&&1
\end{array}
\qquad\qquad
\begin{array}{rrrrrrrr}
1&&1&&1&&1&\\
&2&&3&&2&&3\\
1&&5&&1&&5&\\
&1&&1&&1&&1
\end{array}$$ The first one is self-dual but the second one is not.
An $n$-gon is called [*projectively self dual*]{} if, for some fixed $0\leq\ell\leq{}n-1$, the $n$-gon $(v_{i+\ell}^*)$ is projectively equivalent to $(v_i)$. Note that $\ell$ is a parameter in the definition (so, more accurately, one should say “$\ell$-self dual”); see [@FT].
The determinantal formulas {#DeTS}
==========================
In this section, we give explicit formulas for the Gale transform. It turns out that one can solve the equation (\[REq\]) and obtain explicit formulas for the coefficients of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern. Let us mention that the determinant formulas presented here already appeared in the classical literature on difference equations in the context of “André method of solving difference equations”; see [@And; @Jor].
Calculating the entries of the frieze patterns {#DeTSOne}
----------------------------------------------
Recall that we constructed an isomorphism between the spaces of difference equations ${\mathcal E}_{k+1,n}$ and frieze patterns ${\mathcal F}_{k+1,n}$. We associated an ${{\mathrm {SL}}}_{k+1}$-frieze pattern of width $w$ to every difference equation (\[REq\]). The entries $d_{i,i+j}$ of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern (also denoted by $\alpha_{i-1}^{w-j}$, see (\[friezeGG\]), (\[Corresp\])) were defined non-explicitly by (\[ConstEq\]).
\[DetExpr\] The entries of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern associated to a difference equation (\[REq\]) are expressed in terms of the coefficients $a_i^j$ by the following determinants.
\(i) If $0\leq j \leq k-1$ and $j<w$, then $$\label{DetEq}
d_{i,i+j}=
\left|
\begin{array}{llllll}
a_{i}^1&1&\\[8pt]
a_{i+1}^2&a_{i+1}^1&1&\\[8pt]
\vdots&\ddots&\ddots&\;1\\[6pt]
a_{i+j}^{j+1}&\cdots&a_{i+j}^{2}&a_{i+j}^{1}
\end{array}
\right|.$$
\(ii) If $k-1< j <w$, then $$\label{DetEqDva}
d_{i,i+j}=
\left|
\begin{array}{cccclcc}
a_{i}^1&1&\\[8pt]
\vdots&a_{i+1}^1&1&\\[4pt]
a_{i+k-1}^{k}&&\ddots&\ddots\\[6pt]
1&&&\ddots&\ddots\\[4pt]
&\ddots&&&a_{i+j-1}^1&1\\[8pt]
&&1&a_{i+j}^{k}&\ldots&a_{i+j}^1
\end{array}
\right|.$$
We use the Gale transform ${{\mathcal G}}(F)$ of the frieze $F$ associated to . In ${{\mathcal G}}(F)$, the following diagonals of length $w+1$ $$\begin{blockarray}{cccccccccccccc}
\eta_{i-w-2} & \eta_{i-w-1} & & \eta_{i-w+j}&&\eta_{i-2}& \eta_{i-1} & \\[10pt]
\begin{block}{(ccccccccccccc)c}
a_{i}^1&1&\\[8pt]
a_{i+1}^2&a_{i+1}^1&\ddots&\\[8pt]
\vdots&\vdots&&\;1\\[6pt]
a_{i+j}^{j+1}&a_{i+j}^{j}&&a_{i+j}^{1}&\ddots\\
\vdots&\vdots& & \vdots& &1&\\[4pt]
\vdots&\vdots&&\vdots&&a_{i+w}^1&1\\[6pt]
\end{block}
\end{blockarray}\;,$$ satisfy the recurrence relation $$\eta_{i-1}=\alpha_{i-1}^1\eta_{i-2}-\ldots +(-1)^{w-j-1}\alpha_{i-1}^{w-j}\eta_{i-w+j-1}+\ldots
+(-1)^{w-1}\ \alpha_{i-1}^{w}\eta_{i-w-1} +(-1)^{w}\ \eta_{i-w-2},$$ that can be written in terms of vectors and matrices as $$\eta_{i-1}=
\left(
\begin{array}{ccc}
\eta_{i-w-2}, &\ldots, & \eta_{i-2}
\end{array}\right)
\left(
\begin{array}{l}
(-1)^{w}\\
(-1)^{w-1}\alpha_{i-1}^w\\
\vdots\\
(-1)^0\alpha_{i-1}^1
\end{array}\right).$$ The coefficients $\alpha_{i-1}^{w-j}$ can be computed using the Cramer rule $$\label{CramEq}
\alpha_{i-1}^{w-j}=\dfrac{(-1)^{w-j-1}|\eta_{i-w-2}, \ldots, \eta_{i-w+j}, \;
\eta_{i-1}, \;\eta_{i-w+j+2}, \ldots, \eta_{i-2}|}{|\eta_{i-w-2}, \ldots, \eta_{i-2}|}.$$ The denominator is 1 since ${{\mathcal G}}(F)$ is an ${{\mathrm {SL}}}_{w+1}$-frieze, and the numerator simplifies to or to accordingly after decomposing by the $\eta_{i-1}$-th column. The coefficient $\alpha_{i-1}^{w-j}$ is in position $d_{i,i+j}$ in the frieze $F$.
Equivalent formulas {#DeTSTri}
-------------------
There is another, alternative, way to calculate the entries of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern.
\[DetExprDva\] (i) If $j+k\geq w$ then $$d_{i,i+j}=
\left|
\begin{array}{ccrrc}
a_{i-w+j-1}^k&a_{i-w+j-1}^{k-1}&\ldots&\ldots&a_{i-w+j-1}^{k-w+j+1}\\[8pt]
1&a_{i-w+j}^k&\ldots&\ldots&a_{i-w+j}^{k-w+j}\\[8pt]
&1&&&\vdots\\[8pt]
&&\ddots&\ddots&\vdots\\[12pt]
&&&\;1&a_{i-2}^k\\[8pt]
\end{array}
\right|.$$
\(ii) If $j+k<w$ then $$\label{DetEqBis}
d_{i,i+j}=
\left|
\begin{array}{cccccc}
a_{i-w+j-1}^k&\ldots&a_{i-w+j-1}^1&\;\;1&&\\[8pt]
1&a_{i-w+j}^k&\ldots&a_{i-w+j}^1&\!\!1&\\[8pt]
&1&\ldots&\ldots&&\!\!1\\[8pt]
&&\ddots&&\ddots&\vdots\\[12pt]
&&&\;1&a_{i-3}^k&a_{i-3}^{k-1}\\[8pt]
&&&&\!\!\!\!1&a_{i-2}^k
\end{array}
\right|.$$
These formulas are obtained in the same way as (\[DetEq\]) and (\[DetEqDva\]).
\[HillEx2\]
\(a) Hill’s equations $V_{i}=a_{i}V_{i-1}-V_{i-2}$ with antiperiodic solutions correspond to Coxeter’s frieze patterns with the entries $$d_{i,i+j}=
\left|
\begin{array}{cccccc}
a_{i}&1&&&\\[4pt]
1&a_{i+1}&1&&\\[4pt]
&\ddots&\ddots&\ddots&\\[4pt]
&&1&a_{i+j-2}&1\\[4pt]
&&&1&a_{i+j-1}
\end{array}\right|.$$ The corresponding geometric space is the moduli space (see Theorem \[TriThm\]) of $n$-gons in the projective line known (in the complex case) under the name of moduli space $\mathcal{M}_{0,n}$.
\(b) In the case of third-order difference equations, $V_{i}=a_{i}V_{i-1}-b_iV_{i-2}+V_{i-3}$, we have: $$d_{i,i+j}=
\left|
\begin{array}{llllll}
a_{i}&b_{i+1}&1&&&\\[4pt]
1&a_{i+1}&b_{i+2}&1&&\\[4pt]
&\;\ddots&\; \ddots& \;\ddots&\; \ddots&\\[4pt]
&&1&a_{i+j-3}&b_{i+j-3}&1\\[4pt]
&&&1&a_{i+j-2}&b_{i+j-2}\\[4pt]
&&&&1&a_{i+j-1}
\end{array}
\right|.$$ This case is related to the moduli space ${{\mathcal C}}_{3,n}$ of $n$-gons in the projective plane studied in [@MGOT].
Determinantal formulas for the Gale transform {#DeTSFour}
---------------------------------------------
Formulas from Propositions \[DetExpr\] and \[DetExprDva\] express the entries of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern (\[friezeGG\]) as minors of the Gale dual ${{\mathrm {SL}}}_{w+1}$-frieze pattern (\[friezeG\]). One can reverse the formula and express the entries of (\[friezeG\]), i.e., the coefficients of the equation (\[REq\]) as minors of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern (\[friezeGG\]).
For instance the Gale dual of is $$\label{DuDeT}
a_{i-1}^{k-j}=
\dfrac{(-1)^{k-j-1}|\eta_{i-k-2}, \ldots, \eta_{i-k+j}, \;
\eta_{i-1}, \;\eta_{i-k+j+2}, \ldots, \eta_{i-2}|}{|\eta_{i-k-2}, \ldots, \eta_{i-2}|},$$ where $\eta$’s are the South-East diagonals of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern . Note that the denominator is equal to $1$.
For $j<k$ we obtain an analog of the formula : $$\label{DuDeTBis}
a_{i-1}^{k-j}=
\left|
\begin{array}{cccc}
d_{i+1,i+w}&1&\\[8pt]
\vdots&\ddots&\;1\\[6pt]
d_{i+j+1,i+w}&\cdots&d_{i+j+1,i+j+w}
\end{array}
\right|,$$ and similarly for .
Frieze patterns and the dual equations {#DeTSTwo}
--------------------------------------
\[DualDiag\] The North-East diagonals $(\mu_i)$ of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern corresponding to the difference equation (\[REq\]) satisfy the projectively dual difference equation (\[DualREq\]).
This is a consequence of the tameness of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern corresponding to and can be easily showed using the explicit determinantal formulas that will be established in the next section. We first consider the case $j=i+w-k-2$. We know that $(k+2)\times (k+2)$-determinant vanishes, thus $$\left|
\begin{array}{ccccc}
d_{j-w+1,j}&1&\\[8pt]
d_{j-w+2,j}&d_{j-w+2,j+1}&1\\[12pt]
\vdots&&\ddots&\;1\\[6pt]
d_{i,j}&\cdots&&d_{i,i+w-1}
\end{array}
\right|=0.$$ Decomposing the determinant by the first column gives the recurrence relation $$d_{i,j}=a_{i-2}^kd_{i-1,j}-a_{i-3}^{k-1}d_{i-1,j}+
\cdots+(-1)^{k-1}a_{i-k-1}^1d_{i-k,j}+(-1)^{k}d_{i-k-1,j},$$ where the coefficients are computed using . The recurrence relation then propagates inside the frieze due to the tameness property, and hence will hold for all $j$. This proves that the North-East diagonals satisfy after renumbering $\mu'_i:=\mu_{i+k+1}$.
The Gale transform and the representation theory {#RePSS}
================================================
In this section, we give another description of the combinatorial Gale transform ${{\mathcal G}}$ in terms of representation theory of the Lie group ${{\mathrm {SL}}}_{n}$. Let $N\subset{{\mathrm {SL}}}_{n}$ be the subgroup of upper unitriangular matrices. $$A=\left(
\begin{array}{rccl}
1&*&\ldots&*\\
&\ddots&\ddots&\vdots\\
&&\ddots&*\\
&&&1
\end{array}
\right).$$ We will associate a unitriangular $n\times{}n$-matrix to every ${{\mathrm {SL}}}_{k+1}$-frieze pattern. This idea allows us to apply in our situation many tools of the theory of matrices, as well as more sophisticated tools of representation theory. We make just one small step in this direction: we show that the combinatorial Gale transform ${{\mathcal G}}$ coincides with the restriction of the anti-involution on $N$ introduced in [@BFZ].
From frieze patterns to unitriangular matrices
----------------------------------------------
Given an ${{\mathrm {SL}}}_{k+1}$-frieze pattern $F$ of width $w$, as in formula (\[FREq\]), cutting out a piece of the frieze, see Figure \[CutFig\], we associate to $F$ a unitriangular $n\times n$-matrix $$\label{TheMA}
A_F=\left(
\begin{array}{rcccccc}
1&d_{1,1}&\cdots&d_{1,w}&1&&\\
&\ddots&\ddots&&\ddots&\ddots&\\
&&\ddots&\ddots&&\ddots&1\\
&&&\ddots&\ddots&&d_{n-w,n-1}\\
&&&&\ddots&\ddots&\vdots\\
&&&&&\ddots&d_{n-1,n-1}\\[4pt]
&&&&&&1
\end{array}
\right)$$ with $w+2$ non-zero diagonals. As before, $n=k+w+2$.
Note that the matrix $A_F$ contains all the information about the ${{\mathrm {SL}}}_{k+1}$-frieze pattern. Indeed, the frieze pattern can be reconstructed by even a much smaller $(k+1)\times{}n$-matrix (\[TheEmbM\]). However, $A_F$ is not defined uniquely. Indeed, it depends on the choice of the first element $d_{1,1}$ in the first line of the frieze. A different cutting gives a different matrix.
The combinatorial Gale transform as an anti-involution on $N$
-------------------------------------------------------------
Denote by $^{\iota}$ the anti-involution of $N$ defined for $x\in{}N$ by $$\label{IoT}
x^\iota=Dx^{-1}D,$$ where $D$ is the diagonal matrix (\[DDiagM\]). Recall that the term “anti-involution” means an involution that is an anti-homomorphism, i.e., $(xy)^\iota=y^\iota{}x^\iota$. This anti-involution was introduced in [@BFZ] in order to study the canonical parametrizations of $N$. We will explain the relation of $^\iota$ to the classical representation theory in Section \[ClSec\].
\(i) The operation $A_F\mapsto (A_F)^{\iota}$ associates to a matrix (\[TheMA\]) of a tame ${{\mathrm {SL}}}_{k+1}$-frieze pattern a matrix of a tame ${{\mathrm {SL}}}_{w+1}$-frieze pattern.
\(ii) The corresponding map $$^\iota:{{\mathcal E}}_{k+1,n}\to{{\mathcal E}}_{w+1,n}$$ coincides with the composition of the Gale transform and the projective duality: $$^\iota={{\mathcal G}}\circ*.$$
Let us denote by $S=\{1, 2,\ldots, n\}$ the index set of the rows, resp. columns, of a matrix $x\in N$. For two subsets $I,J\subset{}S$ of the same cardinality we denote by $\Delta_{I,J}(x)$ the minor of the matrix $x$ taken over the rows of indices in $I$ and the columns of indices in $J$. We have the following well-known relation $$\Delta_{i,j}(x^{\iota})=\Delta_{S-\{ j\}, S-\{i\}}(x),$$ where $\Delta_{i,j}(x^{\iota})$ is simply the entry in position $(i,j)$ in $x^{\iota}$. Taking into account that the matrix $x$ belongs to $N$, whenever $j> i$, the $(n-1)\times (n-1)$-minor in the above right hand side simplifies to a $(j-i)\times (j-i)$-minor $$\Delta_{i,j}(x^{\iota})=\Delta_{[ i,j-1], [i+1,j]}(x),$$ where $[a,b]$ denotes the interval $\{a,a+1,\ldots,b\}$. Let us use this relation to compute the entry in position $(i,j)$ in $(A_F)^{\iota}$. One has: $$\Delta_{i,j}(A_F^{\iota})=
\left|
\begin{array}{ccrrc}
d_{i,i}&d_{i,i+1}&\ldots&\ldots&d_{i,j-1}\\[8pt]
1&d_{i+1,i+1}&\ldots&\ldots&d_{i+1,j-1}\\[8pt]
&1&&&\vdots\\[8pt]
&&\ddots&\ddots&\vdots\\[12pt]
&&&\;1&d_{j-1,j-1}\\[8pt]
\end{array}
\right|=
\left|
\begin{array}{ccrrc}
\alpha_{i-1}^{w}&\alpha_{i-1}^{w-1}&\ldots&\ldots&\alpha_{i-1}^{w-j+i-1}\\[8pt]
1&\alpha_{i}^w&\ldots&\ldots&\alpha_{i}^{w-j+i}\\[8pt]
&1&&&\vdots\\[8pt]
&&\ddots&\ddots&\vdots\\[12pt]
&&&\;1&\alpha_{j-2}^w\\[8pt]
\end{array}
\right|.$$ According to the determinantal formulas of Section \[DeTS\], this is precisely the entry $d_{j,i+k}$ of the frieze ${{\mathcal G}}(F)$, and hence the result.
Elements of the representation theory
-------------------------------------
Our next goal is to explain the relation of the involution $^\iota$ with Schubert cells in the Grassmannians. We have to recall some basic notions of representation theory.
The [*Weyl group*]{} of ${{\mathrm {SL}}}_{n}$ is the group ${{\mathcal S}}_{n}$ of permutations over $n$ letters that we think of as the set of integers $\{1, 2, \ldots, n\}$. The group ${{\mathcal S}}_{n}$ is generated by $(n-1)$ elements denoted by $s_i$, $1\leq i\leq n-1$, representing the elementary transposition $i \leftrightarrow i+1$. The relations between the generators are as follows: $$\begin{array}{rcl}
s_{i}s_{i+1}s_{i}&=&s_{i+1}s_{i}s_{i+1},\\[4pt]
s_is_j&=&s_js_i, \quad |i-j|> 1.
\end{array}$$ A decomposition of ${{\sigma}}\in{{\mathcal S}}_{n}$ as $${{\sigma}}=s_{i_1}s_{i_2}\cdots s_{i_p}$$ is called *reduced* if it involves the least possible number of generators. Equivalently, we call the sequence ${{\mathbf {i}}}=(i_1, \ldots, i_p)$ a *reduced word* for ${{\sigma}}$.
The group ${{\mathcal S}}_{n}$ can be viewed as a subgroup of ${{\mathrm {SL}}}_{n}$ using the following lift of the generators $$s_i=\begin{pmatrix}
\ddots&&&\\
&0&1&\\
&-1&0&&\\
&&&\ddots
\end{pmatrix}.$$
Let us now describe the [*standard parametrization*]{} of the unipotent subgroup $N$. Consider the following one-parameter subgroups of $N$: $$x_i(t)=
\begin{pmatrix}
\ddots&&&\\
&1&t&\\
&&1&&\\
&&&\ddots
\end{pmatrix}, \quad
1\leq i \leq n-1,
\quad
t\in{{\mathbb R}},$$ where $t$ is in position $(i,i+1)$. The matrices $x_i(t)$ are called the elementary Jacobi matrices, these are generators of $N$.
The next notion we need is that of the [*Schubert cells*]{}. Denote by $B^-$ the Borel subgroup of lower triangular matrices of ${{\mathrm {SL}}}_{n}$. Fix an arbitrary element ${{\sigma}}\in{{\mathcal S}}_{n}$, and consider the set $$N^{{\sigma}}:=N\cap B^-{{\sigma}}{}B^-$$ which is known as an open dense subset of a Schubert cell. It is well known that generically elements of $N^{{\sigma}}$ can be represented as $$\label{ThexEq}
x_{{\mathbf {i}}}({{\mathbf {t}}}) =x_{i_1}(t_1)x_{i_2}(t_2)\cdots x_{i_p}(t_p),$$ where ${{\mathbf {t}}}=(t_1,\ldots, t_p)\in {{\mathbb R}}^p$ and ${{\mathbf {i}}}=(i_1,\ldots, i_p)$ is an arbitrary reduced word for ${{\sigma}}$.
For the following choice $$\label{Thesigma}
{{\sigma}}=s_{k+1}\cdots s_{n-1}\, s_k\cdots s_{n-2}\, \cdots \, s_{1}\cdots s_{n-k-1},$$ the set $N^{{\sigma}}$ is identified with an open subset of the Grassmannian ${{\mathrm {Gr}}}_{k+1, n}$. See, e.g., [@Spr Chap. 8], for more details.
The anti-involution $^{\iota}$ and the Grassmannians {#ClSec}
----------------------------------------------------
It was proved in [@BFZ] that the anti-involution $^{\iota}$ on $N$ can be written in terms of the generators as follows. Set $x_i(t)^{\iota}=x_i(t)$ and for $x$ as in (\[ThexEq\]), one has: $$x^\iota=
x_{i_p}(t_p)x_{i_{p-1}}(t_{p-1})\cdots{}x_{i_1}(t_1).$$ This map is well-defined, i.e., it is independent of the choice of the decomposition of $x$ into a product of generators since it coincides with (\[IoT\]).
Restricted to $N^{{\sigma}}$, where ${{\sigma}}$ is given by (\[Thesigma\]), the anti-involution reads: $
^{\iota}:N^{{\sigma}}\to{}N^{{{\sigma}}^{-1}}.
$ In particular, it sends an open subset of the Grassmannian ${{\mathrm {Gr}}}_{k+1, n}$ to an open subset of ${{\mathrm {Gr}}}_{w+1, n}$.
We have already defined the embedding (\[TheEmbM\]) of the space of ${{\mathrm {SL}}}_{k+1}$-frieze patterns into the Grassmannian. Quite obviously, one also has an embedding into $N^{{\sigma}}$, so that $${\mathcal F}_{k+1,n}\subset{}N^{{\sigma}}\subset{{\mathrm {Gr}}}_{k+1,n}.$$ It can be shown that the image of the involution $^{\iota}$ restricted to ${\mathcal F}_{k+1,n}$ belongs to ${\mathcal F}_{w+1,n}$. However, the proof is technically involved and we do not dwell on the details here. The following example illustrates the situation quite well.
The case of ${{\mathrm {Gr}}}_{2,5}$. Fix ${{\sigma}}=s_2s_3s_4s_1s_2s_3$ and consider the following element of $N^{{\sigma}}$: $$x=x_2(t_1)x_3(t_2)x_4(t_3)x_1(t_4)x_2(t_5)x_3(t_6)=
\begin{pmatrix}
1 & t_4& t_4t_5&t_4t_5t_6&0\\[4pt]
&1&t_5+t_1&t_1t_2+t_1t_6+t_5t_6&t_1t_2t_3\\[4pt]
&&1&t_6+t_2&t_2t_3\\[4pt]
&&&1&t_3\\[4pt]
&&&&1
\end{pmatrix}.$$ One then has $$x^{\iota}=x_3(t_6)x_2(t_5)x_1(t_4)x_4(t_3)x_3(t_2)x_2(t_1)=
\begin{pmatrix}
1 & t_4& t_4t_1&0&0\\[4pt]
&1&t_5+t_1&t_2t_5&0\\[4pt]
&&1&t_6+t_2&t_6t_3\\[4pt]
&&&1&t_3\\[4pt]
&&&&1
\end{pmatrix}.$$ If now $x\in{{\mathcal F}}_{2,5}$, so that every $2\times2$-minor equals 1, then one has after an easy computation: $$t_1t_4=1,
\quad
t_1t_2t_4t_5=1,
\quad
t_1t_2t_3t_4t_5t_6=1,
\quad
t_1t_2t_3=1,$$ and this implies that $x^{\iota}\in{{\mathcal F}}_{3,5}$.
Periodic rational maps from frieze patterns {#PerSec}
===========================================
The periodic rational maps described in this section are a simple consequence of the isomorphism ${{\mathcal E}}_{k+1,n}\simeq{{\mathcal F}}_{k+1,n}$ and of periodicity condition. However, the maps are of interest. The simplest example is known as the [*Gauss map*]{}, see [@Gau]. This map is given explicitly by $$(c_1,c_2)\mapsto\left(c_2,\frac{1+c_1}{c_1c_2-1}\right),$$ where $c_1,c_2$ are variables. Gauss proved that this map is $5$-periodic. In our terminology, Gauss’ map consists in the index shift: $(c_1,c_2)\mapsto(c_2,c_3)$ in a Hill equation $V_i=c_iV_{i-1}-V_{i-2}$ with $5$-antiperiodic solutions.
The maps that we calculate in Section \[FPerS\], are related to so-called Zamolodchikov periodicity conjecture. They can be deduced from the simplest $A_k\times{}A_w$-case; the periodicity was proved in this case in [@Vol] by a different method. The general case of the conjecture was recently proved in [@Kel]. It would be interesting to investigate an approach based on linear difference equations in this case.
Finally, the maps that we calculate in Section \[SPerS\], correspond to self-dual difference equations. They do not enter into the framework of Zamolodchikov periodicity conjecture and seem to be new.
Periodicity of ${{\mathrm {SL}}}_{k+1}$-frieze patterns and generalized Gauss maps {#FPerS}
----------------------------------------------------------------------------------
An important property of ${{\mathrm {SL}}}_{k+1}$-frieze patterns is their periodicity.
\[periodic\] Tame ${{\mathrm {SL}}}_{k+1}$-frieze patterns of width $w$ are $n$-periodic in the horizontal direction: $d_{i,j}=d_{i+n,j+n}$ for all $i,j$, where $n=k+w+2$.
This statement is a simple corollary of Theorem \[TriThm\], Part (i). Note that, in the simplest case $k=1$, the above statement was proved by Coxeter [@Cox].
Let us introduce the notation: $$U(a_1,a_2,\dots,a_k):=\left|\begin{array}{cccccc}
a_{1}&1&&&\\[4pt]
1&a_{2}&1&&\\[4pt]
&\ddots&\ddots&\ddots&\\[4pt]
&&1&a_{k-1}&1\\[4pt]
&&&1&a_{k}
\end{array}\right|.$$ for the simplest determinants from Section \[DeTS\], see Example \[HillEx2\], part (a). We obtain the following family of rational periodic maps.
\[main\] Let the rational map $F: {{\mathbb R}}^{n-3} \to {{\mathbb R}}^{n-3}$ be given by the formula $$\label{FMap}
F:(a_1,a_2,\dots,a_{n-3})\mapsto
\left(a_2,a_3,\dots,a_{n-3}, P(a_1,a_2,\dots,a_{n-3})\right),$$ where $$P(a_1,a_2,\dots,a_{n-3}) = \frac{1+U(a_1,\dots,a_{n-4})}{U(a_1,\dots,a_{n-3})}.$$ Then $F^n=\mathrm{id}$.
Consider the periodic Hill equation $V_i=a_iV_{i-1}-V_{i-2}$, or equivalently ${{\mathrm {SL}}}_2$-frieze pattern whose first row consists of ones, and the second row is the bi-infinite sequence $(a_i)$. The entries of $k$th row of this frieze pattern are $$U(a_i,\dots,a_{i+k-2}),\qquad i\in{{\mathbb Z}},$$ see Example \[HillEx2\] and [@Cox; @MGOT].
Assume that all the solutions of the Hill equation are $n$-antiperiodic. Then the frieze pattern is closed of width $n-3$ and its rows are $n$-periodic, see [@Cox] and Corollary \[periodic\]. Furthermore, the $(n-2)$th row of a closed ${{\mathrm {SL}}}_2$-frieze pattern consists in $1$’s. Therefore $$U(a_i,a_{i+1},\dots,a_{i+n-3})=1,$$ for all $i$.
On the other hand, decomposing the determinant $U(a_i,a_{i+1},\dots,a_{i+n-3})$ by the last row, we find $$\label{rec1}
a_{i+n-3}=P(a_i,\dots,a_{i+n-4}).$$ We can choose $a_1,\dots,a_{n-3}$ arbitrarily and then consecutively define $a_{n-2}, a_{n-1},\dots$ using formula (\[rec1\]) for $i=1,2,\dots$. That is, we reconstruct the sequence $(a_i)$ from the “seed" $\{a_1,\dots,a_{n-3}\}$ by iterating the map $F$. By periodicity assumption, the result is an $n$-periodic sequence.
Introduce another notation: $$V(a_1,b_1,\dots,b_{k-1},a_k):=\left|
\begin{array}{llllll}
a_{1}&b_{1}&1&&&\\[4pt]
1&a_{2}&b_{2}&1&&\\[4pt]
&\;\ddots&\; \ddots& \;\ddots&\; \ddots&\\[4pt]
&&1&a_{k-2}&b_{k-2}&1\\[4pt]
&&&1&a_{k-1}&b_{k-1}\\[4pt]
&&&&1&a_k
\end{array}
\right|,$$ see Example \[HillEx2\], part (b).
\[mainBis\] Let the rational map $\Phi: {{\mathbb R}}^{2n-8} \to {{\mathbb R}}^{2n-8}$ be given by the formula $$\label{PhiMap}
\Phi:(a_1,b_1,a_2,b_2,\dots,a_{n-4},b_{n-4})\mapsto\left(b_1,a_2,b_2,\dots,b_{n-4},Q(a_1,b_1,\dots,b_{n-4},a_{n-4})\right),$$ where $$Q(a_1,b_1,\dots,a_{n-4},b_{n-4})=
\frac{1+b_{n-4}V(a_1,b_1,\dots,a_{n-5})-V(a_1,b_1,\dots,a_{n-6})}
{V(a_1,b_1,\dots,a_{n-4})}.$$ Then $\Phi^{2n}=\mathrm{id}$.
The arguments are similar to those of the above proof, but we will also use the notion of a projectively dual equation.
Consider the difference equation $V_i=a_iV_{i-1}-b_iV_{i-2}+V_{i-3}$ and assume that all its solutions (and therefore all its coefficients) are $n$-periodic. Consider the dual equation, see Example \[DeEx\], but “read” it from right to left: $$V^*_{i-3}=a_iV^*_{i-2}-b_{i+1}V^*_{i-1}+V^*_i.$$
The map $\Phi$ associates to a “seed” $\{a_1,b_1,\dots,a_{n-4},b_{n-4}\}$ of the initial equation the same “seed” of the dual equation.
Recall finally that the double iteration of the projective duality is a shift: $i\to{}i+1$. Therefore, $\Phi^2:(a_i,b_i,\ldots)\mapsto(a_{i+1},b_{i+1},\ldots)$, which is $n$-periodic by assumption.
\[penta\] [For $n=5$, the maps from Corollaries \[main\] and \[mainBis\] are as follows: $$F(a_1,a_2)=\left(a_2,\frac{1+a_1}{a_1a_2-1}\right), \qquad
\Phi(b,a)=\left(a,\frac{a+1}{b}\right).$$ The first one is the classical $5$-periodic Gauss map, the second, which looks even more elementary, is $10$-periodic. ]{}
Periodic maps in the self-dual case {#SPerS}
-----------------------------------
Let us now consider a version of the rational periodic maps that correspond to self-dual third-order $n$-periodic equations, see Section \[SDS\].
\[selfd\] (i) Let $n=2m-1$, and let the rational map $G_{o}: {{\mathbb R}}^{n-4} \to {{\mathbb R}}^{n-4}$ be given by the formula $$G_{o}(a_1,b_1,a_2,b_2,\dots,a_{m-2},b_{m-2})=\left(b_1,a_2,b_2,\dots,b_{m-2}, R_o(a_1,b_1,\dots,a_{m-2},b_{m-2})\right),$$ where $$\begin{array}{l}
R_o(a_1,b_1,\dots,a_{m-2},b_{m-2}) = \\[6pt]
\displaystyle
\qquad
\qquad
\qquad
\frac{V(a_2,b_2,\dots,a_{m-2}) + b_{m-2} V(a_1,b_1,\dots,a_{m-3})-V(a_1,b_1,\dots,a_{m-4})}{V(a_1,b_1,\dots,a_{m-2})}.
\end{array}$$ Then $G_o^n=\mathrm{id}$.
\(ii) Let $n=2m$, and let the rational map $G_{e}: {{\mathbb R}}^{n-4} \to {{\mathbb R}}^{n-4}$ be given by the formula $$G_e(a_1,b_1,a_2,b_2,\dots,a_{m-2},b_{m-2})=
\left(
b_1,a_2,b_2,\dots,b_{m-2}, R_e(a_1,b_1,\dots,a_{m-2},b_{m-2})\right),$$ where $$\begin{array}{l}
R_e(a_1,b_1,\dots,a_{m-2},b_{m-2}) = \\[6pt]
\displaystyle
\qquad
\qquad
\qquad
\frac{V(b_2,a_3,\dots,a_{m-2}) + b_{m-2} V(a_1,b_1,\dots,a_{m-3})-V(a_1,b_1,\dots,a_{m-4})}{V(a_1,b_1,\dots,a_{m-2})}.
\end{array}$$ Then $G_e^n=\mathrm{id}$.\
Let us consider part (i); the other case is similar.
As before, consider $n$-periodic equations $V_i=a_iV_{i-1}-b_iV_{i-2}+V_{i-3}$. As before, both sequences $(a_i)$ and $(b_i)$ are $n$-periodic, so that the sequence $$\dots,b_1,a_1,b_2,a_2,\dots,b_n,a_n,\dots$$ is $2n$-periodic. However, if the equation is self-dual, then this sequence is, actually, $n$-periodic. More precisely, $$\left\{
\begin{array}{ll}
b_{i+m}=a_i, & n=2m-1;\\[4pt]
b_{i+m}=b_i,\, a_{i+m}=a_i , & n=2m.
\end{array}
\right.$$
We are concerned with the shift $$(a_1,b_1,a_2,b_2,\dots,a_{m-2},b_{m-2}) \mapsto
(b_1,a_2,b_2,\dots,a_{m-2},b_{m-2},a_{m-1}),$$ and we need to express $a_{m-1}$ as a function of $a_1,b_1,a_2,b_2,\dots,a_{m-2},b_{m-2}$.
To this end, we express, in two ways, the same entry of the ${{\mathrm {SL}}}_3$-frieze pattern. Consider the South-East diagonal through the entry $a_1$ of the top non-trivial row and the South-West diagonal through the entry $a_{m-1}$ of the same row. The entry at their intersection is $V(a_1,b_1,\dots,a_{m-1})$, see [@MGOT].
Since the difference equation is self-dual, the bottom non-trivial row of the ${{\mathrm {SL}}}_3$-frieze pattern is identical to the top one. The North-East diagonal through the entry $a_2$ of this bottom non-trivial row intersects the North-West diagonal through the entry $a_{m-2}$ of the same row at the same point as above, and the entry there is $V(a_2,b_2,\dots,a_{m-2})$.
Therefore $$V(a_1,b_1,\dots,a_{m-1})=V(a_2,b_2,\dots,a_{m-2}),$$ and it remains to solve this equation for $a_{m-1}$. This yields the formula for the rational function $R$.
[If $n=6$, we obtain the map $$G_e(a,b)=\left(b,\frac{2b}{a}\right)$$ that indeed has order 6. If $n=7$, we obtain the map $$G_o(a_1,b_1,a_2,b_2)=\left(b_1,a_2,b_2,\frac{a_2+a_1b_2-1}{a_1a_2-b_1}\right)$$ that has order 7. ]{}
Another expression for $2n$-periodic maps
-----------------------------------------
Let us sketch a derivation of the map $\Phi$ from a geometrical point of view. The formula is: $$\Phi(x_1,\dots,x_{2n-8})=
\left(x_2,\dots,x_{2n-8}, R(x_1,\dots,x_{2n-8})\right),$$ where $$R(x_1,\dots,x_{2n-8})=
\frac{O^{2n-7}_{-1}}{O^{2n-9}_{-1}-x_{2n-8}x_{2n-9}\,O^{2n-11}_{-1}},$$ and where $O_a^b$ is defined by recurrence relation $$O_a^b=O_a^{b-2}-x_{b-2}O_a^{b-4}+x_{b-2}x_{b-3}x_{b-4}O_a^{b-6},
\qquad
a=b-4,b-6,\ldots$$ with the initial conditions $O_b^{b}=O_{b-2}^{b}=1$. We will see below why $\Phi^{2n}$ is the identity.
The first non-trivial example is $$\Phi(x_1,x_2)=\left(
x_2,\frac{1-x_1}{1-x_1x_2}
\right),$$ which is the Gauss map. The next example is: $$\Phi(x_1,x_2,x_3,x_4)=\left(
x_2,x_3,x_4,\frac{1-x_1-x_3+x_1x_2x_3}{1-x_1-x_3x_4}
\right),$$ which is $12$-periodic.
These formulas give the expression of the map $\Phi$ in so-called corner coordinates on the moduli space of $n$-gons in ${{\mathbb {P}}}^2$. Let $LL'$ denotes the intersection of lines $L$ and $L'$ and let $PP'$ denote the line containing $P$ and $P'$. We have the [*inverse cross-ratio*]{} $$\label{inverse-cross-ratio}
[a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}.$$
A [*polygonal ray*]{} is an infinite collection of points $P_{-7},P_{-3},P_{+1},...$, with indices congruent to $1$ mod $4$, normalized so that $$P_{-7}=(0,0,1), \hskip 15 pt
P_{-3}=(1,0,1), \hskip 15 pt
P_{+1}=(1,1,1), \hskip 15 pt
P_{+5}=(0,1,1).$$ These points determine lines $$L_{-5+k}=P_{-7+k}P_{-3+k}, \hskip 30 pt$$ and also the flags $$F_{-6+k}=(P_{-7+k},L_{-5+k}), \hskip 15 pt
F_{-4+k}=(P_{-3+k},L_{-5+k})$$
We associate [*corner invariants*]{} to the flags, as follows. $$c(F_{0+k})=[P_{-7+k},P_{-3+k},L_{-5+k}L_{3+k},L_{-5+k}L_{7+k}],$$ $$c(F_{2+k})=[P_{9+k},P_{5+k},L_{7+k}L_{-1+k},L_{7+k}L_{-5+k}],$$ All these equations are meant for $k=0,4,8,12,....$ Finally, we define
$$x_k=c(F_{2k}); \hskip 30 pt k=0,1,2,3...$$
The quantities $x_0,x_1,x_2,...$ are known as the corner invariants of the ray. From the exposition here, it would seem more natural to call these invariants [*flag invariants*]{}, though in the past we have called them [*corner invariants*]{}.
We would like to go in the other direction: Given a list $(x_0,x_1,x_2,...)$, we seek a polygonal ray which has this list as its flag invariants. Taking [@Sch1], eq. (20) and applying a suitable projective duality, we get the [*reconstruction formula*]{}
$$\label{reconstruct}
P_{9+2k}=
\begin{bmatrix}1&-1&x_0x_1 \\ 1&0&0 \\ 1&0&x_0x_1\end{bmatrix}
\begin{bmatrix}O^{3+k}_{-1} \\ O^{3+k}_{+1} \\ O^{3+k}_{+3}\end{bmatrix}
\hskip 30 pt
k=0,2,4,...$$
Multiplying through by the matrix $M^{-1}$, where $M$ is the matrix in equation , we get an alternate normalization. Setting $$\label{norm11}
Q_{-7}=\begin{bmatrix}0\\ x_0x_1\\ 1\end{bmatrix},\hskip 15 pt
Q_{-3}=\begin{bmatrix}0\\ 0\\ x_0x_1\end{bmatrix},\hskip 15 pt
Q_{1}=\begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix},\hskip 15 pt
Q_{5}=\begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}$$ we have $$\label{norm12}
Q_{9+2k}=\begin{bmatrix}O^{3+k}_{-1} \\ O^{3+k}_{+1} \\ O^{3+k}_{+3}\end{bmatrix}
\hskip 30 pt
k=0,2,4,...$$
In case we have a closed $n$-gon, we have $$\label{repeat}
\begin{bmatrix}0\\0\\1\end{bmatrix}=[Q_{-3}]=[Q_{4n-3}]=[Q_{9+2(2n-6)}]=
\begin{bmatrix}O^{2n-3}_{-1}\\ O^{2n-3}_{+1} \\ O^{2n-3}_{+3}\end{bmatrix}.$$ Here $[\cdot ]$ denotes the equivalence class in the projective plane. Equation yields $O^{2n-3}_{-1}=O^{2n-3}_{+1}=0$. Shifting the vertex labels of our polygon by $1$ unit has the effect of shifting the flag invariants by $2$ units. Doing all cyclic shifts, we get $$\label{rel1}
O_a^b=0 \hskip 30 pt b-a=2n-4,2n-2, \hskip 30pt
a,b\ {\rm odd\/}.$$ Given a polygon $P$ with flag invariants $x_1,x_2,...$ we consider the dual polygon $P^*$. The polygon $P^*$ is such that a projective duality carries the lines extending the edges of $P^*$ to the points of $P$, and [*vice versa*]{}. When suitably labeled, the flag invariants of $P^*$ are $x_2,x_3,\ldots$. For this reason, equation also holds when both $a$ and $b$ are even. In particular, we have the $2n$ relations: $$\label{rel2}
O_a^b=0, \hskip 30 pt b-a=2n-4.$$ Equation tells us that $O_{-1}^{2n-5}=0$. But now our basic recurrence relation gives $$x_{2n-7}x_{2n-8}x_{2n-9}O_{-1}^{2n-11}-x_{2n-7}O_{-1}^{2n-9}+O_{-1}^{2n-7}=0.$$ Note that $x_0$ does not occur in this equation. Solving for $x_{2n-7}$, we get $$x_{2n-7}=R_n(x_1,...,x_{2n-8}),$$ where $R_n$ is the expression that occurs in the map $\Phi$ above. Thanks to equation , these equations hold when we shift the indices cyclically by any amount. Thus $$x_{2n-7+k}=R_n(x_{k+1},...,x_{2n-8+k}), \hskip 30 pt k=1,...,2n.$$ This is an explanation of why $\Phi^{2n}$ is the identity.
Relation between the spaces ${{\mathcal E}}_{w+1,n},{{\mathcal F}}_{w+1,n}$ and ${{\mathcal C}}_{k+1,n}$ {#TriThmS}
========================================================================================================
In this section, we give more details about the relations between the main spaces studied in this paper and complete the proof of Theorem \[TriThm\].
Proof of Theorem \[TriThm\], Part (i) {#TriThmP1S}
-------------------------------------
Let us prove that the map ${\mathcal E}_{k+1,n}\longrightarrow{\mathcal F}_{k+1,n}$ constructed in Section \[1stIso\] is indeed an isomorphism of algebraic varieties.
[**A**]{}. Let us first prove that this map is a bijection. By the (anti)periodicity assumption (\[APeriod\]), after $n-k-2$ numbers on each North-East diagonal, there appears 1, followed by $k$ zeros. Thus we indeed obtain an array of numbers bounded by a row of ones and $k$ rows of zeros.
Consider the determinants $D_{i,j}$ defined by (\[DEq\]) for $j\geq{}i-1$. We show that $D_{i,j}=1$ by induction on $j$, assuming $i$ is fixed. By construction, $k+1$ consecutive North-East diagonals of the frieze give a sequence $(V_j)_{j\in {{\mathbb Z}}}$ of vectors in ${{\mathbb R}}^{k+1}$ satisfying the difference equation , where $$V_j=
\begin{pmatrix}
d_{i,j}\\
d_{i+1,j}\\
\vdots\\
d_{i+k,j}
\end{pmatrix}.$$ One has $D_{i,j}=\left|V_j,V_{j+1}, \cdots, V_{j+k}\right|$. Now we compute the first determinant $$D_{i,i-1}=
\left\vert
\begin{array}{cccc}
1&d_{i,i}&\ldots&d_{i,i+k-1}\\[4pt]
0&1&\ldots&d_{i+1,i+k-1}\\[4pt]
\ldots& \ldots&& \ldots\\[4pt]
0&0&\ldots&1
\end{array}
\right\vert=1.$$ Since the last coefficient in the equation satisfied by the sequence of the vectors $(V_j)$ is $a_{i}^{k+1}=(-1)^{k}$, one easily sees that $D_{i,j}=D_{i,j+1}$. Therefore $D_{i,j+1}=D_{i,j}=\cdots=1$. We have proved that the array $d_{i,j}$ is an ${{\mathrm {SL}}}_{k+1}$-frieze pattern.
The defined frieze pattern is tame.
The space of solutions of the difference equation (\[REq\]) is $k+1$-dimensional, therefore every $(k+2)\times (k+2)$-block cut from $k+2$ consecutive diagonals gives a sequence of linearly dependent vectors. Hence every $(k+2)\times (k+2)$-determinant vanishes.
Conversely, consider a tame ${{\mathrm {SL}}}_{k+1}$-frieze pattern. Let $\eta_j=(\ldots,d_{i,j},d_{i+1,j},\ldots)$ be the $j$th South-East diagonal. We claim that, for every $j$, the diagonal $\eta_{j}$ is a linear combination of $\eta_{j-1},\dots,\eta_{j-k-1}$: $$\label{etarel}
\eta_{j}=a_{j}^{1} \eta_{j-1} - a_{j}^{2} \eta_{j-2} + \cdots + (-1)^{k-1} a_{j}^{k} \eta_{j-k}+(-1)^{k}a_{j}^{k+1} \eta_{j-k-1}.$$ Indeed, consider a $(k+2)\times (k+2)$-determinant whose last column is on the diagonal $\eta_{j}$. Since the determinant vanishes, the last column is a linear combination of the previous $k+1$ columns. To extend this linear relation to the whole diagonal, slide the $(k+2)\times (k+2)$-determinant in the $\eta$-direction; this yields (\[etarel\]).
Next, we claim that $a_{j}^{k+1}=1$. To see this, choose a $(k+1)\times (k+1)$-determinant whose last column is on the diagonal $\eta_{j}$. By definition of ${{\mathrm {SL}}}_{k+1}$-frieze pattern, this determinant equals 1. On the other hand, due to the relation (\[etarel\]), this determinant equals $a_{j}^{k+1}$ times a similar $(k+1)\times (k+1)$-determinant whose last column is on the diagonal $\eta_{j-1}$. The latter also equals 1, therefore $a_{j}^{k+1}=1$.
We have shown that each North-East diagonal of the ${{\mathrm {SL}}}_{k+1}$-frieze pattern consists of solutions of the linear difference equation (\[etarel\]) with $a_{j}^{k+1}=1$, that is, the difference equation (\[REq\]). By definition of an ${{\mathrm {SL}}}_{k+1}$-frieze pattern, these solutions are (anti)periodic. Hence the coefficients are periodic as well.
We proved that the map ${\mathcal E}_{k+1,n}\to{\mathcal F}_{k+1,n}$ constructed in Section \[1stIso\] is one-to-one.
[**B**]{}. Let us now show that this map is a morphism of algebraic varieties defined in Sections \[AlgVarS\] and \[SubSGr\].
Recall that the structure of algebraic variety on ${{\mathcal E}}_{k+1,n}$ is defined by polynomial equations on the coefficients resulting from the (anti)periodicity. More precisely, these relations can be written in the form: $$d_{i,i+w}=1,
\qquad
d_{i,j}=0, \quad w<j<n,$$ where $d_{i,j}$ are defined by (\[ConstEq\]) and calculated according to the formulas (\[DetEq\]) and (\[DetEqDva\]) that also make sense for $j\geq{}w$. These polynomial equations guarantee that the solutions of the equation (\[REq\]) are $n$-(anti)periodic. In other words, if $M$ is the monodromy operator of the equation (which is, as well-known, an element of the group ${{\mathrm {SL}}}_{k+1}$), then the (anti)periodicity condition means that $M=(-1)^k\mathrm{Id}$. Note that exactly $k(k+2)$ of these equations are algebraically independent, since this is the dimension of ${{\mathrm {SL}}}_{k+1}$.
The structure of algebraic variety on the space ${{\mathcal F}}_{k+1,n}$ is given by the embedding into ${{\mathrm {Gr}}}_{k+1,n}$. The Grassmannian itself is an algebraic variety defined by the Plücker relations, and the embedding ${{\mathcal F}}_{k+1,n}\subset{{\mathrm {Gr}}}_{k+1,n}$ is defined by the conditions that some of the Plücker coordinates are equal to each other.
We claim that the map ${\mathcal E}_{k+1,n}\to{\mathcal F}_{k+1,n}$, constructed in Section \[1stIso\], is a morphism of algebraic varieties (i.e., a birational map). Indeed, the coefficients $a_i^j$ of the equation (\[REq\]) are pull-backs of rational functions in Plücker coordinates, see formula . Moreover, all the determinants in these formulas are equal to $1$ when restricted to ${{\mathcal E}}_{k+1,n}$. Conversely, the Plücker coordinates restricted to ${{\mathcal F}}_{k+1,n}$ are polynomial functions in $a_i^j$. This follows from the formulas and and from the fact that the Plücker coordinates are polynomial in $d_{i,j}$.
This completes the proof of Theorem \[TriThm\], Part (i).
Second isomorphism, when $n$ and $k+1$ are coprime {#CoPSeC}
--------------------------------------------------
Let us prove that the spaces of difference equations (\[REq\]) with (anti)periodic solutions and the moduli space of $n$-gons in ${{\mathbb {RP}}}^k$ are isomorphic algebraic varieties, provided the period $n$ and the dimension $k+1$ have no common divisors.
[**A**]{}. We need to check that the map (\[MaPP\]) is, indeed, a one-to-one correspondence between ${\mathcal E}_{k+1,n}$ and ${\mathcal C}_{k+1,n}$. Let us construct the inverse map to (\[MaPP\]). Consider a non-degenerate $n$-gon $(v_i)$ in ${{\mathbb {RP}}}^k$. Choose an arbitrary lift $(\tilde V_i)\in{{\mathbb R}}^{k+1}$ of the vertices. The $k+1$ coordinates $\tilde V_i^{(1)},\ldots,\tilde V_i^{(k+1)}$ of the vertices of this $n$-gon are solutions to some (and the same) difference equation (\[REq\]) if and only if the determinant (\[detconst\]) is constant (i.e., independent of $i$). We thus wish to define a new lift $V_i=t_i \tilde V_i$ such that $$t_i t_{i+1}\cdots t_{i+k}
\left|
\tilde V_i, \tilde V_{i+1},\ldots,\tilde V_{i+k}
\right|=1.$$ This system of equations on $t_1,\ldots, t_n$ has a unique solution if and only if $n$ and $k+1$ are coprime. Finally, two projectively equivalent $n$-gons correspond to the same equation. Thus the map (\[MaPP\]) is a bijection.
[**B**]{}. The structures of algebraic varieties are in full accordance since the projection from ${{\mathrm {Gr}}}_{k+1,n}$ to ${\mathcal C}_{k+1,n}$ is given by the projection with respect to the $\mathbb{T}^{n-1}$-action which is an algebraic action of an algebraic group.
Theorem \[TriThm\], Part (ii) is proved.
Proof of Proposition \[NewProp\] {#ProProSec}
--------------------------------
Consider finally the case where $n$ and $k+1$ have common divisors. Suppose that $\gcd(n,k+1)=q \neq 1$. In this case, the constructed map is not injective and its image is a subvariety in the moduli space of polygons.
As before, we assign an $n$-gon $V_1,\ldots,V_n \in {{\mathbb R}}^{k+1}$ to a difference equation. Given numbers $t_0,\ldots,t_{q-1}$ whose product is 1, we can rescale $$V_j \mapsto t_{j\ {\rm mod}\ q}\ V_j,\quad j=1,\ldots,n,$$ keeping the determinants $\left|V_i, V_{i+1},\ldots,V_{i+k}\right|$ intact. This action of $({{{\mathbb R}}^*})^{q-1}$ does not affect the projection of the polygons to ${{\mathbb {RP}}}^k$. Thus this projection has at least $q-1$-dimensional fiber.
To find the dimension of the fiber, we need to consider the system of equations $$t_i t_{i+1}\cdots t_{i+k}=1,\quad i=1,\ldots,n,$$ where, as usual, the indices are understood cyclically mod $n$. Taking logarithms, this is equivalent to a linear system with the circulant matrix $$\left(
\begin{array}{ccccccccc}
1&1&\ldots&1&1&0&\ldots&0&0\\0&1&\ldots&1&1&1&0&\ldots&0\\[4pt]
&&\ddots&&&&\ddots\\[4pt]
0&0&\ldots&0&1&1&1&\ldots&1\\1&0&\ldots&0&0&1&1&\ldots&1\\[4pt]
&&\ddots&&&&\ddots\\[4pt]
1&1&\ldots&1&0&0&\ldots&0&1\\
\end{array}
\right)$$ with $k+1$ ones in each row and column.
The eigenvalues of such a matrix are given by the formula $$1+\omega_j+\omega_j^2 + \dots + \omega_j^k,\quad j=0,1,\ldots,n-1,$$ where $\omega_j = \exp(2\pi {\bf i} j/n)$ is $n$th root of 1, see [@Da]. If $j>0$, the latter sum equals $$\frac{\omega_j^{k+1}-1}{\omega_j -1},$$ and it equals 0 if and only if $j(k+1) = 0 \mod n$. This equation has $q-1$ solutions, hence the circulant matrix has corank $q-1$. Proposition \[NewProp\] is proved.
The ${{\mathrm {SL}}}_2$-case: relations to Teichmüller theory
--------------------------------------------------------------
Now we give a more geometric description of the image of the map (\[MaPP\]) in the case $k=1$. If $n$ is odd then this map is one-to-one, but if $n$ is even then its image has codimension 1. To describe this image, we need some basic facts from decorated Teichmüller theory [@Pen; @Pen2].
Consider ${{\mathbb {RP}}}^1$ as the circle at infinity of the hyperbolic plane. Then a polygon in ${{\mathbb {RP}}}^1$ can be thought of as an ideal polygon in the hyperbolic plane $\mathcal{H}^2$. A decoration of an ideal $n$-gon is a choice of horocycles centered at its vertices.
Choose a decoration and define the side length of the polygon as the signed hyperbolic distance between the intersection points of the respective horocycles with this side; the convention is that if the two consecutive horocycles are disjoint then the respective distance is positive (one can always assume that the horocycles are “small" enough). Denoting the side length by $\delta$, the lambda length is defined as $\lambda=\exp{(\delta/2)}$.
Let $n$ be even. Define the alternating perimeter length of an ideal $n$-gon: choose a decoration and consider the alternating sum of the side lengths. The alternating perimeter length of an ideal even-gon does not depend on the decoration: changing a horocycle adds (or subtracts) the same length to two adjacent sides of the polygon and does not change the alternating sum.
\[perim\] The image of the map (\[MaPP\]) with $k=1$ and $n$ even consists of polygons with zero alternating perimeter length.
Let $(v_i)$ be a polygon in ${{\mathbb {RP}}}^1$. Let $x_i=[v_{i-1},v_i,v_{i+1},v_{i+2}]$ be the cross-ratio of the four consecutive vertices. Of six possible definitions of cross-ratio, we use the following one: $$[t_1,t_2,t_3,t_4]=\frac{(t_1-t_3)(t_2-t_4)}{(t_1-t_2)(t_3-t_4)}.$$ This is the reciprocal of the formula in equation .
We claim that a $2n$-gon is in the image of (\[MaPP\]) if and only if $$\label{altprod}
\prod_{i\ {\rm odd}} x_i = \prod_{i\ {\rm even}} x_i.$$
Indeed, let $(V_i)\subset {{\mathbb R}}^2$ be an anti periodic solution to the discrete Hill’s equation $$V_{i+1}=c_i V_i - V_{i-1}$$ with $|V_i,V_{i+1}|=1$. Then $$x_i=\frac{|V_{i-1},V_{i+1}| |V_i,V_{i+2}|}{|V_{i-1},V_i| |V_{i+1},V_{i+2}|} = c_i c_{i+1}.$$ Therefore (\[altprod\]) holds.
Conversely, let (\[altprod\]) hold. Let $(\tilde V_i)$ be a lift of $(v_i)$ to ${{\mathbb R}}^2$ with $|\tilde V_i,\tilde V_{i+1}|>0$. As before, we want to renormalize these vectors so that the consecutive determinants equal 1. This boils down to solving the system of equations $t_i t_{i+1} |\tilde V_i,\tilde V_{i+1}|=1$. This system has a solution if and only if $$\prod_{i\ {\rm odd}} |\tilde V_i,\tilde V_{i+1}|= \prod_{i\ {\rm even}} |\tilde V_i,\tilde V_{i+1}|.$$ On the other hand, one computes that $$1=\frac{\prod_{i\ {\rm odd}} x_i}{\prod_{i\ {\rm even}} x_i} = \left(\frac{\prod_{i\ {\rm even}} |\tilde V_i,\tilde V_{i+1}|}{\prod_{i\ {\rm odd}} |\tilde V_i,\tilde V_{i+1}|}\right)^2,$$ and the desired rescaling exists.
Finally, one relates cross-ratios with lambda lengths, see [@Pen; @Pen2]: $$[v_{i-1},v_i,v_{i+1},v_{i+2}]=\frac{\lambda_{i-1,i+1} \lambda_{1,i+2}}{\lambda_{i-1,i}\lambda_{i+1,i+2}}.$$ Therefore $$1=\frac{\prod_{i\ {\rm odd}} x_i}{\prod_{i\ {\rm even}} x_i} = \left(\frac{\prod_{i\ {\rm even}} \lambda_{i,i+1}}{\prod_{i\ {\rm odd}} \lambda_{i,i+1}}\right)^2 = e^{\sum (-1)^i \delta_i}.$$ It follows that the alternating perimeter length is zero.
**Acknowledgments**. We are pleased to thank D. Leites and A. Veselov for enlightening discussions. This project originated at the Institut Mathématique de Jussieu (IMJ), Université Paris 6. S. T. is grateful to IMJ for its hospitality. S. M-G. and V. O. were partially supported by the PICS05974 “PENTAFRIZ” of CNRS. R.E.S. was supported by NSF grant DMS-1204471. S. T. was supported by NSF grant DMS-1105442.
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[^1]: Throughout this paper, the “empty” entries of matrices stand for $0$.
[^2]: Throughout the paper $\left|V_i,\ldots,V_{i+k}\right|$ stand for the determinant of the matrix with columns $V_i,\ldots,V_{i+k}$.
|
---
author:
- 'D. Burlon [^1]'
- 'G. Ghirlanda'
- 'G. Ghisellini'
- 'J. Greiner'
- 'A. Celotti'
date: 'Received .. ... .. / Accepted .. ... ..'
title: Time resolved spectral behavior of bright BATSE precursors
---
[To this aim we compare the spectral evolution of the precursor with that of the main GRB event. We also study if and how the spectral parameters, and in particular the peak of the $\nu F_{\nu}$ spectrum of time resolved spectra, correlates with the flux. This allows us to test if the spectra of the precursor and of the main event belong to the same correlation (if any). We searched GRBs with precursor activity in the complete sample of 2704 bursts detected by [[ BATSE]{}]{} finding that 12% of GRBs have one or more precursors. Among these we considered the bursts with time resolved spectral analysis performed by Kaneko et al. 2006, selecting those having at least two time resolved spectra for the precursor.]{} [We find that precursors and main events have very similar spectral properties. The spectral evolution within precursors has similar trends as the spectral evolution observed in the subsequent peaks. Also the typical spectral parameters of the precursors are similar to those of the main GRB events. Moreover, in several cases we find that within the precursors the peak energy of the spectrum is correlated with the flux similarly to what happens in the main GRB event. This strongly favors models in which the precursor is due to the same fireball physics of the main emission episodes. ]{}
Introduction
============
How does a GRB behave before the onset of the main emission is a debated issue. The so–called “precursor” activity has been observationally addressed by e.g. @koshut95 \[hereafter K95\], @lazzati05 \[L05\] and @burlon \[B08\].
K95 searched in the BATSE sample for precursors defined as pulses with a peak intensity lower than that of the main GRB and separated from it by a quiescent phase at least as long as the duration of the main event. They found precursors in $\sim$3% out of a sample of GRBs detected by [[ BATSE]{}]{} up to 1994 May. Their duration appeared weakly correlated with those of the main GRBs and on average shorter than that of the burst. The spectral properties of these precursors showed no systematic difference with respect to those of the main GRB event, being both softer and harder. However, the comparison of the spectral properties of the precursors and of the main event were based on the hardness ratio which is only a proxy of the real shape of burst spectra.
L05 searched for precursors as weak events *preceding* the [[ BATSE]{}]{} trigger. He found, within a sample of 133 bright GRBs, that $\sim$20% showed precursor activity. These precursors were on average extremely dimmer than the main GRB event, and their durations are weakly correlated with that of the main event. In contrast with the results of K95, the precursors studied by L05 were softer than the main event. Also in this analysis, however, the spectral characterization of the precursors were based on the fluence hardness ratio. However, given the typically extreme low fluence of most of the precursors found by L05, a better spectral characterization (e.g. through model fits of a high resolution [[ BATSE]{}]{} spectrum) was almost impossible. A difference is how the precursor–to–burst separation is measured: K95 consider the time difference between the peak of the precursor and that of the main event, while L05 measure the precursor–to–main event separation from the end of the precursor to the start of the GRB.
B08 searched for precursors in the sample of 105 [[*Swift*]{}]{} GRBs with measured redshifts. In $\sim$15% of the sample a precursor was found. The definition of precursors adopted in B08 is similar to that used by K95. The main difference, however, is that B08 did not require that the precursor precedes the main event by an amount of time comparable to the duration of the main event. The novelty of B08 was to search and study precursors found in a sample of bursts with known redshifts. This allowed, for the first time, to characterize the precursor energetics and to study how they compare with the main event energetics, also as a function of the rest–frame time separation between the precursors and the main events. The results of B08 suggest that precursors’ spectra are consistent with those of the main event. Moreover, regardless of the rest frame duration of the quiescence (i.e. the time interval separating the precursor and the burst), precursors carry a significant fraction of the total energy ($\approx$30%) of the main event (see Fig. 1 therein). The conclusions of B08 point to a common origin for both precursor and main event. Namely, they are nothing but two episodes of the same emission process.
Theoretical models for precursors can be separated into three classes: the “fireball precursor” models [@li07; @lyutikov03; @meszaros00; @daigne02; @ruffini01]; the “progenitor precursor” models [@ramirezruiz02; @lazzati05b] and the “two step engine” model ([@wang07] \[W07\], [@lipunova09] \[L09\]). In the first class the precursor is associated to the initially trapped fireball radiation being released when transparency is reached. In the second class, based on the collapsar scenario, the precursor is identified with the interaction of a weakly relativistic jet with the stellar envelope. A strong terminal shock breaking out of the envelope is expected to produce transient emission. In both classes of models the precursors emission is predicted to be thermal, characterized by a black–body spectrum. As for the third class in W07 the progenitor collapse leads to the formation of a neutron star whose emission would be responsible for the precursor, while the star shrinks; subsequent accretion onto the neutron star causes its collapse onto a black hole, originating the GRB prompt. Conversely, in L09 the precursor is produced when a collapsing “spinar” halts at the centrifugal barrier, whereas the main emission is due to a spin–down mechanism. Thus, in L09 accretion is not invoked in either steps.
One of the main limitations of K95 and L05 analyses is the poor spectral characterization of precursors. They used the hardness ratio HR, i.e. the ratio of the counts (or fluences reported in the [[ BATSE]{}]{} catalogue) measured over broad energy channels. However, it is well established that the broad band spectra of GRBs can be fitted by empirical models (e.g. Band et al. 1993) composed by low and high spectral power–laws with different slopes. The HR is only a proxy of the real spectral properties of GRB spectra (e.g. [@ghirla09]), in particular for GRBs with vastly different [$E_{\rm peak}$]{}. The other main limitation of these studies, based on the [[ BATSE]{}]{} GRB catalogue, is the lack of redshifts. Indeed, this motivated the study of B08 of [[*Swift*]{}]{} GRBs with precursors of known redshifts. Nonetheless, the spectral analysis of B08 of [[*Swift*]{}]{}-BAT spectra was limited by the narrow spectral range (15–150keV): most [[*Swift*]{}]{} spectra of precursors could be fitted by a single power–law (i.e. the peak energy of the $\nu F_{\nu}$ spectrum is unknown) and in all cases no time resolved spectral analysis of the precursor could be performed.
The latter point is particularly important: the information carried by the strong spectral evolution of GRB spectra (e.g. [@ryde05]; [@ghirla02], [@kaneko06] \[K06\]) is completely averaged out when time integrated spectra are considered (integrated over the duration of the burst or over the duration of single emission episodes, like the precursor and the main event in B08). An interesting feature found by time resolved analysis of GRB spectra is that there could be a positive trend between the spectral peak energy [$E_{\rm peak}$]{} and the flux $P$ within single emission episodes of GRBs [@liang04] \[L04\]. Interestingly, this trend appears similar [@firmani09] \[F09\] to that found between the rest frame GRB peak energies and their isotropic equivalent luminosities, when considering different GRBs with measured $z$ (i.e. so called “Yonetoku" correlation; [@yonetoku04]).
For these reasons we consider, in this paper, a still unanswered question: how does the spectrum of the precursor evolve and how does it compare with the evolution of the associated main event? In order to answer this question we compare the time evolution of the spectral parameters of precursors and main events. We also want to test if a possible correlation between the peak energy and the flux, i.e. [$E^{\rm obs}_{\rm peak}$–P]{} within the precursors exists. If this correlation is due to the physics of the emission process or to that of the central engine is still to be understood, but if the precursors and the main event do follow a similar correlation, this would be another piece of the puzzle (in addition to the results of B08) suggesting that precursors are nothing else than the first emission episodes of the GRB. To this aims spectral data with high time and spectral resolution are necessary. [[ BATSE]{}]{} provides the best data for this purpose.
The paper is organized as follows: in Sec. 2 we describe the sample selection and global properties; in Sec. 3 we present the spectral comparison between the precursor and the main event within single GRBs and we draw our conclusions in Sec. 4.
The sample
==========
The Compton Gamma Ray Observatory satellite ([*[CGRO]{}*]{}) had on board the Burst Alert and Transient Source Experiment ([[ BATSE]{}]{}, [@fishman89]), which provided the largest sample of GRBs, detected during the 9 yr lifetime. By applying different precursor definitions, K95 and L05 searched for [[ BATSE]{}]{} bursts showing a precursor activity. A common feature of these studies is that a precursor is a peak separated (i.e. preceding) by a time interval and with a lower count rate with respect to the main GRB event. The definition of a precursor is somewhat subjective and can easily bias the sample. L05, by excluding precursors that triggered [[ BATSE]{}]{} selected the faintest precursors. K95 instead is missing precursors which can be closer than the duration of the rest of the burst. For these reasons, consistently with the definition adopted in B08, we adopted a definition of “precursor” as any peak with a peak flux smaller than the main prompt that follows it and that is separated from the main event by a quiescent period (namely, a time interval during which the background subtracted light curve is consistent with zero). We didn’t assume [*a priori*]{} that precursors can occur only in long GRBs (i.e. duration of the main emission episode be $>2$ sec in the observer’s frame), albeit in B08 we found no short burst with a precursor. We adopted this “loose” definition in order to check, a posteriori, if distinguishing characteristics emerge in the analysis. This definition is subject to find more easily precursors events of the type of K95 in the [[ BATSE]{}]{} sample. Since K95 limited the search to half of the [[ BATSE]{}]{} sample (considering events between 910405 and 940529) and due to the slightly different precursor definition, we searched for precursors in the complete [[ BATSE]{}]{} sample.
The final BATSE GRB sample[^2] contains 2704 GRBs. We found 2121 GRBs out of 2704 total triggers for which there was a 64 ms binned light curve[^3] available. We inspected the background subtracted light curve of each GRB and found 264 GRBs (12.5%) with a precursor. The majority (191) of GRBs showed one precursor, 48 showed double precursors, 19 showed three precursors, 5 showed four precursors and in only one case we found five precursors, according to our definition.
Sample properties
-----------------
From the 64 ms [[ BATSE]{}]{} light curves we calculated the duration of the precursor and main emission event for each of the 264 GRBs with precursors. The duration was defined as in the [[ BATSE]{}]{} GRB catalogue, i.e. T90. This corresponds to an integral measure, being the time interval containing the 90% (from 5% to 95%) of the counts inside each peak considered, either precursor or main event.
We define the time delay between the precursor and the main event as the difference between the beginning of the main event and the end time of the precursor. The mean durations of precursors and main emission episodes are $\sim$15 s and $\sim$24 s respectively. The mean duration of the delays is $\sim$50 s.
In Fig. \[del\_dt\] we show the delays of the precursors versus the duration T90 of the main GRB for the 264 GRB with precursors. The probability of a chance correlation among the duration of the GRBs with a single precursor (open circles and filled star symbols in Fig.\[del\_dt\]) and the corresponding delay is $3.53 \times
10^{-14}$. An even lower chance probability is found including also GRBs with multiple precursors.
Since we do not know the redshift of these GRBs, we cannot exclude that the correlation is at least in part the result of the common redshift dependence of both the delay and the T90. Moreover, Fig. \[del\_dt\] shows no apparent difference between GRBs with single or multiple precursors. This result is somewhat different from that reported by @ramirezruiz01a. By investigating the temporal properties of multi–peaked GRBs (but note that they put no particular emphasis on precursors) they found a strong one–to–one correlation (4$\sigma$ consistency) between the duration of a peak and the duration of the quiescence time interval before it.
In Fig. \[del\_rat\] we show the ratio of the total counts (integrated over T90) of each precursor with respect to the counts in the corresponding main GRB plotted as a function of the delay time. In most cases the precursor total counts are a fraction (of the order 10–20%) of the counts of the main GRB. Also in this case we do not find any difference between single precursors and multiple ones. Not surprisingly, a handful of GRBs show a precursor stronger than the main emission. In these cases, typically the precursor has a duration much larger than that of the main which over-compesates its lower peak flux, thus giving a higher integral count number for the precursor with respect to the main event.
Fig. \[fluence\] shows the total counts of the precursors with respect to the total counts in the main GRBs. In this plane different selection cuts are evident. The selection criterion for defining precursors in this work is evident as the lack of precursors to the left of the equality line (solid).
It is apparent from Fig. \[del\_rat\] and Fig. \[fluence\] that neither the delay times of the precursors with respect to the onset of the main event, nor the integrated counts of the peaks seem to show a specific clustering. Therefore, we can rule out the existence of a sub-class of “real” precursors among the complete sample, given the selection method.
Spectral evolution
==================
In order to study the spectral evolution of the precursors and compare it with that of the main event, we rely on the time resolved spectral catalogue of @kaneko06. K06 analyzed the spectra of selected bright [[ BATSE]{}]{} GRBs. These were selected to have a peak photon flux (on the 256 ms time scale and integrated in the 50–300 keV) greater than 10 photons cm$^{-2}$ s$^{-1}$ or a total energy fluence greater than $2.0\times 10^{-5}$ erg cm$^{-2}$ in the energy range $\sim$20–2000 keV. This mixed criterion ensured K06 to have a minimum number of time resolved spectra distributed within the duration of each GRB so to study the features of its spectral evolution with sufficient details. This led to a sample of 350 GRB. For most GRBs the high energy resolution data of the LAD detectors were analyzed. These data consists of $\sim$ 128 energy channels distributed between $\sim 30$ keV and 2 MeV accumulated during the burst with a minimum time resolution of 128 ms. In some cases also lower energy resolution data (MER) were analyzed. K06 fitted both the time integrated spectra and the time resolved spectra with 5 different spectral models: a simple power–law (PWR), the Band model [@band93] (BAND), a Band model with fixed high energy power law component $\beta$ (BETA), a power–law with an exponential cutoff at high energies (COMP), or a smoothly broken power–law (SBPL). The spectra within a single GRB were accumulated in time according to a minimum S/N ratio (required to be larger than 45 in each time resolved spectrum, integrated over the energy range 30 and 2000 keV). In the final catalogue of K06 the best fit parameters for all the fitted models are given for all the time resolved spectra within a single burst. Through this large data set, it is possible to construct the time evolution of the spectral parameters of the bursts.
We cross–checked the sample of K06 with the 264 GRB with precursors that we have found in the [[ BATSE]{}]{} catalogue. We found 51 GRBs with precursors with time resolved analysis reported in the K06 sample. However, since our aim is to characterize how the spectrum of the precursor evolves in time, we restricted this sample to those GRBs with at least 2 time resolved spectra analyzed by K06 in the time interval of the precursor. This condition reduces the sample to 18 GRBs. All these have a single precursor in their light curve (except for trigger \#6472, that has two precursors). In Figs. \[del\_dt\], \[del\_rat\], and \[fluence\] these 18 events are shown (star symbols): they correspond to the bright end of the distribution of count fluence of the precursors.
For these 18 GRBs with time resolved spectral analysis reported in K06 we show (panel (a) of Fig. \[2156\], Fig. \[7688fg1\] and following even figures) the light curve in counts (and in physical units as obtained by the spectral analysis) and the time evolution of the best fit parameters. It has been shown that when analyzing time resolved [[ BATSE]{}]{} spectra, especially for S/N ${ \lower .75ex\hbox{$\sim$} \llap{\raise .27ex \hbox{$<$}} }$ 80 (e.g. K06), the best fit model is often a cutoff power–law. This might be due to the difficulty of constraining the best fit parameter of the BAND model (i.e. the high energy spectral index of the power–law) when the fluence of the spectrum is low (as systematically expected in time resolved spectra with respect to time integrated ones). For this reason we decided to plot for all the 18 GRBs the spectral results given by K06 of the fit with the COMP model. In some cases this is not the best fit model of the time resolved spectra but for the aims of the present analysis, i.e. the [*relative*]{} comparison of the spectral evolution of precursors with respect to that of main bursts, any systematic effect due to the fit of the spectra with the COMP model is not affecting our conclusions. We show in Fig. \[2156\] that both the photon spectral index and [$E^{\rm obs}_{\rm peak}$]{} follow a strong soft–to–hard evolution in the rising part of the precursor, and vice-versa in the descending part. In the main emission event both spectral parameters show a general hard–to–soft trend, but inside each peak they *both* follow the same trend shown inside the single peak of the precursor and moreover they track the flux.
The latter consideration is shown in panel (b) of Fig. \[2156\] (see for comparison the lower panel of Fig. \[7688fg2\] and following odd figures) where a correlation between the peak energy [$E^{\rm obs}_{\rm peak}$]{} and the flux P is apparent. Note however that GRB 930201 is the case with best statistics and hence does not necessarily stand for a general behavior. We connected (dashed line) the evolution of the spectral parameters only inside the precursor. The colour code is as in panel (a): namely, the first (last) spectrum is the black (red) one. It has been recently pointed out (e.g. [@borgonovo01], [@liang04], and more recently by [@firmani09] for [[*Swift*]{}]{} GRBs), that when considering the spectral evolution of long GRBs there is a trend between the evolution of the flux $P$ and the peak energy [$E^{\rm obs}_{\rm peak}$]{}i.e. approximately $P \propto {\rm E}_{\rm{peak, obs}}^{\gamma}$. In particular Firmani et al. (2009) show that 84% of the K06 sample have $\gamma \sim 2$ at the 3$\sigma$ level. In addition, the correlation is not biased systematically by the value of $P$, though its uncertainty increases with decreasing flux. We can fiducially extrapolate this evidence to precursors, keeping open the question of identifying the hidden physical mechanism that determines the value of $\gamma$.
Intriguingly, this is similar to the correlation between the peak luminosity and the peak energy (time integrated over the duration of the burst) in GRBs with measured redshifts (so called “Yonetoku” correlation). A similar result was reached by Liang et al. (2004) based on the spectral evolution of the brightest [[ BATSE]{}]{} GRBs but for which no redshift was measured. Again, when studying the correlation between the luminosity and the peak energy within the few GRBs detected by [[ BATSE]{}]{} and with known $z$, @firmani09 finds that the correlation is present. The existence of a correlation within a single GRB similar to the Yonetoku correlation could be indicative of a physical origin for the quadratic link between the flux and the peak energy.
We can test if and how such a correlation holds in the GRBs with precursors that we have considered and/or if the [$E^{\rm obs}_{\rm peak}$]{} and $P$ of the precursor are consistent with the correlation defined by the prompt. If this correlation is due to the physics of the emission process or to that of the central engine is still to be understood, but if the precursors and the main event do follow a similar correlation, this would be another piece of the puzzle suggesting that precursors are nothing else than the first emission episodes of the GRB.
Discussion
==========
-0.5 cm
Figs. \[correla1\] shows the photon spectral indices $\alpha$ versus the peak energy [$E^{\rm obs}_{\rm peak}$]{}for all 51 GRBs with precursor present in K06, while Fig. \[correla2\] shows for the same bursts how $\alpha$ and [$E^{\rm obs}_{\rm peak}$]{} behave with the flux $P$. Different symbols (and colors, in the electronic edition) marks the precursor and the main event points. Filled symbols correspond to the 18 GRBs with at least two spectra for the precursors. Red triangles mark the remaining precursors in K06 with just one spectrum. Empty black dots correspond to the spectral parameters of the main events.
Fig. \[correla1\] shows that on average precursors and main GRB emission episodes span the same parameter space, while Fig. \[correla2\] shows that they follow similar correlations with the flux.
The distributions of the low energy photon indices $\alpha$ of the precursors and the main events are roughly consistent (the Kolmogorov-Smirnov KS null hypothesis probability is $\simeq 10^{-2}$). Fitting the two distributions (see Fig. \[distrib\], upper panel) with gaussian profiles we find $\langle \alpha_{\rm prec}\rangle = -1.03\pm 0.27$ and $\langle \alpha_{\rm main}\rangle = -0.94 \pm 0.34$.
Three (\#5486, \#6472, \#7343) of the 18 GRBs studied here present extremely hard spectra. One of them, i.e. GRB 960605 (\#5486, see Fig. \[5486fg2\]), could even be consistent with a black–body spectrum at the very beginning of the precursor. These few cases populate the upper part of Figs. \[correla1\] and \[correla2\] (upper panel). We have re-extracted the LAD data for this burst and reanalyzed them. We confirm the findings of K06. The finding of a precursor with a spectrum consistent with a black–body should not be taken as a proof of a radical difference with the main event, since it has been already pointed out (e.g., [@ghirla03]) that a non–negligible fraction of GRB ($\sim$5%) start their emission with a black body spectrum.
Comparing the distributions of $\log$([$E^{\rm obs}_{\rm peak}$]{}) we find that they are somewhat different (K–S null hypothesis probability $\sim 10^{-4}$). Fitting again with gaussian profiles the two distributions in Fig. \[distrib\] (lower panel) we find the mean value and 1$\sigma$ scatter for precursors: $\log$([$E^{\rm obs}_{\rm peak}$]{}) = 2.49$\pm0.35$ to be compared to $\log$([$E^{\rm obs}_{\rm peak}$]{}) = 2.60$\pm 0.24$ for the main emission events. The distribution of [$E^{\rm obs}_{\rm peak}$]{} for the precursors is slightly softer than the one of the main prompt emission. This result is not surprising when looking at the bottom panel of Fig. \[correla2\]: the peak energy of precursors seem to follow the trend (when [$E^{\rm obs}_{\rm peak}$]{} is plotted with respect to flux) drawn by the GRB main emission, but at the lower left end of the track.
In the 7 precursors with more time resolved spectra (\#2156, \#7688, \#5486, \#6472, \#3481, \#3241, \#1676), [$E^{\rm obs}_{\rm peak}$]{} shows a strong evolution but nonetheless is always consistent with the correlation drawn by the main event (as shown in Fig. \[2156\]–b (see lower panel in Figs. \[7688fg2\], \[5486fg2\], \[6472fg2\], \[3481fg2\], \[3241fg2\], \[1676fg2\]). Note that these similar trends in the evolution of [$E^{\rm obs}_{\rm peak}$]{} do not depend upon the delay, as these vary among $\sim$9 s (for \#1676) and $\sim$75 s (for \#7688). Note that at odds with B08, this consideration is based only on observed time intervals, because the redshift $z$ is unknown for all GRBs in this work. Two of them, namely \#2156 and \#1676, also show consistent evolution in $\alpha$ between the precursor and the main event (see upper panels of Figs. \[2156\]–b and \[1676fg2\]). The other 5 GRBs (of this group of 7) show an evolution in $\alpha$ which is different in the precursor and in the main event: in two cases (\#5486 and \#6472) $\alpha$ starts extremely hard and evolves to softer values (see upper panel of Figs. \[5486fg2\] and \[6472fg2\]). In the last three cases (\#3481, \#3241 and \#7688) either the photon spectral index evolves in a different way with respect to the one of the main emission episode (as in Figs. \[3241fg2\] and \[7688fg2\], upper panels), or it lies in a different region of the parameter space (see upper panel of Fig. \[3481fg2\]).
The remaining 11 GRBs of our sample have more coarsely sampled precursor spectra. The trend of [$E^{\rm obs}_{\rm peak}$]{} of the precursor is consistent with that of the main event in 8 cases. In \#3253, \#6454, \#3057, \#4368, \#1157, \#6629, \#3301, \#7343 (see upper panels of Figs. \[3253fg2\], \[6454fg2\], \[3057fg2\], \[4368fg2\], \[1157fg2\], \[6629fg2\], \[3301fg2\], \[7343fg2\]) the peak energy in the spectra of precursors follow the same correlation with the flux drawn by the main emission. Notwithstanding, the number of spectra extracted by K06 in the precursor varies between five and two, thus preventing any more confident claim. Among these 8 GRBs, in the latter 3 the photon spectral indices $\alpha$ of the precursors do not track the trend drawn by the main emission event (see upper panels of Figs. \[6629fg2\], \[3301fg2\], and \[7343fg2\]), being always softer (with the exception of the onset of the precursor in \#7343, which has $\alpha \simeq 0$). In the former badly sampled 5 GRBs, also $\alpha$ is consistent with the trend described by the spectra of the main impulse. Note that also in these 8 cases the delay does not represent a distinguishing feature, as it can vary from 7 s (e.g. \#3253) up to $> 100$ s (\#3663).
The last three GRBs, namely \#3663, \#2700, and \#3448 present hardly distinguishable spectral characteristics (i.e., both $\alpha$ and [$E^{\rm obs}_{\rm peak}$]{}). This is due either to the extremely low number of spectra extracted in the precursor, or in the main impulse, or both at the same time (see Figs. \[3663fg2\], \[2700fg2\], \[3448fg2\]). In our opinion this prevents any further claim.
Conclusion
==========
In this work we presented, for the first time, a time resolved spectral analysis of bright precursors based on spectral parameters, namely the photon spectral indices $\alpha$ and the observed peak energy [$E^{\rm obs}_{\rm peak}$]{}. This was done by using High Energy Resolution spectra extracted by K06 in a sample of 350 bright GRBs out of the complete sample of 2704 confirmed GRBs observed by the [[ BATSE]{}]{} instrument. Of the 51 bursts with precursor present in K06, we selected a sample of 18 GRBs having at least two time resolved spectra of the precursor.
The comparison with the main emission episode has three outcomes. The first is that the photon spectral indices of precursors and main events are consistent, while the peak energies of the precursors are mildly softer (see Fig. \[distrib\]). Secondly, both $\alpha$ and [$E^{\rm obs}_{\rm peak}$]{} do show an evolution (extreme in a handful of cases) that defines a relation between the flux $P$ and the spectral parameters (note that the $P-E_{\rm{peak,obs}}^{\gamma}$ correlation was recently reported (e.g. F09) regardless the presence of precursors). Finally we showed that delays do not represent a distinguishing feature in the trend of $\alpha$ or [$E^{\rm obs}_{\rm peak}$]{}.
We found one GRB (out of 18) in which the onset of the emission of the precursor is consistent with black–body emission (i.e., \#5486 – see Fig. \[5486fg2\]). This was expected, since @ghirla03 showed that 5% of [[ BATSE]{}]{} GRBs show extremely hard emission at the onset of the first impulse.
Moreover, comparing the integrated counts in the peaks of precursors with respect to the ones of the main impulses, we confirmed the results of B08 (see Fig. \[fluence\]). Indeed precursors carry a significant fraction of the energy of the main emission episode, regardless the duration of the time interval of quiescence.
These results, in addition to B08, point strongly to the conclusion that the onset of emission of GRBs (called precursor), even if separated from the main emission episode by hundreds of seconds (in the observers frame), is indistinguishable from that of the main event. Moreover the delay remains a puzzling issue. This suggests that we should reconsider the idea of what a precursor is. Since our result is partially in contrast with L05 we cannot rule out the possibility that “real precursors” belong to another class of very dim pulses of different origin. Nonetheless, both kind of precursors can show very long delays, thus tackling any theoretical model for GRB prompt emission.
We acknowledge Marco Nardini and Lara Nava for stimulating discussion. D.B. is supported through DLR 50 OR 0405. This research was partially supported by ASI-INAF I/088/06/0 and MIUR. We acknowledge the use of public data from the [[ BATSE]{}]{} data archive. D.B. thanks the OAB for the kind ospitality during the completion of this work.
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[^1]: burlon$@$mpe.mpg.de
[^2]: http://heasarc.gsfc.nasa.gov/docs/cgro/batse/BATSE\_Ctlg/basic.html
[^3]: http://\[...\]/batse/batseburst/sixtyfour\_ms/index.html
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---
abstract: 'Polar molecules in selected quantum states can be guided, decelerated, and trapped using electric fields created by microstructured electrodes on a chip. Here we explore how non-adiabatic transitions between levels in which the molecules are trapped and levels in which the molecules are not trapped can be suppressed. We use $^{12}$CO and $^{13}$CO ($a\, ^3\Pi_1, v=0$) molecules, prepared in the upper $\Lambda$-doublet component of the $J=1$ rotational level, and study the trap loss as a function of an offset magnetic field. The experimentally observed suppression (enhancement) of the non-adiabatic transitions for $^{12}$CO ($^{13}$CO) with increasing magnetic field is quantitatively explained.'
author:
- 'Samuel A. Meek'
- Gabriele Santambrogio
- 'Boris G. Sartakov'
- Horst Conrad
- Gerard Meijer
title: 'Suppression of non-adiabatic losses of molecules from chip-based microtraps'
---
Introduction
============
The manipulation and control of polar molecules above a chip using electric fields produced by microstructured electrodes on the chip surface is a fascinating new research field [@meek09a]. Miniaturization of the electric field structures enables the creation of large field gradients, i.e., large forces and tight potential wells for polar molecules. A fundamental assumption that is made when considering the force imposed on the molecules is that their potential energy only depends on the electric field strength. This is usually a good assumption, since the molecules will reorient themselves and follow the new quantization axis when the field changes direction and their potential energy will change smoothly when the strength of the field changes. This approximation can break down, however, when the quantum state that is used for manipulation couples to another quantum state that is very close in energy. If the energy of the quantum state changes at a rate that is fast compared to the energetic splitting, transitions between these states are likely to occur. For trapped molecules in so-called low-field-seeking states in a static electric potential, such transitions are particularly disastrous when they end up in high-field-seeking states or in states that are only weakly influenced by the electric fields, as this results in a loss of the molecules from the trap. This effect has been investigated previously for ammonia molecules in a Ioffe-Pritchard type electrostatic trap with a variable field minimum. In this macroscopic electrostatic trap, losses due to non-adiabatic transitions were observed on a second time scale when the electric field at the center of the trap was zero; with a non-zero electric field minimum at the center of the trap, these losses could be avoided [@kirste09]. Non-adiabatic transitions have recently also been investigated in a “conventional”, i.e. macroscopic, Stark decelerator in which electric fields are rapidly switched between two different configurations. There, these transitions have been found to lead to significant losses of molecules when they are in low electric fields [@wall10]. Similar trap losses will be much more pronounced on a microchip, where the length scales are much shorter and where the electric field vectors change much faster. For atoms in a three-dimensional magnetic quadrupole trap, the trap losses due to spin flip (or Majorana) transitions has been shown to be inversely proportional to the square of the diameter of the atom cloud [@petrich95]. On atom-chips, where paramagnetic atoms are manipulated above a surface using magnetic fields produced by current carrying wires, trap losses due to Majorana transitions are therefore well-known but can be conveniently prevented by using an offset magnetic field [@fortagh07]. Due to the geometry of the molecule chip, however, applying a static offset electric field is not possible, and other solutions must be sought.
We have recently demonstrated that metastable CO molecules, laser-prepared in the upper $\Lambda$-doublet component of the $J=1$ level of the $a\, ^3\Pi_1, v=0$ state can be guided, decelerated and trapped on a chip. In these experiments, non-adiabatic losses have been observed for $^{12}$C$^{16}$O. In this most abundant carbon monoxide isotopologue, the level that is low-field-seeking becomes degenerate with a level that is only weakly influenced by an electric field when the electric field strength goes to zero. Every time that the trapped molecules pass near the zero field region at the center of a micro-trap, they can make a transition between these levels and thereby be lost from the trap. This degeneracy is lifted in $^{13}$C$^{16}$O due to the hyperfine splitting (the $^{13}$C nucleus has a nuclear spin $|\vec I| = 1/2$), and the low-field-seeking levels never come closer than 50 MHz to the non-trappable levels. Therefore, changing from $^{12}$C$^{16}$O to $^{13}$C$^{16}$O (referred to as $^{12}$CO and $^{13}$CO from now on) in the experiment greatly improves the efficiency with which the molecules can be guided and decelerated over the surface and enables trapping of the latter molecules in stationary traps on the chip [@meek09a].
Although it is evident that the 50 MHz splitting between the low-field-seeking and non-trapped levels in $^{13}$CO is beneficial, it is not *a priori* clear whether a smaller splitting would already be sufficient or if a still larger splitting would actually be needed to prevent all losses. While the hyperfine splitting in $^{13}$CO cannot be varied, the degeneracy can be lifted by a variable amount in the normal $^{12}$CO isotopologue by using a magnetic field. If a magnetic field is applied in addition to the electric field, a splitting can be induced between the low-field-seeking and non-trappable levels of $^{12}$CO that depends on the strength of the applied magnetic field; in $^{13}$CO, a magnetic field will actually decrease the splitting between the low-field-seeking and non-trappable levels.
In this paper, we present measurements of the efficiency with which CO molecules are transported over the chip — while they are confined in electric field minima that are traveling at a constant velocity — as a function of magnetic field strength. It is observed that, in the case of $^{12}$CO, the losses due to non-adiabatic transitions can be completely suppressed in sufficiently high magnetic fields; for $^{13}$CO on the other hand, the trap losses increase with increasing magnetic field. A theoretical model that can quantitatively explain these observations is also presented. Although the chip is ultimately used to decelerate molecules to a standstill, measuring the efficiency with which molecules are guided at a constant velocity provides a detailed insight into the underlying trap-loss mechanism. Limiting the experiments to constant velocity guiding also makes them more tractable: bringing molecules to a standstill and subsequently detecting them has thus far required five separate phases of acceleration, which greatly complicates efforts to understand the details of the loss mechanism using numerical calculations [@meek09a]. While deceleration at a constant rate to a non-zero final velocity is possible, the measurable signal in such experiments is significantly lower than in constant velocity guiding. Despite the fact that we only measure at constant velocity, we nonetheless apply the model to examine the non-adiabatic losses expected during deceleration. This work thereby furthers the goal of extending trapping on the molecule chip to a wider range of molecules.
Experimental setup {#expsetup}
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A scheme of the experimental setup is shown in Figure \[beamline\]. A mixture of 20% CO in krypton is expanded into vacuum from a pulsed valve (General Valve, Series 99), cooled to a temperature of 140 K. In this way, a molecular beam with a mean velocity of 300 m/s and with a full-width-half-maximum spread of the velocity distribution of approximately 50 m/s is produced. This beam passes through two 1 mm diameter skimmers and two differential pumping stages (the valve and first skimmer are not shown in the figure) before entering the chamber in which the molecule chip is mounted. Just in front of the second skimmer, the ground state CO molecules are excited to the upper $\Lambda$-doublet component of the $J=1$ level of the metastable $a\, ^3\Pi_1$, $v = 0$ state, using narrow-band pulsed laser radiation around 206 nm (1 mJ in a 5 ns pulse with a bandwidth of about 150 MHz). The metastable CO molecules are subsequently guided in traveling potential wells that move at a constant speed of 300 m/s parallel to the surface of the molecule chip. A uniform magnetic field is applied to the region around the chip using a pair of 30 cm diameter planar coils separated by 23 cm (not shown in the figure). The coils are oriented such that the magnetic field is parallel to the long axis of the chip electrodes, i.e. along the $z$-axis, ensuring that the magnetic field is always perpendicular to the electric field (*vide infra*). The CO molecules that have been stably transported over the chip will pass through the $50$ m high exit slit and enter the ionization detection region a short distance further downstream. There, the metastable CO molecules are resonantly excited to selected rotational levels in the $b\, ^3\Sigma^+$, $v'=0$ state using pulsed laser radiation at 283 nm (4 mJ in a 5 ns pulse with a 0.2 cm$^{-1}$ bandwidth). A second photon from the same laser ionizes the molecules and the parent ions are mass-selectively detected in a compact linear time-of-flight setup using a micro-channel plate (MCP) detector. This detection scheme has been implemented in addition to the Auger detection scheme that we have used in earlier studies [@meek09a; @meek08; @meek09] as it is more versatile and can also be applied to detect other molecules. In addition, the detection sensitivity of the ion detector is less affected by the magnetic field than that of the Auger detector.
The molecule chip and its operation principle have been described in detail before [@meek08; @meek09], and only the features that are essential for understanding of the present experiment are discussed here. The active area of the chip consists of an array of 1254 equidistant electrodes, each 10 m wide and 4 mm long, with a center-to-center distance of 40 m. An edge-on view of the chip electrodes (with the 4 mm dimension of the electrodes perpendicular to plane of the figure) is shown in Figure \[electrodes\]. The potential (in volts) applied to an electrode at a given moment in time is indicated directly above the electrode; these six potentials are repeated periodically on the electrodes on either side of those drawn here. Because the electrodes are much longer than the period length of the array, the electric field distribution can be regarded as two-dimensional, i.e. the component of the electric field along the $z$-axis can be neglected. This is of importance for the present experiments, because only in this case the applied magnetic field is always perpendicular to the electric field. The calculated contour lines of equal electric field strength in the free space above the chip show electric field minima that are separated by 120 m, i.e., there are two electric field minima per period, centered about 25 m above the surface of the chip. By applying sinusoidal waveforms with a frequency $\nu$ to the electrodes, these minima can be translated parallel to the surface with a speed given by $v = 120\, \textrm{\textmu m}\cdot \nu$. When these waveforms are perfectly harmonic and have the correct amplitude, offset, and phase, the minima move with a constant velocity at a constant height above the surface, and the shape of the field strength distribution does not change in time. Because an electric field strength minimum acts as a trap for molecules in low-field-seeking states, these fields act as tubular moving traps that can be used to guide the molecules over the surface of the chip. The tubular traps are closed at the end by the fringe fields caused by the neighboring electrodes. Near the ends of the about 4 mm long traps, the electric field will necessarily have a component along the $z$-axis, i.e. parallel to the applied magnetic field. In the present study, where the molecules are guided at 300 m/s over the chip and are therefore on the chip for less than 200 s, these end-effects are neglected.
The region near an electric field minimum at $(x,y) = (x_0,y_0)$ is a quadrupole, with an electric potential given by $$V = \frac{\alpha}{2} r^2 \cos(2 \phi - \phi_0)$$ where $x - x_0 = r \cos\phi$ and $y - y_0 = r \sin\phi$. In the current experiment, sinusoidal waveforms with an amplitude of 180 V are applied to the electrodes, yielding a value of $\alpha = 0.054\, \textrm{V}/\textrm{\textmu m}^2$ [@meek09]. The resulting electric field is given by $$\vec{E} = -\alpha r (\cos(\phi_0 - \phi) \hat{x} + \sin(\phi_0 - \phi) \hat{y}).$$ The strength of the electric field, $|\vec{E}| = \alpha r$, depends only on the $r$ coordinate, but the direction of the field vector, $\phi' = \phi_0 - \phi + \pi$, depends on the coordinate $\phi$ and on the phase factor $\phi_0$. While the direction of the field vector changes as a result of the motion of the molecule in the quadrupole field (and thus changing $\phi$), the direction of the field at any given position relative to the minimum also rotates when the minimum is translated over the chip. It is seen from the electric field vectors shown in the insets of Figure \[electrodes\] that the frequency of this rotation is 1.5 times the frequency of the applied waveforms and that the direction of the rotation is clockwise, i.e. the rotation vector points along the positive $z$-axis and $\phi_0$ increases linearly in time. To guide the molecules over the chip at 300 m/s, harmonic waveforms with a frequency $\nu$ of 2.5 MHz must be applied, resulting in a rotation frequency of 3.75 MHz.
For CO molecules in the low-field-seeking component of the $J=1$ level, the depth of the tubular traps above the chip is about 60 mK. This implies that CO molecules with a speed of up to 6 m/s relative to the center of the trap can be captured. The oscillation frequency of the molecules in the radial direction of the tubular traps is in the 100–250 kHz range. These parameters are of importance for the non-adiabatic transitions as these determine with which velocity, and how often per second the CO molecules pass by the zero-field region of the traps.
It turns out that, in the actual experiment, the tubular traps do not move perfectly smoothly over the chip. Due to imperfections in the amplitude, offset and phase of the waveforms that we have used, the tabular traps are jittering at rather high velocities. The motion of the center of the traps relative to the ideal, constant velocity motion can be calculated by measuring the real waveforms applied to the chip and using these to compute the position of the minimum for each point in time. The upper row of Figure \[jitter\] shows the motion of a minimum when the waveforms that we will refer to as the “standard waveforms” have been used. The range of this motion extends over $\pm2$ m in the $x$ and $y$ coordinates. Although this is considerably smaller than the size of the trapping region, this motion significantly enlarges the effective region in which non-adiabatic transitions can occur. Moreover, the entire path is traced out periodically every $T = 2/\nu = 800$ ns; because the motion occurs on such a short time scale, the speed with which the trap center moves, and therefore the relative speed with which the molecules encounter the trap center, can be as high as 100 m/s. To improve the waveforms, we inserted an LC filter in the output stage of the amplifiers, thereby reducing the harmonic distortion. The resulting motion using these “improved waveforms” is shown in the lower row of Figure \[jitter\]. It is seen that not only the range of the motion is now contracted but that also the speed with which the trap center jitters is reduced by about a factor of two.
Theoretical Model {#theo}
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Eigenenergies in combined fields
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In order to describe the non-adiabatic transitions in CO, we must first derive the energy levels of CO in combined electric and magnetic fields. For this, the field-free Hamiltonian is expanded with Stark and Zeeman contributions, i.e., $$\hat{H}=\hat{H}_{\Lambda,\textrm{hfs}}+\hat{H}_{S}+\hat{H}_{Z}.$$ Here, $\hat{H}_{\Lambda,\textrm{hfs}}$ describes the $\Lambda$-doubling of the $a\, ^3\Pi_1$, $v=0$, $J=1$ level for either $^{12}$CO or $^{13}$CO and also includes the hyperfine splitting of each $\Lambda$ component into $F=1/2$ and $F=3/2$ hyperfine sublevels for $^{13}$CO; $\hat{H}_{S}=-\hat{\vec{\mu}}_E\cdot\vec{E}$ is the Stark interaction Hamiltonian and $\hat{H}_{Z}=-\hat{\vec{\mu}}\cdot\vec{B}$ is the Zeeman interaction Hamiltonian, where $\hat{\vec{\mu}}_E$ and $\hat{\vec{\mu}}$ are the electric and magnetic dipole moment operators, respectively, and $\vec{E}$ and $\vec{B}$ are the (time-dependent) electric and magnetic field vectors.
The spectroscopic parameters of the $a\, ^3\Pi_1$, $v=0$ state of CO that are used in the field-free Hamiltonian are given elsewhere [@meek10; @wicke72; @gammon71; @carballo88; @yamamoto88; @wada00; @field72; @warnerphd]. The $\Lambda$-doublet splitting between the positive parity component (upper) and the negative parity component (lower) of the $J=1$ level is about 400 MHz while the hyperfine splitting of each parity level of $^{13}$CO into $F=1/2$ (lower) and $F=3/2$ (upper) sublevels is about one order of magnitude smaller. The body-fixed electric dipole moment $\mu_E = |\vec{\mu}_E|$ in the electronically excited metastable state is 1.3745 Debye for both $^{12}$CO and $^{13}$CO [@wicke72; @gammon71]. The magnetic moment of the molecule can be expressed as $\hat{\vec{\mu}}=-\mu_B\,(g_L\cdot \hat{\vec{L}} +
g_S\cdot \hat{\vec{S}})$, where $\mu_B$ is the Bohr magneton, $\hat{\vec{L}}$ is the electron orbital angular momentum operator, $\hat{\vec{S}}$ is the electron spin operator, and where the magnetic $g$-factors are fixed at the values of the bare electron, $g_L =1.0$ and $g_S = 2.0023$.
A detailed description of the formalism used to calculate the eigenenergies of the various components of the $J=1$ level in the $a\, ^3\Pi_1$, $v=0$ state of both $^{12}$CO and $^{13}$CO in combined electric and magnetic fields is presented in the Appendix. The formalism has been set up for mutually orthogonal static electric and magnetic fields. As will be discussed below, this is also adequate to treat the actual situation, in which the electric field rotates with a constant frequency in a plane perpendicular to the magnetic field. In Figure \[colevel\] we only show the outcome of these calculations in the form of plots of the energy levels for the upper $\Lambda$-doublet component of $^{12}$CO and $^{13}$CO as a function of electric field strength in the absence (left column) or in the presence of a 50 Gauss magnetic field (right column). At low electric field strengths, the Stark shift is quadratic, but as the electric field strength increases and $|H_S|$ becomes much larger than $|H_{\Lambda,\textrm{hfs}}|$ and $|H_Z|$, the Stark energy shows a linear dependence on the electric field strength, i.e. $\Delta E_{S} \propto -\Omega M_J^E \mu_E E$, and the product of the projection of the electronic angular momentum along the internuclear axis $\Omega$ and the projection $M_J^E$ of the angular momentum $\hat{\vec{J}}$ on the electric field vector $\vec{E}$ becomes an approximately good quantum number. If a weak magnetic field such as that present in the experiment is applied in the absence of an electric field, each of the zero-field levels splits into $(2J + 1)$ (or in $(2F + 1)$) separate levels based on their $M_J^B$ (or $M_F^B$) quantum number. The Zeeman energy is linear in magnetic field and can be calculated using first order perturbation theory as $\Delta E_Z \approx \mu_{\textrm{eff}} M B$ where $M=M_J^B$ (or $M_F^B$) and $\mu_{\textrm{eff}}\sim \mu_B$ describes the effective magnetic moment of a particular parity and, in the case of $^{13}$CO, particular $F$ component. When a strong electric field is applied to the molecule in addition to the magnetic field, such that $|H_S| \gg |H_{\Lambda,\textrm{hfs}}|,|H_Z|$, the energy levels can again be characterized with the approximately good quantum number $\Omega M_J^E$ and show a linear Stark shift.
The behavior of the eigenenergies in static electric and magnetic fields is quite instructive in describing the non-adiabatic transitions — and thus losses — of molecules from the low-field-seeking states ($\Omega M_J^E = -1$) to non-trappable ($M_J^E = 0$) states. Near the edge of the trap, where the electric field is as high as 4.2 kV/cm, the Stark effect provides an energy gap of about $U_{\textrm{trap}}= 60\, \textrm{mK}\simeq 1.3\, \textrm{GHz}$ between low-field-seeking states and non-trappable states. Since this is much larger than both the frequency of the motion of the molecules in the traps and the frequency of the applied waveforms, non-adiabatic losses will not occur near the edge of the trap. In the vicinity of the trap center, however, this argument no longer holds, and the eigenfunctions can change from $M_J^E$-type wavefunctions to $M_J^B$-type (or $M_F^B$-type) at a rate faster than the energy gap $E_{\textrm{gap}}$ in that region. In the case of a $^{12}$CO molecule, the energy gap at the center of the trap goes to zero in the absence of a magnetic field. If a molecule in a low-field-seeking state flies near the trap center with a velocity $v$ high enough, or with a distance of closest approach $b$ small enough, that the corresponding interaction time with the trap center $\tau=b/v$ no longer fulfills the adiabaticity condition, i.e. when the condition $E_{\textrm{gap}} \gg h/\tau$ no longer holds, then the probability of transitions to non-trappable states can become significant. In the absence of a magnetic field, $^{13}$CO molecules are much safer from such non-adiabatic losses due to the energy gap of 50 MHz between the $F=3/2$ level (which becomes low-field-seeking in an electric field) and the $F=1/2$ level (which correlates with non-trappable $M_J^E=0$ states).
Calculating rates for non-adiabatic transitions
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The basic idea underlying the calculation of the non-adiabatic losses for the molecules in low-field-seeking states is that, since the transitions to non-trappable states happen primarily as the molecules pass the zero field region at the center of a microtrap, the overall loss probability can be estimated by first calculating the loss probability in a single pass. For simplicity, the trajectory of the CO molecules is assumed to have a constant velocity; this is reasonable, since the forces on the molecules approach zero at the center of the trap due to the $\Lambda$-doubling. The transition probability $P_{i,j}(v, B, b)$ of a molecule making a transition from a low-field-seeking state $i$ to a non-trappable state $j$ for a single pass by the trap center depends on the speed of the molecule relative to the center of the trap $v$, on the strength of the magnetic field $B$, and on the distance of closest approach $b$. Due to the rotation of the electric field, there is a difference between positive and negative values of $b$, which is why we do not refer to it as an “impact parameter” here.
To calculate the probability $P_{i,j}(v, B, b)$, the time-dependent Schrödinger equation, which describes the evolution of the quantum states, must be solved: $$i \hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi.
\label{tds}$$ This equation is solved numerically as an initial value problem on a set of coupled first-order ordinary differential equations using the basis vectors given in the Appendix. The Hamiltonian $\hat{H}$ depends on the electric field vector $\vec{E}$, which, for a molecule moving in the trap, is a function of both position $\vec{r}(t)$ (due to the inhomogeneous field distribution) and time $t$ (due to the rotation of the field vectors relative to the trap center), as discussed in Section \[expsetup\]. The time dependent Hamiltonian that the molecule experiences is calculated by assuming a position in time given by $r(t) = (v t - x_{\textrm{init}}) \hat{x} + b \hat{y}$, which corresponds to the molecule moving with a constant velocity $v$ and approaching the trap center with a minimum distance $b$. The variable $x_{\textrm{init}}$ must be chosen such that the initial position $r(0) = -x_{\textrm{init}} \hat{x} + b \hat{y}$ is sufficiently far from the center such that at $r(0)$ the adiabaticity condition is still satisfied.
The initial state of the molecule could be chosen as a low-field-seeking eigenstate of the instantaneous Hamiltonian at the initial position and time. If the electric field vector is rotating, however, the wavefunction of the molecule will immediately accumulate amplitude in other quantum states, even if the magnitude of the electric field is constant. The amplitude that appears in other states will only be negligible if the energy splitting between them and the initial state is much larger than the rotation frequency. Alternatively, one can choose an initial state that is a stationary state in the rotating system, as will be discussed below. The calculation of the transition probability as the molecule flies past the field minimum can then be started at lower fields, i.e. with a smaller value of $x_{\textrm{init}}$.
In the current system, the magnetic field vector is oriented along the $\hat{z}$ axis and the electric field vector lies in the $xy$ plane. If the angle of the electric field vector with respect to the $+\hat{x}$ axis (toward the $+\hat{y}$ axis) is given by $\phi'$, the Hamiltonian of the molecule can be computed by rotating the physical system around the $\hat{z}$ axis through an angle $-\phi'$, operating on it with a Hamiltonian $\hat{H}'$ which corresponds to a system with the same magnetic field vector and in which the electric field has the same magnitude but is directed along the $+\hat{x}$ axis, and then rotating the physical system back through an angle $\phi'$. In operator form, this can be written as $$\hat{H} = e^{-i \hat{F}_z \phi'} \hat{H}' e^{i \hat{F}_z \phi'}
\label{HprimeH}$$ where $\hat{F}_z$ is the total angular momentum of the molecule along the $\hat{z}$ axis. Using this form of $\hat{H}$, equation (\[tds\]) can be rewritten as $$i \hbar \frac{\partial \psi_Q}{\partial t} = \Bigl(\hat{H}' - \hat{F}_z \hbar \frac{\partial \phi'}{\partial t}\Bigr) \psi_Q
\label{qeh}$$ where $\psi_Q = e^{i \hat{F}_z \phi'} \psi$. This equation has the same form as equation (\[tds\]) and can be solved in the same way as the time independent Schrödinger equation if the magnetic field is constant, the electric field strength is constant, and the electric field vector rotates at a constant frequency. The eigenvalues that result are called “quasienergies” and $\hat{H}_Q = \hat{H}' - \hat{F}_z \hbar \frac{\partial \phi'}{\partial t}$ is the quasienergy Hamiltonian [@zeldovich67; @ritus67]. A similar approach has been used in the recent work by Wall et al. [@wall10].
In section II it was shown that the angle of the electric field in the $xy$ plane $\phi' = \phi_0 - \phi + \pi$ contains contributions from both the angular coordinate of the molecule with respect to the trap center $\phi$ and a phase that results from the constant rotation of the field vectors as the traps move over the chip $\phi_0 = 2\pi \frac{3\nu}{2}\,t$. Because $\phi_0$ increases linearly in time, the contribution of $\phi_0$ to the quasienergy Hamiltonian is time independent, having the form $-h \frac{3\nu}{2} \hat{F}_z$. This operator is diagonal in the basis sets used for both $^{12}$CO and $^{13}$CO, and in the case of $^{12}$CO, it is exactly equivalent to the Zeeman interaction for each energy eigenstate. As the rotation frequency $\frac{3\nu}{2}$ is 3.75 MHz when the molecules are guided with 300 m/s over the chip, its contribution to the quasienergy Hamiltonian is equivalent to a magnetic field of about $-8$ Gauss. It should be understood, however, that we are not dealing with a real magnetic field produced by the rotating electric field here; the effect of a rotating coordinate system merely produces a shift to the quasienergies that resembles the Zeeman Hamiltonian but that does not depend on the magnetic moment. In the case of $^{13}$CO, the rotation frequency is not equivalent to a Zeeman interaction since, as stated in the definition of $\hat{\vec{\mu}}$, the gyromagnetic factors of the orbital angular momentum and the electron spin are different.
For the subsequent calculations, the quasienergy eigenvector instead of the normal Hamiltonian eigenvector is chosen as the initial state. After choosing an initial state $i$ at the position $r(0) = -x_{\textrm{init}} \hat{x} + b \hat{y}$, the quasienergy vector is propagated in time using equation (\[qeh\]) until the molecule reaches the position $r(t_\textrm{final}) = x_{\textrm{init}} \hat{x} + b \hat {y}$. The final state is then expressed in terms of quasienergy eigenvectors at the final position and time, and the probability of the molecule ending up in a state $j$, $P_{i,j} (v, B, b)$, is calculated. For all calculations, the population is assumed to be initially distributed equally over all low-field-seeking levels, two for $^{12}$CO and four for $^{13}$CO. In $^{12}$CO, the calculation of $P_{i,j} (v, B, b)$ only needs to be carried out for the initial state $i$ corresponding to the $M_J^B=+1$ low-field-seeking level, since the $M_J^B=0$ low-field-seeking level is completely stable against non-adiabatic transitions (see the Appendix). For $^{13}$CO, none of the four low-field-seeking levels is stable against non-adiabatic transitions at all magnetic fields, so $P_{i,j} (v, B, b)$ must be calculated for each initial level $i$.
In the end, we are interested in the probability $T(B)$ of a molecule remaining in a low-field-seeking state for the duration of its time in the microtrap, as this is the quantity that is measured in the experiment. For $^{12}$CO molecules in the ideal case, i.e. when the electric field is perfectly perpendicular to the magnetic field and the traps move perfectly smoothly over the chip, the survival probability for a single molecule in a single state is given by the product of its survival probabilities after each individual encounter with the trap center. The overall transmission probability $T(B)$ is then calculated by averaging this over both low-field-seeking states and over $N$ molecules. $$\begin{split}
T(B) = \frac{1}{N} \sum_{n=1}^N \Biggl[
&\frac{1}{2} \prod_{k=1}^K P_{M_J^B=0,M_J^B=0}(v_{n,k}, B, b_{n,k}) +\\
&\frac{1}{2} \prod_{k=1}^K P_{M_J^B=+1,M_J^B=+1}(v_{n,k}, B, b_{n,k})
\Biggr]
\end{split}$$ The total number of passes $K$ of each molecule and the speed $v_{n,k}$ and closest approach distance $b_{n,k}$ of the $k$th pass of the $n$th molecule are determined using simulations of the classical trajectory of a molecule in the trap, as described elsewhere [@meek09]. Since the $M_J^B=0$ state is completely stable, $P_{M_J^B=0,M_J^B=0}(v_{n,k}, B, b_{n,k}) = 1$, and the transmission probability $T(B)$ in the ideal case can never be less than $1/2$. For $^{13}$CO, calculating $T(B)$ is somewhat more complicated, since the $F=3/2, M_F^B = +3/2$ low-field-seeking level and the $F=3/2, M_F^B = -1/2$ low-field-seeking level are coupled, as are the $F=3/2, M_F^B = +1/2$ and $F=3/2, M_F^B = -3/2$ levels. During each encounter with the trap center, a molecule in a particular low-field-seeking state can transition not only to a non-trappable state but also to one other low-field-seeking state. To calculate the transmission probability of a single $^{13}$CO molecule, the population in each low-field-seeking state after an encounter with the trap center is computed based on the population distribution before the encounter, and the total population still in a low-field-seeking state after the last pass is recorded. As in $^{12}$CO, $T(B)$ is obtained by averaging the result of this calculation over a large number of molecules.
To accurately describe the experimental data, the theoretical calculations must be extended to include the jittering motion of the traps and the nonperpendicularity of the electric and magnetic fields. The jittering motion is accounted for by including the full motion of the center of the trap as shown in Figure \[jitter\] in the calculation of the transition probability $P_{i,j} (v, B, b, t, \phi)$. As in the ideal case, the transition probability depends on speed $v$, magnetic field $B$, and closest approach distance $b$ (although this distance is now defined relative to the average position of the minimum instead of the actual position). Additionally, the transition probability now also depends on the time $t$ at which the molecule arrives relative to the jittering cycle and the direction $\phi$ from which it comes. If the electric and magnetic fields are not exactly perpendicular, low-field-seeking states that are normally decoupled can mix. In $^{12}$CO, the $M_J^B = 0$ and $M_J^B = +1$ states, which converge asymptotically at high electric fields, then become coupled. As a result, the population can partially redistribute between these two levels while the molecule is in a region of high electric field between successive encounters with the trap center. To account for this effect, it is assumed in the calculations that, after each encounter with the trap center, a molecule’s population $n$ in each of these two low-field-seeking states is redistributed such that $$\begin{aligned}
n'_{M_J^B = +1} &= (1-m) n_{M_J^B = +1} + m n_{M_J^B = 0}\\
n'_{M_J^B = 0} &= (1-m) n_{M_J^B = 0} + m n_{M_J^B = +1}\end{aligned}$$ where $n'$ is the new population distribution. The parameter $m$ describes the degree of the redistribution; at the extremes, a value of $0$ indicates that no redistribution occurs while a value of $1/2$ corresponds to complete redistribution. Its exact value is difficult to predict and should actually depend on the trajectory of the molecule. For simplicity, $m$ is determined by fitting it to the data; note that this is the only fitting parameter used. For $^{13}$CO, remixing can occur at high electric fields between the $F=3/2, M_F^B = -3/2$ and the $F=3/2, M_F^B = -1/2$ levels and also between the $F=3/2, M_F^B = +1/2$ and the $F=3/2, M_F^B = +3/2$ levels. The remixing coefficient $m$ can be different for each of these pairs of levels, and thus for $^{13}$CO, two fitting parameters are necessary.
Experimental results
====================
To measure the effectiveness of the magnetic field at suppressing non-adiabatic losses, metastable CO molecules are guided over the full length of the chip at a constant velocity of 300 m/s and are subsequently detected using laser ionization. Measurements are carried out for both positive and negative magnetic fields, where the direction of positive magnetic field coincides with the $+\hat{z}$-axis (see Figure \[beamline\]). To compensate for long term drifts in the intensity of the molecular beam, the parent ion signal is measured with the magnetic field on and with the magnetic field off, and the ratio between these two measurements is recorded. In Figure \[data\], the thus recorded relative number of $^{12}$CO (upper panel) and $^{13}$CO (lower panel) molecules that are guided over the chip is shown as a function of the applied magnetic field. Both the measurements with the “standard waveforms” and with the “improved waveforms” are shown in the upper panel.
It is clear from the data shown in Figure \[data\] that, as the magnetic field strength increases, the number of $^{12}$CO molecules reaching the detector increases. This is as expected because the splitting between the low-field-seeking levels of $^{12}$CO and the non-trappable level increases with the magnetic field strength, thereby increasingly suppressing non-adiabatic losses. The measurements with the “improved waveforms” show that at magnetic fields more negative than $-40$ Gauss and more positive than $+50$ Gauss, the number of guided molecules becomes constant, indicating that all non-adiabatic losses are suppressed under these conditions. The data have been scaled vertically such that the saturation observed at high magnetic field strengths corresponds to a transmission of unity. The ratio between the signal at high magnetic field and low magnetic field is smaller than when the “standard waveforms” are used, implying that the “improved waveforms” also reduce the losses without a magnetic field. The number of $^{13}$CO molecules reaching the ionization detector, on the other hand, is seen to decrease as a function of magnetic field. The vertical scale in this case is based on the results of the theoretical calculations at low magnetic field strengths (*vide infra*). The lower panels of Figure \[colevel\] show that the $M_F^B=+1/2$ level of the non-trappable $F = 1/2$ state increases in energy in a magnetic field, while the $M_F^B=-3/2$ and $M_F^B=-1/2$ levels of the low-field-seeking $F = 3/2$ state are at the same time lowered in energy. This reduction of the energy gap between the low-field-seeking and non-trappable levels enhances the non-adiabatic losses. Both the $^{12}$CO and $^{13}$CO data are slightly asymmetric for positive and negative magnetic fields. The data for $^{12}$CO indeed seem to be symmetric around a magnetic field of about $+8$ Gauss, as expected from the theoretical model.
The results of the theoretical models are shown as solid curves in Figure \[data\] as well. The thick solid curves in the upper and in the lower panel result from the theoretical model for the ideal case, i.e. when the jittering of the traps would be absent and no remixing would occur between decoupled states. The width of the theoretically predicted transmission minimum for $^{12}$CO around $+8$ Gauss depends sensitively on the relative velocity between the CO molecule and the center of the trap as they pass, and gets larger with increasing velocities. In the case of $^{13}$CO, two narrow transmission minima are expected around $-65$ Gauss and $+90$ Gauss, corresponding to fields at which the the $F = 1/2, M_F^B = +1/2$ and $F = 3/2, M_F^B = -3/2$ levels cross, and between these two minima, no losses are expected. As in $^{12}$CO, the width of the transmission minima increases as the relative velocity between the molecules and the trap center increases. It is clear from the comparison of these theoretical curves with the experimental data, that the observed measurements can not be quantitatively explained if the traps are assumed to move smoothly over the chip; the jittering of the traps must be taken into account.
The transmission probabilities calculated for the case in which the jittering is explicitly taken into account are seen to almost quantitatively agree with the measurements. In particular the theoretical curves reproduce the asymmetry between the intensity of the guided $^{12}$CO molecules at positive and negative magnetic fields as well as the narrowing of the profiles when the waveforms are improved. In the calculations for $^{12}$CO, a partial redistribution after each pass of 18% ($m = 0.09$) has been assumed for the “standard waveforms” and 12% ($m = 0.06$) for the “improved waveforms”. It can not be excluded that the jittering of the traps was still slightly more severe during the actual experiments than shown for the “standard waveforms” in Figure \[jitter\], which would explain the experimentally observed additional broadening for that case. In the $^{13}$CO calculations, the best agreement with the experimental data was found when assuming no remixing between the $M_F^B = +3/2$ and the $M_F^B = +1/2$ levels, and 30% remixing ($m = 0.15$) between the $M_F^B = -1/2$ and the $M_F^B = -3/2$ level. The maximum transmission at about $+10$ Gauss is not sensitive to the remixing coefficients, since the transition probability for each of the low-field-seeking states is about the same in this region. It can be reliably inferred from the theoretical calculations that, even at low magnetic fields, about $1/3$ of the $^{13}$CO molecules are lost to non-adiabatic transitions while being guided over the chip. Based on this, the $^{13}$CO data shown in the lower panel of Figure \[data\] have been scaled vertically such that the transmission at zero magnetic field is 2/3.
The theoretical model used to explain the guiding data can also be applied to predict the non-adiabatic losses that are expected to occur during linear deceleration. The red dashed curves in the upper (lower) panel of Figure \[data\] show the survival probability of $^{12}$CO ($^{13}$CO) molecules decelerated from 300 m/s to zero velocity in 250 s. In these calculations, it is assumed that the jittering motion at 300 m/s is that of the “standard waveforms”. The velocity of the jittering motion is assumed to be proportional to the frequency of the applied waveforms while the latter is reduced from 2.5 MHz to zero. For $^{12}$CO at low magnetic fields, the survival probability during deceleration is only 1/4 of the survival probability of $^{12}$CO guided at a constant velocity of 300 m/s. The magnetic field needed to suppress losses is smaller, however. While the survival probability for guiding is symmetric around a magnetic field of +8 Gauss, the symmetry point for deceleration is shifted closer to zero field, to +4 Gauss, due to the lower rotation frequency of the electric field vectors in the trap at lower velocities. For $^{13}$CO, the model predicts that the transmission probability during deceleration is larger than for guiding at all magnetic field strengths.
The differences in transmission probability between guiding and deceleration can result from various effects that either enhance or suppress losses as the deceleration of the trap increases. The smaller spatial acceptance of a strongly accelerated trap results in a larger fraction of the trapped cloud being in the jittering region at any given time, which enhances the losses during deceleration [@meek09]. Losses are also enhanced due to the molecules spending a longer time on the chip. On the other hand, since the average velocity of a decelerating trap is lower than that of a trap at constant velocity, the velocity of the jittering motion is reduced, suppressing non-adiabatic losses. While it is difficult to predict through simple arguments the relative importance of these effects, the outcome of the calculations is corroborated by previous measurements at zero magnetic field, in which it was shown that $^{13}$CO molecules can be decelerated to a standstill while $^{12}$CO molecules are rapidly lost with increasing deceleration [@meek09a].
Conclusions
===========
In this paper we have studied the losses due to non-adiabatic transitions in metastable CO molecules — laser-prepared in the upper $\Lambda$-doublet component of the $J=1$ level in the $a\, ^3\Pi_1$, $v=0$ state — guided at a constant velocity in microtraps over a chip. Transitions between levels in which the molecules are trapped and levels in which the molecules are not trapped can be suppressed (enhanced) when the energetic splitting between these levels is increased (decreased) by the application of a static magnetic field. For a quantitative understanding of this effect, the energy level structure of $^{12}$CO and $^{13}$CO molecules in combined magnetic and electric fields has been analyzed in detail. When the CO molecules are guided over the chip, they are in an electric field that rotates with a constant frequency; the direction of the externally applied magnetic field is perpendicular to the plane of the electric field. The probability with which either $^{12}$CO or $^{13}$CO molecules are transmitted over the chip, i.e. the probability that the molecules stay in a trapped level for the complete duration of the flight over the chip, has been measured as a function of the magnetic field. The observed transmission probability can be quantitatively explained.
To reduce trap losses in future experiments, it will be important to improve the applied waveforms. This will not only reduce losses due to non-adiabatic transitions caused by the jittering, but it will also reduce losses due to mechanical heating. Mechanical losses might also have been present in the current experiment, but because the measurements always compared the guiding efficiency with magnetic field on and off, we have not been sensitive to these losses. Alternatively, it might be possible to avoid the need for improved waveforms by moving the minimum on an orbit that is much larger than the amplitude of the jittering motion, creating a large region around the effective trap center through which the minimum never passes. Such a trap, known as a time orbiting potential (TOP) trap, prevents non-adiabatic losses but is much shallower than a static trap [@petrich95].
The intrinsic difficulties with making the waveforms required for the experiments discussed here should be stressed; with present day technology these waveforms can hardly be made better than we have them now, in particular because, in order to bring molecules to a standstill, we want to be able to rapidly chirp the frequency down from 2.5 MHz to zero. While the LC filter used to produce the “improved waveforms” reduces the total harmonic distortion of the amplitudes from 7% to 3%, it also makes producing a constant amplitude frequency chirp more complicated. We are nevertheless optimistic that the jittering can be reduced by another factor of two to three relative to the best waveforms that we have used so far. In the case of $^{12}$CO, for instance, a magnetic field of 10 Gauss, applied in the right direction, would then already completely avoid losses due to non-adiabatic transitions. With the present waveforms, trap losses can only be avoided when the applied magnetic fields are made sufficiently high. One should realize that there is an upper limit to these fields, however, as at some point transitions to the lower $\Lambda$-doublet components can be induced, opening up a new loss-channel.
The extreme sensitivity to the details of the applied voltages results from the fact that the electric field minima above the chip originate from the vectorial cancellation of rather large electric field terms. Design studies are in progress to find an electrode geometry that is less sensitive to imperfections in the applied waveforms. A modified electrode geometry is also required to avoid trap losses at the ends of the tubular traps. Although the ends are closed in the present geometry by the fringe fields of adjacent electrodes, the electric field near the ends has components along the long axis of the trap, presumably leading to non-adiabatic losses even in the presence of the offset magnetic field.
The design of the electronics by G. Heyne, V. Platschkowski and T. Vetter has been crucial for this work. This research has been funded by the European Community’s Seventh Framework Program FP7/2007-2013 under grant agreement 216 774, and ERC-2009-AdG under grant agreement 247142-MolChip. G.S. gratefully acknowledges the support of the Alexander von Humboldt foundation.
The $\boldsymbol{a\, ^3\Pi_1}$, $\boldsymbol{v = 0}$, $\boldsymbol{J = 1}$ Hamiltonian {#appham}
======================================================================================
In this Appendix, the formalism that has been used to calculate the energies of the $M$-components of the $J=1$ level in the $a\, ^3\Pi_1$, $v=0$ state of both $^{12}$CO and $^{13}$CO in combined, but mutually orthogonal, static electric and magnetic fields is presented. In the coordinate system used here, the magnetic field vector is oriented along the $\hat{z}$ axis and the electric field vector is in the $xy$ plane. In this case, the molecular Hamiltonian is invariant under reflection in the $xy$-plane. It is thus possible to separate the basis states into two uncoupled sets, consisting of wavefunctions that are either symmetric or antisymmetric under reflection in the $xy$-plane, thereby reducing the computational complexity. The magnetic field can only couple states of the same parity and the same $M_F^B$ quantum number, where $M_F^B$ is the projection of the total angular momentum including nuclear spin along the $+\hat{z}$ axis. In the case of $^{12}$CO, there is no hyperfine interaction and thus $F \equiv J$ and $M_F^B \equiv M_J^B$. As the electric field vector lies in the plane perpendicular to the quantization axis it can only couple states of opposite parity with $M_F^B$ differing by $\pm 1$. The two resulting sets of uncoupled basis states for $^{12}$CO are given by:
1. M$_J^B = -1, \pm$
2. M$_J^B = 0, \mp$
3. M$_J^B = 1; \pm$
The $+$ and $-$ sign at the end describe the parity of the basis state. All states with the upper (lower) sign belong to one set. For $^{13}$CO there are two sets containing six basis states each:
1. $F = 3/2, M_F^B = -3/2, \pm$
2. $F = 1/2, M_F^B = -1/2, \mp$
3. $F = 3/2, M_F^B = -1/2, \mp$
4. $F = 1/2, M_F^B = 1/2, \pm$
5. $F = 3/2, M_F^B = 1/2, \pm$
6. $F = 3/2, M_F^B = 3/2, \mp$
Again, all states with the upper (lower) parity belong to one set.
Based on this formalism and using the zero-field spectroscopic parameters and matrix elements given in references [@meek10; @wicke72; @gammon71; @carballo88; @yamamoto88; @wada00; @field72; @warnerphd], the corresponding Hamiltonian matrices can be calculated. Without loss of generality, the electric field vector is taken to be oriented along the $\hat{x}$ axis, i.e. $\vec{E} = E \hat{x}$. The Hamiltonian matrices for other orientations of the electric field vector in the $xy$ plane can be computed using the unitary transformation given in equation (\[HprimeH\]); the matrices given here correspond to $\hat{H}'$ in this equation. The origin of the energy scale for each isotopologue is defined to be the lowest energy field-free state in the upper $\Lambda$-doublet component, as shown in Figure \[colevel\].
For $^{12}$CO, the matrices $\hat{H}_{\textrm{upper}}$ and $\hat{H}_{\textrm{lower}}$ for the set of basis states with the upper and lower parity, respectively, are given by: $$\hat{H}_{\textrm{upper}} =
\begin{pmatrix}
-R_1& S& 0\\
S& -\Lambda& S\\
0& S& R_1\\
\end{pmatrix}$$ and $$\hat{H}_{\textrm{lower}} =
\begin{pmatrix}
-\Lambda - R_2& S& 0\\
S& 0& S\\
0& S& -\Lambda + R_2\\
\end{pmatrix}$$ where $\Lambda = 394.066~\textrm{MHz}$\
$R_1 = \langle 1, + | \hat{H}_Z | 1, + \rangle = 0.3332\mu_B B $\
$R_2 = \langle 1, - | \hat{H}_Z | 1, - \rangle = 0.3406\mu_B B $\
$S = \langle 0, + | \hat{H}_S | 1, - \rangle = 0.3513\mu_E E$,\
and the bra and ket vectors have the form $|M_J^B, \textrm{parity}\rangle$. It is clear from these matrices, that the energy level labeled as $M_J^B$=0 in the case of $^{12}$CO (upper right panel of Figure \[colevel\]) is only directly coupled to the $M_J^B$=$\pm$1 levels of the lower $\Lambda$-doublet component and that there is no direct coupling to the nearby $M_J^B$=$\pm$1 levels of the upper $\Lambda$-doublet component. Provided that the electric and magnetic fields are exactly perpendicular, this $M_J^B$=0 level is therefore stable against non-adiabatic transitions.
For $^{13}$CO, the corresponding matrices for the sets of basis states with the upper and lower parity are given by:
$$\hat{H}_{\textrm{upper}} =
\begin{pmatrix}
E_1 -3 R_5& \sqrt{3} S_3& \frac{\sqrt{3}}{2} S_4& 0& 0& 0\\
\sqrt{3} S_3& -E_3 -R_2& R_4& S_1& -S_3& 0\\
\frac{\sqrt{3}}{2} S_4& R_4& -E_2 -R_6& S_2& S_4& 0\\
0& S_1& S_2& R_1& R_3& -\sqrt{3} S_2\\
0& -S_3& S_4& R_3& E_1 + R_5& \frac{\sqrt{3}}{2} S_4\\
0& 0& 0& -\sqrt{3} S_2& \frac{\sqrt{3}}{2} S_4& -E_2 + 3 R_6\\
\end{pmatrix}$$
and $$\hat{H}_{\textrm{lower}} =
\begin{pmatrix}
-E_2 -3 R_6& \sqrt{3} S_2& \frac{\sqrt{3}}{2} S_4& 0& 0& 0\\
\sqrt{3} S_2& -R_1& R_3& S_1& -S_2& 0\\
\frac{\sqrt{3}}{2} S_4& R_3& E_1 -R_5& S_3& S_4& 0\\
0& S_1& S_3& -E_3 +R_2& R_4& -\sqrt{3} S_3\\
0& -S_2& S_4& R_4& -E_2 +R_6& \frac{\sqrt{3}}{2} S_4\\
0& 0& 0& -\sqrt{3} S_3& \frac{\sqrt{3}}{2} S_4& E_1 + 3 R_5\\
\end{pmatrix}$$
where $E_1 = 58.412~\textrm{MHz}$\
$E_2 = 309.340~\textrm{MHz}$\
$E_3 = 346.346~\textrm{MHz}$\
$R_1 = \langle 1/2, 1/2, + | \hat{H}_Z | 1/2, 1/2, + \rangle = 0.2264\mu_B B $\
$R_2 = \langle 1/2, 1/2, - | \hat{H}_Z | 1/2, 1/2, - \rangle = 0.2309\mu_B B$\
$R_3 = \langle 1/2, 1/2, + | \hat{H}_Z | 3/2, 1/2, + \rangle = 0.1600\mu_B B$\
$R_4 = \langle 1/2, 1/2, - | \hat{H}_Z | 3/2, 1/2, - \rangle = 0.1635\mu_B B$\
$R_5 = \langle 3/2, 1/2, + | \hat{H}_Z | 3/2, 1/2, + \rangle = 0.1132\mu_B B $\
$R_6 = \langle 3/2, 1/2, - | \hat{H}_Z | 3/2, 1/2, - \rangle = 0.1155\mu_B B $\
$S_1 = \langle 1/2, -1/2, + | \hat{H}_S | 1/2, 1/2, -\rangle = 0.3313\mu_E E$\
$S_2 = \langle 3/2, -1/2, - | \hat{H}_S | 1/2, 1/2, + \rangle = 0.1172\mu_E E$\
$S_3 = \langle 3/2, -1/2, + | \hat{H}_S | 1/2, 1/2, - \rangle = 0.1172\mu_E E$\
$S_4 = \langle 3/2, -1/2, + | \hat{H}_S | 3/2, 1/2, - \rangle = 0.3313\mu_E E$,\
and the basis vectors have the form $|F,M_F^B,\textrm{parity}\rangle$.
If the magnetic field is not perpendicular to the electric field, additional non-zero matrix elements will appear in the Hamiltonian that couple the states of $\hat{H}_{\textrm{upper}}$ and $\hat{H}_{\textrm{lower}}$.
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|
---
abstract: 'Imprecise continuous-time Markov chains are a robust type of continuous-time Markov chains that allow for partially specified time-dependent parameters. Computing inferences for them requires the solution of a non-linear differential equation. As there is no general analytical expression for this solution, efficient numerical approximation methods are essential to the applicability of this model. We here improve the uniform approximation method of [@2016Krak] in two ways and propose a novel and more efficient adaptive approximation method. For ergodic chains, we also provide a method that allows us to approximate stationary distributions up to any desired maximal error.'
author:
- |
Alexander Erreygers alexander.erreygers@ugent.be\
Ghent University, SMACS Research Group\
Jasper De Bock jasper.debock@ugent.be\
Ghent University - imec, IDLab, ELIS\
bibliography:
- 'extended-references.bib'
title: |
Imprecise Continuous-Time Markov Chains:\
Efficient Computational Methods with Guaranteed Error Bounds
---
=1
Imprecise continuous-time Markov chain; lower transition operator; lower transition rate operator; approximation method; ergodicity; coefficient of ergodicity.
Introduction {#sec:Intro}
============
Markov chains are a popular type of stochastic processes that can be used to model a variety of systems with uncertain dynamics, both in discrete and continuous time. In many applications, however, the core assumption of a Markov chain—i.e., the Markov property—is not entirely justified. Moreover, it is often difficult to exactly determine the parameters that characterise the Markov chain. In an effort to handle these modelling errors in an elegant manner, several authors have recently turned to imprecise probabilities [@decooman2009; @2013Skulj; @2012Hermans; @2015Skulj; @2016Krak; @2017DeBock].
As [@2016Krak] thoroughly demonstrate, making inferences about an imprecise continuous-time Markov chain—determining lower and upper expectations or probabilities—requires the solution of a non-linear vector differential equation. To the best of our knowledge, this differential equation cannot be solved analytically, at least not in general. [@2016Krak] proposed a method to numerically approximate the solution of the differential equation, and argued that it outperforms the approximation method that [@2015Skulj] previously introduced. One of the main results of this contribution is a novel approximation method that outperforms that of [@2016Krak].
An important property—both theoretically and practically—of continuous-time Markov chains is the behaviour of the solution of the differential equation as the time parameter recedes to infinity. If regardless of the initial condition the solution converges, we say that the chain is ergodic. We show that in this case the approximation is guaranteed to converge as well. This constitutes the second main result of this contribution and serves as a motivation behind the novel approximation method. Furthermore, we also quantify a worst-case convergence rate for the approximation. This unites the work of [@2015Skulj], who studied the rate of convergence for discrete-time Markov chains, and [@2017DeBock], who studied the ergodic behaviour of continuous-time Markov chains from a qualitative point of view. One of the uses of our worst-case convergence rate is that it allows us to approximate the limit value of the solution up to a guaranteed error.
This paper is an extended preprint of [@2017erreygers]. Recently, it has come to our attention that one of the results in that paper, namely \[prop:CoeffOfErgod:ErgodicUpperBound\], is false. Fortunately, none of the other results in [@2017erreygers]—and hence also in this preprint—depend on \[prop:CoeffOfErgod:ErgodicUpperBound\] and the main conclusions and contributions of the paper therefore remain intact. For that reason, we have only made the following two modifications with respect to the previous version: we have omitted the proof of \[prop:CoeffOfErgod:ErgodicUpperBound\], and we have added a counterexample to show that the statement is indeed incorrect.
To ensure the readability of the main text, we have gathered the proofs of all the results in the Appendix. In this Appendix, we also discuss the ergodicity of both discrete and continuous-time Markov chains more thoroughly.
Mathematical preliminaries {#sec:Preliminaries}
==========================
Throughout this contribution, we denote the set of real, non-negative real and strictly positive real numbers by ${\mathbb{R}}$, ${{\mathbb{R}}_{\geq 0}}$ and ${{\mathbb{R}}_{> 0}}{}$, respectively. The set of natural numbers is denoted by ${\mathbb{N}}$, if we include zero we write ${{\mathbb{N}}_{0}}\coloneqq {\mathbb{N}}\cup \{ 0 \}$. For any set $S$, we let ${\left\vert {S} \right\vert}$ denote its cardinality. If $a$ and $b$ are two real numbers, we say that $a$ is lower (greater) than $b$ if $a \leq b$ ($a \geq b$), and that $a$ is strictly lower (greater) than $b$ if $a < b$ ($a > b$).
Gambles and norms
-----------------
We consider a finite *state space* ${\mathcal{X}}$, and are mainly concerned with real-valued functions on ${\mathcal{X}}$. All of these real-valued functions on ${\mathcal{X}}$ are collected in the set ${{\mathcal{L}}({\mathcal{X}})}$, which is a vector space. If we identify the state space ${\mathcal{X}}$ with $\{1, \dots, {\left\vert {{\mathcal{X}}} \right\vert}\}$, then any function $f \in {{\mathcal{L}}({\mathcal{X}})}$ can be identified with a vector: for all $x\in{\mathcal{X}}$, the $x$-component of this vector is $f(x)$. A special function on ${\mathcal{X}}$ is the indicator ${\mathbb{I}_{A}}$ of an event $A$. For any $A \subseteq {\mathcal{X}}$, it is defined for all $x \in {\mathcal{X}}$ as ${{\mathbb{I}_{A}}(x)} = 1$ if $x \in A$ and ${{\mathbb{I}_{A}}(x)} = 0$ otherwise. In order not to obfuscate the notation too much, for any $y \in {\mathcal{X}}$ we write ${\mathbb{I}_{y}}$ instead of ${\mathbb{I}_{\{y\}}}$. If it is required from the context, we will also identify the real number $\gamma \in {\mathbb{R}}$ with the map $\gamma$ from ${\mathcal{X}}$ to ${\mathbb{R}}$, defined as $\gamma(x) = \gamma$ for all $x \in {\mathcal{X}}$.
We provide the set ${{\mathcal{L}}({\mathcal{X}})}$ of functions with the standard maximum norm ${\left\Vert {\cdot} \right\Vert}$, defined for all $f\in{{\mathcal{L}}({\mathcal{X}})}$ as ${\left\Vert {f} \right\Vert} \coloneqq \max \left\{ {\left\vert {f(x)} \right\vert} \colon x \in {\mathcal{X}}\right\}$. A seminorm that captures the variation of $f \in {{\mathcal{L}}({\mathcal{X}})}$ will also be of use; we therefore define the variation seminorm ${\left\Vert {f} \right\Vert}_{v} \coloneqq \max f - \min f$. Since the value ${\left\Vert {f} \right\Vert}_{v} / 2$ occurs often in formulas, we introduce the shorthand notation ${\left\Vert {f} \right\Vert}_{c} \coloneqq {\left\Vert {f} \right\Vert}_{v}/2$.
Non-negatively homogeneous operators
------------------------------------
An operator $A$ that maps ${{\mathcal{L}}({\mathcal{X}})}$ to ${{\mathcal{L}}({\mathcal{X}})}$ is *non-negatively homogeneous* if for all $\mu \in {{\mathbb{R}}_{\geq 0}}$ and all $f \in {{\mathcal{L}}({\mathcal{X}})}$, $A (\mu f) = \mu A f$. The maximum norm ${\left\Vert {\cdot} \right\Vert}$ for functions induces an operator norm: $${\left\Vert {A} \right\Vert}
\coloneqq \sup \{ {\left\Vert {A f} \right\Vert} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, {\left\Vert {f} \right\Vert} = 1 \}.$$ If for all $\mu \in {\mathbb{R}}$ and all $f,g \in {{\mathcal{L}}({\mathcal{X}})}$, $A(\mu f + g) = \mu A f + A g$, then the operator $A$ is *linear*. In that case, it can be identified with a matrix of dimension ${\left\vert {{\mathcal{X}}} \right\vert}\times{\left\vert {{\mathcal{X}}} \right\vert}$, the $(x,y)$-component of which is $[A {\mathbb{I}_{y}}](x)$. The identity operator $I$ is an important special case, defined for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ as $I f \coloneqq f$.
Two types of non-negatively homogeneous operators play a vital role in the theory of imprecise Markov chains: lower transition operators and lower transition rate operators.
\[def:LowerTransitionOperator\] An operator ${\underline{T}}{}$ from ${{\mathcal{L}}({\mathcal{X}})}$ to ${{\mathcal{L}}({\mathcal{X}})}$ is called a *lower transition operator* if for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ and all $\mu \in {{\mathbb{R}}_{\geq 0}}$:
1. \[def:LTO:DominatesMin\] ${\underline{T}}f \geq \min f$;
2. \[def:LTO:SuperAdditive\] ${\underline{T}}(f + g) \geq {\underline{T}}f + {\underline{T}}g$;
3. \[def:LTO:NonNegativelyHom\] ${\underline{T}}(\mu f) = \mu {\underline{T}}f$.
Every lower transition operator ${\underline{T}}$ has a conjugate upper transition operator ${\overline{T}}$, defined for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ as ${\overline{T}}f \coloneqq - {\underline{T}}(- f)$.
\[def:LowerTransitionRateOperator\] An operator ${\underline{Q}}{}\,$ from ${{\mathcal{L}}({\mathcal{X}})}$ to ${{\mathcal{L}}({\mathcal{X}})}$ is called a *lower transition rate operator* if for any $f,g \in {{\mathcal{L}}({\mathcal{X}})}$, any $\mu \in {{\mathbb{R}}_{\geq 0}}$, any $\gamma\in{\mathbb{R}}$ and any $x,y \in {\mathcal{X}}$ such that $x\neq y$:
1. \[def:LTRO:Constant\] ${\underline{Q}}\gamma = 0$;
2. \[def:LTRO:SuperAdditive\] ${{\underline{Q}}(f + g) \geq {\underline{Q}}f + {\underline{Q}}g}$;
3. \[def:LTRO:NonNegativelyHom\] ${\underline{Q}}(\mu f) = \mu {\underline{Q}}f$;
4. \[def:LTRO:Sign\] $[{\underline{Q}}{\mathbb{I}_{x}}](y) \geq 0$.
The conjugate lower transition rate operator ${\overline{Q}}$ is defined for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ as ${\overline{Q}}f \coloneqq - {\underline{Q}}(-f)$. As will become clear in Section \[sec:MCs\], lower transition operators and lower transition rate operators are tightly linked. For instance, we can use a lower transition rate operator to construct a lower transition operator. One way is to use Eqn. further on. Another one is given in the following proposition, which is a strengthened version of [@2017DeBock Proposition 5].
\[prop:IPlusDeltaQLowTranOp\] Consider any lower transition rate operator ${\underline{Q}}$ and any $\delta \in {{\mathbb{R}}_{\geq 0}}$. Then the operator $(I + \delta {\underline{Q}})$ is a lower transition operator if and only if $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$.
We end this section with the first—although minor—novel result of this contribution. The norm of a lower transition rate operator is essential for all the approximation methods that we will discuss. The following proposition supplies us with an easy formula for determining it.
\[prop:LTRO:PropositionNorm\] Let ${\underline{Q}}$ be a lower transition rate operator. Then ${\left\Vert {{\underline{Q}}} \right\Vert} = 2 \max \{ {\left\vert {[{\underline{Q}}{\mathbb{I}_{x}}](x)} \right\vert} \colon x \in {\mathcal{X}}\}$.
\[binex:LTRO\] Consider a binary state space ${\mathcal{X}}= \{0, 1\}$ and two closed intervals $[{\underline{q}_{0}}, {\overline{q}_{0}}] \subset {{\mathbb{R}}_{\geq 0}}$ and $[{\underline{q}_{1}}, {\overline{q}_{1}}] \subset {{\mathbb{R}}_{\geq 0}}$. Let $${\underline{Q}}f
\coloneqq \min \left\{
\begin{bmatrix}
q_0 (f(1) - f(0)) \\
q_1 (f(0) - f(1))
\end{bmatrix}
\colon q_0 \in [{\underline{q}_{0}}, {\overline{q}_{0}}], q_1 \in [{\underline{q}_{1}}, {\overline{q}_{1}}] \right\}
\text{ for all } f \in {{\mathcal{L}}({\mathcal{X}})}.$$ Then one can easily verify that ${\underline{Q}}$ is a lower transition rate operator. [@2016Krak] also consider a running example with a binary state space, but they let ${\mathcal{X}}\coloneqq \{ \texttt{healthy}, \texttt{sick} \}$. We here identify `healthy` with $0$ and `sick` with $1$. In [@2016Krak Example 18], they propose the following values for the transition rates: $[{\underline{q}_{0}}, {\overline{q}_{0}}] \coloneqq [1/52, 3/52]$ and $[{\underline{q}_{1}}, {\overline{q}_{1}}] \coloneqq [1/2,2]$. It takes @2016Krak a lot of work to determine the exact value of the norm of ${\underline{Q}}$, see [@2016Krak Example 19]. We simply use Proposition \[prop:LTRO:PropositionNorm\]: $\smash{{\left\Vert {{\underline{Q}}} \right\Vert} = 2 \max\{ 3/52, 2 \} = 4}$.
Imprecise continuous-time Markov chains {#sec:MCs}
=======================================
For any lower transition rate operator ${\underline{Q}}$ and any $f \in {{\mathcal{L}}({\mathcal{X}})}$, [@2015Skulj] has shown that the differential equation $$\label{eqn:TDLTO:FunctionDifferentialEquation}
\frac{\mathrm{d}}{\mathrm{d} t} {{\underline{T}}_{t}} f = {\underline{Q}}{{\underline{T}}_{t}} f
\vspace{3pt}$$ with initial condition ${{\underline{T}}_{0}} f \coloneqq f$ has a unique solution for all $t \in {{\mathbb{R}}_{\geq 0}}$. Later, [@2017DeBock] proved that the time-dependent operator ${{\underline{T}}_{t}}$ itself satisfies a similar differential equation, and that it is a lower transition operator. Finding the unique solution of Eqn. is non-trivial. Fortunately, we can approximate this solution, as by [@2017DeBock Proposition 10] $$\label{eqn:TDLTO:LimitFormula}
{{\underline{T}}_{t}}
= \lim_{n \to \infty} \left( I + \frac{t}{n} {\underline{Q}}\right)^{n}.$$
\[binex:AnalyticalExpressionsForAppliedLTO\] In the simple case of Example \[binex:LTRO\], we can use Eqn. to obtain analytical expressions for the solution of Eqn. . Assume that ${\underline{q}_{0}} + {\overline{q}_{1}}> 0$ and fix some $t \in {{\mathbb{R}}_{\geq 0}}$. Then $$\begin{aligned}
[{{\underline{T}}_{t}} f](0)
= f(0) + {\underline{q}_{0}} h(t)
~~\text{and}~~
[{{\underline{T}}_{t}} f](1)
= f(1) - {\overline{q}_{1}} h(t)
~~\text{for all $f\in{{\mathcal{L}}({\mathcal{X}})}$ with $f(0)\leq f(1)$,}
\end{aligned}$$ where $h(t) \coloneqq {\left\Vert {f} \right\Vert}_{v} ({\underline{q}_{0}} + {\overline{q}_{1}})^{-1} \big(1 - e^{-t ({\underline{q}_{0}} + {\overline{q}_{1}})}\big)$. The case $f(0) \geq f(1)$ yields similar expressions.
For a linear lower transition rate operator ${\underline{Q}}$—i.e., if it is a transition rate matrix $Q$—Eqn. reduces to the definition of the matrix exponential. It is well-known—see [@1991Anderson]—that this matrix exponential $T_t = e^{t Q}$ can be interpreted as the transition matrix at time $t$ of a time-homogeneous or stationary continuous-time Markov chain: the $(x,y)$-component of $T_t$ is the probability of being in state $y$ at time $t$ if the chain started in state $x$ at time $0$. Therefore, it follows that the expectation of the function $f \in {{\mathcal{L}}({\mathcal{X}})}$ at time $t \in {{\mathbb{R}}_{\geq 0}}$ conditional on the initial state $x \in {\mathcal{X}}$, denoted by ${\mathrm{E}}(f(X_{t})|X_{0} = x)$, is equal to $[T_t f](x)$. As Eqn. is a non-linear generalisation of the definition of the matrix exponential, we can interpret ${{\underline{T}}_{t}}$ as the non-linear generalisation of the matrix exponential $T_t = e^{t Q}$. Extending this parallel, we might interpret ${{\underline{T}}_{t}}$ as the non-linear generalisation of the transition matrix—i.e., as the lower transition operator—at time $t$ of a generalised continuous-time Markov chain. In fact, [@2016Krak] prove that this is indeed the case. They show that—under some conditions on ${\underline{Q}}$—$[{{\underline{T}}_{t}} f](x)$ can be interpreted as the tightest lower bound for ${\mathrm{E}}(f(X_{t})|X_{0} = x)$ with respect to a set of—not necessarily Markovian—stochastic processes that are consistent with ${\underline{Q}}$. [@2016Krak] argue that, just like a transition rate matrix $Q$ characterises a (precise) continuous-time Markov chain, a lower transition rate operator ${\underline{Q}}$ characterises a so-called imprecise continuous-time Markov chain.
The main objective of this contribution is to determine ${{\underline{T}}_{t}} f$ for some $f \in {{\mathcal{L}}({\mathcal{X}})}$ and some $t \in {{\mathbb{R}}_{> 0}}$. Our motivation is that, from an applied point of view on imprecise continuous-time Markov chains, what one is most interested in are tight lower and upper bounds on expectations of the form ${\mathrm{E}}(f(X_t)|X_{0} = x)$. As explained above, the lower bound is given by ${{\underline{{\mathrm{E}}}}(f(X_t)|X_0 = x)} = [{{\underline{T}}_{t}} f](x)$. Similarly, the upper bound is given by ${{\overline{{\mathrm{E}}}}(f(X_t)|X_0 = x)} = -[{{\underline{T}}_{t}} (-f)](x)$. Note that the lower (or upper) probability of an event $A \subseteq {\mathcal{X}}$ conditional on the initial state $x$ is a special case of a lower (or upper) expectation: $\underline{{\mathrm{P}}}(X_{t} \in A | X_0 = x) = {{\underline{{\mathrm{E}}}}({{\mathbb{I}_{A}}(X_t)}|X_0 = x)}$ and similarly for the upper probability. Hence, for the sake of generality we can focus on ${{\underline{T}}_{t}} f$ and forget about its interpretation. As in most cases analytically solving Eqn. is infeasible or even impossible, we resort to methods that yield an approximation up to some guaranteed maximal error.
Approximation methods {#sec:EfficientComputation}
=====================
[@2015Skulj] was, to the best of our knowledge, the first to propose methods that approximate the solution ${{\underline{T}}_{t}} f$ of Eqn. . He proposes three methods: one with a uniform grid, a second with an adaptive grid and a third that is a combination of the previous two. In essence, he determines a step size $\delta$ and then approximates ${{\underline{T}}_{t + \delta}} f$ with $e^{\delta Q} {{\underline{T}}_{t}} f$, where $Q$ is a transition rate matrix determined from . One drawback of this method is that it needs the matrix exponential $e^{\delta Q}$, which—in general—needs to be approximated as well. [@2015Skulj] mentions that his methods turn out to be quite computationally heavy, even if the uniform and adaptive methods are combined.
We consider two alternative approximation methods—one with a uniform grid and one with an adaptive grid—that both work in the same way. First, we pick a small step $\delta_1 \in {{\mathbb{R}}_{\geq 0}}$ and apply the operator $(I + \delta_1 {\underline{Q}})$ to the function $g_0 = f$, resulting in a function $g_1 \coloneqq (I + \delta_1 {\underline{Q}}) f$. Recall from Proposition \[prop:IPlusDeltaQLowTranOp\] that if we want $(I + \delta_1 {\underline{Q}})$ to be a lower transition operator, then we need to demand that $\delta_1 {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. Next, we pick a (possibly different) small step $\delta_2 \in {{\mathbb{R}}_{\geq 0}}$ such that $\delta_2 {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ and apply the lower transition operator $(I + \delta_2 {\underline{Q}})$ to the function $g_1$, resulting in a function $g_2 \coloneqq (I + \delta_2 {\underline{Q}}) g_1$. If we continue this process until the sum of all the small steps is equal to $t$, then we end up with an approximation for ${{\underline{T}}_{t}} f$. More formally, let $s \coloneqq (\delta_1, \dots, \delta_k)$ denote a sequence in ${{\mathbb{R}}_{\geq 0}}$ such that, for all $i \in \{ 1, \dots, k \}$, $\delta_i {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. Using this sequence $s$ we define the *approximating lower transition operator* $$\Phi(s)
\coloneqq (I + \delta_k {\underline{Q}}) \cdots (I + \delta_1 {\underline{Q}}).\vspace{3pt}$$ What we are looking for is a convenient way to determine the sequence $s$ such that the error ${\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}$ is guaranteed to be lower than some desired maximal error $\epsilon \in {{\mathbb{R}}_{> 0}}$.
Using a uniform grid {#ssec:UniformGrid}
--------------------
[@2016Krak] provide one way to determine the sequence $s$. They assume a uniform grid, in the sense that all elements of the sequence $s$ are equal to $\delta$. The step size $\delta$ is completely determined by the desired maximal error $\epsilon$, the time $t$, the variation norm of the function $f$ and the norm of ${\underline{Q}}$; [@2016Krak Proposition 8.5] guarantees that the actual error is lower than $\epsilon$. Algorithm \[alg:Uniform\] provides a slightly improved version of [@2016Krak Algorithm 1]. The improvement is due to Proposition \[prop:IPlusDeltaQLowTranOp\]: we demand that $n \geq t {\left\Vert {{\underline{Q}}} \right\Vert} / 2$ instead of $n \geq t {\left\Vert {{\underline{Q}}} \right\Vert}$.
$g_{0} \gets f$ $g_{n}$
More formally, for any $t \in {{\mathbb{R}}_{\geq 0}}$ and any $n \in {\mathbb{N}}$ such that $t {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2 n$, we consider the *uniformly approximating lower transition operator* $$\Psi_{t}(n)
\coloneqq \left(I + \frac{t}{n} {\underline{Q}}\right)^{n}.\vspace{3pt}$$ As a special case, we define $\Psi_{t}(0) \coloneqq I$. The following theorem then guarantees that the choice of $n$ in Algorithm \[alg:Uniform\] results in an error ${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert}$ that is lower than the desired maximal error $\epsilon$.
\[the:UniformApproximationWithError\] Let ${\underline{Q}}$ be a lower transition rate operator and fix some $f\in{{\mathcal{L}}({\mathcal{X}})}$, $t \in {{\mathbb{R}}_{\geq 0}}$ and $\epsilon \in {{\mathbb{R}}_{> 0}}$. If we use Algorithm \[alg:Uniform\] to determine $n$, $\delta$ and $g_{0}, \dots, g_n$, then we are guaranteed that $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert}
= {\left\Vert {{{\underline{T}}_{t}} f - g_n} \right\Vert}
\leq \epsilon'
\coloneqq
\delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{i=0}^{n-1} {\left\Vert {g_{i}} \right\Vert}_{c}
\leq \epsilon.$$
Theorem \[the:UniformApproximationWithError\] is an extension of [@2016Krak Proposition 8.5]. We already mentioned that the demand $n \geq t {\left\Vert {{\underline{Q}}} \right\Vert}$ can be relaxed to $n \geq t {\left\Vert {{\underline{Q}}} \right\Vert} / 2$. Furthermore, it turns out that we can compute an upper bound $\epsilon'$ on the error that is (possibly) lower than the desired maximal error $\epsilon$. If we want to determine this $\epsilon'$ while running Algorithm \[alg:Uniform\], we simply need to add $\epsilon' \gets 0$ to line \[line:Uniform:First\] and insert $\epsilon' \gets \epsilon' + \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {g_{i}} \right\Vert}_{c}$ just before line \[line:Uniform:IncrementOfG\].
\[binex:UniformApproximation\] We again consider the simple case of Example \[binex:LTRO\] and illustrate the use of Theorem \[the:UniformApproximationWithError\] with a numerical example based on [@2016Krak Example 20]. [@2016Krak] use Algorithm \[alg:Uniform\] to approximate ${{\underline{T}}_{1}} {\mathbb{I}_{1}}$, and find that $n = \num{8000}$ guarantees an error lower than the desired maximal error $\epsilon \coloneqq \num{1e-3}$. As reported in Table \[tab:ComparisonOfCompDuration\], we use Theorem \[the:UniformApproximationWithError\] to compute $\epsilon'$. We find that $\epsilon' \approx \num{0.430e-3}$, which is approximately a factor two smaller than the desired maximal error $\epsilon$.
[rS\[table-format=4.\]S\[table-format=1.4\]S\[table-format=1.4\]S\[table-format=1.4\]S\[table-format=1.4\]]{} & [$N$]{} & [$D_{\epsilon}$]{} & [$D_{\epsilon'}$]{} & [$\epsilon' \times 10^3$]{} & [$\epsilon_a \times 10^3$]{}\
Uniform & 8000 & 0.0345 & 0.0574 & 0.430 & 0.0335\
Uniform & 250 & 0.00171 & 0.0264 & 13.8 & 1.07\
Adaptive with $m = 1$ & 3437 & 0.0371 & 0.0428 & 1.000 & 0.108\
Adaptive with $m = 20$ & 3456 & 0.0143 & 0.0254 & 0.992 & 0.107\
Uniform ergodic with $m = 1$ & 6133 & 0.0264 & 0.0449 & 0.560 & 0.0437\
In this case, since we know the analytical expression for ${{\underline{T}}_{1}} {\mathbb{I}_{1}}$ from Example \[binex:AnalyticalExpressionsForAppliedLTO\], we can determine the actual error $\epsilon_{a} = {\left\Vert {{{\underline{T}}_{1}} {\mathbb{I}_{1}} - \Psi_{1}(8000) {\mathbb{I}_{1}}} \right\Vert}$. Quite remarkably, the actual error is approximately $\num{3.35e-5}$, which is roughly 30 times smaller than the desired maximal error. This leads us to think that the number of iterations used by the uniform method is too high. In fact, we find that using as few as iterations—roughly —already results in an actual error that is approximately equal to the desired one: ${\left\Vert {{{\underline{T}}_{1}} {\mathbb{I}_{1}} - \Psi_{1}(250) {\mathbb{I}_{1}}} \right\Vert} \approx \num{1.07e-3}$.
Using an adaptive grid {#ssec:AdaptiveGrid}
----------------------
In Example \[binex:UniformApproximation\], we noticed that the maximal desired error was already satisfied for a uniform grid that was much coarser than that constructed by Algorithm \[alg:Uniform\]. Because of this, we are led to believe that we can find a better approximation method than the uniform method of Algorithm \[alg:Uniform\].
To this end, we now consider grids where, for some integer $m$, every $m$ consecutive time steps in the grid are equal. In particular, we consider a sequence $\delta_1, \dots, \delta_n$ in ${{\mathbb{R}}_{\geq 0}}$ and some $k \in {\mathbb{N}}$ such that $1 \leq k \leq m$ and, for all $i \in \{ 1, \dots, n \}$, $\delta_i {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. From such a sequence, we then construct the *$m$-fold approximating lower transition operator*: $$\Phi_{m,k}(\delta_1, \dots, \delta_n)
\coloneqq (I + \delta_n {\underline{Q}})^{k} (I + \delta_{n-1} {\underline{Q}})^{m} \cdots (I + \delta_{1} {\underline{Q}})^{m},$$ where if $n = 1$ only $(I + \delta_1 {\underline{Q}})^{k}$ remains and if $n = 2$ only $(I + \delta_2 {\underline{Q}})^{k} (I + \delta_{1} {\underline{Q}})^{m}$ remains.
The uniform approximation method of before is a special case of the $m$-fold approximating lower transition operator; a more interesting method to construct an $m$-fold approximation is Algorithm \[alg:Adaptive\]. In this algorithm, we re-evaluate the time step every $m$ iterations, possibly increasing its length.
$(g_{(0,m)}, \Delta, i) \gets (f, t, 0)$ $g_{(n,k)}$
From the properties of lower transition operators, it follows that for all $\smash{i \in \{ 2, \dots, n-1 \}}$, $\smash{{\left\Vert {g_{(i-1,m)}} \right\Vert}_{c} \leq {\left\Vert {g_{(i-2,m)}} \right\Vert}_{c}}$. Hence, the re-evaluated step size $\delta_i$ is indeed larger than (or equal to) the previous step size $\delta_{i-1}$. The only exception to this is the final step size $\delta_n$: it might be that the remaining time $\Delta$ is smaller than $m \delta_n$, in which case we need to choose $k$ and $\delta_n$ such that $k \delta_n = \Delta$.
Theorem \[the:AdaptiveApproximation\] guarantees that the adaptive approximation of Algorithm \[alg:Adaptive\] indeed results in an actual error lower than the desired maximal error $\epsilon$. Even more, it provides a method to compute an upper bound $\epsilon'$ of the actual error that is lower than the desired maximal error. Finally, it also states that the adaptive method of Algorithm \[alg:Adaptive\] needs at most an equal number of iterations than the uniform method of Algorithm \[alg:Uniform\].
\[the:AdaptiveApproximation\] Let ${\underline{Q}}$ be a lower transition rate operator, $f\in{{\mathcal{L}}({\mathcal{X}})}$, $t \in {{\mathbb{R}}_{\geq 0}}$, $\epsilon \in {{\mathbb{R}}_{> 0}}$ and $m \in {\mathbb{N}}$. We use Algorithm \[alg:Adaptive\] to determine $n$ and $k$, and if applicable also $k_i$, $\delta_{i}$ and $g_{(i,j)}$. If ${\left\Vert {f} \right\Vert}_{c} = 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} = 0$ or $t = 0$, then ${\left\Vert {{{\underline{T}}_{t}} f - g_{(n,k)}} \right\Vert} = 0$. Otherwise, we are guaranteed that $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi_{m,k}(\delta_1 \dots, \delta_n) f} \right\Vert}
=
{\left\Vert {{{\underline{T}}_{t}} f - g_{(n,k)}} \right\Vert}
\leq \epsilon'
&\coloneqq \sum_{i = 1}^{n} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j=0}^{k_i - 1} {\left\Vert {g_{(i,j)}} \right\Vert}_{c}
\leq \epsilon
\end{aligned}$$ and that the total number of iterations has an upper bound: $$\sum_{i=1}^{n} k_i
= (n-1) m + k
\leq \left\lceil \max \left\{ {\left\Vert {{\underline{Q}}} \right\Vert} t/2, t^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}/ \epsilon \right\} \right\rceil.$$
Again, we can determine $\epsilon'$ while running Algorithm \[alg:Adaptive\]. An alternate—less tight—version of $\epsilon'$ can be obtained by replacing the sum of ${\left\Vert {g_{(i,j)}} \right\Vert}_{c}$ for $j$ from $0$ to $k_{i}-1$ by $k_i {\left\Vert {g_{(i,0)}} \right\Vert}_{c} = k_i {\left\Vert {g_{(i-1,m)}} \right\Vert}_{c}$. Determining this alternative $\epsilon'$ while running Algorithm \[alg:Adaptive\] adds negligible computational overhead compared to the $\epsilon'$ of Theorem \[the:AdaptiveApproximation\], as ${\left\Vert {g_{(i-1,m)}} \right\Vert}_{c}$ is needed to re-evaluate the step size anyway.
The reason why we only re-evaluate the step size $\delta$ after every $m$ iterations is twofold. First and foremost, all we currently know for sure is that for all $\delta \in {{\mathbb{R}}_{\geq 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$, all $m \in {\mathbb{N}}$ and all $f\in{{\mathcal{L}}({\mathcal{X}})}$, ${\left\Vert {(I + \delta {\underline{Q}})^{m} f} \right\Vert}_{c} \leq {\left\Vert {f} \right\Vert}_{c}$. Re-evaluating the step size every $m$ iterations is therefore only justified if a priori we are certain that $\smash{{\left\Vert {(I + \delta_{i} {\underline{Q}})^{m} g_{(i-1,m)}} \right\Vert}_{c} < {\left\Vert {g_{(i-1,m)}} \right\Vert}_{c}}$. We come back to this in Section \[sec:ergodicity\]. A second reason is that there might be a trade-off between the time it takes to re-evaluate the step size and the time that is gained by the resulting reduction of the number of iterations. The following numerical example illustrates this trade off.
\[binex:AdaptiveApproximation\] Recall that in Example \[binex:UniformApproximation\] we wanted to approximate ${{\underline{T}}_{1}} {\mathbb{I}_{1}}$ up to a maximal desired error $\epsilon = \num{1e-3}$. Instead of using the uniform method of Algorithm \[alg:Uniform\], we now use the adaptive method of Algorithm \[alg:Adaptive\] with $m = 1$. The initial step size is the same as that of the uniform method, but because we re-evaluate the step size we only need iterations, as reported in Table \[tab:ComparisonOfCompDuration\]. We also find that in this case $\epsilon' = \num{1.00e-3}$, which is a coincidence. Nevertheless, the actual error of the approximation is $\num{0.108e-3}$, which is about ten times smaller than what we were aiming for.
However, fewer iterations do not necessarily imply a shorter duration of the computations. Qualitatively, we can conclude the following from Table \[tab:ComparisonOfCompDuration\]. First, keeping track of $\epsilon'$ increases the duration, as expected. Second, the adaptive method is faster than the uniform method, at least if we choose $m$ large enough. And third, both methods yield an actual error that is at least an order of magnitude lower than the desired maximal error.
Ergodicity {#sec:ergodicity}
==========
Let $\Phi_{m,k}(\delta_1, \dots, \delta_n) f$ be an approximation constructed using the adaptive method of Algorithm \[alg:Adaptive\]. Re-evaluating the step size is then only justified if a priori we are sure that $$\nicefrac{1}{2} {\left\Vert {(I + \delta_{i} {\underline{Q}})^m \Phi_{i-1} f} \right\Vert}_{v} = {\left\Vert {g_{(i,m)}} \right\Vert}_{c} < {\left\Vert {g_{(i-1,m)}} \right\Vert}_{c} = \nicefrac{1}{2} {\left\Vert {\Phi_{i-1} f} \right\Vert}_{v} \text{ for all } i \in \{ 1, \dots, n-1 \},
\vspace{2pt}$$ where $\Phi_0 \coloneqq I$ and $\Phi_{i} \coloneqq (I + \delta_i {\underline{Q}})^m \Phi_{i-1}$. As $(\Phi_{i-1} f) \in {{\mathcal{L}}({\mathcal{X}})}$, this is definitely true if we require that $$\label{eqn:Ergodicity:IntroEqn}
(\forall \delta \in \{ \delta_1, \dots, \delta_{n-1} \}) (\forall f \in {{\mathcal{L}}({\mathcal{X}})}) ~ {\left\Vert {(I + \delta {\underline{Q}})^{m} f} \right\Vert}_{v} < {\left\Vert {f} \right\Vert}_{v}. \vspace{2pt}$$ In fact, since this inequality is invariant under translation or positive scaling of $f$, it suffices if $$(\forall \delta \in \{ \delta_1, \dots, \delta_{n-1} \}) (\forall f \in {{\mathcal{L}}({\mathcal{X}})}\colon 0 \leq f \leq 1) ~ {\left\Vert {(I + \delta {\underline{Q}})^{m} f} \right\Vert}_{v} < 1.
\vspace{1pt}$$ Readers that are familiar with (the ergodicity of) imprecise discrete-time Markov chains—see [@2012Hermans] or —will probably recognise this condition, as it states that the (weak) coefficient of ergodicity of $\smash{(I + \delta {\underline{Q}})^{m}}$ should be strictly smaller than 1. For all lower transition operators ${\underline{T}}$, [@2013Skulj] define this (weak) *coefficient of ergodicity* as $$\label{eqn:CoeffOfErgod}
{{\rho}({\underline{T}})}
\coloneqq \max \left\{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\}.$$
Ergodicity of lower transition rate operators
---------------------------------------------
As will become apparent, whether or not combinations of $m \in {\mathbb{N}}$ and $\delta \in {{\mathbb{R}}_{\geq 0}}$ exist such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ and ${{\rho}((I + \delta {\underline{Q}})^m)} < 1$ is tightly connected with the behaviour of ${{\underline{T}}_{t}} f$ for large $t$. [@2017DeBock] proved that for all lower transition rate operators ${\underline{Q}}$ and all $f \in {{\mathcal{L}}({\mathcal{X}})}$, the limit $\lim_{t \to \infty} {{\underline{T}}_{t}} f$ exists. An important case is when this limit is a constant function for all $f$.
The lower transition rate operator ${\underline{Q}}$ is *ergodic* if for all $f\in{{\mathcal{L}}({\mathcal{X}})}$, $\lim_{t \to \infty} {{\underline{T}}_{t}} f$ exists and is a constant function.
As shown by [@2017DeBock], ergodicity is easily verified in practice: it is completely determined by the signs of $[{\overline{Q}}{\mathbb{I}_{x}}](y)$ and $[{\underline{Q}}{\mathbb{I}_{A}}](z)$, for all $x,y \in {\mathcal{X}}$ and certain combinations of $z \in {\mathcal{X}}$ and $A \subset {\mathcal{X}}$. It turns out that an ergodic lower transition rate operator ${\underline{Q}}$ does not only induce a lower transition operator ${{\underline{T}}_{t}}$ that converges, it also induces discrete approximations—of the form $(I + \delta_{k} {\underline{Q}}) \cdots (I + \delta_1 {\underline{Q}})$—with special properties. The following theorem, which we consider to be one of the main results of this contribution, highlights this.
\[the:ContinuousErgodicity:CoefficientOfErgodicityOfApproximation\] The lower transition rate operator ${\underline{Q}}$ is ergodic if and only if there is some $n<{\left\vert {{\mathcal{X}}} \right\vert}$ such that ${{\rho}(\Phi(\delta_1,\dots,\delta_{k}))} < 1$ for one (and then all) $k \geq n$ and one (and then all) sequence(s) $\delta_1, \dots, \delta_k$ in ${{\mathbb{R}}_{> 0}}$ such that $\delta_i {\left\Vert {{\underline{Q}}} \right\Vert} < 2$ for all $i \in \{1, \dots, k\}$.
Ergodicity and the uniform approximation method {#ssec:Ergodicity:UniformImprovement}
-----------------------------------------------
Theorem \[the:ContinuousErgodicity:CoefficientOfErgodicityOfApproximation\] guarantees that the conditions that were discussed at the beginning of this section are satisfied. In particular, if the lower transition rate operator is ergodic, then there is some $n < {\left\vert {{\mathcal{X}}} \right\vert}$ such that ${{\rho}((I + \delta {\underline{Q}})^{m})} < 1$ for all $m \geq n$ and all $\delta \in {{\mathbb{R}}_{> 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. Consequently, if we choose $m \geq {\left\vert {{\mathcal{X}}} \right\vert}-1$ then re-evaluating the step size $\delta$ will—except maybe for the last re-evaluation—result in a new step size that is strictly greater than the previous one. Therefore, we conclude that if the lower transition rate operator is ergodic, then using the adaptive method of Algorithm \[alg:Adaptive\] is certainly justified; it will result in fewer iterations, provided we choose a large enough $m$.
Another nice consequence of the ergodicity of a lower transition rate operator ${\underline{Q}}$ is that we can prove an alternate a priori guaranteed upper bound for the error of uniform approximations.
\[prop:UniformApproximationErgodicError\] Let ${\underline{Q}}$ be a lower transition rate operator and fix some $f \in {{\mathcal{L}}({\mathcal{X}})}$, $m, n \in {\mathbb{N}}$ and $\delta \in {{\mathbb{R}}_{> 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. If $\beta \coloneqq {{\rho}((I + \delta {\underline{Q}})^{m})} < 1$, then $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
\leq \epsilon_{e} \coloneqq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c} \frac{1 - \beta^{k}}{1 - \beta}
\leq \epsilon_{d} \coloneqq \frac{m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}}{1 - \beta},$$ where $t \coloneqq n \delta$ and $k \coloneqq \lceil \nicefrac{n}{m} \rceil$. The same is true for $\beta = {{\rho}({{\underline{T}}_{m\delta}})}$.
Interestingly enough, the upper bound $\epsilon_{d}$ is not dependent on $t$ (or $n$) at all! This is a significant improvement on the upper bound of Theorem \[the:UniformApproximationWithError\], as that upper bound is proportional to $t^2$.
By Theorem \[the:ContinuousErgodicity:CoefficientOfErgodicityOfApproximation\], there always is an $m < {\left\vert {{\mathcal{X}}} \right\vert}$ such that ${{\rho}((I + \delta {\underline{Q}})^{m})} < 1$ for all $\delta \in {{\mathbb{R}}_{> 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. Thus, given such an $m$, we can easily improve Algorithm \[alg:Uniform\]. After we have determined $n$ and $\delta$ with Algorithm \[alg:Uniform\], we can simply determine the upper bound of Proposition \[prop:UniformApproximationErgodicError\]. If $m (1 - \beta^k) < n (1 - \beta)$ (or $m < n(1 - \beta)$), then this upper bound is smaller than the desired maximal error $\epsilon$, and we have found a tighter upper bound on the actual error. We can even go the extra mile and replace line \[line:Uniform:DetermineN\] with a method that looks for the smallest possible $n \in {\mathbb{N}}$ that yields $$m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c} (1 - \beta^{k}) \leq (1 - \beta) \epsilon,
\vspace{2pt}$$ where $k=\lceil \nicefrac{n}{m} \rceil$ and $\delta=\nicefrac{t}{n}$—and therefore also $\beta$—are dependent of $n$. This method could yield a smaller $n$, but the time we gain by having to execute fewer iterations does not necessarily compensate the time lost by looking for a smaller $n$. In any case, to actually implement these improvements we need to be able to compute $\beta\coloneqq{{\rho}((I + \delta {\underline{Q}})^m)}$.
\[binex:UniformErgodic\] For the simple case of Example \[binex:LTRO\], we can derive an analytical expression for ${{\rho}((I + \delta {\underline{Q}}))}$ that is valid for all $\delta\in{\mathbb{R}}_{\geq0}$ such that $\delta{\left\Vert {{\underline{Q}}} \right\Vert}\leq 2$. Therefore, we can use Proposition \[prop:UniformApproximationErgodicError\] to a priori determine an upper bound for the error. If we choose $m = 1$, then $\epsilon_{e} = \num{0.767e-3}$ and $\epsilon_{d} = \num{1.79e-3}$. Note that $\epsilon_{e} < \epsilon$, so we can probably decrease the number of iterations $n$. As reported in Table \[tab:ComparisonOfCompDuration\], we find that $n = \num{6133}$ still suffices, and that this results in an approximation correct up to $\epsilon' = \num{0.560e-3}$, roughly two times smaller than the desired maximal error $\epsilon$. The actual error is , roughly ten times smaller than $\epsilon$.
Approximating the coefficient of ergodicity {#ssec:CoeffOfErgod:Approximation}
-------------------------------------------
Unfortunately, determining the exact value of ${{\rho}((I + \delta {\underline{Q}})^{m})}$—and of ${{\rho}({\underline{T}})}$ in general—turns out to be non-trivial and is often even impossible. Nevertheless, the following theorem gives some—actually computable—lower and upper bounds for the coefficient of ergodicity.
\[the:CoeffOfErgod:Approximation\] Let ${\underline{T}}$ be a lower transition operator. Then $$\begin{aligned}
{{\rho}({\underline{T}})}
&\leq \max \big\{ \max \{ [{\overline{T}}{\mathbb{I}_{A}}](x) - [{\underline{T}}{\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\big\}, \label{eqn:CoeffOfErgod:UpperBound} \\
{{\rho}({\underline{T}})}
&\geq \max \big\{ \max \{ [{\underline{T}}{\mathbb{I}_{A}}](x) - [{\underline{T}}{\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\big\}. \label{eqn:CoeffOfErgod:LowerBound}
\end{aligned}$$
The upper bound in Theorem \[the:CoeffOfErgod:Approximation\] is particularly useful in combination with Proposition \[prop:UniformApproximationErgodicError\], as it allows us to replace $\beta\coloneqq{{\rho}((I + \delta {\underline{Q}})^{m})}$ with a guaranteed upper bound. Of course, this only makes sense if this upper bound is strictly smaller than one. In the previous versions of this pre-print, we claimed that for ergodic lower transition rate operators ${\underline{Q}}$, this is always the case. Unfortunately—and to our great regret—we have since then discovered that this result is in fact incorrect. We have nonetheless included the (incorrect) statement so that we can easily refer to it, and have added a counterexample that demonstrates that it is indeed incorrect.
\[prop:CoeffOfErgod:ErgodicUpperBound\] Let ${\underline{Q}}$ be an ergodic lower transition rate operator. Then there is some $n < {\left\vert {{\mathcal{X}}} \right\vert}$ such that, for all $k \geq n$ and $\delta_{1}, \dots, \delta_{k}$ in ${{\mathbb{R}}_{> 0}}$ such that $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$ for all $i \in \{ 1, \dots, k \}$, the upper bound for ${{\rho}(\Phi(\delta_{1}, \dots, \delta_{k}))}$ that is given by Eqn. is strictly smaller than one.
Consider the lower transition rate operator defined in \[binex:LTRO\], with ${\underline{q}_{0}} = 0 = {\underline{q}_{1}}$, ${\overline{q}_{0}} > 0$ and ${\overline{q}_{1}} > 0$. One can easily verify that this lower transition rate operator is ergodic.
Note that if \[prop:CoeffOfErgod:ErgodicUpperBound\] were to be true, then for all $\delta \in {{\mathbb{R}}_{> 0}}{}$ such that $\delta {\left\Vert {{\underline{Q}}{}} \right\Vert} < 2$, $$\max \big\{ \max \{ [(I + \delta {\overline{Q}}) {\mathbb{I}_{A}}](x) - [(I + \delta {\underline{Q}}) {\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\big\}
< 1.$$ However, after some straightforward computations we obtain that $$\begin{gathered}
\max \big\{ \max \{ [(I + \delta {\overline{Q}}) {\mathbb{I}_{A}}](x) - [(I + \delta {\underline{Q}}) {\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\big\} \\
\geq [(I + \delta {\overline{Q}}) {\mathbb{I}_{0}}](0) - [(I + \delta {\underline{Q}}) {\mathbb{I}_{0}}](1)
= 1.
\end{gathered}$$
Approximating limit values
--------------------------
The results that we have obtained earlier in this section naturally lead to a method to approximate ${{\underline{T}}_{\infty}} f \coloneqq \lim_{t \to \infty} {{\underline{T}}_{t}} f$ up to some maximal error. This is an important problem in applications; for instance, [@2015Troffaes] try to determine ${{\underline{T}}_{\infty}}f$ for an ergodic lower transition rate operator that arises in their specific reliability analysis application. The method they use is rather ad hoc: they pick some $t$ and $n$ and then determine the uniform approximation $\Psi_{t}(n) f$. As ${\left\Vert {\Psi_{t}(n) f} \right\Vert}_{v}$ is small, they suspect that they are close to the actual limit value. They also observe that $\Psi_{2t}(4n) f$ only differs from $\Psi_{t}(n) f$ after the fourth significant digit, which they regard as further empirical evidence for the correctness of their approximation. While this ad hoc method seemingly works, the initial values for $t$ and $n$ have to be chosen somewhat arbitrarily. Also, this method provides no guarantee that the actual error is lower than some desired maximal error.
Theorem \[the:ContinuousErgodicity:CoefficientOfErgodicityOfApproximation\], Proposition \[prop:UniformApproximationErgodicError\], Theorem \[the:CoeffOfErgod:Approximation\] and the following stopping criterion allow us to propose a method that corrects these two shortcomings.
\[prop:StoppingCriterionWithErgodicity\] Let $\smash{{\underline{Q}}}$ be an ergodic lower transition rate operator and let $f \in {{\mathcal{L}}({\mathcal{X}})}$, and $\epsilon \in {{\mathbb{R}}_{> 0}}$. Let $s$ denote a sequence $\delta_1, \dots, \delta_k$ in ${{\mathbb{R}}_{\geq 0}}$ such that $\sum_{i = 1}^{k} \delta_i = t$ and, for all $i \in \{1,\dots,k\}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. If ${\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert} \leq\nicefrac{\epsilon}{2}$ and ${\left\Vert {\Phi(s) f} \right\Vert}_{c} \leq\nicefrac{\epsilon}{2}$, then for all $\Delta \in {{\mathbb{R}}_{\geq 0}}$: $$\begin{aligned}
{\left\vert {{{\underline{T}}_{t + \Delta}} f - \frac{\max \Phi(s) f + \min \Phi(s) f}{2}} \right\vert}
\leq \epsilon
~~~\text{and }~~
{\left\vert {{{\underline{T}}_{\infty}} f - \frac{\max \Phi(s) f + \min \Phi(s) f}{2}} \right\vert}
\leq \epsilon.
\end{aligned}$$
Without actually stating it, we mention that a similar—though less useful—stopping criterion can be proved for non-ergodic transition rate matrices as well.
Our method for determining ${{\underline{T}}_{\infty}} f$ is now relatively straightforward. Let ${\underline{Q}}$ be an ergodic lower transition rate operator and fix some $f \in {{\mathcal{L}}({\mathcal{X}})}$. We can then approximate ${{\underline{T}}_{\infty}} f$ up to any desired maximal error $\epsilon \in {{\mathbb{R}}_{> 0}}$ as follows. First, we look for some $m \in {\mathbb{N}}$ and some—preferably large—$\delta \in {{\mathbb{R}}_{> 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert}<2$ and $$2m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c} \leq (1 - \beta)\epsilon,
$$ where $\beta \coloneqq {{\rho}((I + \delta {\underline{Q}})^{m})}$. From Theorem \[the:ContinuousErgodicity:CoefficientOfErgodicityOfApproximation\], we know that a possible starting point for $m$ is ${\left\vert {{\mathcal{X}}} \right\vert} - 1$. If we do not have an analytical expression for ${{\rho}((I + \delta {\underline{Q}})^{m})}$, then we can instead use the guaranteed upper bound of Theorem \[the:CoeffOfErgod:Approximation\]—provided it is strictly smaller than one. If no such $m$ and $\delta$ exist—for instance because the guaranteed upper bound on $\beta$ is too conservative—then this method does not work. If on the other hand we do find such an $m$ and $\delta$, then we can keep on running the iterative step (line \[line:Uniform:IncrementOfG\]) of Algorithm \[alg:Uniform\] until we reach the first index $i \in {\mathbb{N}}$ such that ${\left\Vert {g_{i}} \right\Vert}_{c} \leq \nicefrac{\epsilon}{2}$. By Propositions \[prop:UniformApproximationErgodicError\] and \[prop:StoppingCriterionWithErgodicity\], we are now guaranteed that $(\max g_{i} + \min g_{i}) / 2$ is an approximation of ${{\underline{T}}_{\infty}} f$ up to a maximal error $\epsilon$.
Alternatively, we can fix a step size $\delta$ ourselves and use the method of Theorem \[the:UniformApproximationWithError\] to compute $\epsilon'$. In that case, we simply need to run the iterative scheme until we reach the first index $i$ such that ${\left\Vert {g_{i}} \right\Vert}_{c} \leq \epsilon'$. By Proposition \[prop:StoppingCriterionWithErgodicity\], we are then guaranteed that the error $(\max g_{i} + \min g_{i})/2$ is an approximation of ${{\underline{T}}_{\infty}} f$ up to a maximal error $\epsilon=2 \epsilon'$. The same is true if we replace $\epsilon'$ by the error $\epsilon_{e}$ that is used in Proposition \[prop:UniformApproximationErgodicError\].
Using the analytical expressions of Example \[binex:AnalyticalExpressionsForAppliedLTO\], we obtain ${{\underline{T}}_{\infty}} {\mathbb{I}_{1}} \approx \num{9.5238095e-3}$.
We want to approximate ${{\underline{T}}_{\infty}} {\mathbb{I}_{1}}$ up to a maximum error $\epsilon \coloneqq \num{1e-6}$. We observe that $m = \num{1}$ and $\delta \approx \num{3.485e-8}$ yield an $\epsilon_{d}$ that is lower than $\nicefrac{\epsilon}{2}$. After iterations, the norm of the approximation is sufficiently small, resulting in the approximation ${{\underline{T}}_{\infty}} {\mathbb{I}_{1}} = \num{9.524(1)e-3}$. Alternatively, choosing $\delta = \num{1e-7}$ and continuing until ${\left\Vert {g_{i}} \right\Vert}_{c} \leq \epsilon'$ yields the approximation ${{\underline{T}}_{\infty}} {\mathbb{I}_{1}} = \num{9.5242(8)e-3}$ after only iterations.
Mimicking [@2015Troffaes], we also tried the heuristic method of increasing $t$ and $n$ until we observe empirical convergence. After some trying, we find that $t = \num{7}$ and $n = 7 \cdot \num{250} = 1750$ already yield an approximation with sufficiently small error: ${\left\Vert {{{\underline{T}}_{\infty}} {\mathbb{I}_{1}} - \Psi_{7}(1750) {\mathbb{I}_{1}}} \right\Vert} \approx \num{7e-7} < \epsilon$. Note however that for non-binary examples, where ${{\underline{T}}_{\infty}} f$ cannot be computed analytically, this heuristic approach is unable to provide a guaranteed bound.
Conclusion {#sec:Conclusion}
==========
We have improved an existing method and proposed a novel method to approximate ${{\underline{T}}_{t}} f$ up to any desired maximal error, where ${{\underline{T}}_{t}}f$ is the solution of the non-linear differential equation that plays an essential role in the theory of imprecise continuous-time Markov chains. As guaranteed by our theoretical results, and as verified by our numerical examples, our methods outperform the existing method by [@2016Krak], especially if the lower transition rate operator is ergodic. For these ergodic lower transition rate operators, we also proposed a method to approximate $\lim_{t \to \infty} {{\underline{T}}_{t}} f$ up to any desired maximal error.
For the simple case of a binary state space, we observed in numerical examples that there is a rather large difference between the theoretically required number of iterations and the number of iterations that are empirically found to be sufficient. Similar differences can—although this falls beyond the scope of our present contribution—also be observed for the lower transition rate operator that is studied in [@2015Troffaes]. The underlying reason for these observed differences remains unclear so far. On the one hand, it could be that our methods are still on the conservative side, and that further improvements are possible. On the other hand, it might be that these differences are unavoidable, in the sense that guaranteed theoretical bounds come at the price of conservatism. We leave this as an interesting line of future research. Additionally, the performance of our proposed methods for systems with a larger state space deserves further inquiry.
Extra material and proofs for Section \[sec:Preliminaries\] {#app:Preliminaries}
===========================================================
An operator ${\left\Vert {\cdot} \right\Vert}$ on a linear vector space ${\mathcal{L}}$ is a *norm* if it maps ${\mathcal{L}}$ to ${{\mathbb{R}}_{\geq 0}}$ and if for all $a, b \in {\mathcal{L}}$ and all $\mu \in {\mathbb{R}}$,
1. \[def:Norm:ScalarMult\] ${\left\Vert {\mu a} \right\Vert} = {\left\vert {\mu} \right\vert} {\left\Vert {a} \right\Vert}$,
2. \[def:Norm:TriangleInequality\] ${\left\Vert {a + b} \right\Vert} \leq {\left\Vert {a} \right\Vert} + {\left\Vert {b} \right\Vert}$,
3. \[def:Norm:NormZeroOnly\] ${\left\Vert {a} \right\Vert} = 0 \Leftrightarrow a = 0$.
If an operator only satisfies \[def:Norm:ScalarMult\] and \[def:Norm:TriangleInequality\], then it is called a *seminorm*.
It can be immediately checked that the maximum norm ${\left\Vert {\cdot} \right\Vert}$ on ${{\mathcal{L}}({\mathcal{X}})}$ is a proper norm, and similarly for the induced operator norm on non-negatively homogeneous operators from ${{\mathcal{L}}({\mathcal{X}})}$ to ${{\mathcal{L}}({\mathcal{X}})}$. For all $f \in {{\mathcal{L}}({\mathcal{X}})}$ we define the variation seminorm ${\left\Vert {\cdot} \right\Vert}_{v}$ and the centred seminorm ${\left\Vert {\cdot} \right\Vert}_{c}$ as $$\label{eqn:VariationNorm}
{\left\Vert {f} \right\Vert}_{v}
\coloneqq {\left\Vert {f - \min{f}} \right\Vert}
= \max \{ {\left\vert {f(x) - \min{f}} \right\vert} \colon x \in {\mathcal{X}}\}
= \max f - \min f$$ and $$\label{eqn:CentredNorm}
{\left\Vert {f} \right\Vert}_{c}
\coloneqq {\left\Vert {f - {\tilde{f}}} \right\Vert}
= \max \left\{ {\left\vert {f(x) - {\tilde{f}}} \right\vert} \colon x \in {\mathcal{X}}\right\}
= (\max f - \min f) / 2,$$ where ${\tilde{f}} \coloneqq (\max{f} + \min{f})/2$. Verifying that ${\left\Vert {\cdot} \right\Vert}_{v}$ and ${\left\Vert {\cdot} \right\Vert}_{c}$ are seminorms and not norms is straightforward.
\[prop:norms:properties\] For all $f\in{{\mathcal{L}}({\mathcal{X}})}$, all $\mu\in{\mathbb{R}}$ and any non-negatively homogeneous operator $A$,
1. \[prop:norm:CenteredEqVar\] ${\left\Vert {f} \right\Vert}_{c} = {\left\Vert {f} \right\Vert}_{v} / 2$,
2. \[prop:norm:CenteredLeqNormal\] ${\left\Vert {f} \right\Vert}_{c} \leq {\left\Vert {f} \right\Vert}$,
3. \[prop:norm:VarAddConstant\] ${\left\Vert {f + \mu} \right\Vert}_{v} = {\left\Vert {f} \right\Vert}_{v}$,
4. \[prop:norms:BoundOnNormOf\_Af\] ${\left\Vert {A f} \right\Vert} \leq {\left\Vert {A} \right\Vert} {\left\Vert {f} \right\Vert}$,
5. \[prop:norms:NormOfAB\] ${\left\Vert {A B} \right\Vert} \leq {\left\Vert {A} \right\Vert} {\left\Vert {B} \right\Vert}$.
Properties \[prop:norm:CenteredEqVar\], \[prop:norm:CenteredLeqNormal\] and \[prop:norm:VarAddConstant\] follow almost immediately from the definitions of the centred and variation seminorms. Proofs for \[prop:norms:BoundOnNormOf\_Af\] and \[prop:norms:NormOfAB\] can be found in [@2017DeBock].
The following properties of lower transition operators will turn out to be useful in the proofs.
\[prop:LowerTransitionOperator:Properties\] Let ${\underline{T}}$, ${{\underline{T}}_{1}}$, ${{\underline{T}}_{2}}$, $\underline{S}_{1}$ and $\underline{S}_{2}$ be lower transition operators. Then for all $f, g \in {{\mathcal{L}}({\mathcal{X}})}$ and all $\mu \in {\mathbb{R}}$:
1. \[prop:LTO:BoundedByMinAndMax\] $\min f \leq {\underline{T}}f \leq {\overline{T}}f \leq \max f$;
2. \[prop:LTO:AdditionOfConstant\] ${\underline{T}}(f + \mu) = {\underline{T}}(f) + \mu$;
3. \[prop:LTO:Monotonicity\] $f \geq g \Rightarrow {\underline{T}}f \geq {\underline{T}}g$ and ${\overline{T}}f \geq {\overline{T}}g$;
4. ${\left\vert {{\underline{T}}f - {\underline{T}}g} \right\vert} \leq {\overline{T}}({\left\vert {f - g} \right\vert})$;
5. \[prop:LTO:NormLowerThan1\] ${\left\Vert {{\underline{T}}} \right\Vert} \leq 1$;
6. \[prop:LTO:NonExpansiveness\] ${\left\Vert {{\underline{T}}f - {\underline{T}}g} \right\Vert} \leq {\left\Vert {f - g} \right\Vert}$;
7. \[prop:LTO:BoundOnNormTBTB\] ${\left\Vert {{\underline{T}}A - {\underline{T}}B} \right\Vert} \leq {\left\Vert {A - B} \right\Vert}$;
8. \[prop:LTO:VarNormTf\] ${\left\Vert {{\underline{T}}f} \right\Vert}_{v} \leq {\left\Vert {f} \right\Vert}_{v}$;
<!-- -->
1. \[prop:LTO:CompositionIsAlsoLTO\] ${{\underline{T}}_{1}} {{\underline{T}}_{2}}$ is a lower transition operator;
2. \[prop:LTO:DifferenceIsNonNegativeHomogeneous\] $({{\underline{T}}_{1}} - {{\underline{T}}_{2}})$ is a non-negatively homogeneous operator;
3. \[prop:LTO:BoundOnDifferenceTfSf\] ${\left\Vert {{{\underline{T}}_{1}} f - \underline{S}_{1} f} \right\Vert}_{c} \leq {\left\Vert {{{\underline{T}}_{1}} f - \underline{S}_{1} f} \right\Vert} \leq {\left\Vert {{{\underline{T}}_{1}} - \underline{S}_{1}} \right\Vert} {\left\Vert {f} \right\Vert}_{c}$;
4. \[prop:LTO:BoundOnDifferenceTTfSSf\] ${\left\Vert {{{\underline{T}}_{1}} {{\underline{T}}_{2}} f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert}_{c} \leq {\left\Vert {{{\underline{T}}_{1}} {{\underline{T}}_{2}} f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert} \leq {\left\Vert {{{\underline{T}}_{2}} f - \underline{S}_{2} f} \right\Vert} + {\left\Vert {{{\underline{T}}_{1}} - \underline{S}_{1}} \right\Vert} {\left\Vert {\underline{S}_{2} f} \right\Vert}_{c}$.
Proofs for \[prop:LTO:BoundedByMinAndMax\]–\[prop:LTO:BoundOnNormTBTB\] and \[prop:LTO:CompositionIsAlsoLTO\] can be found in [@2017DeBock]. \[prop:LTO:VarNormTf\] follows almost immediately from \[prop:LTO:BoundedByMinAndMax\] and Eqn. : $${\left\Vert {{\underline{T}}f} \right\Vert}_{v}
= \max {\underline{T}}f - \min {\underline{T}}f
\leq \max f - \min f
= {\left\Vert {f} \right\Vert}_{v}.$$
Note that for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ and all $\gamma \in {{\mathbb{R}}_{\geq 0}}$, $$({{\underline{T}}_{1}} - {{\underline{T}}_{2}})(\gamma f)
= {{\underline{T}}_{1}}(\gamma f) - {{\underline{T}}_{2}} (\gamma f)
= \gamma ({{\underline{T}}_{1}} f - {{\underline{T}}_{2}} f)
= \gamma ({{\underline{T}}_{1}} - {{\underline{T}}_{2}}) (f),$$ which proves \[prop:LTO:DifferenceIsNonNegativeHomogeneous\].
Next, we prove \[prop:LTO:BoundOnDifferenceTfSf\]. The first inequality follows from \[prop:norm:CenteredLeqNormal\]. By \[prop:LTO:DifferenceIsNonNegativeHomogeneous\], $({{\underline{T}}_{1}} - \underline{S}_{1})$ is a non-negatively homogeneous operator, such that $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{1}} f - \underline{S}_{1} f} \right\Vert}
&= {\left\Vert {{{\underline{T}}_{1}} f - {\tilde{f}} - \underline{S}_{1} f + {\tilde{f}}} \right\Vert}
= {\left\Vert {{{\underline{T}}_{1}} (f - {\tilde{f}}) - \underline{S}_{1} (f - {\tilde{f}})} \right\Vert} \\
&= {\left\Vert {({{\underline{T}}_{1}} - \underline{S}_{1}) (f - {\tilde{f}})} \right\Vert}
\leq {\left\Vert {{{\underline{T}}_{1}} - \underline{S}_{1}} \right\Vert} {\left\Vert {f - {\tilde{f}}} \right\Vert}
= {\left\Vert {{{\underline{T}}_{1}} - \underline{S}_{1}} \right\Vert} {\left\Vert {f} \right\Vert}_{c},
\end{aligned}$$ where the second equality follows from \[prop:LTO:AdditionOfConstant\], the inequality follows from \[prop:LTO:DifferenceIsNonNegativeHomogeneous\] and \[prop:norms:BoundOnNormOf\_Af\] and the last equality follows from Eqn. .
\[prop:LTO:BoundOnDifferenceTTfSSf\] can be proved similarly. Again, the first inequality of \[prop:LTO:BoundOnDifferenceTTfSSf\] follows from \[prop:norm:CenteredLeqNormal\]. To prove the second inequality of \[prop:LTO:BoundOnDifferenceTTfSSf\], we observe that $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{1}} {{\underline{T}}_{2}} f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert}
&= {\left\Vert {{{\underline{T}}_{1}} {{\underline{T}}_{2}} f - {{\underline{T}}_{1}} \underline{S}_2 f + {{\underline{T}}_{1}} \underline{S}_2 f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert} \\
&\leq {\left\Vert {{{\underline{T}}_{1}} {{\underline{T}}_{2}} f - {{\underline{T}}_{1}} \underline{S}_2 f} \right\Vert} + {\left\Vert {{{\underline{T}}_{1}} \underline{S}_2 f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert} \\
&\leq {\left\Vert {{{\underline{T}}_{2}} f - \underline{S}_2 f} \right\Vert} + {\left\Vert {{{\underline{T}}_{1}} \underline{S}_2 f - \underline{S}_{1} \underline{S}_{2} f} \right\Vert} \\
&\leq {\left\Vert {{{\underline{T}}_{2}} f - \underline{S}_2 f} \right\Vert} + {\left\Vert {{{\underline{T}}_{1}} - \underline{S}_{1}} \right\Vert} {\left\Vert {\underline{S}_{2} f} \right\Vert}_{c},
\end{aligned}$$ where the first inequality follows from \[def:Norm:TriangleInequality\], the second inequality follows from \[prop:LTO:NonExpansiveness\] and the third inequality follows from \[prop:LTO:BoundOnDifferenceTfSf\].
A linear lower transition rate operator ${\underline{Q}}$—one for which \[def:LTRO:SuperAdditive\] holds with equality—can be identified with a matrix $Q$ of dimension ${\left\vert {{\mathcal{X}}} \right\vert} \times {\left\vert {{\mathcal{X}}} \right\vert}$. This matrix is called a *transition rate matrix*, the $(x,y)$-component $Q(x,y)$ of which is equal to $[{\underline{Q}}{\mathbb{I}_{y}}](x)$.
\[lem:BoundsOnElementsOfTransitionMatrix\] Let $Q$ be a transition rate matrix. Then for all $x,y \in {\mathcal{X}}$ such that $x \neq y$,
1. $Q(x,y) \geq 0$, \[lem:RateMatrix:XY\]
2. $Q(x,x) = - \sum_{y\neq x} Q(x,y)$. \[lem:RateMatrix:XX\]
Also, $${\left\Vert {Q} \right\Vert} = 2 \max \left\{ {\left\vert {Q(x,x)} \right\vert} \colon x\in{\mathcal{X}}\right\}.$$
Note that \[lem:RateMatrix:XY\] follows immediately from \[def:LTRO:Sign\]. From \[def:LTRO:Constant\], we find that for all $x\in{\mathcal{X}}$, $[Q {\mathbb{I}_{{\mathcal{X}}}}](x) = 0$. Using the linearity and \[def:LTRO:Constant\] yields $$Q(x,x)
= [Q {\mathbb{I}_{x}}](x)
= \left[Q \left(1 - \sum_{y \neq x} {\mathbb{I}_{y}}\right)\right](x)
= - \sum_{y \neq x} [Q {\mathbb{I}_{y}}](x) = \sum_{y \neq x} Q(x,y).$$
It is a matter of straightforward verification to prove that $${\left\Vert {Q} \right\Vert} = \max \left\{ \sum_{y \in {\mathcal{X}}} {\left\vert {Q(x,y)} \right\vert} \colon x\in{\mathcal{X}}\right\} = 2 \max \left\{ {\left\vert {Q(x,x)} \right\vert} \colon x\in{\mathcal{X}}\right\}. \qedhere$$
Let ${\underline{Q}}$ be a lower transition rate operator. The associated set of dominating rate matrices ${\mathcal{Q}_{{\underline{Q}}}}$, defined as $${\mathcal{Q}_{{\underline{Q}}}}\coloneqq \left\{ Q \text{ a transition rate matrix} \colon (\forall f \in {{\mathcal{L}}({\mathcal{X}})})~{\underline{Q}}f \leq Q f \right\},$$ is non-empty and bounded, and for all $f\in{{\mathcal{L}}({\mathcal{X}})}$ there is some $Q\in{\mathcal{Q}_{{\underline{Q}}}}$ such that ${\underline{Q}}f = Q f$.
\[lem:NormRateMatixLowerThanNormRateOperator\] Let ${\underline{Q}}$ be a lower rate operator, then for any $Q \in {\mathcal{Q}_{{\underline{Q}}}}$, ${\left\Vert {Q} \right\Vert} \leq {\left\Vert {{\underline{Q}}} \right\Vert}$.
\[prop:LowerTransitionRateOperator:Properties\] Let ${\underline{Q}}$ be a lower transition rate operator. Then for all $f\in{{\mathcal{L}}({\mathcal{X}})}$, all $\mu \in {\mathbb{R}}$ and all $x,y\in{\mathcal{X}}$ such that $x\neq y$:
1. \[prop:LTRO:LowUp\] ${\underline{Q}}f \leq {\overline{Q}}f$;
2. \[prop:LTRO:AdditionOfConstant\] ${\underline{Q}}(f + \mu) = {\underline{Q}}f$;
3. \[prop:LTRO:Ixx\] $- {\left\Vert {{\underline{Q}}} \right\Vert} / 2 \leq [{\underline{Q}}{\mathbb{I}_{x}}](x) \leq [{\overline{Q}}{\mathbb{I}_{x}}](x) \leq 0$;
4. $0 \leq \sum_{y \neq x} [{\underline{Q}}{\mathbb{I}_{x}}](y) \leq {\left\Vert {{\underline{Q}}} \right\Vert} / 2$;
5. \[prop:LTRO:Norm\] ${\left\Vert {{\underline{Q}}} \right\Vert} = 2 \max \{ {\left\vert {[{\underline{Q}}{\mathbb{I}_{x}}](x)} \right\vert} \colon x \in {\mathcal{X}}\}$.
The properties \[prop:LTRO:LowUp\] and \[prop:LTRO:AdditionOfConstant\] are proved in [@2017DeBock]. Hence, we only prove the remaining properties.
1. By the conjugacy of ${\underline{Q}}$ and ${\overline{Q}}$, $$\begin{aligned}
[{\overline{Q}}{\mathbb{I}_{x}}](x)
&= \left[{\overline{Q}}\left( 1 - \sum_{z \neq x} {\mathbb{I}_{z}} \right)\right](x)
= - \left[{\underline{Q}}\left( -1 + \sum_{z \neq x} {\mathbb{I}_{z}} \right)\right](x) \\
&\leq - [{\underline{Q}}(-1)](x) - \sum_{z \neq x} [{\underline{Q}}{\mathbb{I}_{z}}](x),
\end{aligned}$$ where the inequality follows from \[def:LTRO:SuperAdditive\]. By \[def:LTRO:Constant\] the first term is zero, such that $$[{\overline{Q}}{\mathbb{I}_{x}}](x) \leq - \sum_{z \neq x} [{\underline{Q}}{\mathbb{I}_{z}}](x) \leq 0,$$ where the second inequality follows from \[def:LTRO:Sign\].
Recall that there is some $Q \in {\mathcal{Q}_{{\underline{Q}}}}$ such that ${\underline{Q}}{\mathbb{I}_{x}} = Q {\mathbb{I}_{x}}$. It holds that $$\begin{aligned}
[{\underline{Q}}{\mathbb{I}_{x}}](x) = [Q {\mathbb{I}_{x}}](x) = Q(x,x) \geq -\frac{{\left\Vert {Q} \right\Vert}}{2} \geq -\frac{{\left\Vert {{\underline{Q}}} \right\Vert}}{2},
\end{aligned}$$ where for the first inequality we used Lemma \[lem:BoundsOnElementsOfTransitionMatrix\] and for the second inequality we used Lemma \[lem:NormRateMatixLowerThanNormRateOperator\].
The property now follows by combining the obtained lower bound for $[{\underline{Q}}{\mathbb{I}_{x}}](x)$ and the obtained upper bound for $[{\overline{Q}}{\mathbb{I}_{x}}](x)$ with \[prop:LTRO:LowUp\].
2. Recall from \[def:LTRO:Sign\] that $[{\underline{Q}}{\mathbb{I}_{y}}](x)$ is non-negative if $y \neq x$, such that $\sum_{y \neq x} [{\underline{Q}}{\mathbb{I}_{y}}](x)$ is non-negative. Some manipulations yield $$\begin{aligned}
0
\leq \sum_{y \neq x} [{\underline{Q}}{\mathbb{I}_{y}}](x)
\leq \left[{\underline{Q}}\left(\sum_{y \neq x} {\mathbb{I}_{y}}\right)\right](x)
&= - \left[{\overline{Q}}\left(- \sum_{y \neq x} {\mathbb{I}_{y}}\right)\right](x) \\
&= - \left[{\overline{Q}}\left(1 - \sum_{y \neq x} {\mathbb{I}_{y}}\right)\right](x)
= - [{\overline{Q}}{\mathbb{I}_{x}}](x) \\
&\leq - [{\underline{Q}}{\mathbb{I}_{x}}](x),
\end{aligned}$$ where the second inequality follows from \[def:LTRO:SuperAdditive\], the first equality follows from conjugacy, the second equality follows from \[prop:LTRO:AdditionOfConstant\], and the final inequality follows from \[prop:LTRO:Ixx\]. Also by \[prop:LTRO:Ixx\], we know that $- [{\underline{Q}}{\mathbb{I}_{x}}](x)$ is non-negative and bounded above by ${\left\Vert {{\underline{Q}}} \right\Vert}/2$, hence $$0 \leq \sum_{y \neq x} [{\underline{Q}}{\mathbb{I}_{y}}](x) \leq \frac{{\left\Vert {{\underline{Q}}} \right\Vert}}{2}.$$
3. Let ${\underline{Q}}$ be a lower transition rate operator. From [@2017DeBock R9] it follows that $${\left\Vert {{\underline{Q}}} \right\Vert} \leq 2 \max_{x \in {\mathcal{X}}} {\left\vert { [{\underline{Q}}{\mathbb{I}_{x}}](x) } \right\vert}.$$ From \[prop:LTRO:Ixx\], however, we know that for all $x \in {\mathcal{X}}$, ${\left\vert { [{\underline{Q}}{\mathbb{I}_{x}}](x) } \right\vert} \leq {\left\Vert {{\underline{Q}}} \right\Vert} / 2$. Combining these two inequalities yields ${\left\Vert {{\underline{Q}}} \right\Vert} = 2 \max \{{\left\vert {[{\underline{Q}}{\mathbb{I}_{x}}](x)} \right\vert} \colon x \in {\mathcal{X}}\}$.
Fix some lower transition rate operator ${\underline{Q}}$ and some $\delta \in {{\mathbb{R}}_{\geq 0}}$. We first prove that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ implies that the operator $(I + \delta {\underline{Q}})$ is a lower transition operator. The operator $(I + \delta {\underline{Q}})$ trivially satisfies \[def:LTO:SuperAdditive\] and \[def:LTO:NonNegativelyHom\], such that we only need to prove that it satisfies \[def:LTO:DominatesMin\]. In order to do so, we fix some arbitrary $x \in {\mathcal{X}}$ and $f \in {{\mathcal{L}}({\mathcal{X}})}$. It holds that $$\begin{aligned}
[(I + \delta {\underline{Q}})f](x)
&= f(x) + \delta [{\underline{Q}}f](x) \\
&= f(x) + \delta [{\underline{Q}}(f - \min f)](x) \\
&= f(x) + \delta \left[{\underline{Q}}\left(\sum_{y\in{\mathcal{X}}} (f(y) - \min f) {\mathbb{I}_{y}}\right)\right](x) \\
&\geq f(x) + \delta (f(x) - \min f) [{\underline{Q}}{\mathbb{I}_{x}}](x) + \delta \sum_{y \neq x} (f(y) - \min f) [{\underline{Q}}{\mathbb{I}_{y}}](x) \\
&\geq f(x) + \delta (f(x) - \min f) [{\underline{Q}}{\mathbb{I}_{x}}](x) \\
&\geq f(x) - \delta (f(x) - \min f) \frac{{\left\Vert {{\underline{Q}}} \right\Vert}}{2},
\intertext{ where the second equality follows \ref{prop:LTRO:AdditionOfConstant}, the first inequality follows from \ref{def:LTRO:SuperAdditive}, the second inequality follows from \ref{def:LTRO:Sign} and the third inequality follows from \ref{prop:LTRO:Ixx}.
Recall that by assumption $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$, and therefore
}
[(I + \delta {\underline{Q}})f](x)
&\geq \min f.
\end{aligned}$$
Next, we prove the reverse implication. Assume that $(I + \delta {\underline{Q}})$ is a transition rate operator. By \[prop:LTRO:Ixx\] and \[prop:LTRO:Norm\], there is some $x \in {\mathcal{X}}$ such that $[{\underline{Q}}{\mathbb{I}_{x}}](x) = - {\left\Vert {{\underline{Q}}} \right\Vert}/2$. Hence, $$\begin{aligned}
[(I + \delta {\underline{Q}}) {\mathbb{I}_{x}}](x)
&= {\mathbb{I}_{x}}(x) + \delta [{\underline{Q}}{\mathbb{I}_{x}}](x)
= 1 - \delta \frac{{\left\Vert {{\underline{Q}}} \right\Vert}}{2}.
\intertext{
If we now assume that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} > 2$, then
}
[(I + \delta {\underline{Q}}) {\mathbb{I}_{x}}](x)
&< 0 \leq \min {\mathbb{I}_{x}},
\end{aligned}$$ which, by \[def:LTO:DominatesMin\], contradicts the initial assumption that $(I + \delta {\underline{Q}})$ is a lower transition operator. This allows us to conclude that if $(I + \delta {\underline{Q}})$ is a lower transition operator, then $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ .
This proposition simply states \[prop:LTRO:Norm\] of Proposition \[prop:LowerTransitionRateOperator:Properties\].
We can immediately verify that ${\underline{Q}}$ satisfies \[def:LTRO:Constant\]–\[def:LTRO:Sign\], such that it is indeed a lower transition rate operator.
Extra material for Section \[sec:MCs\]
======================================
We here give a slightly more detailed description of the differential equation of interest. Recall from the beginning of Section \[sec:MCs\] that [@2015Skulj] proved that for any lower transition rate operator ${\underline{Q}}$ and any $f \in {{\mathcal{L}}({\mathcal{X}})}$, the differential equation $$\frac{\mathrm{d}}{\mathrm{d} t} f_{t} = {\underline{Q}}f_{t}$$ with initial condition $f_{0} \coloneqq f$ has a unique solution for all $t \in {{\mathbb{R}}_{\geq 0}}$. As mentioned by [@2017DeBock], this differential equation actually determines a time-dependent operator ${{\underline{T}}_{t}}$: for all $t \in {{\mathbb{R}}_{\geq 0}}$, ${{\underline{T}}_{t}} f \coloneqq f_{t}$. Even more, [@2017DeBock Proposition 9] states that for all $t \in {{\mathbb{R}}_{\geq 0}}$, the time-dependent operator ${{\underline{T}}_{t}}$ itself satisfies the differential equation $$\label{eqn:TDLTO:DifferentialEquation}
\frac{\mathrm{d} }{\mathrm{d} t} {{\underline{T}}_{t}} = {\underline{Q}}{{\underline{T}}_{t}}$$ with initial condition ${{\underline{T}}_{0}} \coloneqq I$. [@2017DeBock] also shows that this operator ${{\underline{T}}_{t}}$ is a lower transition operator, and that it satisfies the semi-group property: for all $t_1, t_2 \in{{\mathbb{R}}_{\geq 0}}$, $$\label{eqn:TDLTO:SemiGroup}
{{\underline{T}}_{t_1+t_2}} = {{\underline{T}}_{t_1}} {{\underline{T}}_{t_2}}.$$
For a transition rate matrix, Eqn. reduces to the linear differential equation $$\frac{\mathrm{d}}{\mathrm{d} t} T_t = Q T_t$$ with initial condition $T_{0} \coloneqq I$. This differential equation is essential to precise continuous-time Markov chains, and is often referred to as the *forward Kolmogorov* equation. The solution to this differential equation is called the *matrix exponential*, and is denoted by $T_t = e^{t Q}$.
Fix any $\delta \in {{\mathbb{R}}_{\geq 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$, and let $f$ be an arbitrary element of ${{\mathcal{L}}({\mathcal{X}})}$. We immediately obtain that if $f(0) \geq f(1)$, then $$\begin{aligned}
[\Phi(\delta) f](0)
&= f(0) - \delta {\overline{q}_{0}} (f(0) - f(1))
= f(0) - \delta {\overline{q}_{0}} {\left\Vert {f} \right\Vert}_{v}, \\
[\Phi(\delta) f](1)
&= f(1) + \delta {\underline{q}_{1}} (f(0) - f(1))
= f(1) + \delta {\underline{q}_{1}} {\left\Vert {f} \right\Vert}_{v}.
\intertext{Similarly, if $f(0) \leq f(1)$, then}
[\Phi(\delta) f](0)
&= f(0) + \delta {\underline{q}_{0}} {\left\Vert {f} \right\Vert}_{v}, \\
[\Phi(\delta) f](1)
&= f(1) - \delta {\overline{q}_{1}} {\left\Vert {f} \right\Vert}_{v}.
\end{aligned}$$ Therefore, if $f(0) \geq f(1)$ then $$\begin{aligned}
[\Phi(\delta) f](0) - [\Phi(\delta) f](1)
&= {\left\Vert {f} \right\Vert}_{v} (1 - \delta ({\overline{q}_{0}} + {\underline{q}_{1}})),
\intertext{and similarly if $f(0) \leq f(1)$, then}
[\Phi(\delta) f](1) - [\Phi(\delta) f](0)
&= {\left\Vert {f} \right\Vert}_{v} (1 - \delta ({\underline{q}_{0}} + {\overline{q}_{1}})).
\end{aligned}$$ Consequently $$\begin{aligned}
f(0) \geq f(1) &\Rightarrow
\begin{cases}
[\Phi(\delta) f](0) \geq [\Phi(\delta) f](1) &\text{if } \delta ({\overline{q}_{0}} + {\underline{q}_{1}}) \leq 1, \\
[\Phi(\delta) f](0) \leq [\Phi(\delta) f](1) &\text{if } \delta ({\overline{q}_{0}} + {\underline{q}_{1}}) \geq 1,
\end{cases}
\intertext{and}
f(0) \leq f(1) &\Rightarrow
\begin{cases}
[\Phi(\delta) f](0) \leq [\Phi(\delta) f](1) &\text{if } \delta ({\underline{q}_{0}} + {\overline{q}_{1}}) \leq 1, \\
[\Phi(\delta) f](0) \geq [\Phi(\delta) f](1) &\text{if } \delta ({\underline{q}_{0}} + {\overline{q}_{1}}) \geq 1.
\end{cases}
\end{aligned}$$
Fix some $f \in {{\mathcal{L}}({\mathcal{X}})}$, some $t \in {{\mathbb{R}}_{\geq 0}}$ and let $n \in {\mathbb{N}}$ such that $$\begin{aligned}
t ({\overline{q}_{0}} + {\underline{q}_{1}}) \leq n,
t ({\underline{q}_{0}} + {\overline{q}_{1}}) \leq n
~\text{and}~
t {\left\Vert {{\underline{Q}}} \right\Vert}\leq 2 n.
\end{aligned}$$ In this case, we can use the results obtained above to obtain an analytical expression for $\Psi_{t}(n) f$. If $f(0) \geq f(1)$, then $$\begin{aligned}
[\Psi_{t}(n) f](0)
&= f(0) - \frac{t}{n} {\overline{q}_{0}} {\left\Vert {f} \right\Vert}_{v} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{i}, \\
[\Psi_{t}(n) f](1)
&= f(1) + \frac{t}{n} {\underline{q}_{1}} {\left\Vert {f} \right\Vert}_{v} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{i}.
\intertext{Similarly, if $f(0) \leq f(1)$, then}
[\Psi_{t}(n) f](0)
&= f(0) + \frac{t}{n} {\underline{q}_{0}} {\left\Vert {f} \right\Vert}_{v} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\underline{q}_{0}} + {\overline{q}_{1}})\right)^{i}, \\
[\Psi_{t}(n) f](1)
&= f(1) - \frac{t}{n} {\overline{q}_{1}} {\left\Vert {f} \right\Vert}_{v} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\underline{q}_{0}} + {\overline{q}_{1}})\right)^{i}.
\end{aligned}$$
We now use Eqn. to derive analytical expressions for the components of ${{\underline{T}}_{t}} f$. If $f(0) \geq f(1)$, then $$\begin{aligned}
[{{\underline{T}}_{t}} f](0)
&= \lim_{n \to \infty} [\Psi_{t}(n) f](0) \\
&= \lim_{n \to \infty}\Bigg( f(0) - \frac{t}{n} {\overline{q}_{0}} {\left\Vert {f} \right\Vert}_{v} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{i} \Bigg)\\
&= f(0) - {\overline{q}_{0}} {\left\Vert {f} \right\Vert}_{v} \lim_{n \to \infty} \frac{t}{n} \sum_{i = 0}^{n-1} \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{i}.
\intertext{ Let us now assume that ${\overline{q}_{0}} + {\underline{q}_{1}}>0$.
If $t\neq0$ and $n$ is greater than the lower bounds mentioned above, the expression inside the parenthesis is bounded below by $0$ and strictly bounded above by $1$.
Therefore,
}
[{{\underline{T}}_{t}} f](0)
&= f(0) - {\overline{q}_{0}} {\left\Vert {f} \right\Vert}_{v} \lim_{n \to \infty} \frac{t}{n} \frac{1 - \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{n}}{1 - \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)} \\
&= f(0) - \frac{{\overline{q}_{0}}}{{\overline{q}_{0}} + {\underline{q}_{1}}} {\left\Vert {f} \right\Vert}_{v} \lim_{n \to \infty} \left(1 - \left(1 - \frac{t}{n} ({\overline{q}_{0}} + {\underline{q}_{1}})\right)^{n} \right) \\
&= f(0) - \frac{{\overline{q}_{0}}}{{\overline{q}_{0}} + {\underline{q}_{1}}} {\left\Vert {f} \right\Vert}_{v} \left(1 - e^{-t ({\overline{q}_{0}} + {\underline{q}_{1}})} \right),
\intertext{and}
[{{\underline{T}}_{t}} f](1)
&= f(1) + \frac{{\underline{q}_{1}}}{{\overline{q}_{0}} + {\underline{q}_{1}}} {\left\Vert {f} \right\Vert}_{v} \left(1 - e^{-t ({\overline{q}_{0}} + {\underline{q}_{1}})} \right).
\end{aligned}$$ If $t=0$, the obtained expressions hold trivially. Completely analogous, if ${\underline{q}_{0}} + {\overline{q}_{1}}>0$, the case $f(0) \leq f(1)$ yields $$\begin{aligned}
[{{\underline{T}}_{t}} f](0)
&= f(0) + \frac{{\underline{q}_{0}}}{{\underline{q}_{0}} + {\overline{q}_{1}}} {\left\Vert {f} \right\Vert}_{v} \left(1 - e^{-t ({\underline{q}_{0}} + {\overline{q}_{1}})} \right) \\
[{{\underline{T}}_{t}} f](1)
&= f(1) - \frac{{\overline{q}_{1}}}{{\underline{q}_{0}} + {\overline{q}_{1}}} {\left\Vert {f} \right\Vert}_{v} \left(1 - e^{-t ({\underline{q}_{0}} + {\overline{q}_{1}})} \right). \qedhere
\end{aligned}$$
Extra material and proofs for Section \[sec:EfficientComputation\] {#app:EfficientComputation}
==================================================================
In many of the following proofs, we frequently use the following lemma.
\[lem:BoundForErrorOfIPlusDeltaQ\] Let ${\underline{Q}}$ be a lower transition rate operator. For any $\delta \in {{\mathbb{R}}_{\geq 0}}$, ${\left\Vert {{{\underline{T}}_{\delta}} - (I + \delta {\underline{Q}})} \right\Vert} \leq \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2$.
\[lem:ExplicitErrorBound\] Let ${\underline{Q}}$ be a lower transition rate operator, $f\in{{\mathcal{L}}({\mathcal{X}})}$ and $t \in {{\mathbb{R}}_{\geq 0}}$. Let $s \coloneqq (\delta_1, \dots, \delta_k)$ be any sequence in ${{\mathbb{R}}_{\geq 0}}$ such that $\sum_{i = 1}^{k} \delta_i = t$ and, for all $i \in \{ 1, \dots, k \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. Then $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
&\leq \sum_{i = 1}^{k} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}
\intertext{and}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
&\leq \sum_{i = 1}^{k} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {{{\underline{T}}_{\Delta_{i-1}}} f} \right\Vert}_{c},
\end{aligned}$$ where $\Phi_{0} \coloneqq I$ and $\Delta_{0} = 0$, and for all $i \in \{1, \dots, k \}$, $\Phi_{i} \coloneqq (I + \delta_i {\underline{Q}}) \Phi_{i-1}$ and $\Delta_{i} \coloneqq \Delta_{i-1} + \delta_i$.
By the semi-group property of Eqn. , $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
&= {\left\Vert {{{\underline{T}}_{\delta_k}} {{\underline{T}}_{t - \delta_k}} f - (I + \delta_k {\underline{Q}}) \Phi_{k-1} f} \right\Vert}.
\intertext{ By Proposition~\ref{prop:IPlusDeltaQLowTranOp}, the operator $(I + \delta_{i} {\underline{Q}})$ is a lower transition operator for all $i \in \{ 1, \dots, k \}$.
Even more, \ref{prop:LTO:CompositionIsAlsoLTO} implies that the operator $\Phi_{i-1}$ is a lower transition transition operator for all $i \in \{ 1, \dots, k \}$.
Recall that ${{\underline{T}}_{\delta_k}}$ and ${{\underline{T}}_{t-\delta_{k}}}$ are lower transition operators by definition, such that using \ref{prop:LTO:BoundOnDifferenceTTfSSf} and Lemma~\ref{lem:BoundForErrorOfIPlusDeltaQ} yields
}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
&\leq {\left\Vert {{{\underline{T}}_{\delta_k}} - (I + \delta_k {\underline{Q}})} \right\Vert} {\left\Vert {\Phi_{k-1} f} \right\Vert}_{c} + {\left\Vert {{{\underline{T}}_{t - \delta_k}} f - \Phi_{k-1} f} \right\Vert} \\
&\leq \delta_k^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{k-1} f} \right\Vert}_{c} + {\left\Vert {{{\underline{T}}_{t - \delta_k}} f - \Phi_{k-1} f} \right\Vert}.
\intertext{Repeated application of the same trick yields}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
&\leq \sum_{i = 1}^{k} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}.
\end{aligned}$$
The second inequality of the statement can be proved in a completely similar manner.
\[lem:LTRO:SpecialNoApproximationCase\] Let ${\underline{Q}}$ be a lower transition rate operator, $t \in {{\mathbb{R}}_{\geq 0}}$ and $f \in {{\mathcal{L}}({\mathcal{X}})}$. If ${\left\Vert {f} \right\Vert}_{c} = 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} = 0$ or $t = 0$, then ${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(0) f} \right\Vert}={\left\Vert {{{\underline{T}}_{t}} f - f} \right\Vert} = 0$.
If ${\left\Vert {f} \right\Vert}_{c} = 0$, then $\min f = \max f$, or equivalently $f$ is a constant function. From \[prop:LTO:BoundedByMinAndMax\] it follows that in this case ${{\underline{T}}_{t}} f = f$ for all $t \in {{\mathbb{R}}_{\geq 0}}$. If ${\left\Vert {{\underline{Q}}} \right\Vert} = 0$, then ${\underline{Q}}g = 0$ for all $g \in {{\mathcal{L}}({\mathcal{X}})}$. Therefore $$\frac{\mathrm{d}}{\mathrm{d} t} {{\underline{T}}_{t}} f = {\underline{Q}}{{\underline{T}}_{t}} f = 0 \text{ for all } t \in {{\mathbb{R}}_{\geq 0}}.$$ Consequently, ${{\underline{T}}_{t}} f = {{\underline{T}}_{0}} f = I f = f$. If $t = 0$, then we can simply use the initial condition: ${{\underline{T}}_{t}} f = {{\underline{T}}_{0}} f = I f = f$.
In all three cases we find that ${{\underline{T}}_{t}} f = f$, and hence $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(0) f} \right\Vert} = {\left\Vert {{{\underline{T}}_{t}} f - f} \right\Vert} = {\left\Vert {f-f} \right\Vert} = 0. \qedhere$$
\[lem:UniformApproximationWithError\] Let ${\underline{Q}}$ be a lower transition rate operator, $f\in{{\mathcal{L}}({\mathcal{X}})}$, $t \in {{\mathbb{R}}_{\geq 0}}$, $\epsilon \in {{\mathbb{R}}_{> 0}}$ and $n \in {\mathbb{N}}$, and define $\delta \coloneqq t / n$. If $$n \geq \max \left\{ \frac{t {\left\Vert {{\underline{Q}}} \right\Vert}}{2}, \frac{t^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c} }{\epsilon} \right\},$$ then we are guaranteed that $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert}
\leq \epsilon'
\coloneqq \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{i=0}^{n-1} {\left\Vert {\left(I + \delta {\underline{Q}}\right)^{i} f} \right\Vert}_{c}
\leq \epsilon.$$
By Proposition \[prop:IPlusDeltaQLowTranOp\], the operator $(I + \delta {\underline{Q}})$ is a lower transition operator if and only if $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$, or equivalently if and only if $$\label{eqn:UniformApproximationWithError:Ineq1}
n \geq \frac{t {\left\Vert {{\underline{Q}}} \right\Vert}}{2}.$$ From now on, we assume that $n$ satisfies this inequality. Therefore, we may use Lemma \[lem:ExplicitErrorBound\] to yield $$\label{eqn:UniformApproximationWithError:UpperBoundError}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert}
\leq \sum_{i = 0}^{n-1} \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {(I+\delta {\underline{Q}})^{i} f} \right\Vert}_{c}.$$ Note that for any $i \in \{0, \dots, n-1\}$, $(I + \delta {\underline{Q}})^{i}$ is a lower transition operator by \[prop:LTO:CompositionIsAlsoLTO\]; hence it follows from \[prop:LTO:VarNormTf\] that ${\left\Vert {(I+\delta{\underline{Q}})^{i} f} \right\Vert}_{c} \leq {\left\Vert {f} \right\Vert}_{c}$. Therefore $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert}
\leq \sum_{i = 0}^{n-1} \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c} = \frac{t^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}}{n}.
\end{aligned}$$ It is now obvious that if $$\label{eqn:UniformApproximationWithError:Ineq2}
n \geq \frac{t^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}}{\epsilon},$$ then ${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n) f} \right\Vert} \leq \epsilon$. It also follows almost immediately from Eqn. that if $n$ satisfies both Eqns. and , then $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_t(n) f} \right\Vert}
\leq \epsilon'
\coloneqq \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{i=0}^{n-1} {\left\Vert {(I + \delta {\underline{Q}})^{i} f} \right\Vert}_{c}
\leq \epsilon. \qedhere$$
First, we assume $t = 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} = 0$ or ${\left\Vert {f} \right\Vert}_{c} = 0$. In this case, $n = 0$ and $\delta = 0$. By Lemma \[lem:LTRO:SpecialNoApproximationCase\], we find that $${\left\Vert {{{\underline{T}}_{t}} f - g_{(0)}} \right\Vert} = {\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(0) f} \right\Vert} = 0 < \epsilon.$$
Next, we assume $t > 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} > 0$ and ${\left\Vert {f} \right\Vert}_{c} > 0$. In this case, the integer $n$ that is determined on line \[line:Uniform:DetermineN\] of Algorithm \[alg:Uniform\] is just the lowest natural number that satisfies the requirement of Lemma \[lem:UniformApproximationWithError\], from which the stated follows immediately.
\[lem:AdaptiveApproximation\] Let ${\underline{Q}}$ be a lower transition operator, $f \in {{\mathcal{L}}({\mathcal{X}})}$, $t' \in {{\mathbb{R}}_{\geq 0}}$, $\epsilon \in {{\mathbb{R}}_{> 0}}$, $n, m, k \in {\mathbb{N}}$ and let $\delta_{1}, \dots, \delta_{n}$ be a sequence in ${{\mathbb{R}}_{\geq 0}}$. If (i) $k \leq m$, (ii) $k \delta_{n} + \sum_{i = 1}^{n-1} m \delta_i = t'$, and (iii) for all $i \in \{ 1, \dots, n \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ and $$t' {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c} \delta_i \leq \epsilon,$$ where $\Phi_0 \coloneqq I$ and for all $i \in \{ 1, \dots, n-1 \}$, $\Phi_i \coloneqq (I + \delta_i {\underline{Q}})^{m} \Phi_{i-1}$; then $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t'}} f - \Phi_{m,k}(\delta_1,\dots,\delta_n) f} \right\Vert}
&\leq \epsilon'
\coloneqq \sum_{i = 1}^{n} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j=0}^{k_i - 1} {\left\Vert {(I + \delta_{i} {\underline{Q}})^{j} \Phi_{i-1} f} \right\Vert}_{c} \\
&\leq \sum_{i = 1}^{n} k_{i} \delta_{i}^{2} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}
\leq \epsilon,
\end{aligned}$$ where $k_i \coloneqq m$ for all $i \in \{ 1, \dots, n-1 \}$ and $k_{n} \coloneqq k$.
Assume that (i) $1 \leq k \leq m$, (ii) $k \delta_{n} + \sum_{i=1}^{n-1} m \delta_{i} = t'$, and (iii) for all $i \in \{ 1, \dots, n \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. Observe that by Proposition \[prop:IPlusDeltaQLowTranOp\] and \[prop:LTO:CompositionIsAlsoLTO\], the operators $\Phi_0, \dots, \Phi_{n-1}$ are all lower transition operators. From Lemma \[lem:ExplicitErrorBound\], it follows that $$\begin{aligned}
\label{eqn:AdaptiveApproximation:BoundOnError}
{\left\Vert {{{\underline{T}}_{t'}} f - \Phi_{m,k}(\delta_1,\dots,\delta_n) f} \right\Vert}
\leq \sum_{i = 1}^{n} \delta_{i}^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j = 0}^{k_{i}-1} {\left\Vert {(I + \delta_{i} {\underline{Q}})^{j} \Phi_{i-1} f} \right\Vert}_{c}.
\end{aligned}$$ Hence, it is obvious that the contribution of the $i$-th approximation step to (the upper bound of) the error is $$\label{eqn:AdaptiveApproximation:UpperBoundOnError}
\delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j= 0}^{k_{i}-1} {\left\Vert {(I + \delta_{i} {\underline{Q}})^{j} \Phi_{i-1} f} \right\Vert}_{c}
\leq k_{i} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert { \Phi_{i-1} f} \right\Vert}_{c},$$ where the inequality follows from \[prop:LTO:VarNormTf\]. We want that the contribution of the $i$-th approximation step to the error is proportional to its length $k_{i} \delta_i$. Therefore, we demand that the contribution of the $i$-th approximation step is bounded above by $k_{i} \delta_i \epsilon / t'$, which yields the condition $$\label{eqn:AdaptiveApproximation:ConditionOnDelta}
t' \delta_i {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c} \leq \epsilon.$$ It is obvious that the conditions we have imposed on $\delta_{1}, \dots, \delta_{n}$ are those of the statement. Combining Eqns. , and yields $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi_{m,k}(\delta_1,\dots,\delta_n) f} \right\Vert}
&\leq \epsilon' \coloneqq \sum_{i = 1}^{n} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j = 0}^{k_{i}-1} {\left\Vert {(I + \delta_{i} {\underline{Q}})^{j} \Phi_{i-1} f} \right\Vert}_{c} \\
&\leq \sum_{i = 1}^{n} k_{i} \delta_{i}^{2} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}
\leq \epsilon. \qedhere
\end{aligned}$$
We use Algorithm \[alg:Adaptive\] to determine $n$ and $k$, and if applicable also $k_i$, $\delta_{i}$ and $g_{(i,j)}$. If ${\left\Vert {f} \right\Vert}_{c} = 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} = 0$ or $t = 0$, then by Lemma \[lem:LTRO:SpecialNoApproximationCase\] $${\left\Vert {{{\underline{T}}_{t}} f - g_{(0,m)}} \right\Vert} = {\left\Vert {{{\underline{T}}_{t}} f - f} \right\Vert} = 0 < \epsilon.$$
We therefore assume that ${\left\Vert {f} \right\Vert}_{c} > 0$, $\smash{{\left\Vert {{\underline{Q}}} \right\Vert} > 0}$ and $t > 0$, and let $\delta_1, \dots, \delta_n \in {{\mathbb{R}}_{> 0}}$ and $k \in {\mathbb{N}}$ be determined by running Algorithm \[alg:Adaptive\]. Let $t' \coloneqq k \delta_n + \sum_{i=1}^{n-1} m \delta_{i} \leq t$. It is then a matter of straightforward verification that $\delta_1,\dots, \delta_n$ and $k$ satisfy the requirements of Lemma \[lem:AdaptiveApproximation\]: (i) $1 \leq k \leq m$, (ii) $k \delta_n + \sum_{j = 1}^{n-1} m \delta_j = t'$, and (iii) for all $i \in \{ 1, \dots, n\}$, $\delta_i{\left\Vert {{\underline{Q}}} \right\Vert}\leq2$ and $$t' \delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}\leq t \delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c} = t \delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {g_{(i-1,m)}} \right\Vert}_{c} \leq \epsilon.$$ Therefore, $${\left\Vert {{{\underline{T}}_{t'}} f - g_{(n,k)}} \right\Vert}
\leq \sum_{i=1}^{n} \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 \sum_{j = 0}^{k_i - 1} {\left\Vert {g_{(i,j)}} \right\Vert}_{c}
\leq \sum_{i=1}^{n} k_i \delta_i^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {g_{(i-1,m)}} \right\Vert}_{c}
\leq \epsilon.\label{eq:tprimeformula}$$ If $t'=t$, this concludes the proof of the first part of the statement. If $t'<t$, we have that ${\left\Vert {g_{(n,k)}} \right\Vert}_{c} = 0$, which implies that there is some $\mu\in{\mathbb{R}}$ such that $g_{(n,k)}=\mu$. Hence, it follows that $${\left\Vert {{{\underline{T}}_{t}}f-g_{(n,k)}} \right\Vert}
=
{\left\Vert {{{\underline{T}}_{t}}f-\mu} \right\Vert}
={\left\Vert {{{\underline{T}}_{t-t'}}{{\underline{T}}_{t'}}f-{{\underline{T}}_{t-t'}}\mu} \right\Vert}
\leq{\left\Vert {{{\underline{T}}_{t'}}f-\mu} \right\Vert}
={\left\Vert {{{\underline{T}}_{t'}}f-g_{(n,k)}} \right\Vert},$$ where the second equality follows from Eqn. and \[prop:LTO:AdditionOfConstant\] and where the inequality follows from \[prop:LTO:NonExpansiveness\]. Combined with Eqn. , this again implies the first part of the statement.
To prove the final part of the statement, we assume that ${\left\Vert {f} \right\Vert}_{c} > 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} > 0$ and $t > 0$, and let $\delta_1, \dots, \delta_n \in {{\mathbb{R}}_{> 0}}$ and $k \in {\mathbb{N}}$ be constructed by running Algorithm \[alg:Adaptive\]. We let $n_{u}$ denote the number of iterations of the uniform method: $$n_{u}
\coloneqq \left\lceil \max \left\{ \frac{t {\left\Vert {{\underline{Q}}} \right\Vert}}{2}, \frac{t^2 {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}}{\epsilon} \right\} \right\rceil.$$ If we let $\delta_u \coloneqq t / n_{u}$, then obviously $$0 < \delta_{u} \leq \min \left\{ t, \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}} \right\}.$$
We now consider two cases: $n = 1$ and $n > 1$. We start with the case $n=1$. Let $$\delta_{1}^{*} \coloneqq \min \left\{ t, \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}} \right\}.$$ Since $n=1$, it then holds that $t \leq m \delta_{1}^{*}$ and/or ${\left\Vert {g_{(1,m)}} \right\Vert}_{c} = 0$. We first assume that $t \leq m \delta_{1}^{*}$. Note that $\delta_{1}^{*}$ is strictly positive as we have assumed that ${\left\Vert {f} \right\Vert}_{c}$, ${\left\Vert {{\underline{Q}}} \right\Vert}$ and $t$ are strictly positive. We let $k \coloneqq \left\lceil t / \delta_{1}^{*} \right\rceil$ and $\delta_{1} \coloneqq t / k$, such that $$k = \left\lceil \frac{t}{\delta_{1}^{*}} \right\rceil = \left\lceil \max \left\{ 1, \frac{t {\left\Vert {{\underline{Q}}} \right\Vert}}{2}, \frac{t^{2} {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}}{\epsilon} \right\} \right\rceil.$$ As in this case the definitions of $n_{u}$ and $k$ are equivalent, we find that $k + m (n-1) = k = n_{u}$.
Next, we assume that $n = 1$ but $t > m \delta_{1}^{*}$. This can only be the case if ${\left\Vert {g_{(1,m)}} \right\Vert}_{c} = 0$ and $$\delta_{1} \coloneqq \delta_{1}^{*} = \min \left\{ \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {f} \right\Vert}_{c}} \right\}.$$ Therefore $\delta_{u} \leq \delta_{1}$, such that $n_u \geq t / \delta_{1} > m$. As the total number of iterations is $k = m$, it immediately follows that $m (n-1) + k = m < n_u$.
Next, we consider the case $n > 1$. For all $i \in \{ 1, \dots, n-1 \}$, $$\delta_{i}
\coloneqq \min \left\{ \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}} \right\},$$ where $\Phi_0 \coloneqq I$ and $\Phi_{i} \coloneqq (I + \delta_{i}{\underline{Q}})^{m} \Phi_{i-1}$ and this definition is valid because we previously assumed that ${\left\Vert {f} \right\Vert}_{c} > 0$, ${\left\Vert {{\underline{Q}}} \right\Vert} > 0$ and $t > 0$. Note that our definition of $\delta_{i}$ differs from that of line \[line:Adaptive:delta\] in Algorithm \[alg:Adaptive\]: we have left out the upper bound $\Delta = t - \sum_{j = 1}^{i-1} m \delta_{j}$ because this upper bound only plays a part for the final step $\delta_{n}$. As by \[prop:LTO:BoundedByMinAndMax\] ${\left\Vert {\Phi_{i} f} \right\Vert}_{c} \leq {\left\Vert {\Phi_{i-1} f} \right\Vert}_{c}$, we find that $$\delta_{u} \leq \delta_1 \leq \delta_2 \leq \cdots \leq \delta_{n-1},$$ where the first inequality follows from the definition of $\delta_u$. As the step sizes that are used are all larger than the uniform step size, we intuitively expect that the number of necessary iterations will be bounded above by $n_{u}$. To formally prove this, we again distinguish two sub-cases: $k \delta_{n} + \sum_{i = 1}^{n-1} m \delta_{i} < t$ and $k \delta_{n} + \sum_{i = 1}^{n-1} m \delta_{i} = t$.
We first consider the sub-case $k \delta_{n} + \sum_{i = 1}^{n-1} m \delta_{i} < t$. This can only occur if ${\left\Vert {g_{(n,m)}} \right\Vert}_{c} = 0$ and $k = m$. As $m \delta_{n} < t - \sum_{i = 1}^{n-1} m \delta_{i}$ and ${\left\Vert {g_{(n-1,m)}} \right\Vert}_{c} = {\left\Vert {\Phi_{n-1} f} \right\Vert}_{c} > 0$, $$\delta_{n} = \left\{ \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^2 {\left\Vert {\Phi_{n-1} f} \right\Vert}_{c}} \right\} \geq \delta_{n-1},$$ where the inequality follows from ${\left\Vert {\Phi_{n-2} f} \right\Vert}_{c} \geq {\left\Vert {\Phi_{n-1} f} \right\Vert}_{c}$. Note that $$m n \delta_{1} = \left(k + m (n-1)\right) \delta_{1} \leq k \delta_{n} + \sum_{i=1}^{n-1} m \delta_{i} < t = n_{u} \delta_{u},$$ where the first inequality follows from the increasing character of $\delta_1, \dots, \delta_n$. If we divide both sides of the inequality by $\delta_{1}$, then we find that $m n < n_{u} \delta_{u} / \delta_{1}$. Using that $\delta_{u} \leq \delta_{1}$ now yields that the total number of iterations $k + (n-1) m = m n$ is strictly smaller than $n_{u}$.
Next, we consider the sub-case $k \delta_{n} + \sum_{i = 1}^{n-1} m \delta_{i} = t$. Because $1 \leq k \leq m$ and $\delta_{n}>0$, $\sum_{i=1}^{n-1} m \delta_{i} < t = n_{u} \delta_{u}$. Hence, there is some $n_{u}' < n_{u}$ such that $n_{u}' \delta_{u} < \sum_{i=1}^{n-1} m \delta_{i} \leq (n_{u}' + 1) \delta_{u}$. The final step size $\delta_{n}$ is derived from the remaining time $$\begin{aligned}
t - \sum_{i=1}^{n-1} m \delta_{i}
\eqqcolon \Delta
&\geq n_{u} \delta_{u} - (n_{u}' + 1) \delta_{u} = (n_{u} - n_{u}' - 1) \delta_{u}, \\
\Delta
&< n_{u} \delta_{u} - n_{u}' \delta_{u} = (n_{u} - n_{u}') \delta_{u},
\end{aligned}$$ where the first inequality follows from $\sum_{i=1}^{n-1} m\delta_{i} \leq (n_{u}' + 1) \delta_{u}$ and the second inequality follows from $\sum_{i=1}^{n-1} m\delta_{i} > n_{u}' \delta_{u}$. We first determine the maximal allowable final step size $$\delta_{n}^{*}
\coloneqq \min \left\{ \Delta, \frac{2}{{\left\Vert {{\underline{Q}}} \right\Vert}}, \frac{\epsilon}{t {\left\Vert {{\underline{Q}}} \right\Vert}^{2} {\left\Vert {\Phi_{n-1} f} \right\Vert}_{c}} \right\},$$ and then determine the actual final step size as $\delta_{n} \coloneqq \Delta / k$, with $1 \leq k \coloneqq \lceil \Delta / \delta_{n}^{*} \rceil \leq m$.
If $(n_{u} - n_{u}' - 1) > 0$, then $\Delta \geq (n_{u} - n_{u}' - 1) \delta_{u} \geq \delta_{u}$. Therefore, and because the two other upper bounds of $\delta_{n}^{*}$ are also greater than $\delta_{u}$, we find that $\delta_{n}^{*} \geq \delta_{u}$. From this, we infer that $k = \left\lceil \nicefrac{\Delta}{\delta_{n}^{*}} \right\rceil \leq \left\lceil \nicefrac{\Delta}{\delta_{u}} \right\rceil$. As $\Delta < (n_{u} - n_{u}') \delta_{u}$, we now find that $k \leq (n_{u} - n_{u}')$. Note that $$m(n-1) \delta_{1} \leq \sum_{i = 1}^{n-1} m \delta_{i} \leq (n_{u}' + 1) \delta_{u},$$ where the first inequality follows from the non-decreasing character of $\delta_{1}, \dots, \delta_{n-1}$. Dividing both sides of the inequality by $\delta_{1}$ and using $\delta_{u} \leq \delta_{1}$ yields $m (n-1) \leq n_{u}' + 1$.
If $m (n-1) < n_{u}' + 1$, then combining this strict inequality with the obtained upper bound for $k$ yields $$k + m(n-1) < (n_{u} - n_{u}') + (n_{u}' + 1) = n_{u} + 1,$$ which implies that $k+m(n-1)\leq n_u$, as desired.
If $m (n-1) = n_{u}' + 1$, then $$ \Delta = t - \sum_{i = 1}^{n-1} m \delta_{i} \leq t - \sum_{i = 1}^{n-1} m \delta_{u} = (n_{u} - n_{u}' - 1) \delta_{u},$$ where the inequality is allowed because $\delta_{u} \leq \delta_{1}, \dots, \delta_{n-1}$. As we previously proved that $\Delta \geq (n_{u} - n_{u}' - 1) \delta_{u}$, we obtain that $m (n-1) = n_{u}' + 1$ implies that $\Delta = (n_{u} - n_{u}' - 1) \delta_{u}$. As $\delta_{n}^{*} \geq \delta_{u}$, in this case we are guaranteed that $k = \lceil \nicefrac{\Delta}{\delta_{n}^{*}} \rceil = \lceil \nicefrac{(n_{u} - n_{u}' - 1) \delta_{u}}{\delta_{n}^{*}} \rceil \leq (n_{u} - n_{u}' - 1)$. Hence, we again find that $$k + m(n-1) \leq (n_{u} - n_{u}' - 1) + (n_{u}' + 1) = n_{u},$$ as desired.
If $(n_{u} - n_{u}' - 1) = 0$, then $\Delta < (n_{u} - n_{u}') \delta_{u} = \delta_{u}$. As the two other upper bounds on $\delta_{n}^{*}$ are greater than $\delta_{u}$, this implies that $\delta_{n}^{*} = \Delta$. Consequently, $k = \lceil \nicefrac{\Delta}{\delta_{n}^{*}} \rceil = \lceil \nicefrac{\Delta}{\Delta} \rceil = 1$. Note that $$m(n-1) \delta_{1} \leq \sum_{i = 1}^{n-1} m \delta_{i} < n_{u} \delta_{u},$$ from which it follows that $m (n-1) < n_{u}$. Hence, we find that $k + m (n-1) < 1 + n_{u}$, and therefore also, once more, that $k + m (n-1) \leq n_{u}$. This concludes the proof.
A more thorough look at ergodicity {#app:Extra:QualitativeErgodicity}
==================================
Before we prove the results of Section \[sec:ergodicity\], we need to properly introduce the ergodicity of lower transition (rate) operators. We explicitly chose not to do this in the main text, as the main focus of this contribution is approximating ${{\underline{T}}_{t}} f$. Nevertheless, we now give a brief overview of the relevant literature, limiting ourselves to the qualitative point of view of [@decooman2009], [@2012Hermans] and [@2017DeBock].
Qualitatively characterising ergodicity of lower transition operators
---------------------------------------------------------------------
Recall that a lower transition rate operator is ergodic if and only if ${{\underline{T}}_{t}} f$ converges to a constant function for all $f \in {{\mathcal{L}}({\mathcal{X}})}$. [@2012Hermans] say something similar for lower transition operators.
A lower transition operator ${\underline{T}}$ is *ergodic* if, for all $f \in {{\mathcal{L}}({\mathcal{X}})}$, the limit $\lim_{n \to \infty} {\underline{T}}^{n} f$ exists and is a constant function.
The condition of this definition can, in general, not be checked in practice. Nonetheless, @2012Hermans provide a necessary and sufficient condition for the ergodicity of a lower transition operator, based on the following definition.
\[def:LowTranOp:RegularlyAbsorbing\] The lower transition operator ${\underline{T}}$ is *regularly absorbing* if it is (i) *top class regular*, i.e. $${{\mathcal{X}}_{\mathit{PA}}}
\coloneqq \left\{ x \in {\mathcal{X}}\colon (\exists n \in {\mathbb{N}})(\forall y \in {\mathcal{X}})~[{\overline{T}}^n {\mathbb{I}_{x}}](y) > 0 \right\} \neq 0,$$ and (ii) *top class absorbing*, i.e. $$(\forall y \in {\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{PA}}})(\exists n \in {\mathbb{N}})~[{\underline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y) > 0.$$
\[prop:DiscreteErgodicity:NecAndSuff\] The lower transition operator ${\underline{T}}$ is ergodic if and only if it is regularly absorbing.
[@decooman2009] mention an equivalent way of looking at top class regularity that uses the ternary accessibility relation $\cdot {\overset{\cdot}{\rightsquigarrow}} \cdot$.
\[def:LowTranOp:PossiblyAccesible\] Let ${\underline{T}}$ be any lower transition operator. For all $x,y \in {\mathcal{X}}$ and all $n \in {{\mathbb{N}}_{0}}$, we say that *$x$ is possibly accessible from $y$ in $n$ steps*, denoted by $y {\overset{n}{\rightsquigarrow}} x$, if and only if $[{\overline{T}}^n {\mathbb{I}_{x}}](y)>0$. If there is some $n \in {{\mathbb{N}}_{0}}$ such that $y {\overset{n}{\rightsquigarrow}} x$, then the state $x$ is simply said to be *possibly accessible* from the state $y$, denoted by $y {\rightsquigarrow}x$.
\[lem:LowTranOp:PossiblyAccesibleSequence\] Let ${\underline{T}}$ be a lower transition operator, $x, y \in {\mathcal{X}}$ and $n \in {\mathbb{N}}$. Then $y {\overset{n}{\rightsquigarrow}} x$ if and only if there is a sequence $y = x_0, \dots, x_{n} = x$ in ${\mathcal{X}}$ such that for all $k \in \{ 1,\dots, n \}$, $[{\overline{T}}{\mathbb{I}_{x_{k}}}](x_{k-1})>0$.
Follows immediately from [@2012Hermans Proposition 4].
It can be almost immediately verified—for instance using Lemma \[lem:LowTranOp:PossiblyAccesibleSequence\]—that $\cdot {\overset{\cdot}{\rightsquigarrow}} \cdot$ satisfies the three defining properties of a ternary accessibility relation:
1. $(\forall x,y \in {\mathcal{X}})~x {\overset{0}{\rightsquigarrow}} y \Leftrightarrow x = y$,
2. \[def:AccesRelation:xyz\] $(\forall x,y,z \in {\mathcal{X}}) (\forall n,m \in {{\mathbb{N}}_{0}})~x {\overset{n}{\rightsquigarrow}} y \text{ and } y {\overset{m}{\rightsquigarrow}} z \Rightarrow x {\overset{n+m}{\rightsquigarrow}} z$,
3. $(\forall x \in {\mathcal{X}})(\forall n \in {\mathbb{N}})(\exists y \in {\mathcal{X}}) x {\overset{n}{\rightsquigarrow}} y$.
The following proposition is the reason why we introduced the accessibility relation $\cdot {\overset{\cdot}{\rightsquigarrow}} \cdot$.
\[prop:LowTranOp:AlternativeDefinitionOfTopClassRegularity\] The lower transition operator ${\underline{T}}$ is top class regular if and only if $${{\mathcal{X}}_{\mathit{PA}}} = \{ x\in{\mathcal{X}}\colon (\exists n \in {\mathbb{N}})(\forall k \geq n)(\forall y \in {\mathcal{X}})~y {\overset{k}{\rightsquigarrow}} x \} \neq \emptyset.$$
\[lem:LowTranOp:TopClassRegularSpecialValues\] If the lower transition operator ${\underline{T}}$ is top class regular, then for all $x \in {{\mathcal{X}}_{\mathit{PA}}}$, all $y \in {{\mathcal{X}}_{\mathit{PA}}}^{c}$ and all $k \in {\mathbb{N}}$, $$\begin{aligned}
[{\overline{T}}^k {\mathbb{I}_{y}}](x)
&= 0
& \text{and} & &
[{\underline{T}}^k {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x)
&= 1.
\end{aligned}$$
Let ${\underline{T}}$ be a top class regular lower transition operator with regular top class ${{\mathcal{X}}_{\mathit{PA}}}$. We first prove the first equality. To this end, we fix some arbitrary $x \in {{\mathcal{X}}_{\mathit{PA}}}$ and $y \in {{\mathcal{X}}_{\mathit{PA}}}^{c}$. Assume ex-absurdo that there is some $k \in {\mathbb{N}}$ such that $[{\overline{T}}^{k} {\mathbb{I}_{y}}](x) > 0$. By Definition \[def:LowTranOp:PossiblyAccesible\], this assumption is equivalent to $x {\overset{k}{\rightsquigarrow}} y$. By Proposition \[prop:LowTranOp:AlternativeDefinitionOfTopClassRegularity\], there is some $n \in {\mathbb{N}}$ such that for all $n \leq \ell \in {\mathbb{N}}$ and $z \in {\mathcal{X}}$, $z {\overset{\ell}{\rightsquigarrow}} x$. As a consequence of \[def:AccesRelation:xyz\], we find that for all $z \in {\mathcal{X}}$, $z {\overset{\ell+k}{\rightsquigarrow}} y$, which in turn implies that $y \in {{\mathcal{X}}_{\mathit{PA}}}$. However, this obviously contradicts our initial assumption, such that for all $k \in {\mathbb{N}}$, $[{\overline{T}}^{k} {\mathbb{I}_{y}}](x) = 0$.
Next, we prove the second statement. From the conjugacy of ${\underline{T}}$ and ${\overline{T}}$ and \[prop:LTO:AdditionOfConstant\], it follows that $$\begin{aligned}
{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}
&= - {\overline{T}}(-{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}})
= 1 - {\overline{T}}(1 - {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}})
= 1 - {\overline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}^{c}}}.
\end{aligned}$$ From the conjugacy of ${\underline{T}}$ and ${\overline{T}}$ and \[def:LTO:SuperAdditive\], it follows that $${\overline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}^{c}}}
\leq \sum_{z \in {{\mathcal{X}}_{\mathit{PA}}}^{c}} {\overline{T}}{\mathbb{I}_{z}}.$$ From the—already proven—first equality of the statement, we know that $\sum_{z \in {{\mathcal{X}}_{\mathit{PA}}}^{c}} [{\overline{T}}{\mathbb{I}_{z}}](x) = 0$, hence $$[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) = 1 - [{\overline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}^{c}}}](x)
\geq 1 - \sum_{z \in {{\mathcal{X}}_{\mathit{PA}}}^{c}} [{\overline{T}}{\mathbb{I}_{z}}](x) = 1.$$ Note that by \[prop:LTO:BoundedByMinAndMax\], $[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) \leq \max {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} = 1$. By combining the two obtained inequalities, we find that the the second equality of the statement holds for $k = 1$: $[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) = 1$. Next, fix some $k > 1$, and assume that the second equality holds for all $1 \leq \ell \leq k-1$. Then by the induction hypothesis and \[prop:LTO:BoundedByMinAndMax\], ${\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} \leq {\underline{T}}^{k-1} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}$. By \[prop:LTO:Monotonicity\], this implies that ${\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} \leq {\underline{T}}^{k} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}$. As by the induction hypothesis $[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) = 1$, we find that $[{\underline{T}}^{k} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) \geq 1$. It immediately follows from \[prop:LTO:BoundedByMinAndMax\] and \[prop:LTO:CompositionIsAlsoLTO\] that ${\underline{T}}^k {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} \leq 1$. Hence, we have shown that $[{\underline{T}}^{k} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) = 1$, which finalises the proof.
The following proposition is an altered statement of [@2012Hermans Proposition 6].
\[prop:LowTranOp:TopClassAbsorbingWithRecursion\] Let ${\underline{T}}$ be a top class regular lower transition operator. Then ${\underline{T}}$ is top class absorbing if and only if $B_{n} = {\mathcal{X}}$, where $\{ B_k \}_{k\in{{\mathbb{N}}_{0}}}$ is the sequence defined by the initial condition $B_0 \coloneqq {{\mathcal{X}}_{\mathit{PA}}}$ and, for all $k \in {{\mathbb{N}}_{0}}$, by the recursive relation $$B_{k+1}
\coloneqq B_{k} \cup \big\{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \big\} = \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\},$$ and where $n \leq {\left\vert {{\mathcal{X}}\setminus{{\mathcal{X}}_{\mathit{PA}}}} \right\vert}$ is the first index such that $B_{n} = B_{n+1}$.
Let ${\underline{T}}$ be a top class regular lower transition operator with regular top class ${{\mathcal{X}}_{\mathit{PA}}}$. By [@2012Hermans Proposition 6], ${\underline{T}}$ is top class absorbing if and only if $A_n = \emptyset$, where $A_n$ is the set determined by the initial condition $A_0 \coloneqq {\mathcal{X}}\setminus{{\mathcal{X}}_{\mathit{PA}}}$ and, for all $k \in {{\mathbb{N}}_{0}}$, by the recursive relation $$\begin{aligned}
A_{n+1}
\coloneqq \{ x \in A_{k} \colon [{\overline{T}}{\mathbb{I}_{A_{k}}}](x) = 1 \},
\end{aligned}$$ and where $n \leq {\left\vert {{\mathcal{X}}\setminus{{\mathcal{X}}_{\mathit{PA}}}} \right\vert}$ is the first index for which $A_{n} = A_{n+1}$. For any $k \in {{\mathbb{N}}_{0}}$, $${\overline{T}}{\mathbb{I}_{A_{k}}} = - {\underline{T}}(- {\mathbb{I}_{A_{k}}}) =1 - {\underline{T}}(1 - {\mathbb{I}_{A_{k}}}) = 1 - {\underline{T}}{\mathbb{I}_{{\mathcal{X}}\setminus A_{k}}},$$ where the first equality follows from the conjugacy of ${\underline{T}}$ and ${\overline{T}}$ and the second equality follows from \[prop:LTO:AdditionOfConstant\]. Therefore, for all $x \in A_k$, $[{\overline{T}}{\mathbb{I}_{A_{k}}}](x) = 1$ if and only if $[{\underline{T}}{\mathbb{I}_{{\mathcal{X}}\setminus A_{k}}}](x) = 0$. Observe that $A_{k+1} \subseteq A_{k}$ and define $B_{k} \coloneqq {\mathcal{X}}\setminus A_{k}$ for all $k \in {{\mathbb{N}}_{0}}$. Note that for all $k \in {{\mathbb{N}}_{0}}$, $B_{k} \subseteq B_{k+1}$ and $$B_{k+1} \setminus B_{k}
= A_{k} \setminus A_{k+1}
= \{ x \in A_{k} \colon [{\underline{T}}{\mathbb{I}_{{\mathcal{X}}\setminus A_{k}}}](x) > 0 \}
= \{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \}.$$ Observe that $B_0 = {\mathcal{X}}\setminus A_0 = {{\mathcal{X}}_{\mathit{PA}}}$ and by the previous equality, for all $k \in {{\mathbb{N}}_{0}}$, $$B_{k+1}
= B_{k} \cup \{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \}.$$
We now prove by induction that $$B_{k+1}
= \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\} ~\text{for all}~k \in {{\mathbb{N}}_{0}}.$$ First, we consider the case $k = 0$. Recall from Lemma \[lem:LowTranOp:TopClassRegularSpecialValues\] that $[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x_0) > 0$ for all $x_0 \in {{\mathcal{X}}_{\mathit{PA}}}$. Hence, $$\begin{aligned}
B_{1}
&= B_{0} \cup \{ x \in {\mathcal{X}}\setminus B_{0} \colon [{\underline{T}}{\mathbb{I}_{B_{0}}}](x) > 0 \} \\
&= \{ x \in B_{0} \colon [{\underline{T}}{\mathbb{I}_{B_{0}}}](x) > 0 \} \cup \{ x \in {\mathcal{X}}\setminus B_{0} \colon [{\underline{T}}{\mathbb{I}_{B_{0}}}](x) > 0 \} \\
&= \{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{0}}}](x) > 0 \}.
\end{aligned}$$ Next, we fix some $i \in {\mathbb{N}}$ and assume that the equality holds for all $k < i$. We now prove that the equality then also holds for $k = i$. Observe that $B_{k-1} \subseteq B_{k}$ implies ${\mathbb{I}_{B_{k-1}}} \leq {\mathbb{I}_{B_{k}}}$, which by \[prop:LTO:Monotonicity\] implies that ${\underline{T}}{\mathbb{I}_{B_{k-1}}} \leq {\underline{T}}{\mathbb{I}_{B_{k}}}$. Therefore, for all $x\in B_{k}$, since the induction hypothesis implies that $[{\underline{T}}{\mathbb{I}_{B_{k-1}}}](x) > 0$, we find $[{\underline{T}}{\mathbb{I}_{B_{k}}}](x) \geq [{\underline{T}}{\mathbb{I}_{B_{k-1}}}](x) > 0$. Hence, $$\begin{aligned}
B_{k+1}
&= B_{k} \cup \{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \} \\
&= \{ x \in B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \} \cup \{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \} \\
&= \{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \}. \qedhere
\end{aligned}$$
The observant reader might have noticed that our definitions of top class regularity and top class absorption differ slightly from those in [@2012Hermans], but they are actually entirely equivalent. For top class regularity, we demand that there is some $n\in{\mathbb{N}}$ such that ${\overline{T}}^n {\mathbb{I}_{x}} > 0$. By \[prop:LTO:BoundedByMinAndMax\], for any $k \geq n$ it then holds that $\smash{{\overline{T}}^k {\mathbb{I}_{x}} > 0}$, which is what @2012Hermans demand. For top class absorption, @2012Hermans demand that $$(\forall y \in {{\mathcal{X}}_{\mathit{PA}}}^{c})(\exists n\in{\mathbb{N}})~[{\overline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}^{c}}}](y) < 1,$$ where ${{\mathcal{X}}_{\mathit{PA}}}^{c}\coloneqq{\mathcal{X}}\setminus{{\mathcal{X}}_{\mathit{PA}}}$. Note that $[{\overline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}^{c}}}](y) = 1 - [{\underline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y)$, such that their demand is equivalent to our demand $$(\forall y \in {{\mathcal{X}}_{\mathit{PA}}}^{c})(\exists n\in{\mathbb{N}})~[{\underline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y) > 0.$$ By Lemma \[lem:LowTranOp:TopClassRegularSpecialValues\], for all $n\in{\mathbb{N}}$ and all $y\in{{\mathcal{X}}_{\mathit{PA}}}$, $[{\underline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y) > 0$, such that we could actually demand that $$(\forall y \in {\mathcal{X}})(\exists n\in{\mathbb{N}})~[{\underline{T}}^n {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y) > 0.$$
Qualitatively characterising ergodicity of lower transition rate operators
--------------------------------------------------------------------------
We now turn to the ergodicity of imprecise continuous-time Markov chains. A first and thorough study of the quantitative aspects concerning ergodicity was conducted by @2017DeBock. We only recall the definitions and results from [@2017DeBock] that will be relevant to us in the remainder.
\[def:LowRateOp:UpperReachable\] A state $x\in{\mathcal{X}}$ is upper reachable from the state $y\in{\mathcal{X}}$, denoted by $y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x$, if (i) $x = y$, or (ii) there is some sequence $y=x_0, \dots, x_{n}=x$ in ${\mathcal{X}}$ of length $n+1 \geq 2$ such that for all $k\in\{1,\dots,n\}$, $[{\overline{Q}}{\mathbb{I}_{x_{k}}}](x_{k-1}) > 0$.
Note that a state $x$ is always upper reachable from itself! Rather remarkably, this definition of upper reachability is strikingly similar to the alternative condition of Lemma \[lem:LowTranOp:PossiblyAccesibleSequence\] for possible accessibility. The links between these two definition will be made more explicit later.
\[lem:LowRateOp:UpperReachableShorterSequence\] Let ${\underline{Q}}$ be a lower rate operator, and $x,y\in{\mathcal{X}}$ such that $x \neq y$. Then $x$ is upper reachable from $y$ if and only if there is some sequence $y=x_0, \dots, x_{n}=x$ in ${\mathcal{X}}$ in which every state occurs at most once and for all $k\in\{1,\dots,n\}$, $[{\overline{Q}}{\mathbb{I}_{x_{k}}}](x_{k-1}) > 0$. Consequently, $n < {\left\vert {{\mathcal{X}}} \right\vert}$.
The forward implication follows almost immediately from Definition \[def:LowRateOp:UpperReachable\]. Assume that $y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x$, then by Definition \[def:LowRateOp:UpperReachable\] there is some sequence $y=x_0, \dots, x_{n} = x$ in ${\mathcal{X}}$ such that for all $k\in\{1,\dots,n\}$, $[{\overline{Q}}{\mathbb{I}_{x_{k}}}](x_{k-1}) > 0$. Assume that there is a state $z \in {\mathcal{X}}$ that occurs more than once in this sequence. Then we can simply delete every element of the sequence from right after the the first occurrence of $z$ up to and including the last occurrence of $z$, and still have a valid sequence. If we continue this way, then we end up with a sequence in which every state occurs at most once. As every state occurs at most once, the length $n+1$ of the sequence is lower than or equal to ${\left\vert {{\mathcal{X}}} \right\vert}$. Consequently, $n < {\left\vert {{\mathcal{X}}} \right\vert}$.
The reverse implication follows from the fact that the requirements if Definition \[def:LowRateOp:UpperReachable\] are trivially satisfied.
\[lem:LowRateOp:PathOfArbitraryLength\] Let ${\underline{Q}}$ be a lower transition rate operator, and $x,y \in {\mathcal{X}}$ such that $y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x$. Then there is an integer $n < {\left\vert {{\mathcal{X}}} \right\vert}$ such that for all $k \geq n$ and all $\delta_1, \dots, \delta_k \in {{\mathbb{R}}_{> 0}}$ such that $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$ for all $i \in \{ 1, \dots, k \}$, there is a sequence $y = x_0, \dots, x_{k} = x$ in ${\mathcal{X}}$ such that $[(I + \delta_{i} {\overline{Q}}) {\mathbb{I}_{x_{i}}}](x_{i-1}) > 0$ for all $i \in \{ 1, \dots, k \}$.
We first consider the special case $x = y$. For all $\delta \in {{\mathbb{R}}_{> 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} < 2$, $$[(I + \delta {\overline{Q}}) {\mathbb{I}_{x}}](x)
= {\mathbb{I}_{x}}(x) + \delta [{\overline{Q}}{\mathbb{I}_{x}}](x)
= 1 + \delta [{\overline{Q}}{\mathbb{I}_{x}}](x)
> 0,$$ where the inequality follows from \[prop:LTRO:Ixx\]. Therefore, for all $k \in {\mathbb{N}}$ and all $\delta_1, \dots, \delta_k \in {{\mathbb{R}}_{> 0}}$ such that for all $i \in \{ 1, \dots, k \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$, we find that $[(I + \delta_{i} {\overline{Q}}){\mathbb{I}_{x}}](x) > 0$ for all $i \in \{ 1, \dots, k \}$.
Next, we consider the case $y \neq x$. From Lemma \[lem:LowRateOp:UpperReachableShorterSequence\] we know that there is a sequence $S_y \coloneqq (y = x_0, \dots, x_{n} = x)$ in ${\mathcal{X}}$ such that every state occurs at most once—i.e. $n < {\left\vert {{\mathcal{X}}} \right\vert}$—and for all $i \in \{ 1,\dots,n \}$, $[{\overline{Q}}{\mathbb{I}_{x_{i}}}](x_{i-1}) > 0$. We fix an arbitrary $k \geq n$ and an arbitrary sequence $\delta_1, \dots, \delta_{k}$ in ${{\mathbb{R}}_{> 0}}$ such that for all $i \in \{ 1, \dots, k \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. Note that for all $i \in \{ 1, \dots, n \}$, $$0 < \delta_i [{\overline{Q}}{\mathbb{I}_{x_{i}}}](x_{i-1})
= {\mathbb{I}_{x_{i}}}(x_{i-1}) + \delta_{i} [{\overline{Q}}{\mathbb{I}_{x_{i}}}](x_{i-1})
= [(I + \delta_i {\overline{Q}}) {\mathbb{I}_{x_{i}}}](x_{i-1}),$$ where the inequality follows from $0 < \delta_i$ and the first equality is true because—by construction—$x_{i} \neq x_{i-1}$. Also, from the previous we know that for all $i \in \{ n+1, \dots, k \}$, $[(I + \delta_{i} {\overline{Q}}){\mathbb{I}_{x}}](x) > 0$. Hence, appending the sequence $S_y$ with $(k - n)$ times $x$ yields a sequence $y = x_0, \dots, x_{k} = x$ in ${\mathcal{X}}$ such that for all $i \in \{ 1, \dots, k \}$, $[(I + \delta_{i} {\overline{Q}}) {\mathbb{I}_{x_{i}}}](x_{i-1}) > 0$.
\[def:LowRateOp:LowerReachable\] A (non-empty) set of states $A \subseteq {\mathcal{X}}$ is lower reachable from the state $x$, denoted by $x {\mathrel{\ooalign{\hss${\underline{Q}}$\hss\cr$\longrightarrow$}}}A$, if $x \in B_n$, where $\{ B_k \}_{k \in {{\mathbb{N}}_{0}}}$ is the sequence that is defined by the initial condition $B_0 \coloneqq A$ and for all $k \in {{\mathbb{N}}_{0}}$ by the recursive relation $$B_{k+1}
\coloneqq B_{k} \cup \left\{ y \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{Q}}{\mathbb{I}_{B_{k}}}](y) > 0 \right\},$$ and $n \leq {\left\vert {{\mathcal{X}}\setminus A} \right\vert}$ is the first index for which $B_{k} = B_{k+1}$.
Again, remark the striking similarity between Definition \[def:LowRateOp:LowerReachable\] and Proposition \[prop:LowTranOp:TopClassAbsorbingWithRecursion\].
\[def:LowRateOp:RegularlyAbsorbing\] A lower transition rate operator ${\underline{Q}}$ is *regularly absorbing* if it is (i) *top class regular*, i.e. $${{\mathcal{X}}_{\mathit{R}}}
\coloneqq \left\{ x \in {\mathcal{X}}\colon (\forall y \in {\mathcal{X}})~y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x \right\} \neq 0,$$ and (ii) *top class absorbing*, i.e. $$(\forall y \in {\mathcal{X}}\setminus{{\mathcal{X}}_{\mathit{R}}})~y {\mathrel{\ooalign{\hss${\underline{Q}}$\hss\cr$\longrightarrow$}}}{\mathcal{X}}_R.$$
\[the:ContinuousErgodicity:NecessaryAndSufficient\] A lower transition rate operator ${\underline{Q}}$ is ergodic if and only if it is regularly absorbing.
Not surprisingly, these necessary and sufficient conditions for the ergodicity of lower rate matrices are rather similar to the necessary and sufficient conditions for ergodicity of lower transition operators given in Proposition \[prop:DiscreteErgodicity:NecAndSuff\].
Extra material and proofs for Section \[sec:ergodicity\] {#app:ergodicity}
========================================================
Before we give any proofs, we first define the coefficient of ergodicity of an upper transition operator ${\overline{T}}$: $$\label{eqn:CoeffOfErgod:UpTranOpDefinition}
{{\rho}({\overline{T}})}
\coloneqq \max \{ {\left\Vert {{\overline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \}.$$
\[prop:CoeffOfErgod:Properties\] Let ${\underline{T}}$ and $\underline{S}$ be lower transition operators. For any $f \in {{\mathcal{L}}({\mathcal{X}})}$,
1. \[prop:CoeffOfErgod:Bounds\] $0 \leq {{\rho}({\underline{T}})} \leq 1$,
2. \[prop:CoeffOfErgod:BoundOnNormTf\] ${\left\Vert {{\underline{T}}f} \right\Vert}_{v} \leq {{\rho}({\underline{T}})} {\left\Vert {f} \right\Vert}_{v}$,
3. \[prop:CoeffOfErgod:LowerEqualsUpper\] ${{\rho}({\overline{T}})} = {{\rho}({\underline{T}})}$,
4. \[prop:CoeffOfErgod:Composition\] ${{\rho}({\underline{T}}\, \underline{S})} \leq {{\rho}({\underline{T}})} {{\rho}(\underline{S})}$,
<!-- -->
1. Follows immediately from \[prop:LTO:BoundedByMinAndMax\].
2. If ${\left\Vert {f} \right\Vert}_{v} = 0$, then by \[prop:LTO:BoundedByMinAndMax\] ${\left\Vert {{\underline{T}}f} \right\Vert}_{v} = 0$, such that the stated holds. Therefore, we now assume—without loss of generality—that ${\left\Vert {f} \right\Vert}_{v} > 0$. Note that $0 \leq (f - \min{f})/{\left\Vert {f} \right\Vert}_{v} \leq 1$. Combining this with—in that order—\[prop:norm:VarAddConstant\], \[prop:LTO:AdditionOfConstant\], \[def:LTO:NonNegativelyHom\], \[def:Norm:ScalarMult\] and Eqn. , we find that $$\begin{aligned}
{\left\Vert {{\underline{T}}f} \right\Vert}_{v}
&= {\left\Vert {{\underline{T}}f - \min{f}} \right\Vert}_{v}
= {\left\Vert {{\underline{T}}(f - \min{f})} \right\Vert}_{v}
= {\left\Vert {{\left\Vert {f} \right\Vert}_{v} {\underline{T}}\left(\frac{f - \min{f}}{{\left\Vert {f} \right\Vert}_{v}}\right)} \right\Vert}_{v} \\
&= {\left\Vert {{\underline{T}}\left(\frac{f - \min{f}}{{\left\Vert {f} \right\Vert}_{v}}\right) } \right\Vert}_{v} {\left\Vert {f} \right\Vert}_{v} \\
&\leq {{\rho}({\underline{T}})} {\left\Vert {f} \right\Vert}_{v}.
\end{aligned}$$
3. By Eqn. , $$\begin{aligned}
{{\rho}({\underline{T}})}
&= \max \left\{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ {\left\Vert {1 - {\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ {\left\Vert {1 + {\overline{T}}(-f)} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ {\left\Vert {{\overline{T}}(1 - f)} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ {\left\Vert {{\overline{T}}g} \right\Vert}_{v} \colon g \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq g \leq 1 \right\} \\
&= {{\rho}({\overline{T}})},
\end{aligned}$$ where the second equality follows from \[prop:norm:VarAddConstant\], the third equality follows from the conjugacy of ${\underline{T}}$ and ${\overline{T}}$, the fourth equality follows from \[prop:LTO:AdditionOfConstant\], the fifth equality follows from the fact that $0 \leq f \leq 1$ if and only if $0 \leq 1 - f \leq 1$, and the final equality follows from Eqn. .
4. By Eqn. and \[prop:CoeffOfErgod:BoundOnNormTf\], $$\begin{aligned}
{{\rho}({\underline{T}}\, \underline{S})}
&= \max \{ {\left\Vert {{\underline{T}}\, \underline{S} f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&\leq \max \{ {{\rho}({\underline{T}})} {\left\Vert {\underline{S} f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&= {{\rho}({\underline{T}})} \max \{ {\left\Vert {\underline{S} f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} = {{\rho}({\underline{T}})} {{\rho}(\underline{S})}. \qedhere
\end{aligned}$$
Theorem 21 in [@2013Skulj] highlights the usefulness of the coefficient of ergodicity.
\[the:CoeffOfErg:StrictlySmallerThanOneIsNecAndSuffForErg\] A lower transition operator ${\underline{T}}$ is ergodic if and only if there is some $k\in{\mathbb{N}}$ such that ${{\rho}({\underline{T}}^k)} < 1$.
\[prop:CoeffOfErgod:AlternativeFunctions\] Let ${\underline{T}}$ be a a lower transition operator. Then $$\begin{aligned}
{{\rho}({\underline{T}})}
&= \max \left\{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f\in{{\mathcal{L}}({\mathcal{X}})}, \max f = 1, \min f = 0 \right\} \label{eqn:CoeffOfErgod:WithMax} \\
&= \max \left\{ {\left\Vert {{\underline{T}}f} \right\Vert}_{c} \colon f\in{{\mathcal{L}}({\mathcal{X}})}, -1 \leq f \leq 1 \right\} \label{eqn:CoeffOfErgod:WithCentered} \\
&= \max \left\{ {\left\Vert {{\underline{T}}f} \right\Vert}_{c} \colon f\in{{\mathcal{L}}({\mathcal{X}})}, \max f = 1, \min f = -1 \right\}. \label{eqn:CoeffOfErgod:WithCenteredAndMax}
\end{aligned}$$
Because of Eqn. , there is some $g\in{{\mathcal{L}}({\mathcal{X}})}$ such that $0 \leq g \leq 1$ and ${\left\Vert {{\underline{T}}g} \right\Vert}_{v} = {{\rho}({\underline{T}})}$. By \[prop:CoeffOfErgod:BoundOnNormTf\], ${\left\Vert {{\underline{T}}g} \right\Vert}_{v} \leq {{\rho}({\underline{T}})} {\left\Vert {g} \right\Vert}_{v}$, such that ${\left\Vert {g} \right\Vert}_{v} = 1$, or equivalently $\max g = 1$ and $\min g = 0$. Hence, it follows from Eqn. that $$\begin{aligned}
{{\rho}({\underline{T}})}
&= \max \{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, \max f = 1, \min f = 0 \}.
\intertext{Next, manipulating Eqn.~\eqref{eqn:CoeffOfErgod} yields}
{{\rho}({\underline{T}})}
&= \max \{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&= \max \left\{ {\left\Vert {{\underline{T}}\left(f - \frac{1}{2}\right)} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ \frac{2}{2} {\left\Vert {{\underline{T}}\left(f - \frac{1}{2}\right)} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&= \max \left\{ \frac{1}{2} {\left\Vert {{\underline{T}}\left(2 f - 1\right)} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right \} \\
&= \max \left\{ {\left\Vert {{\underline{T}}\left(2 f - 1 \right)} \right\Vert}_{c} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\},
\intertext{ where the second equality follows from \ref{prop:norm:VarAddConstant} and \ref{prop:LTO:AdditionOfConstant}, the fourth equality follows from \ref{def:Norm:ScalarMult} and \ref{def:LTO:NonNegativelyHom}, and the final equality follows from Eqn.~\eqref{eqn:CentredNorm}.
Note that for all $f\in{{\mathcal{L}}({\mathcal{X}})}$, $0 \leq f \leq 1$ is equivalent to $-1 \leq (2f - 1) \leq 1$.
Hence,
}
{{\rho}({\underline{T}})}
&= \max \{ {\left\Vert {{\underline{T}}f} \right\Vert}_{c} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, -1 \leq f \leq 1 \}.
\end{aligned}$$
The proof of the final equality of the statement is now similar to that of the first.
The following lemma is a more general version of Lemma \[lem:LowTranOp:PossiblyAccesibleSequence\].
\[lem:LowTranOp:SignOfCompositionWithIndicator\] Let $k \in {\mathbb{N}}$ and $x,y\in {\mathcal{X}}$. For all arbitrary upper transition operators ${{\overline{T}}_{1}}, \dots, {{\overline{T}}_{k}}$, we define ${{\overline{T}}_{1:k}} \coloneqq {{\overline{T}}_{k}} \cdots {{\overline{T}}_{1}}$. Then $$[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) \geq [{{\overline{T}}_{1}} {\mathbb{I}_{z_{1}}}](z_{2}) \cdots [{{\overline{T}}_{k}} {\mathbb{I}_{z_{k}}}](z_{k+1}),$$ for any sequence $y = z_{k+1}, \dots, z_{1} = x$ in ${\mathcal{X}}$. Furthermore, $[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) > 0$ if and only if there is some sequence $y = z_{k+1}, \dots, z_{1} = x$ in ${\mathcal{X}}$ such that for all $i \in \{ 1, \dots, k \}$, $[{{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}](z_{i+1}) > 0$.
This proof is a straightforward generalisation of the proof of Proposition 4 in [@2012Hermans]. Fix some $k \in {\mathbb{N}}$, some $x, y \in {\mathcal{X}}$ and some arbitrary upper transition operators ${{\overline{T}}_{1}}, \dots, {{\overline{T}}_{k}}$. We also define ${{\overline{T}}_{1:k}} \coloneqq {{\overline{T}}_{k}} \cdots {{\overline{T}}_{1}}$, and note that by \[prop:LTO:CompositionIsAlsoLTO\] this is also an upper transition operator.
To prove the first part of the statement, we note that for all $i \in \{ 1, \dots, k \}$ and all $z_{i}, z_{i+1} \in {\mathcal{X}}$, $${{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}
= \sum_{z \in {\mathcal{X}}} [{{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}](z) {\mathbb{I}_{z}}
\geq [{{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}](z_{i-1}) {\mathbb{I}_{z_{i+1}}},$$ where the inequality is allowed because by \[prop:LTO:BoundedByMinAndMax\] the sum contains only non-negative terms. We fix any $z_{2} \in {\mathcal{X}}$, and use \[prop:LTO:Monotonicity\] and this inequality to yield $$\begin{aligned}
{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}
&= {{\overline{T}}_{1:k-1}} {{\overline{T}}_{1}} {\mathbb{I}_{x}}
\geq {{\overline{T}}_{1:k-1}} \left([{{\overline{T}}_{1}} {\mathbb{I}_{x}}](z_{2}) {\mathbb{I}_{z_{2}}}\right)
= [{{\overline{T}}_{1}} {\mathbb{I}_{x}}](z_{2}) {{\overline{T}}_{1:k-1}} {\mathbb{I}_{z_{2}}},
\end{aligned}$$ where ${{\overline{T}}_{1:k-1}} \coloneqq {{\overline{T}}_{k}} \cdots {{\overline{T}}_{2}}$—which by \[prop:LTO:CompositionIsAlsoLTO\] is also an upper transition operator—and the final equality follows from \[def:LTO:NonNegativelyHom\] and \[prop:LTO:BoundedByMinAndMax\]. Repeated application of the same reasoning yields $$[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y)
\geq [{{\overline{T}}_{1}} {\mathbb{I}_{z_1}}](z_{2}) \cdots [{{\overline{T}}_{k}} {\mathbb{I}_{z_{k}}}](z_{k+1}),$$ where $z_{k+1} \coloneqq y$, $z_{1} \coloneqq x$, and $z_{2}, \dots, z_{k}$ are arbitrary elements of ${\mathcal{X}}$. This proves the first part of the statement.
The reverse implication of the second part of the statement follows immediately from the first part. We therefore only need to prove that the forward implication holds as well. To that end, we first note that $$[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) = \left[{{\overline{T}}_{k}} \left( \sum_{z_{k} \in {\mathcal{X}}} [ {{\overline{T}}_{2:k}} {\mathbb{I}_{x}}](z_{k}) {\mathbb{I}_{z_{k}}} \right) \right](y)
\leq \sum_{z_{k} \in {\mathcal{X}}} [{{\overline{T}}_{2:k}} {\mathbb{I}_{x}}](z_{k}) [{{\overline{T}}_{1}} {\mathbb{I}_{z_{k}}}](y),$$ where ${{\overline{T}}_{2:k}} \coloneqq {{\overline{T}}_{k}} \cdots {{\overline{T}}_{2}}$ and the inequality follows from \[def:LTO:SuperAdditive\]. Repeating this same reasoning another $(k-2)$ times yields $$[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) \leq \sum_{z_{2} \in {\mathcal{X}}} \sum_{z_{3} \in {\mathcal{X}}} \cdots \sum_{z_{k} \in {\mathcal{X}}} [{{\overline{T}}_{1}} {\mathbb{I}_{x}}](z_{2}) [{{\overline{T}}_{2}} {\mathbb{I}_{z_{2}}}](z_{3}) \cdots [{{\overline{T}}_{k}} {\mathbb{I}_{z_{k}}}](y).$$ If now $[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) > 0$, then—because all terms are non-negative due to \[prop:LTO:BoundedByMinAndMax\]—at least one of the terms of the sum on the right hand side has to be strictly positive. Therefore, $[{{\overline{T}}_{1:k}} {\mathbb{I}_{x}}](y) > 0$ implies that there is at least one sequence $y = z_{k+1}, \dots, z_{1} = x$ in ${\mathcal{X}}$ such that for all $i \in \{ 1, \dots, k\}$, $[{{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}](z_{i+1}) > 0$.
\[lem:LowTranOp:SignOfCompositionWithEvent\] Let $k \in {\mathbb{N}}$ and $A \subseteq {\mathcal{X}}$. For all arbitrary lower transition operators ${{\underline{T}}_{1}}, \dots, {{\underline{T}}_{k}}$, we define ${{\underline{T}}_{1:k}} \coloneqq {{\underline{T}}_{k}} \cdots {{\underline{T}}_{1}}$. Then $$c_1 \cdots c_k {\mathbb{I}_{A_k}} \leq {{\underline{T}}_{1:k}} {\mathbb{I}_{A}} \leq {\mathbb{I}_{A_k}}.$$ In this expression, $A_k \subseteq {\mathcal{X}}$ is derived from the initial condition $A_{0} \coloneqq A$ and, for all $i \in \{ 1, \dots, k \}$, from the recursive relation $$A_{i}
\coloneqq \{ x \in {\mathcal{X}}\colon [{{\underline{T}}_{i}} {\mathbb{I}_{A_{i-1}}}](x) > 0 \}.$$ The non-negative real numbers $c_1, \dots, c_{k}$ are defined as $$c_{i}
\coloneqq \min \left\{ [{{\underline{T}}_{i}} {\mathbb{I}_{A_{i-1}}}](x) \colon x \in A_{i} \right\} \text{ for all } i \in \{ 1, \dots, k \},$$ with the convention that the minimum of an empty set is zero. Also, $A_k = \emptyset$ if and only if $c_i = 0$ for some $i \in \{ 1, \dots, k \}$.
Let ${\underline{T}}$ be an arbitrary lower transition operator, and fix an arbitrary $A \subset {\mathcal{X}}$. We define the set $A' \coloneqq \{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{A}}](x) > 0 \}$. On the one hand, from \[prop:LTO:BoundedByMinAndMax\] it follows that ${\underline{T}}{\mathbb{I}_{A}} \leq {\mathbb{I}_{A'}}$. On the other hand, ${\underline{T}}{\mathbb{I}_{A}} \geq c {\mathbb{I}_{A'}}$, where we let $$c
\coloneqq \min \left\{ [{\underline{T}}{\mathbb{I}_{A}}](x) \colon x \in A' \right\},$$ with the convention that the minimum of an empty set is zero. Note that by \[prop:LTO:BoundedByMinAndMax\], $0 \leq c \leq 1$. Combining these two inequalities yields $c {\mathbb{I}_{A'}} \leq {\underline{T}}{\mathbb{I}_{A}} \leq {\mathbb{I}_{A'}}$. Proving the first part of the statement is now fairly trivial; we simply need to apply both inequalities and \[prop:LTO:Monotonicity\] $k$ times.
To prove the second part of the statement, we observe that $c_i = 0$ is equivalent to $A_{i} = \emptyset$. Therefore, we assume that there is some $i \in \{ 1, \dots, k \}$ for which $c_i = 0$ and $A_{i} = \emptyset$. If $c_k = 0$, then obviously $A_k = \emptyset$ and the stated is true. We therefore assume that $i < k$, and observe that by \[prop:LTO:BoundedByMinAndMax\], ${{\underline{T}}_{i+1}} {\mathbb{I}_{A_{i}}} = {{\underline{T}}_{i+1}} {\mathbb{I}_{\emptyset}} = 0$, and therefore $A_{i+1} = \emptyset$. Repeating the same reasoning, we find that $A_{j} = \emptyset$ and $c_{j} = 0$ for all $j \in \{ i, \dots, k \}$, which proves the stated.
The following lemma is an alternate, slightly extended version of Proposition \[prop:LowTranOp:TopClassAbsorbingWithRecursion\].
\[lem:LowTranOp:StrongerNecAndSuffTopClassAbsorption\] Let ${\underline{T}}$ be a top class regular lower transition operator. Then ${\underline{T}}$ is top class absorbing if and only if $B_{n} = {\mathcal{X}}$, where $\{ B_{i} \}_{i \in {{\mathbb{N}}_{0}}}$ is the sequence defined by the initial condition $B_{0} \coloneqq {{\mathcal{X}}_{\mathit{PA}}}$ and the recursive relation $$B_{i}
= B_{i-1} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i-1} \colon [{\underline{T}}{\mathbb{I}_{B_{i-1}}}](x) > 0 \right\} ~~\text{for all}~i \in {\mathbb{N}},$$ and where $n \leq {\left\vert {{\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{PA}}}} \right\vert}$ is the first index such that $B_{n} = B_{n+1}$. Alternatively, ${\underline{T}}$ is top class absorbing if and only if there is some $m \in {{\mathbb{N}}_{0}}$ such that ${\underline{T}}^{m} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} > 0$, and in this case $n$ is the lowest such $m$.
We first prove the forward implication. By Proposition \[prop:LowTranOp:TopClassAbsorbingWithRecursion\], if ${\underline{T}}$ is top class absorbing then $B_{n} = {\mathcal{X}}$, where the sequence $\{B_{i}\}_{i \in {{\mathbb{N}}_{0}}}$ is defined from the initial condition $B_{0} \coloneqq {{\mathcal{X}}_{\mathit{PA}}}$ and, for all $i \in {\mathbb{N}}$, from the recursive relation $$B_{i} \coloneqq B_{i-1} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i-1} \colon [{\underline{T}}{\mathbb{I}_{B_{i-1}}}](x) > 0 \right\} = \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{i-1}}}](x) > 0 \right\},$$ and where $n \leq {\left\vert {{\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{PA}}}} \right\vert}$ is the first index such that $B_{n} = B_{n+1}$. We can immediately verify that ${{\mathcal{X}}_{\mathit{PA}}} = B_{0} \subseteq B_{1} \subseteq \cdots \subseteq B_{n} = {\mathcal{X}}$ and $B_{i} \setminus B_{i-1} \neq \emptyset$ for all $i \in \{ 1, \dots, n \}$.
Observe that the sequence $B_{0}, \dots, B_{n}$ satisfies the conditions of Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\], such that or all $i \in \{ 1, \dots, n \}$, $$c_{1} \cdots c_{i} {\mathbb{I}_{B_{i}}} \leq {\underline{T}}^{i} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} \leq {\mathbb{I}_{B_{i}}},$$ where $c_1, \dots, c_{n}$ are strictly positive real numbers because $\emptyset \neq B_{1}, \dots, B_{n}$. From this we infer that $\min {\underline{T}}^{i} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} > 0$ if and only if $B_{i} = {\mathcal{X}}$. As $B_{n} = {\mathcal{X}}$ and $B_{0}, \dots, B_{n-1} \neq B_{n}$, this confirms that indeed ${\underline{T}}^{n} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} > 0$ and that $n$ is the lowest non-negative natural number for which this holds. Next, we prove the reverse implication. Let $B_{0}, \dots, B_{n}$ and $n$ be defined as in the statement. From the definition, it is obvious that $B_{i-1} \subseteq B_{i}$ for all $i \in {\mathbb{N}}$. Also, if $n$ is the first index such that $B_{n} = B_{n+1}$, then $B_{i-1} \neq B_{i}$ for all $i \in \{ 1, \dots, n \}$ and $B_{n} = B_{n+i}$ for all $i \in {\mathbb{N}}$. From $B_{0} = {{\mathcal{X}}_{\mathit{PA}}}$ and $B_{i} \setminus B_{i-1} \neq \emptyset$ for all $i \in \{ 1, \dots, n \}$, we infer that indeed $n \leq {\left\vert {{\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{PA}}}} \right\vert}$. If $B_{n} = {\mathcal{X}}$, then the sequence $B_0, \dots, B_{n}$ satisfies the conditions of Proposition \[prop:LowTranOp:TopClassAbsorbingWithRecursion\], such that ${\underline{T}}$ is indeed top class absorbing.
Let $B_{0}, \dots, B_{n},\dots$ be the sequence as defined in the statement. Similar to what we did in the proof of Proposition \[prop:LowTranOp:TopClassAbsorbingWithRecursion\], we now verify using induction that $$B_{i} = \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{i-1}}}](x) > 0 \right\} ~\text{for all}~i \in{\mathbb{N}}.$$ We first consider the case $i = 1$. By Lemma \[lem:LowTranOp:TopClassRegularSpecialValues\], we know that $[{\underline{T}}{\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](x) > 0$ for all $x \in {{\mathcal{X}}_{\mathit{PA}}}$. Hence, $$\begin{aligned}
B_{1}
&= B_{0} \cup \left\{ x \in {\mathcal{X}}\setminus B_{0} \colon [{{\underline{T}}_{{\mathbb{I}_{B_{0}}}}}](x) > 0 \right\} \\
&= \left\{ x \in B_{0} \colon [{{\underline{T}}_{{\mathbb{I}_{B_{0}}}}}](x) > 0 \right\} \cup \left\{ x \in {\mathcal{X}}\setminus B_{0} \colon [{{\underline{T}}_{{\mathbb{I}_{B_{0}}}}}](x) > 0 \right\} \\
&= \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{0}}}](x) > 0 \right\},
\end{aligned}$$ where the second equality follows from the initial condition $B_{0} = {{\mathcal{X}}_{\mathit{PA}}}$. Fix some $k \in \{ 1, \dots, n-1\}$, and assume that the alternate definition holds for all $i \leq k$. We now argue that in that case the stated also holds for $i = k+1$. By the induction hypothesis, $B_{k}$ contains all $x \in {\mathcal{X}}$ for which $[{\underline{T}}{\mathbb{I}_{B_{k-1}}}](x) > 0$. Also, it holds by definition that $B_{k-1} \subseteq B_{k}$. Using \[prop:LTO:Monotonicity\], we infer from ${\mathbb{I}_{B_{k}}} \geq {\mathbb{I}_{B_{k-1}}}$ that $[{\underline{T}}{\mathbb{I}_{B_{k}}}](x) \geq [{\underline{T}}{\mathbb{I}_{B_{k-1}}}](x) > 0$ for all $x\in B_{k}$. Hence, $$\begin{aligned}
B_{k+1}
&= B_{k} \cup \left\{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\} \\
&= \left\{ x \in B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\} \cup \left\{ x \in {\mathcal{X}}\setminus B_{k} \colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\} \\
&= \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{k}}}](x) > 0 \right\}.
\end{aligned}$$
Now that we know that $$B_{i} = \left\{ x \in {\mathcal{X}}\colon [{\underline{T}}{\mathbb{I}_{B_{i-1}}}](x) > 0 \right\} ~\text{for all}~i \in {\mathbb{N}},$$ we observe that this equivalent definition of the sequence satisfies the conditions of the sequence in Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\]. Moreover, as $\emptyset \neq B_{0} \subseteq B_{1} \subseteq \dots$, it follows from the second part of Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\] that $c_{i} > 0$ for all $i \in {\mathbb{N}}$. Also from Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\], we know that $$c_1 \cdots c_{i} {\mathbb{I}_{B_{i}}} \leq {\underline{T}}^{i} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} \leq {\mathbb{I}_{B_{i}}} ~\text{for all}~ i \in {\mathbb{N}}.$$ Assume now that there is some $m \in {{\mathbb{N}}_{0}}$ such that ${\underline{T}}^{m} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}} > 0$, and let $n$ be the lowest such $m$. Then for all $y \in {\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{PA}}}$, $[{\underline{T}}^{n} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}](y) > 0$, such that the second condition of Definition \[def:LowTranOp:RegularlyAbsorbing\] is satisfied and ${\underline{T}}$ is indeed top class absorbing. If $n = 0$, then ${{\mathcal{X}}_{\mathit{PA}}} = {\mathcal{X}}= B_{0}$, and $n$ is indeed the first index for which $B_{n} = B_{n+1}$. If $n > 0$, then from the strict positivity of $c_{1}, \dots, c_{n}$ and the lower and upper bound for ${\underline{T}}^{i} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{PA}}}}}$ we infer that $B_{1}, \dots, B_{n-1} \neq {\mathcal{X}}$ and $B_{n} = {\mathcal{X}}$. We deduce from the recursive relation between $B_{0}, \dots, B_{n}, B_{n+1}$ that $n$ is indeed the first index for which $B_{n} = B_{n+1}$, which finalises this proof.
We first prove the forward implication. To this end, we let ${\underline{Q}}$ be an ergodic lower transition rate operator, and $n \coloneqq {\left\vert {{\mathcal{X}}} \right\vert} - 1$—we ignore the case ${\left\vert {{\mathcal{X}}} \right\vert} = 1$, as this case is trivially ergodic. We furthermore fix some $k \geq n$ and some $\delta_1, \dots, \delta_k$ in ${{\mathbb{R}}_{> 0}}$ such that for all $i \in \{ 1, \dots, k \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. For all $i \in \{ 1, \dots, k\}$, we define ${{\underline{T}}_{i}} \coloneqq (I + \delta_{i} {\underline{Q}})$. By Proposition \[prop:IPlusDeltaQLowTranOp\], the operators ${{\underline{T}}_{1}}, \dots, {{\underline{T}}_{k}}$ are lower transition operators, such that by \[prop:LTO:CompositionIsAlsoLTO\] their composition ${{\underline{T}}_{1:k}} \coloneqq {{\underline{T}}_{k}} \cdots {{\underline{T}}_{1}}$ is also a lower transition operator. Note that the same holds for their conjugate upper transition operators, defined as ${{\overline{T}}_{i}} \coloneqq (I + \delta_i {\overline{Q}})$ and ${{\overline{T}}_{1:k}} \coloneqq {{\overline{T}}_{k}} \cdots {{\overline{T}}_{1}}$. We now assume ex-absurdo that ${{\rho}(\Phi(\delta_{1}, \dots, \delta_{k}))} = {{\rho}({{\underline{T}}_{1:k}})} = 1$. As a consequence of Proposition \[prop:CoeffOfErgod:AlternativeFunctions\], there is some $f^{*} \in {{\mathcal{L}}({\mathcal{X}})}$ with $\min f^{*} = 0$ and $\max f^{*} = 1$ such that ${\left\Vert {{{\underline{T}}_{1:k}} f^{*}} \right\Vert}_{v} = 1$. By construction and \[prop:LTO:BoundedByMinAndMax\], there are now some $y_0, y_1 \in {\mathcal{X}}$ such that $[{{\underline{T}}_{1:k}} f^{*}](y_0) = 0$ and $[{{\underline{T}}_{1:k}} f^{*}](y_1) = 1$.
We define the—obviously non-empty—set $${\mathcal{X}}^{*}
\coloneqq \left\{ x \in {\mathcal{X}}\colon f^{*}(x) = 0 \right\},$$ and distinguish two cases: either ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} \neq \emptyset$ or ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} = \emptyset$.
We first consider the case ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} \neq \emptyset$, and fix any arbitrary $x^{*} \in {{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*}$. Note that, by construction, ${\mathbb{I}_{x^{*}}} \leq 1 - f^{*}$. Using the conjugacy of ${{\underline{T}}_{1:k}}$ and ${{\overline{T}}_{1:k}}$ and \[prop:LTO:Monotonicity\], we find that $${{\overline{T}}_{1:k}} {\mathbb{I}_{x^{*}}} \leq {{\overline{T}}_{1:k}} (1 - f^{*}) = 1 + {{\overline{T}}_{1:k}} (- f^{*}) = 1 - {{\underline{T}}_{1:k}} f^{*},$$ where the first equality follows from \[prop:LTO:AdditionOfConstant\] and the second equality follows from the conjugacy. From the previous inequality and \[prop:LTO:BoundedByMinAndMax\], it follows that $$0 \leq [{{\overline{T}}_{1:k}} {\mathbb{I}_{x^{*}}}](y_1) \leq 1 - [{{\underline{T}}_{1:k}} f^{*}](y_1) = 0,$$ and hence $[{{\overline{T}}_{1:k}} {\mathbb{I}_{x^{*}}}](y_1) = 0$. From Lemma \[lem:LowTranOp:SignOfCompositionWithIndicator\], it now follows that that $$\label{eqn:ErogidictyOfApproximation:Contradiction1}
0 = [{{\overline{T}}_{1:k}} {\mathbb{I}_{x^{*}}}](y_1) \geq \prod_{i=1}^{k} {{\overline{T}}_{i}} {\mathbb{I}_{z_{i}}}(z_{i+1}) = \prod_{i=1}^{k} [(I + \delta_{i} {\overline{Q}}) {\mathbb{I}_{z_{i}}}](z_{i+1})$$ for any arbitrary sequence $y_1 = z_{k+1}, z_2, \dots, z_1 = x^{*}$ in ${\mathcal{X}}$. On the other hand, as $k \geq n = {\left\vert {{\mathcal{X}}} \right\vert} - 1$ and $x^{*} \in {{\mathcal{X}}_{\mathit{R}}}$ it follows from Lemma \[lem:LowRateOp:PathOfArbitraryLength\] that there exists a sequence $y_1 = x_{k+1}, x_{k}, \dots, x_{1} = x^{*}$ in ${\mathcal{X}}$ such that $[(I + \delta_{i} {\overline{Q}}) {\mathbb{I}_{x_{i}}}](x_{i+1}) > 0$ for all $i \in \{1, \dots, k \}$. This obviously contradicts Eqn. .
Next, we consider the case ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} = \emptyset$. In this case, $c {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}} \leq f^{*}$, where we let $$c \coloneqq \min \{ f^{*}(x) \colon x \in {{\mathcal{X}}_{\mathit{R}}} \} > 0.$$ From Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\], we know that $$c_{1} \cdots c_{k} {\mathbb{I}_{A_k}} \leq {{\underline{T}}_{1:k}} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}},$$ where $A_{0} \coloneqq {{\mathcal{X}}_{\mathit{R}}}$ and, for all $i \in \{ 1, \dots, k \}$, $$\begin{aligned}
A_{i}
&\coloneqq \{ x \in {\mathcal{X}}\colon [{{\underline{T}}_{i}} {\mathbb{I}_{A_{i-1}}}](x) > 0\}
&\text{and} & &
c_{i}
&\coloneqq \min \{ [{{\underline{T}}_{i}} {\mathbb{I}_{A_{i-1}}}](x) \colon x \in A_{i} \}.
\end{aligned}$$ As $c > 0$ and $c {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}} \leq f^{*}$, it follows from \[def:LTO:NonNegativelyHom\] and \[prop:LTO:Monotonicity\] that $c {{\underline{T}}_{1:k}} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}} \leq {{\underline{T}}_{1:k}} f^{*}$. Combining the two obtained inequalities yields $$c c_1 \cdots c_{k} {{\mathbb{I}_{A_{k}}}(y_0)} \leq c [{{\underline{T}}_{1:k}} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}}](y_0) \leq [{{\underline{T}}_{1:k}} f^{*}](y_0) = 0.$$ From the second part of Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\], it now follows that $y_{0} \notin A_{k}$.
Nonetheless, we now prove that $A_{k} = {\mathcal{X}}$, an obvious contradiction. To that end, observe that for all $i \in \{ 1, \dots, k \}$, $$\begin{aligned}
A_{i}
&= \{ x \in {\mathcal{X}}\colon [(I + \delta_{i} {\underline{Q}}) {\mathbb{I}_{A_{i-1}}}](x) > 0 \} \\
&= \{ x \in A_{i-1} \colon [(I + \delta_{i} {\underline{Q}}) {\mathbb{I}_{A_{i-1}}}](x) > 0 \} \cup \{ x \in {\mathcal{X}}\setminus A_{i-1} \colon [(I + \delta_{i} {\underline{Q}}) {\mathbb{I}_{A_{i-1}}}](x) > 0 \}.
\intertext{ Note that for all $x_{i-1} \in A_{i-1}$, ${\mathbb{I}_{A_{i-1}}} \geq {\mathbb{I}_{x_{i-1}}}$.
Also, from \ref{prop:LTRO:Ixx} it follows that $[(I + \delta_i {\underline{Q}}) {\mathbb{I}_{x_{i-1}}}](x_{i-1}) > 0$.
Using \ref{prop:LTO:Monotonicity} allows us to conclude that for all $x_{i-1} \in A_{i-1}$, $[(I + \delta_{i} {\underline{Q}}) {\mathbb{I}_{A_{i-1}}}](x_{i-1}) > 0$.
Therefore,
}
A_{i}
&= A_{i-1} \cup \{ x \in {\mathcal{X}}\setminus A_{i-1} \colon [(I + \delta_{i} {\underline{Q}}) {\mathbb{I}_{A_{i-1}}}](x) > 0 \} \\
&= A_{i-1} \cup \{ x \in {\mathcal{X}}\setminus A_{i-1} \colon {{\mathbb{I}_{A_{i-1}}}(x)} + \delta_{i} [{\underline{Q}}{\mathbb{I}_{A_{i-1}}}](x) > 0 \} \\
&= A_{i-1} \cup \{ x \in {\mathcal{X}}\setminus A_{i-1} \colon [{\underline{Q}}{\mathbb{I}_{A_{i-1}}}](x) > 0 \},
\end{aligned}$$ where the third equality is allowed because $\delta_i > 0$. From this recursive relation, it is obvious that ${{\mathcal{X}}_{\mathit{R}}} \subseteq A_{k}$. Even more, we can prove that ${{\mathcal{X}}_{\mathit{R}}}^{c} \subseteq A_{k}$, which implies that ${{\mathcal{X}}_{\mathit{R}}} \cup {{\mathcal{X}}_{\mathit{R}}}^{c} = {\mathcal{X}}\subseteq A_{k} \subseteq {\mathcal{X}}$, and consequently $A_{k} = {\mathcal{X}}$. Indeed, note that the sequence $A_{0}, \dots, A_{k}$ is equal to the first $(k+1)$ terms of the sequence $\{B_{i}\}_{i \in {{\mathbb{N}}_{0}}}$ that is defined in Definition \[def:LowRateOp:LowerReachable\] for $B_{0} = {{\mathcal{X}}_{\mathit{R}}}$. As ${\underline{Q}}$ was assumed to be ergodic and $k \geq {\left\vert {{\mathcal{X}}} \right\vert} - 1 \geq {\left\vert {{\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{R}}}} \right\vert}$, it follows from Definitions \[def:LowRateOp:LowerReachable\] and \[def:LowRateOp:RegularlyAbsorbing\] and Theorem \[the:ContinuousErgodicity:NecessaryAndSufficient\] that ${{\mathcal{X}}_{\mathit{R}}}^{c} \subseteq B_{k}$.
For both ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} \neq \emptyset$ and ${{\mathcal{X}}_{\mathit{R}}} \cap {\mathcal{X}}^{*} = \emptyset$ we have obtained a contradiction, such that the ergodicity of ${\underline{Q}}$ indeed implies the stated.
Next, we prove the reverse implication. Fix some lower transition rate operator ${\underline{Q}}$, and assume that there is some $k < {\left\vert {{\mathcal{X}}} \right\vert}$ and some $\delta_1, \dots, \delta_k \in {{\mathbb{R}}_{> 0}}$ such that $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$ for all $i \in \{ 1, \dots, k \}$ and $${{\rho}(\Phi(\delta_{1}, \dots, \delta_{k}))} < 1.$$ By Proposition \[the:CoeffOfErg:StrictlySmallerThanOneIsNecAndSuffForErg\] this implies that the lower transition operator ${{\underline{T}}_{1:k}} \coloneqq (I + \delta_{k} {\underline{Q}}) \cdots (I + \delta_{1} {\underline{Q}})$ is ergodic. By Proposition \[prop:DiscreteErgodicity:NecAndSuff\], the ergodicity of ${{\underline{T}}_{1:k}}$ is equivalent to ${{\underline{T}}_{1:k}}$ being regularly absorbing, in the sense that
1. ${\mathcal{X}}_{1:k} \coloneqq \left\{ x \in {\mathcal{X}}\colon (\exists n \in {\mathbb{N}})(\forall y \in {\mathcal{X}})~[({{\overline{T}}_{1:k}})^{n} {\mathbb{I}_{x}}](y) > 0 \right\} \neq \emptyset$;
2. $(\forall y \in {\mathcal{X}}\setminus {\mathcal{X}}_{1:k})(\exists n \in {\mathbb{N}})~[({{\underline{T}}_{1:k}})^{n} {\mathbb{I}_{{\mathcal{X}}_{1:k}}}](y) > 0$.
Fix some $x^{*} \in {\mathcal{X}}_{1:k}$, and let $m \in {\mathbb{N}}$ such that $({{\overline{T}}_{1:k}})^{m} {\mathbb{I}_{x^{*}}} > 0$. Fix an arbitrary $y \in {\mathcal{X}}$. Then by Lemma \[lem:LowTranOp:SignOfCompositionWithIndicator\], there exists a sequence $y = x_{m+1}, \dots, x_{1} = x^{*}$ in ${\mathcal{X}}$ such that for all $i \in \{1, \dots, m\}$, $[{{\overline{T}}_{1:k}} {\mathbb{I}_{x_{i}}}](x_{i+1}) > 0$. Again using Lemma \[lem:LowTranOp:SignOfCompositionWithIndicator\], this implies that for all $i \in \{ 1, \dots, m \}$ there is a sequence $x_{i+1} = x_{i,k+1}, \dots, x_{i,1} = x_{i}$ in ${\mathcal{X}}$ such that for all $j \in \{ 1, \dots, k \}$, $$[(I + \delta_{j} {\overline{Q}}) {\mathbb{I}_{x_{i,j}}}](x_{i,j+1}) > 0.$$ As such, we have now constructed one long sequence $$y = x_{m, k+1}, x_{m,k} \dots, x_{m,1} = x_{m-1, k+1}, x_{m-1,k}, \dots, x_{m-1,1} = x_{m-2, k+1} \dots, x_{1, 1} = x$$ in ${\mathcal{X}}$. From this sequence we remove all “loops” (as we previously did in the proof of Lemma \[lem:LowRateOp:UpperReachableShorterSequence\]), and denote this shortened sequence by $y = z_{n'+1}, \dots, z_{1} = x^{*}$ with corresponding time steps $\delta_{n'}', \dots, \delta_{1}'$. Then for all $i \in \{ 1, \dots, n' \}$, $$0 < [(I + \delta_{i}' {\overline{Q}}) {\mathbb{I}_{z_{i}}}](z_{i+1}) = {\mathbb{I}_{z_{i}}}(z_{i+1}) + \delta_{i}' [{\overline{Q}}{\mathbb{I}_{z_{i}}}](z_{i+1}) = \delta_{i}' [{\overline{Q}}{\mathbb{I}_{z_{i}}}](z_{i+1}).$$ As all $\delta_{i}'$ are strictly positive, we find that for all $i \in \{ 1, \dots, n' \}$, $[{\overline{Q}}{\mathbb{I}_{z_{i}}}](z_{i+1}) > 0$. By Definition \[def:LowRateOp:UpperReachable\], this means that $y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x^{*}$. As $y$ was an arbitrary element of ${\mathcal{X}}$ and $x^{*}$ an arbitrary element of ${\mathcal{X}}_{1:k}$, ${\mathcal{X}}_{1:k} \subseteq {{\mathcal{X}}_{\mathit{R}}}$ and hence ${\underline{Q}}$ is top class regular. Furthermore, we can show that ${{\mathcal{X}}_{\mathit{R}}} \subseteq {\mathcal{X}}_{1:k}$, such that ${{\mathcal{X}}_{\mathit{R}}} = {\mathcal{X}}_{1:k}$. To that end, assume that ${{\mathcal{X}}_{\mathit{R}}} \setminus {\mathcal{X}}_{1:k} \neq \emptyset$ and fix some arbitrary $x^{*} \in {{\mathcal{X}}_{\mathit{R}}} \setminus {\mathcal{X}}_{1:k}$. Then by Definition \[def:LowRateOp:RegularlyAbsorbing\], $y {\mathrel{\ooalign{\hss${\overline{Q}}$\hss\cr$\longrightarrow$}}}x^{*}$ for all $y \in {\mathcal{X}}$. By Lemmas \[lem:LowRateOp:PathOfArbitraryLength\] and \[lem:LowTranOp:SignOfCompositionWithIndicator\], for all $y \in {\mathcal{X}}$ there is an integer $n_{y}$ such that for all $\ell \geq n_{y}$, $[({{\overline{T}}_{1:k}})^{\ell} {\mathbb{I}_{x^{*}}}](y) > 0$. Hence, if we let $m \coloneqq \max \{ n_{y} \colon y \in {\mathcal{X}}\}$, then $[({{\overline{T}}_{1:k}})^{m} {\mathbb{I}_{x^{*}}}](y) > 0$ for all $y \in {\mathcal{X}}$. By Definition \[def:LowTranOp:RegularlyAbsorbing\], this implies that $x^{*} \in {\mathcal{X}}_{1:k}$. However, this contradicts our assumption that $x^{*} \in {{\mathcal{X}}_{\mathit{R}}} \setminus {\mathcal{X}}_{1:k}$, such that ${{\mathcal{X}}_{\mathit{R}}} \setminus {\mathcal{X}}_{1:k} = \emptyset$ and hence indeed ${{\mathcal{X}}_{\mathit{R}}} \subseteq {\mathcal{X}}_{1:k}$.
We now show that (ii) implies that ${\underline{Q}}$ is top class absorbing. Since ${{\underline{T}}_{1:k}}$ is top class regular and absorbing, and because ${\mathcal{X}}_{1:k}={\mathcal{X}}_{R}$, it follows from Lemma \[lem:LowTranOp:StrongerNecAndSuffTopClassAbsorption\] that there is some $m \in {{\mathbb{N}}_{0}}$ such that $({{\underline{T}}_{1:k}})^{m} {\mathbb{I}_{{{\mathcal{X}}_{\mathit{R}}}}} > 0$. Also, we know that $B_{m} = {\mathcal{X}}$, where $B_{0} = {{\mathcal{X}}_{\mathit{R}}}$ and $$B_{i+1}
\coloneqq B_{i} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i} \colon [{{\underline{T}}_{1:k}} {\mathbb{I}_{B_{i}}}](x) > 0 \right\} ~\text{for all}~i \in \{ 0, \dots, m-1 \}.$$ For any $i \in \{ 0, \dots, m-1 \}$ and any $x \in {\mathcal{X}}$, it follows from Lemma \[lem:LowTranOp:SignOfCompositionWithEvent\] that $[{{\underline{T}}_{1:k}} {\mathbb{I}_{B_{i}}}](x) > 0$ if and only if $x \in B_{i,k}$, where $B_{i,k}$ is derived from the initial condition $B_{i,0} \coloneqq B_{i}$ and, for all $j \in \{ 1, \dots, k \}$, from the recursive relation $$\begin{aligned}
B_{i,j}
&= \left\{ x \in {\mathcal{X}}\colon [(I + \delta_{j} {\underline{Q}}) {\mathbb{I}_{B_{i,j-1}}}](z) > 0 \right\}.
\intertext{Similar to what we did before, we can rewrite this recursive relation as}
B_{i,j}
&= \left\{ x \in B_{i,j-1} \colon [(I + \delta_{j} {\underline{Q}}) {\mathbb{I}_{B_{i,j-1}}}](z)>0 \right\} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i,j-1} \colon [(I + \delta_{j} {\underline{Q}}) {\mathbb{I}_{B_{i,j-1}}}](z) > 0 \right\} \\
&= \left\{ x \in B_{i,j-1} \colon 1 + \delta_{j} [{\underline{Q}}{\mathbb{I}_{B_{i,j-1}}}](z) > 0 \right\} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i,j-1} \colon \delta_{j} [{\underline{Q}}{\mathbb{I}_{B_{i,j-1}}}](z) > 0 \right\}.
\intertext{As before, we can verify that $1 + \delta_{j} [{\underline{Q}}{\mathbb{I}_{B_{i,j-1}}}](z) > 0$ for all $x \in B_{i,j-1}$.
Hence,}
B_{i,j}
&= B_{i,j-1} \cup \left\{ x \in {\mathcal{X}}\setminus B_{i,j-1} \colon [{\underline{Q}}{\mathbb{I}_{B_{i,j-1}}}](z) > 0 \right\}.
\end{aligned}$$ This way, we have constructed a sequence of sets $$B_{0} = B_{0, 0}, B_{0, 1}, \dots, B_{0,k} = B_1 = B_{1,0}, B_{1,1}, \dots, B_{1,k} = B_{2} = B_{2,0}, \dots, B_{m-1, k} = B_{m}$$ with $B_{0} = {{\mathcal{X}}_{\mathit{R}}}$ and $B_{m} = {\mathcal{X}}$. Denote this sequence by $A_{0}, \dots, A_{mk + 1}$ and let $A_{mk+2}\coloneqq{\mathcal{X}}$. Then $A_{0} = {{\mathcal{X}}_{\mathit{R}}}$, $A_{mk + 1} =A_{mk + 2}={\mathcal{X}}$ and for all $i \in \{0, \dots, mk+1\}$, $$A_{i+1} = A_{i} \cup \left\{ x \in {\mathcal{X}}\setminus A_{i} \colon [{\underline{Q}}{\mathbb{I}_{A_{i}}}](z) > 0 \right\}.$$ Let $n\in\{0,\dots,mk+1\}$ be the first index for which $A_{n} = A_{n+1}$. From the recursive relation between $A_{n}, \dots, A_{m k + 1},A_{m k + 2}$, we infer that $A_{n} = A_{n+1} = \cdots = A_{m k + 2} = {\mathcal{X}}$. Fix an arbitrary $y{*} \in {\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{R}}}$. Then the sequence ${{\mathcal{X}}_{\mathit{R}}} = A_{0}, \dots, A_{n}, A_{n+1}$ satisfies the recursive relation of Definition \[def:LowRateOp:LowerReachable\] and $y^{*}\in{\mathcal{X}}=A_n$, so $y^{*} {\mathrel{\ooalign{\hss${\underline{Q}}$\hss\cr$\longrightarrow$}}}{{\mathcal{X}}_{\mathit{R}}}$. As $y^{*}$ was an arbitrary element of ${\mathcal{X}}\setminus {{\mathcal{X}}_{\mathit{R}}}$, it follows that ${\underline{Q}}$ is top class absorbing.
We have proven that if there is some $k < {\left\vert {{\mathcal{X}}} \right\vert}$ and some sequence $\delta_1, \dots, \delta_k$ in ${{\mathbb{R}}_{> 0}}$ such that $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} < 2$ for all $i \in \{ 1, \dots, k \}$ and ${{\rho}(\Phi(\delta_{1},\dots,\delta_{k}))} < 1$, then ${\underline{Q}}$ is both top class regular and top class absorbing. As an immediate consequence of Theorem \[the:ContinuousErgodicity:NecessaryAndSufficient\], this implies that ${\underline{Q}}$ is ergodic.
From the requirements on $\delta$, \[prop:LTO:CompositionIsAlsoLTO\] and Proposition \[prop:IPlusDeltaQLowTranOp\], it follows that $(I + \delta {\underline{Q}})^{i}$ is a lower transition operator for all $i \in {\mathbb{N}}$. By Lemma \[lem:ExplicitErrorBound\], $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
&\leq \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} \sum_{i = 0}^{n-1} {\left\Vert {(I + \delta {\underline{Q}})^{i} f} \right\Vert}_{c} \\
&= \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} \sum_{i = 0}^{k-1} \sum_{j = 0}^{m-1} {\left\Vert {(I + \delta {\underline{Q}})^{j}(I + \delta {\underline{Q}})^{m i} f} \right\Vert}_{c}.
\intertext{ We use \ref{prop:LTO:VarNormTf} to yield
}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
&\leq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} \sum_{i = 0}^{k-1} {\left\Vert {(I + \delta {\underline{Q}})^{m i} f} \right\Vert}_{c}.
\intertext{Next, we simply use \ref{prop:CoeffOfErgod:BoundOnNormTf} and \ref{prop:CoeffOfErgod:Composition} to yield}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
&\leq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} {\left\Vert {f} \right\Vert}_{c} \sum_{i = 0}^{k-1} {{\rho}((I + \delta {\underline{Q}})^{m})}^{i}.
\end{aligned}$$ For any $a \in [0, 1)$ and any $\ell \in {\mathbb{N}}$, it is well known that $$\sum_{i = 0}^{\ell} a^{i} = \frac{1 - a^{\ell+1}}{1 - a} \leq \frac{1}{1-a}.$$ If $\beta \coloneqq {{\rho}((I + \delta {\underline{Q}})^{m})} < 1$, then we can use this well-known relation to yield $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
&\leq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} {\left\Vert {f} \right\Vert}_{c} \frac{1 - \beta^{k}}{1 - \beta}
\leq \frac{m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} {\left\Vert {f} \right\Vert}_{c}}{1 - \beta}.
\end{aligned}$$
The proof for $\beta = {{\rho}({{\underline{T}}_{m \delta}})}$ is entirely analoguous. We can use the second inequality of Lemma \[lem:ExplicitErrorBound\], the semi-group property and \[prop:LTO:VarNormTf\], which yields $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
\leq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} \sum_{i = 0}^{k-1}{\left\Vert {({{\underline{T}}_{m \delta}})^{i} f} \right\Vert}_{c}.$$ Next, we again use \[prop:CoeffOfErgod:BoundOnNormTf\] and \[prop:CoeffOfErgod:Composition\] to yield $${\left\Vert {{{\underline{T}}_{t}} f - \Psi_{t}(n)} \right\Vert}
\leq m \delta^2 {\left\Vert {{\underline{Q}}} \right\Vert}^{2} {\left\Vert {f} \right\Vert}_{c} \sum_{i = 0}^{k-1} {{\rho}({{\underline{T}}_{m \delta}})}^{i}. \qedhere$$
Let $\delta \in {{\mathbb{R}}_{\geq 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$. Using Proposition \[prop:CoeffOfErgod:AlternativeFunctions\] yields $$\begin{aligned}
{{\rho}(\Phi(\delta))}
&= \max \{ {\left\Vert {\Phi(\delta) f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, \max f = 1, \min f = 0 \}.
\end{aligned}$$ In the special case of a binary state space, only two functions satisfy this requirement: ${\mathbb{I}_{0}}$ and ${\mathbb{I}_{1}}$. Therefore $$\begin{aligned}
{{\rho}(\Phi(\delta))}
&= \max \big\{ {\left\vert {[\Phi(\delta) {\mathbb{I}_{0}}](0) - [\Phi(\delta) {\mathbb{I}_{0}}](1)} \right\vert}, {\left\vert {[\Phi(\delta) {\mathbb{I}_{1}}](0) - [\Phi(\delta) {\mathbb{I}_{1}}](1)} \right\vert} \big\}.
\end{aligned}$$ Recall that in the Proof of Example \[binex:AnalyticalExpressionsForAppliedLTO\] we proved that for all $\delta \in {{\mathbb{R}}_{\geq 0}}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$ and all $f \in {{\mathcal{L}}({\mathcal{X}})}$, $$[\Phi(\delta) f](0) - [\Phi(\delta) f](1)
= \begin{cases}
{\left\Vert {f} \right\Vert}_{v} (1 - \delta ({\overline{q}_{0}} + {\underline{q}_{1}})) &\text{if } f(0) \geq f(1), \\
{\left\Vert {f} \right\Vert}_{v} (1 - \delta ({\underline{q}_{0}} + {\overline{q}_{1}})) &\text{if } f(0) \leq f(1).
\end{cases}$$ As ${\left\Vert {{\mathbb{I}_{0}}} \right\Vert}_{v} = 1 = {\left\Vert {{\mathbb{I}_{1}}} \right\Vert}_{v}$, this yields $${{\rho}(I + \delta {\underline{Q}})}
= {{\rho}(\Phi(\delta))}
= \max \left\{ {\left\vert {1 - \delta ({\overline{q}_{0}} + {\underline{q}_{1}})} \right\vert}, {\left\vert {1 - \delta ({\underline{q}_{0}} + {\overline{q}_{1}})} \right\vert} \right\}. \qedhere$$
For the proof of Theorem \[the:CoeffOfErgod:Approximation\], we need some definitions and results from the theory of imprecise probabilities. The reason for this is that, as [@decooman2009] already mention, the functional $[{\underline{T}}\cdot](x)$ is actually a coherent (conditional) lower expectation. For a more thorough discussion of coherent lower expectations—often also called coherent lower previsions—we refer to the seminal work of [@1991Walley] and the more recent treatment of [@2014LowPrev].
A functional ${\underline{{\mathrm{E}}}}$ that maps ${{\mathcal{L}}({\mathcal{X}})}$ to ${\mathbb{R}}$ is a *coherent lower expectation* if for all $f, g \in {{\mathcal{L}}({\mathcal{X}})}$ and all $\mu \in {{\mathbb{R}}_{\geq 0}}$:
1. ${{\underline{{\mathrm{E}}}}(f)} \geq \min f$; \[def:CLP:BoundedByMin\]
2. ${{\underline{{\mathrm{E}}}}(f + g)} \geq {{\underline{{\mathrm{E}}}}(f)} + {{\underline{{\mathrm{E}}}}(g)}$; \[def:CLP:subadditive\]
3. ${{\underline{{\mathrm{E}}}}(\mu f)} = \mu {{\underline{{\mathrm{E}}}}(f)}$. \[def:CLP:PositiveHomogeneity\]
The conjugate *coherent upper expectation* is defined for all $f \in {{\mathcal{L}}({\mathcal{X}})}$ as $${{\overline{{\mathrm{E}}}}(f)} = - {{\underline{{\mathrm{E}}}}(-f)}.$$ If for all $f \in {{\mathcal{L}}({\mathcal{X}})}$, ${{\overline{{\mathrm{E}}}}(f)} = {{\underline{{\mathrm{E}}}}(f)} = {{\mathrm{E}}(f)}$, then we call ${\mathrm{E}}$ a *linear expectation*. The reason for this terminology is that the inequality in \[def:CLP:subadditive\] can then be replaced by an equality, and the condition $\mu \in {{\mathbb{R}}_{\geq 0}}$ for \[def:CLP:PositiveHomogeneity\] can be relaxed to $\mu \in {\mathbb{R}}$.
The following corollary highlights the link between the components of a lower transition operator and coherent lower previsions.
\[cor:LTOisCLP\] Let ${\underline{T}}$ be a lower transition operator and $x \in {\mathcal{X}}$. Then the functional $[{\underline{T}}\cdot](x) \colon f \in {{\mathcal{L}}({\mathcal{X}})}\mapsto [{\underline{T}}f](x)$ is a coherent lower prevision.
The operator $[{\underline{T}}\cdot](x)$ indeed maps ${{\mathcal{L}}({\mathcal{X}})}$ to ${\mathbb{R}}$. Furthermore, \[def:CLP:BoundedByMin\] follows from \[def:LTO:DominatesMin\], \[def:CLP:subadditive\] follows from \[def:LTO:SuperAdditive\] and \[def:CLP:PositiveHomogeneity\] follows from \[def:LTO:NonNegativelyHom\]. Hence, the operator is indeed a coherent lower prevision.
For any coherent lower expectation ${\underline{{\mathrm{E}}}}$, the set ${{\mathcal{M}}({\underline{{\mathrm{E}}}})}$ of dominating linear expectations, defined as $${{\mathcal{M}}({\underline{{\mathrm{E}}}})} \coloneqq \{ {\mathrm{E}}\text{ a linear expectation operator} \colon (\forall f \in {{\mathcal{L}}({\mathcal{X}})})~{{\underline{{\mathrm{E}}}}(f)} \leq {{\mathrm{E}}(f)} \},$$ is non-empty. Moreover, from [@1991Walley Section 3.3.3] it follows that ${\underline{{\mathrm{E}}}}$ is the lower envelope of ${{\mathcal{M}}({\underline{{\mathrm{E}}}})}$, in the sense that for all $f \in {{\mathcal{L}}({\mathcal{X}})}$, $${{\underline{{\mathrm{E}}}}(f)} = \min \{ {{\mathrm{E}}(f)} \colon {\mathrm{E}}\in {{\mathcal{M}}({\underline{{\mathrm{E}}}})} \}.$$
\[lem:MaxDifferenceBetweenLinearPrevisionsForIndicators\] If ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ are two linear expectation operators, then $$\max \{ {\mathrm{E}}_{1}(f) - {\mathrm{E}}_{2}(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \}
= \max \{ {\mathrm{E}}_{1}({\mathbb{I}_{A}}) - {\mathrm{E}}_{2}({\mathbb{I}_{A}}) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, \emptyset \neq A \subset {\mathcal{X}}\}.$$
Let ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be any two linear expectation operators on ${{\mathcal{L}}({\mathcal{X}})}$. Then $$\begin{aligned}
&\max \{ {\mathrm{E}}_{1}(f) - {\mathrm{E}}_{2}(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&\qquad = \max \left\{ \sum_{x \in {\mathcal{X}}} ({\mathrm{E}}_{1}({\mathbb{I}_{x}}) - {\mathrm{E}}_{2}({\mathbb{I}_{x}})) f(x) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \right\} \\
&\qquad = \sum_{x \in A^*} ({\mathrm{E}}_{1}({\mathbb{I}_{x}}) - {\mathrm{E}}_{2}({\mathbb{I}_{x}}))
= {\mathrm{E}}_{1}({\mathbb{I}_{A^*}}) - {\mathrm{E}}_{2}({\mathbb{I}_{A^*}}),
\end{aligned}$$ where $A^* \subset {\mathcal{X}}$ is defined as $A^* \coloneqq \left\{ x \in {\mathcal{X}}\colon {\mathrm{E}}_{1}({\mathbb{I}_{x}}) > {\mathrm{E}}_{2}({\mathbb{I}_{x}}) \right\}$.
\[lem:MaxDifferenceBetweenLowerPrevisionsUpperBoundWithIndicators\] If ${\underline{{\mathrm{E}}}}_{1}$ and ${\underline{{\mathrm{E}}}}_{2}$ are two coherent lower expectations on ${{\mathcal{L}}({\mathcal{X}})}$, then $$\max \{ {\underline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \}
\leq \max \{ {\overline{{\mathrm{E}}}}_1({\mathbb{I}_{A}}) - {\underline{{\mathrm{E}}}}_2({\mathbb{I}_{A}}) \colon 0 \neq A \subset {\mathcal{X}}\}.$$
Define ${\mathcal{M}}_1 \coloneqq {{\mathcal{M}}({\underline{{\mathrm{E}}}}_1)}$ and ${\mathcal{M}}_2 \coloneqq {{\mathcal{M}}({\underline{{\mathrm{E}}}}_2)}$. Note that for all $f \in {{\mathcal{L}}({\mathcal{X}})}$, $$0 ={\underline{{\mathrm{E}}}}_{1}(0) = {\underline{{\mathrm{E}}}}_{1}(f - f) \geq {\underline{{\mathrm{E}}}}_{1}(f) + {\underline{{\mathrm{E}}}}_{1}(-f),$$ where the first equality follows from \[def:CLP:PositiveHomogeneity\]—with $\mu = 0$ and $f = 0$—and the first inequality follows from \[def:CLP:subadditive\]. Bringing the second term to the left hand side and using the conjugacy relation between ${\underline{{\mathrm{E}}}}_{1}$ and ${\overline{{\mathrm{E}}}}_{1}$, we find ${\overline{{\mathrm{E}}}}_{1}(f) \geq {\underline{{\mathrm{E}}}}_{1}(f)$. Hence $${\underline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f)
\leq {\overline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f),$$ and consequently $$\max \{ {\underline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \}
\leq \max \{ {\overline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \}.$$ Recall that for any $f \in {{\mathcal{L}}({\mathcal{X}})}$, ${\underline{{\mathrm{E}}}}_i(f) = \min_{{\mathrm{E}}_i \in {\mathcal{M}}_{i}} {\mathrm{E}}_i(f)$, so $$\begin{aligned}
{\overline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f)
&= \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} {\mathrm{E}}_1(f) - \min_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} {\mathrm{E}}_2(f)
= \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} {\mathrm{E}}_1(f)-{\mathrm{E}}_2(f).
\end{aligned}$$ We use the previous equality to rewrite the right hand side of the previous inequality: $$\begin{aligned}
&\max \{ {\overline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&\qquad = \max \big\{ \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} ({\mathrm{E}}_{1}(f) - {\mathrm{E}}_{2}(f)) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \big\} \\
&\qquad = \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} \max \big\{ ({\mathrm{E}}_{1}(f) - {\mathrm{E}}_{2}(f)) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \big\}.
\end{aligned}$$ Next, we use Lemma \[lem:MaxDifferenceBetweenLinearPrevisionsForIndicators\] to yield $$\begin{aligned}
&\max \{ {\underline{{\mathrm{E}}}}_1(f) - {\underline{{\mathrm{E}}}}_2(f) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&\qquad \leq \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} \max \big\{ ({\mathrm{E}}_{1}(f) - {\mathrm{E}}_{2}(f)) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \big\} \\
&\qquad = \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} \max \big\{ ({\mathrm{E}}_{1}({\mathbb{I}_{A}}) - {\mathrm{E}}_{2}({\mathbb{I}_{A}})) \colon 0 \neq A \subset {\mathcal{X}}\big\} \\
&\qquad = \max \big\{ \max_{{\mathrm{E}}_1 \in {\mathcal{M}}_{1}} \max_{{\mathrm{E}}_2 \in {\mathcal{M}}_{2}} ({\mathrm{E}}_{1}({\mathbb{I}_{A}}) - {\mathrm{E}}_{2}({\mathbb{I}_{A}})) \colon 0 \neq A \subset {\mathcal{X}}\big\} \\
&\qquad = \max \{ {\overline{{\mathrm{E}}}}_{1}({\mathbb{I}_{A}}) - {\underline{{\mathrm{E}}}}_{2}({\mathbb{I}_{A}}) \colon 0 \neq A \subset {\mathcal{X}}\}. \qedhere
\end{aligned}$$
Fix some lower transition operator ${\underline{T}}$. The lower bound on ${{\rho}({\underline{T}})}$ follows from the fact that for any $\emptyset \neq A \subset {\mathcal{X}}$, $0 \leq {\mathbb{I}_{A}} \leq 1$. Recall from Corollary \[cor:LTOisCLP\] that for any $x\in{\mathcal{X}}$, $[{\underline{T}}\cdot](x)$ is a coherent lower prevision. Therefore, we can use Lemma \[lem:MaxDifferenceBetweenLowerPrevisionsUpperBoundWithIndicators\] to yield the upper bound: $$\begin{aligned}
{{\rho}({\underline{T}})}
&= \max \{ {\left\Vert {{\underline{T}}f} \right\Vert}_{v} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \\
&= \max \big\{ \max \{ [{\underline{T}}f](x) - [{\underline{T}}f](y) \colon x,y \in {\mathcal{X}}\} \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \big\} \\
&= \max \big\{ \max \{ [{\underline{T}}f](x) - [{\underline{T}}f](y) \colon f \in {{\mathcal{L}}({\mathcal{X}})}, 0 \leq f \leq 1 \} \colon x,y \in {\mathcal{X}}\big\} \\
&\leq \max \big\{ \max \{ [{\overline{T}}{\mathbb{I}_{A}}](x) - [{\underline{T}}{\mathbb{I}_{A}}](y) \colon \emptyset \neq A \subseteq {\mathcal{X}}\} \colon x,y \in {\mathcal{X}}\big\} \\
&= \max \big\{ \max \{ [{\overline{T}}{\mathbb{I}_{A}}](x) - [{\underline{T}}{\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subseteq {\mathcal{X}}\big\}. \qedhere
\end{aligned}$$
We first verify that ${\underline{Q}}$ is ergodic. Since $[{\overline{Q}}{\mathbb{I}_{1}}](0) = {\overline{q}_{0}} > 0$ and $[{\overline{Q}}{\mathbb{I}_{0}}](1) = {\overline{q}_{1}} > 0$, it follows from \[def:LowRateOp:UpperReachable,def:LowRateOp:RegularlyAbsorbing\] that ${{\mathcal{X}}_{\mathit{R}}} = {\mathcal{X}}$, such that ${\underline{Q}}$ is regularly absorbing. Hence, by also invoking \[the:ContinuousErgodicity:NecessaryAndSufficient\] we can conclude that ${\underline{Q}}$ is ergodic.
Next, we fix some $\delta \in {{\mathbb{R}}_{> 0}}{}$ such that $\delta {\left\Vert {{\underline{Q}}} \right\Vert} < 2$. Recall from \[prop:IPlusDeltaQLowTranOp\] that $(I + \delta {\underline{Q}})$ is a lower transition operator. Consequently, we can use \[eqn:CoeffOfErgod:UpperBound\] to compute an upper bound for ${{\rho}(I + \delta {\underline{Q}})}$. In this case, there are clearly only two possibilities for $A$ in the optimisation of \[eqn:CoeffOfErgod:UpperBound\]: $A = \{0\}$ and $A = \{1\}$. For $A = \{0\}$, some straightforward calculations yield $$\begin{aligned}
[(I + \delta {\overline{Q}}) {\mathbb{I}_{0}}](0)
&= 1, &
[(I + \delta {\overline{Q}}) {\mathbb{I}_{0}}](1)
&= \delta {\overline{q}_{1}}, \\
[(I + \delta {\underline{Q}}) {\mathbb{I}_{0}}](0)
&= 1 - \delta {\overline{q}_{0}}, &
[(I + \delta {\underline{Q}}) {\mathbb{I}_{0}}](1)
&= 0.
\end{aligned}$$ This implies that $$\begin{gathered}
\max \big\{ \max \{ [(I + \delta {\overline{Q}}) {\mathbb{I}_{A}}](x) - [(I + \delta {\underline{Q}}) {\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\big\} \\
\geq [(I + \delta {\overline{Q}}) {\mathbb{I}_{0}}](0) - [(I + \delta {\underline{Q}}) {\mathbb{I}_{0}}](1)
= 1.
\qedhere
\end{gathered}$$
If the lower transition operator is linear, then the lower and upper bounds of Theorem \[the:CoeffOfErgod:Approximation\] are equal. Moreover, from this special case we can immediately verify that the ergodic coefficient we use is a proper generalisation of an ergodic coefficient—the delta coefficient $\delta$ of [@1991Anderson], which is equivalent to $\tau_1$, one of the proper coefficients of ergodicity discussed by [@1981Seneta]—used in the study of precise Markov chains.
\[the:LowTranOp:ApproximationOfCoeffOfErgod\] Let $T$ be a transition matrix, then $$\begin{aligned}
{{\rho}(T)}
&= \max \left\{ \frac{1}{2} \sum_{z \in {\mathcal{X}}} {\left\vert {T(x,z) - T(y,z)} \right\vert} \colon x,y \in {\mathcal{X}}\right\}.
\end{aligned}$$
For a transition matrix, the the upper bound of Eqn. and the lower bound of Eqn. in Theorem \[the:CoeffOfErgod:Approximation\] are equal. Therefore $$\begin{aligned}
{{\rho}(T)}
&= \max \left\{ \max \{ [T {\mathbb{I}_{A}}](x) - [T {\mathbb{I}_{A}}](y) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\right\} \\
&= \max \left\{ \max \left\{ \frac{1}{2} [T (2 {\mathbb{I}_{A}})](x) - \frac{1}{2} [T (2 {\mathbb{I}_{A}})](y) \colon x,y \in {\mathcal{X}}\right\} \colon \emptyset \neq A \subset {\mathcal{X}}\right\} \\
&= \max \left\{ \max \left\{ \frac{1}{2} [T (2 {\mathbb{I}_{A}} - 1)](x) - [T (2 {\mathbb{I}_{A}} - 1)](y) \colon x,y \in {\mathcal{X}}\right\} \colon \emptyset \neq A \subset {\mathcal{X}}\right\},
\intertext{ where the first equality follows from Theorem~\ref{the:CoeffOfErgod:Approximation}, the second equality follows from \ref{def:LTO:NonNegativelyHom} and the third equality follows from \ref{prop:LTO:AdditionOfConstant}.
From the linearity of $T$, it follows that $[T f](x) = \sum_{z \in {\mathcal{X}}} f(z) [T {\mathbb{I}_{z}}](x) = \sum_{z \in {\mathcal{X}}} f(z) T(x,z)$, such that
}
{{\rho}(T)}
&= \max \left\{ \max \left\{ \frac{1}{2} \sum_{z \in {\mathcal{X}}} [2 {\mathbb{I}_{A}} - 1](z) \left(T(x,z) - T(y,z) \right) \colon x,y \in {\mathcal{X}}\right\} \colon \emptyset \neq A \subset {\mathcal{X}}\right\}\\
&= \max \left\{ \max \left\{ \frac{1}{2} \sum_{z \in {\mathcal{X}}} [2 {\mathbb{I}_{A}} - 1](z) \left(T(x,z) - T(y,z) \right) \colon \emptyset \neq A \subset {\mathcal{X}}\right\} \colon x,y \in {\mathcal{X}}\right\}.
\intertext{ Solving the inner maximisation problem for some fixed $x, y \in {\mathcal{X}}$ is trivial: the maximising $A$ is $\{ z \in {\mathcal{X}}\colon T(x,z) \geq T(y,z) \}$ as for all $z \in {\mathcal{X}}$, $[2{\mathbb{I}_{A}} - 1](z)$ is $1$ if $z \in A$ or $-1$ if $z \notin A$.
This results in
}
{{\rho}(T)}
&= \max \left\{ \frac{1}{2} \sum_{z \in {\mathcal{X}}} {\left\vert {T(x,z) - T(y,z)} \right\vert} \colon x,y \in {\mathcal{X}}\right\},
\end{aligned}$$ which proves that ${{\rho}(T)}$ is indeed equal to $\delta(T)$ of [@1991Anderson] or $\tau_1(T)$ of [@1981Seneta].
Linear transition operators are not the only lower transition operators for which the lower bound of Theorem \[the:CoeffOfErgod:Approximation\] is the actual value of the coefficient of ergodicity. [@2013Skulj] show that this is also the case for lower transition operators defined using Choquet integrals. Let $\{ L_x \}_{x\in{\mathcal{X}}}$ be a family of Choquet capacities, and assume that for all $x\in{\mathcal{X}}$, $[{\underline{T}}\cdot ](x)$ is the Choquet integral with respect to $L_x$. By [@2013Skulj Corollary 23], $$\begin{aligned}
{{\rho}({\underline{T}})}
&= \max \big\{ \max\{ L_{x}(A) - L_{y}(A) \colon x,y \in {\mathcal{X}}\} \colon 0 \neq A \subset {\mathcal{X}}\big\}.
\label{eqn:CoeffOfErgod:Choquet}\end{aligned}$$ This result allows us to exactly compute ${{\rho}({\underline{T}})}$. However, we are often interested in ${{\rho}({\underline{T}}^{k})}$, where $k > 1$ is an integer. Let $k\in{\mathbb{N}}$ and $x\in{\mathcal{X}}$, then we define the Choquet capacity $L^{k}_{x}$ for all $A \subseteq {\mathcal{X}}$ as $L^{k}_{x}(A) \coloneqq [{\underline{T}}^k {\mathbb{I}_{A}}](x)$. In general, the coherent lower expectation $[{\underline{T}}^k \cdot](x)$ is *not* a Choquet integral with respect to the Choquet capacity $L^{k}_{x}$, a fact that is seemingly overlooked in [@2013Skulj Section 5.5]. What is definitely true is that $$\max \{ \max \{ L^{k}_{x}(A) - L^{k}_{y}(A) \colon x,y \in {\mathcal{X}}\} \colon \emptyset \neq A \subset {\mathcal{X}}\}
$$ is a lower bound of ${{\rho}({\underline{T}}^{k})}$, as it is equal to the lower bound of Theorem \[the:CoeffOfErgod:Approximation\].
\[lem:StoppigCriterionWithConvergence\] Let ${\underline{Q}}$ be a lower transition rate operator and assume that $f$ is an element of ${{\mathcal{L}}({\mathcal{X}})}$ such that ${{\underline{T}}_{\infty}} f \coloneqq \lim_{t \to \infty} {{\underline{T}}_{t}} f$ is a constant function. We let $t \in {{\mathbb{R}}_{\geq 0}}$, $\epsilon \in {{\mathbb{R}}_{> 0}}$ and $\delta_1, \dots, \delta_k \in {{\mathbb{R}}_{\geq 0}}$ such that $\sum_{i=1}^{k} \delta_i = t$ and for all $i \in \{ 1, \dots, k \}$, $\delta_{i} {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2$, and define $g \coloneqq \Phi(\delta_1,\dots,\delta_k) f$. If ${\left\Vert {{{\underline{T}}_{t}} f - g} \right\Vert} \leq \epsilon$ and ${\left\Vert {g} \right\Vert}_{c} \leq \epsilon$, then $${\left\Vert {{{\underline{T}}_{\infty}} f - \tilde{g}} \right\Vert} \leq 2 \epsilon$$ and for all $\Delta \in {{\mathbb{R}}_{\geq 0}}$, $${\left\Vert {{{\underline{T}}_{t+\Delta}} f - \tilde{g}} \right\Vert} \leq 2 \epsilon,$$ where $\tilde{g} \coloneqq (\max \Phi(\delta_1, \dots, \delta_k) f + \min \Phi(\delta_1, \dots, \delta_k) f) / 2$.
Note that by \[prop:LTO:BoundedByMinAndMax\], $$\min {{\underline{T}}_{t}} f \leq \min {{\underline{T}}_{t + \Delta}} f \leq {{\underline{T}}_{\infty}} f \leq \max {{\underline{T}}_{t + \Delta}} f \leq {{\underline{T}}_{t}} f.$$ If we let $g \coloneqq \Phi(\delta_1,\dots,\delta_k) f$ and assume that ${\left\Vert {{{\underline{T}}_{t}} f - g } \right\Vert} \leq \epsilon$, then $$\min g - \epsilon \leq \min {{\underline{T}}_{t}} f \leq \min {{\underline{T}}_{t + \Delta}} \leq {{\underline{T}}_{\infty}} f \leq \max {{\underline{T}}_{t + \Delta}} f \leq {{\underline{T}}_{t}} f \leq \max g + \epsilon.$$ Hence, $${{\underline{T}}_{\infty}} f - \tilde{g} = {{\underline{T}}_{\infty}} f - \max g + \frac{\max g - \min g}{2} \leq \epsilon + {\left\Vert {g} \right\Vert}_{c},$$ and $${{\underline{T}}_{\infty}} f - \tilde{g} = {{\underline{T}}_{\infty}} f - \min g - \frac{\max g - \min g}{2} \geq - \epsilon- {\left\Vert {g} \right\Vert}_{c},$$ where $\tilde{g} \coloneqq (\max g + \min g) / 2$. Therefore, if ${\left\Vert {g} \right\Vert}_{c} \leq \epsilon$, then $${\left\Vert {{{\underline{T}}_{\infty}} f - \tilde{g}} \right\Vert} \leq 2 \epsilon,$$ which proves the first inequality of the statement. The proof of the second inequality of the statement is almost entirely similar.
If ${\underline{Q}}$ is ergodic, then by definition $\lim_{t \to \infty} {{\underline{T}}_{t}} f$ is a constant function for all $f \in {{\mathcal{L}}({\mathcal{X}})}$. Therefore, the stated follows immediately from Lemma \[lem:StoppigCriterionWithConvergence\].
In Example \[binex:AdaptiveApproximation\], we have observed that keeping track of $\epsilon'$ increases the duration of the computations. The following proposition shows that, even if one is not really interested in the value of $\epsilon'$, there is still a reason why one nevertheless would want to keep track of $\epsilon'$: it could be that we can stop the approximation because we have already attained the desired maximal error.
\[prop:StoppingCriterionWithRemainingCost\] Let ${\underline{Q}}$ be a lower transition rate operator, $f \in {{\mathcal{L}}({\mathcal{X}})}$ and $t, \epsilon \in {{\mathbb{R}}_{> 0}}$. Let $s$ denote some sequence $\delta_1, \dots, \delta_k$ in ${{\mathbb{R}}_{\geq 0}}$ such that $\smash{t' \coloneqq \sum_{i = i}^{k} \delta_i \leq t}$ and, for all $i \in \{ 1, \dots, k \}$, $\smash{\delta_i {\left\Vert {{\underline{Q}}} \right\Vert} \leq 2}$. If $\epsilon' \leq \epsilon$ is an upper bound for ${\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert}$ and ${\left\Vert {\Phi(s) f} \right\Vert}_{v} \leq \epsilon - \epsilon'$, then $${\left\Vert {{{\underline{T}}_{t}} f - \Phi(s) f} \right\Vert}
\leq \epsilon.$$
First, note that by the semi-group property ${{\underline{T}}_{t}} f = {{\underline{T}}_{t - t'}} {{\underline{T}}_{t'}} f$. Using \[prop:LTO:BoundedByMinAndMax\] yields $$\min {{\underline{T}}_{t'}} f \leq {{\underline{T}}_{t}} f \leq \max {{\underline{T}}_{t'}} f.$$ Hence $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s)f} \right\Vert}
&= \max \{ {\left\vert {[{{\underline{T}}_{t}} f](x) - [\Phi(s)f](x) } \right\vert} \colon x \in {\mathcal{X}}\} \\
&\leq \max \{ \max \{{\left\vert {\max {{\underline{T}}_{t'}} f - [\Phi(s)f](x) } \right\vert}, {\left\vert {\min {{\underline{T}}_{t'}} f - [\Phi(s)f](x) } \right\vert} \} \colon x \in {\mathcal{X}}\},
\end{aligned}$$ where the inequality follows from the obtained bounds on ${{\underline{T}}_{t}} f$. Let $x^{+} \in {\mathcal{X}}$ such that $[{{\underline{T}}_{t'}} f](x^{+}) = \max {{\underline{T}}_{t'}} f$. Then for all $x \in {\mathcal{X}}$, $$\begin{aligned}
{\left\vert {\max {{\underline{T}}_{t'}} f - [\Phi(s)f](x) } \right\vert}
&= {\left\vert {[{{\underline{T}}_{t'}} f](x^{+}) - [\Phi(s)f](x) - [\Phi(s)f](x^{+}) + [\Phi(s)f](x^{+}) } \right\vert} \\
&\leq {\left\vert {[{{\underline{T}}_{t'}} f](x^{+}) - [\Phi(s)f](x^{+}) } \right\vert} + {\left\vert {[\Phi(s)f](x) - [\Phi(s)f](x^{+})} \right\vert} \\
&\leq {\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert} + {\left\Vert {\Phi(s) f} \right\Vert}_{v}.
\intertext{Similarly, }
{\left\vert {\min {{\underline{T}}_{t'}} f - [\Phi(s)f](x) } \right\vert}
&\leq {\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert} + {\left\Vert {\Phi(s) f} \right\Vert}_{v}.
\end{aligned}$$ Therefore, $$\begin{aligned}
{\left\Vert {{{\underline{T}}_{t}} f - \Phi(s)f} \right\Vert}
&\leq \max \{ \max \{{\left\vert {\max {{\underline{T}}_{t'}} f - [\Phi(s)](x) } \right\vert}, {\left\vert {\min {{\underline{T}}_{t'}} f - [\Phi(s)](x) } \right\vert} \} \colon x \in {\mathcal{X}}\} \\
&\leq \max \{ {\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert} + {\left\Vert {\Phi(s) f} \right\Vert}_{v} \colon x \in {\mathcal{X}}\} \\
&= {\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert} + {\left\Vert {\Phi(s) f} \right\Vert}_{v}.
\end{aligned}$$
If we now assume that ${\left\Vert {{{\underline{T}}_{t'}} f - \Phi(s) f} \right\Vert} \leq \epsilon' \leq \epsilon$ and ${\left\Vert {\Phi(s) f} \right\Vert}_{v} \leq \epsilon - \epsilon'$, then $${\left\Vert {{{\underline{T}}_{t}} f - \Phi(s)f} \right\Vert}
\leq \epsilon. \qedhere$$
|
---
abstract: 'Let $X$ be an algebraic toric set in a projective space over a finite field. We study the vanishing ideal, $I(X)$, of $X$ and show some useful degree bounds for a minimal set of generators of $I(X)$. We give an explicit combinatorial description of a set of generators of $I(X)$, when $X$ is the algebraic toric set associated to an even cycle or to a connected bipartite graph with pairwise vertex disjoint even cycles. In this case, a formula for the regularity of $I(X)$ is given. We show an upper bound for this invariant, when $X$ is associated to a (not necessarily connected) bipartite graph. The upper bound is sharp if the graph is connected. We are able to show a formula for the length of the parameterized linear code associated with any graph, in terms of the number of bipartite and non-bipartite components.'
address:
- 'CMUC, Department of Mathematics, University of Coimbra 3001-454 Coimbra, Portugal. '
- |
Departamento de Matemática\
Instituto Superior Técnico\
Universidade Técnica de Lisboa\
Avenida Rovisco Pais, 1\
1049-001 Lisboa, Portugal
- |
Departamento de Matemáticas\
Centro de Investigación y de Estudios Avanzados del IPN\
Apartado Postal 14–740\
07000 Mexico City, D.F.
author:
- Jorge Neves
- Maria Vaz Pinto
- 'Rafael H. Villarreal'
title: Vanishing ideals over graphs and even cycles
---
[^1]
Introduction
============
Let $\mathbb{P}^{s-1}$ be a projective space over a finite field $\mathbb{F}_q$. An [*evaluation code*]{}, also known as a [*generalized Reed-Muller code*]{}, is a linear code obtained by evaluating the linear space of homogeneous $d$-forms on a set of points $X\subset {\mathbb{P}}^{s-1}$ (see Definition \[definition: evaluation code\]). A linear code obtained in this way, denoted by $C_X(d)$, has length ${\left | X \right | }$. Evaluation codes have been the object of much attention in recent years. To describe their basic parameters (length, dimension and minimum distance), many authors have been using tools coming from Algebraic Geometry and Commutative Algebra, see [@delsarte-goethals-macwilliams; @duursma-renteria-tapia; @gold-little-schenck; @GRT; @ci-codes; @sorensen; @tohaneanu]. Let ${\mathbb{T}}^{s-1}$ be a projective torus in $\mathbb{P}^{s-1}$. A [*parameterized linear code*]{} is a special type of generalized Reed-Muller code obtained when $X\subset
{\mathbb{T}}^{s-1}\subset {\mathbb{P}}^{s-1}$ is parameterized by a set of monomials (see Definition \[definition: parameterized linear code\]), in this case $X$ is called an [*algebraic toric set*]{} because it generalizes the notion of a projective torus. Parameterized linear codes were introduced and studied in [@algcodes]. The extra structure on $X$ yields alternative methods to compute the basic parameters of $C_X(d)$.
In this article we focus on linear codes parameterized by the edges of a graph ${\mathcal{G}}$ (see Definition \[definition: parameterized code associated to a graph\]). For the study of algebraic toric sets parameterized by the edges of a [*clutter*]{}, which is a natural generalization of the concept of graph, we refer the reader to [@ci-codes; @vanishing]. Not much is known about the parameterized linear codes associated to a general graph. The first results in this direction appear in [@GR], where the length, dimension and minimum distance of the codes associated to complete bipartite graphs are computed. In [@algcodes], one can find a formula for the length of the code associated to a connected graph (see this formula in Proposition \[proposition: order of X for G connected\]) and also a bound for the minimum distance of the code associated to a connected non-bipartite graph.
An important algebraic invariant associated to a parameterized linear code is the regularity of the ring $S/I(X)$, where $S$ is the coordinate ring of ${\mathbb{P}}^{s-1}$, [*i.e.*]{}, a polynomial ring in $s$ variables, and $I(X)$ is the vanishing ideal of $X$ (see Definition \[definition: regularity\]). The knowledge of the regularity of $S/I(X)$ is important for applications to coding theory: for $d\geq {\operatorname{reg}}S/I(X)$ the code $C_X(d)$ coincides with the underlying vector space $\mathbb{F}_q^{|X|}$ and has, accordingly, minimum distance equal to $1$. In [@deg-and-reg Corollary 2.31] the authors give bounds for the regularity of $S/I(X)$, when $X$ is the algebraic toric set associated to a connected bipartite graph. In [@codes; @over; @cycles] a bound is given for the minimum distance of the codes associated to a graph isomorphic to a cycle of even length, as well as another bound for ${\operatorname{reg}}S/I(X)$ in this case.
The contents of this paper are as follows. In Section \[sec: prelimiaries\], we recall the necessary background. To the best of our knowledge, there is no information available on the parameterized codes arising from disconnected graphs. If ${\mathcal{G}}$ is an arbitrary graph, in Section \[length\], Theorem \[theorem: computation of the length for any graph\], we show our first main result, an explicit formula for the length of $C_X(d)$ in terms of the number of bipartite and non-bipartite connected components of the graph.
An earlier result of [@algcodes] shows that the vanishing ideal $I(X)$ is minimally generated by a finite set of homogeneous binomials. In Section \[sec: generators of I(X)\], we study $I(X)$ for an arbitrary algebraic toric set $X$ and show some useful degree bounds for a minimal set of generators of $I(X)$ (see Theorem \[lemma: first reduction\] and Proposition \[proposition: bound on the degrees of generators\]). If the graph ${\mathcal{G}}$ is an even cycle, another main result of this article is an explicit combinatorial description of a generating set for $I(X)$ consisting of binomials (see Theorem \[theorem: the generators of I(X)\]). This result is generalized to any connected bipartite graph whose cycles are vertex disjoint (see Theorem \[theorem: I(X) for almost general graph\]). We give examples of bipartite graphs not satisfying this assumption for which $I(X)$ is not generated by the set prescribed in Theorem \[theorem: I(X) for almost general graph\] (see Example \[example: two offending graphs\]).
If the graph ${\mathcal{G}}$ is an even cycle of length $2k$, using our description of a generating set for $I(X)$, we derive the following formula for the regularity: $${\operatorname{reg}}S/I(X) = (q-2)(k-1)$$ (see Theorem \[theorem: regularity of even cycles\]). Then, we give the following upper bound for the regularity of $S/I(X)$ for a general (not necessarily connected) bipartite graph with $s$ edges and $m$ cycles, with disjoint edge sets, of orders $2k_1,\dots,2k_m$: $${\operatorname{reg}}S/I(X) \leq (q-2)\big(s-{\textstyle \sum}_{i=1}^m k_i -1\big)$$ (see Theorem \[theorem: bound on regularity for a general graph\]). In Corollary \[corollary: regularity for almost general graphs\], we show that this estimate is the actual value of ${\operatorname{reg}}S/I(X)$ if ${\mathcal{G}}$ is a connected bipartite graph with $s$ edges and with exactly $m$ even cycles, with disjoint vertex sets, of orders $2k_1,\dots,2k_m$.
The computational algebra techniques of [@algcodes] played an important role in discovering some of the results, conjectures, and examples of this paper. Using the computer algebra system *Macaulay*$2$ [@mac2] and the results of [@algcodes], one can compute the reduced Gröbner basis, the degree and the regularity of a vanishing ideal $I(X)$ of an algebraic toric set $X$ over a finite field $\mathbb{F}_q$. This allows us to study and to gain insight on the algebraic invariants of a vanishing ideal that are useful in algebraic coding theory.
For all unexplained terminology and additional information, we refer to [@Boll] (for graph theory), [@EisStu] (for the theory of binomial ideals), [@eisenbud-syzygies; @harris; @Sta1] (for commutative algebra and the theory of Hilbert functions), and [@stichtenoth; @tsfasman] (for the theory of linear codes and evaluation codes).
Preliminaries {#sec: prelimiaries}
=============
Let $K=\mathbb{F}_q$ be a finite field of order $q$ and fix $s$ a positive integer. Recall that the [*projective space*]{} of dimension $s-1$ over $K$, denoted by $\mathbb{P}^{s-1}$, is the quotient space $(K^{s}\setminus\{0\})/\sim $ where two vectors ${\mathbf{x}}_1$, ${\mathbf{x}}_2$ in $K^{s}\setminus\{0\}$ are equivalent if ${\mathbf{x}}_1=\lambda{{\mathbf{x}}_2}$ for some $\lambda\in
K^*=K\setminus {\left\{ 0 \right \} }$. Denote by ${\mathbb{T}}^{s-1}$ the subset of ${\mathbb{P}}^{s-1}$ given by $${\mathbb{T}}^{s-1}={\left\{ [{\mathbf{x}}]=[(x_1,\dots,x_s)]\in {\mathbb{P}}^{s-1}: x_1\cdots x_s
\not =0 \right \} },$$ where $[{\mathbf{x}}]$ is the equivalent class of ${\mathbf{x}}$. The *projective torus* ${\mathbb{T}}^{s-1}$ is an Abelian group under componentwise multiplication and is isomorphic to the standard $(s-1)$-dimensional torus, $(K^*)^{s-1}$, over $K$.
Consider , a polynomial ring over the field $K$ with the standard grading. Given a nonempty set of points $X={\left\{ [{\mathbf{x}}_1],\dots,[{\mathbf{x}}_m] \right \} }\subset
{\mathbb{T}}^{s-1}\subset {\mathbb{P}}^{s-1}$ and letting $f_0=t_1$, consider, for each $d$, the map: ${\rm ev}_d\colon S_d\rightarrow K^{{\left | X \right | }}$ given by $$\label{eq: ev-map}
f\mapsto
\left(\frac{f({\mathbf{x}}_1)}{f_0^d({\mathbf{x}}_1)},\ldots,
\frac{f({\mathbf{x}}_m)}{f_0^d({\mathbf{x}}_m)}\right),\quad
\forall\:{f \in S_d}.$$ For each $d\geq 0$, ${\rm ev}_d$ is a linear map of $K$-vector spaces. Its image is denoted by $C_X(d)$.
\[definition: evaluation code\] The *evaluation code of order $d$* associated to $X$ is the linear subspace of $K^{{\left | X \right | }}$ given by $C_X(d)$, for $d\geq 0$.
Notice that if $q=2$ then ${\mathbb{T}}^{s-1}$ is a point and, accordingly, $C_X(d)=K$, for all $d$. For this reason, throughout this article we assume that $q>2$.
Clearly an evaluation code is a linear code, [*[i.e.]{}*]{}, it is a linear subspace of $K^{{\left | X \right | }}$. Accordingly, one defines the *dimension* of the code as its dimension as a vector space, [*[i.e.]{}*]{}, as $\dim_K
C_X(d)$, its *length* as the dimension of the ambient vector space, which, for evaluation codes, coincides with ${\left | X \right | }$ and, finally, its *minimum distance*, is defined as: $$\delta_X(d)=\min\{\|{\mathbf{w}}\|
\colon 0\neq {\mathbf{w}}\in C_X(d)\},$$ where $\|{\mathbf{w}}\|$ is the number of nonzero coordinates of ${\mathbf{w}}$. The basic parameters of $C_X(d)$ are related by the [*Singleton bound*]{} for the minimum distance: $$\delta_X(d)\leq |X|-\dim_K C_X(d)+1.$$
Two of the basic parameters of $C_X(d)$, the dimension and the length, can be expressed using the Hilbert function of the quotient of $S$ by a particular homogeneous ideal. This ideal is the *vanishing ideal* of $X$, [*[i.e.]{}*]{}, the ideal of $S$ generated by the homogeneous polynomials of $S$ that vanish on $X$. Denote it by $I(X)$. Recall that the *Hilbert function* of $S/I(X)$ is given by $$H_X(d):=\dim_K (S/I(X))_d=\dim_K S_d/I(X)_d=\dim_K C_X(d),$$ see [@Sta1]. The unique polynomial $h_X(t)=\sum_{i=0}^{k-1}c_it^i\in
\mathbb{Q}[t]$ of degree such that $h_X(d)=H_X(d)$ for $d\gg 0$ is called the [*Hilbert polynomial*]{} of $S/I(X)$. The , denoted by $\deg S/I(X)$, is called the [*degree*]{} or [*multiplicity*]{} of $S/I(X)$. In our $h_X(t)$ is a nonzero constant because $S/I(X)$ has dimension $1$. Furthermore for $d\geq |X|-1$, see [@harris Lecture 13] and [@geramita-cayley-bacharach]. This means that $|X|$ is equal to the [*degree*]{} of $S/I(X)$.
A good parameterized code should have large $|X|$ together with $\dim_K C_X(d)/|X|$ and $\delta_X(d)/|X|$ as large as possible. Here, another algebraic invariant gives an indication of where to look for nontrivial evaluation codes.
\[definition: regularity\] The *index of regularity* of $S/I(X)$, denoted by ${\operatorname{reg}}S/I(X)$, is the least integer $\ell\geq 0$ such that $h_X(d)=H_X(d)$ for $d\geq \ell$.
As $S/I(X)$ is a $1$-dimensional Cohen-Macaulay graded algebra [@geramita-cayley-bacharach], the index of regularity of $S/I(X)$ is the Castelnuovo-Mumford regularity of $S/I(X)$ [@eisenbud-syzygies]. We will refer to ${\rm reg}(S/I(X))$ simply as the [*regularity*]{} of $S/I(X)$. The regularity is related to the degrees of a minimal generating set of $I(X)$.
Let $f_1,\ldots,f_r$ be a minimal homogeneous generating set of $I(X)$. The [*big degree*]{} of $I(X)$ is defined as ${\rm bigdeg}\, I(X)=
\max_i\{\deg(f_i)\}$.
From the definition of the Castelnuovo-Mumford regularity of $S/I(X)$ [@eisenbud-syzygies], one has:
\[castelnuovo-vs-bigdegree\] ${\rm bigdeg}\, I(X)-1\leq{\rm reg}(S/I(X))$.
Since $\dim_K C_X(d) = H_X(d)$ and the Hilbert polynomial of $S/I(X)$ is a constant polynomial with constant term equal to the dimension of the ambient vector space, $K^{{\left | X \right | }}$, we deduce that for $d\geq {\operatorname{reg}}S/I(X)$ the linear code $C_X(d)$ coincides with $K^{{\left | X \right | }}$. This can also be expressed by $\delta_X(d)=1$ for all $d\geq {\operatorname{reg}}S/I(X)$. We conclude that the potentially good codes $C_X(d)$ can occur only if $1\leq d < {\operatorname{reg}}(S/I(X))$.
For a particular class of evaluation codes, called *parameterized linear codes*, the ideal $I(X)$ has been studied to an extent that it is possible to use algebraic methods, based on elimination theory and Gröbner bases, to compute the dimension and the length of $C_X(d)$, see [@algcodes]. Let us briefly describe the notion of a parameterized linear code.
Given an $n$-tuple of integers, $\nu=(r_1,\dots,r_n)\in {\mathbb{Z}}^n$, and a vector ${\mathbf{x}}=(x_1,\dots,x_n)\in (K^*)^n$, we set ${\mathbf{x}}^\nu = x_1^{r_1}\cdots x_n^{r_n}\in K^*$. Let $\nu_1,\dots,\nu_s\in{\mathbb{Z}}^n$ and let $X^*\subset (K^*)^s$ be the set given by: $$X^* = {\left\{ ({\mathbf{x}}^{\nu_1},\dots,{\mathbf{x}}^{\nu_s}):{\mathbf{x}}\in (K^*)^n \right \} }.$$ Consider the multiplicative group structure of $(K^*)^s$ and let $\pi
\colon (K^*)^s {\rightarrow}{\mathbb{T}}^{s-1}$ be the quotient map by the [*diagonal subgroup*]{} $\Lambda={\left\{ (\lambda,\dots,\lambda)\in (K^*)^s : \lambda \in K^* \right \} }$. Notice that ${\mathbb{T}}^{s-1} = (K^*)^s/\Lambda$ is the projective torus in ${\mathbb{P}}^{s-1}$.
\[definition: parameterized linear code\] Let $\nu_1,\dots,\nu_s\in {\mathbb{N}}^n$. The set of points given by $X =
\pi(X^*)$ is called an *algebraic toric set* parameterized by $\nu_1,\dots,\nu_s\in {\mathbb{N}}^n$. The evaluation codes $C_X(d)$ obtained from an algebraic toric set $X$ are called *parameterized linear codes*.
It is clear that $X^*$ is a subgroup of $(K^*)^s$, since it is the image of the group homomorphism $(K^*)^n {\rightarrow}(K^*)^s$ given by ${\mathbf{x}}\mapsto ({\mathbf{x}}^{\nu_1},\dots,{\mathbf{x}}^{\nu_s})$. Denote by $\theta \colon (K^*)^n {\rightarrow}X^*$ and by $\widetilde{\pi} \colon X^* {\rightarrow}X$ the restrictions of the corresponding homomorphisms. Thus, we have the following sequence: $$\label{eq: structural exact sequence}
(K^*)^n \stackrel{\theta}{{\longrightarrow}} X^* \stackrel{\widetilde{\pi}}{{\longrightarrow}} X {\longrightarrow}1.$$
For a parameterized algebraic toric set $X$, the vanishing ideal $I(X)$ carries extra structure. We know that, in this situation, $I(X)$ is $1$-dimensional Cohen-Macaulay lattice ideal [@algcodes]. In particular $I(X)$ is a binomial ideal, [*[i.e.]{}*]{}, it is generated by binomials. Recall that a binomial in $S$ is a polynomial of the form $t^a-t^b$, where $a,b\in {\mathbb{N}}^s$ and where, if , we set $$t^a=t_1^{a_1}\cdots t_s^{a_s}\in S.$$ A binomial of the form $t^a-t^b$ is usually referred to as a [*pure binomial*]{} [@EisStu], although here we are dropping the adjective “pure”.
Let ${\mathcal{G}}$ be a simple graph with vertex set $V_{\mathcal{G}}={\left\{ v_1,\dots,v_n \right \} }$ and edge set $E_{\mathcal{G}}={\left\{ e_1,\dots,e_s \right \} }$. Throughout the remainder of this article, when dealing with a graph, we shall reserve the use of $n$ and $s$ for the number of vertices and the number of edges of the graph in question. For an edge $e_i={\left\{ v_j,v_k \right \} }$, where $v_j,v_k\in V_{\mathcal{G}}$, let $\nu_i
= \mathbf{e}_j+\mathbf{e}_k\in {\mathbb{N}}^n$, where, for $1\leq j\leq n$, $\mathbf{e}_j$ is the $j$-th element of the canonical basis of ${\mathbb{Q}}^n$.
\[definition: parameterized code associated to a graph\] The *algebraic toric set associated to ${\mathcal{G}}$* is the toric set parameterized by the $n$-tuples , obtained from the edges of ${\mathcal{G}}$. If $X$ is the parameterized toric set associated to ${\mathcal{G}}$ we call its associated linear code $C_X(d)$ *the parameterized code associated to ${\mathcal{G}}$* and we refer to the vanishing ideal of $X$ as the *vanishing ideal over ${\mathcal{G}}$*.
If ${\mathbf{x}}=(x_1,\dots,x_n) \in (K^*)^n$ and $e_i={\left\{ v_j,v_k \right \} }$ is an edge of ${\mathcal{G}}$, we set ${\mathbf{x}}^{e_i}={\mathbf{x}}^{\mathbf{e}_j + \mathbf{e}_k}=x_jx_k$, so that the structural map $\theta\colon (K^*)^n {\rightarrow}X^*$ is given by ${\mathbf{x}}\mapsto ({\mathbf{x}}^{e_1},\dots,{\mathbf{x}}^{e_s})$. It is clear that if ${\mathcal{G}}$ contains isolated vertices, then the associated algebraic toric set $X$ coincides with the algebraic toric set associated to the subgraph of ${\mathcal{G}}$ obtained by removing these vertices. If ${\mathcal{G}}$ has a second edge through two vertices, then $X$ is isomorphic to its projection away from the coordinate point of ${\mathbb{P}}^{s-1}$ corresponding to that edge; which, in turn, coincides with the algebraic toric set defined by the graph obtained from ${\mathcal{G}}$ by removing the multiple edge. Hence, from the point of view of the algebraic toric set $X$, the existence of multiple edges in ${\mathcal{G}}$ is not interesting. If ${\mathcal{G}}$ has only one edge then is easy to see that $X={\mathbb{P}}^{s-1}$ is a point, $I(X)=0$ and $C_X(d)=K^*$. Thus throughout the remainder of this article we shall assume that ${\mathcal{G}}$ is a simple graph with no isolated vertices and with $s\geq 2$.
If ${\mathcal{G}}$ is a connected graph, the length of $C_X(d)$ has been determined.
\[proposition: order of X for G connected\] Let ${\mathcal{G}}$ be a connected graph and $X$ its associated algebraic toric set. Then ${\left | X \right | }=(q-1)^{n-1}$ if ${\mathcal{G}}$ is non-bipartite and ${\left | X \right | }=(q-1)^{n-2}$ if ${\mathcal{G}}$ is bipartite.
In particular, since $X\subset {\mathbb{T}}^{s-1}\subset {\mathbb{P}}^{s-1}$ and ${\left | {\mathbb{T}}^{s-1} \right | }=(q-1)^{s-1}$ we see that if ${\mathcal{G}}$ is a connected non-bipartite graph with $n=s$, then the algebraic toric set parameterized by the edges of ${\mathcal{G}}$ coincides with ${\mathbb{T}}^{s-1}$. In this situation, the vanishing ideal of ${\mathbb{T}}^{s-1}$, its invariants and all of the parameters of $C_X(d)$ are known, and are summarized in the following proposition.
[([@GRH Theorem 1, Lemma 1], [@ci-codes Corollary 2.2, Theorem 3.5])]{} If $\mathbb{T}^{s-1}$ is the projective torus in $\mathbb{P}^{s-1}$, then
1. $I(\mathbb{T}^{s-1})=\bigr(\{t_i^{q-1}-t_1^{q-1}\}_{i=2}^s\bigl)$;
2. $ F_{\mathbb{T}^{s-1}}(t)=(1-t^{q-1})^{s-1}/(1-t)^s$;
3. ${\rm reg}(S/I(\mathbb{T}^{s-1}))=(s-1)(q-2)$ and ${\rm
deg}(S/I(\mathbb{T}^{s-1}))={\left | {\mathbb{T}}^{s-1} \right | }=(q-1)^{s-1}$;
4. $\dim_K C_{{\mathbb{T}}^{s-1}}(d)=\sum_{j=0}^{\left \lfloor{d}/{(q-1)}
\right \rfloor}(-1)^j\binom{s-1}{j}\binom{s-1+d-j(q-1)}{s-1}$;
5. $\delta_{{\mathbb{T}}^{s-1}}(d) = (q-1)^{s-(k+2)}(q-1-\ell)$ for all $d<{\rm reg}(S/I(\mathbb{T}^{s-1}))$, where $k\geq 0$ and $1\leq
\ell\leq q-2$ are the unique integers such that $d=k(q-2)+\ell$.
In the statement of the result, $F_{{\mathbb{T}}^{s-1}} (t) =
\sum_{i=0}^\infty H_{{\mathbb{T}}^{s-1}} (i) t^i$ is the Hilbert series of $S/I({\mathbb{T}}^{s-1})$. The fact that the vanishing ideal in the case of the torus is a complete intersection plays a crucial part in the proof of these results. We know that in practice the vanishing ideal associated to a general graph is far from being a complete intersection. Indeed, by [@ci-codes Corollary 4.5] for an algebraic toric set $X$ associated to a graph (or more generally a *clutter*—see [@ci-codes] for a definition), $I(X)$ is a complete intersection if and only if $X={\mathbb{T}}^{s-1}$.
The length of parameterized codes of graphs {#length}
===========================================
We continue to use the notation and definitions used in Section \[sec: prelimiaries\]. In this section, we show an explicit formula for the length of any parameterized code associated to an arbitrary graph.
Let ${\mathcal{G}}$ be a simple graph with vertex set $V_{\mathcal{G}}={\left\{ v_1,\dots,v_n \right \} }$ and edge set $E_{\mathcal{G}}={\left\{ e_1,\dots,e_s \right \} }$. Denote by ${\mathcal{G}}_1,\dots,{\mathcal{G}}_m$ the connected components of $\mathcal{G}$. For each $1\leq j\leq m$, let $n_j$ and $s_j$ denote the number of vertices and edges of ${\mathcal{G}}_j$, respectively; so that $n=n_1+\cdots + n_m$ and $s=s_1+\cdots + s_m$. Denote the edges of ${\mathcal{G}}_j$ by ${\left\{ e_{j1},\dots,e_{js_j} \right \} }$, let $X_j\subset {\mathbb{P}}^{s_j-1}$ be the algebraic toric set parameterized by ${\mathcal{G}}_j$ and let $$(K^*)^{n_j} \stackrel{\theta_j}{{\longrightarrow}} X^*_j \stackrel{\widetilde{\pi}_j}{{\longrightarrow}} X_j {\longrightarrow}1$$ be the corresponding structural sequences. Since for fixed distinct $j_1\not = j_2$ the edges $e_{j_1k_1}$ and $e_{j_2k_2}$ have no vertex in common and thus ${\mathbf{x}}^{e_{j_1k_1}}$ and ${\mathbf{x}}^{e_{j_2k_2}}$ involve disjoint sets of coordinates of the vector ${\mathbf{x}}$, we deduce that $\theta \colon (K^*)^n {\rightarrow}X^*$ is isomorphic to $$\theta_1\times\cdots \times \theta_m \colon (K^*)^{n_1}\times \cdots
\times (K^*)^{n_m} {\rightarrow}X^*_1\times \cdots \times X^*_m.$$ In particular ${\left | X^* \right | }=\prod_{j=1}^m {| X^*_j | }$. We need to find the order of the kernel of the maps $\widetilde{\pi}_j$.
\[lemma: the order of the kernel of pi\] Let ${\mathcal{G}}$ be a connected graph. If ${\mathcal{G}}$ is non-bipartite, then ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }=\frac{q-1}{2}$ if $q$ is odd and ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }=q-1$ if $q$ is even. If ${\mathcal{G}}$ is bipartite, then ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }=q-1$.
Let ${\mathbf{x}}\in (K^*)^{n}$. Then implies that ${\mathbf{x}}^e=1$ for all $e\in E_{{\mathcal{G}}}$. Suppose ${\mathcal{G}}$ is non-bipartite. Then ${\mathcal{G}}$ contains an odd cycle. We assume, without loss of generality, that the edges in this cycle are $$e_1={\left\{ v_1,v_2 \right \} },\dots,e_{2k-1}={\left\{ v_{2k-1},v_1 \right \} },$$ where $v_1\dots,v_{2k-1}\in V_{{\mathcal{G}}}$. We deduce that $x_1x_2=\cdots=x_{2k-1}x_1=1$, which, in turn, implies that with $u^2=1$. Now, let $v_r\in V_{{\mathcal{G}}}$ be any vertex of ${\mathcal{G}}$. Then, there exists a path $${\left\{ v_1,v_{\ell_1} \right \} },{\left\{ v_{\ell_1},v_{\ell_2} \right \} },\dots,{\left\{ v_{\ell_k},v_r \right \} }$$ connecting $x_1$ with $x_r$. Since $x_1x_{j_1}=x_{j_1}x_{j_2}=\cdots=x_{j_k}x_r=1$, we deduce that $x_r=u$. Hence, either ${\mathbf{x}}=(1,\dots,1)$ or ${\mathbf{x}}=(-1,\dots,-1)$, from which we conclude that ${\left | {\operatorname{Ker}}\theta \right | }=2$ if $q$ is odd and ${\left | {\operatorname{Ker}}\theta \right | }=1$ if $q$ even. Suppose now that ${\mathcal{G}}$ is bipartite, and, without loss of generality, denote the bipartition of $V_{\mathcal{G}}$ by ${\left\{ v_1,\dots,v_\ell \right \} }\cup {\left\{ v_{\ell+1},\dots,v_n \right \} }$. Let $v_r$ be any vertex and let $${\left\{ v_1,v_{j_1} \right \} },{\left\{ v_{j_1},v_{j_2} \right \} },\ldots,{\left\{ v_{j_k},v_r \right \} }$$ be a path connecting $v_1$ with $v_r$. Notice that ${\left\{ v_{j_1},v_{j_3},\dots \right \} }$ is a subset of ${\left\{ v_{\ell+1},\dots,v_n \right \} }$ and ${\left\{ v_{j_2},v_{j_4},\dots \right \} }$ is a subset of ${\left\{ v_1,\dots,v_\ell \right \} }$. From $x_1x_{j_1}=x_{j_1}x_{j_2}=\cdots = x_{j_k}x_r=1$ we deduce that $x_r=x_1$ if $v_r\in {\left\{ v_1,\dots,v_\ell \right \} }$ or $x_r=x_1^{-1}$ otherwise. Hence $x=(x_1,\dots,x_1,x_1^{-1},\dots,x_1^{-1})$, [*[i.e.]{}*]{}, the $\ell$ first coordinates of ${\mathbf{x}}$ are equal to $x_1$ and the remaining ones are equal to $x_1^{-1}$. Conversely, it is easy to see that any element of $(K^*)^n$ of the form $(u,\dots,u,u^{-1},\dots,u^{-1})$ belongs to ${\operatorname{Ker}}\theta$. We deduce that in this case ${\left | {\operatorname{Ker}}\theta \right | }=q-1$. The proof now follows easily from Proposition \[proposition: order of X for G connected\]. Indeed, we know that the order of $X$ is $(q-1)^{n-1}$, if ${\mathcal{G}}$ is non-bipartite and $(q-1)^{n-2}$ otherwise. Hence, ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }=\frac{q-1}{2}$ if ${\mathcal{G}}$ is non-bipartite and $q$ is odd, ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }={q-1}$ if ${\mathcal{G}}$ is non-bipartite and $q$ is even, and ${\left | {\operatorname{Ker}}\widetilde{\pi} \right | }={q-1}$ if ${\mathcal{G}}$ is bipartite.
We come to the main result of this section.
\[theorem: computation of the length for any graph\] Suppose ${\mathcal{G}}$ has $m$ connected components, of which $\gamma$ are non-bipartite. Then, $$\renewcommand{\arraystretch}{1.3}
{\left | X \right | } = \left \{
\begin{array}{l}
\bigr(\frac{1}{2}\bigl)^{\gamma-1}(q-1)^{n-m+\gamma-1}{}, \text{ if }
\gamma\geq 1\text{ and }q \text{ is odd}, \\
(q-1)^{n-m+\gamma-1}{}, \text{ if }
\gamma\geq 1\text{ and }q \text{ is even}, \\
(q-1)^{n-m-1}, \text{ if }\gamma=0.
\end{array}
\right.$$
As in the discussion above, let $X_1,\dots,X_m$ be the parameterized toric sets associated to the connected components of ${\mathcal{G}}$. Then ${\left | X^* \right | }=\prod_{j=1}^m {| X^*_j | }$, which, by Lemma \[lemma: the order of the kernel of pi\], is given by $$\renewcommand{\arraystretch}{1.3}
{\left | X^* \right | } = \left \{
\begin{array}{l}
\bigr(\frac{1}{2}\bigl)^{\gamma}(q-1)^{n-m+\gamma}{}, \text{ if } q \text{ is odd}, \\
(q-1)^{n-m+\gamma}{}, \text{ if } q \text{ is even}.
\end{array}
\right.$$ From the proof of Lemma \[lemma: the order of the kernel of pi\], it is seen that the kernel of the map $\widetilde{\pi}\colon X^*\rightarrow X$ is equal to $\Lambda$, the diagonal subgroup of $(K^*)^s$, if $\gamma=0$, and it is equal to $\Lambda^2=\{(\lambda^2,\ldots,\lambda^2)\vert\, \lambda\in
F_q^*\}$ if $\gamma\geq 1$. The subgroup $\Lambda$ has order $q-1$. The subgroup $\Lambda^2$ has order $q-1$ if $q$ is even and has order $(q-1)/2$ if $q$ is odd (this follows readily using the map $\lambda\mapsto (\lambda^2,\ldots,\lambda^2)$). As $|X|=|X^*|/|{\rm
Ker}\, \widetilde{\pi}|$, the result follows.
Let $G$ be the graph whose connected components are a triangle and a square. Thus, $n=7$, $m=2$, $\gamma=1$. Using the formula of Theorem \[theorem: computation of the length for any graph\], we get: (a) $|X|=1024$ if $q=5$, and (b) $|X|=243$ if $q=2^2$.
Degree bounds for the generators of $I(X)$ {#sec: generators of I(X)}
==========================================
We continue to use the notation and definitions used in Section \[sec: prelimiaries\]. In this section $X\subset {\mathbb{P}}^{s-1}$ is the algebraic toric set parameterized by $\nu_1,\ldots,\nu_s\in\mathbb{N}^n$ and $I(X)\subset S=K[t_1,\dots,t_s]$ is the vanishing ideal of $X$. We show some degree bounds for a minimal set of generators of $I(X)$ consisting of binomials.
Recall that by [@algcodes] we know that $I(X)$ is generated by homogeneous binomials $t^a-t^b$, with $a,b\in {\mathbb{N}}^s$. There are a number of elementary observations to be made. Let $f=t^a-t^b$ be a nonzero binomial of $S$. Firstly, since $X\subset {\mathbb{T}}^{s-1}$, evidently $I({\mathbb{T}}^{s-1})\subset I(X)$, hence $t_i^{q-1}-t_j^{q-1}\in I(X)$, for all $1\leq i,j\leq s$. Secondly, if $\gcd(t^a,t^b)\not = 1$, then we can factor the greatest common divisor $t^c$ from both $t^a$ and $t^b$ to obtain , for some $a',b'\in {\mathbb{N}}^s$. Since $t^c$ is never zero on ${\mathbb{T}}^{s-1}$, for any $c\in {\mathbb{N}}^s$, we deduce that if and only if $t^{a'}-t^{b'}\in I(X)$. Therefore, when looking for “binomial generators” of $I(X)$ we may ourselves to those binomials $t^a-t^b$ such that $t^a$ and $t^b$ have no common divisors. Given $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$, we set $|a|= a_1+\cdots + a_s$ and . Then, clearly, $t^a$ and $t^b$ have no common divisors if and only if ${\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)=\emptyset$.
A subgroup of $\mathbb{Z}^s$ is called a [*lattice*]{}. A [*lattice ideal*]{} is an ideal of the form $$I(\mathcal{L})=(\{t^{a}-t^{b}\, \colon\,
a-b\in\mathcal{L}\mbox{ and }{\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)=\emptyset\})\subset S$$ for some lattice $\mathcal{L}$ of $\mathbb{Z}^s$.
\[l-lemma\] Let $L\subset S$ be a lattice ideal generated by $\mathcal{B}=\{t^{a_i}-t^{b_i}\}_{i=1}^r$. Then, [(a)]{} $L=I(\mathcal{L})$, where $\mathcal{L}$ is the subgroup of $\mathbb{Z}^s$ generated by $\{a_i-b_i\}_{i=1}^r$, and [(b)]{} if $t^{a_i}-t^{b_i}$ is homogeneous for all $i$ and $f=t^a-t^b\in L$, then $f$ is homogeneous.
Part (a) follows from [@cca Lemma 7.6]. To show (b) notice that, from part (a), $f\in I(\mathcal{L})$. Then, $a-b$ is a linear combination of $\{a_i-b_i\}_{i=1}^r$. Thus, if $\mathbf{1}=(1,\ldots,1)$, we get that $|a|-|b|$ is equal to $\langle\mathbf{1},a-b\rangle=0$ because $\langle\mathbf{1},a_i-b_i\rangle=0$ for all $i$. Thus, $|a|=\deg(t^a)=\deg(t^b)=|b|$.
\[h-lemma\] If $f=t^a-t^b\in I(X)$, then $f$ is homogeneous.
According to [@algcodes Theorem 2.1], $I(X)$ is lattice ideal generated by homogeneous binomials. Thus, the lemma follows from Lemma \[l-lemma\].
\[lemma: reduction of the degree in each variable\] Let $f=t^a-t^b \in I(X)$, where $a,b \in {\mathbb{N}}^s$ and ${\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)=\emptyset$. Suppose that there exists $i$ such that $t_i^{q-1}$ divides $t^a$ and ${\operatorname{supp}}(b)\not = \emptyset$. Then, there exists a binomial $g \in I(X)$, with $\deg(g)<\deg(f)$, and there exists $j$, such that $f-t_{j}g \in I({\mathbb{T}}^{s-1})$.
Write $t^a=t_i^{q-1}t^{a'}$, with $a'\in {\mathbb{N}}^s$. Since ${\operatorname{supp}}(b)\not =\emptyset$, there exists $j$ such that $t_j$ divides $t^b$. Write $t^b=t_jt^{b'}$, for some $b'\in {\mathbb{N}}^s$. Then, $$t^a-t^b= t_i^{q-1}t^{a'}-t_j t^{b'} = (t_i^{q-1}-t_j^{q-1})t^{a'} + t_j(t_j^{q-2}t^{a'}-t^{b'}).$$ Set $g=t_j^{q-2}t^{a'}-t^{b'}$. Then, since $t_i^{q-1}-t_j^{q-1}\in I(X)$, we see that $g\in I(X)$ and, moreover, it is clear that if $g\not = 0$ then .
\[lemma: first reduction\] There exists a set of generators of $I(X)$ which consists of the toric relations $t_i^{q-1}-t_j^{q-1}$ plus a finite set of homogeneous binomials $t^a-t^b$ with ${\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)= \emptyset$ and such that the degree of $t^a-t^b$ in each of the variables $t_i$ is $\leq q-2$.
We know that $I(X)$ is generated by binomials [@algcodes]. If ${\left\{ f_1,\dots,f_r \right \} }$ is a set of binomials generating $I(X)$, then so is the set $$\mathcal{B}={\left\{ f_1,\dots,f_r \right \} }\cup \{t^{q-1}_i-t^{q-1}_j :
1\leq i,j\leq s\}.$$ If $f_i\in I(\mathbb{T}^{s-1})$, we have $(\mathcal{B})=(\{{\left\{ f_1,\dots,f_r \right \} }\setminus\{f_i\}\}\cup \{t^{q-1}_i-t^{q-1}_j :
1\leq i,j\leq s\})$. Thus, we may assume that $\mathcal{B}$ is a generating set of $I(X)$ with $f_i\notin I(\mathbb{T}^{s-1})$ for all $i$. By the discussion above we may also assume that each $f_i$ is of the form $t^a-t^b$ with ${\operatorname{supp}}(a)\cap{\operatorname{supp}}(b)=\emptyset$. We can write $f_1=t^a-t^b$, with $a,b\in{\mathbb{N}}^s$. Suppose that there exists $i$ such that $t_i^{q-1}$ divides $t^a$ or $t^b$. Hence, since $f_1$ is homogeneous by Lemma \[h-lemma\], we deduce that the sets ${\operatorname{supp}}(a)$ and ${\operatorname{supp}}(b)$ are both nonempty. Then, from Lemma \[lemma: reduction of the degree in each variable\], there exists $j$ and a homogeneous binomial $g_1'\in I(X)$ such that $\deg(g_1')<\deg(f_1)$ and $f_1-t_jg_1'\in I({\mathbb{T}}^{s-1})$. We can write $g_1'=t^cg_1$ for some $c\in\mathbb{N}^s$, where $g_1$ is a binomial in $I(X)$ whose terms have disjoint support. Clearly, $$I(X)=(\mathcal{B})=\bigl({\left\{ g_1,f_2,\dots,f_r \right \} }\cup \{t^{q-1}_i-t^{q-1}_j :
1\leq i,j\leq s\}\bigr)$$ and $g_1\notin I({\mathbb{T}}^{s-1})$. If there exists $i$ such that $t_i^{q-1}$ divides one of the terms of $g_1$, we repeat the previous procedure with $g_1$ playing the role of $f_1$ and obtain a binomial $g_2$, and so on. Thus, by iterating the previous procedure, we obtain a sequence of homogeneous binomials $f_1,g_1,\dots,g_m$, with decreasing degrees, such that $$\label{nov29-11}
I(X)=(\mathcal{B})=\bigl({\left\{ g_m,f_2,\dots,f_r \right \} }\cup \{t^{q-1}_i-t^{q-1}_j :
1\leq i,j\leq s\}\bigr)$$ and $g_m\notin I({\mathbb{T}}^{s-1})$. Thus, using the previous procedure enough times, we obtain a binomial $g_m=t^{a'}-t^{b'}$ none of whose terms $t^{a'}$ or $t^{b'}$ is divisible by any $t_i^{q-1}$, for $1\leq i\leq s$. If we proceed in this manner, with each of the remaining $f_2,\dots,f_r$, we reach a generating set satisfying the condition in the statement.
The next proposition is intended mainly for practical applications. It gives a bound on the degrees of a minimal set of generators of $I(X)$. It is a valuable tool to use when implementing the calculation of $I(X)$ in a computer algebra software.
\[proposition: bound on the degrees of generators\] Set $k=\left \lfloor \frac{s}{2} \right\rfloor$. If $k\geq 2$, then the vanishing ideal of $X$ has a generating set whose elements have degree $\leq k(q-2)$.
Let $t^a-t^b\in I(X)$ be a homogeneous binomial. Write $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$ and $b=(b_1,\dots,b_s)\in {\mathbb{N}}^s$. By Theorem \[lemma: first reduction\], we may assume that ${\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)=\emptyset$ and that $0\leq a_i,b_j\leq q-2$. Let $r={\left | {\operatorname{supp}}(a) \right | }$ and $\ell={\left | {\operatorname{supp}}(b) \right | }$. Then, either $r$ or $\ell$ is $\leq k$, for otherwise: $$r+\ell\geq 2k+2 = 2\left \lfloor {s}/{2} \right\rfloor +2 \geq s+1,$$ which is impossible. Assume $r\leq k$. Then, $\deg(t^a-t^b)=a_1+\cdots +a_s \leq r(q-2)\leq k(q-2)$.
If $X$ is the algebraic toric set associated to a cycle ${\mathcal{G}}$ of order $s=2k$, then, by Corollary \[remark: on the degrees of the generators of I(X)\], $I(X)$ is generated in degrees $\leq (k-1)(q-2)+1$. Hence for this restricted class of vanishing ideals our estimate is not sharp. On the other hand, for $q=3$, the estimate that $I(X)$ is generated in degrees $\leq k$ is sharp, as the following example shows.
(200,70)(4,20) (20,80)[$\bullet$]{} (10,85)[$1$]{} (7,50)[$e_1$]{} (22.5,82)[(0,-1)[60]{}]{} (20,20)[$\bullet$]{} (10,15)[$2$]{} (47,28)[$e_2$]{} (22.2,22.2)[(5,3)[50]{}]{} (70,50)[$\bullet$]{} (80,50)[$3$]{} (47,72)[$e_3$]{} (72.2,52.7)[(-5,3)[50]{}]{} (130,50)[$\bullet$]{} (120,50)[$4$]{} (147,72)[$e_4$]{} (132.2,52.2)[(5,3)[50]{}]{} (180,80)[$\bullet$]{} (190,85)[$5$]{} (182.5,82)[(0,-1)[60]{}]{} (190,50)[$e_5$]{} (180,20)[$\bullet$]{} (190,15)[$6$]{} (147,28)[$e_6$]{} (184,22)[(-5,3)[50]{}]{}
\[example: two triangles\] Let ${\mathcal{G}}$ be the graph in Figure \[fig: graph with large degree generators\] and assume that $q=3$. Then, using *Macaulay*$2$ [@mac2], we found that $I(X)$ is generated by the (minimal) set of binomials: $$\begin{array}{c}
t_5^2-t_6^2,\quad t_4^2-t_6^2,\quad t_3^2-t_6^2,\quad t_2^2-t_6^2,\quad t_1^2-t_6^2,\\
t_3t_4t_5-t_1t_2t_6, \quad t_2t_4t_5-t_1t_3t_6,\quad
t_1t_4t_5-t_2t_3t_6,\quad t_2t_3t_5-t_1t_4t_6,\quad t_1t_3t_5-t_2t_4t_6,\\
t_1t_2t_5-t_3t_4t_6, \quad t_2t_3t_4-t_1t_5t_6,\quad
t_1t_3t_4-t_2t_5t_6,\quad t_1t_2t_4-t_3t_5t_6,\quad t_1t_2t_3-t_4t_5t_6.
\end{array}$$
Generators of $I(X)$ for even cycles and certain bipartite graphs {#sec: generators of I(X) for
graphs}
=================================================================
We keep the notation of Section \[length\]: $X\subset {\mathbb{P}}^{s-1}$ is the algebraic toric set parameterized by a graph ${\mathcal{G}}$ and $I(X)\subset S=K[t_1,\dots,t_s]$ is the vanishing ideal of $X$. This section is devoted to giving an explicit description of a binomial generating set for $I(X)$, when ${\mathcal{G}}={\mathcal{C}}_{2k}$ is a cycle of even order, or when ${\mathcal{G}}$ is a bipartite graph whose cycles are pairwise vertex disjoint.
\[proposition: nonoccuring variables\] Let $f=t^a-t^b\in I(X)$, with $a=(a_1,\ldots,a_s)$ and $b=(b_1,\ldots,b_s)$, such that and $a_j,b_j\leq q-2$ for all $j$. [(a)]{} If $G$ is a connected bipartite graph and $e_i$ is an edge of ${\mathcal{G}}$ which does not belong to any cycle of ${\mathcal{G}}$, then $a_i=b_i=0$. [(b)]{} If ${\mathcal{G}}$ is any graph and ${\mathcal{G}}$ has an edge $e_i$ with a degree $1$ incident vertex, then $a_i=b_i=0$.
\(a) Assume, without loss of generality that, $e_i={\left\{ v_1,v_2 \right \} }$. In what follows we use the symbol $\sqcup$ to denote a disjoint union of objects. Since ${\mathcal{G}}$ is bipartite there exist a bipartition $V_G=A\sqcup B$ with, say, $v_1\in A$ and $v_2\in B$. Since $e_i$ does not belong to a cycle of ${\mathcal{G}}$, the removal of edge $e_i$ produces a disconnected graph ${\mathcal{G}}_1\sqcup {\mathcal{G}}_2$, with $v_1\in V_{{\mathcal{G}}_1}$ and $v_2\in V_{{\mathcal{G}}_2}$. Let $u\in K^*$ be a generator of the multiplicative group of $K$. Let us label the vertices of ${\mathcal{G}}$ with one of the elements $u$, $u^{-1}$ or $1$, according to the rule that we now explain. Let $v_r$ be any vertex. If $v_r\in V_{{\mathcal{G}}_1}$ label $v_r$ with $1$, if $v_r\in V_{{\mathcal{G}}_2}\cap A$ label $v_r$ with $u^{-1}$, and if $v_r\in V_{{\mathcal{G}}_2}\cap B$ label $v_r$ with $u$. Consider ${\mathbf{x}}=(x_1,\dots,x_n)\in (K^*)^{n}$ where, for $1\leq r\leq n$, the coordinate $x_r$ takes on the value of the label of $v_r$. Then ${\mathbf{x}}^{e_j}=1$ if $j\not = i$ and ${\mathbf{x}}^{e_i}=u$. Assume that $a_i>0$, then $b_i=0$ because $a$ and $b$ have disjoint support. Thus $f({\mathbf{x}}^{e_1},\dots,{\mathbf{x}}^{e_s})=0$, implies that $u^{a_i}-1=0$, a contradiction because $1\leq a_i\leq q-2$. Similarly if $b_i>0$ we derive a contradiction. Hence, we deduce that $a_i=b_i=0$. (b) This part follows using a similar argument.
\[example: two triangles joined at the hip\] For non-bipartite graphs Proposition \[proposition: nonoccuring variables\](a) does not hold. Let ${\mathcal{G}}$ be the graph in Figure \[fig: graph – two triangles joined at the hip\] and assume that $q=5$. Then, using *Macaulay*$2$ [@mac2], we found that the binomial $t_1t_2t_4^2t_7-t_3t_5^2t_6t_8$ is in a minimal generating set of $I(X)$. In this monomial the variables $t_4$ and $t_5$, which are not in any cycle of ${\mathcal{G}}$, occur.
(200,70)(33,20) (20,80)[$\bullet$]{} (10,85)[$1$]{} (7,50)[$e_3$]{} (27,50)[$\scriptstyle 1$]{} (22.5,82)[(0,-1)[60]{}]{} (20,20)[$\bullet$]{} (10,15)[$3$]{} (47,28)[$e_2$]{} (43,40)[$\scriptstyle 1$]{} (22.2,20.5)(2.5,1.5)[21]{}[$\cdot$]{} (70,50)[$\bullet$]{} (72,58)[$2$]{} (47,72)[$e_1$]{} (43,60)[$\scriptstyle 1$]{} (72.2,49)(-2.5,1.5)[21]{}[$\cdot$]{} (72.2,50.25)(3,0)[21]{}[$\cdot$]{} (98.5,57.5)[$e_4$]{} (101,42.5)[$\scriptstyle 2$]{} (130,50)[$\bullet$]{} (130,58)[$4$]{} (132,53.5)[(1,0)[60]{}]{} (158,57.5)[$e_5$]{} (160,42.5)[$\scriptstyle 2$]{} (187.5,58)[$5$]{} (190,50)[$\bullet$]{} (207,72)[$e_6$]{} (218,60)[$\scriptstyle 1$]{} (192.2,52.2)[(5,3)[50]{}]{} (240,80)[$\bullet$]{} (250,85)[$6$]{} (241,81)(0,-3)[21]{}[$\cdot$]{} (250,50)[$e_7$]{} (234,50)[$\scriptstyle 1$]{} (240,20)[$\bullet$]{} (250,15)[$7$]{} (207,28)[$e_8$]{} (218,40)[$\scriptstyle 1$]{} (244,22)[(-5,3)[50]{}]{}
\[corollary: supports add up to all\] Suppose that ${\mathcal{G}}={\mathcal{C}}_{2k}$ is a cycle of even order. Let $f=t^a-t^b$ be a nonzero homogeneous binomial in $I(X)$, with $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$ and $b=(b_1,\dots,b_s)\in {\mathbb{N}}^s$ such that and $0\leq a_i,b_j\leq q-2$. Then ${\operatorname{supp}}(a)\cup{\operatorname{supp}}(b)={\left\{ 1,\dots,s \right \} }$.
Assume, without loss of generality that $s\not \in {\operatorname{supp}}(a)\cup
{\operatorname{supp}}(b)$. Then, $f$ is a polynomial in the variables $t_1,\dots,t_{s-1}$ which vanishes along the projection of $X$ onto the first $s-1$ coordinates. The algebraic toric set obtained after projecting is none other that the algebraic toric set associated with the graph obtained from ${\mathcal{G}}={\mathcal{C}}_{2k}$ by removing the edge $e_s$, which is a tree. Hence, by Proposition \[proposition: nonoccuring variables\], none of the remaining variables $t_1,\dots,t_{s-1}$ occurs in $f$, in other words, $f=0$, which is a contradiction.
From now on, until otherwise stated, we will restrict to the case of ${\mathcal{G}}={\mathcal{C}}_{2k}$, a cycle of order $2k$ with $k\geq 2$. Let $V_{{\mathcal{C}}_{2k}}={\left\{ v_1,\dots,v_{2k} \right \} }$ and $e_i={\left\{ v_i,v_{i+1} \right \} }$ for $1\leq i \leq 2k-1$ and . We are now ready to give a combinatorial description of the generators of $I(X)$ other than those coming from the toric relations. From Theorem \[lemma: first reduction\] and Corollary \[corollary: supports add up to all\] we know that there is a set of generators of $I(X)$ consisting of the toric generators $t_i^{q-1}-t_j^{q-1}$ plus a set of binomials of the type $t^a-t^b$ where $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$, $b=(b_1,\dots,b_s)\in {\mathbb{N}}^s$ are such that and $0\leq a_i,b_j\leq q-2$. Hence to any such binomial one can associate a partition of ${\left\{ 1,\dots,s \right \} }$. For the remainder of this article, given $r\in
{\left\{ 1,\dots,q-2 \right \} }$ we will fix the following notation: $$\hat{r}=q-1-r.$$
\[definition: recursive function\] Let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ and fix $r\in {\left\{ 1,\dots,q-2 \right \} }$. Define a function , recursively, by setting $\rho_\sigma^r(1)=r$ and, $$\label{eq: recursive relation}
\begin{cases}
\rho_\sigma^r(i+1)=\widehat{\rho_\sigma^r(i)}, & \text{if
${\left\{ i,i+1 \right \} }\subset A$ or ${\left\{ i,i+1 \right \} }\subset B$}\\
\rho_\sigma^r(i+1)=\rho_\sigma^r(i), & \text{otherwise,}
\end{cases}$$ for every $1\leq i\leq s-1$.
Notice that, for every $i\in
{\left\{ 1,\dots,s-2 \right \} }$, $\rho_\sigma^r(i)=\rho_\sigma^r(i+2)$ if and only if $i$ and $i+2$ are in the same partition. Since $s$ is even, we deduce that $\rho_\sigma^r(1)=\rho_\sigma^r(s-1)$ if and only if $1$ and $s-1$ are in the same partition. This implies that $\rho_\sigma^r(1)$ can be defined from $\rho_\sigma^r(s)$ using the same recursive formula. Indeed, working in ${\left\{ 1,\dots,s \right \} }$ modulo $s$, the function $\rho_\sigma^r$ can be recovered recursively, using the above rule, from $\rho^r_\sigma(k)$, for any $k\in {\left\{ 1,\dots,s \right \} }$. The following lemma will be used in the proofs of some of the results below.
\[lemma: technical lemma\] Let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ and $r\in
{\left\{ 1,\dots,q-2 \right \} }$. Consider $i\in A$ and $\sigma'=A'\sqcup B'$ where $A'=A\setminus {\left\{ i \right \} }$ and $B'=B\cup {\left\{ i \right \} }$. Let $\rho\colon {\left\{ 1,\dots,s \right \} }{\rightarrow}{\left\{ r,\hat{r} \right \} }$ be given by $\rho(j)=\rho_\sigma^r(j)$ for every $j\not=i$ and $\rho(i)=\widehat{\rho_\sigma^r(i)}$. Then $\rho=\rho_{\sigma'}^r$, if $i>1$ or $\rho=\rho_{\sigma'}^{\hat{r}}$, if $i=1$.
We will look first at the case $i=1$. In this case, $\rho(1)=\widehat{\rho_\sigma^r(1)}=\hat{r}=\rho_{\sigma'}^{\hat{r}}(1)$. If $2 \in A$, then $\rho(2)=\rho_{\sigma}^r(2)=\hat{r}$, according to the definition of the function $\rho$, to Definition \[definition: recursive function\] and to the fact that $1 \in A$. But if $2 \in A$, then $2 \in A'$, and $\rho_{\sigma'}^{\hat{r}}(2)=\hat{r}$ since $1 \in B'$. If $2 \in B$, then $\rho(2)=\rho_{\sigma}^r(2)=r$; but if $2 \in B$, then $2 \in B'$, and $\rho_{\sigma'}^{\hat{r}}(2)=r$. In any case, $\rho(2)=\rho_{\sigma'}^{\hat{r}}(2)$. Let $j \in {\left\{ 3,\dots,s \right \} }$. By definition, $\rho(j)=\rho_{\sigma}^r(j)$; and $\rho_{\sigma}^r(j)$ is determined by $\sigma$, by $\rho_{\sigma}^r(2)$ and by Eq. (\[eq: recursive relation\]). Since $\rho_{\sigma'}^{\hat{r}}(j)$ is determined by $\sigma'$, by $\rho_{\sigma'}^{\hat{r}}(2)$ and by Eq. (\[eq: recursive relation\]), since $\rho_{\sigma}^r(2)=\rho_{\sigma'}^{\hat{r}}(2)$, and since the partitions $\sigma$ and $\sigma'$ agree in ${\left\{ 2,\dots,s \right \} }$, we conclude that $\rho(j)=\rho_{\sigma}^r(j)=\rho_{\sigma'}^{\hat{r}}(j)$. Therefore, $\rho=\rho_{\sigma'}^{\hat{r}}$. For the case $i>1$, we use a similar argument to show that $\rho=\rho_{\sigma'}^{r}$.
Given any $\sigma = A \sqcup B$, a partition of ${\left\{ 1,\dots,s \right \} }$, if, without loss in generality, we choose $1 \in A$, it is clear that given any $r\in{\left\{ 1,\dots,q-2 \right \} }$, there exist unique $a$ and $b$ in ${\mathbb{N}}^s$ such that ${\operatorname{supp}}(a)=A$, ${\operatorname{supp}}(b)=B$, $a_i=\rho_\sigma^r(i)$ if $i\in {\operatorname{supp}}(a)$ and $b_j=\rho_\sigma^r(j)$ if $j\in {\operatorname{supp}}(b)$.
Let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ with $1\in A$ and let $r\in{\left\{ 1,\dots,q-2 \right \} }$. We denote by $f_\sigma^r$ the unique binomial $t^a-t^b$ such that ${\operatorname{supp}}(a)=A$, ${\operatorname{supp}}(b)=B$, $a_i=\rho_\sigma^r(i)$ if $i\in {\operatorname{supp}}(a)$ and $b_j=\rho_\sigma^r(j)$ if $j\in {\operatorname{supp}}(b)$.
(150,140)(-70,-65)
(-60,0)[$\bullet$]{} (-71,0)[$v_1$]{} (-56.5,2)[(2,-5)[17]{}]{} (-45,-20)[$6$]{} (-60,-20)[$e_1$]{}
(-42.4,-42.4)[$\bullet$]{} (-50,-48)[$v_2$]{} (-41.4,-42.4)(2.5,-1)[18]{}[$\cdot$]{} (-19,-45)[$6$]{} (-22,-60)[$e_2$]{}
(0,-60)[$\bullet$]{} (0,-68)[$v_3$]{} (4,-56.5)[(5,2)[40]{}]{} (19,-45)[$6$]{} (19,-60)[$e_3$]{}
(42.4,-42.4)[$\bullet$]{} (48.4,-46.4)[$v_4$]{} (45,-41.4)(1,2.5)[18]{}[$\cdot$]{} (45,-20)[$6$]{} (60,-20)[$e_4$]{}
(60,0)[$\bullet$]{} (67,0)[$v_5$]{} (62,4)[(-2,5)[17]{}]{} (45,18)[$6$]{} (60,18)[$e_5$]{}
(42.4,42.4)[$\bullet$]{} (48.4,46.4)[$v_6$]{} (43.4,46.4)[(-5,2)[40]{}]{} (19,45)[$1$]{} (19,60)[$e_6$]{}
(0,60)[$\bullet$]{} (0,68)[$v_7$]{} (2,60.5)(-2.5,-1)[18]{}[$\cdot$]{} (-19,45)[$1$]{} (-22,60)[$e_7$]{}
(-42.4,42.4)[$\bullet$]{} (-52,49)[$v_8$]{} (-41.4,43.4)(-1,-2.5)[19]{}[$\cdot$]{} (-45,18)[$6$]{} (-63,18)[$e_8$]{}
The combinatorial data that give rise to a binomial $f_\sigma^r=t^a-t^b$ is clarified by representing it in the graph ${\mathcal{G}}$, by putting a label $r$ or $\widehat{r}$ to each edge. Figure \[fig: labeling of the graph\] illustrates the map $\rho_\sigma^6$ when $q=8$, $r=6$, $s=8$ and $\sigma={\left\{ 1,3,5,6 \right \} }\sqcup{\left\{ 2,4,7,8 \right \} }$. The labels of the edges correspond to the exponents of the variables in the corresponding binomial of $I(X)$. Thus, .
\[lemma: transferring an element from one part to the other\] Let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ and let $r\in {\left\{ 1,\dots,q-2 \right \} }$. Suppose that $1\in A$ and that there exists $i\in A$ such that $i>2$ and $i-1\not \in A$. Let $\sigma'$ be the partition given by $A'\sqcup B'$ where $A'=(A\setminus{\left\{ i \right \} })\cup {\left\{ i-1 \right \} }$ and $B'=(B\setminus{\left\{ i-1 \right \} })\cup {\left\{ i \right \} }$. Then $f_\sigma^r\in I(X)$ if and only if $f_{\sigma'}^r \in I(X)$.
Let $f_\sigma^r = t^a - t^b$. Using the assumption, we can write $t^a=t_i^{c}t^{a'}$ and $t^b=t_{i-1}^{c}t^{b'}$, where $c=a_i=b_{i-1}$ and $a',b'\in{\mathbb{N}}^s$. Then: $$\renewcommand{\arraystretch}{1.3}
\begin{array}{rcl}
(t_{i-1}t_i)^{\hat{c}}f^r_\sigma & = &t_{i-1}^{\hat{c}}t_i^{q-1}t^{a'} -t_{i-1}^{q-1}t_i^{\hat{c}}t^{b'}\\
& =& t_{i-1}^{\hat{c}}t_i^{q-1}t^{a'}-t_{i-1}^{\hat{c}}t_{i-1}^{q-1}t^{a'}+t_{i-1}^{\hat{c}}t_{i-1}^{q-1}t^{a'}
-t_{i-1}^{q-1}t_i^{\hat{c}}t^{b'} \\
& =& t_{i-1}^{\hat{c}}t^{a'}(t_i^{q-1}-t_{i-1}^{q-1})+(t_{i-1}^{\hat{c}}t^{a'}
-t_i^{\hat{c}}t^{b'})t_{i-1}^{q-1}.
\end{array}$$ Since $t_j$ is never zero on $X$ we get: $$f^r_\sigma \in I(X) \Leftrightarrow(t_{i-1}t_i)^{\hat{c}}f^r_\sigma
\in I(X) \Leftrightarrow
(t_{i-1}^{\hat{c}}t^{a'} -t_i^{\hat{c}}t^{b'})t_{i-1}^{q-1} \in
I(X)\Leftrightarrow t_{i-1}^{\hat{c}}t^{a'} -t_i^{\hat{c}}t^{b'}\in I(X).$$ Now let $a^\sharp,b^\sharp \in {\mathbb{N}}^s$ be such that $t^{a^\sharp}=t_{i-1}^{\hat{c}}t^{a'}$ and $t^{b^\sharp}=t_i^{\hat{c}}t^{b'}$. Then, $\sigma'={\operatorname{supp}}(a^\sharp)\sqcup {\operatorname{supp}}(b^\sharp)$ is the partition of ${\left\{ 1,\dots,s \right \} }$ obtained from interchanging $i-1$ and $i$ in $A\sqcup B$. Applying Lemma \[lemma: technical lemma\] twice, we deduce that $f^r_{\sigma'}=t_{i-1}^{\hat{c}}t^{a'} -t_i^{\hat{c}}t^{b'}$.
\[lemma: condition on homogeneity\] Let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ with $1\in A$ and let $r\in{\left\{ 1,\dots,q-2 \right \} }$. If $f^r_\sigma\in I(X)$ then ${\left | A \right | }={\left | B \right | }$.
Let $\ell={\left | A \right | }$. sufficiently many times Lemma \[lemma: transferring an element from one part to the other\], we may assume that $\sigma$ is the partition ${\left\{ 1,\dots,\ell \right \} }\sqcup
{\left\{ \ell+1,\dots,s \right \} }$. $$f_\sigma^r=t_1^r t_2^{\hat{r}}\cdots t_\ell^{r'}-t_{\ell+1}^{r'}\cdots t_{s-1}^{\hat{r}}t_s^r,$$ where $r'\in {\left\{ r,\hat{r} \right \} }$. Now $\deg(t^{{r}}_{j}\cdots )$, for a monomial consisting of a product of variables with consecutive exponents alternating in ${\left\{ r,\hat{r} \right \} }$, is a strictly increasing function with respect to the number of variables involved. Since $f^r_\sigma$ is homogeneous (by Lemma \[h-lemma\]) we deduce that ${\left | A \right | }=\ell=s-\ell={\left | B \right | }$.
We come to one of the main results of this section, a combinatorial description of a generating set for a vanishing ideals over an even cycle.
\[theorem: the generators of I(X)\] Let $I(X)$ be the vanishing ideal of the algebraic toric set $X$ associated to an even cycle ${\mathcal{G}}={\mathcal{C}}_{2k}$. Then, $I(X)$ is generated by the binomials $t_i^{q-1}-t_j^{q-1}$, $1\leq i,j\leq s=2k$, and the binomials $f_\sigma^r$ obtained from all $r\in {\left\{ 1,\dots,q-2 \right \} }$ and all partitions $\sigma=A\sqcup B$ of ${\left\{ 1,\dots,s \right \} }$ with ${\left | A \right | }={\left | B \right | }$.
By Theorem \[lemma: first reduction\] and Corollary \[corollary: supports add up to all\], we know that $I(X)$ is generated by the binomials of the form $t_i^{q-1}-t_j^{q-1}$, $1\leq i,j\leq s=2k$, and the homogeneous binomials $f=t^a-t^b$ with $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$ and $b=(b_1,\dots,b_s)\in {\mathbb{N}}^s$ such that ${\operatorname{supp}}(a)\sqcup {\operatorname{supp}}(b)={\left\{ 1,\dots,s \right \} }$ and $0\leq a_i,b_j \leq q-2$. Let $f$ be a binomial of the latter type. We may assume that $1\in{\rm supp}(a)$, for we can always replace $f$ by $-f$ in a generating set of $I(X)$. Set $\sigma = {\operatorname{supp}}(a)\sqcup {\operatorname{supp}}(b)$ and let $r=a_1$. Let us show that $f=f_\sigma^r$, i.e., let us show that $a_{i}=\rho_\sigma^r(i)$, for every $i\in {\operatorname{supp}}(a)\setminus {\left\{ 1 \right \} }$ and $b_j=\rho_\sigma^r(j)$ for every $j\in{\operatorname{supp}}(b)$. Let $i\in {\operatorname{supp}}(a)\setminus {\left\{ 1 \right \} }$ and let $u\in K^*$ be a generator of the multiplicative group of $K$. Consider ${\mathbf{x}}\in (K^*)^n$ given by setting $x_i=u$ and $x_j=1$ for all $j\not = i$. Then, $f({\mathbf{x}}^{\nu_1},\ldots,{\mathbf{x}}^{\nu_s})=0$ implies that $u^{a_{i-1}}u^{a_{i}}=1$, if $i-1\in {\operatorname{supp}}(a)$ or $u^{a_i}=u^{b_{i-1}}$ if $i-1\in {\operatorname{supp}}(b)$. We get, in the first case, $a_i=q-1-a_{i-1}=\rho_\sigma^r(i)$, and, in the second case, $a_i=b_{i-1}=\rho_\sigma^r(i)$. Similarly, if $j\in {\operatorname{supp}}(b)$, then $b_j=\rho_\sigma^r(j)$. Since $f^r_\sigma \in I(X)$, by Lemma \[lemma: condition on homogeneity\], ${\left | A \right | }={\left | B \right | }$.
To complete the proof let $\sigma=A\sqcup B$ be a partition of ${\left\{ 1,\dots,s \right \} }$ with ${\left | A \right | }={\left | B \right | }$, and let us show that $f^r_\sigma \in I(X)$. By Lemma \[lemma: transferring an element from one part to the other\], we may assume that $\sigma$ is the partition ${\left\{ 1,\dots,k \right \} }\sqcup
{\left\{ k+1,\dots,s \right \} }$ and $f_\sigma^r=t_1^r t_2^{\hat{r}}\cdots t_k^{r'}-t_{k+1}^{r'}\cdots t_{s-1}^{\hat{r}}t_s^r$, where $r'\in {\left\{ r,\hat{r} \right \} }$. Now, let ${\mathbf{x}}\in (K^*)^n$. Then $f_\sigma^r({\mathbf{x}}^{\nu_1},\ldots,{\mathbf{x}}^{\nu_s})=x_1^rx_{k+1}^{r'}-x_{k+1}^{r'}
x_1^r = 0$, [*[i.e.]{}*]{}, $f_\sigma^r\in I(X)$.
Te following conjecture has been verified in a number of examples using [*[Macaulay]{}*]{}$2$ [@mac2].
\[remark: on the degree of generators for even cycles\] Let $X$ be the algebraic toric set associated to an even cycle ${\mathcal{G}}={\mathcal{C}}_{2k}$ and let $\lambda$ be the partition ${\left\{ 1,3,\dots,2k-1 \right \} }\sqcup {\left\{ 2,4,\dots,2k \right \} }$. If $k\geq 2$, then the set of binomials $$\renewcommand{\arraystretch}{1.3}
\begin{array}{c}
\mathcal{B}= \{\{f^r_\sigma\colon \sigma=A\sqcup B\mbox{ is a partition
of } \{1,\ldots,s\}\mbox{ with }|A|=|B|, 1\in A\mbox{ and }1\leq r\leq q-2 \}\\
\cup\{t_i^{q-1}-t_s^{q-1}\colon\, 1\leq i \leq s-1\}\}\setminus\{f^r_\lambda\colon\,2\leq r \leq q-2\}
\end{array}$$ is a minimal set of generators and a Gröbner basis of $I(X)$ with respect to the reverse lexicographic order.
\[rmk: About the generating set of I(X)\] By Theorem \[theorem: the generators of I(X)\] and since, for each $2\leq r \leq q-2$, there exists $g_r\in S$ such that $f^r_\lambda = g_rf^1_{\lambda}$, we get that $\mathcal{B}$ is a generating set for $I(X)$.
\[remark: on the degrees of the generators of I(X)\] Let ${\mathcal{G}}={\mathcal{C}}_{2k}$ be an even cycle.
- If $f=t^a-t^b$ is an element of $\mathcal{B}$, then $\deg(f)$ is at most $(q-2)(k-1)+1$.
- Any subset of $\mathcal{B}$ that is also generating set of $I(X)$ contains an element of the form $f^{q-2}_\sigma$, with $\deg(f_\sigma^{q-2})=(q-2)(k-1)+1$.
We set $M=(q-2)(k-1)+1$. (a) If $f=t_i^{q-1}-t_j^{q-1}$ for some $i,j$, then $\deg(f)\leq M$ because $k\geq 2$. Assume that $f$ is not of this form. Then, $f=f_\sigma^r$ for some $1\leq
r\leq q-2$, where $\sigma$ is the partition $\sigma=A\sqcup B$ and $A$, $B$ are the supports of $a$, $b$ respectively, ${\left | A \right | }={\left | B \right | }=k$ and $1\in A$. Let $i$ be the cardinality of the set $\{j\in A\colon\,f_\sigma^r(j)=r\}$. Then, $\deg(f)=ir+(k-i)\widehat{r}$. If $r=\widehat{r}$, then $r=(q-1)/2$ and $\deg(f)=k(q-1)/2\leq M$. We may now assume $r\neq\widehat{r}$. If $i=k$, then $$f^r_\sigma = f^1_\lambda = t_1t_3\cdots t_{2k-1}-t_2t_4\cdots t_{2k}.$$ Hence, $\deg(f_\sigma^r)=k\leq M$. To complete the proof we may now assume that $1\leq i\leq k-1$. In this case, we have $$\begin{aligned}
\deg(f_\sigma^r)&=&ir+(k-i)\widehat{r}=i(r-\widehat{r})+k\widehat{r}
\\
&\leq&(k-1)(r-\widehat{r})+k\widehat{r}=(k-1)r+\widehat{r}=r(k-2)+(q-1)\\
&\leq&(q-2)(k-2)+(q-1)=(k-1)(q-2)+1.\end{aligned}$$ Thus, $\deg(f_\sigma^r)\leq M$, as required. To prove (b) consider $\mathcal{B}'\subset \mathcal{B}$ be a generating set of $I(X)$. Let , then $$f_\sigma^{q-2}=t_1^{q-2}t_3^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}-t_2^{q-2}t_4^{q-2}\cdots
t_{2(k-1)}^{q-2}t_{2k-1}$$ is in $\mathcal{B}$ and has degree $M$. We will show that $f_\sigma^{q-2} \in \; \mathcal{B}'$. Since $\mathcal{B}'$ is a generating set of $I(X)$, $f_\sigma^{q-2}$ is a linear combination, with coefficients in $S$, of binomials in $\mathcal{B}'$. These are binomials of the form $f_\rho^m$, $1\leq m\leq
q-2$, $\deg(f_\rho^{m}) \leq M$, and of the form $t_i^{q-1}-t_{2k}^{q-1}$ for some $1\leq i\leq 2k-1$. It is seen that there is a binomial in $\mathcal{B}'$, $f_\rho^{m}=t^{a'}-t^{b'}$, such that $$t_1^{q-2}t_3^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}=
t^ct^{a'}$$ for some monomial $t^c$. Since ${\operatorname{supp}}(a')\subset {\left\{ 1,3,\dots,2k-3,2k \right \} }$ and $t^{a'}-t^{b'}$ cannot be of the form $t_1^{q-1}-t_{2k}^{q-1}$, because of its degree, we deduce that $\sigma=\rho$. From the equality $$t_1^{q-2}t_3^{q-2}\cdots t_{2k-3}^{q-2}t_{2k}=t^ct^{a'}=
t^c(t_1^{m}t_3^{m}\cdots t_{2k-3}^{m}t_{2k}^{\widehat{m}})$$ we conclude that $\widehat{m}=1$, that is, $m=q-2$. Thus, $f_\sigma^{q-2}=f_\rho^m$.
Consider the general case when ${\mathcal{G}}$ is any graph. Suppose that ${\mathcal{G}}$ contains a subgraph , isomorphic to an even order cycle. Assume without loss of generality that $t_1,\dots,t_{2k}$ are the variables of $S$ corresponding to the edges of $\mathcal{H}$. Then, given $r\in {\left\{ 1,\dots,q-2 \right \} }$ and a partition $\sigma=A\sqcup B$ of ${\left\{ 1,\dots, 2k \right \} }$ with ${\left | A \right | }={\left | B \right | }=k$ and $1 \in A$, the homogeneous binomial $f^r_\sigma\in K[t_1,\dots,t_{2k}]\subset S$ clearly vanishes on the algebraic toric set associated to ${\mathcal{G}}$. One could conjecture that together with the binomials $t_i^{q-1}-t_j^{q-1}$, for $1\leq i,j\leq s$, the binomials obtained in this way, going through all the even cycles of ${\mathcal{G}}$, would form a generating set of $I(X)$. This is not true, even for bipartite graphs, as is shown by Example \[example: two offending graphs\]. This conjecture is true if we restrict to bipartite graphs the cycles of which are vertex disjoint; as we show in Theorem \[theorem: I(X) for almost general graph\].
Suppose ${\mathcal{G}}$ is a bipartite graph the cycles of which have disjoint vertex sets. Let ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$ be the subgraphs of ${\mathcal{G}}$ isomorphic to some even order cycle, i.e., such that ${\mathcal{H}}_i\cong {\mathcal{C}}_{2k_i}$. Let $t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}\in S$ be the variables associated to the edges, $e^i_1,\dots,e^i_{2k_i}$ of ${\mathcal{H}}_i$. Accordingly, set $$S_i=K\bigl[t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}\bigr]\subset S.$$ Finally, denote by $I_i(X)$ the intersection $I(X)\cap S_i$. Then, $I_i(X)\subset S_i$ is equal to $I(X_i)$, the vanishing ideal of the algebraic toric set $X_i$ associated to ${\mathcal{H}}_i$.
\[theorem: I(X) for almost general graph\] Let ${\mathcal{G}}$ be a connected bipartite graph, whose [(]{}even[)]{} cycles ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$ have disjoint vertex sets. Let $X$ be the algebraic toric set associated to ${\mathcal{G}}$. Then $I(X)$ is generated by the union of the set $\{t_i^{q-1}-t_j^{q-1}:1\leq i,j\leq s\}$ with the set $I_1(X)\cup\cdots\cup I_m(X)$.
By Theorem \[lemma: first reduction\], it suffices to show that if $f=t^a-t^b\in I(X)$, with $a=(a_1,\dots,a_s)\in {\mathbb{N}}^s$, $b=(b_1,\dots,b_s)\in {\mathbb{N}}^s$, such that ${\operatorname{supp}}(a)\cap {\operatorname{supp}}(b)=\emptyset$ and $1\leq a_i,b_j\leq
q-2$, then $f$ belongs to the ideal generated by $$\mathcal{J}=\{t_i^{q-1}-t_j^{q-1}:1\leq i,j\leq s\} \cup I_1(X)\cup\cdots\cup I_m(X).$$ Recall that $f$ is homogeneous by Lemma \[h-lemma\]. By Proposition \[proposition: nonoccuring variables\], we know that ${\operatorname{supp}}(a)\cup{\operatorname{supp}}(b)$ is contained in the union of the sets of indices of the variables corresponding to edges of the cycles of ${\mathcal{G}}$. In other words, if $e_i$ is an edge not in any edge set of ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$ then $i\not \in {\operatorname{supp}}(a)\cup{\operatorname{supp}}(b)$. As above, denote by $t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}$ the variables associated to ${\mathcal{H}}_i$. We proceed by induction on $$\mu_f={\left\{ i\in {\left\{ 1,\dots,m \right \} } : ({\operatorname{supp}}(a)\cup {\operatorname{supp}}(b))\cap
{\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \} }
\not = \emptyset \right \} }.$$
Let $i\in{\left\{ 1,\dots,m \right \} }$ be such that $({\operatorname{supp}}(a)\cup{\operatorname{supp}}(b))\cap{\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \} }\not = \emptyset$. Consider such that ${\operatorname{supp}}(a^\sharp)\cup {\operatorname{supp}}(b^\sharp)\subset {\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \} }$, $({\operatorname{supp}}(a^\flat)\cup {\operatorname{supp}}(b^\flat))\cap
{\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \} }=\emptyset$, $$t^a=t^{a^\sharp} t^{a^\flat}\quad \text{and}\quad t^b=t^{b^\sharp} t^{b^\flat}.$$ By Corollary \[corollary: supports add up to all\], . Since we are assuming ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$ have disjoint vertex sets, setting $t_\ell=1$ for all $\ell\not\in {\left\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \right \} }$ is equivalent to setting in , $x_\ell=1$ for all $\ell\not \in V_{{\mathcal{H}}_i}$. Hence, making these substitutions and running the argument of the proof of Theorem \[theorem: I(X) for almost general graph\], we see that $t^{a^\sharp}-t^{b^\sharp}=
f_\sigma^r$, where , (assuming that $\epsilon^i_1\in
{\operatorname{supp}}(a^\sharp)$), and where $\sigma$ is the partition
Suppose that $\mu_f=1$. Then $a^\flat=b^\flat=0\in{\mathbb{N}}^s$, $f^r_\sigma$ is homogeneous and we are done.
Suppose that every binomial $g=t^a-t^b\in I(X)$ with $\mu_g\leq m'< m$ is in the ideal generated by $\mathcal{J}$. Let $f=t^a-t^b\in I(X)$ be a binomial with $\mu_f=m'+1$. Let $i\in{\left\{ 1,\dots,m \right \} }$ be such that $({\operatorname{supp}}(a)\cup{\operatorname{supp}}(b))\cap{\{ \epsilon^i_1,\dots,\epsilon^i_{2k_i} \} }\not = \emptyset$. Consider, as above, such that $t^a=t^{a^\sharp} t^{a^\flat}$ and $t^b=t^{b^\sharp} t^{b^\flat}$. Repeating the previous argument we deduce that $t^{a^\sharp}-t^{b^\sharp}=f^r_\sigma$ where, $r=(a^\sharp)_{\epsilon^i_1}$ and $\sigma={\operatorname{supp}}(a^\sharp)\sqcup {\operatorname{supp}}(b^\sharp)$. However, notice that in this case $f^r_\sigma$ is not necessarily homogeneous. Assume that . Let $\delta\in {\mathbb{N}}^s$ be such that $\epsilon^i_1\not \in {\operatorname{supp}}(\delta)\subset {\operatorname{supp}}(a^\sharp)$, $\delta_\ell=a^\sharp_\ell$ for all $\ell\in {\operatorname{supp}}(\delta)$ and ${\left | {\operatorname{supp}}(a^\sharp -\delta) \right | }=k_i$ (where $2k_i$ is the order of $\mathcal{H}_i$). Set $h={\left | {\operatorname{supp}}(\delta) \right | }$, $a'=a^\sharp - \delta$ and let $b'\in {\mathbb{N}}^s$ be obtained by applying $h$ times Lemma \[lemma: transferring an element from one part to the other\] to $\sigma={\operatorname{supp}}(a^\sharp)\sqcup {\operatorname{supp}}(b^\sharp)$. Then $b'=b^\sharp + \hat\delta$, where $\hat\delta$ has the same support as $\delta$ and $(\hat\delta)_\ell=q-1-\delta_\ell$, for every $\ell\in {\operatorname{supp}}(\hat\delta)$. Set $\sigma' = {\operatorname{supp}}(a')\sqcup {\operatorname{supp}}(b')$. Then $f_{\sigma'}^r=t^{a'}-t^{b'}$ is homogeneous and belongs to $I_i(X)$. Moreover, $$\begin{array}{c}
f = t^a-t^b = t^{a'}t^\delta t^{a^\flat} - t^{b^\sharp}t^{b^{\flat}}=
t^{a'}t^\delta t^{a^\flat}-t^{b'}t^\delta t^{a^\flat}+t^{b'}t^\delta
t^{a^\flat} - t^{b^\sharp}t^{b^{\flat}}\\
=f^r_{\sigma'} t^\delta t^{a^\flat} +
t^{b^\sharp}(t^{\widehat{\delta}} t^\delta t^{a^\flat} -t^{b^\flat}).
\end{array}$$ Now $(\hat{\delta})_\ell+\delta_\ell=q-1$, for all $\ell\in {\operatorname{supp}}(\delta)$ and since $f$ is homogeneous, $h={| {\operatorname{supp}}(\delta) | }>{| {\operatorname{supp}}(b^\flat) | }$. Choose $\ell_1,\dots,\ell_h\in {\operatorname{supp}}(b^\flat)$, $h$ distinct indices. Let $\gamma \in {\mathbb{N}}^s$ to be such that ${\operatorname{supp}}(\gamma)={\left\{ \ell_1,\dots,\ell_h \right \} }$ and $(\gamma)_{\ell_j}=q-1$, for $j=1,\dots,h$. Then $t^{\delta}t^{\widehat{\delta}}-t^\gamma$ is in the ideal of $S$ generated by $\mathcal{J}$, since it is in the ideal of the torus. We have $$f= f^r_{\sigma'} t^\delta t^{a^\flat} +
t^{b^\sharp}(t^{\delta}t^{\widehat{\delta}} t^{a^\flat} -t^{b^\flat})
=f^r_{\sigma'} t^\delta t^{a^\flat} +
t^{b^\sharp}t^{a^{\flat}}(t^{\delta}t^{\widehat{\delta}}-t^\gamma)
+t^{b^\sharp}( t^\gamma t^{a^\flat} -t^{b^\flat}).$$ Let $\gamma^\sharp\in {\mathbb{N}}^s$ be such that ${\operatorname{supp}}(\gamma^\sharp)={\left\{ \ell_1,\dots,\ell_h \right \} }$ and $(\gamma^\sharp)_{\ell_j}=(b^\flat)_{\ell_j}$, for $j=1,\dots,h$ and set $\gamma^\flat = \gamma - \gamma^\sharp$ and $b^\natural = b^\flat - \gamma^\sharp$. Then, $$f= f^r_{\sigma'} t^\delta t^{a^\flat} +
t^{b^\sharp}t^{a^{\flat}}(t^{\delta^*}-t^\gamma)
+t^{b^\sharp}t^{\gamma^\sharp}( t^{\gamma^\flat} t^{a^\flat} -t^{b^\natural}),$$ where $g= t^{\gamma^\flat} t^{a^\flat} -t^{b^\natural}$ is a homogeneous binomial with $\mu_g\leq m'$. Hence, by induction, $g$, and therefore $f$, are in the ideal generated by $\mathcal{J}$.
In Example \[example: two offending graphs\], we show that Theorem \[theorem: I(X) for almost general graph\] does not hold for general connected bipartite graphs.
(200,130)(-90,-140)
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(100,-140)[$\bullet$]{} (100,-150)[$4$]{} (102,-137)[(-1,1)[60]{}]{} (74,-107)[$\scriptstyle 1$]{} (62.5,-115)[$e_4$]{} (102,-138)[(1,1)[60]{}]{} (127,-107)[$\scriptstyle 1$]{} (135,-115)[$e_3$]{}
\[example: two offending graphs\] Let ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be the two graphs in Figure \[fig: two offending graphs\] (from left to right) and assume that $q=5$. Notice that we are identifying the two vertices, labeled by $1$, in the representation of ${\mathcal{G}}_1$. Thus, ${\mathcal{G}}_1$ is a bipartite graph with six vertices and eight edges. Denote by $X_1$ and $X_2$, respectively, the corresponding algebraic toric sets. Then, using *Macaulay*$2$ [@mac2], we found that the binomial $t_1t_4t_6t_7-t_2t_3t_5t_8$ is in a minimal generating set of $I(X_1)$. In this case, the argument of the proof of Theorem \[theorem: I(X) for almost general graph\] does not work, to the extent that if we set $t_1,t_2,t_3,t_4$ equal to $1$, the resulting binomial, $t_6t_7-t_5t_8$, albeit homogeneous, is not of the type $f^r_\sigma$ for any partition $\sigma$ of ${\left\{ 5,6,7,8 \right \} }$. The same can be said for the binomial resulting from substituting to $1$ the variables $t_5,t_6,t_7,t_8$. As to the vanishing ideal of $X_2$, we found that there exists a minimal generating set containing $t_1t_2t_5^2-t_3t_4t_5^2$, which, when restricted to any of the $3$ cycles in ${\mathcal{G}}_2$ is not of the type $f^r_\sigma$ for any partition of the corresponding index set.
The regularity of $R/I(X)$ {#sec: regularity}
==========================
In this section we address the question of computing the regularity of $S/I(X)$ for an algebraic toric set $X$ parameterized by a bipartite graph. Theorem \[theorem: bound on regularity for a general graph\] gives an upper bound for the regularity of $S/I(X)$ for a general bipartite graph. If $X$ is the algebraic toric set parameterized by an even cycle of length $2k$, by Proposition \[castelnuovo-vs-bigdegree\] and Corollary \[remark: on the degrees of the generators of I(X)\], we get $${\rm bigdeg}\, I(X)-1=(q-2)(k-1)\leq {\rm reg}\, S/I(X).$$ This inequality is already known in the literature, see [@codes; @over; @cycles Corollary 3.1] and [@deg-and-reg Corollary 2.19]. We will show that the regularity of $S/I(X)$ is in fact equal to $(q-2)(k-1)$, and generalize this result by giving a formula for the regularity of any connected bipartite graph whose cycles have disjoint vertex sets. In the proof of Theorem \[theorem: regularity of even cycles\], we show the inequality above as an easy consequence of the description of the generators of the ideal $I(X)$.
\[lemma: symmetries on the generating set\] Let $1\leq i\leq s-2$. Consider the $K$-automorphism $\sigma_{i}\colon S {\rightarrow}S$ defined by exchanging $t_i$ with $t_{i+2}$ and leaving all other variables fixed. Then, $\sigma_{i}$ permutes the elements of the set of all $f^r_\sigma\in I(X)$, for $r\in {\left\{ 1,\dots,q-2 \right \} }$ and $\sigma=A\sqcup B$ a partition of ${\left\{ 1,\dots,s \right \} }$ with ${\left | A \right | }={\left | B \right | }$.
Let $f^r_\sigma$ be a binomial associated to $r\in{\left\{ 1,\dots,q-2 \right \} }$ and $\sigma=A\sqcup B$ a partition of ${\left\{ 1,\dots,s \right \} }$. Thus, $f^r_\sigma = t^a-t^b$ where $A={\operatorname{supp}}(a)$, $B={\operatorname{supp}}(b)$, $a_\ell=\rho_\sigma^r(\ell)$, for all $\ell\in {\operatorname{supp}}(a)$ and $b_\ell=\rho^r_\sigma(\ell)$, for all $\ell\in {\operatorname{supp}}(b)$. As $\rho_\sigma^r(\ell)=\rho_\sigma^r(\ell+2)$ if and only if $\ell$ and $\ell+2$ are in the same part of the partition, if $i$ and $i+2$ are in the same part of the partition then $\sigma_i(f^r_\sigma)=f^r_\sigma$. Suppose that $i$ and $i+2$ are in different parts of the partition and therefore that $\rho_\sigma^r(i+2)=\widehat{\rho_\sigma^r(i)}$. Without loss in generality we may write $f^r_\sigma=t_i^{a_i}t^{a'}-t_{i+1}^{\widehat{a_i}}t^{b'}$, where ${\operatorname{supp}}(a')={\operatorname{supp}}(a)\cup{\left\{ i \right \} }$ and ${\operatorname{supp}}(b')={\operatorname{supp}}(b)\cup{\left\{ i+2 \right \} }$. In this situation, we apply Lemma \[lemma: transferring an element from one part to the other\] twice, transferring $i$ to the part it does not belong to, and proceeding similarly with $i+2$. Let $\sigma'$ be the partition of ${\left\{ 1,\dots,s \right \} }$ obtained in this way and consider the resulting binomial $f^r_{\sigma'}$. By Lemma \[lemma: transferring an element from one part to the other\] we see that $f^r_{\sigma'}=t_{i+2}^{a_i}t^{a'}-t_{i}^{\widehat{a_i}}t^{b'}=\sigma_i(f^r_\sigma)$.
\[theorem: regularity of even cycles\] Let $X$ be the algebraic toric set associated to an even order cycle ${\mathcal{G}}={\mathcal{C}}_{2k}$. Then ${\operatorname{reg}}S/I(X) =(q-2)(k-1)$.
Recall that $k\geq 2$. Denote by $R$ the graded ring $S/I(X)$. Consider $t_1\in S$. Since $t_1$ is regular on $R$, we have the following exact sequence of graded $S$-modules: $$\label{eq: SES of graded modules}
0\longrightarrow R[-1]\stackrel{t_1}{\longrightarrow}R\longrightarrow
R/(t_1)\longrightarrow 0,$$ where $R[-1]$ is the graded $S$-module obtained by a shift in the graduation, [*[i.e.]{}*]{}, $R[-1]_i=R_{i-1}$. Recall that $H_X(d)$ is, by definition, $\dim_K (S/I(X))_d$, and since $S/I(X)$ is a $1$-dimensional ring, the regularity of $S/I(X)$ is the least integer $l$ for which $H_X(d)$ is equal to some constant (indeed equal to ${\left | X \right | }$) for all $d\geq l$. Now, from (\[eq: SES of graded modules\]) we get $H_X(d)-H_X(d-1)=\dim_K
(R/(t_1))_d$. Hence ${\operatorname{reg}}S/I(X)= {\operatorname{reg}}R/(t_1)-1$. For $d \geq 0$, we define $$h_d:= \dim_K
(R/(t_1))_d = H_X(d)-H_X(d-1) .$$ We start by showing that ${\operatorname{reg}}S/I(X)\leq (q-2)(k-1)$. If we show that $h_d=0$, for $d\geq (q-2)(k-1)+1$, then $H_X(d-1)=H_X(d)$, for $d-1\geq (q-2)(k-1)$, and our result follows. Set $S'=K[t_2,\dots,t_s]$. There is a surjection of graded $S'$-modules $$\varphi\colon S'\longrightarrow
S/(I(X),t_1)\cong R/(t_1)$$ defined by $\varphi (f)=f+(I(X),t_1)$, for every $f\in S'$. Set $I'(X)=\mbox{Ker}(\varphi)$, so that $$S'/I'(X) \cong {S}/(I(X),t_1).$$ Then, $I'(X)$ is a monomial ideal generated by the monomials obtained by setting $t_1=0$ in the generators of $I(X)$; in particular it is generated by $t_j^{q-1}$, for $2\leq j\leq s$ and by the monomials $t^b$ in some $f^r_\sigma=t^a-t^b$, for $r\in {\left\{ 1,\dots,q-2 \right \} }$ and $\sigma$ a partition of ${\left\{ 1,\dots,s \right \} }$ into $2$ parts of equal cardinality. To show that $h_d=0$, for $d\geq (q-2)(k-1)+1$, it is enough to show that every monomial $M$ in $S'$ of degree $\geq (q-2)(k-1)+1$ belongs to $I'(X)$. Since $t_j^{q-1}\in I'(X)$ for all $2\leq j\leq s$, we may assume that there is no $j$ for which $t_j^{q-1}$ divides the monomial $M$ in question. Let us write it in the following way: $$M=t_2^{b_1}t_4^{b_2}\cdots t_{2k}^{b_k}\,t_3^{c_1}t_5^{c_2}\cdots t_{2k-1}^{c_{k-1}},$$ with $0\leq b_i,c_j\leq q-2$. We want to show that there exists $f^r_\sigma=t^a-t^b\in I(X)$ such that $t^b$ divides $M$. By Lemma \[lemma: symmetries on the generating set\], if $t^b$ divides $M$ and there exists $r,\sigma$ such that $f^r_\sigma=t^a-t^b$, then, for all $i\in {\left\{ 2,\dots,s-2 \right \} }$, $\sigma_i(t^b)$ divides $\sigma_i(M)$ and there exists $\sigma'$ such that $f^r_{\sigma'}=t^{a'}-\sigma_i(t^b)$. Hence, we may assume that $c_1\leq c_2\leq \cdots \leq c_{k-1}$ and that $b_1\geq b_2 \geq \cdots \geq b_{k}$. There are two cases. If $b_{k}>0$, then $M$ is divisible by $t_2t_4\cdots t_{2k}$, which belongs to $I'(X)$, since for $\sigma={\left\{ 1,3,\dots,2k-1 \right \} }\sqcup {\left\{ 2,4,\dots,2k \right \} }$, we have $f^1_\sigma=t_1t_3\cdots t_{2k-1}-t_2 t_4\cdots t_{2k}$. The second case is for $b_k=0$. In this case, from $$\deg M=\sum_{i=1}^{k-1}(b_i+c_i)\geq (q-2)(k-1)+1$$ we deduce that there exists $j \in \{1, \ldots, k-1\}$ such that $b_j+c_j \geq q-1$. Since $c_{j}\leq q-2$ we get $b_j\geq 1$. Set $r=b_j$. Notice that then $c_j\geq q-1-b_j=q-1-r=\widehat{r}$. Consider the set given by and let $\sigma=A \sqcup B$ be the partition of ${\left\{ 1,\dots,s \right \} }$ that it determines. Then: $$f^r_\sigma=
(t_1 t_3\cdots t_{2j-1})^r (t_{2j+2}\cdots t_{2k-2} t_{2k})^{\widehat{r}} -
(t_2 t_4\cdots t_{2j})^r (t_{2j+1} t_{2j+3}\cdots t_{2k-1})^{\widehat{r}}\in I(X).$$ Accordingly, $(t_2 t_4\cdots t_{2j})^r (t_{2j+1} t_{2j+3}\cdots t_{2k-1})^{\widehat{r}}\in I'(X)$. Since $b_l\geq b_j=r$, for all $1\leq l\leq j$, we deduce that $t_{2l}^r$ divides $M$, for all $1\leq l\leq j$. Since $\widehat{r}\leq c_{j}\leq c_l$, for all $j\leq l \leq k-1$, we deduce that $t_{2l+1}^{\widehat{r}}$ divides $M$, for all $j\leq l \leq k-1$. In conclusion, $(t_2 t_4\cdots t_{2j})^r (t_{2j+1} t_{2j+3}\cdots t_{2k-1})^{\widehat{r}}$ divides $M$ and hence $M\in I'(X)$.
Let us now show that ${\operatorname{reg}}S/I(X)\geq (q-2)(k-1)$. If we show that $h_d\neq0$, i.e., $h_d>0$, for $d= (q-2)(k-1)$, then $H_X(d-1)<H_X(d)$, for $d= (q-2)(k-1)$, and our result follows. It suffices to produce a monomial $M$ of degree $d=(q-2)(k-1)$, such that $M \in (S')_d$ but $M \notin (I'(X))_d$. Consider $$M = (t_2\cdots t_{k})^{q-2} \; \in \;(S')_d.$$ Suppose $M \in I'(X)$. Then, as we have seen above, $$M = \sum_{j=2}^s g_j t_j^{q-1} + \sum_{\sigma,r} h_{\sigma,r} t^b$$ where $g_j, h_{\sigma,r} \in S'$, $f^r_\sigma=t^a-t^b$ and the second summation runs over all partitions $\sigma$ of ${\left\{ 1,\dots,s \right \} }$ into $2$ parts of equal cardinality and $r\in {\left\{ 1,\dots,q-2 \right \} }$. Since $M$ is a monomial and its degree in each one of the variables is $q-2$, we deduce that $M$ must be a monomial of the form $$M= h_{\sigma,r}t^b$$ for $h_{\sigma,r} \in S'$, one partition $\sigma$ of ${\left\{ 1,\dots,s \right \} }$ into $2$ parts of equal cardinality, one $r\in {\left\{ 1,\dots,q-2 \right \} }$ and $f^r_\sigma=t^a-t^b$. But this is not possible because the monomial $M$ has $k-1$ variables, while $h_{\sigma,r}t^b$ has at least $k$ variables. We conclude that $M \notin I'(X)$.
\[theorem: bound on regularity for a general graph\] Let ${\mathcal{G}}$ be a bipartite graph. Let ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$ be subgraphs of ${\mathcal{G}}$ isomorphic to $($even$)$ cycles ${\mathcal{H}}_i\cong {\mathcal{C}}_{2k_i}$ that have disjoint edge sets. Then $$\textstyle {\operatorname{reg}}S/I(X) \leq (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl).$$
For all $1\leq i\leq m$, let $t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}$ be the variables associated to the edges of ${\mathcal{H}}_i$. Without loss of generality, assume that $t_1=t_{\epsilon^1_1}, \;t_2=t_{\epsilon^2_1},\; \ldots,
\;t_m=t_{\epsilon^m_1}$.
Denote by $R$ the quotient $S/I(X)$ and, for $1\leq i\leq m$, let $$R_i = R /(t_1,\dots,t_i).$$ Since $t_1$ is a regular element of $R$, we have the following short exact sequence of graded $S$-modules: $$\label{eq: SES}
0\longrightarrow R[-1] \stackrel{t_1}{\longrightarrow} R \longrightarrow R_1 \longrightarrow 0 .$$ Furthermore, for all $1\leq i \leq m-1$, we have exact sequences of graded $S$-modules: $$\label{eq: amputated SESs}
R_i[-1] \stackrel{t_{i+1}}{\longrightarrow} R_i \longrightarrow R_{i+1} \longrightarrow 0 .$$
For all $1\leq i\leq m$, $t_j^{q-1}=0$ in $R_i$, for all $1\leq j\leq s$.
Since $t_j^{q-1}-t_i^{q-1}\in I(X)$ and $t_i^{q-1}=0$ in $R_i$, we deduce that $t_j^{q-1}=0$ in $R_i$, for all $1\leq j\leq s$.
If there exists a nonnegative integer $\ell$ such that $(R_{i+1})_d=0$, for all $d\geq \ell$, then $(R_i)_d=0$ for all $d\geq \ell+q-2$, where $1\leq i\leq m-1$.
If $(R_{i+1})_d=0$, for $d\geq \ell$ then from (\[eq: amputated SESs\]) we deduce that for all $d\geq \ell$ the maps $(R_i)_{d-1} \stackrel{t_{i+1}}{\longrightarrow} (R_i)_d$ are surjective, *i.e.*, $(R_i)_d = t_{i+1}(R_i)_{d-1}$, for all $d\geq
\ell$. Iterating and using Claim 1, we get: $(R_i)_{d+q-2} = t_{i+1}^{q-1}(R_i)_{d-1}=0$, *i.e.*, $(R_i)_d=0$ for all $d\geq \ell+q-2$.
Let $t^a$ be a monomial in $S$. Suppose that the degree of $t^a$ in the variables associated to ${\mathcal{H}}_i$ is $\geq (q-2)(k_i-1)+1$. Then $t^a=0$ in $R_i$.
We may assume that $t_i$ does not divide $t^a$. Defining $$S_i:= K\big[t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}\big] ,$$ we have $I(X_i) \subset S_i$, where $X_i \subseteq {\mathbb{P}}^{2k_i-1}$ is the set of points parameterized by the edges of the cycle ${\mathcal{H}}_i$. It is straightforward to check that $I(X_i)\subset I(X) \subset S$. Let $t^a=t^bt^c$, where $t^b$ is a monomial in $t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}$. It suffices to show $t^b=0$ in $S_i/(I(X_i)+t_i)$, but since $t^b$ has degree $\geq (q-2)(k_i-1)+1$, we can run the same argument as in the proof of Theorem \[theorem: regularity of even cycles\].
Let $\ell_0=(q-2)\bigr(\sum_{i=1}^m(k_i-1)\bigl) + (q-2)\bigr(s-\sum_{i=1}^m 2k_i\bigl) + 1$. Then $(R_m)_d =0$, $\forall\; d\geq \ell_0$.
Let $t^a$ be a monomial of degree $d\geq \ell_0$. In view of Claim 3, we may assume that the degree of $t^a$ in the variables associated to ${\mathcal{H}}_i$ is $\leq
(q-2)(k_i-1)$. Then, the degree of $t^a$ in the remaining $s-\sum_{i=1}^m 2k_i$ variables is $\geq (q-2)\bigr(s-\sum_{i=0}^m 2k_i\bigl) + 1$ which implies that one of them is raised to a power $\geq q-1$ and therefore, by Claim 1, $t^a=0$ in $R_m$.
We now finish the proof of the theorem. Notice that $\ell_0=(q-2)\bigr( s -\sum_{i=1}^m (k_i+1)\bigr)+1$. Combining Claim 2 with Claim 4 we deduce that $(R_1)_d = 0$, for all $d\geq \ell_0+(m-1)(q-2)$. Now $\ell_0+(m-1)(q-2) = (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl)+1$ and using (\[eq: SES\]) we see that $(R)_{d-1}\stackrel{t_1}{\longrightarrow} (R)_d$ is an isomorphism for all $d\geq (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl)+1$. This means that the Hilbert function of $R$ satisfies: $H_X(d-1)=H_X(d)$, for $d-1 \geq (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl)$. Hence, ${\operatorname{reg}}R\leq (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl)$.
Notice we do not assume that ${\mathcal{G}}$ is connected nor do we assume that any $2$ cycles, ${\mathcal{H}}_1$ and ${\mathcal{H}}_2$, in ${\mathcal{G}}$ have disjoint edge or vertex sets. In fact, we can apply the bound of Theorem \[theorem: bound on regularity for a general graph\] to both graphs in Figure \[fig: two offending graphs\]. For ${\mathcal{G}}_1$, on the left, we should use both cycles of order $4$. We obtain . Using [*Macaulay*]{}$2$ [@mac2], for $q=5$, we checked that this is the actual value of the regularity. For ${\mathcal{G}}_2$, on the right, we may only use one of the cycles. Then, Theorem \[theorem: bound on regularity for a general graph\] yields ${\operatorname{reg}}S/I(X_2) \leq (q-2)(6-2-1)=3(q-2)$, which, for $q=5$, is not sharp, as the value of ${\operatorname{reg}}S/I(X_2)$ is $6$. The inequality of Theorem \[theorem: bound on regularity for a general graph\] is an improvement of the inequality given in [@deg-and-reg Corollary 2.31].
\[corollary: regularity for almost general graphs\] Let ${\mathcal{G}}$ be a connected bipartite graph, the $($even$)$ cycles of which, ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$, with ${\mathcal{H}}_i\cong {\mathcal{C}}_{2k_i}$, have disjoint vertex sets. Then $$\textstyle {\operatorname{reg}}S/I(X)= (q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl).$$
Let $t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}\hspace{-.2cm}\in S$ be the set of variables associated to the edges, $e^i_1,\dots,e^i_{2k_i}$ of the even cycle ${\mathcal{H}}_i$. We set $$S_i=K\bigl[t_{\epsilon^i_1},\dots,t_{\epsilon^i_{2k_i}}\bigr]\subset
S,$$ and denote by $I_i(X)$ the intersection $I(X)\cap S_i$. Then, $I_i(X)\subset S_i$ is the vanishing ideal of the algebraic toric set associated to ${\mathcal{H}}_i$. By Theorem \[theorem: I(X) for almost general graph\], $I(X)$ is generated by the set $$\mathcal{J}=\{t_i^{q-1}-t_j^{q-1}:1\leq i,j\leq s\} \cup I_1(X)\cup\cdots\cup I_m(X).$$ We proceed by induction on the number of edges of ${\mathcal{G}}$. If ${\mathcal{G}}$ is an even cycle, the result follows from Theorem \[theorem: regularity of even cycles\]. We may assume that $e_s$ is an edge of ${\mathcal{G}}$ that does not lie on any cycle of ${\mathcal{G}}$ and that $t_s$ is the variable that corresponds to $e_s$. For simplicity of notation, we identify the edge $e_i$ with the variable $t_i$ for $i=1,\ldots,s$ and refer to $t_i$ as an edge of the graph ${\mathcal{G}}$. Consider the graph ${\mathcal{G}}_1$ whose edge set is $\{e_1, \ldots, e_{s-1}\}$ (the edge set of ${\mathcal{G}}$ minus the edge $e_s$), and whose vertex set is the set of endpoints of the edges $e_1, \ldots, e_{s-1}$. Let $X_1$ be the algebraic toric set parameterized by the edges of ${\mathcal{G}}_1$. Clearly ${\mathcal{G}}_1$ is a bipartite graph whose (even) cycles are again ${\mathcal{H}}_1,\dots,{\mathcal{H}}_m$.
Case (I): The graph ${\mathcal{G}}_1$ is connected. Let $A(X_1)=K[t_1,\ldots,t_{s-1}]/I(X_1)$ be the coordinate ring of $X_1$ and let $F_{X_1}(t)$ be the Hilbert series of $A(X_1)$. The Hilbert series can be uniquely written as $F_{X_1}(t)=g_1(t)/(1-t)$, where $g_1(t)$ is a polynomial of degree equal to the regularity of $A(X_1)$. Because ${\mathcal{G}}_1$ is a connected bipartite graph and has the same cycles as ${\mathcal{G}}$, by Theorem \[theorem: I(X) for almost general graph\], the vanishing ideal $I(X_1)$ is generated by the set $$\mathcal{J}_1=\{t_i^{q-1}-t_j^{q-1}:1\leq i,j\leq s-1\} \cup
I_1(X)\cup\cdots\cup I_m(X)$$ (notice that $I_j(X)=I_j(X_1)$, for $j= 1, \ldots, m$). Hence, there is an exact sequence
$$0\rightarrow A(X_1)[-(q-1)]\stackrel{\scriptstyle \hspace{.1cm}t_1^{q-1}}{\longrightarrow}
A(X_1)\longrightarrow
C=K[t_1,\ldots,t_{s-1}]/(I_1(X),\ldots,I_m(X),t_1^{q-1},\ldots,t_{s-1}^{q-1})
\rightarrow 0.$$
As a consequence, we get that the Hilbert series $F(C,t)$ of $C$ is given by $$F(C,t)=F_{X_1}(t)(1-t^{q-1})=g_1(t)(1+t+\cdots+t^{q-2}),$$ and $\textstyle \deg\, F(C,t)=(q-2)+{\rm reg}\, A(X_1)$. Since ${\mathcal{G}}_1$ is a connected bipartite graph ant its even cycles have disjoint vertex sets, by induction we get ${\rm reg}\, A(X_1)=(q-2)\bigr(s-1-\sum_{i=1}^m k_i-1\bigl)$, and therefore, $$\label{nov2-11}
\textstyle \deg\, F(C,t)=(q-2)\bigr(s-\sum_{i=1}^m k_i-1\bigl).$$ From the exact sequence $$0\rightarrow (S/I(X))[-1]\stackrel{t_s\ }{\longrightarrow}
S/I(X)\longrightarrow
S/(t_s,I(X))\rightarrow 0,$$ we get that $F_X(t)=F(S/(t_s,I(X)),t)/(1-t)$. Thus ${\rm reg}(S/I(X))=
\deg\, F(S/(t_s,I(X)),t)$. Using the isomorphism $$S/(t_s,I(X))\simeq K[t_1,\ldots,t_{s-1}]/(t_1^{q-1},\ldots,t_{s-1}^{q-1},
I_1(X),\ldots,I_m(X)),$$ we obtain that $C\simeq S/(t_s,I(X))$. Hence, by Eq. (\[nov2-11\]), the desired formula follows.
Case (II): The graph ${\mathcal{G}}_1$ is disconnected. It is not hard to show that ${\mathcal{G}}_1$ has exactly two connected components ${\mathcal{G}}_1'$, ${\mathcal{G}}_1''$. Let $E_1'$, $E_1''$ be the edge sets of ${\mathcal{G}}_1'$, ${\mathcal{G}}_1''$ respectively and let $X_1'$, $X_1''$ be the algebraic toric sets parameterized by the edges of ${\mathcal{G}}_1'$, ${\mathcal{G}}_1''$ respectively. We may assume that ${\mathcal{H}}_1,\ldots,{\mathcal{H}}_{r}$ are the cycles of ${\mathcal{G}}_1'$ and ${\mathcal{H}}_{r+1},\ldots,{\mathcal{H}}_{m}$ are the cycles of ${\mathcal{G}}_1''$. By Theorem \[theorem: I(X) for almost general graph\], we have that $I(X_1')$ and $I(X_1'')$ are generated by $$\begin{aligned}
\mathcal{J}_1'=\{t_i^{q-1}-t_j^{q-1}:\, t_i,t_j\in E_1'\} \cup
I_1(X)\cup\cdots\cup I_r(X)\ \mbox{ and }\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ & &\\
\mathcal{J}_1''=\{t_i^{q-1}-t_j^{q-1}:\, t_i,t_j\in E_1''\} \cup
I_{r+1}(X)\cup\cdots\cup I_m(X), & &\end{aligned}$$ respectively. We set $$C_1'=K[E_1']/(\{t_i^{q-1}\}_{t_i\in
E_1'},I_1(X),\ldots,I_r(X)),\ \ C_1''=K[E_1'']/(\{t_i^{q-1}\}_{t_i\in
E_1''},I_{r+1}(X),\ldots,I_m(X)).$$ By the arguments that we used to prove Case (I), and using the induction hypothesis, we get $$\textstyle \deg F(C_1',t)=(q-2)\bigr(|E_1'|-\sum_{i=1}^r
k_i\bigl),\quad \deg F(C_1'',t)=(q-2)\bigr(|E_1''|-\sum_{i=r+1}^m k_i\bigl).$$ Since $K[E_1']$ and $K[E_1'']$ are polynomial rings in disjoint sets of variables $E_1'$ and $E_1''$, according to [@monalg Proposition 2.2.20, p. 42], we have an isomorphism $$C_1'\otimes_K
C_1''\simeq
K[t_1,\ldots,t_{s-1}]/(t_1^{q-1},\ldots,t_{s-1}^{q-1},I_1(X),\ldots,I_m(X))=S/(t_s,I(X)).$$ Altogether, as $F(C_1'\otimes_K C_1'',t)=F(C_1',t)F(C_1'',t)$ (see [@monalg p. 102]), we obtain $$\begin{aligned}
{\operatorname{reg}}S/I(X)&=&\deg\, F(S/(t_s,I(X)),t)=\deg F(C_1'\otimes_K
C_1'',t)=\deg F(C_1',t)+\deg F(C_1'',t)\\
&=&\textstyle(q-2)\bigr(|E_1'|+|E_1''|-\sum_{i=1}^m
k_i\bigl)=(q-2)\bigr(s-\sum_{i=1}^m
k_i-1\bigl),\end{aligned}$$ as required. This completes the proof of case (II).
[10]{}
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[^1]: The first author was partially supported by CMUC and FCT (Portugal), through European program COMPETE/FEDER. The second author is a member of the Center for Mathematical Analysis, Geometry and Dynamical Systems. The third author was partially supported by SNI
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abstract: 'The incorporation of bulk viscosity process to General Relativity leads to the appearance of nonsingular backgrounds that, at early and late times, depict an accelerated universe. These backgrounds could be analytically calculated [and]{} mimicked, in the context of General Relativity, by a single scalar field whose potential could also be obtained analytically. We will show that, we can build viable backgrounds that, at early times, depict an inflationary universe leading [to]{} a power spectrum of cosmological perturbations [which]{} match with current observational data, [and]{} after leaving the inflationary phase, the universe suffers a phase transition needed to explain the reheating of the universe via gravitational particle production, and finally, at late times, it enters [into the]{} de Sitter phase that [can]{} explain the current cosmic acceleration.'
author:
- 'Jaume Haro$^{a,}$[^1] and Supriya Pan$^{b,}$[^2]'
title: Bulk viscous quintessential inflation
---
[*Introduction.–*]{} Recently, bulk viscosity phenomena (see, for instance, [@Zimdahl1] for a detailed description) applied to cosmology have attracted considerable interest since [this]{} dissipative mechanism in the Friedmann-Lema[î]{}tre-Robertson-Walker space-time [is]{} able to explain [the]{} inflation [@bo; @bt] and the current cosmic acceleration [@beno; @begt].
It is well-known that, in the context of bulk viscous cosmology, [our universe attains two accelerated phases at early and late times (see [@bcno] for a review), where essentially the universe starts in an unstable de Sitter solution (early accelerated phase) and ends in a stable one (current accelerating phase).]{} Unfortunately, the early time acceleration proposed in current models do not correspond to an inflationary phase [@pan1].
For this reason, our main goal in the present letter, is to [pledge]{} simple models containing an isotropic viscous fluid filling the universe with a linear Equation of State (EoS) whose pressure was positive, depicting nonsingular backgrounds (without the big bang singularity) that have, at early times, an inflationary phase that ends in a sudden phase transition in order to produce enough particles to reheat the universe, and, at late times, [renders]{} an accelerated universe. With these nonsingular backgrounds, using the reconstruction method, we find inflationary quintessential potentials, that predict, at early times, a spectral index of scalar cosmological perturbations with running and its corresponding ratio of tensor to scalar perturbations that fit well with recent observational data. Moreover, these potentials have an absolute minimum at the corresponding de Sitter solution, meaning that [the de]{} Sitter solution is a late time attractor, and thus, the background will depict the current cosmic acceleration.
The units used throughout the letter are $\hbar=c=8\pi G=1$.
[*Bulk viscous cosmology.–*]{} In cosmology, the simplest effective way to incorporate [the]{} bulk viscosity, is to use Eckart theory [@eckart] (see also [@Zimdahl1]), where basically the pressure $p$ is replaced by $p- 3 H \xi(H)$, where $\xi(H)$ is the so-called coefficient of bulk viscosity [@bg], which could depend on the Hubble parameter $H$, or, equivalently, of the energy density of the universe. Hence, in the flat Friedmann-Lema[î]{}tre-Robertson-Walker (FLRW) spacetime, the classical Friedmann and Raychaudhuri’s equations are modified as $$\begin{aligned}
\rho = 3H^2,\quad
\dot{H}=- \frac{1}{2} (p+\rho)+ \frac{3}{2} H \xi(H),\label{bulk-friedmann2}\end{aligned}$$ where $\rho$ is the energy density.
Now, assuming that the universe is filled with a barotropic fluid with Equation of State (EoS): $p= (\gamma- 1)\rho$, where the dimensionless parameter $\gamma$ has been chosen greater or equal to $1$, in order to have a non-negative pressure. Hence, Raychaudhuri’s equation can be written as $$\begin{aligned}
\dot{H} &=& -\frac{3}{2}(1+w_{eff}(H)) H^2\label{bulk-friedmann3},\end{aligned}$$ where we have introduced the effective EoS parameter $$\begin{aligned}
w_{eff}(H)&\equiv& -1-\frac{2\dot{H}}{3H^2}= -1+ \gamma \left(1-\frac{\xi(H)}{\gamma H}\right).\label{eff-eos}\end{aligned}$$
The idea is to find models, i.e., to find the coefficient of bulk viscosity, such that, the equation $1+w_{eff}(H)=0$ has two positive roots, namely $H_+>H_-$, that will correspond to different de Sitter solutions, one of them will be an attractor and the other one must be a repeller. When $H_+$, which eventually could be $+\infty$, is the repeller and $H_-$ is an attractor, we will have a nonsingular background with $w_{eff}(H)> -1$, that starts at $H_+$ and ends at $H_-$, depicting an accelerated universe at early and late times. This is a candidate to depict our universe. However, to check its viability, one has to show that at early times, this accelerated phase is an inflationary one. That is what we will do in the next sections.
[*The model and its dynamical analysis.–* ]{} Due to the equivalence between bulk viscous and open cosmology, where isentropic particle production is allowed [@Prigogine98], our first attempt is to choose a coefficient of bulk viscosity [@pan] $$\begin{aligned}
\xi(H)=-\xi_0+mH+\frac{n}{H},\end{aligned}$$ where $\xi_0$, $m$ and $n$ are some positive parameters.
As a consequence of the entropy growth in irreversible processes of energy dissipation, $\xi(H)$ must be positive (see section $49$ of [@ll]), then taking into account its minimum [achieved at $\sqrt{\frac{n}{m}}$, its positivity condition is satisfied]{} when $\frac{\xi_0^2}{4m}\leq n$.
After a dynamical analysis, one can show that, in order to have a nonsingular dynamics without the big bang singularity where the universe depicts two accelerated phases, one at early times, and one at late-times (current accelerating phase) the parameters $\xi_0$, $m$ and $n$ must belong [to the following]{} domain of ${{\mathbb R}}^3$ [ $$\begin{aligned}
W=\{\xi_0>0, m\geq \gamma, n>0, \mbox{and,}~\frac{4(\gamma-m)n}{\xi_0^2}>-1 \}.\end{aligned}$$]{} In that domain, the critical points (de Sitter solutions) of the system, i.e., the values of $H$ that are the solutions of the equation $1+w_{eff}(H)=0$, are $$\begin{aligned}
H_{\pm}=\frac{\xi_0}{2(m-\gamma)}\left(1\pm \sqrt{1+\frac{4(\gamma-m)n}{\xi_0^2} }\right),\end{aligned}$$ where $H_+$ is a repeller and $H_-$ is an attractor. Then, the dynamics going from $H_+$ to $H_-$ becomes nonsingular. Moreover, since at the critical points $w_{eff}=-1$, one will have a universe starting and ending in an accelerated phase.
[Assuming that, $n\ll \xi_0^2\min\left(m-\gamma,\frac{1}{m-\gamma} \right)$, one obtains $H_+\cong \frac{\xi_0}{m-\gamma}$, $H_-\cong \frac{n}{\xi_0}$, with $0<H_-\ll H_+$,]{} and consequently, the universe could start at very high energies and ends at low ones. In particular, for $m=\gamma$, one has $H_+=\infty$, but there is no big bang singularity, because in that case, $\dot{H}=-\frac{3}{2}(\xi_0H-n)$, [which for large values of $H$ gives]{} $$\begin{aligned}
\dot{H}\cong -\frac{3}{2}\xi_0H\Longleftrightarrow H(t)\cong H_ie^{-\frac{3}{2}\xi_0(t-t_i)},\end{aligned}$$ meaning that $H$ diverges only when $t=-\infty$.
In fact, for our model, the [Raychaudhuri equation (\[bulk-friedmann3\])]{} can be analytically solved for $m>\gamma$ as $$\begin{aligned}
H(t)=\frac{H_+e^{-\frac{3}{4}(H_+-H_-)(m-\gamma)t}+
H_-e^{\frac{3}{4}(H_+-H_-)(m-\gamma)t}}{2\cosh{\left(\frac{3}{4}(H_+-H_-)(m-\gamma)t\right)}},\end{aligned}$$ [and,]{} $$\begin{aligned}
H(t)=\xi_0e^{-\frac{3}{2}\xi_0 t}+\frac{n}{\xi_0},\end{aligned}$$ for $m=\gamma$.
[*Inflationary quintessential potential.–*]{} In this section, we will see [under which conditions a scalar field $\varphi$ with potential $V (\varphi)$ could mimic the dynamics of a perfect fluid with bulk viscosity in order to provide viable backgrounds that could depict our universe correctly.]{}
It is well-known that the energy density, namely $\rho_{\varphi}$, and pressure, namely $p_{\varphi}$, of the scalar field minimally coupled with gravity are given by $$\begin{aligned}
\rho_{\varphi}= \frac{1}{2} \dot{\varphi}^2+ V (\varphi),\quad
p_{\varphi}= \frac{1}{2} \dot{\varphi}^2- V (\varphi),\label{pdensity1}\end{aligned}$$
To show the equivalence with the bulk viscous system (\[bulk-friedmann2\]), we perform the replacement $$\begin{aligned}
\rho\longrightarrow \rho_{\varphi},\quad
p-3H\xi(H) \longrightarrow& p_{\varphi}\label{pdensity2},\end{aligned}$$ to recover the standard Friedmann and Raychaudhuri equations for a universe filled by an scalar field $$\begin{aligned}
3H^2=\rho_{\varphi}, \quad 2\dot{H}=-\dot{\varphi}^2.\label{friedmann2}\end{aligned}$$
Note that, since the equations in (\[friedmann2\]) are the usual equations for a single scalar field, this means that the dynamics driven by a fluid with bulk viscosity with an effective EoS parameter greater than $-1$, could be mimicked by a single scalar field in the context of General Relativity.
Combining the Friedmann and Raychaudhuri equations (\[bulk-friedmann2\]), (\[bulk-friedmann3\]) and (\[friedmann2\]), one easily obtains $$\begin{aligned}
\dot{\varphi}= \sqrt{-2\dot{H}}=\sqrt{3\gamma H^2 \left(1- \frac{\xi(H)}{\gamma H}\right)}~,\label{scalarfield2}\\
V (\varphi)= \frac{3H^2}{2}\left[(2- \gamma)+ \frac{\xi(H)}{ H}\right] ~.\label{potential}\end{aligned}$$
Then, to reconstruct the potential, the first step is to integrate (\[scalarfield2\]). [Therefore,]{} performing the change of variable $dt=\frac{dH}{\dot{H}}$, we obtain $$\begin{aligned}
\varphi=-\int\sqrt{-~\frac{2}{\dot{H}}}~dH =-\frac{2}{\sqrt{3}}\int \frac{dH}{\sqrt{\gamma H^2-\xi(H)H}}.\end{aligned}$$
[For our model, where]{} $\xi(H)= -\xi_0+ m H+ n/H$, one has $$\begin{aligned}
\varphi=-\frac{2}{\sqrt{3}}\int \frac{dH}{\sqrt{(\gamma-m)H^2+\xi_0H-n}},\end{aligned}$$ [and,]{} when the parameters belong to the domain $W$, for $m<\gamma$ one has (see formula $2.261$ of [@gr]) $$\begin{aligned}
\label{sinus}
\varphi=\frac{2}{\sqrt{3(m-\gamma)}}\arcsin\left(\frac{2(\gamma-m)H+\xi_0}{\sqrt{\xi_0^2+4(\gamma-m)n}}\right),\end{aligned}$$ defined in $\left( \frac{-\pi}{\sqrt{3(m-\gamma)}}, \frac{\pi}{\sqrt{3(m-\gamma)}}\right)$, and when $m=\gamma$ $$\begin{aligned}
\varphi=-\frac{4}{\sqrt{3}\xi_0}\sqrt{\xi_0H-n},\end{aligned}$$ which is defined in the domain $(-\infty,0)$.
Finally, isolating $H$ and inserting it in (\[potential\]), one obtains the corresponding potentials. Once the potential has been reconstructed, one has the corresponding conservation equation $$\begin{aligned}
\label{KG}
\ddot{\varphi}+\sqrt{3}\sqrt{\frac{\dot{\varphi}^2}{2}+V(\varphi)}\dot{\varphi}+V_{\varphi}(\varphi)=0,\end{aligned}$$ whose solutions depict different backgrounds, and where one of them is the solution of (\[bulk-friedmann3\]).
For example, in the simplest case $m=\gamma$, one easily obtains $$\begin{aligned}
\label{example}
V(\varphi)=\frac{27}{256}\xi_0^2\varphi^4+\frac{9}{8}\left(n-\frac{\xi_0^2}{4} \right)\varphi^2+\frac{3n^2}{\xi_0^2}.\end{aligned}$$
[Recall that,]{} the positivity of $\xi(H)$ implies $\frac{\xi_0^2}{4\gamma}\leq n$, and note that, when $n\geq\frac{\xi_0^2}{4}$, the unique minimum of the potential (\[example\]) is [achieved]{} at $\varphi=0$. On the contrary, when $n<\frac{\xi_0^2}{4}$, the potential has a maximum at $\varphi=0$. Then, since we want that $H_-$ must be an attractor, we will have to choose $n\geq\frac{\xi_0^2}{4}$. Moreover, [as the]{} fluid has positive pressure ($m=\gamma\geq 1$), this condition is compatible with the positivity of $\xi(H)$ because $\frac{\xi_0^2}{4\gamma}\leq \frac{\xi_0^2}{4}$, [which]{} means that the critical point satisfies $H_-=\frac{n}{\xi_0}\geq \frac{\xi_0}{4}$. Unfortunately, [the]{} models we have chosen has two important defaults: 1.- As we well see, the main slow-roll parameter is of the order $\frac{\xi_0}{H}$, this means that, in order to match with current observational data, the observable modes must leave the Hubble radius at scales of the order $H\sim 10^{3}\xi_0\leq 10^3 H_-$. However, since $H_-$ has to be close to the current value of the Hubble parameter, the condition $H\leq 10^3 H_-$ is compatible with the fact that [the]{} observational modes must leave the Hubble radius at high energy densities (few orders below Planck’s one).
2.- There is no mechanism to reheat the universe, because neither oscillations nor abrupt phase transitions at high scales, that breakdown the adiabaticity to produce enough amount of particles that thermalize the universe after inflation, occur in that models. A simple way to overpass both problems consists of introducing a phase transition at early times when the universe ceases to accelerate. More precisely, we will choose the phase transition when the universe is radiation dominated.
Taking $\gamma=\frac{4}{3}\Longleftrightarrow p=\frac{1}{3}\rho$, $n=\frac{3\xi_0^2}{16}\Longrightarrow \xi(H)\geq 0$, our continuous coefficient of bulk viscosity is improved as follows $$\begin{aligned}
\label{model1}
\xi(H)=\left\{\begin{array}{ccc}
-\xi_0+\frac{4}{3}H+\frac{3\xi_0^2}{16H},& \mbox{for}& H\geq H_E\\
\xi_1,&\mbox{ for}& H\leq H_E,
\end{array}\right.\end{aligned}$$ [where $0<\xi_1\ll \xi_0$, and,]{} $$H_E=\frac{3}{8}(\xi_1+\xi_0)\left(1+\sqrt{1-\frac{\xi_0^2}{(\xi_1+\xi_0)^2}} \right)\cong \frac{3\xi_0}{8}.$$
The corresponding potential has the form $$\begin{aligned}
\label{potential3}
V(\varphi)=\left\{\begin{array}{ccc}
\frac{27\xi_0^2}{256}(\varphi^4-\frac{2}{3}\varphi^2+1),&\mbox{for}& \varphi\leq \varphi_E \\
H(\varphi)\left(H(\varphi)+\frac{3\xi_1}{2}\right),&\mbox{for}& \varphi\geq \varphi_E,
\end{array}\right.\end{aligned}$$ where $\varphi_E=-\sqrt{\frac{16H_E}{3\xi_0}-1}\cong -1$ and $$\begin{aligned}
H(\varphi)=\frac{A^2e^{-\frac{\sqrt{3}}{2}(\varphi-\varphi_E)}+ 9\xi_1^2e^{\frac{\sqrt{3}}{2}(\varphi-\varphi_E)}}{16A}+\frac{3\xi_1}{8}\nonumber\\
\cong \frac{3\xi_0}{8}e^{-\frac{\sqrt{3}}{2}(\varphi+1)}+ \frac{3\xi_1^2}{32\xi_0}e^{\frac{\sqrt{3}}{2}(\varphi+1)}+\frac{3\xi_1}{8},\end{aligned}$$ with $$\begin{aligned}
A\equiv 4\sqrt{4H_E^2-3\xi_1H_E}+8H_E-3\xi_1\cong 6\xi_0.\end{aligned}$$
The corresponding conservation equation (\[KG\]) provides backgrounds that could depict our universe, and one of them is the solution of (\[bulk-friedmann3\]) $$\begin{aligned}
\label{nonsingular}
H(t)=\left\{\begin{array}{ccc}
\left(H_E-\frac{3\xi_0}{16}\right)e^{-\frac{3\xi_0}{2}(t-t_E)} + \frac{3\xi_0}{16}~, &\mbox{for}& t\leq t_E, \\
\frac{\frac{3\xi_1}{4}H_E}{H_E-(H_E-\frac{3\xi_1}{4} )e^{-\frac{3\xi_1}{2}(t-t_E)} }~, &\mbox{for}& t\geq t_E,
\end{array}\right.\end{aligned}$$ where $t_E$ is the phase transition time, and whose viability will be checked in the next section.
Finally, note that, at late time, the system has a critical point at $H_-=\frac{3\xi_1}{4}\Longleftrightarrow \varphi=\varphi_E+\frac{2}{\sqrt{3}}\ln\left(\frac{A}{3\xi_1}\right)$. To show that it is an attractor, we have to calculate, applying the chain rule, $V_{\varphi\varphi}$ at this critical point, leading to $$\begin{aligned}
\label{model}
V_{\varphi\varphi}=\frac{2}{\varphi_H^2}-{3\xi_1}\frac{\varphi_{HH}}{\varphi_H^3}=\frac{9}{8}\left(\frac{8H}{3}-\xi_1\right)=\frac{9\xi_1}{8}>0,\end{aligned}$$ meaning that the potential has a minimum at the critical point, and consequently, it is an attractor.
A final remark is in order: When one considers the case $m\gtrsim \gamma=\frac{4}{3}$, and assumes a phase transition as in the model (\[model1\]), the bulk viscous Raychaudhuri equation (
\[bulk-friedmann3\]) leads to a nonsingular solution that starts at $H_+\cong \frac{\xi_0}{m-\frac{4}{3}}$ and ends at $H_-=\frac{3\xi_1}{4}$. Then, the corresponding quintessential inflationary potential will have a maximum at $H_+$ (unstable) and a minimum at $H_-$ (attractor), that is, some backgrounds ([models which, at early time, are close to our nonsingular background (\[nonsingular\])]{}) given by (\[KG\]), [leave the de Sitter phase $H_+$ at early-times, and suffer]{} a sudden phase transition when the universe starts to decelerate, and finally, enters [into the stable]{} de Sitter phase $H_-$. In fact, the shape of the potential can easily be imagined: [From equation (\[sinus\]), one can deduce that, before the phase transition ($\varphi<\varphi_E$), the potential has a sinusoidal form with period $\frac{4\pi}{\sqrt{3m-4}}$,]{} and, after the phase transition, it has the same shape as (\[potential3\]).
[*Slow roll phase.–*]{} We know that, at early time, our background (\[nonsingular\]) satisfy $w_{eff}(H)\cong -1$, this means that the universe is quasi de Sitter, and [we aim to check whether this]{} background could lead to a power spectrum of cosmological perturbations that fit well with current observational data [@Ade].
[Hence,]{} we consider the slow roll parameters [@btw] $$\begin{aligned}
\epsilon=-\frac{\dot{H}}{H^2}, \quad \eta=2\epsilon-\frac{\dot{\epsilon}}{2H\epsilon},\end{aligned}$$ that allows us to calculate the spectral index, its running and the ratio of tensor to scalar perturbations $$\begin{aligned}
n_s-1=-6\epsilon+2\eta, \quad \alpha_s=\frac{H\dot{n}_s}{H^2 +\dot{H}},\quad
r=16\epsilon.\end{aligned}$$
At early times, i.e., when $H>H_E$, introducing the notation $x\equiv \frac{3\xi_0}{2H}$, one has $$\begin{aligned}
\epsilon=x\left(1-\frac{x}{8}\right),\quad \eta=\epsilon+\frac{x}{2},\end{aligned}$$ and, as a consequence, $
n_s-1=-3x+\frac{x^2}{2}.$ Conversely, $$\begin{aligned}
x=3\left(1-\sqrt{1-\frac{2(1-n_s)}{9}}\right).\end{aligned}$$
Then, given the observational values of the spectral index, one can obtain the range of $x$. [Recent PLANCK+WP 2013 data (see table $5$ of [@Ade]) predicts the spectral index at 1$\sigma$ Confidence Level (C.L.) to be $n_s=0.9583\pm 0.0081$, which means that, at $2\sigma$ C.L., one has $0.0085\leq x \leq 0.0193$, and thus, $0.1344\leq r=16\epsilon\leq 0.3072$.]{}
Since PLANCK+WP 2013 data provides the constrain $r\leq 0.25$, at $95.5\%$ C.L., then when $0.0085\leq x \leq 0.0156$, i.e., for the modes that leave the Hubble radius at scales $ 2\times 10^5\xi_0^2 \lesssim \rho \lesssim 9\times 10^5\xi_0^2$, the spectral index belongs to the 1-dimensional marginalized $95.5\%$ C.L., and also $r\leq 0.25$, at $95.5\%$ C.L.
For the running at 1$\sigma$ C.L., PLANCK+WP 2013 data gives $\alpha_s=-0.021\pm 0.012$, and our background (\[nonsingular\]) leads to the theoretical value $\alpha_s= \frac{x\epsilon}{4(1-\epsilon)}$. Consequently, at the scales we are dealing with, $10^{-5}\leq \alpha_s\leq 6\times 10^{-5}$, and thus, the running also belongs to the 1-dimensional marginalized $95.5\%$ C.L.
Note also that, we have the relation $w_{eff}(H)=-1+\frac{2}{3}\epsilon$. Therefore, if we assume that [the slow-roll ends when $\epsilon=1$,]{} and let $H_{end}$ be the value of the Hubble parameter when [the]{} slow roll ends, then the slow roll will end when $w_{eff}(H_{end})=-\frac{1}{3}$, i.e., when the universe will start to decelerate.
On the other hand, the number of e-folds from observable scales exiting the Hubble radius to the end of inflation, namely $N(H)$, could be calculated using the formula $N(H)=-\int_{H_{end}}^H\frac{H}{\dot{H}}dH$, leading to $$\begin{aligned}
N(x)=\frac{1}{x}-\frac{1}{x_{end}}+\frac{1}{16}\ln\left(\frac{16-x}{16-x_{end}} \right),\end{aligned}$$ where $x_{end}=4(1-\sqrt{1/2})\cong 1.1715$, is the value of the parameter $x$ when inflation ends. For our values of $x$ that allow to fit well with the theoretical value of the spectral index, its running and the tensor/scalar ratio with their observable values, we will obtain $64\leq N\leq 117$.
To determinate the value of $\xi_0$, one has to take into account the theoretical [@btw] and the observational [@bld] value of the power spectrum $$\begin{aligned}
{\mathcal P}\cong \frac{H^2}{8\pi^2\epsilon}=\frac{9\xi_0^2}{32\pi^2\epsilon x}=\frac{18\xi_0^2}{\rho_{pl}\epsilon x}\cong 2\times 10^{-9},\end{aligned}$$ where we have explicitly introduced the Planck’s energy density, which in our units is $\rho_{pl}=64\pi^2$. Using the values of $x$ in the range $[0.0085,0.0156]$, we can conclude that $$\begin{aligned}
10^{-7}\sqrt{\rho_{pl}}\leq \xi_0\leq 10^{-6}\sqrt{\rho_{pl}}.\end{aligned}$$
Summing up, the observable modes in our model leave the Hubble radius at scales $$\begin{aligned}
2\times 10^{-9} \rho_{pl}\lesssim \rho \lesssim 9\times 10^{-7} \rho_{pl},\end{aligned}$$ and since the sudden transition occurs at $H_E\cong \frac{3\xi_0}{8}\Longrightarrow \rho_E\sim 10^{-1}\xi_0^2$, one can deduce that the universe pre-heats, due to the gravitational particle production, at scales (the same result was obtained in formula $(15)$ of [@pv]) $$\begin{aligned}
10^{-15} \rho_{pl}\lesssim \rho \lesssim 10^{-13} \rho_{pl}.\end{aligned}$$
These particles will interact exchanging gauge bosons [@spokoiny] to reach a thermal equilibrium temperature which in the radiation dominated era, is of the order $T_R\sim 10^3 $ GeV
(see for details [@pv]), that matches with the hot Friedmann universe, and finally, at late times, due to the expansion of the universe, the energy density of the matter will become sub-dominant and the scalar field will come back to dominate the evolution of the universe, leading to the current cosmic acceleration.
[*Acknowledgments.–*]{} The investigation of J. Haro has been supported in part by MINECO (Spain), project MTM2014-52402-C3-1-P. SP is partially supported by the Council of Scientific and Industrial Research (CSIR), Govt. of India, by the research grants (File No. 09/096 (0749)/2012-EMR-I).
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[^1]: E-mail: jaime.haro@upc.edu
[^2]: E-mail: span@research.jdvu.ac.in
|
---
abstract: |
Hamiltonian tridiagonal matrices characterized by multi-fractal spectral measures in the family of Iterated Function Systems can be constructed by a recursive technique here described. We prove that these Hamiltonians are almost-periodic. They are suited to describe quantum lattice systems with nearest neighbours coupling, as well as chains of linear classical oscillators, and electrical transmission lines.
We investigate numerically and theoretically the time dynamics of the systems so constructed. We derive a relation linking the long-time, power-law behaviour of the moments of the position operator, expressed by a scaling function $\beta$ of the moment order $\alpha$, and spectral multi-fractal dimensions, $D_q$, via $
\beta(\alpha) = D_{1-\alpha}.
$ We show cases in which this relation is exact, and cases where it is only approximate, unveiling the reasons for the discrepancies.
author:
- |
Giorgio Mantica\
Istituto di Scienze Matematiche, Università di Milano a Como,\
Via Lucini 3, I-22100 COMO, ITALY\
[mantica@mi.infn.it]{}\
date:
title: ' Quantum Intermittency in Almost-Periodic Lattice Systems Derived from their Spectral Properties'
---
6.25in 9.5in .5in 1.1in -4.1cm -2.5cm
[*This paper is dedicated to the memory of Professor Joseph Ford,*]{}\
[*teacher, original researcher, and founder of Physica D*]{}\
PACS numbers: 05.45.+b, 02.30.-f, 71.30.+h, 71.55.Jv\
1991 [*Mathematics Subject Classification:*]{} 28A80, 58F11, 81Q10\
[*Keywords and Phrases:*]{} Self-similar measures, iterated function systems, quantum intermittency, almost-periodic systems, multi-fractal dimensions.\
Introduction
============
Usually, the study of almost/quasi-periodic systems starts by assigning a suitable rule for building a quantum Hamiltonian operator, and then proceeds to the determination of its spectral quantities [@magn] (which is frequently a hard task) and of the time dynamics it generates. In so doing, [*multi-fractal*]{} energy spectra have been frequently observed, and anomalous characteristics of the time evolution have been exhibited [@koh; @yam; @geis1; @ig1; @igm].
These findings raise the question if multi-fractal spectra are typical in almost/quasi-periodic systems [@gen], and, vice versa, if almost/quasi-periodicity is always associated with singular continuous spectral measures. The relations between this pair (spectral multi-fractality and Hamiltonian almost-periodicity) and the time dynamics generated via Schrödinger’s equation are also interesting, and intricate: do the former always imply anomalous scaling relations of physical quantities like, for instance, the expectation value of the position operator ? Can we make any quantitative statement to this effect ?
In this paper, we employ a new algorithm for deriving a Hamiltonian operator (called a [*Jacobi matrix*]{} because of its mathematical nature) with a pre-assigned spectral measure in the vast class of [*Iterated Function Systems*]{} (I.F.S.). This technique provides us with an ideal patient for our surgical table, who can be fully dissected and analyzed. In particular, we provide evidence that Jacobi matrices associated with I.F.S. are almost periodic, and we argue that this is likely to be the typical case in a large class of measures with fractal support.
The analysis which can be carried out in this example permits us to compute exactly the asymptotic behaviour of the wave–function projections, for short and long times. By introducing a renormalization approach in the theory of orthogonal polynomials, we also derive a relation linking the asymptotic power-law growth of the moments of the position operator and multi-fractal generalized dimensions. This theory explains the phenomenon that we have termed [*quantum intermittency*]{}.
The specific properties of I.F.S. are crucial for our theory; Yet, since particular I.F.S.’s can be found so to approximate arbitrarily well (in a technical sense) any “fractal” measure [@gio2; @gio; @vrs; @gio-guzzi], the results obtained in the I.F.S. class may have a much wider generality.
We shall present our results as follows: in Section II we introduce the general formalism of I.F.S. and of Jacobi matrices, employed to solve the [*inverse problem*]{} of finding an Hamiltonian with a given spectrum. This formalism is then applied in Sect. III to derive a stable solution algorithm. The almost-periodic properties of the Hamiltonian so determined are studied numerically in Sect. IV, and the intermittent quantum dynamics it generates is then discussed in Sect. V and VI. The Conclusions summarize the work and present some previews on further investigations.
I.F.S. and their Jacobi Matrices
================================
Systems of linear iterated functions [@hut; @dia; @dem; @ba2] are finite collections of maps $$\label{mappi}
\phi_{i} (x) := \delta_{i} x + \beta_{i}, \;\; i = 1, \ldots, M ,$$ where $\delta_{i},\beta_{i}$ are real constants, and where the contraction rates $\delta_{i}$ have modulus less than one. Without loss of generality, we may assume that each $\phi_i$ maps $[0,1]$ into itself, and that $\phi_1(0) = 0$.
A probability, $\pi_i$, is associated with each map: $\pi_{i} > 0$, $\sum_{i} \pi_{i} = 1$. Employing these probabilities, a measure over $[0,1]$ can be defined as the unique positive measure satisfying the balance property $$\label{bala}
\int _{0}^1 f \; d\mu \; =
\sum_{i=1}^{M}
\; \pi_{i}
\; \int _{0}^1 \;
(f \circ \phi_{i}) \; d\mu ,$$ for any continuous function $f$. This measure is supported on $A$, the subset of $[0,1]$ which solves the equation $$\label{attra}
A=\bigcup_{i=1,\ldots ,M}\;\phi_i(A),$$ The set $A$ is invariant under the action of shrinking it to smaller copies of itself, and glueing them together. Because of eq. (\[attra\]), the geometry of this set is typically fractal (except for special choices of the map parameters); In turn, the balance relation (\[bala\]) is responsible for the multi-fractal properties of the measure $\mu$. In fact, let us consider a [*disconnected*]{} I.F.S., that is to say, one for which the sets $\phi_i(A)$ do not intersect each other. Under these circumstances, the multi-fractal properties of the balanced measure are easily computable: the spectrum of generalized dimensions $D_q$ follows from the equation $$\sum_{j=1}^{M}
\pi_j^q \delta_j^{-\tau} = 1 ,
\label{multi1}$$ whose unique real solution defines $\tau$ as a function of $q$, and leads to $D_q = \frac{\tau(q)}{q-1}$. In virtue of this relation, one can tune the map parameters to obtain various multi-fractal spectra.
The problem of determining a Hamiltonian possessing $\mu$ as spectral measure can be solved [@kac] considering the set of associated orthonormal polynomials, $\{p_n\}$: $$\int p_i(x) p_k(x) \; d\mu(x) = \delta_{i,k}.$$ In fact, any such set of polynomials is characterized by a three-terms recurrence relation which can be written $$\label{nor2}
x p_j (x) = r_{j+1} p_{j+1}(x) + A_j p_j(x) + r_{j} p_{j-1}(x),$$ or, in matrix form $$\label{jac2}
H p (x) = x p (x).$$ In the above, $p(x)$ is the vector whose components are the orthonormal polynomials evaluated at site $x$, and $H$ is the Jacobi matrix, which is constructed as the real, symmetric, tridiagonal matrix whose diagonal and outer diagonals are the vectors $A_j$ and $r_j$, respectively: $$\label{jac11}
H_{i,i} = A_i , \; \; \; H_{i+1,i} = H_{i,i+1} = r_{i+1} , \;
\; i = 0,1,\ldots .$$ $H$ defines a nearest neighbours lattice system, with site energies $A_i$ and hopping constants $r_i$. Similarly, $H$ can describe a linear array of masses coupled by springs, and also an electrical transmission line, whose characteristics vary from one element to the next.
Standard theory proves that, letting the Jacobi matrix $H$ act in $l_2$ (the space of square summable sequences, whose canonical basis will be indicated by $\{e_0,e_1,\ldots\}$) the spectral measure of $H$ with respect to the vector $e_0$ (the [*local density of states*]{} of physical jargon) is precisely $\mu$: in fact, one has $$\label{jac8}
(e_0, g(H) e_0) = \int g(x) \; d\mu(x),$$ for well-behaved functions $g$. This is the theoretical solution of the inverse problem we have proposed. In order to translate it into a [*practical*]{} solution, we need to compute the Jacobi matrix coefficients starting from the measure $\mu$, i.e. from the map parameters defining the I.F.S.
A Stable Technique for Computing I.F.S. Jacobi Matrices
=======================================================
The problem of constructing the Jacobi matrix associated with I.F.S. measures is hard, and the usual techniques of [*polynomial sampling*]{} [@gax; @steve; @gaut] are plagued by exponentially increasing errors which allow only computation of very few Jacobi matrix coefficients [@cap]. Alternatively, the sole technique available so far has been an [*algebraic*]{} procedure programmed in [*MAPLE*]{} by Vrscay [@vrs2]. Yet, it is severely limited by memory and time requirements. To overcome these difficulties we have devised a direct algorithm applicable to I.F.S. measures.
We first observe that, for any $n$, $$\label{le1}
p_n(\phi_i(x))=\sum_{l=0}^n\Gamma _{i,l}^n\;p_l(x),
\;\;\; i = 1,\ldots, M.$$ This is immediate, since $p_n(\phi_i(x))$ is an $n$-th degree polynomials which can be expanded on the first $n$ orthogonal polynomials. Less immediate is to derive a recursive rule for the coefficients $\Gamma _{i,l}^n$, $l=0,\ldots ,n$. It turns out that, at fixed $n$, they can be determined from the map parameters, and from the Jacobi matrix entries $A_j$, for $j=0,1,\ldots ,n-1$, and $r_m$, for $m=0,1,\ldots ,n$. In fact, (dropping for simplicity the map index $i$) we have that $p_0(\delta x+\beta )=p_0(x)$, and hence $\Gamma _0^0=1$. Suppose now that $\Gamma^{k}_l$ is known for $k=0,\ldots,n-1$ and all relative $l$’s: from eq. (\[nor2\]) we obtain the complete decomposition of $p_n(\delta x + \beta)$ over $p_l$, $l=1,\ldots ,n$: $$\label{rec4}
\begin{array}{ll}
r_np_n(\delta x+\beta )
=& (\beta -A_{n-1})\sum_{l=0}^{n-1}\Gamma_l^{n-1}p_l(x) + \\
& +\delta \sum_{l=0}^{n-1}\Gamma_l^{n-1}(r_{l+1}p_{l+1}(x)+A_lp_l(x)
+r_lp_{l-1}(x)) + \\
& -r_{n-1}\sum_{l=0}^{n-2} \Gamma _l^{n-2}p_l(x). \\
\end{array}$$ Equation (\[rec4\]) allows now the determination of the coefficients $\Gamma _l^n$.
We observe that the highest order polynomial, $p_n$, appears twice in the above equation, always in the form of the product $r_n p_n$: hence, the coefficients in the expansion of the polynomial $r_n p_n$ can be determined [*without*]{} knowing $r_n$. Therefore, if we let $\tilde{p}_n(x) = r_n p_n(x)$, a second decomposition can be written as $$\label{le1b}
\tilde{p}_n(\phi_i(x)) = \tilde{\Gamma}^n_{i,n} \; \tilde{p}_n (x)
+ \sum_{l=0}^{n-1} \tilde{\Gamma}^n_{i,l} \; p_l(x) ,$$ where the coefficients $\tilde{\Gamma}$ can be computed recursively from eq. (\[rec4\]), on the basis of the knowledge of only $A_j$, $r_j$, for $j=0,1,\ldots,n-1$.
We can now compute the non-diagonal entries of the Jacobi matrix: from eq. (\[nor2\]) we write $$\label{rec8}
r_n^2 = \int \tilde{p}_n(x) x p_{n-1}(x) \; d\mu .$$ Hence, using the balance property (\[bala\]) and eqs. (\[le1\]),(\[le1b\]) this becomes $$\label{rec9}
r_n^2 = \sum_{i=1}^M \pi_i \int (\delta_i x + \beta_i) [
\sum_{m=0}^{n-1} \sum_{l=0}^{n-1} \tilde{\Gamma}^n_{i,m} \Gamma^{n-1}_{i,l}
p_m(x) p_l(x) + \sum_{l=0}^{n-1} \tilde{\Gamma}^n_{i,n} \Gamma^{n-1}_{i,l}
\tilde{p}_n(x) p_l(x) ] d\mu .$$ Again, we can use the recurrence relations (\[nor2\]), to get $$\label{rec10}
r_n^2 = \sum_{i=1}^M \pi_i \; (B_i + C_i + D_i) ,$$ where we have put: $$\label{rec11}
B_i = \sum_{l=0}^{n-1} (\beta_i + \delta_i A_l)
\tilde{\Gamma}^n_{i,l} \Gamma^{n-1}_{i,l} ,$$ $$\label{rec12}
C_i = \delta_i \sum_{l=0}^{n-2} r_{l+1} ( \tilde{\Gamma}^n_{i,l}
\Gamma^{n-1}_{i,l+1} + \tilde{\Gamma}^n_{i,l+1} \Gamma^{n-1}_{i,l}) ,$$ and $$\label{rec13}
D_i = \delta_i \tilde{\Gamma}^n_{i,n} \Gamma^{n-1}_{i,n-1}
r_n^2 .$$ Because of contractivity of the maps, $|D_i|r_n^{-2} < 1$. Therefore, $r_n^2$ (and hence $r_n>0$) can be computed from eq. (\[rec10\]), on the basis of the knowledge of the coefficients in the expansions (\[le1\]) of order $n-1$, of order $n$ in (\[le1b\]), of the map parameters, and of the matrix entries $A_j$, $r_j$, for $j=0,1,\ldots ,n-1$.
A similar trick allows the computation of the diagonal entries $A_n$; We use eqs. (\[bala\]) and (\[nor2\]) (integrals are taken with respect to $\mu$): $$\label{rec6}
A_n = \int x p^2_n(x) = \sum_{i=1}^M \pi_i \int (\delta_i x +
\beta_i) p_n^2(\delta_i x + \beta_i) \; = \; \sum_{i=1}^M \pi_i \int
(\delta_i x + \beta_i) \sum_{m,l=0}^{n} \Gamma^n_{i,l} \Gamma^n_{i,m} p_l(x)
p_m(x) .$$ Using the orthonormality properties of the sequence $p_n$, and the recurrence relation, eq. (\[nor2\]), we get $$\label{rec7}
A_n = \sum_{i=1}^M \pi_i [ \sum_{m=0}^{n} (\Gamma^n_{i,m})^2 \;
(\beta_i + \delta_i A_m) + \sum_{m=0}^{n-1} \Gamma^n_{i,m} \Gamma^n_{i,m+1}
\delta_i (r_m+r_{m+1}) ],$$ thereby determining $A_n$ as a function of the coefficients in eq. (\[le1\]) of order $n$ fixed, of the map parameters, and of the matrix entries $A_j$, for $j=0,1,\ldots ,n-1$, and $r_m$, for $m=0,1,\ldots ,n$.
These results can be properly chained into an iterative construction of the Jacobi matrix $H$: The algorithm is structured as follows:
- [*Initialization*]{}. At the first step, we have $A_0=\mu_1$, $r_0=0$, $\Gamma^0_0=1$. The first order moment of $\mu$, $\mu_1$, can be simply computed from eq. (\[bala\]).
- [*Iteration*]{}. Suppose that $A_l$, $r_l$, and $\Gamma^l$ are known for $l=0,1,\ldots ,n-1$. Then we:
- [*Compute $\tilde \Gamma ^n$*]{}. We use equations (\[rec4\] - \[le1b\]).
- [*Compute $r_n$*]{}. We use eqs. (\[rec8\] - \[rec13\]).
- [*Compute $\Gamma ^n$*]{}. This is immediate at this stage.
- [*Compute $A_n$*]{}. We use eqs. (\[rec6\] - \[rec7\]). Then we iterate the procedure.
Graphically: $$\label{recu}
\left(
\Gamma^{n-1}, \; \;
\begin{array}{l}
r_0,\ldots,r_{n-1} \\
A_0,\ldots,A_{n-1} \\
\end{array} \right)
\Rightarrow \tilde{\Gamma}^n \Rightarrow r_n \Rightarrow \Gamma^n
\Rightarrow A_n \Rightarrow \left( \Gamma^{n}, \; \;
\begin{array}{l}
r_0,\ldots,r_{n} \\
A_0,\ldots,A_{n} \\
\end{array} \right)$$
In a separate work [@cap] we have analyzed the reasons of the failure of classical polynomial sampling [@gax; @gaut] when applied to singular measures, and assessed the numerical stability of the recursive algorithm presented above. We have observed a polynomial error propagation with respect to matrix order for the recursive algorithm, while using the classical algorithms the error growth was found to be exponential.
Almost Periodicity of I.F.S. Jacobi Matrices
============================================
Having devised a stable solution of the Hamiltonian inverse problem, we can study the properties of large Jacobi matrices. Fig. 1 shows an I.F.S. measure, one of its orthogonal polynomials, and the begining of the sequence of $r_n$ coefficients. Let us focus our attention on the last.
We can clearly observe a zero frequency (the average value), a $\pi$ frequency (flipping up and down), and clearly other frequencies are present in the sequence. A Fourier analysis is simply effected writing $$r_n = \sum_k F_k e^{i n \omega_k} .
\label{fu1}$$ This sum may not converge in the usual sense, and it might have to be replaced by an integral in the case of a continuous component in the “spectrum” of the sequence $r_n$. If the continuous component is absent, the system is almost-periodic. Within this case, if the set of frequencies $\omega_k$ can be derived from a finite set of periods, the sequence $r_n$ is [*quasi-periodic*]{}: that is, this is the case if there exist suitable $\Omega_1, \ldots ,
\Omega_p$ such that for all $k$ the frequency $\omega_k$ can be written $\omega_k = n_1 \Omega_1 + \ldots + n_p \Omega_p$, for integer $n_1,\ldots,n_p$.
A numerical, fast Fourier analysis of the sequence $r_n$ is presented in Fig. 5, where peaks in the distribution of $|F_k|^2$ with a clear hierarchical structure are observed. These peaks seem to suggest the presence of a point component in the spectrum of this sequence. Yet, care has always to be exerted to assess this fact numerically. To obtain a further piece of evidence we performed an analysis of the phase of $F_k$ around these peaks, like that shown in Fig. 2, and found a $\pi$ discontinuity, which indicates [@andrei] that they are indeed related to a point component. The sequence $r_n$ is therefore almost periodic.
Since no simple rational relation among the peak sequences seems to hold, numerical evidence seems to suggest that the sequence is not quasi-periodic. Our numerical investigations have shown that these characteristics are typical in the class of Hamiltonian associated with I.F.S. measures, supported on Cantor sets. In view of the approximation properties of I.F.S. measures, this result is likely to be much more general: indeed, in the family of Jacobi matrices associated with real Julia sets [@danbel],[@barn], which can be well approximated by I.F.S., limit periodicity of the sequence $r_n$ has been proven directly [@danbak]. The problem of a formal proof is therefore open.
Quantum Dynamics of Almost Periodic Lattice Systems
===================================================
Jacobi matrices generate a quantum dynamics in $l_2$ via Schrödinger’s equation, $$i \frac{d \psi}{d t} = H \psi, \;\;
\psi(0) = e_0 := (1,0,\ldots) .
\label{ev1}$$ The initial state of the evolution, $e_0$, is the zeroth lattice state. In oscillator terms, this corresponds to a situation where the first mass is displaced from its equilibrium position, while all the other masses are at rest in their equilibria. In electrical terms, the current (or the voltage) is non-zero only in the first element of the transmission line described by the Jacobi matrix $H$.
The solution of Schrödinger equation can be formally obtained as [@dan; @parl] $$c_n(t) := (e_n, e^{-itH} e_0) = \int
e^{-itx} p_n(x) \; d\mu(x) ,
\label{crux}$$ where $c_n(t)$ is the component of $\psi(t)$ at the $n$-th lattice state. Equation (\[crux\]) shows that this component is the Fourier transform of the orthogonal polynomial $p_n$ with respect to the spectral measure $\mu$. This fact allows us to derive important results.
Firstly, the asymptotic behaviour for small $t$ can be controlled as follows: $|c_n(t)|^2 \sim t^{2n}$. In fact, $$c_n(t) =
\int d\mu(x) \; p_n(x)
\sum_{l=0}^{\infty} \frac{(-it)^l}{l!} x^l =
\sum_{l=n}^{\infty} \frac{(-it)^l}{l!} \int d\mu(x) p_n(x) x^l .
\label{rev10}$$ Because of the orthogonality properties of the set $p_n$ this expansion begins with $l=n$, which proves the result.
Secondly, in the infinite time limit, denoting by $\overline{S}_n (T)$ the time average of $|c_n|^2$ up to time $T$, $$\overline{S}_n(T)
= \frac{1}{2T} \int_{-T}^{T} |c_n|^2 (t) \; dt ,$$ we have that $$\overline{S}_n(T)
\sim T^{-D_2}
\label{decay}$$ for all $n$, a result which involves the correlation dimension $D_2$ of the fractal measure $\mu$. The case with $n=0$ is implicitly contained in Bessis [*et al.*]{} [@turca], and was originally proposed in the present context by Ketzmerick [*et al.*]{} [@geis1]. Successively, it has attracted a lot of attention, mainly from cultors of mathematical rigour. Our generalization has the advantage of requiring a simple proof, via the usage of the Mellin transform, as in [@turca]. In fact, we write $$\overline{S}_n(T) =
\int d\mu(x) \int d\mu(y) \;
\frac{\sin \; (x-y)T}{(x-y)T} \;
p_n(x) p_n(y) .
\label{rev8}$$ To find the asymptotic behaviour of eq. (\[rev8\]), we take the Mellin transform, $M_n(z)$, of $\overline{S}_n(T) $: $$M_n(z) = \int T^{z-1} \overline{S}_n(T) dT = G \times
\int d\mu(x) \int d\mu(y) \;
\frac{p_n(x) p_n(y)}
{|x-y|^{z}} = G(z) \times E_n(z) ,
\label{rev9}$$ where $G(z) = \Gamma(z-1) \sin (\frac{\pi}{2}(z-1))$, and where $E_n(z)$ is defined implicitly by the last equality. The dominating power law in the long time behaviour of $S_n$ is determined by the divergence abscissa of $M_n(z)$: that is to say, $\overline{S}_n(T) \sim T^{-w}$, where $w$ is the largest real $z$ for which $M_n(z)$, hence $E_n(z)$ converges. It is apparent from eq. (\[rev9\]) that the divergence of $E_n$ is piloted by the small scale structure of the measure $\mu$. Because the polynomials $p_n$ are smooth functions, with bounded derivatives on the support of $\mu$, the divergence abscissa of $E_n$ is the same for all $n$, and, in particular, it coincides with that of $E_0$. $E_0(z)$ is known as the [*generalized electrostatic energy*]{} of the measure $\mu$ and its divergence abscissa is known to be $D_2$ [@turca], the correlation dimension of the measure $\mu$.
It is important to remark that the domains of validity of the asymptotic expansions just derived are not uniform in $n$. This adds to the difficulty of the problem to be discussed in the next Section.
Renormalization Theory of Quantum Intermittency
===============================================
An important characteristics of the quantum motion introduced in the previous section is the way it spreads over the $l_2$ lattice basis, $\{e_n\}$. In fact, in oscillator terms, spreading corresponds to energy transmission along the linear chain, be it mechanical or electrical. In quantum mechanical terms, it corresponds to unbounded motion of the lattice particle, of the kind treated only qualitatively by R.A.G.E. theorems. To gauge this phenomenon, we define the moments of the position operator $\hat{n}$: $$\nu_\alpha(t) := (\psi(t), \hat{n}^\alpha
\psi(t)) = \sum_n n^\alpha |c_n(t)|^2.
\label{galas}$$ Their asymptotic behaviour follows a power law, $$\nu_\alpha(t) \sim t^{\alpha \beta},
\label{sca1}$$ where $\beta$ is a non-trivial function of the moment order $\alpha$. In [@igm; @parl] we found that $\beta$ is convex, non-decreasing, and non-constant even in the case of a one-scale Cantor set, characterized by trivial thermodynamics: this is what we call [*quantum intermittency*]{}. Corrections to eq. (\[sca1\]) can also be observed in the form of log-periodic oscillations of $\nu_\alpha(t)$, super-imposed to its leading behavior. They can be explained by the Mellin-type analysis presented in the previous section.
We can estimate the function $\beta(\alpha)$ on the basis of simple renormalization group considerations. For simplicity, let us consider an I.F.S. with $M$ maps, of equal probability $\pi_i=\frac{1}{M}$. Let this I.F.S. be non-overlapping. Then, let $I$ be the smallest interval containing $A$, the I.F.S. attractor, and let $I_l$ be the image of $I$ under the map $\phi_l$. Clearly, $I_l \cap I_m = \emptyset$ if $l \neq m$, and the measure $\mu$ restricted to $I_l$ is a linearly rescaled copy of the original. Then, as a first approximation, we can assume that the orthogonal polynomials of the restricted measure are also obtained by linear rescaling of the original polynomials: $$p_{Mn} (\phi_l(x)) = \sum_{k=0}^{Mn} \Gamma_{l,k}^{Mn} p_k(x)
\simeq \sigma^n_l p_n(x) ,
\label{est1}$$ where $\sigma^n_l = \pm 1$. In other words, we assume a very simple form for the coefficients $\Gamma^{Mn}$, which amounts to making a renormalization ansatz.
Let us now consider $\overline{S}_{Mn}(T)$, as defined above. Because of the balance property (\[bala\]), it can be written $$\overline{S}_{Mn}(T) =
% \frac{1}{M^2}
\sum_{l,m=1}^{M}
\pi_l \pi_m
\int d\mu(x) \int d\mu(y) \;
\frac{\sin T(\phi_l(x)-\phi_m(y))}{T(\phi_l(x)-\phi_m(y))}
p_{Mn}(\phi_l(x)) p_{Mn}(\phi_m(y)) .
\label{est3}$$ In the previous equation, $\phi_l(x)$ and $\phi_m(y)$ belong to $I_l$ and $I_m$, respectively. If $l \neq m$, these intervals are separated by a finite gap. As $T$ tends to infinity, these contributions tend to zero as $T^{-1}$. We can therefore retain only the diagonal terms in eq. (\[est3\]).
If we now employ the approximate estimate (\[est1\]) in the r.h.s. of eq. (\[est3\]) we can write $$\overline{S}_{Mn}(T) =
%\frac{1}{M^2}
\sum_{l=1}^{M}
\pi_l^2
\overline{S}_{n}(\delta_l T).
\label{est5}$$ This too is a sort of renormalization equation which links the wave-function component at site $Mn$ and time $T$ to the component at site $n$ and at shorter times $\delta_l T$. When inserted in eqs. (\[galas\]), eq. (\[est5\]) implies that the growth exponent $\beta$ associated with the averaged moments $\overline{\nu}_\alpha$ via eq. (\[sca1\]) must satisfy the relation $$1 = M^{\alpha-1} \; \sum_{l=1}^{M} \delta_l^{\alpha \beta}.
\label{est9}$$ Comparing this result with eq. (\[multi1\]) we obtain the crucial equation $$\beta (\alpha) = D_{1-\alpha} ,
\label{multap}$$ which links multi-fractal properties and time dynamics. In particular, eq. (\[multap\]) implies that $\beta(0) = D_1$, which is consistent with the rigorous result $\beta(0) \geq D_1$ [@ig1]. Notice that $\beta(0)$ can be defined by a limiting procedure on $\beta(\alpha)$, or by the evolution of the logarithmic moment. We have also $\beta(1) = D_0$.
Because of the rough approximation involved in eq. (\[est1\]), and because for the validity of eq. (\[est5\]) both $c_{Mn}$ and $c_n$ need to be in their asymptotic regimes, we do not expect eq. (\[multap\]) to be always exact. Indeed, in Fig. 4 we have considered a family of I.F.S. measures, characterized by $M=2$, $\delta_2 = \frac{2}{5}$, $\beta_1=0$, $\beta_2=\frac{3}{5}$, $\pi_1=\frac{3}{5}$, and $\pi_2=\frac{2}{5}$. The contraction rate $\delta_1$ is allowed to vary in the range $[.2,.5]$, which implies a significant variation both in the structure of the support of the balanced measure and in its multi-fractal properties. Plotted in Fig. 4 are the scaling exponents $\beta(0)$ and $\beta(1)$, compared with the multi-fractal dimensions $D_1$ and $D_0$, respectively. We observe a substantial agreement between the two data sets, dynamical and multi-fractal. Numerically, the discrepancy is always less than five percent. We can therefore conclude that the relation (\[multap\]) catches some essential part of the physics. Yet, the situation is more complicated, as the following pair of examples show.
The first is a magnificent counter-example. Let us consider a new class of I.F.S. measures (and related Hamiltonians) characterized by $M=2$ and by a particular choice of the weights: $$\label{gold1}
\pi_j = \delta_j^{D}, \;\; j=1,2$$ where $D$ is the (constant) value $\frac{\log 2}{\log 5 - \log 2}$. This choice originates what is called a [*uniform Gibbs measure*]{}. The first of such I.F.S. is that of Figs. 1 to 3, and $D$ is its fractal dimension. Indeed, all I.F.S. with the property (\[gold1\]) are characterized by the same flat thermodynamic function $D_q=D$. Clearly, because of eq. (\[gold1\]), and because $\pi_1+\pi_2=1$, only one parameter among the map weights and contraction rates is left free. By varying this parameter we can construct different I.F.S. measures, with the same flat thermodynamics. What are then the corresponding dynamical exponents $\beta(\alpha)$ ? The approximate relation (\[multap\]) predicts $\beta(\alpha) \simeq D$ for all $\alpha$.
In Fig. 5 we have considered: [**a:**]{} The I.F.S. with $\delta_1=\delta_2=\frac{2}{5}$, $\pi_1=\pi_2=\frac{1}{2}$, which is a “pure” Cantor Set. [**b:**]{} The I.F.S. with $\delta_1=.5090$, $\delta_2=.2978$, and $\pi_1 = \frac{3}{5}$. [**c:**]{} The I.F.S. with $\delta_1=.5293$, $\delta_2=.2802$, and $\pi_1 = .6180$. [**d:**]{} The I.F.S. with $\delta_1=.6033$, $\delta_2=.2196$, and $\pi_1 = .6823$. The first observation we can draw from this figure is that $\beta$ is not flat, as shown in [@parl], even if the [*intermittency range*]{} in the $[0,5]$ interval is very narrow. The second, is that the prediction $\beta = D = .7565$ is correct within two percent at $\alpha=0$ and about five percent at $\alpha=5$. The third, and most important, that the scaling function $\beta$ is roughly invariant from case to case.
These results are intriguing: the coincidence of the curves in Fig. 5 suggests that the spectrum of generalized dimensions $D_q$ must play some rôle in determining $\beta(\alpha)$: the fractal measures [**a**]{} – [**d**]{} seem to have little in common beyond having the same flat thermodynamics. Nevertheless, precisely because in these cases $D_q$ is flat, neither eq. (\[multap\]), nor any general relation of the kind $\beta(\alpha) = D_{q(\alpha)}$, with $q$ an as yet unknown function of $\alpha$ can hold rigorously.
Let us now come to a favourable example: we can construct a class of measures for which the renormalization eq. (\[est1\]) is [*exact*]{}: these are the equilibrium measures of the Julia sets generated by the polynomials $$\label{ju1}
P(z) = z^2 - \lambda,$$ where $\lambda \geq 2$ is a real constant. As we have already remarked, the Jacobi matrices for these problems can be constructed by a stable recursion algorithm [@danbel], [@barn]. Non-linearity of the I.F.S. maps stemming from eq. (\[ju1\]) as inverse branches of $P(z)$ can be treated by considering sufficiently high iterations $P^{(l)}$, and a theory perfectly analogous to (\[sca1\]-\[multap\]) can be carried out, with the same result.
In fig. 6 we make the usual comparison between the moment scaling function, $\beta(\alpha)$, and thermodynamics [@turcb], $D_{1-\alpha}$: the curves coincide within numerical precision! Therefore, one can conclude that discrepancies from eq. (\[multap\]) are due to the non-exactness of eq. (\[est1\]) when a spectral measure is approximated by I.F.S., except for the case of Julia measures, which are known to have strong algebraic properties.
Incidentally, we note that for Julia sets, the invariant measure coincides with the measure of the asymptotic distribution of the zeros of the associated orthogonal polynomials, the latter being also the physicists’ global density of states. Might it be that the correct quantity entering eq. (\[multap\]) is this second measure ? The analysis of the I.F.S. data presented here seems to exclude this case, although we cannot exclude that this rôle is played by yet another spectral measure still to be determined.
Conclusions
===========
We have presented a stable algorithm for the determination of lattice Hamiltonian operators possessing a given spectral measure, in the class of linear I.F.S. This algorithm consists of a recursive determination of the associated Jacobi matrix, in the framework of the theory of orthogonal polynomials.
The Hamiltonian operators determined in this way are characterized by almost periodic coefficients: since I.F.S. measures approximate arbitrary well any measure supported on a Cantor set, this fact might lead to a proof that almost periodicity is always associated with this kind of spectra.
In a quantum mechanical context, the Jacobi matrices studied here can be employed as models of almost-periodic systems: the dynamical properties of such systems can be studied in their essence, having extracted the crucial information on the related spectral measures. We have shown that connections between spectral properties and dynamics go far beyond the conventional RAGE theorems: in particular, delocalization of particle’s position along the lattice basis can be described by a scaling function $\beta$ governing the moments of order $\alpha$ of the position operator. Non-constancy of this function translates mathematically the phenomenon of quantum interference.
We have derived an intriguing relation, $\beta(\alpha) = D_{1-\alpha}$, linking dynamics and the thermodynamical properties of the spectral measure: considering the Jacobi matrices associated with Julia sets we have constructed a family of quantum systems for which the relation is exact, and we have discussed the reasons for the discrepancies present in the general case. We believe that a further refinement of the results presented in this paper will lead to a profound understanding of the mathematical and physical properties of almost-periodic quantum systems.
Finally, we remark that the Jacobi Hamiltonians considered in this paper are not simple exotic curiosities, but can also describe time-resolved energy absorption in externally perturbed quantum systems, as well as electron dynamics in solid-state eterostructures like super-lattices [@sup1], where by varying an alloy concentration along a deposition axis different spectral structures can be found [@sup2]. Here, our results may become relevant in several problems, like –for instance– the design of lasers and radiation detectors.
Fig. 1.\
Orthogonal polynomial $p_8(x)$ of the I.F.S. measure with maps $(\delta_i,\beta_i,\pi_i)$ $=$ $(\frac{2}{5},0,\frac{1}{2})$, $(\frac{2}{5},\frac{3}{5},\frac{1}{2})$, with a finite-resolution representation of the support of the measure obtained by plotting a large number of points on the attractor. Because of the finite size of points, this latter appears as a sequence of dashes. Only the symmetrical half is shown. In the inset, the beginning of the sequence of $r_n$. The vertical scale ranges from zero to $\frac{1}{2}$. Lines are merely to guide the eye.
Fig. 2.\
Discrete Fourier transform of the $r_n$ sequence ($n=1,\ldots,2^{13}$) associated with the I.F.S. of Fig. 1. The constant and $\pi$ frequencies exceed the vertical scale, and are not reported.
Fig. 3.\
Plot of the phase $\Phi$ of the discrete Fourier transform of the sequence $r_n$ associated with the I.F.S. of Fig. 1 and 2, to show the $\pi$ discontinuity close to the value of the main peak of Fig. 2.
Fig. 4.\
Multi-fractal dimensions $D_0$ (full diamonds) and $D_1$ (full squares) and dynamical exponents $\beta(0)$ (open squares) and $\beta(1)$ (open diamonds) versus contraction rate $\delta_1$, for the family of I.F.S. described in the text.
Fig. 5.\
Scaling functions $\beta(\alpha)$ for the four I.F.S.’s [**a**]{} - [**d**]{} described in the text: [**a**]{}: circles; [**b**]{}: squares; [**c**]{}: triangles; [**d**]{}: diamonds.
Fig. 6.\
Scaling function $\beta(\alpha)$ for the Julia set measure with $\lambda = 2.2$ (diamonds) and thermodynamical dimensions $D_{1-\alpha}$ (crosses).
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---
abstract: 'We consider vacuum solutions of four dimensional general relativity with $\Lambda < 0$. We numerically construct stationary solutions that asymptotically approach a boundary metric with differential rotation. Smooth solutions only exist up to a critical rotation. We thus argue that increasing the differential rotation by a finite amount will cause the curvature to grow without bound. This holds for both zero and nonzero temperature, and both compact and noncompact boundaries. However, the boundary metric always develops an ergoregion before reaching the critical rotation, which probably means that the energy is unbounded from below for these counterexamples to cosmic censorship.'
author:
- Toby Crisford
- 'Gary T. Horowitz'
- 'Jorge E. Santos'
bibliography:
- 'all.bib'
title: |
Attempts at vacuum counterexamples to\
cosmic censorship in AdS
---
Introduction and Summary
========================
Cosmic censorship [@Penrose:1969pc] remains one of the most important open problems in classical general relativity. We will be interested in the weak form of this conjecture which roughly says that one cannot generically form regions of arbitrarily large curvature that are visible to infinity. Although it is usually discussed in the context of four-dimensional asymptotically flat spacetimes, there has been recent interest in higher dimensions and other boundary conditions. We will stay in four-dimensions, but consider asymptotically anti-de Sitter (AdS) spacetimes. This is motivated by gauge/gravity duality which relates gravity with this boundary condition to a nongravitational gauge theory [@Aharony:1999ti].
In this context, a class of counterexamples to cosmic censorship has recently been found if one adds a Maxwell field [@Horowitz:2016ezu; @Crisford:2017zpi]. In asymptotically AdS spacetimes, one is free to specify a boundary (conformal) metric and boundary values of any matter fields that might be present. If one assumes certain static profiles for the asymptotic vector potential and multiplies them by an overall amplitude $a$, it turns out that static, nonsingular, $T=0$ solutions only exist up to a maximum amplitude $a_{\max}$ (which depends on the profile). These are static self-gravitating electric fields which become singular as $ a\to a_{\max}$. It was then shown [@Crisford:2017zpi] that if one considers a time dependent boundary condition where the amplitude starts with $a<a_{\max}$ and ends with $a>a_{\max}$, the electric field and hence the curvature grow as a power law in time over a large region visible from infinity. Although the singularity does not form in finite time, these examples clearly violate the spirit of cosmic censorship.
In this paper, we attempt to construct vacuum analogs of these counterexamples. Instead of adding a Maxwell field, we will add differential rotation to the boundary metric and construct smooth stationary solutions which approach this asymptotic geometry. A similar setup was studied in [@Markeviciute:2017jcp], where a dipolar differential rotation was added at the conformal boundary using global coordinates. We will show that if one keeps the profile of the differential rotation fixed, but increases the overall amplitude, smooth solutions only exist up to a finite maximum amplitude $a_{\max}$. As before, we expect that in the time dependent case where the amplitude is increased from $a<a_{\max}$ to $a>a_{\max}$, the curvature will again increase without bound.
We consider both $T = 0$ and $T>0$ solutions, and boundaries that are both compact and noncompact. In all cases the results are qualitatively the same. There is a finite amplitude, $a_{\max}$, beyond which smooth stationary solutions do not exist. However, before reaching $a_{\max}$, both the boundary metric and bulk spacetime develop an ergoregion, *i.e.*, a region of spacetime where the time translation Killing vector becomes spacelike[^1]. If $a_{{\rm ergo}}$ denotes the amplitude at which the ergoregion first forms, we will see that $a_{\max} -a_{{\rm ergo}}$ can be made as small as one likes by varying the profile or temperature, but it is always positive. The existence of an ergoregion causes two problems which we now discuss.
First, spacetimes with ergoregions in AdS may be unstable due to superradiant scattering. This is known to happen when the ergoregion surrounds a spherical black hole. Certain modes can scatter off the black hole and return with greater amplitude. They then reflect off infinity and scatter off the black hole repeatedly, leading to an instability. The endpoint of this instability is not known although there has been some remarkable recent numerical progress [@Chesler:2018txn]. It may in fact violate weak cosmic censorship in vacuum [@Dias:2015rxy; @Niehoff:2015oga]. In our case, the ergoregion is in the asymptotic region and it is not clear if a superradiant instability exists, since ingoing waves are partially absorbed by the horizon and return with smaller amplitude. This can compensate for the enhanced scattering off the ergoregion. However, given the results in [@Green:2015kur], it is likely that our solutions are also unstable.
A more serious problem is that the energy is likely to be unbounded from below. The existing proofs of positive energy in AdS [@Gibbons:1982jg; @Cheng:2005wk; @Xie:2007qp; @Chrusciel:2018lpj] do not apply to boundary metrics with ergoregions. We will discuss this in section V and give arguments that the energy is probably not bounded from below. Thus these vacuum counterexamples to cosmic censorship are less interesting than the electromagnetic counterexamples, and do not have the same status.
This is good news for the suggested connection between cosmic censorship and the weak gravity conjecture [@ArkaniHamed:2006dz]. For the electromagnetic counterexamples, it was shown that adding a charged scalar field with mass $m$ and charge $q$ causes the Einstein-Maxwell solutions that violate cosmic censorship to become unstable if $q/m$ is large enough [@Crisford:2017gsb]. Furthermore, the instability results in a nonzero scalar field and one can no longer violate cosmic censorship. Surprisingly, the minimum value of $q/m$ to preserve cosmic censorship turns out to be precisely that predicted by the weak gravity conjecture adapted to AdS [@Crisford:2017gsb]. Since our current vacuum counterexamples to cosmic censorship probably have a similar effect on the geometry but do not involve any electromagnetic fields, they could not be removed by invoking the weak gravity conjecture.
Our stationary solutions with $T > 0$ have a standard black hole horizon in the interior. But the infrared behavior of the $T=0$ solutions depends on the fall-off of the differential rotation. If it falls-off faster than $1/r$, the effects of the rotation die off as one moves into the bulk and the solution has a standard Poincaré horizon. If it falls off like $1/r$, there is a new extremal horizon which we will describe explicitly. If the profile is exactly $1/r$, the solution has an additional scaling symmetry and can be written analytically. We call this the “spinning top" solution since the angular momentum density is concentrated at the origin. It can be viewed as the vacuum analog of the analytic “point charge" solution found in [@Horowitz:2014gva].
In many cases, the boundary metrics and bulk black holes that we construct will be axisymmetric as well as stationary. This raises the possibility that there may be a nonaxisymmetric, stationary black holes with $a > a_{\max}$. This is not possible in asymptotically flat spacetimes, but might occur in AdS [@Dias:2015rxy]. To check this, we will study some cases where the only symmetry of the boundary metric is the stationary Killing field. We again find there is a maximum amplitude for smooth solutions.
In the next section we briefly review how to numerically construct stationary vacuum solutions. Sections III and IV give some further details on the construction and contain our main results for stationary solutions with rotating planar (III) or compact (IV) boundary metrics. We show there is a maximum amplitude and describe some properties of the solutions. In the last section we give arguments that the energy is unbounded from below, and discuss some implications for the dual field theory.
\[sec:turck\]Constructing general rotating defects
==================================================
We will be interested in finding stationary, asymptotically AdS solutions of Einstein’s equation: $$R_{ab}+\frac{3}{L^2}g_{ab}=0\,,
\label{eq:einstein}$$ where $L$ is the AdS length scale and $R_{ab}$ the four-dimensional Ricci tensor associated with the metric $g_{ab}$. Throughout this manuscript we work with $G_4=1$.
In order to find solutions to numerically we will use the so called DeTurck method, which was first presented in [@Headrick:2009pv] and reviewed in great detail in [@Wiseman:2011by; @Dias:2015nua]. The idea is to consider the following modification of Eq (\[eq:einstein\]) $$R_{ab}+\frac{3}{L^2}g_{ab}-\nabla_{(a}\xi_{b)}=0\,,
\label{eq:einsteindeturck}$$ where $\xi^a = [\Gamma(g)^{a}_{bc}-\Gamma(\bar{g})^{a}_{bc}]g^{bc}$ is the so called DeTurck vector, $\Gamma(\mathfrak{g})$ is the Levi-Civita connection associated to a metric $\mathfrak{g}$ and $\bar{g}$ is a reference metric which will be related to our choice of gauge. In terms of spacetime coordinates, we have $\xi^a = -\Box x^a+H^a$, where $H^a \equiv -\Gamma(\bar{g})^{a}_{bc}g^{bc}$ does not explicitly depend on derivatives of $g$. Solutions of the Einstein equation (\[eq:einstein\]) will be solutions of the Einstein-DeTurck equation (\[eq:einsteindeturck\]) with $\bar{g}=g$, however, the converse might not always be true. That is to say, it is not clear whether solutions of (\[eq:einsteindeturck\]) will necessarily be solutions of (\[eq:einstein\]), *i.e.* solutions with $\xi^a\neq0$ might exist.
However, most of the boundary metrics we will consider are stationary, axially symmetric and have a $(t,\phi)\to-(t,\phi)$ reflection symmetry, all of which extend into the bulk. Under these symmetries, Figueras and Wiseman have shown in [@Figueras:2016nmo] that $\xi$ must vanish on solutions of (\[eq:einsteindeturck\]). The advantage of solving Eq. (\[eq:einsteindeturck\]) instead of Eq. (\[eq:einstein\]) is immense, since the former represents a system of elliptic equations which can be readily solved using a standard relaxation procedure. The gauge, which is dynamically determined during the numerical procedure, is given by $\xi=0\Rightarrow \Box x^a = H^a$.
Planar solutions
================
In this section we discuss stationary, axisymmetric solutions to with boundary metrics of the form $$\dd s^2_\partial = -\dd t^2+\dd r^2+r^2 [\dd\phi - \omega(r) \dd t]^2\;,
\label{eq:bndmetric}$$ with (r) = ap(r). These metrics describe geometries with differential rotation with an amplitude $a$ and profile $p(r)$. We will demand that $p(r) \to 0$ as $r\to \infty$.
Zero temperature solutions
--------------------------
We start by considering solutions at zero temperature. It is clear from that $\omega(r)$ must have (mass) dimension one. So if it falls off like $a/r^n$, the dimension of $a$ must be $1-n$. Thus for $n > 1$, turning on $a$ represents an irrelevant deformation of the boundary metric and the solution should have a standard Poincaré horizon. We will see below that this is indeed the case. For $n < 1$, turning on $a$ is a relevant deformation and the solution will be very different in the infrared. For $n =1$, $a$ is dimensionless and corresponds to a marginal deformation. In this case, the extremal horizon is deformed in a way that we now describe.
### \[sec:spinning\]Holographic spinning top
When $\omega(r) = a/r$ everywhere, the boundary metric has a new scaling symmetry: $t \to \lambda t, \ r\to \lambda r$. In fact, the conformal metric is invariant under an $SO(2,1)\times SO(2)$ subgroup of the full $SO(3,2)$ conformal group of flat space. The corresponding bulk solution also has this extended symmetry and can be described analytically. In fact, it can be obtained by a double Wick rotation of a hyperbolic Taub-Nut black hole in AdS$_4$ [@Chamblin:1998pz], which is an algebraically special solution in the Petrov classification.
The resulting solution can be written $$\label{eq:taubnut}
\mathrm{d}s^2=\frac{L^2}{\eta^2}\left[H(\eta)\left(-\rho^2\mathrm{d}t^2+\frac{\mathrm{d}\rho^2}{\rho^2}\right)+\frac{y_+^2\,H(\eta)\mathrm{d}\eta^2}{(1-\eta)G(\eta)}+\frac{G(\eta)(1-\eta)}{H(\eta)}(\mathrm{d}\phi-2n\rho\mathrm{d}t)^2\right]$$ with $$G(\eta)=n^2 \left(3 n^2-1\right) \eta^3+\left(6 n^2-1\right) \eta^2 y_+^2+\left(1+\eta+\eta^2\right) y_+^4\,,\quad\text{and}\quad H(\eta)=y_+^2+n^2\,\eta^2\,.$$ Here, $\eta\in(0,1]$ with $\eta = 0$ being the asymptotic boundary and $\eta =1$ is the axis of rotation for $\partial_\phi$. The first term in parenthesis on the right is AdS$_2$ with a horizon at $\rho=0$. This null surface defines a degenerate horizon for the full four dimensional spacetime, so the solution has zero temperature. The solution depends on two parameters, $y_+$ and $n$, and for generic values of them, there is a conical singularity along the rotation axis $\eta =1$. This can be readily avoided by demanding $$n=\frac{\sqrt{(1-\epsilon\, y_+) (1+3 \epsilon \,y_+)}}{\sqrt{3}}
\label{eq:nspecial}$$ with $\epsilon^2=1$. For $\epsilon=-1$, we need to restrict $y_+\leq1/3$, but this implies that $G(\eta)$ would have an additional root smaller than unity. We are thus left with $\epsilon=1$ to avoid any conical singularities, which also means $0<y_+\leq1$. The solution with $y_+=1$ has $n=0, \ G(\eta) = 1+\eta$, and corresponds to pure AdS$_4$.
Following [@deHaro:2000vlm] one can compute the resulting holographic stress energy tensor analytically, and fix the conformal frame by demanding that the boundary metric has fixed $g^{tt}_{\partial}=-1$. To accomplish this, we first change to Fefferman-Graham coordinates via the following asymptotic expansion $$\begin{aligned}
&\eta =\frac{y_+ z}{r} \left[1-\frac{\left(2-5 n^2\right)}{2}\frac{z^2}{r^2}+\frac{\left(1-y_+\right) \left(1-12 y_+^2\right)}{9}\frac{z^3}{r^3}+\mathcal{O}(z^4)\right]\,,
\\
&\rho = \frac{1}{r}\left[1-\frac{1}{2}\frac{z^2}{r^2}+\frac{3 \left(1-n^2\right)}{8}\frac{z^4}{r^4}+\mathcal{O}(z^5)\right]\,,\end{aligned}$$ which brings the metric into the following asymptotic form $$\mathrm{d}s^2=\frac{L^2}{z^2}\left[\mathrm{d}z^2 + \mathrm{d}s^2_\partial +z^2 \mathrm{d}s^2_2+z^3\mathrm{d}s^2_3+\mathcal{O}(z^4)\right]\,.$$ In the above expression we have
$$\mathrm{d}s^2_\partial=-\dd t^2+\dd r^2+r^2 \left(\dd\phi - \frac{2\,n}{r} \dd t\right)^2
\label{eq:profilespecial}$$
and $$\mathrm{d}s^2_2=-\frac{\left(4-15 n^2\right) \mathrm{d}t^2}{10 r^2}-\frac{3 n^2}{2 r^2}\mathrm{d}r^2+\frac{5 n^2}{2} \left[\mathrm{d}\phi-\frac{2\left(1-5 n^2\right)}{5 n r}\mathrm{d}t\right]^2\,.$$
Eq. (\[eq:profilespecial\]) allows us to identify $\omega(r)=2n/r$ for this particular profile, and thus $a=2\,n$, with $n$ given by Eq. (\[eq:nspecial\]). Since $0<y_+\leq1$, $a$ has a maximum value at $y_+ = 1/3$ which corresponds to $a_{\max} = 4/3$. For $a=a_{\max}$, the solution exhibits no bulk curvature singularity, but we believe this is due to some special fine tuning induced by this very special profile. It is significant that $a_{\max} > 1$. For $a=1$, $\partial_t$ is null everywhere on the boundary and for $a>1$, it is spacelike. Thus the bulk solution develops an ergoregion before reaching $a_{\max}$. $a=1$ is reached when $y_+=1/3+\sqrt{7}/6$, corresponding to $n=1/2$.
As $y_+$ increases from zero to one, $a$ does not change monotonically. In Fig. \[fig:special\] we plot $a$ as a function of $y_+$ and mark both the onset of the ergoregion, $a=1$, and $a=a_{\max}$.
![$a$ as a function of $y_+$: the horizontal red dashed like corresponds to $a=a_{\max}=4/3$ and the vertical blue dotted line to $a=1$, beyond which $g_{tt}$ becomes everywhere spacelike at the boundary.[]{data-label="fig:special"}](graphics/a1overr.pdf){width="45.00000%"}
The holographic stress energy tensor is given in terms of $\mathrm{d}s^2_3$ by $$\langle T_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu\rangle =\frac{3}{16 \pi}\mathrm{d}s^2_3= -\frac{\left(1-y_+\right) \left(1-12 y_+^2\right)}{24\pi r^3}\left[-\mathrm{d}t^2+\mathrm{d}r^2-2r^2\left(\mathrm{d}\phi-\frac{2\,n}{r}\mathrm{d}t\right)^2\right]\,,
\label{eq:stressspinning}$$ where Greek indices run over the boundary spacetime directions. It is easy to see that the angular momentum density, which is proportional to ${{T^t}_\phi}$, vanishes for all $r > 0$, so it might at first seem that the total angular momentum will be zero. However, in deriving \[eq:stressspinning\] we have completely neglected the fact that $g_{t\phi}$ is singular at $r=0$. If we were to take that into account, \[eq:stressspinning\] would have a $\delta(r)$ contribution to the angular momentum density. Instead of keeping track of this contribution, we use the methods developed in [@Magnon:1985sc] to compute the total angular momentum. It turns out to be given by: $$J = -\frac{\left(1-3 y_+\right) \sqrt{\left(2-3 y_+\right) y_++1}}{4 \sqrt{3}}\,.
\label{eq:momentum}$$ Note that $J\geq0$ only for $y_+\in[1/3,1]$ and becomes negative for smaller $y_+$. This change in sign occurs precisely at $a_{\max}$. (See Fig. \[fig:angular\] for a plot of $J$ as a function of $a$.) At present we have no understanding of why this is the case.
![$J$ as a function of $a$, with $a=a_{\max}=4/3$ corresponding to a change of sign in $J$.[]{data-label="fig:angular"}](graphics/J_holographic_spinning.pdf){width="45.00000%"}
This dependence of $J$ on the amplitude turns out to be independent of the details of the differential rotation $\omega(r)$, and only depends on the fact that $\omega = a/r$ asymptotically. We will recover Fig. \[fig:angular\] (for $J>0$ ) when we construct zero-temperature solutions whose profile decays at large $r$ as $1/r$, but is regular as $r\to0$. In fact, the entire solution is universal in the sense that it provides the near horizon geometry for this class of profiles.
### \[sec:irr\]Metric *ansatz*
We first discuss the metric ansatz for profiles that fall off faster than $1/r$ asymptotically, which correspond to irrelevant deformations of the boundary. These solutions should have standard Poincaré horizons in the IR. We will use the coordinates first described in [@Horowitz:2014gva] and recently used in [@Crisford:2017zpi; @Crisford:2017gsb]. We start with AdS written in Poincaré coordinates $$\mathrm{d}s^2=\frac{L^2}{z^2}\left(-\mathrm{d}t^2+\mathrm{d}z^2+\mathrm{d}r^2+r^2\mathrm{d}\phi^2\right)\,,
\label{eq:poincare}$$ where $z\in[0,+\infty)$, with $z=0$ marking the location of the conformal boundary, and $z=+\infty$ the Poincaré horizon. Furthermore, $(r,\phi)$ are standard polar coordinates in $\mathbb{R}^2$, with $r=0$ marking the axis of rotation. We now introduce two new coordinates $(\tilde{x},\tilde{y})$ which compactify both $r$ and $z$ in the following form $$(z,r)=\frac{\tilde{y}\sqrt{2-\tilde{y}^2}}{1-\tilde{y}^2}(1-\tilde{x}^2,\tilde{x}\sqrt{2-\tilde{x}^2})\,,$$ with $(\tilde{x},\tilde{y})\in[0,1]^2$. In terms of the $(t,\tilde{x},\tilde{y},\phi)$ coordinates, the metric on Poincaré AdS reads $$\mathrm{d}s^2=\frac{L^2}{(1-\tilde{x}^2)}\left[-\frac{\left(1-\tilde{y}^2\right)^2}{\tilde{y}^2 \left(2-\tilde{y}^2\right)}\mathrm{d}t^2+\frac{4 \mathrm{d}\tilde{y}^2}{\tilde{y}^2 \left(1-\tilde{y}^2\right)^2 \left(2-\tilde{y}^2\right)^2}+\frac{4 \mathrm{d}\tilde{x}^2}{2-\tilde{x}^2}+\tilde{x}^2(2-\tilde{x}^2)\mathrm{d}\phi^2\right]\,.$$ The Poincaré horizon is now at $\tilde{y}=1$, where the above metric reveals an AdS$_2$ like throat characteristic of zero temperature horizons. The boundary is located at $\tilde{x}=1$ and $\tilde{y}=0$ ($\tilde{y}=0$ just corresponds to the point $z = r= 0$), and $\tilde{x}=0$ marks the axis of rotation. We note that, at the boundary, the relation between $\tilde{y}$ and $r$, reduces to $$r=\frac{\tilde{y}\sqrt{2-\tilde{y}^2}}{1-\tilde{y}^2}\,.
\label{eq:rboundary}$$
In order to use the DeTurck method we need to write down the most general line element compatible with our symmetries. Recall that our boundary metric has two commuting Killing fields, $\partial_t$ and $\partial_\phi$ and an additional discrete symmetry $(t,\phi)\to-(t,\phi)$. We assume that these symmetries extend smoothly into the bulk. The most general line element compatible with general diffeomorphisms along the $(\tilde{x},\tilde{y})$ directions takes the following form $$\begin{gathered}
\mathrm{d}s^2=\frac{L^2}{(1-\tilde{x}^2)^2}\Bigg[-\frac{\left(1-\tilde{y}^2\right)^2}{\tilde{y}^2 \left(2-\tilde{y}^2\right)}\,q_1\mathrm{d}t^2+\frac{4\,q_2\mathrm{d}\tilde{y}^2}{\tilde{y}^2 \left(1-\tilde{y}^2\right)^2 \left(2-\tilde{y}^2\right)^2}\\+\frac{4\,q_4}{2-\tilde{x}^2}\left(\mathrm{d}\tilde{x}+\frac{q_3}{1-\tilde{y}^2}\mathrm{d}\tilde{y}\right)^2+\tilde{x}^2(2-\tilde{x}^2)\,q_5(\mathrm{d}\phi-q_6 \mathrm{d}t)^2\Bigg]\,,
\label{eq:ansatz}\end{gathered}$$ where $q_i$, with $i\in\{1,\ldots,6\}$, are the functions of $(\tilde{x},\tilde{y})$ we wish to determine.
Four our reference metric we take $$q_1=q_2=q_4=q_5=1\,,\quad q_3 = 0\,,\quad \text{and}\quad q_6=g(\tilde{y})\,,
\label{eq:ref0}$$ where $g(\tilde{y})$ will control our chosen boundary profile for the differential rotation.
The boundary conditions are now easily obtained by requiring regularity at $\tilde{x}=0$, which in turn implies \_ q\_1=\_ q\_2= q\_3=\_ q\_4=\_ q\_5=\_ q\_6=0,q\_4=q\_5. At the conformal boundary, that is to say, at $\tilde{y}=0$ and $\tilde{x}=1$ we demand q\_1=q\_2=q\_4=q\_5=1,q\_3 = 0,q\_6=g(), and finally, since we expect these solutions to have a standard Poincaré horizon, at $\tilde{y}=1$ we have q\_1=q\_2=q\_4=q\_5=1,q\_3 = 0,q\_6=0. Note that consistency of our boundary conditions imposes $g(1)=0$.
The case of marginal deformations ($\omega \sim 1/r$) is different, since we expect the IR to be deformed away from pure AdS into the family of exact solutions discussed in section \[sec:spinning\]. The metric *ansatz* remains as in Eq. \[eq:ansatz\] except we set $q_6 = (1-\tilde{y}^2) \hat{q}_6$, and express all boundary conditions in terms of $\hat{q}_6$. Note that this means at the boundary (both at $\tilde{x}=1$ and $\tilde{y}=0$) we want $\hat{q}_6=a$, and that at the symmetry axis we still have $\partial_{\tilde{x}} \hat{q}_6=0$. The only significant change comes at the would be Poincaré horizon, where we impose $$q_1=q_2\,,\quad \partial_{\tilde{y}} q_1=\partial_{\tilde{y}} q_2=\partial_{\tilde{y}} q_4=\partial_{\tilde{y}} q_5=\partial_{\tilde{y}} \hat{q}_6=q_3=0\,,
\label{eq:bcsmarginal}$$ which are enforced via regularity in ingoing Eddington-Finkelstein coordinates.
### Results {#sec:resultsT0}
We will first focus on the following class of profiles for the differential rotation: $$\omega(r)=\frac{A}{\displaystyle\left(1+\frac{r^2}{\sigma^2}\right)^{n/2}}\,,
\label{eq:pro1}$$ where $A$ is the boundary profile amplitude, $\sigma$ is a length scale, and $n$ is a positive integer[^2]. At large $r$, these profiles decay like $1/r^n$. Since the boundary metric is only determined up to conformal rescalings, we can always rescale $A$ and $\sigma$ such that the only meaningful quantity is $a\equiv A\sigma$. We will fix $\sigma=1$ in the numerics, and so $A=a$. In terms of the $\tilde{y}$ coordinate (\[eq:rboundary\]), we have $$\omega(r)= \frac{a}{(1+r^2)^{n/2}} = g(\tilde{y})=a\,(1-\tilde{y}^2)^n\,.
\label{eq:pro1a}$$
We start with results for $n>1$. (The special case $n=1$ will be discussed shortly.) For each fixed value of $n$, we construct the solutions numerically by increasing $a$ starting with $a=0$. In all cases we find a critical value $a=a_{\max}$, at which the solution becomes singular. This maximum amplitude always lies past the point where an ergoregion develops on the boundary, $a_{{\rm ergo}}$. For $2 \leq n \leq 8$, this is shown in Fig. \[fig:amaxpro1\] where we have plotted both $a=a_{\max}$ (represented by the blue disks with error bars[^3]) and $a_{{\rm ergo}}$ (represented by the red squares) for various profiles. All of these solutions have a standard Poincaré horizon as expected.
![$a_{\max}$ (represented by the blue disks) and $a_{ergo}$ (represented by the red squares) as a function of $n$. The error bars in determining $a_{\max}$ are computed via the failure of our code to find solutions for the upper range of $a$.[]{data-label="fig:amaxpro1"}](graphics/amax.pdf){width="45.00000%"}
For the above class of profiles, $a_{{\rm ergo}}$ is given by a\_[[ergo]{}]{}= ()\^. For $a> a_{{\rm ergo}}$, the ergoregion is an annular region around the origin on the boundary, and extends into the bulk. For $a= a_{{\rm ergo}}$ the ergoregion collapses to a single circle and has been called an evanescent ergoregion [@Gibbons:2013tqa]. It does not extend into the bulk. Note that both $a_{\max}$ and $a_{{\rm ergo}}$ increase with $n$. So when the differential rotation on the boundary falls off faster, solutions exist for a larger amplitude. It is also clear from Fig. \[fig:amaxpro1\] that the difference $a_{\max}-a_{{\rm ergo}}$ increases with $n$.
To show the formation of a singularity as we approach $a_{\max}$, we monitor the square of the Weyl tensor $C_{abcd}C^{abcd}$ throughout spacetime. Let \[eq:cmax\] C\_|C\_[abcd]{}C\^[abcd]{}|, where $\mathcal{M}$ denotes our spacetime manifold. In Fig. \[fig:C2max\] we plot $C_{\max}$ as a function of $a$ for $n=8$. The rapid growth as $a\to a_{\max}$ is clearly visible. To gain more information about where the singularity appears, in Fig. \[fig:weyl\] we plot $C_{abcd}C^{abcd}$ for $a=a_{\max}$ and $n=2$ (left panel) and $n=8$ (right panel). Since the rotation axis corresponds to $\tilde{x}=0$, it is clear that the large curvature is occurring away from this axis. One might wonder if it always occurs inside the ergoregion. To check this, we have denoted the boundary of the ergoregion by a solid black line in Fig. \[fig:weyl\]. It is clear that the maximum curvature is not always inside the ergoregion.
![Maximum value of $C_{abcd}C^{abcd}$ over spacetime, computed with $n=8$, as a function of $a$. []{data-label="fig:C2max"}](graphics/max_c2.pdf){width="45.00000%"}
![The Weyl tensor squared computed with $a=a_{\max}$ for $n=2$ (left panel) and $n=8$ (right panel). Note that the maximum curvature does not appear on the axis ($\tilde{x}=0$). The ergoregion lies inside the black line, and does not always contain the maximum curvature. (In these coordinates, the asymptotic boundary is $\tilde{x} =1$.)[]{data-label="fig:weyl"}](graphics/weyl.pdf){width="90.00000%"}
We next investigate physical quantities like the energy density $\rho$ and angular momentum density $j$. These are defined in terms of the holographic stress tensor by
$$\rho \equiv -\langle T^{t}_{\phantom{t}t}\rangle\,,$$
and $$j \equiv \langle T^{t}_{\phantom{t}\phi}\rangle\,.$$ \[eq:thermo\]
Within our symmetry class, the holographic stress energy tensor has four non-zero components. In addition, it should be traceless and conserved, which gives two constraints amongst these four components. Thus, the full stress energy tensor is determined by $\rho$ and $j$.
Two important questions are whether $\rho$ and $j$ change their behavior qualitatively after the ergoregion forms on the boundary, and whether they diverge as we approach $a_{\max}$. To check this, we computed the holographic stress energy tensor using [@deHaro:2000vlm], and following *mutatis mutandis* section \[eq:holeplanar\]. We find that the answer to both questions is no: the formation of the ergoregion does not dramatically affect these quantities and they appear to remain finite. This is illustrated in Fig. \[fig:holo\] for $n=2$ and several values of $a$, including $a = a_{\max}$ (most right column). Note that the scales on the vertical axis are different in the six plots, and the maximum values of $|\rho|$ and $|j|$ tend to increase with $a$. Curiously, although $\rho $ remains finite, for $a=a_{\max}$ it reaches a maximum precisely at the edge of the ergoregion. We see this happening for all values of $n$, and for different profiles. Note that even though we have imposed a differential rotation $\omega(r)$ that is positive everywhere, the induced angular momentum density $j$ takes both positive and negative values. In fact, the total angular momentum, $J$, in the spacetime turns out to be exactly zero (to machine precision). This is directly analogous to what was found in [@Horowitz:2014gva] where static solutions of Einstein-Maxwell were discussed. There it was shown that a localized positive chemical potential that falls off faster than $1/r$ produces regions of both positive and negative charge density, but the total charge remains exactly zero.
![The holographic energy density (top row) and holographic angular momentum density (bottom row) computed with $n=2$. The dashed horizontal line marks $0$, and the blue region indicates the location of the boundary ergoregion. From left to right, in each of the rows, we have $a=0.9,2.1,3$. (The last value corresponds to $a_{\max}$.)[]{data-label="fig:holo"}](graphics/holographic_stress_energy_tensor.pdf){width="90.00000%"}
The total energy is plotted in Fig. \[fig:energy\] for $n=8$. We note that the energy is always positive even after the formation of the boundary ergoregion, denoted by the vertical dashed line *i.e.* $a=a_{{\rm ergo}}$. The behaviour of the energy as we approach $a_{\max}$ is puzzling to us, *i.e.* we do not understand why it is not monotonic with increasing $a$.
![Total energy $E$ as a function of $a$, for $n=8$. The verticle dashed line corresponds to $a=a_{{\rm ergo}}$ and the curve ends at $a_{\max}$.[]{data-label="fig:energy"}](graphics/energy.pdf){width="45.00000%"}
It is clear from Fig. (\[fig:amaxpro1\]) that the difference $a_{\max} - a_{{\rm ergo}}$ depends on the profile for the differential rotation. It turns out that one can make this difference arbitrarily small by a judicious choice of profile[^4]. One way to do this is to choose a profile that is very sharply peaked at the origin. Consider $$\omega(r)=\frac{a}{\displaystyle\left(b^2+{r^2}\right)^{1/2}\left(1+{r^2}\right)^{1/2}}
\label{eq:pro2}$$ We have added a new parameter $b$, which controls the height and thickness of the profile around $r=0$. In terms of the coordinate (\[eq:rboundary\]), this profile reads $$\omega(r)=a\,\frac{(1-\tilde{y}^2)^2}{\left[b^2(1-\tilde{y}^2)^2+\tilde{y}^2(2-\tilde{y}^2)\right]^{1/2}}
\label{eq:pro2a}$$
To see the effect of $b$, we set $a=1$ and decrease $b$. Then $g_{tt} = -1 + r^2 \omega^2(r) < 0 $ everywhere for $b>0$, but vanishes at $r=0$ when $b=0$. So the ergoregion first forms at the origin in this case. Note that for $b=0$, the profile looks like $a/r$ near the origin, just like the spinning top solution discussed in section \[sec:spinning\]. We now compute the maximum curvature $C_{\max}$ as a function of $b$ to see when a singularity forms. The results are shown in Fig. \[fig:kretextreme\], where we see that solutions exist for all $b >0$, but the $b\to0$ limit appears to be singular. So for this profile, stationary solutions cease to exist precisely when the ergoregion first forms.
![$C_{\max}$ as a function of $b$ for $a=1$, $n=2$ and profile \[eq:pro2\]: the blow up as $b\to0$ suggests that the solution stops existing precisely when an ergoregion first forms.[]{data-label="fig:kretextreme"}](graphics/c_max_b.pdf){width="45.00000%"}
To see if we could find situations where $a_{\max}$ is reached before an ergoregion exists, we considered a class of rotating boundary geometries without ergoregions: $$\dd s^2_\partial = -\dd t^2+\dd r^2+r^2 \dd\phi^2 -2 r^2\omega(r) \dd t \dd\phi\;,
\label{eq:bndmetric2}$$ These solutions can be constructed exactly as before (section \[sec:irr\]), except that we change our reference metric to be such that no ergoregion is present. For the reference metric we take all functions as in Eq. (\[eq:ref0\]), except for $q_1$ which is now given by $$q_1 = 1+g(\tilde{y})^2\frac{\tilde{y}^2(2-\tilde{y}^2)}{(1-\tilde{y}^2)^2}\,,$$ The form of $\omega(r)$ is again given by . We studied in great detail the cases with $n=2,4,6,8$ and we found no upper bound on $a$. In all these cases, we were able to reach $a\sim 20$ without seeing any indication that the solution is becoming singular. In Fig. \[fig:kret\] we plot $C_{\max}$ as a function of $a$ for $n=2$. Contrary to the case where the ergoregion is present, the maximum now occurs along the axis of rotation.
![$C_{\max}$ as a function of $a$ for boundary metrics without ergoregions: there is no indication that the metric is becoming singular at any finite value of $a$.[]{data-label="fig:kret"}](graphics/c_max.pdf){width="45.00000%"}
Finally, we briefly discuss the case $n=1$. There are two qualitative differences from the $n>1$ solutions. First, the IR geometry is not given by a standard Poincaré horizon, but rather by the extremal horizon of a member of the holographic spinning top solution discussed in section \[sec:spinning\]. In other words, any profile that asymptotically behaves like $\omega(r) \sim a/r$ has the same IR geometry as the solution where $\omega(r) = a/r$ everywhere. Note that our boundary conditions (\[eq:bcsmarginal\]) do not impose this as a Dirichlet condition. Instead, this emerges as the natural IR solution.
Second, the total angular momentum is no longer zero. Since our $n=1$ boundary profile (r) = =a(1-\^2), is not singular at $r=0$, we can compute the total angular momentum just by integrating the angular momentum density. The result is depicted in Fig. \[fig:match\], where we also superimpose our exact result (\[eq:momentum\]).
![Total angular momentum as as a function of $a$, for the marginal case ($n=1$). The solid red line is the result for the spinning top shown in Fig. \[fig:angular\].[]{data-label="fig:match"}](graphics/angular_momentum.pdf){width="45.00000%"}
We see that the angular momentum of the regular profile agrees with that of the spinning top with the same coefficient of the $1/r$ fall-off.
### Stability analysis
For $a < a_{{\rm ergo}}$, we expect our solutions to be stable, but for $a > a_{{\rm ergo}}$, they may be unstable to nonaxisymmetric perturbations due to the superradiant instability. To investigate this, instead of studying the full gravitational perturbations, we will study perturbations by a massless scalar field. So we add a field $\Phi$ satisfying the massless wave equation $$\Box \Phi =0\,.
\label{eq:scalar}$$ Since the background is stationary and axisymmetric, we can Fourier decompose $\Phi$ as $$\Phi = e^{-i\,\omega\,t+i\,m\,\phi}\,\widehat{\Phi}(\tilde{x},\tilde{y})\,,
\label{eq:sepa}$$ and find the quasinormal mode spectrum, *i.e.* the complex values of $\omega$ for which $\widehat{\Phi}$ is normalisable at the conformal boundary, and regular at the horizon. We are primarily interested in finding the onset of the instability, which occurs for $\omega=0$. We can then interpret Eq. (\[eq:scalar\]) as an eigenvalue equation for $m^2$ for a given value of $a$. Of course, we want $m\in\mathbb{Z}$, since $\phi$ is chosen to have period $2\pi$.
We now have to discuss the thorny issue of boundary conditions. The best way to do this is to expand Eq. (\[eq:scalar\]) around each of our integration boundaries and use Frobenius’s method to extract the leading non-analytic behaviour. For $\omega=0$, we find the following behaviour: (1-\^2)\^3 \^[|m|]{} (2-\^2)\^[|m|/2]{} \^[|m|+2 p+3]{} (2-\^2)\^[ (|m|+2 p+3)]{}C\_+, where $p$ is an integer and $C_+$ is a smooth function around $\tilde{y}=0$. In terms of the original coordinates $(r,z)$ of the line element (\[eq:poincare\]) this reads \_+(r,z), where $\widehat{C}_+$ is a smooth function of $r^2+z^2$. Regularity at the origin thus demands $p=0$. Note that the factor $r^{|m|}$ is needed to cancel the non-analytic behaviour of $e^{im\phi}$ included in Eq. (\[eq:sepa\]). We thus perform the following change of variables = (1-\^2)\^3 \^[|m|]{} (2-\^2)\^[|m|/2]{} \^[|m|+3]{} (2-\^2)\^[ (|m|+3)]{} and solve for $\widetilde{\Phi}$ numerically. All we are missing is a choice of boundary conditions. At $\tilde{x}=0$ and $\tilde{x}=1$ we find $\partial_{\tilde{x}} \widetilde{\Phi}=0$, while at $\tilde{y}=0$ we have $\partial_{\tilde{y}} \widetilde{\Phi}=0$. Finally at the Poincaré horizon we find $\widetilde{\Phi}=0$.
In Fig. \[fig:onset\] we show the results of our stability analysis. Our solutions all become unstable before we reach $a_{\max}$. We plot the amplitude for the onset of the instability for a variety of modes, $5 \leq m\leq 18$, and for several profiles, $2\leq n\leq 8$. In all cases, the onset occurs for $a > a_{{\rm ergo}}$ as expected. Independent of profile, the onset of the instability monotonically decreases with $m$, and appears to approach $a_{ergo}$ as $m\to \infty$. This is similar to the results found for Kerr AdS. It supports the idea that an ergoregion is needed to have an instability, and the shortest modes become unstable first.
![Onset of superradiance for each mode $m$, around our profile (\[eq:pro1a\]) for several values of $n$.[]{data-label="fig:onset"}](graphics/onset.pdf){width="45.00000%"}
### Scalar condensate
We now ask if there is a possible stationary endpoint for this instability. In the analogous problem involving gravity coupled to a Maxwell field, it was found that if one adds a charged scalar field, the solutions also become unstable before $a_{\max}$. However, it was shown that there is a stationary solution with nonzero scalar field for all amplitudes [@Crisford:2017gsb], so there is a natural endpoint to the instability. We now check if the same is true for our vacuum black holes coupled to a scalar field.
It turns out to be convenient to work with a complex massless scalar field so we consider the Einstein-scalar action $$S=\frac{1}{16\pi G_4}\int_{\mathcal{M}}\mathrm{d}^4x\,\sqrt{-g}\left(R+\frac{6}{L^2}-2\nabla_a \Phi\,\nabla^a \Phi^*\right)\,,$$ where $^{*}$ denotes complex conjugation. We again use the Einstein-DeTurck equation, but including a scalar field. That is to say, we solve
$$\begin{aligned}
&R_{ab}+\frac{3}{L^2}g_{ab}-\nabla_{(a}\xi_{b)}=\nabla_a \Phi\nabla_b \Phi^*+\nabla_b \Phi\nabla_a \Phi^*\,,
\\
&\Box \Phi =0\,.\end{aligned}$$
To ease our numerical calculations, we consider a scalar field with a definite quantum number $m$: $$\Phi = e^{i\,m\,\phi}\widehat{\Phi}\quad\text{and}\quad \Phi^* = e^{-i\,m\,\phi}\widehat{\Phi}\,,$$ with $\widehat{\Phi}$ being real. The matter sector breaks axisymmetry, but the metric does not since the stress energy tensor only involves $\Phi$ in the combination $\nabla_{(a} \Phi\nabla_{b)} \Phi^*$. This is similar in spirit to the black holes with a single Killing field of [@Dias:2011at; @Herdeiro:2014goa] and holographic Q-lattices of [@Donos:2013eha]. As before, we choose an *ansatz* for our scalar field of the form: =(1-\^2)\^3 \^[| m| ]{} (2-\^2)\^ \^[| m| +3]{} (2-\^2)\^q\_7, while our metric *ansatz* remains as in Eq. (\[eq:ansatz\]).
Our results are rather surprising and very different from the electromagnetic case. Stationary solutions with nonzero scalar field indeed branch off from the onset of the instability. However, they now extend towards [*smaller*]{} values of $a$. Eventually, these solutions become singular and terminate. This is depicted in Fig. \[fig:condensate\] for $n=2$ and $m=6$. To judge the size of the scalar field, we use the maximum of the expectation value of the dual scalar operator over the boundary, which is essentially the coefficient of the leading term in $\Phi$ as one approaches the boundary. This is perhaps similar to the results found in [@Dias:2015rxy], where the black resonators and black holes with a single Killing field of [@Dias:2011at] extend to smaller values of the angular velocity. Since our vacuum solutions are stable for these values of the amplitude, it is likely that these new solutions with nonzero scalar field are unstable[^5]. More importantly, there are no stationary configurations for the larger amplitude solutions to settle down to, even including the scalar field.
![Solutions with a scalar condensate only exist for amplitudes [*less*]{} than the onset. The data shown here gives the maximum value of the condensate along the boundary for the $n=2$ profile and $m=6$ mode. The curve terminates at the left when the solution becomes singular. []{data-label="fig:condensate"}](graphics/condensate.pdf){width="45.00000%"}
\[sec:blacknon\]Black Holes
---------------------------
We now extend our results to nonzero temperature, to see what happens if we start with a black hole rather than the vacuum. The boundary metric will again be given by , and the differential rotation $\omega(r)$, will again take the form .
### \[eq:holeplanar\]Metric *ansatz*
We will start by describing our choice of reference metric. First, we take the usual planar black hole written in the familiar Schwarzschild coordinates $(r,Z)$ s\^2=, \[eq:planar\] where the horizon is the null hypersurface $Z=Z_+$ with the associated Hawking temperature $T = {3}/{4\pi\,Z_+}$. According to the gauge/gravity duality, the Hawking temperature will be identified with the field theory temperature $T$ [@Witten:1998qj].
We now introduce new coordinates r= Z = Z\_+(1-y\^2), in terms of which the line element (\[eq:planar\]) can be written as s\^2={-G(y)y\_+\^2y\^2t\^2++y\_+\^2}, where $G(y)=3-3y^2+y^4$ and $y_+\equiv1/Z_+$. In terms of these new coordinates, the profile (\[eq:pro1a\]) reduces to (r)= g(x)=a(1-x\^2)\^n. \[eq:prox\]
We now propose the following *ansatz* for our metric $$\begin{gathered}
\mathrm{d}s^2=\frac{L^2}{(1-y^2)^2}\Bigg\{-G(y)\,y_+^2\,y^2\,q_1\mathrm{d}t^2+\frac{4\,q_2}{G(y)}\left[\mathrm{d}y+\frac{q_3\dd x}{(1-x^2)^2}\right]^2+\\
y_+^2\left[\frac{4\,q_4\mathrm{d}x^2}{(2-x^2)(1-x^2)^4}+\frac{x^2(2-x^2)}{(1-x^2)^2}\,q_5\left(\mathrm{d}\phi-y^2\,q_6\,\dd t\right)^2\right]\Bigg\}\,.
\label{eq:angen}\end{gathered}$$ For the reference metric we take Eq. (\[eq:angen\]) with $q_1=q_2=q_3=q_5=1$, $q_3=0$ and $q_6=g(x)$.
We now discuss the issue of boundary conditions. Infinitely far away from the fixed points of $\partial_\phi$, *i.e.* at $x=1$, we demand $q_1=q_2=q_3=q_5=1$ and $q_3=q_6=0$, which is consistent with our choice of profile (\[eq:prox\]). At the centre, located at $x=0$, regularity demands \_x q\_1=\_x q\_2=\_x q\_4=\_x q\_5=\_x q\_6=q\_3=0.
At the conformal boundary, located at $y=1$, we choose the line element (\[eq:angen\]) to approach the reference metric, *i.e.* q\_1=q\_2=q\_3=q\_5=1,q\_3=0, q\_6=g(x). Finally, at the horizon, located at $y=0$, regularity in ingoing Eddington-Finkelstein coordinates imposes \_y q\_1=\_y q\_2=\_y q\_4=\_y q\_5=\_y q\_6=q\_3=0q\_1=q\_2, with the later condition fixing the black hole temperature to be T = y\_+. \[eq:hawking\]
Lastly, we discuss how to extract the holographic stress energy tensor, following [@deHaro:2000vlm]. First, we solve (\[eq:einsteindeturck\]) in a series expansion around conformal boundary $y=1$. This is done via the rather intricate expansion q\_i = \_[j=0]{}\^[+]{}q\_i\^[(j)]{}(x)(1-y)\^j+ (1-y)\^[(3+)/2]{}\_[j=0]{}\^[+]{}\_i\^[(j)]{}(x)(1-y)\^j+(1-y)\^4(1-y)\_[j=0]{}\^[+]{}\_i\^[(j)]{}(x)(1-y)\^j. The nonanalytic terms, which will affect the convergence of our numerical method, were first uncovered in [@Santos:2012he] and [@Donos:2014yya].
Once the expansion is sorted out, the idea is to then change from our coordinates $(x,y)$ to Fefferman-Graham coordinates via a new asymptotic expansion
$$\begin{aligned}
&x =\sqrt{1-\frac{1}{\sqrt{1+r^2}}}+\alpha_1(r)\,z+\alpha_2(r)\,z^2+\alpha_3(r)\,z^3+\alpha_4(r)\,z^4+{
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(z^4)
\\
&y=1+\beta_1(r)\,z+\beta_2(r)\,z^2+\beta_3(r)\,z^3+\beta_4(r)\,z^4+{
\mathchoice
{{\scriptstyle\mathcal{O}}} {{\scriptstyle\mathcal{O}}} {{\scriptscriptstyle\mathcal{O}}} {\scalebox{.7}{$\scriptscriptstyle\mathcal{O}$}} }(z^4)\end{aligned}$$
and determine the coefficients $\alpha_i$ and $\beta_i$ by requiring the line element (\[eq:angen\]) to be in the Fefferman-Graham form, *i.e.* s\^2=, where $\dd s_i^2$, with $i\in\{0,2,3\}$, only has components along the boundary directions. For instance, we find \_1=0\_1=-.
After recasting the metric is in this form, the holographic stress energy tensor is recovered via [@deHaro:2000vlm] T\_x\^x\^= s\_3\^2.
### Results {#results}
We have followed the same procedure as before, increasing the amplitude $a$ for fixed temperature $T$, and for several different profiles: $n=2,4,6,8$. In all cases, we again find that there is a maximum amplitude we can attain before our solution becomes singular[^6]. As a typical example, in Fig. \[fig:cmaxt\] we plot the maximum value of the square of the Weyl tensor, $C_{\max}$ for the case $n=4$ and $T = .9/4\pi$. The apparent kink in Fig. \[fig:cmaxt\] results from the crossing of two local maxima, analogous to the absolute maximum going through a first order phase transition.
![$C_{\max}$ as a function of $a$, computed for $n=4$ and $T=.9/4\pi$.[]{data-label="fig:cmaxt"}](graphics/c_max_finite_t_non_compact.pdf){width="45.00000%"}
As in the $T=0$ case, $a_{\max}$ increases with $n$. This is illustrated in Fig. \[fig:amaxseveraln\], where we plot $a_{\max}$ at fixed $T = 3/4\pi$, for several values of $n$. For all profiles, $a_{\max} > a_{{\rm ergo}}$.
![$a_{\max}$ as a function of $n$, computed for $T = 3/4\pi$.[]{data-label="fig:amaxseveraln"}](graphics/a_max_several_n.pdf){width="45.00000%"}
Next, we examine how $a_{\max}$ changes when we turn up the temperature. We find that $a_{\max}$ decreases rapidly from its $T=0$ value and settles down to $a_{{\rm ergo}}$. To illustrate this, in Fig. \[fig:amaxt0\] we plot $a_{\max}$ as a function of $T$, for fixed $n=4$. The black dot is the $T=0$ value, the dotted red line is $a_{{\rm ergo}}$, and the blue dots are the numerical values of $a_{\max}$ we extracted at finite temperature.
![$a_{\max}$ as a function of $T$, computed for $n=4$. The black star is the $T=0$ result and the dotted red line is $a_{{\rm ergo}}$.[]{data-label="fig:amaxt0"}](graphics/amax_finite_t_noncompact.pdf){width="45.00000%"}
Despite appearances in Fig. \[fig:amaxt0\], we do not expect $a_{\max} = a_{{\rm ergo}}$ at finite $T$, but only to approach it from above as $T\to\infty$. This is because one can construct stationary black holes with large $T$ for all $a < a_{{\rm ergo}}$ using holography, and in particular, the fluid/gravity correspondence [@Bhattacharyya:2008jc; @Bhattacharyya:2008xc; @Hubeny:2011hd]. When the scale of curvature is much larger than the thermal wavelength, one expects the dual field theory will be well described by a fluid. In this case, the fluid/gravity correspondence constructs a bulk solution by associating a piece of a boosted planar black hole to boundary regions that are smaller than the curvature scale but larger than the thermal wavelength, and suitably patching them together. For $a<a_{{\rm ergo}}$ one can indeed construct stationary bulk black holes using this procedure which agree very well with our numerical solutions. However one cannot obtain stationary solutions this way when there is an ergoregion on the boundary, since the Killing field becomes spacelike and can no longer define a local rest frame for the fluid. One can presumably construct nonstationary black holes by picking a slowly varying unit timelike vector on the boundary to use as the fluid four-velocity.
Finally, we have studied how the energy density $\rho$ and angular momentum density $j$ depend on the temperature. This is shown in Fig. \[fig:holot\] for $4\pi T/3 =0.0,0.1,0.5,1.0$ at fixed $n=4$ and $a=3.1$, which is close to $a_{\max}$. The shaded regions correspond to the location of the boundary ergoregion. The horizontal solid black line in the holographic energy density plots indicate the value that these quantities take for a planar Schwarzschild black hole (\[eq:planar\]). Note that $j$ vanishes identically for the planar Schwarzschild black hole (\[eq:planar\]). The figure shows that $\rho$ and $j$ differ significantly from their Schwarzschild values in the vicinity of the ergoregion. The curves corresponding to $T=0$ were taken from the analysis of the previous section (corresponding to the two plots on the left column of Fig. \[fig:holot\]). The total angular momentum $J$ is now nonzero and grows with $T$. Unlike the total energy $E$, $J$ appears to diverge as $a\to a_{\max}$.
![The holographic energy density (top row) and holographic momentum density (bottom row) computed for $n=4$ and fixed $a=3.1$ at four distinct temperatures. The shaded region denotes the ergoregion. From left to right we have $4\pi T/3=0,0.1,0.5,1$. The horizontal solid black line in the holographic energy density plots indicate the value that these quantities take for a planar Schwarzschild black brane (\[eq:planar\]).[]{data-label="fig:holot"}](graphics/stress_energy_tensor_finite_t.pdf){width="\textwidth"}
Compact solutions
=================
In this section we consider a different class of solutions where the boundary has topology $\mathbb{T}^2\times \mathbb{R}$ and we add rotation around the circles. We will mostly focus on the case where we add rotation around one circle, so the boundary metric takes the form \[eq:rotcircle\] s\_\^2=-t\^2+[X\^2]{}+\^2, where $X$ and $W$ are both periodic with periods $\ell_X = 2\pi/k_X$ and $\ell_W=2\pi/k_W$ respectively, and (X)=ak\_X X. \[eq:protor\] Note that if $a>1$, there is a boundary ergoregion, whereas if $a<1$ there is none, so $a_{{\rm ergo}}= 1$. The case $a=1$ corresponds to the situation when we have an evanescent ergosurface. Unlike the non-compact case, $a$ now always has conformal dimension $0$, so it corresponds to a marginal deformation of the boundary metric.
At the end of this section, we will briefly comment on what happens if we add rotation to both circles and the boundary metric takes the form: \[eq:rotcircle2\] s\_\^2=-t\^2+\^2 +\^2, with (W)=ak\_W W. \[eq:protorW\]
Zero-temperature solutions
--------------------------
We first discuss zero-temperature solutions. Even though our ansatz for the boundary metric has three parameters $a,k_X,k_W$, there is only a one parameter family of inequivalent solutions labelled by $a$. This is because we can use scale invariance to set $k_X = 1$, and since our boundary metric is independent of $W$, our solution will be independent of $W$ and $\ell_W$ will only appear as an overall factor.
Since our boundary metrics are compact, we have a couple of possibilities for the IR behaviour of our solutions. Namely, one can have a solitonic solution, with no horizons, where a spatially compact direction smoothly caps off spacetime [@Horowitz:1998ha], or we can try to compactify the Poincaré horizon. If one starts with the Poincaré patch of AdS and makes the spacelike directions of the Minkowski slices compact by periodically identifying them, the horizon develops a conical singularity and is no longer smooth. This is because the translational symmetries have a fixed point there.
In a canonical ensemble, the solution that is likely to dominate at $T=0$ is the solitonic one, since this is true without the rotation. However, we are interested in studying these solutions from a microcanonical perspective, since that is appropriate when evolving at fixed energy. Furthermore, we are interested in the $T\to 0$ limit of the black holes we will construct in part [**B**]{}, so we will focus on the solutions with a compactified Poincaré horizon. We leave the construction of the solitonic solutions to a future endeavour.
### Metric ansatz
Again, we use the DeTurck method which we outlined in section \[sec:turck\] to construct solutions. We start with pure AdS written in familiar Fefferman-Graham coordinates s\^2=(-t\^2+X\^2+W\^2+Z\^2), \[eq:almostfinal\] where again $X$ and $W$ are periodic coordinates with period $\ell_X\equiv2\pi/k_X$ and $\ell_W\equiv2\pi/k_W$. We now introduce coordinates $$Z=\frac{y\sqrt{2-y^2}}{1-y^2}\,,\qquad X = \frac{x}{k_X}\,,\qquad \text{and}\qquad W=\frac{w}{k_W}\,,$$ which brings Eq. (\[eq:almostfinal\]) into the following form s\^2=. \[eq:almostfinal2\] The form of (\[eq:almostfinal2\]) is ideal to introduce the DeTurck trick, since $y=1$ marks the Poincaré horizon and $y=0$ the location of the conformal boundary while the remaining two boundary coordinates have period $2\pi$. In this section we will only study rotation profiles along the $W$ direction, which explicitly depend on $X$. Our line element for the DeTurck method reads $$\begin{gathered}
\mathrm{d}s^2=\frac{L^2}{y^2(2-y^2)}\Bigg\{(1-y^2)^2\left[-q_1\,\dd t^2+q_4\,\left(\frac{\dd x}{k_X}+q_3\dd y\right)^2+q_5\left(\frac{\dd w}{k_W}-q_6(1 - y^2)^2 \dd t\right)^2\right]\\+\frac{4\,q_2\dd y^2}{(2-y^2)(1-y^2)^2}\Bigg\}\,,
\label{eq:almostfinal3}\end{gathered}$$ which is invariant under general reparametrizations of $(x,y)$. For the reference metric we will take q\_1=q\_2=q\_4=q\_5=1,q\_3=0,q\_6 = ax. \[eq:referencezerocompact\]
The boundary conditions at the IR of the theory, that is to say at the horizon located at $y=1$, are simply \_y q\_1=\_y q\_4=\_y q\_5=0,q\_2=1q\_3=q\_6=0, while at the conformal boundary we demand our physical bulk spacetime metric to approach the reference metric (\[eq:referencezerocompact\]). We shall see that the IR will depend on $a$, but in a trivial way. In particular, the IR will always be Poincaré, but $g_{tt}$, $g_{ww}$ and $g_{xx}$ will appear renormalised along the RG flow as we move from the UV to the IR. This is to be expected, since from the perspective of the UV theory, $a$ is a marginal deformation. The boundary conditions above are compatible with such IR behaviour. We should *a posteriori* check that $q_1$, $q_4$ and $q_5$ are independent of $x$ when $y=1$, which will turn out to be the case for all the runs we have made.
### Results {#results-1}
Just as in the noncompact case, there is a maximum amplitude $a_{\max}$ beyond which the solutions develop a curvature singularity. To find $a_{\max}$, we again monitor the maximum value of the square of the Weyl tensor, $C_{\max}$, as a function of $a$ and determine where it diverges. This is plotted in Fig. [\[fig:cmaxzerotcompact\]]{}. Like the non-compact zero-temperature solutions of section \[sec:resultsT0\], solutions exists even when $a>a_{{\rm ergo}}=1$. In fact we find $a_{\max}\approx1.28$.
![$C_{\max}$ as a function of $a$ for the $T=0$ compact case.[]{data-label="fig:cmaxzerotcompact"}](graphics/c_max_toroidal_zero.pdf){width="50.00000%"}
We have computed other quantities such as $\rho$ and $j$, but they behave just like in the non-compact case, so we will not present them here. One of the quantities of interest that we can extract from these is the energy density $E/\ell_W$ as a function of $a$. This is presented in Fig. \[fig:energytotalcompact\], where we again see $E$ increasing monotonically even past $a=a_{{\rm ergo}}$, but reaching a maximum value just before $a_{\max}$. Just like for the non-compact case, we have no current understanding of this behaviour.
![The energy density $E/\ell_W$ as a function of $a$, with the non-monotonic behaviour starting at around $a\approx1.24422$.[]{data-label="fig:energytotalcompact"}](graphics/total_energy_compact.pdf){width="45.00000%"}
Black Holes
-----------
We next discuss black hole solutions with boundary conditions or . Since the temperature $T$ is a new dimensionful parameter, inequivalent solutions can now depend on the dimensionless ratios $T/k_X$ and $T/k_W$. Increasing the wavenumbers at fixed temperature will have the same effect as decreasing the temperature. Combined with the dimensionless amplitude $a$, our moduli space is thus either two or three-dimensional depending on which boundary condition we impose.
### Metric ansatz
We will again use the DeTurck method. We will start by recalling the line element of a Schwarzschild black brane with toroidal spatial cross sections s\^2=, \[eq:planartorus\] and we are interested in the case where both $X$ and $W$ are periodic with periods $\ell_X\equiv 2\pi/k_X$ and $\ell_W\equiv 2\pi/k_W$, respectively. Next, we change to new variables Z=Z\_+(1-y\^2),X=W= in terms of which (\[eq:planartorus\]) can be recast as s\^2=, \[eq:planartorus\] where $G(y)=3-3\,y^2+y^4$, $y_+\equiv 1/Z_+$ and $x$, $w$ are periodic coordinates with period $2\pi$. The horizon is the null hypersurface $y=0$, and has Hawking temperature $$T = \frac{3\,y_+}{4\pi}\,.
\label{eq:hawkingto}$$
We can now detail the *ansatz* we used in the DeTurck method. We recall that this *ansatz* should be compatible with diffeomorphism invariance in the $(x,y)$ directions. The line element reads s\^2={-G(y)y\_+\^2y\^2q\_1t\^2++q\_4(x+q\_3y)\^2+y\_+\^2q\_5(-y\^2q\_6t)\^2}, where all six functions $q_i$ are functions of $(x,y)$ only. For the reference metric in the DeTurck method, we will use the line element above with q\_1=q\_2=q\_4=q\_5=1,q\_3=0,q\_6=(x). \[eq:refto\]
The boundary conditions are determined by requiring regularity across the event horizon, which demands \_y q\_1=\_y q\_2=\_y q\_4=\_y q\_5=\_y q\_6=0,q\_3=0q\_1=q\_2. Note that the last boundary condition ensures that the black hole temperature is given as in (\[eq:hawkingto\]). At the boundary, we give Dirichlet boundary conditions and demand the metric to approach the reference metric (\[eq:refto\])
Finally, we will also briefly discuss the case where we have boundary deformations in both the $x$ and the $w$ directions. This corresponds to a full three dimensional problem, where the black hole has a single Killing isometry corresponding to time translations $\partial/\partial t$. The most general line element compatible with such reduced symmetries reads $$\begin{gathered}
\dd s^2=\frac{L^2}{(1-y^2)^2}\Bigg[-G(y)\,y_+^2\,y^2\,Q_1\,\mathrm{d}t^2+\frac{4\,Q_2}{G(y)}\left(\mathrm{d}y+y^2Q_7 \mathrm{d}t\right)^2+y_+^2 Q_3\left(\frac{\mathrm{d}x}{k_X}-y^2Q_5\mathrm{d}t+Q_8\mathrm{d}y\right)^2\\
+y_+^2\,Q_4\left(\frac{\mathrm{d}w}{k_W}-y^2Q_6\,\dd t+Q_9\dd y+Q_{10}\dd x\right)^2\Bigg]\end{gathered}$$ where all ten functions $Q_i$ are functions of $(x,y,w)$. For the reference metric we now choose $$\begin{aligned}
&Q_i=1, \qquad\text{for}\qquad i\in\{1,2,3,4\}\,,\nonumber
\\
&Q_i=0, \qquad\text{for}\qquad i\in\{7,8,9,10\}\,,\label{eq:refcrazy}
\\
&Q_5=a_x\,\cos w\quad\text{and}\quad Q_6=a_w\,\cos x\,.\nonumber\end{aligned}$$
The boundary conditions at the horizon again following from requiring regularity across the event horizon $$\begin{aligned}
&\partial_y Q_i=1, \qquad\text{for}\qquad i\in\{1,2,3,4,5,6\}\,,\nonumber
\\
&Q_i=0, \qquad\text{for}\qquad i\in\{7,8,9,10\}\,,
\\
&Q_1=Q_2\,,\nonumber\end{aligned}$$ with the later fixing the temperature to be given by (\[eq:hawkingto\]). Finally, at the conformal boundary, we demand the bulk physical metric to approach the reference metric (\[eq:refcrazy\]).
### Results {#results-2}
We start by adding rotation around one circle. Just as in section \[sec:blacknon\] we find that solutions exist only up to a maximum value $a_{\max}$, which strongly depends on the ratio $T/k_X$. We first fix $T/k_X$ and increase $a$ until the curvature, $C_{\max}$, appears to diverge. This is depicted in Fig. \[fig:cmaxtoroidal\] for $T/k_X=0.239$ (top curve) and $T/k_X=0.0119$ (bottom curve).
![$C_{\max}$ as a function of $a$, depicted for $T/k_X=0.2387$ (top curve) and $T/k_X=0.0119$ (bottom curve). The kink in the bottom curve corresponds to the interchange of two local maxima.[]{data-label="fig:cmaxtoroidal"}](graphics/c_max_toroidal.pdf){width="45.00000%"}
In the next step, we investigate how $a_{\max}$ depends on $T/k_X$ by repeating the same calculation that leads to Fig. \[fig:cmaxtoroidal\] for many values of $T/k_X$. The results are plotted in Fig. \[fig:amaxt0c\]. Again we see that $a_{\max}$ decreases rapidly from the $T=0$ result computed in the previous section to $a_{{\rm ergo}} =1$ in the fluid limit (corresponding to the high-temperature regime).
![$a_{\max}$ as a function of $T/k_X$. The black star is the $T=0$ result obtained in the previous section and the dotted red line is $a=a_{{\rm ergo}}=1$.[]{data-label="fig:amaxt0c"}](graphics/amax_finite_t_compact.pdf){width="45.00000%"}
We also studied how $\rho$ and $j$ depend on $T/k_X$ at fixed $a=1.1008$, which can be seen in Fig. \[fig:holont\] for $T/k_X=0.0239,0.0477,0.0716$. Since $a >1$ there is an ergoregion on the boundary, but for these low temperatures, $\rho$ and $j$ change only modestly with $T$.
![The holographic energy density (top row) and holographic momentum density (bottom row) computed at fixed $a=1.1008$ at three distinct temperatures. From left to right we have $T/k_X=0.0239,0.0477,0.0716$.[]{data-label="fig:holont"}](graphics/stress_energy_tensor_toroidal_finite_t_1.pdf){width="90.00000%"}
At high temperatures, the behaviour of $\rho$ and $j$ is dramatically different from the one depicted in Fig. \[fig:holont\]. In Fig. \[fig:holont1\] we show both $\rho$ and $j$ computed for $T/k_X\approx 1.43$ and $a=0.975$. Although the boundary metric now does not have an ergoregion, if $a$ is increased slightly an ergoregion forms at $k_X X = 0,\pi$. Note that both the energy and momentum densities develop large features precisely at the location of the would be ergoregion. This is in perfect agreement with the fluid gravity calculation, which indicates a similar feature.
![The holographic energy density (left panel) and holographic momentum density (right panel) computed for $a=0.975$ and $T/k_X\approx 1.43$.[]{data-label="fig:holont1"}](graphics/stress_energy_tensor_toroidal_finite_t_2.pdf){width="90.00000%"}
In Fig. (\[fig:holont2\]) we show the analytic curve derived in the fluid approximation (represented as a dashed line) and our numerical data (represented as blue disks). The agreement even at these modest values of $T/k_X$ is very reassuring.
![The ratio $j/\rho$ as a function of $X k_X$, computed using $a=0.975$ and $T/k_X\approx 1.43$. The dashed black line represents the fluid calculation and the blue disks our numerical data.[]{data-label="fig:holont2"}](graphics/stress_energy_tensor_toroidal_finite_t_3.pdf){width="45.00000%"}
If we fix $T/k_X$ and increase $a$, the area of the event horizon increases rapidly as $a\to a_{\max}$. We show this behaviour in Fig. \[fig:entropy\_density\] where we plot the entropy density $S/\ell_W$ as a function of $a$ and using $T/k_X\approx0.2387$. Other values of $T/k_X$ behave similarly.
![The entropy density $S/\ell_W$ as a function of $a$, computed using $T/k_X\approx0.2387$. The vertical dashed red line marks $a=1$, where a boundary ergoregion forms. (This is very close to $a_{\max}$ and our numerics cannot distinguish them.)[]{data-label="fig:entropy_density"}](graphics/entropy_density.pdf){width="45.00000%"}
Finally, we briefly discuss what happens if we add rotation around both circles and use boundary metric . We will assume equal amplitude for the two rotations: $a = \tilde a$. The calculations are, of course, much more time consuming since we are now solving ten coupled three-dimensional nonlinear partial differential equations. Nevertheless, we reach similar conclusions. Again we find that solutions exist up to a maximum value $a_{\max}$ and that this can be larger than $a_{{\rm ergo}}\equiv1/\sqrt{2}$. The ergoregion now consists of disconnected disks centred at $X k_X = 0,\pi$ and $Wk_W = 0,\pi$ Perhaps the most interesting quantities to plot are now the energy density $\rho$ and the remaining components of the stress energy tensor. These are displayed in Fig. \[fig:giant\_3d\] for $T/k_W=0.239$, $k_X/k_W=1$, and $a=0.6$. The behaviour is very similar for other values of $a$ we have studied, except that the extrema get more noticeable as one approaches $a=a_{\max}$.
![All the components of the holographic stress energy tensor $\langle T^{\mu}_{\;\;\;\nu}\rangle$ when there is rotation about both circles. These plots are for $T/k_W=0.239$, $k_X =k_W$, and $a=0.6$[]{data-label="fig:giant_3d"}](graphics/giant_3d.pdf){width="\textwidth"}
Discussion
==========
We have numerically constructed stationary, asymptotically AdS solutions of Einstein’s equation with $\Lambda < 0$ with rotating boundary metrics. If we fix the profile of the differential rotation and increase the overall amplitude $a$, we find a maximum value $a_{\max}$ where the solution becomes singular. This happens both at zero and nonzero temperature, and for compact or noncompact boundaries. We expect that in the time dependent problem where $a$ is increased from $a < a_{\max}$ to $a > a_{\max}$ the curvature will grow without bound violating weak cosmic censorship.
However, the boundary metrics all develop ergoregions before reaching $a_{\max}$ and since one can extract energy from an ergoregion, it is natural to ask if there is a positive energy theorem for these boundary conditions. The following argument suggests that the answer is no. One can clearly place test particles in the ergoregion and boost them so that their energy is arbitrarily negative. We now want to replace the test particle by a small black hole. There are gluing theorems which ensure that one can add a small black hole in the ergoregion to initial data on a constant $t$ surface [@Isenberg:2005xp]. An $O(1)$ boost of this black hole will cause it to contribute negatively to the total energy, and should not result any singularities in the initial data. Moving the black hole farther into the asymptotic region increases its negative contribution to the energy without bound.
Using gauge/gravity duality, there is another argument that the total energy is unbounded from below. Consider the planar case with $a_{{\rm ergo}} < a< a_{\max}$. The boundary metric is an asymptotically flat spacetime with an ergoregion and no horizon. Consider first classical or free quantum fields. Classical fields on such spacetimes are known [@Friedman1978] to be unstable since one can construct negative energy solutions by exciting fields in the ergoregion. Since stationary solutions must have zero energy and the energy radiated to infinity is always positive, if the energy is negative initially, it will continue to decrease. Free quantum fields on such a spacetime exhibit a similar instability: it has been shown [@Ashtekar1975; @Kang:1997uw] that there is no Fock vacuum that is time translation invariant. In other words, there is particle creation in all states. It is also clear that there is no lower bound on the energy for free quantum fields in such a spacetime. This is because excitations localized in the ergoregion can have negative energy, and one can give them arbitrarily large occupation number.
Even at strong coupling, a CFT on a spacetime with ergoregion and no horizon cannot have a minimum energy state[^7]. Start with the ground state in Minkowski space and act with a unitary operator in a finite region $A$. This creates a state with $E>0$ that looks like the vacuum outside $A$. By scale invariance, we can make $A$ as small as we want. Now consider our boundary metric and pick a small locally flat region inside the ergoregion. As long as $A$ is small enough, we can insert the above state into this geometry. We can then boost it to give it arbitrarily negative energy.
In addition to the instability associated with the ergoregion, there is another potential instability in the dual field theory if the scalar curvature is negative over a large enough region. Conformally invariant scalars in such backgrounds can be unstable. However this is not a problem for the boundary metrics we consider. In the compact case, the scalar curvature of or is nonnegative. In the noncompact case, although the scalar curvature of can become negative, it is confined to a small area.
We conclude with a comment about another possible class of solutions. In the electromagnetic case, there is a family of static, $T=0$ solutions for any amplitude that describe hovering black holes [@Horowitz:2014gva]. These are extremal spherical black holes that hover above the Poincaré horizon since the usual attraction to the horizon is balanced by an electrostatic attraction to the boundary. This family of solutions did not play any role in our counterexamples to cosmic censorship since if we only have a Maxwell field, there is no charged matter, and no way to form a charged black hole. It is natural to ask if an analogous hovering black hole could form in the vacuum case and provide a stationary endpoint for any amplitude. It appears the answer is no. We have seen that the singularity arises off the axis so it actually forms a ring. This could not be enclosed by a spherical black hole unless the black hole was quite large and unlikely to be supported by any spin-spin forces.
**Acknowledgements**
It is a pleasure to thank D. Marolf and J. Markevičiūtė for discussions. GH was supported in part by NSF grant PHY-1504541. JES was supported in part by STFC grants PHY-1504541 and ST/P000681/1. .5 cm
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[^1]: Note that the existence of an ergoregion is a conformally invariant property of the boundary metric.
[^2]: This should not be confused with the $n$ which appeared in section \[sec:spinning\] which will not be referred to again.
[^3]: The error bars in determining $a_{\max}$ simply reflect the fact that a solution exists at the blue dot but not at the upper end of the error bar (using a uniform grid over $a$). In the following, when we quote results for $a_{\max}$, we mean this largest value for which we have found a solution, and not literally the singular solution.
[^4]: We will see in section \[sec:blacknon\] that another way to make this difference arbitrarily small is to go to high temperature.
[^5]: Any such solution will be unstable to higher $m$ perturbations. We are saying here that even for fixed $m$, these solutions are likely to be unstable.
[^6]: This was not true in the analogous electromagnetic problem, where static solutions were found with $T>0$ for any amplitude of the chemical potential [@Horowitz:2016ezu].
[^7]: We thank D. Marolf for suggesting this argument.
|
---
abstract: 'We study the distribution of eigenvalues of the Schrödinger operator with a complex valued potential $V$. We prove that if $|V|$ decays faster than the Coulomb potential, then all eigenvalues are in a disc of a finite radius.'
author:
- Oleg Safronov
title: Estimates for eigenvalues of the Schrödinger operator with a complex potential
---
Introduction
============
We consider the Schrödinger operator $H = -\Delta + V$ with a complex potential $V$ and then we study the distribution of eigenvalues of $H$ in the complex plane.
Our work in this direction was motivated by the question of E.B. Davies about an integral estimate for eigenvalues of $H$ (see [@D1] and [@AN]). If $d = 1$ then all eigenvalues ¸ of $H$ which do not belong to ${\Bbb R}_+=[0,\infty)$ satisfy $$|\lambda|\leq \frac14\Bigl(\int_{\Bbb R}|V(x)|dx\Bigr)^2.$$ The question is whether something similar holds in dimension $d\geq2$. We prove the following result related directly to this matter.
\[main\] Let $V:{\Bbb R}^d\mapsto {\Bbb C}$ satisfy the condition $$|V(x)|\leq \frac L{(1+|x|^2)^{p/2}}, \qquad 1<p<3,$$ with a constant $L>0$. Let $\varkappa=(p-1)/2$ and let $\epsilon>0$ be an arbitrarily small number that belongs to the intersection of the intervals $(0,(1-\varkappa)/2)\cap(0, 1/2)$. Then any eigenvalue $\lambda\notin {\Bbb R}_+$ of $H$ with $\Re\lambda>0$ satisfies one of the conditions: $${\rm 1) \,\, either} \,\,\,\,\, |\lambda|\leq1$$ or $${\rm 2)} \quad 1 \leq C L\Bigl(|\Re\lambda|^{(\varkappa+2\varepsilon-1)/2}+
|\lambda|^{\varepsilon-1/2}+ \frac {1+|\lambda|^{\varepsilon}}{(|\lambda|-1)}\Bigr)$$ where the constant $C$ depends on the dimension $d$ and on the parameters $p$ and $\varepsilon$. In particular, it means that all non-real eigenvalues are in a disc of a finite radius.
The study of eigenvalue estimates for operators with a complex potential already has a bibliography. Besides [@D1] and [@AN], we would like to mention the papers [@FLLS] and [@LS]. The main result of [@FLLS] tells us, that for any $t>0$, the eigenvalues $z_j$ of $H$ lying outside the sector $\{z:\ \ |\Im z|<t\ \Re z\}$ satisfy the estimate $$\sum |z_j|^\gamma\leq C\int |V(x)|^{\gamma+d/2}dx,\qquad \gamma\geq 1,$$ where the constant $C$ depends on $t, \gamma$ and $d$ (see also [@LT] for the case when $V$ is real).
The paper [@LS] deals with natural question that appears in relation to the main result of [@FLLS]: what estimates are valid for the eigenvalues situated inside the conical sector $\{z:\ |\Im z|<t \Re z\}$, where the eigenvalues might be close to the positive half-line? Theorems of the article [@LS] provide some information about the rate of accumulation of eigenvalues to the set ${\Bbb R}_+=[0,\infty)$. Namely, [@LS] gives sufficient conditions on $V$ that guarantee convergence of the sum $$\sum_{a<\Re z_j<b}|\Im z_j|^\gamma<\infty$$ for $0\leq a<b<\infty$. Moreover, the following result is also proven in [@LS]:
\[LS\] Let $V$ be a function from $L^p({\Bbb R}^d)$, where $p\geq d/2$, if $d\geq3$>; $p>1$, if $d=2$, and $p\geq1$, if $d=1$. Then every eigenvalue $\lambda$ of the operator $H=-\Delta+V$ with the property $\Re \lambda> 0$ satisfies the estimate $$\label{21t3}
|\Im \lambda|^{p-1}\leq |\lambda|^{d/2-1}C\int_{{\Bbb R}^d} |V|^pdx.$$ The constant $C$ in this inequality depends only on $d$ and $p$. Moreover, $C=1/2$ for $p=d=1$.
Proof of Theorem \[main\]
=========================
Consider first the case $L=1$. For the sake of convenience we introduce the notations $W=|V|^{1/2}$ and $l=p/2$. According to the Birman-Schwinger principle, a number $\lambda\notin {\mathbb R}_+$ is an eigenvalue of the operator $H=-\Delta+V(x)$ if and only if the number $-1$ is an eigenvalue of the operator $$X_0=W(-\Delta-\lambda)^{-1}W\frac{V}{|V|}.$$ Therefore if $\lambda$ is a point of the spectrum of the operator $H$, then $||X_0||\geq1$. On the other side, since multiplication by the function $\frac{V}{|V|}$ represents a unitary operator, the condition $||X_0||\geq1$ implies that the norm of the operator $$X=W(-\Delta-\lambda)^{-1}W$$ is also not less than 1.
In order to estimate the norm of the operator $X$ from above, we consider its kernel $$(2\pi)^{-d}W(x)\int\frac{e^{i\xi(x-y)}}{\xi^2-\lambda}d\xi\,W(y)$$ It follows from this formula that $X$ can be represented in the form $$X=\int_0^\infty\frac{\Gamma_\rho^*\Gamma_\rho}{\rho^2-\lambda}d\rho,$$ where $\Gamma_\rho$ is the operator mapping $L^2({\mathbb R}^d)$ into $L^2({\mathbb S}_\rho)$, and ${\mathbb S}_\rho$ is the sphere of radius $\rho$ with the center at the point $0$: $$\Gamma_\rho
u(\theta)=(2\pi)^{-d/2}\int_{{\mathbb R}^d}e^{-i\rho(\theta x)}W(x)u(x)dx$$ The main properties of this operator follow from Sobolev’s embedding theorems. Suppose that $W(x)\leq (1+|x|^2)^{-l/2}$ and $u\in L^2({\mathbb R}^d)$. Then the Fourier transformation of the function $W(x)u(x)$ belongs to the class $H^l({\mathbb R}^d)$, moreover the norm $||\hat{Wu}||_{H^l}$ is estimated by the norm $||u||_{L^2}$. According to Sobolev’s theorems, the embedding of the class $H^l({\mathbb R}^d)$ into the class $L^2({\mathbb S}_\rho)$ is continuous under the condition $l>1/2$. Moreover, the norm of the embedding operator depends in a weak manner on the parameter $\rho\geq1$. Indeed, suppose that the inequality $$\int_{{\mathbb S}_1}|\phi(\theta)|^2
d\theta\leq C\int_{{\mathbb R}^d}\Bigl(|\nabla^l\phi|^2+|\phi|^2\Bigr)dx$$ holds for any function $\phi\in H^l({\mathbb R}^d)$. Then setting $\phi(x)=u(\rho x)$ we obtain that $$\int_{{\mathbb S}_1}|u(\rho\theta)|^2
d\theta\leq C\int_{{\mathbb R}^d}\Bigl(\rho^{2l}|\nabla^l u(\rho x)|^2+|u(\rho x)|^2\Bigr)dx.$$ Multiplying both sides of this inequality by $\rho^{d-1}$, we obtain that $$\int_{{\mathbb S}_\rho}|u(x)|^2
dS\leq C\int_{{\mathbb R}^d}\Bigl(\rho^{2l-1}|\nabla^l u( x)|^2+\rho^{-1}|u( x)|^2\Bigr)dx.$$
If $l$ is close to $1/2$ then $\rho^{2l-1}$ practically behaves as a constant. Anyway, without loss of generality we can assume that for $\rho>1$ $$\int_{{\mathbb S}_\rho}|u(x)|^2
dS\leq C_\varepsilon\rho^{2\varepsilon}||u||^2_{H^l}$$ where $\varepsilon>0$ is an arbitrary small number. It implies that $$\label{1}
||\Gamma_\rho||\leq C_\varepsilon \rho^\varepsilon,\qquad \rho\geq 1.$$ Moreover, $\Gamma_\rho$ depends continuously on the parameter $\rho$ in the following sense. Let us introduce the operator $U_\rho$ that transforms functions on the sphere ${\mathbb S}_\rho$ into functions on the sphere ${\mathbb S}={\mathbb S}_1$. according to the rule $$U_\rho u(\theta)=u(\rho \theta) \rho^{(d-1)/2}.$$ This operator is unitary and therefore its norm equals 1. Define now the operator $Y_\rho=U_\rho \Gamma_\rho$. Our statement is that $$||Y_{\rho'}-Y_\rho||\leq C |\rho'-\rho|^{\alpha}\rho^\delta(\rho^\varepsilon+(\rho')^\varepsilon)$$ where $\alpha<l-1/2$, $\ \delta=l-\alpha-1/2$ and $\rho'>\rho\geq1$. Our arguments are similar to those we used in the proof of the inequality . If we assume that the inequality $$\int_{{\mathbb S}_1}|\phi((1+h)\theta)-\phi(\theta)|^2
d\theta\leq Ch^{2\alpha}\int_{{\mathbb R}^d}\Bigl(|\nabla^l\phi|^2+|\phi|^2\Bigr)dx$$ holds for any function $\phi\in H^l({\mathbb R}^d)$. Then the substitution $\phi(x)=u(\rho x)$ will lead to the inequality $$\int_{{\mathbb S}_1}|u((1+h)\rho\theta)-u(\rho\theta)|^2
d\theta\leq Ch^{2\alpha}\int_{{\mathbb R}^d}\Bigl(\rho^{2l}|\nabla^l u(\rho x)|^2+|u(\rho x)|^2\Bigr)dx.$$ Multiplying both sides of this inequality by $\rho^{d-1}$ and denoting $\rho'=(1+h)\rho$, we obtain that $$\int_{{\mathbb S}_\rho}|u(\rho^{-1}\rho'x)-u(x)|^2
dS\leq C|\rho'-\rho|^{2\alpha}\int_{{\mathbb R}^d}\Bigl(\rho^{2\delta}|\nabla^l u( x)|^2+\rho^{-2l}|u( x)|^2\Bigr)dx.$$ provided that $\rho'>\rho\geq1$. This leads to $$||\Bigl(\frac{\rho}{\rho'}\Bigr)^{(d-1)/2}Y_{\rho'}-Y_{\rho}||\leq C|\rho'-\rho|^{\alpha}\rho^{\delta}.$$ We apply now the triangle inequality to estimate the norm of the difference $Y_{\rho'}-Y_{\rho}$ for $\rho'>\rho\geq 1$ $$||Y_{\rho'}-Y_{\rho}||\leq \Bigl|\Bigl(\frac{\rho}{\rho'}\Bigr)^{(d-1)/2}-1\Bigr|\,
||Y_{\rho'}||+C|\rho'-\rho|^{\alpha}\rho^{\delta}\leq C_0 (\rho^\varepsilon+ (\rho')^\varepsilon)|\rho'-\rho|^{\alpha}\rho^{\delta}.$$ To be more convincing, we mention that $$\Bigl|\Bigl(\frac{\rho}{\rho'}\Bigr)^{(d-1)/2}-1\Bigr|\leq \min\{2^{-1}(d-1)|\rho'-\rho|,\, 2\}.$$ Introduce now the notation $G_\rho=\Gamma^*_\rho \Gamma_\rho$. Obviously, $G_\rho$ aslo has representation $G_\rho =Y^*_\rho Y_\rho $. Consequently, $$||G_{\rho'}-G_{\rho}||\leq ||Y^*_{\rho'}-Y^*_{\rho}||\cdot||Y_{\rho'}||+||Y^*_{\rho}||\cdot||Y_{\rho'}-Y_{\rho}||\leq C( \rho^\varepsilon+ (\rho')^\varepsilon)^2|\rho'-\rho|^{\alpha}\rho^{\delta}.$$ Let us summarize the results. The operator $X$ can be written in the form $$X=\int_0^\infty \frac{G_\rho d\rho}{\rho^2-\lambda},$$ where $$||G_\rho||\leq C\rho^{2\varepsilon},\qquad \rho\geq1,$$ and $$||G_{\rho'}-G_{\rho}||\leq C ( \rho^\varepsilon+ (\rho')^\varepsilon)^2|\rho'-\rho|^{\alpha}\rho^{\delta},\qquad \rho'>\rho\geq1.$$ Now, since the integral representation for the operator $X$ can be also rewritten in the form $$X=\int_1^\infty \frac{(G_\rho-G_\tau )d\rho}{\rho^2-\lambda}+\int_1^\infty \frac{G_\tau d\rho}{\rho^2-\lambda}+ W(-\Delta-\lambda)^{-1}E[0,1]W$$ where $\tau=|\Re\lambda|^{1/2}$ and $E[0,1]$ is the spectral projection of the operator $-\Delta$ corresponding to the interval $[0,1]$, we obtain that $$||X||\leq \int_1^\infty \frac{||G_\rho-G_\tau || d\rho}{|\rho^2-\lambda|}+\frac{\pi||G_\tau ||}{2|\lambda|^{1/2}}+ \frac{||V||_{L^\infty}+||G_\tau||}{(|\lambda|-1)},$$ for $ |\lambda|>1.$ Consequently, $$||X||\leq
C\int_0^\infty
\frac {|\rho-\tau|^\alpha ( \rho^\delta+|\Re\lambda|^{\delta/2})( \rho^\varepsilon+|\Re\lambda|^{\varepsilon/2})^2}
{|\rho^2-\Re\lambda|}d\rho+
C \frac {\tau^{2\varepsilon }}{|\lambda|^{1/2}}+ \frac {||V||_{L^\infty}+C\tau^{2\varepsilon}}{(|\lambda|-1)},$$ which leads to $$1\leq ||X||\leq C \Bigl(|\Re\lambda|^{(\alpha+\delta+2\varepsilon-1)/2}+
|\lambda|^{\varepsilon-1/2}+ \frac {1+|\lambda|^{\varepsilon}}{(|\lambda|-1)}\Bigr).$$ We proved the statement of the theorem for the case $L=1$. If $L\neq1$ then this inequality takes the form $$1 \leq C L\Bigl(|\Re\lambda|^{(\alpha+\delta+2\varepsilon-1)/2}+
|\lambda|^{\varepsilon-1/2}+ \frac {1+|\lambda|^{\varepsilon}}{(|\lambda|-1)}\Bigr).$$ The proof is completed. $\,\,\,\,\,\Box$
[99]{}
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Davies, E.B., Nath, J.: [*Schrödinger operators with slowly decaying potentials*]{}. J. Comput. Appl. Math. [**148**]{} (1), 1–28 (2002)
Frank, Rupert L.; Laptev, Ari; Lieb, Elliott H.; Seiringer, Robert [*Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials*]{}. Lett. Math. Phys. [**77**]{} (2006), no. 3, 309–316.
Laptev, A. and Safronov, O: [*Eigenvalue estimates for Schrödinger operators with complex potentials*]{} submitted.
Lieb, E. H. and Thirring, W.:[*Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities*]{}, in Studies in Mathematical Physics (Essays in Honor of Valentine Bargmann), 269–303. Princeton Univ. Press, Princeton, NJ, 1976.
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abstract: 'We show that the number of solutions of Schroedinger Maxwell system on a smooth bounded domain $\Omega\subset\mathbb{R}^{3}$. depends on the topological properties of the domain. In particular we consider the Lusternik-Schnirelmann category and the Poincaré polynomial of the domain.'
address:
- 'Dipartimento di Matematica, Università di Pisa, via F. Buonarroti 1/c, 56127 Pisa, Italy'
- 'Dipartimento di Matematica, Università di Pisa, via F. Buonarroti 1/c, 56127 Pisa, Italy'
author:
- Marco Ghimenti
- Anna Maria Micheletti
title: Low energy solutions for the semiclassical limit of Schroedinger Maxwell systems
---
[**Dedicated to our friend Bernhard**]{}
Introduction
============
Given real numbers $q>0$, $\omega>0$ we consider the following Schroedinger Maxwell system on a smooth bounded domain $\Omega\subset\mathbb{R}^{3}$. $$\left\{ \begin{array}{cc}
-\varepsilon^{2}\Delta u+u+\omega uv=|u|^{p-2}u & \text{ in }\Omega\\
-\Delta v=qu^{2} & \text{ in }\Omega\\
u,v=0 & \text{ on }\partial\Omega
\end{array}\right.\label{eq:sms}$$
This paper deals with the semiclassical limit of the system (\[eq:sms\]), i.e. it is concerned with the problem of finding solutions of (\[eq:sms\]) when the parameter $\varepsilon$ is sufficiently small. This problem has some relevance for the understanding of a wide class of quantum phenomena. We are interested in the relation between the number of solutions of (\[eq:sms\]) and the topology of the bounded set $\Omega$. In particular we consider the Lusternik Schnirelmann category $\operatorname{cat}\Omega$ of $\Omega$ in itself and its Poincaré polynomial $P_{t}(\Omega)$.
Our main results are the following.
\[thm:1\]Let $4<p<6$. For $\varepsilon$ small enough there exist at least $\operatorname{cat}(\Omega)$ positive solutions of (\[eq:sms\]).
\[thm:2\]Let $4<p<6$. Assume that for $\varepsilon$ small enough all the solutions of problem (\[eq:sms\]) are non- degenerate. Then there are at least $2P_{1}(\Omega)-1$ positive solutions.
Schroedinger Maxwell systems recently received considerable attention from the mathematical community. In the pioneering paper [@BF] Benci and Fortunato studied system (\[eq:sms\]) when $\varepsilon=1$ and without nonlinearity. Regarding the system in a semiclassical regime Ruiz [@R] and D’Aprile-Wei [@DW1] showed the existence of a family of radially symmetric solutions respectively for $\Omega=\mathbb{R}^{3}$ or a ball. D’Aprile-Wei [@DW2] also proved the existence of clustered solutions in the case of a bounded domain $\Omega$ in $\mathbb{R}^{3}$.
Recently, Siciliano [@S] relates the number of solution with the topology of the set $\Omega$ when $\varepsilon=1$, and the nonlinearity is a pure power with exponent $p$ close to the critical exponent $6$. Moreover, in the case $\varepsilon=1$, many authors proved results of existence and non existence of solution of (\[eq:sms\]) in presence of a pure power nonlinearity $|u|^{p-2}u$, $2<p<6$ or more general nonlinearities [@AR; @ADP; @AP; @BJL; @DM; @IV; @K; @PS; @WZ].
In a forthcoming paper [@GM1], we aim to use our approach to give an estimate on the number of low energy solutions for Klein Gordon Maxwell systems on a Riemannian manifold in terms of the topology of the manifold and some information on the profile of the low energy solutions.
In the following we always assume $4<p<6$.
Notations and definitions
=========================
In the following we use the following notations.
- $B(x,r)$ is the ball in $\mathbb{R}^{3}$ centered in $x$ with radius $r$.
- The function $U(x)$ is the unique positive spherically symmetric function in $\mathbb{R}^{3}$ such that $$-\Delta U+U=U^{p-1}\text{ in }\mathbb{R}^{3}$$ we remark that $U$ and its first derivative decay exponentially at infinity.
- Given $\varepsilon>0$ we define $U_{\varepsilon}(x)=U\left(\frac{x}{\varepsilon}\right)$.
- We denote by $\text{supp }\varphi$ the support of the function $\varphi$.
- We define $$m_{\infty}=\inf_{\int_{\mathbb{R}^{3}}|\nabla v|^{2}+v^{2}dx=
|v|_{L^{p}(\mathbb{R}^{3})}^{p}}\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}+v^{2}dx
-\frac{1}{p}|v|_{L^{p}(\mathbb{R}^{3})}^{p}$$
- We also use the following notation for the different norms for $u\in H_{g}^{1}(M)$: $$\begin{aligned}
\|u\|_{\varepsilon}^{2}=\frac{1}{\varepsilon^{3}}\int_{M}\varepsilon^{2}|\nabla u|^{2}+u^{2}dx & &
|u|_{\varepsilon,p}^{p}=\frac{1}{\varepsilon^{3}}\int_{\Omega}|u|^{p}dx\\
\|u\|_{H_{0}^{1}}^{2}=\int_{\Omega}|\nabla u|^{2}dx & & |u|_{p}^{p}=\int_{\Omega}|u|^{p}dx\end{aligned}$$ and we denote by $H_{\varepsilon}$ the Hilbert space $H_{0}^{1}(\Omega)$ endowed with the $\|\cdot\|_{\varepsilon}$ norm.
Let $X$ a topological space and consider a closed subset $A\subset X$. We say that $A$ has category $k$ relative to $X$ ($\operatorname{cat}_{M}A=k$) if $A$ is covered by $k$ closed sets $A_{j}$, $j=1,\dots,k$, which are contractible in $X$, and $k$ is the minimum integer with this property. We simply denote $\operatorname{cat}X=\operatorname{cat}_{X}X$.
Let $X_{1}$ and $X_{2}$ be topological spaces. If $g_{1}:X_{1}\rightarrow X_{2}$ and $g_{2}:X_{2}\rightarrow X_{1}$ are continuous operators such that $g_{2}\circ g_{1}$ is homotopic to the identity on $X_{1}$, then $\operatorname{cat}X_{1}\leq\operatorname{cat}X_{2}$ .
Let X be any topological space and let $H_{k}(X)$ denotes its $k$-th homology group with coefficients in $\mathbb{Q}$. The Poincaré polynomial $P_{t}(X)$ of $X$ is defined as the following power series in $t$ $$P_{t}(X):=\sum_{k\ge0}\left(\text{dim}H_{k}(X)\right)t^{k}$$
Actually, if $X$ is a compact space, we have that $\text{dim}H_{k}(X)<\infty$ and this series is finite; in this case, $P_{t}(X)$ is a polynomial and not a formal series.
\[rem:morse\]Let $X$ and $Y$ be topological spaces. If $f:X\rightarrow Y$ and $g:Y\rightarrow X$ are continuous operators such that $g\circ f$ is homotopic to the identity on $X$, then $P_{t}(Y)=P_{t}(X)+Z(t)$ where $Z(t)$ is a polynomial with non-negative coefficients.
These topological tools are classical and can be found, e.g., in [@P] and in [@B].
Preliminary results
===================
Using an idea in a paper of Benci and Fortunato [@BF] we define the map $\psi:H_{0}^{1}(\Omega)\rightarrow H_{0}^{1}(\Omega)$ defined by the equation $$-\Delta\psi(u)=qu^{2}\text{ in }\Omega\label{eq:psi}$$
\[lem:psi\]The map $\psi:H_{0}^{1}(\Omega)\rightarrow H_{0}^{1}(\Omega)$ is of class $C^{2}$ with derivatives $$\begin{aligned}
\psi'(u)[\varphi] & = & i^{*}(2qu\varphi)\label{eq:derprima}\\
\psi''(u)[\varphi_{1},\varphi_{2}] & = & i^{*}(2q\varphi_{1}\varphi_{2})\label{eq:derseconda}\end{aligned}$$ where the operator $i_{\varepsilon}^{*}:L^{p'},|\cdot|_{\varepsilon,p'}\rightarrow H_{\varepsilon}$ is the adjoint operator of the immersion operator $i_{\varepsilon}:H_{\varepsilon}\rightarrow L^{p},|\cdot|_{\varepsilon,p}$.
The proof is standard.
\[lem:Tder\]The map $T:H_{0}^{1}(\Omega)\rightarrow\mathbb{R}$ given by $$T(u)=\int_{\Omega}u^{2}\psi(u)dx$$ is a $C^{2}$ map and its first derivative is $$T'(u)[\varphi]=4\int_{\Omega}\varphi u\psi(u)dx.$$
The regularity is standard. The first derivative is $$T'(u)[\varphi]=2\int u\varphi\psi(u)+\int u^{2}\psi'(u)[\varphi].$$
By (\[eq:derprima\]) and (\[eq:psi\]) we have $$\begin{aligned}
2q\int u\varphi\psi(u) & = & -\int\Delta(\psi'(u)[\varphi])\psi(u)=-\int\psi'(u)[\varphi]\Delta\psi(u)=\\
& = & \int\psi'(u)[\varphi]qu^{2}\end{aligned}$$ and the claim follows.
At this point we consider the following functional $I_{\varepsilon}\in C^{2}(H_{0}^{1}(\Omega),\mathbb{R})$. $$I_{\varepsilon}(u)=\frac{1}{2}\|u\|_{\varepsilon}^{2}+\frac{\omega}{4}G_{\varepsilon}(u)-\frac{1}{p}|u^{+}|_{\varepsilon,p}^{p}\label{eq:ieps}$$ where $$G_{\varepsilon}(u)=\frac{1}{\varepsilon^{3}}\int_{\Omega}u^{2}\psi(u)dx=\frac{1}{\varepsilon^{3}}T(u).$$
By Lemma \[lem:Tder\] we have $$I_{\varepsilon}'(u)[\varphi]=\frac{1}{\varepsilon^{3}}\int_{\Omega}\varepsilon^{2}\nabla u\nabla\varphi+u\varphi+\omega u\psi(u)\varphi-(u^{+})^{p-1}\varphi$$ $$I_{\varepsilon}'(u)[u]=\|u\|_{\varepsilon}^{2}+\omega G_{\varepsilon}(u)-|u^{+}|_{\varepsilon,p}^{p}$$ then if $u$ is a critical points of the functional $I_{\varepsilon}$ the pair of positive functions $(u,\psi(u))$ is a solution of (\[eq:sms\]).
Nehari Manifold
===============
We define the following Nehari set $${\mathcal N}_{\varepsilon}=\left\{ u\in H_{0}^{1}(\Omega)\smallsetminus0\ :\ N_{\varepsilon}(u):=I'_{\varepsilon}(u)[u]=0\right\}$$ In this section we give an explicit proof of the main properties of the Nehari manifold, although standard, for the sake of completeness
${\mathcal N}_{\varepsilon}$ is a $C^{2}$ manifold and $\inf_{{\mathcal N}_{\varepsilon}}\|u\|_{\varepsilon}>0$.
If $u\in{\mathcal N}_{\varepsilon}$, using that $N_{\varepsilon}(u)=0$, and $p>4$ we have $$N'_{\varepsilon}(u)[u]=2\|u\|_{\varepsilon}^{2}+4\omega G_{\varepsilon}(u)-p|u^{+}|_{\varepsilon,p}
=(2-p)\|u\|_{\varepsilon}+(4-p)\omega G_{\varepsilon}(u)<0$$ so ${\mathcal N}_{\varepsilon}$ is a $C^{2}$ manifold.
We prove the second claim by contradiction. Take a sequence $\left\{ u_{n}\right\} _{n}\in{\mathcal N}_{\varepsilon}$ with $\|u_{n}\|_{\varepsilon}\rightarrow0$ while $n\rightarrow+\infty$. Thus, using that $N_{\varepsilon}(u)=0$, $$\|u_{n}\|_{\varepsilon}^{2}+\omega G_{\varepsilon}(u_{n})=|u_{n}^{+}|_{p,\varepsilon}^{p}\le C\|u_{n}\|_{\varepsilon}^{p},$$ so $$1<1+\frac{\omega G_{\varepsilon}(u)}{\|u_{n}\|_{\varepsilon}}\le C\|u_{n}\|_{\varepsilon}^{p-2}\rightarrow0$$ and this is a contradiction.
\[rem:nehari\]If $u\in{\mathcal N}_{\varepsilon}$, then $$\begin{aligned}
I_{\varepsilon}(u) & = & \left(\frac{1}{2}-\frac{1}{p}\right)\|u\|_{\varepsilon}^{2}+\omega\left(\frac{1}{4}-\frac{1}{p}\right)G_{\varepsilon}(u)\\
& = & \left(\frac{1}{2}-\frac{1}{p}\right)|u^{+}|_{p,\varepsilon}^{p}-\frac{\omega}{4}G_{\varepsilon}(u)\end{aligned}$$
It holds Palais-Smale condition for the functional $I_{\varepsilon}$ on ${\mathcal N}_{\varepsilon}$.
We start proving PS condition for $I_{\varepsilon}$. Let $\left\{ u_{n}\right\} _{n}\in H_{0}^{1}(\Omega)$ such that $$\begin{aligned}
I_{\varepsilon}(u_{n})\rightarrow c & &
\left|I'_{\varepsilon}(u_{n})[\varphi]\right|\le\sigma_{n}\|\varphi\|_{\varepsilon}\text{ where }\sigma_{n}\rightarrow0\end{aligned}$$ We prove that $\|u_{n}\|_{\varepsilon}$ is bounded. Suppose $\|u_{n}\|_{\varepsilon}\rightarrow\infty$. Then, by PS hypothesis $$\frac{pI_{\varepsilon}(u_{n})-I'_{\varepsilon}(u_{n})[u_{n}]}{\|u_{n}\|_{\varepsilon}}
=\left(\frac{p}{2}-1\right)\|u_{n}\|_{\varepsilon}+\left(\frac{p}{4}-1\right)\frac{G_{\varepsilon}(u_{n})}{\|u_{n}\|_{\varepsilon}}\rightarrow0$$ and this is a contradiction because $p>4$.
At this point, up to subsequence $u_{n}\rightarrow u$ weakly in $H_{0}^{1}(\Omega)$ and strongly in $L^{t}(\Omega)$ for each $2\le t<6$. Since $u_{n}$ is a PS sequence $$u_{n}+\omega i_{\varepsilon}^{*}(\psi(u_{n})u_{n})-i_{\varepsilon}^{*}\left((u_{n}^{+})^{p-1}\right)\rightarrow0\text{ in }H_{0}^{1}(\Omega)$$ we have only to prove that $i_{\varepsilon}^{*}(\psi(u_{n})u_{n})\rightarrow i_{\varepsilon}^{*}(\psi(u)u)$ in $H_{0}^{1}(\Omega)$, then we have to prove that $$\psi(u_{n})u_{n}\rightarrow\psi(u)u\text{ in }L^{t'}$$ We have $|\psi(u_{n})u_{n}-\psi(u)u|_{\varepsilon,t'}\le\left|\psi(u)(u_{n}-u)\right|_{\varepsilon,t'}
+\left|\left(\psi(u_{n})-\psi(u)\right)u_{n}\right|_{\varepsilon,t'}$. We get $$\int_{\Omega}|\psi(u_{n})-\psi(u)|^{\frac{t}{t-1}}|u_{n}|^{\frac{t}{t-1}}
\le\left(\int_{\Omega}|\psi(u_{n})-\psi(u)|^{t}\right)^{\frac{1}{t-1}}\left(\int_{\Omega}|u_{n}|^{\frac{t}{t-2}}\right)^{\frac{t-2}{t-1}}\rightarrow0,$$ thus we can conclude easily.
Now we prove PS condition for the constrained functional. Let $\left\{ u_{n}\right\} _{n}\in{\mathcal N}_{\varepsilon}$ such that $$\begin{array}{cc}
I_{\varepsilon}(u_{n})\rightarrow c\\
\left|I'_{\varepsilon}(u_{n})[\varphi]-\lambda_{n}N'(u_{n})[\varphi]\right|\le\sigma_{n}\|\varphi\|_{\varepsilon} & \text{ with }\sigma_{n}\rightarrow0
\end{array}$$ In particular $I'_{\varepsilon}(u_{n})\left[\frac{u_{n}}{\|u_{n}\|_{\varepsilon}}\right]
-\lambda_{n}N'(u_{n})\left[\frac{u_{n}}{\|u_{n}\|_{\varepsilon}}\right]\rightarrow0$. Then $$\lambda_{n}\left\{ \left(p-2\right)\|u_{n}\|_{\varepsilon}+\left(p-4\right)\omega\frac{G_{\varepsilon}(u_{n})}{\|u_{n}\|_{\varepsilon}}\right\} \rightarrow0$$ thus $\lambda_{n}\rightarrow0$ because $p>4$. Since $N'(u_{n})=u_{n}-i_{\varepsilon}^{*}(4\omega\psi(u_{n})u_{n})-pi_{\varepsilon}^{*}(|u_{n}^{+}|^{p-1})$ is bounded we obtain that $\left\{ u_{n}\right\} _{n}$ is a PS sequence for the free functional $I_{\varepsilon}$, and we get the claim
\[lem:teps\]For all $w\in H_{0}^{1}(\Omega)$ such that $|w^{+}|_{\varepsilon,p}=1$ there exists a unique positive number $t_{\varepsilon}=t_{\varepsilon}(w)$ such that $t_{\varepsilon}(w)w\in{\mathcal N}_{\varepsilon}$.
We define, for $t>0$ $$H(t)=I_{\varepsilon}(tw)=\frac{1}{2}t^{2}\|w\|_{\varepsilon}^{2}+\frac{t^{4}}{4}\omega G_{\varepsilon}(w)-\frac{t^{p}}{p}.$$ Thus $$\begin{aligned}
H'(t) & = & t\left(\|w\|_{\varepsilon}^{2}+t^{2}\omega G_{\varepsilon}(w)-t^{p-2}\right)\label{eq:Hprimo}\\
H''(t) & = & \|w\|_{\varepsilon}^{2}+3t^{2}\omega G_{\varepsilon}(w)-(p-1)t^{p-2}\label{eq:Hsec}\end{aligned}$$ By (\[eq:Hprimo\]) there exists $t_{\varepsilon}>0$ such that $H'(t_{\varepsilon})$. Moreover, by (\[eq:Hprimo\]), (\[eq:Hsec\]) and because $p>4$ we that $H''(t_{\varepsilon})<0$, so $t_{\varepsilon}$ is unique.
Main ingredient of the proof
============================
We sketch the proof of Theorem \[thm:1\]. First of all, since the functional $I_{\varepsilon}\in C^{2}$ is bounded below and satisfies PS condition on the complete $C^{2}$ manifold ${\mathcal N}_{\varepsilon}$, we have, by well known results, that $I_{\varepsilon}$ has at least $\operatorname{cat}I_{\varepsilon}^{d}$ critical points in the sublevel $$I_{\varepsilon}^{d}=\left\{ u\in H^{1}\ :\ I_{\varepsilon}(u)\le d\right\} .$$ We prove that, for $\varepsilon$ and $\delta$ small enough, it holds $$\operatorname{cat}\Omega\le\operatorname{cat}\left({\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}\right)$$ where $$m_{\infty}:=\inf_{{\mathcal N}_{\infty}}\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}+v^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|v|^{p}dx$$ $${\mathcal N}_{\infty}=\left\{ v\in H^{1}(\mathbb{R}^{3})\smallsetminus\left\{ 0\right\} \ :\ \int_{\mathbb{R}^{3}}|\nabla v|^{2}
+v^{2}dx=\int_{\mathbb{R}^{3}}|v|^{p}dx\right\} .$$ To get the inequality $\operatorname{cat}\Omega\le\operatorname{cat}\left({\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}\right)$ we build two continuous operators $$\begin{aligned}
\Phi_{\varepsilon} & : & \Omega^{-}\rightarrow{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}\\
\beta & : & {\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}\rightarrow\Omega^{+}.\end{aligned}$$ where $$\Omega^{-}=\left\{ x\in\Omega\ :\ d(x,\partial\Omega)<r\right\}$$ $$\Omega^{+}=\left\{ x\in\mathbb{R}^{3}\ :\ d(x,\partial\Omega)<r\right\}$$ with $r$ small enough so that $\operatorname{cat}(\Omega^{-})=\operatorname{cat}(\Omega^{+})=\operatorname{cat}(\Omega)$.
Following an idea in [@BC1], we build these operators $\Phi_{\varepsilon}$ and $\beta$ such that $\beta\circ\Phi_{\varepsilon}:\Omega^{-}\rightarrow\Omega^{+}$ is homotopic to the immersion $i:\Omega^{-}\rightarrow\Omega^{+}$. By the properties of Lusternik Schinerlmann category we have $$\operatorname{cat}\Omega\le\operatorname{cat}\left({\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}\right)$$ which ends the proof of Theorem \[thm:1\].
Concerning Theorem \[thm:2\], we can re-state classical results contained in [@B; @BC2] in the following form.
Let $I_{\varepsilon}$ be the functional (\[eq:ieps\]) on $H^{1}(\Omega)$ and let $K_{\varepsilon}$ be the set of its critical points. If all its critical points are non-degenerate then $$\sum_{u\in K_{\varepsilon}}t^{\mu(u)}=tP_{t}(\Omega)+t^{2}(P_{t}(\Omega)-1)+t(1+t)Q(t)\label{eq:morse1}$$ where Q(t) is a polynomial with non-negative integer coefficients and $\mu(u)$ is the Morse index of the critical point $u$.
By Remark \[rem:morse\] and by means of the maps $\Phi_{\varepsilon}$ and $\beta$ we have that $$P_{t}({\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta})=P_{t}(\Omega)+Z(t)\label{eq:morse2}$$ where $Z(t)$ is a polynomial with non-negative coefficients. Provided that $\inf_{\varepsilon}m_{\varepsilon}=:\alpha>0$, because ${\displaystyle \lim_{\varepsilon\rightarrow0}m_{\varepsilon}=m_{\infty}}$ (see \[eq:mepsminfty\]) , we have the following relations [@B; @BC2] $$P_{t}(I_{\varepsilon}^{m_{\infty}+\delta},I_{\varepsilon}^{\alpha/2})
=tP_{t}({\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta})\label{eq:morse3}$$ $$P_{t}(H_{0}^{1}(\Omega),I_{\varepsilon}^{m_{\infty}+\delta}))
=t(P_{t}(I_{\varepsilon}^{m_{\infty}+\delta},I_{\varepsilon}^{\alpha/2})-t)\label{eq:morse4}$$ $$\sum_{u\in K_{\varepsilon}}t^{\mu(u)}
=P_{t}(H_{0}^{1}(\Omega),I_{\varepsilon}^{m_{\infty}+\delta}))
+P_{t}(I_{\varepsilon}^{m_{\infty}+\delta},I_{\varepsilon}^{\alpha/2})+(1+t)\tilde{Q}(t)\label{eq:morse5}$$ where $\tilde{Q}(t)$ is a polynomial with non-negative integer coefficients. Hence, by (\[eq:morse2\]), (\[eq:morse3\]), (\[eq:morse4\]), (\[eq:morse5\]) we obtain (\[eq:morse1\]). At this point, evaluating equation (\[eq:morse1\]) for $t=1$ we obtain the claim of Theorem \[thm:2\]
The map $\Phi_{\varepsilon}$
=============================
For every $\xi\in\Omega^{-}$ we define the function $$W_{\xi,\varepsilon}(x)=U_{\varepsilon}(x-\xi)\chi(|x-\xi|)$$ where $\chi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ where $\chi\equiv1$ for $t\in[0,r/2)$, $\chi\equiv0$ for $t>r$ and $|\chi'(t)|\le2/r$.
We can define a map $$\begin{aligned}
\Phi_{\varepsilon} & : & \Omega^{-}\rightarrow{\mathcal N}_{\varepsilon}\\
\Phi_{\varepsilon}(\xi) & = & t_{\varepsilon}(W_{\xi,\varepsilon})W_{\xi,\varepsilon}\end{aligned}$$
\[w\]We have that the following limits hold uniformly with respect to $\xi\in\Omega$ $$\begin{aligned}
\|W_{\varepsilon,\xi}\|_{\varepsilon} & \rightarrow & \|U\|_{H^{1}(\mathbb{R}^{3})}\\
|W_{\varepsilon,\xi}|_{\varepsilon,t} & \rightarrow & \|U\|_{L^{t}(\mathbb{R}^{3})}\text{ for all }2\le t\le6\end{aligned}$$
\[lem:stimaGeps\]There exists $\bar{\varepsilon}>0$ and a constant $c>0$ such that $$G_{\varepsilon}(W_{\varepsilon,\xi})=\frac{1}{\varepsilon^{3}}\int_{\Omega}qW_{\varepsilon,\xi}^{2}(x)\psi(W_{\varepsilon,\xi})dx<c\varepsilon^{2}$$
It holds $$\begin{aligned}
\|\psi(W_{\varepsilon,\xi})\|_{H_{0}^{1}(\Omega)}^{2} & =
& \int_{\Omega}qW_{\varepsilon,\xi}^{2}(x)\psi(W_{\varepsilon,\xi})dx
\le q\|\psi(W_{\varepsilon,\xi})\|_{L^{6}(\Omega)}\left(\int_{\Omega}W_{\varepsilon,\xi}^{12/5}dx\right)^{5/6}\\
& \le & c\|\psi(W_{\varepsilon,\xi})\|_{H_{0}^{1}(\Omega)}
\left(\frac{1}{\varepsilon^{3}}\int_{\Omega}W_{\varepsilon,\xi}^{12/5}dx\right)^{5/6}\varepsilon^{5/2}\end{aligned}$$ By Remark \[w\] we have that $\|\psi(W_{\varepsilon,\xi})\|_{H_{0}^{1}(\Omega)}\le\varepsilon^{5/2}$ and the claim follows by applying again Cauchy Schwartz inequality.
\[prop:phieps\]For all $\varepsilon>0$ the map $\Phi_{\varepsilon}$ is continuous. Moreover for any $\delta>0$ there exists $\varepsilon_{0}=\varepsilon_{0}(\delta)$ such that, if $\varepsilon<\varepsilon_{0}$ then $I_{\varepsilon}\left(\Phi_{\varepsilon}(\xi)\right)<m_{\infty}+\delta$.
It is easy to see that $\Phi_{\varepsilon}$ is continuous because $t_{\varepsilon}(w)$ depends continously on $w\in H_{0}^{1}$.
At this point we prove that $t_{\varepsilon}(W_{\varepsilon,\xi})\rightarrow1$ uniformly with respect to $\xi\in\Omega$. In fact, by Lemma \[lem:teps\] $t_{\varepsilon}(W_{\varepsilon,\xi})$ is the unique solution of $$\|W_{\varepsilon,\xi}\|_{\varepsilon}^{2}+t^{2}\omega G_{\varepsilon}(W_{\varepsilon,\xi})-t^{p-2}|W_{\varepsilon,\xi}|_{\varepsilon,p}^{p}=0.$$ By Remark \[w\] and Lemma \[lem:stimaGeps\] we have the claim.
Now, we have $$I_{\varepsilon}\left(t_{\varepsilon}(W_{\varepsilon,\xi})W_{\varepsilon,\xi}\right)
=\left(\frac{1}{2}-\frac{1}{p}\right)\|W_{\varepsilon,\xi}\|_{\varepsilon}^{2}t_{\varepsilon}^{2}
+\omega\left(\frac{1}{4}-\frac{1}{p}\right)t_{\varepsilon}^{4}G_{\varepsilon}(W_{\varepsilon,\xi})$$ Again, by Remark \[w\] and Lemma \[lem:stimaGeps\] we have
$$I_{\varepsilon}\left(t_{\varepsilon}(W_{\varepsilon,\xi})W_{\varepsilon,\xi}\right)
\rightarrow\left(\frac{1}{2}-\frac{1}{p}\right)\|U\|_{H^{1}(\mathbb{R}^{3})}^{2}=m_{\infty}$$ that concludes the proof.
\[rem:limsup\]We set $$m_{\varepsilon}=\inf_{{\mathcal N}_{\varepsilon}}I_{\varepsilon.}$$ By Proposition \[prop:phieps\] we have that
$$\limsup_{\varepsilon\rightarrow0}m_{\varepsilon}\le m_{\infty.}\label{eq:limsup}$$
The map $\beta$
===============
For any $u\in{\mathcal N}_{\varepsilon}$ we can define a point $\beta(u)\in\mathbb{R}^{3}$ by $$\beta(u)=\frac{\int_{\Omega}x|u^{+}|^{p}dx}{\int_{\Omega}|u^{+}|^{p}dx}.$$ The function $\beta$ is well defined in ${\mathcal N}_{\varepsilon}$ because, if $u\in{\mathcal N}_{\varepsilon}$, then $u^{+}\neq0$.
We have to prove that, if $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$ then $\beta(u)\in\Omega^{+}$.
Let us consider partitions of $\Omega$. For a given $\varepsilon>0$ we say that a finite partition ${\mathcal P}_{\varepsilon}=\left\{ P_{j}^{\varepsilon}\right\} _{j\in\Lambda_{\varepsilon}}$ of $\Omega$ is a “good” partition if: for any $j\in\Lambda_{\varepsilon}$ the set $P_{j}^{\varepsilon}$ is closed; $P_{i}^{\varepsilon}\cap P_{j}^{\varepsilon}\subset\partial P_{i}^{\varepsilon}\cap\partial P_{j}^{\varepsilon}$ for any $i\ne j$; there exist $r_{1}(\varepsilon),r_{2}(\varepsilon)>0$ such that there are points $q_{j}^{\varepsilon}\in P_{j}^{\varepsilon}$ for which $B(q_{j}^{\varepsilon},\varepsilon)\subset P_{j}^{\varepsilon}\subset B(q_{j}^{\varepsilon},r_{2}(\varepsilon))
\subset B_{g}(q_{j}^{\varepsilon},r_{1}(\varepsilon))$, with $r_{1}(\varepsilon)\ge r_{2}(\varepsilon)\ge C\varepsilon$ for some positive constant $C$; lastly, there exists a finite number $\nu\in\mathbb{N}$ such that every $x\in\Omega$ is contained in at most $\nu$ balls $B(q_{j}^{\varepsilon},r_{1}(\varepsilon))$, where $\nu$ does not depends on $\varepsilon$.
\[lem:gamma\]There exists a constant $\gamma>0$ such that, for any $\delta>0$ and for any $\varepsilon<\varepsilon_{0}(\delta)$ as in Proposition \[prop:phieps\], given any “good” partition ${\mathcal P}_{\varepsilon}=\left\{ P_{j}^{\varepsilon}\right\} _{j}$ of the domain $\Omega$ and for any function $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$ there exists, for an index $\bar{j}$ a set $P_{\bar{j}}^{\varepsilon}$ such that $$\frac{1}{\varepsilon^{3}}\int_{P_{\bar{j}}^{\varepsilon}}|u^{+}|^{p}dx\ge\gamma.$$
Taking in account that $I'(u)[u]=0$ we have $$\begin{aligned}
\|u\|_{\varepsilon}^{2} & = & |u^{+}|_{\varepsilon,p}^{p}
-\frac{1}{\varepsilon^{3}}\int_{\Omega}\omega u^{2}\psi(u)\le|u^{+}|_{\varepsilon,p}^{p}
=\sum_{j}\frac{1}{\varepsilon^{3}}\int_{P_{j}}|u^{+}|^{p}\\
& = & \sum_{j}|u_{j}^{+}|_{\varepsilon,p}^{p}
=\sum_{j}|u_{j}^{+}|_{\varepsilon,p}^{p-2}|u_{j}^{+}|_{\varepsilon,p}^{2}
\le\max_{j}\left\{ |u_{j}^{+}|_{\varepsilon,p}^{p-2}\right\} \sum_{j}|u_{j}^{+}|_{\varepsilon,p}^{2}\end{aligned}$$ where $u_{j}^{+}$ is the restriction of the function $u^{+}$ on the set $P_{j}$.
At this point, arguing as in [@BBM Lemma 5.3], we prove that there exists a constant $C>0$ such that $$\sum_{j}|u_{j}^{+}|_{\varepsilon,p}^{2}\le C\nu\|u^{+}\|_{\varepsilon}^{2},$$ thus $$\max_{j}\left\{ |u_{j}^{+}|_{\varepsilon,p}^{p-2}\right\} \ge\frac{1}{C\nu}$$ that conludes the proof.
\[prop:conc\]For any $\eta\in(0,1)$ there exists $\delta_{0}<m_{\infty}$ such that for any $\delta\in(0,\delta_{0})$ and any $\varepsilon\in(0,\varepsilon_{0}(\delta))$ as in Proposition \[prop:phieps\], for any function $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$ we can find a point $q=q(u)\in\Omega$ such that $$\frac{1}{\varepsilon^{3}}\int_{B(q,r/2)}(u^{+})^{p}>\left(1-\eta\right)\frac{2p}{p-2}m_{\infty}.$$
First, we prove the proposition for $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\varepsilon}+2\delta}$.
By contradiction, we assume that there exists $\eta\in(0,1)$ such that we can find two sequences of vanishing real number $\delta_{k}$ and $\varepsilon_{k}$ and a sequence of functions $\left\{ u_{k}\right\} _{k}$ such that $u_{k}\in{\mathcal N}_{\varepsilon_{k}}$, $$m_{\varepsilon_{k}}\le I_{\varepsilon_{k}}(u_{k})=\left(\frac{1}{2}-\frac{1}{p}\right)\|u_{k}\|_{\varepsilon_{k}}^{2}
+\omega\left(\frac{1}{4}-\frac{1}{p}\right)G_{\varepsilon_{k}}(u_{k})\le m_{\varepsilon_{k}}
+2\delta_{k}\le m_{\infty}+3\delta_{k}\label{eq:mepsk}$$ for $k$ large enough (see Remark \[rem:limsup\]), and, for any $q\in\Omega$, $$\frac{1}{\varepsilon_{k}^{3}}\int_{B(q,r/2)}(u_{k}^{+})^{p}\le\left(1-\eta\right)\frac{2p}{p-2}m_{\infty}.$$ By Ekeland principle and by definition of ${\mathcal N}_{\varepsilon_{k}}$ we can assume $$\left|I'_{\varepsilon_{k}}(u_{k})[\varphi]\right|\le\sigma_{k}\|\varphi\|_{\varepsilon_{k}}\text{ where }\sigma_{k}\rightarrow0.\label{eq:ps}$$
By Lemma \[lem:gamma\] there exists a set $P_{k}^{\varepsilon_{k}}\in{\mathcal P}_{\varepsilon_{k}}$ such that $$\frac{1}{\varepsilon_{k}^{3}}\int_{P_{k}^{\varepsilon_{k}}}|u_{k}^{+}|^{p}dx\ge\gamma.$$ We choose a point $q_{k}\in P_{k}^{\varepsilon_{k}}$ and we define, for $z\in\Omega_{\varepsilon_{k}}:=\frac{1}{\varepsilon_{k}}\left(\Omega-q_{k}\right)$ $$w_{k}(z)=u_{k}(\varepsilon_{k}z+q_{k})=u_{k}(x).$$
We have that $w_{k}\in H_{0}^{1}(\Omega_{\varepsilon_{k}})\subset H^{1}(\mathbb{R}^{3})$. By equation (\[eq:mepsk\]) we have $$\|w_{k}\|_{H^{1}(\mathbb{R}^{3})}^{2}=\|u_{k}\|_{\varepsilon_{k}}^{2}\le C.$$ So $w_{k}\rightarrow w$ weakly in $H^{1}(\mathbb{R}^{3})$ and strongly in $L_{\text{loc}}^{t}(\mathbb{R}^{3})$.
We set $\psi(u_{k})(x):=\psi_{k}(x)=\psi_{k}(\varepsilon_{k}z+q_{k}):=\tilde{\psi}_{k}(z)$ where $x\in\Omega$ and $z\in\Omega_{\varepsilon_{k}}$. It is easy to verify that $$-\Delta_{z}\tilde{\psi}_{k}(z)=\varepsilon_{k}^{2}qw_{k}^{2}(z).$$ With abuse of language we set $$\tilde{\psi}_{k}(z)=\psi(\varepsilon_{k}w_{k}).$$ Thus $$\begin{aligned}
I_{\varepsilon_{k}}(u_{k}) & = & \frac{1}{2}\|u_{k}\|_{\varepsilon_{k}}^{2}-\frac{1}{p}|u_{k}^{+}|_{\varepsilon_{k},p}^{p}+\frac{\omega}{4}\frac{1}{\varepsilon_{k}^{3}}\int_{\Omega}qu_{k}^{2}\psi(u_{k})=\nonumber \\
& = & \frac{1}{2}\|w_{k}\|_{H^{1}(\mathbb{R}^{3})}^{2}-\frac{1}{p}\|w_{k}^{+}\|_{L^{p}(\mathbb{R}^{3})}^{p}+\frac{\omega}{4}\int_{\Omega_{\varepsilon_{k}}}qw_{k}^{2}\psi(\varepsilon_{k}w_{k})=\label{eq:ik}\\
& = & \frac{1}{2}\|w_{k}\|_{H^{1}(\mathbb{R}^{3})}^{2}-\frac{1}{p}\|w_{k}^{+}\|_{L^{p}(\mathbb{R}^{3})}^{p}+\varepsilon_{k}^{2}\frac{\omega}{4}\int_{\mathbb{R}^{3}}qw_{k}^{2}\psi(w_{k}):=E_{\varepsilon_{k}}(w_{k})\nonumber \end{aligned}$$ By definition of $E_{\varepsilon_{k}}:H^{1}(\mathbb{R}^{3})\rightarrow\mathbb{R},$ we get $E_{\varepsilon_{k}}(w_{k})\rightarrow m_{\infty}$.
Given any $\varphi\in C_{0}^{\infty}(\mathbb{R}^{3})$ we set $\varphi(x)=\varphi(\varepsilon_{k}z+q_{k}):=\tilde{\varphi_{k}}(z)$. For $k$ large enough we have that $\text{supp}\tilde{\varphi}_{k}\subset\Omega$ and, by (\[eq:ps\]), that $E'_{\varepsilon_{k}}(w_{k})[\varphi]=I'_{\varepsilon_{k}}(u_{k})[\tilde{\varphi}_{k}]\rightarrow0.$ Moreover, by definiton of $E_{\varepsilon_{k}}$ and by Lemma \[lem:Tder\] we have $$\begin{aligned}
E'_{\varepsilon_{k}}(w_{k})[\varphi] & =
& \left\langle w_{k},\varphi\right\rangle _{H^{1}(\mathbb{R}^{3})}
-\int_{\mathbb{R}^{3}}|w_{k}^{+}|^{p-1}\varphi+\omega\varepsilon_{k}^{2}\int_{\mathbb{R}^{3}}qw_{k}\psi(w_{k})\varphi+\\
& \rightarrow & \left\langle w,\varphi\right\rangle _{H^{1}(\mathbb{R}^{3})}-\int_{\mathbb{R}^{3}}|w^{+}|^{p-1}\varphi.\end{aligned}$$ Thus $w$ is a weak solution of $$-\Delta w+w=(w^{+})^{p-1}\text{ on }\mathbb{R}^{3}.$$ By Lemma \[lem:gamma\] and by the choice of $q_{k}$ we have that $w\ne0$, so $w>0$.
Arguing as in (\[eq:ik\]), and using that $u_{k}\in{\mathcal N}_{\varepsilon_{k}}$ we have $$\begin{aligned}
I_{\varepsilon_{k}}(u_{k}) & = & \left(\frac{1}{2}-\frac{1}{p}\right)\|u_{k}\|_{\varepsilon_{k}}^{2}
+\omega\left(\frac{1}{4}-\frac{1}{p}\right)\frac{1}{\varepsilon_{k}^{3}}\int_{\Omega}qu_{k}^{2}\psi(u_{k})\label{eq:ikH1}\\
& = & \left(\frac{1}{2}-\frac{1}{p}\right)\|w_{k}\|_{H^{1}(\mathbb{R}^{3})}^{2}
+\varepsilon_{k}^{2}\omega\left(\frac{1}{4}-\frac{1}{p}\right)\int_{\mathbb{R}^{3}}qw_{k}^{2}\psi(w_{k})\rightarrow m_{\infty}\nonumber \end{aligned}$$ and $$\begin{aligned}
I_{\varepsilon_{k}}(u_{k}) & = & \left(\frac{1}{2}-\frac{1}{p}\right)|u_{k}^{+}|_{p,\varepsilon_{k}}^{p}
-\frac{\omega}{4}\frac{1}{\varepsilon_{k}^{3}}\int_{\Omega}qu_{k}^{2}\psi(u_{k})\label{eq:ikLp}\\
& = & \left(\frac{1}{2}-\frac{1}{p}\right)|w_{k}^{+}|_{p}^{p}
-\varepsilon_{k}^{2}\frac{\omega}{4}\int_{\mathbb{R}^{3}}qw_{k}^{2}\psi(w_{k})\rightarrow m_{\infty}.\nonumber \end{aligned}$$ So, by (\[eq:ikH1\]) we have that $\|w\|_{H^{1}(\mathbb{R}^{3})}^{2}=\frac{2p}{p-2}m_{\infty}$ and that $\left(\frac{1}{2}-\frac{1}{p}\right)\|w_{k}\|_{H^{1}(\mathbb{R}^{3})}^{2}\rightarrow m_{\infty}$ and we conclude that $ $$w_{k}\rightarrow w$ strongly in $H^{1}(\mathbb{R}^{3})$.
Given $T>0$, by the definiton of $w_{k}$ we get, for $k$ large enough $$\begin{aligned}
|w_{k}^{+}|_{L^{p}(B(0,T))}^{p} & =
& \frac{1}{\varepsilon_{k}^{3}}\int_{B(q_{k},\varepsilon_{k}T)}|u_{k}^{+}|^{p}dx
\le\frac{1}{\varepsilon_{k}^{3}}\int_{B(q_{k},r/2)}|u_{k}^{+}|^{p}dx\nonumber \\
& \le & \left(1-\eta\right)\frac{2p}{p-2}m_{\infty}.\label{eq:contr}\end{aligned}$$ Then we have the contradiction. In fact, by (\[eq:ikLp\]) we have $\left(\frac{1}{2}-\frac{1}{p}\right)|w_{k}^{+}|_{p}^{p}\rightarrow m_{\infty}$ and this contradicts (\[eq:contr\]). At this point we have proved the claim for $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\varepsilon}+2\delta}$. Now, by the thesis for $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\varepsilon}+2\delta}$ and by (\[eq:ikLp\]) we have $$I_{\varepsilon_{k}}(u_{k})=\left(\frac{1}{2}-\frac{1}{p}\right)|u_{k}^{+}|_{p,\varepsilon_{k}}^{p}
+O(\varepsilon^{2})\ge(1-\eta)m_{\infty}+O(\varepsilon^{2})$$ and, passing to the limit, $$\liminf_{k\rightarrow\infty}m_{\varepsilon_{k}}\ge m_{\infty}.$$ This, combined by (\[eq:limsup\]) gives us that $$\lim_{\varepsilon\rightarrow0}m_{\varepsilon}=m_{\infty}.\label{eq:mepsminfty}$$ Hence, when $\varepsilon,\delta$ are small enough, ${\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}
\subset{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\varepsilon}+2\delta}$ and the general claim follows.
There exists $\delta_{0}\in(0,m_{\infty})$ such that for any $\delta\in(0,\delta_{0})$ and any $\varepsilon\in(0,\varepsilon(\delta_{0})$ (see Proposition \[prop:phieps\]), for every function $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$ it holds $\beta(u)\in\Omega^{+}$. Moreover the composition $$\beta\circ\Phi_{\varepsilon}:\Omega^{-}\rightarrow\Omega^{+}$$ is s homotopic to the immersion $i:\Omega^{-}\rightarrow\Omega^{+}$
By Proposition \[prop:conc\], for any function $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$, for any $\eta\in(0,1)$ and for $\varepsilon,\delta$ small enough, we can find a point $q=q(u)\in\Omega$ such that $$\frac{1}{\varepsilon^{3}}\int_{B(q,r/2)}(u^{+})^{p}>\left(1-\eta\right)\frac{2p}{p-2}m_{\infty}.$$ Moreover, since $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$ we have $$I_{\varepsilon}(u)=\left(\frac{p-2}{2p}\right)|u^{+}|_{p,\varepsilon}^{p}
-\frac{\omega}{4}\frac{1}{\varepsilon^{3}}\int_{\Omega}qu^{2}\psi(u)\le m_{\infty}+\delta.$$ Now, arguing as in Lemma \[lem:stimaGeps\] we have that $$\|\psi(u)\|_{H^{1}(\Omega)}^{2}=q\int_{\Omega}\psi(u)u^{2}\le C\|\psi(u)\|_{H^{1}(\Omega)}\left(\int_{\Omega}u^{12/5}\right)^{5/6},$$ so $\|\psi(u)\|_{H^{1}(\Omega)}\le\left(\int_{\Omega}u^{12/5}\right)^{5/6}$, then $$\begin{aligned}
\frac{1}{\varepsilon^{3}}\int\psi(u)u^{2} & \le
& \frac{1}{\varepsilon^{3}}\|\psi\|_{H^{1}(\Omega)}\left(\int_{\Omega}u^{12/5}\right)^{5/6}
\le C\frac{1}{\varepsilon^{3}}\left(\int_{\Omega}u^{12/5}\right)^{5/3}\\
& \le & C\varepsilon^{2}|u|_{12/5,\varepsilon}^{4}\le C\varepsilon^{2}\|u\|_{\varepsilon}^{4}\le C\varepsilon^{2}\end{aligned}$$ because $\|u\|_{\varepsilon}$ is bounded since $u\in{\mathcal N}_{\varepsilon}\cap I_{\varepsilon}^{m_{\infty}+\delta}$.
Hence, provided we choose $\varepsilon(\delta_{0})$ small enough, we have $$\left(\frac{p-2}{2p}\right)|u^{+}|_{p,\varepsilon}^{p}\le m_{\infty}+2\delta_{0}.$$ So, $$\frac{\frac{1}{\varepsilon^{3}}\int_{B(q,r/2)}(u^{+})^{p}}{|u^{+}|_{p,\varepsilon}^{p}}>\frac{1-\eta}{1+2\delta_{0}/m_{\infty}}$$ Finally, $$\begin{aligned}
|\beta(u)-q| & \le & \frac{\left|\frac{1}{\varepsilon^{3}}\int_{\Omega}(x-q)(u^{+})^{p}\right|}{|u^{+}|_{p,\varepsilon}^{p}}\\
& \le & \frac{\left|\frac{1}{\varepsilon^{3}}\int_{B(q,r/2)}(x-q)(u^{+})^{p}\right|}{|u^{+}|_{p,\varepsilon}^{p}}+\frac{\left|\frac{1}{\varepsilon^{3}}\int_{\Omega\smallsetminus B(q,r/2)}(x-q)(u^{+})^{p}\right|}{|u^{+}|_{p,\varepsilon}^{p}}\\
& \le & \frac{r}{2}+2\text{diam}(\Omega)\left(1-\frac{1-\eta}{1+2\delta_{0}/m_{\infty}}\right),\end{aligned}$$ so, choosing $\eta$, $\delta_{0}$ and $\varepsilon(\delta_{0})$ small enough we proved the first claim. The second claim is standard.
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|
---
author:
- 'Fraser Daly[^1], Fatemeh Ghaderinezhad[^2], Christophe Ley[^3] and Yvik Swan[^4]'
title: Simple variance bounds with applications to Bayesian posteriors and intractable distributions
---
[**Abstract**]{} Using coupling techniques based on Stein’s method for probability approximation, we revisit classical variance bounding inequalities of Chernoff, Cacoullos, Chen and Klaassen. Taking advantage of modern coupling techniques allows us to establish novel variance bounds in settings where the underlying density function is unknown or intractable. Applications include bounds for asymptotically Gaussian random variables using zero-biased couplings, bounds for random variables which are New Better (Worse) than Used in Expectation, and analysis of the posterior in Bayesian statistics.
[**Key words and phrases:**]{} Stein kernel; Stein operator; prior density; stochastic ordering; variance bound.
[**MSC 2010 subject classification:**]{} 60E15; 26D10; 62F15
Introduction {#sec:literature-review}
============
Weighted Poincaré (or isoperimetric) inequalities, giving upper bounds on the variance of a function of a random variable, have a long and rich history, beginning with the work of Chernoff [@Chernoff81]. Chernoff proved that if $X$ has a centred Gaussian distribution with variance $\sigma^2$, then $$\label{eq:14}
\mathrm{Var}[g(X)]\leq \sigma^2\mathbb{E}[(g^\prime(X))^2]\,,$$ for any absolutely continuous function $g:\mathbb{R}\mapsto\mathbb{R}$ such that $g(X)$ has finite variance. This inequality has since been generalized by many authors, including Cacoullos [@C82], Chen [@Chen82] and Klaassen [@Klaassen85]. To accompany these upper variance bounds, many of these authors have also established corresponding lower bounds, in the form of generalized Cramér-Rao inequalities. In particular in the centred Gaussian case we have $$\label{eq:12}
\mathrm{Var}[g(X)]\geq \sigma^2\mathbb{E}[g^\prime(X)]^2,$$ see [@C82]. The above cited works represent early entries in what is now a vast literature; we refer to [@ERS19vb1; @ERSvb2] for recent overviews of this large body of work.
The purpose of the present article is to revisit these classical variance bounding inequalities in light of the coupling techniques at the heart of Stein’s method for probability approximation (see, for example, [@ChGoSh11] and [@ley2017stein] for recent introductions to Stein’s method). These techniques allow us to establish upper and lower variance bounds in a variety of settings, including many in which the density of the underlying random variable is unknown or intractable. Making use, for example, of the zero-biased coupling allows us to establish explicit variance bounds for a wide range of situations in which the underlying random variable is known to be asymptotically Gaussian. In Sections \[sec:appl-stein-fram\]–\[sec:bounds-using-stoch\] we will consider a variety of situations where bounds may be derived using this, and other, couplings. Before doing so, we use the remainder of this section to outline the general coupling techniques we employ from Stein’s method, and how these can be used to establish upper and lower variance bounds in the spirit of Chernoff, Cacoullos, Chen and Klaassen.
Let $W$ be a real random variable on some fixed probability space. Let $\gamma$ be a real-valued function. We say that a pair of random variables $(T_1, T_2)$ (living on the same probability space as $W$) form a Stein coupling for $W$ with respect to $\gamma$ if $$\label{eq:1}
\mathbb{E} \left[ \gamma(W) \phi(W) \right] = \mathbb{E} \left[ T_1 \phi'(T_2) \right]$$ for all test functions $\phi \in C$ with $C\subset C^1({\mathbb R})$ some appropriately chosen class of functions. Although the choice $C=C_0^{\infty}({\mathbb R})$ is always allowed, it will generally be necessary to use $C$ as wide as possible; this fact is often reflected in the literature wherein one rather makes use of the generic expression “where $C$ is the class of functions for which expectations on both sides exist”.
We begin by showing an elementary argument allowing us to use to obtain tight upper variance bounds. To this end, suppose that $\gamma$ is a strictly increasing, differentiable function with exactly one sign change. Then in particular it is invertible and $\gamma^{-1}(0)$ is well-defined. Let $g$ be a real-valued differentiable function such that $\mathrm{Var}[g(W)]$ is finite. Following [@LS16] we write $$\begin{aligned}
\mathrm{Var}[g(W)] & \le \mathbb{E} \left[ \left( g(W) - g(\gamma^{-1}(0))
\right)^2 \right] = \mathbb{E} \left[ \left( \int_0^{\gamma(W)}
\frac{g'(\gamma^{-1}(u))}{\gamma'(\gamma^{-1}(u))}\mathrm{d}u \right)^2
\right] \\
& \le \mathbb{E} \left[ \gamma(W) \int_0^{\gamma(W)}
\left( \frac{g'(\gamma^{-1}(u))}{\gamma'(\gamma^{-1}(u))}
\right)^2\mathrm{d}u \right]\,,
\end{aligned}$$ where the equality follows by differentiability of $g$ and the subsequent inequality via Cauchy-Schwarz. Applying as well as Leibnitz’ rule for differentiating integrals we deduce the general upper variance bound $$\label{eq:2}
\mathrm{Var}[g(W)] \le \mathbb{E} \left[ \frac{T_1}{\gamma'(T_2)} \left(
g'(T_2) \right)^2 \right],$$ which holds as soon as the function $x \mapsto \int_0^{\gamma(x)} \left(
\frac{g'(\gamma^{-1}(u))}{\gamma'(\gamma^{-1}(u))} \right)^2\mathrm{d}u$ belongs to the (so far unspecified) class $C$. Note that inequality also holds if in we replace the equality sign by an increasing inequality.
Identity can also readily be combined with the Cauchy-Schwarz inequality to obtain lower variance bounds. To this end, consider a mean zero function $\gamma$ (this is in any case necessary for relationships such as to hold) for which $(\mathbb{E} \left[ \gamma(W) g(W) \right])^2 = (\mathbb{E} \left[
\gamma(W)(g(W) - \mathbb{E}[g(W)]) \right])^2 \le \mathbb{E}
\left[ \gamma(W)^2 \right] \mathrm{{Var}}[g(W)]$. Then from we deduce $$\label{eq:3}
\mathrm{Var}[g(W)]\ge \frac{(\mathbb{E} \left[ T_1 g'(T_2) \right])^2}{\mathrm{Var}
\left[ \gamma(W) \right]}$$ for all $g \in C$. As above, we note that inequality also holds if in we replace the equality sign by a decreasing inequality.
The rest of this paper is devoted to proposing situations wherein such couplings $W, T_1$ and $T_2$ occur naturally and may be used to establish upper and lower variance bounds. In Section \[sec:appl-stein-fram\] we use the framework of Stein kernels to express suitable couplings. Section \[sec:vb-via-biasing\] makes use of zero-biased couplings to derive variance bounds suitable for random variables which are asymptotically Gaussian. Finally, in Section \[sec:bounds-using-stoch\] we consider random variables satisfying certain stochastic or convex ordering assumptions, which allow us to derive bounds sharper than we would otherwise obtain with our method. Some proofs and additional examples illustrating the results of Section \[sec:appl-stein-fram\] are deferred to the appendices.
Stein kernel and a bound of Cacoullos {#sec:appl-stein-fram}
=====================================
Suppose that the target $W$ has a differentiable density $p$ with interval support. Following, for example, [@CPU94] and [@ERS19vb1], we define the *Stein kernel* of $W$ as the function $\tau$ satisfying $$\label{eq:6}
\mathrm{Cov}\left[W, \phi (W)\right] = \mathbb{E} \left[
\tau(W) \phi'(W) \right]$$ for all functions $\phi$ such that either integral is defined. See [@ERS19vb1] for an extensive discussion of this function. In the notation of Section \[sec:literature-review\], this means that we can take $\gamma(x) = x-\mathbb{E}[W]$, $T_1 = \tau(W)$ and $T_2 = W$ in . Note that $\mathbb{E}[\tau(W)] = \mathrm{Var}[W]$. Applying and , we get for all $g \in L^2(W)$ that $$\label{eq:8}
\frac{ \mathbb{E} \left[
\tau(W) g'(W)
\right]^2}{\mathrm{Var} \left[ W
\right]} \le \mathrm{{Var}}[g(W)]
\le \mathbb{E} \left[ \tau(W) \left( g'(W) \right)^2
\right],$$ which is nothing but a restatement of classical bounds already available in [@C82].
Of course for to be of use it remains to identify situations in which the Stein kernel has an agreeable form. We give several such situations.
Following [@nourdin2013integration], it is easy to see that if $W = n^{-1/2} \sum_{i=1}^n X_i$, where the $X_i$ are centred, independent random variables with Stein kernel $\tau_i(\cdot)$ and common variance $\sigma^2$, then $ \tau_W(w) = \frac{1}{n} \sum_{i=1}^n\mathbb{E}[\tau_i(X_i) \, |
\, W = w]$ is a Stein kernel for $W$. If the $X_i$ are copies of $X_1$ with kernel $\tau_{1}(\cdot)$, becomes $$\frac{\mathbb{E}\left[ \tau_1(X_1) g'(W)\right]^2}{\sigma^2} \le
\mathrm{Var}[g(W)] \le
\mathbb{E}\left[\tau_1(X_1) (g'(W))^2\right]$$ If $W$ and $X_1$ were independent, we could use $\mathbb{E}[\tau_1(X_1)]= \sigma^2$ to recover the Gaussian case stated in and . Here we need to apply a limited development to make independence appear. Let $U \sim \mathrm{Unif}[0,1]$ and recall the mean-value theorem $g'(x+t) = g'(x) + t \mathbb{E}[g''(x+Ut)]$. Let $W^{(1)} = W - n^{-1/2}X_1$. Then, by independence, if $g$ is twice differentiable the lower bound becomes $ \sigma^2\mathbb{E}[g'(W^{(1)})]^2 + \frac{C_1 }{\sqrt n}$ where $C_1 =C_1(g,n)$ is given by $C_1= 2\mathbb{E}[g'(W^{(1)})]\mathbb{E}[ \tau_1(X_1)
X_1g''(W^{(1)}+n^{-1/2}UX_1)] + n^{-1/2}/\sigma^2\mathbb{E}[ \tau_1(X_1)
X_1g''(W^{(1)}+n^{-1/2}UX_1)]^2$. Clearly $\lim_{n\rightarrow\infty} C_1(g,n)/\sqrt{n}=0$ for all $g$. Similar considerations apply for the upper bound. Indeed, recall that for a twice differentiable function $g$ we have $$\label{eq:WstarTVI}
\left|g^\prime(x+t)^2-g^\prime(x)^2\right|\leq2\|
g^\prime
g^{\prime\prime} \| |t|\,,$$ (where $\lVert\cdot\rVert$ is the supremum norm) so that we have $\mathbb{E}[\tau_1(X_1) (g'(W))^2] \le \sigma^2
\mathbb{E}[(g'(W^{(1)}))^2] + \frac{2}{\sqrt n} \| g' g''\| \sigma^2
\mathbb{E}[|X_1|] =: \sigma^2 \mathbb{E}[(g'(W^{(1)}))^2] +
\frac{C_2}{\sqrt n}$. Wrapping up, $$\sigma^2
\mathbb{E}[(g'(W^{(1)}))]^{2} + \frac{C_1}{\sqrt n} \le \mathrm{Var}[g(W)] \le \sigma^2
\mathbb{E}[(g'(W^{(1)}))^2] + \frac{C_2}{\sqrt n},$$ where the proximity with the corresponding inequalities for the Gaussian case are now made explicit.
\[eg:smoothing\] Let $Y$ be a real-valued random variable with $\mathbb{E}[Y]=\mu$. Note that we do not require $Y$ to have a density function, and the bounds of this example apply if, for instance, $Y$ is a discrete random variable. In order to allow us to derive variance bounds for $Y$ using our approach, we smooth it by convolving it with independent Gaussian noise with small variance. We let $Z\sim\mathcal{N}(0,\epsilon^2)$ have a Gaussian distribution, independent of $Y$. Let $\varphi_\epsilon$ and $\Phi_\epsilon$ be the density and distribution functions of $Z$, respectively, and define $$\label{eq:smooth}
\tau_{\epsilon}(x)=\epsilon^2+\frac{\mathbb{E}\left[(Y^\prime-\mu)\bar{\Phi}_\epsilon(x-Y^\prime)\right]}{\mathbb{E}[\varphi_\epsilon(x-Y^\prime)]}\,,$$ where $\bar{\Phi}_\epsilon(y)=1-\Phi_\epsilon(y)$ and $Y^\prime$ is an independent copy of $Y$. Then $\tau_{\epsilon}(x)$ is a Stein kernel for $Y+Z$ (see Appendix \[sec:proofs\]) and applies to all differentiable functions $g:\mathbb{R}\mapsto\mathbb{R}$ such that $\mathrm{Var}[g(Y+Z)]$ is finite. Moreover, the following hold:
- If the mapping $x\mapsto\left(g(x)-\mathbb{E}[g(Y+Z)]\right)^2$ is convex, then $$\mathrm{Var}[g(Y)]\leq\mathbb{E}\left[\tau_{\epsilon}(Y+Z)g^\prime(Y+Z)^2\right]\,.$$
- If the mapping $x\mapsto (g(x)-\mathbb{E}[g(Y)])^2$ is concave, then $$\mathrm{Var}\left[g(Y)\right]\geq\frac{\mathbb{E}\left[\tau_{\epsilon}(Y+Z)g^\prime(Y+Z)\right]^2}{\epsilon^2+\mathrm{Var}[Y]}\,.$$
We defer the proofs of these claims to Appendix \[sec:proofs\].
\[eg:Pearson\]
As is well known, the Pearson family has explicit Stein kernels given by Proposition \[prop:perason\] recalled in the Appendix. Such a result is particularly useful in the following situation inherited from Bayesian statistics. In a Bayesian setting, the initial distribution of the parameter of interest is some prior distribution with density $ \pi_0(\theta)$; upon observing data points $\mathbf{x}=(x_1, \ldots, x_n)$ sampled independently with sampling distribution $\pi(\theta, \mathbf{x})$ we update from the prior to the posterior density given by $\pi_2(\theta) = \kappa_2(\mathbf{x})\pi(\theta,
\mathbf{x})\pi_0(\theta)$. We use the notation $\Theta_0$ to indicate the distribution of the parameter under the prior, $\Theta_2$ its distribution under the posterior, and $X$ a random variable following the same common distribution of the observations. We also write $\Theta_1$ for the parameter under the sampling distribution $\pi_1(\theta) = \kappa_1(\mathbf{x}) \pi(\theta, \mathbf{x})$, which corresponds to a posterior with flat (uninformative) prior. A popular choice of prior is that of a *conjugate* prior for which the mathematical properties of the posterior are the same as those of the sampling distribution; the impact of the data is then visible in the parameters of the posterior distribution who are updated. Restricting our attention to Pearson distributed families, we can apply Proposition \[prop:perason\] and read variance bounds directly from the updated parameters. For instance:
- Gaussian data, inference on mean, Gaussian prior: If $X \sim \mathcal{N}(\theta, \sigma^2)$ with $\theta\in{\mathbb R}$ and fixed $\sigma>0,$ and $\Theta_0 \sim \mathcal{N}(\mu, \delta^2)$ with $\mu\in{\mathbb R},\delta>0$, then $\Theta_2 \sim
\mathcal{N}\left(\frac{\sigma^2\mu+n\delta^2\bar{x}}{n\delta^2+\sigma^2},\frac{\sigma^2\delta^2}{n\delta^2+\sigma^2}\right)$, where $\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i$. The Stein kernel for this Gaussian distribution is $\tau(\theta) = ( \frac{n}{\sigma^2} +
\frac{1}{\delta^2})^{-1}$. Consequently, $$\mathbb{E} \left[g'(\Theta_2) \right]^2 \leq \bigg( \frac{n}{\sigma^2} +
\frac{1}{\delta^2}\bigg) \mathrm{Var}[g(\Theta_2)] \leq \mathbb{E} [g'(\Theta_2)^2]$$ for all suitable $g$, all $n$ and all values of the parameters.
- Gaussian data, inference on variance, Inverse Gamma prior: If $X \sim \mathcal{N}(\mu, \theta)$ with $\theta>0$ and fixed $\mu\in{\mathbb R}$, and $\Theta_0 \sim \mathcal{IG}(\alpha, \beta)$ has an Inverse Gamma distribution with density $$\theta \mapsto \frac{\beta^\alpha}{\Gamma(\alpha)} \theta^{-\alpha -1} \exp \left( -\frac{\beta}{\theta} \right), \, \alpha,\beta>0,$$ then $\Theta_2 \sim \mathcal{IG}\left(\frac{n}{2} + \alpha, \frac{1}{2}
\sum_{i=1}^n (x_i - \mu)^2 + \beta\right)$. The Stein kernel for this Inverse Gamma distribution is $\tau (\theta) =
\frac{\theta^2}{\frac{n}{2}+\alpha-1}$. Consequently, for all suitable $g$, $$\frac{(\frac{n}{2} + \alpha-2)}{(\frac{1}{2} \sum_{i=1}^n (x_i - \mu)^2 + \beta)^2} \mathbb{E} [\Theta_2^2g'(\Theta_2)]^2 \leq \mathrm{Var}[g(\Theta_2)] \leq \frac{1}{\frac{n}{2}+\alpha-1} \mathbb{E} [\Theta_2^2g'(\Theta_2)^2].$$
- Binomial data, inference on proportion, Beta prior: If $X \sim {Bin}(n, \theta)$ with $\theta\in[0,1]$, and $\Theta_0 \sim{Beta}(\alpha, \beta)$ with density $$\theta \mapsto \frac{\theta^{\alpha-1}(1-\theta)^{\beta-1}}{\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}}, \, \alpha,\beta>0,$$ then $\Theta_2 \sim {Beta}\left(x+\alpha,n-x+\beta\right)$, where $x$ denotes the observed number of successes. The Stein kernel for this Beta distribution is $\tau (\theta) = \frac{\theta (1 - \theta)}{n+ \alpha +
\beta}$. Consequently, for all suitable $g$, $$\frac{(n+\alpha +\beta+1)}{(x+\alpha )(n - x + \beta)} \mathbb{E} [\Theta_2 (1-\Theta_2) g'(\Theta_2)]^2 \leq \mathrm{Var}[g (\Theta_2)] \leq \frac{\mathbb{E}[\Theta_2 (1 - \Theta_2)g'(\Theta_2)^2]}{n+\alpha +\beta}.$$
Further examples are provided in Appendix \[sec:more-examples-1\].
Variance bounds from zero-biased couplings {#sec:vb-via-biasing}
==========================================
In this section, we suppose that the target $W$ has mean zero, finite variance $\sigma^2$, and can be coupled to some random variable $W^{\star}$ through $$\label{eq:bias}
\mathbb{E}[W\phi(W)] = \sigma^2 \mathbb{E} [\phi'(W^\star)]$$ for all functions $\phi:\mathbb{R}\mapsto\mathbb{R}$. Such $W^{\star}$ always exists, and its law is unique. It has the $W$-zero-biased distribution; see, e.g., [@ChGoSh11 Section 2.3.3] and references therein for more details. Note that $W^\star$ is a continuous random variable, regardless of whether $W$ is discrete or continuous. Under , we immediately obtain $$\label{eq:zb1}
\sigma^2\mathbb{E}\left[g^\prime(W^\star)\right]^2\le
\mathrm{Var}[g(W)]\leq\sigma^2\mathbb{E}\left[g^\prime(W^\star)^2\right]$$ by using and with $\gamma(x) = x$, $T_1 = \sigma^2$ and $T_2 = W^{\star}$ for all $g:\mathbb{R}\mapsto\mathbb{R}$ for which $\mathrm{Var}[g(W)]$ is finite. Obviously it may be of interest to express in terms of the original variable. Using , we obtain the following result.
Let $W$ have mean zero and finite variance $\sigma^2$, and $W^{\star}$ have the $W$-zero biased distribution. Then $$\label{eq:bdd}
\mathrm{Var}[g(W)]\leq\sigma^2\mathbb{E}\left[g^\prime(W)^2\right]+
2\sigma^2\|g^\prime
g^{\prime\prime} \|\mathbb{E}|W^\star-W|$$ for all twice differentiable functions $g:\mathbb{R}\mapsto\mathbb{R}$ for which $\mathrm{Var}[g(W)]$ exists.
It is classical that the Gaussian distribution is the unique fixed point of the zero-bias transform, in the sense that $W\sim \mathcal{N}(0, \sigma^2)$ if and only if $W = W^{\star}$. Hence $|W^\star - W|$ gives information on the distributional proximity between the law $\mathcal{L}(W)$ of $W$ and $\mathcal{N}(0, \sigma^2)$. Also, it is classical that the Gaussian is characterized by the fact that $ \sigma^2 = \sup_g {\mathrm{Var}[g(W)]}/{\mathbb{E}[g'(W)^2]}$, see, e.g., [@CPU94]. Inequality captures these two essential features of the Gaussian distribution.
Let $X_1,X_2,\ldots,X_n$ be independent mean zero random variables with finite variances $\mathbb{E}[X_i^2] = \sigma^2_i, i=1,\ldots,n$. Set $W = X_1 + \cdots + X_n$ and $\mathbb{E}[W^2] = \sigma^2=\sum_{i=1}^n\sigma_i^2$. Let $I$ be a random index independent of all else such that $P(I = i) = \sigma^2_i/\sigma^2$ and let $W_i
= W-X_i$. Finally let $X_i^{\star}$ be the zero-bias transform of $X_i$. Then $ W^{\star}-W = X_I - X_I^{\star}$ (see Example 2.1 of [@GR97]) so that the bound becomes $$\begin{aligned}
\mathrm{Var}[g(W)] & \le \sigma^2 \mathbb{E}[g'(W)^2]
+ 2 \|g'g''\| \sum_{i=1}^n \sigma^2_i \mathbb{E}[|X_i - X_i^{\star}|] .
\end{aligned}$$ If, furthermore, we suppose the summands to be independent copies of $X$ such that $\sigma^2 = 1$ then $$\begin{aligned}
\mathrm{Var}[g(W)] \le \mathbb{E}[g'(W)^2]
+ 2 \|g'g''\| \mathbb{E}[|X - X^{\star}|]\,.
\end{aligned}$$ To see how this plays out in practice, suppose that $X = (\xi-p)/\sqrt{npq}$ with $\xi$ Bernoulli with success parameter $p$. Following [@ChGoSh11 Corollary 4.1], we obtain $\mathbb{E}[|X - X^{\star}|] = (p^2+q^2)/(2\sqrt{npq})$ and $$\mathrm{Var}[g(W)] \le \sigma^2 \mathbb{E}[g'(W)^2]
+ \|g'g''\| \frac{p^2+q^2}{\sqrt{npq}}.$$ Many other examples can be explicitly worked out along these lines.
Let $(a_{i,j})_{i,j=1}^n$ be an array of real numbers and $\pi$ a uniformly chosen permutation of $\{1,\ldots,n\}$. Let $W=\sum_{i=1}^na_{i,\pi(i)}$. We further define $$a_{\bullet\bullet}=\frac{1}{n^2}\sum_{i,j=1}^na_{i,j}\,,\quad
a_{i\bullet}=\frac{1}{n}\sum_{j=1}^na_{i,j}\,,\quad\mbox{and}\quad
a_{\bullet j}=\frac{1}{n}\sum_{i=1}^na_{i,j}\,,$$ and note that $\mathbb{E}[W]=na_{\bullet\bullet}$ and $$\mathrm{Var}[W]=\sigma^2=\frac{1}{n-1}\sum_{i,j=1}^n\left(a_{i,j}-a_{i\bullet}-a_{\bullet j}+a_{\bullet\bullet}\right)^2\,.$$ See, for example, [@ChGoSh11 Section 4.4]. Letting $Z=\sigma^{-1}(W-na_{\bullet\bullet})$ and $C=\max_{1\leq i,j\leq n}|a_{i,j}-a_{i\bullet}-a_{\bullet
j}+a_{\bullet\bullet}|$, the proof of Theorem 6.1 of [@ChGoSh11] shows that $\mathbb{E}|Z^\star-Z|\leq8C\sigma^{-1}$ for some positive constant $C$, and so we have from (\[eq:bdd\]) that $$\mathrm{Var}[g(Z)]\leq\mathbb{E}\left[g^\prime(Z)^2\right]+\frac{16C}{\sigma}\|g^\prime
g^{\prime\prime}\|\,,$$ for all twice differentiable $g$ such that $\mathrm{Var}[g(Z)]$ is finite.
Variance bounds using stochastic ordering {#sec:bounds-using-stoch}
=========================================
We consider now some further applications in which we do not require explicit knowledge of the density of $W$ in order to derive bounds on $\mathrm{Var}[g(W)]$ using our techniques. Unlike those examples in Section \[sec:vb-via-biasing\], the bounds we obtain here have the same form as in applications where we employ the exact expression for the underlying density, as in Section \[sec:appl-stein-fram\], without any additional ‘remainder’ terms. We may obtain such bounds under natural assumptions on the random variable $W$, which we express in terms of stochastic orderings; the price we pay is in some restriction on the class of functions $g$ for which the bounds apply.
We begin by recalling the definitions of the orderings which we will use. For any random variables $X$ and $Y$, we will say that $X$ is stochastically smaller than $Y$ (denoted $X\leq_{st}Y$) if $\mathbb{P}(X>t)\leq\mathbb{P}(Y>t)$ for all $t$. We will say that $X$ is smaller than $Y$ in the convex order (denoted $X\leq_{cx}Y$) if $\mathbb{E}[\phi(X)]\leq\mathbb{E}[\phi(Y)]$ for all convex functions $\phi$ for which the expectations exist. See [@ss07] for background and many further details.
Zero-biased couplings and the convex order {#sec:ConvexOrder}
------------------------------------------
Let $W$ be a real-valued random variable with mean zero and variance $\sigma^2$. Recall the definition (\[eq:bias\]) of $W^\star$, the zero-biased version of $W$. We note that, from Lemma 2.1(ii) of [@GR97], $W^\star$ is supported on the closed convex hull of the support of $W$ and has density function given by $$\label{eq:density}
p^\star_W(w)=\frac{1}{\sigma^2}\mathbb{E}[WI(W>w)]\,.$$
If we assume that $W^\star\leq_{cx}W$, then we may write $$\label{eq:conv}
\mathbb{E}[W\phi(W)]=\sigma^2\mathbb{E}[\phi^\prime(W^\star)]\leq\sigma^2\mathbb{E}[\phi^\prime(W)]\,,$$ for all differentiable functions $\phi$ such that $\phi^\prime$ is convex. That is, (\[eq:1\]) holds with the equality replaced by an inequality for all such $\phi$, with the choices $\gamma(W)=W$, $T_1=\sigma^2$, and $T_2=W$.
Following the proof of (\[eq:2\]), the inequality (\[eq:conv\]) is sufficient to obtain this upper bound on $\mathrm{Var}[g(W)]$. In proving this bound, we apply (\[eq:conv\]) with $\phi$ such that $\phi^\prime(x)=g^\prime(x)^2$; we must therefore assume that $g^\prime(x)^2$ is convex in order to do this. We thus obtain the following bound.
\[theorem:convex\] Let $W$ have mean 0 and variance $\sigma^2$, and assume that $W^\star\leq_{cx}W$. For all differentiable $g:\mathbb{R}\mapsto\mathbb{R}$ such that $\mathrm{Var}[g(W)]$ exists and $g^\prime(x)^2$ is convex, $$\label{eq:convex_bd}
\mathrm{Var}[g(W)]\leq\sigma^2\mathbb{E}[g^\prime(W)^2]\,.$$
Let $W=X_1+X_2+\cdots+X_n$, where $X_1,X_2,\ldots,X_n$ are independent, mean-zero random variables, with $X_i$ supported on the set $\{-a_i,b_i\}$ for $a_i,b_i>0$, for each $i=1,\ldots,n$. That is, $\mathbb{P}(X_i=-a_i)=p_i=1-\mathbb{P}(X_i=b_i)$ for $1\leq i\leq n$, where $p_i=b_i/(a_i+b_i)$ so that $\mathbb{E}[X_i]=0$. Let $\sigma_i^2=\mathrm{Var}(X_i)$ and $\sigma^2=\sigma_1^2+\cdots+\sigma_n^2$.
A straightforward calculation using (\[eq:density\]) shows that, for each $i=1,\ldots,n$, $X_i^\star$ is uniformly distributed on the interval $[-a_i,b_i]$. Hence, Theorem 3.A.44 of [@ss07] gives that $X_i^\star\leq_{cx}X_i$ for each $i$.
Let $I$ be a random index, chosen independently of all else, with $\mathbb{P}(I=i)=\sigma_i^2/\sigma^2$, for $i=1,\ldots,n$. Now, using Lemma 2.1(v) of [@GR97], $W^\star$ is equal in distribution to $X_I^\star+\sum_{j\not=I}X_j$, which is smaller than $W$ in the convex order for each possible value of $I$ by (3.A.46) of [@ss07]. It then follows from Theorem 3.A.12(b) of [@ss07] that $W^\star\leq_{cx}W$, and hence our upper bound (\[eq:convex\_bd\]) applies.
Equilibrium couplings {#sec:StocOrder}
---------------------
Throughout this section, let $W$ be a non-negative random variable with mean $\lambda^{-1}$. Following, for example, [@PeRo11], we say that a random variable $W^e$ has the equilibrium distribution with respect to $W$ if $$\label{eq:EqCoup}
\mathbb{E}[\phi(W)]-\phi(0)=\lambda^{-1}\mathbb{E}[\phi^\prime(W^e)]\,,$$ for all a.e. differentiable functions $\phi$.
Note that this definition is motivated by the fact that $W$ is Exponential if and only if $W$ and $W^e$ are equal in distribution. Applying the definition to the function $\phi_x(w) = (w-x) \mathbb{I}(w \ge x)$ and integrating by parts we obtain that $\mathbb{P}(W^e>x)=\lambda\int_x^\infty\mathbb{P}(W>y)\,\mathrm{d}y$ for all $x \ge 0$.
In this section we consider random variables that are new better than used in expectation (NBUE) and new worse than used in expectation (NWUE). Recall that $W$ is NBUE if $\lambda\int_x^\infty \mathbb{P}(W>s)\,ds\leq\mathbb{P}(W>x)$ for all $x\geq0$, and that $W$ is NWUE if this holds with the inequality reversed. These properties are well-known in reliability theory; see, for example, [@ss07].
From this definition and the remark above, it is clear that $W$ is NBUE if and only if $W^e\leq_{st}W$, and that $W$ is NWUE if and only if $W\leq_{st}W^e$. For a random variable $W$ which is either NBUE or NWUE, we employ this stochastic ordering in a similar way to the convex ordering we used in Section \[sec:ConvexOrder\] above.
We begin by deriving an inequality analogous to (\[eq:conv\]). For a differentiable function $\phi$, the definition of $W^e$ gives that $$\mathbb{E}[W\phi(W)]=\lambda^{-1}\mathbb{E}[\phi(W^e)+W^e\phi^\prime(W^e)]\,,$$ and hence $$\mathbb{E}[(\lambda W-1)\phi(W)]+\mathbb{E}[\phi(W)]=\mathbb{E}[W^e\phi^\prime(W^e)]+\mathbb{E}[\phi(W^e)]\,.$$ Thus, the inequality $$\label{eq:equilibrium}
\mathbb{E}[(\lambda W-1)\phi(W)]\leq\mathbb{E}[W\phi^\prime(W)]$$ holds if and only if $$\mathbb{E}[\phi(W^e)+W^e\phi^\prime(W^e)]\leq\mathbb{E}[\phi(W)+W\phi^\prime(W)]\,.$$ Therefore, inequality (\[eq:equilibrium\]) holds if $W$ is NBUE and $\phi(x)+x\phi^\prime(x)$ is increasing in $x$. Alternatively, (\[eq:equilibrium\]) also holds if $W$ is NWUE and $\phi(x)+x\phi^\prime(x)$ is decreasing in $x$. Analogously to the use of (\[eq:conv\]) in proving Theorem \[theorem:convex\] above, an upper bound on $\mathrm{Var}[g(W)]$ therefore holds for some functions $g$ under either of these assumptions; see Theorem \[theorem:equilibrium\] below for a precise statement.
Similarly, we may ask when the reversed inequality $\mathbb{E}[(\lambda W-1)\phi(W)]\geq\mathbb{E}[W\phi^\prime(W)]$ holds. By similar reasoning, this holds if either (i) $W$ is NBUE and $\phi(x)+x\phi^\prime(x)$ is decreasing in $x$, or (ii) $W$ is NWUE and $\phi(x)+x\phi^\prime(x)$ is increasing in $x$. Under either of these assumptions, we have a lower variance bound.
We have thus proved the following.
\[theorem:equilibrium\] Let $W$ be a non-negative random variable with mean $\mathbb{E}[W]=\lambda^{-1}$.
1. For a differentiable function $g:\mathbb{R}^+\mapsto\mathbb{R}$ such that $\mathrm{Var}[g(W)]$ exists, let $\phi_g(x)=\int_0^{\lambda x-1}g^\prime(\lambda^{-1}(u+1))\,du$. Assume that either
1. $W$ is NBUE and $\phi_g(x)+x\phi_g^\prime(x)$ is increasing in $x$; or
2. $W$ is NWUE and $\phi_g(x)+x\phi_g^\prime(x)$ is decreasing in $x$.
Then $$\mathrm{Var}[g(W)]\leq\frac{1}{\lambda}\mathbb{E}[Wg^\prime(W)^2]\,.$$
2. For a differentiable function $g:\mathbb{R}^+\mapsto\mathbb{R}$ such that $\mathrm{Var}[g(W)]$ exists, assume that either
1. $W$ is NBUE and $g(x)+xg^\prime(x)$ is decreasing in $x$; or
2. $W$ is NWUE and $g(x)+xg^\prime(x)$ is increasing in $x$.
Then $$\mathrm{Var}[g(W)]\geq\frac{(\mathbb{E}[Wg^\prime(W)])^2}{\lambda^2\mathrm{Var}[W]}\,.$$
Consider the random sum $W=\sum_{i=1}^NX_i$, where $X,X_1,X_2,\ldots$ are independent and identically distributed, continuous, real-valued random variables and $N$ is a counting random variable supported on the non-negative integers. Conditions are known under which $W$ is NWUE. For example, [@Brown90] shows that if $N$ is Geometric, then $W$ is NWUE, regardless of the distribution of $X$. More generally, Corollary 2.1 of [@Willmot05] establishes that if $N$ satisfies $$\label{eq:randomsum}
\sum_{k=0}^\infty\mathbb{P}(N>n+k+1)\geq\mathbb{P}(N>n)\sum_{k=0}^\infty\mathbb{P}(N>k)\,,$$ for all $n=0,1,\ldots$, then $W$ is NWUE. This includes, for example, the case where $N$ is mixed Poisson with a mixing distribution that is itself NWUE; see Corollary 3.1 of [@Willmot05]. Thus, under the condition (\[eq:randomsum\]), the bounds of the NWUE cases of Theorem \[theorem:equilibrium\] apply, with $\lambda^{-1}=\mathbb{E}[N]\mathbb{E}[X]$ and $\mathrm{Var}[W]=(\mathbb{E}[X])^2\mathrm{Var}[N]+\mathbb{E}[N]\mathrm{Var}[X]$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Part of this work was completed while FD and YS were attending the Workshop on New Directions in Stein’s Method, held at the Institute for Mathematical Sciences, National University of Singapore in May 2015. We thank the IMS, and the organisers of that workshop, for their support and hospitality. FD also thanks the University of Liège for supporting a visit there. The research of FG and CL is supported by a BOF Starting Grant of Ghent University.
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Example \[eg:smoothing\]: Proofs of claims {#sec:proofs}
==========================================
We begin by showing that $\tau_{\epsilon}(x)$, as defined in (\[eq:smooth\]), is the Stein kernel of $Y+Z$. To see this, note that $\mathbb{P}(Y+Z\leq t)=\mathbb{E}[\Phi_\epsilon(t-Y)]$, so that $Y+Z$ has density $p_{\epsilon}(t)=\mathbb{E}[\varphi_\epsilon(t-Y)]$. Hence, since $Y+Z$ has expectation $\mu$, its Stein kernel is given by $$\frac{1}{p_{\epsilon}(x)}\int_x^\infty(y-\mu)p_{\epsilon}(y)\,dy
=\frac{1}{p_{\epsilon}(x)}\int_x^\infty\int_{-\infty}^\infty(y-\mu)\varphi_\epsilon(y-t)\,dF(t)\,dy\,,$$ where $F$ is the distribution function of $Y$; see [@CPU94]. Applying Fubini’s theorem, this is equal to $$\frac{1}{p_{\epsilon}(x)}\int_{-\infty}^\infty\int_{x-t}^\infty(s+t-\mu)\varphi_\epsilon(s)\,ds\,dF(t)
=\frac{1}{p_{\epsilon}(x)}\mathbb{E}\left[\epsilon^2\varphi_\epsilon(x-Y)+(Y-\mu)\bar{\Phi}_\epsilon(x-Y)\right]\,,$$ since $\int_y^\infty s\varphi_\epsilon(s)\,ds=\epsilon^2\varphi_\epsilon(y)$. This Stein kernel is easily seen to be equal to $\tau_{\epsilon}(x)$ given in (\[eq:smooth\]).
Now, to prove claim (i), we firstly note that $Y\leq_{cx}Y+Z$ (see Theorem 3.A.34 of [@ss07]), so that $\mathbb{E}[\phi(Y)]\leq\mathbb{E}[\phi(Y+Z)]$ for any convex function $\phi$. Noting that the function $f(\alpha)=\mathbb{E}[(g(Y)-\alpha)^2]$ is minimized at $\alpha=\mathbb{E}[g(Y)]$, we have $$\mathrm{Var}[g(Y)]=\mathbb{E}\left[\left(g(Y)-\mathbb{E}[g(Y)]\right)^2\right]\leq
\mathbb{E}\left[\left(g(Y)-\mathbb{E}[g(Y+Z)]\right)^2\right]\leq\mathrm{Var}[g(Y+Z)]\,,$$ where the final inequality follows from the assumption in (i) that the mapping $x\mapsto\left(g(x)-\mathbb{E}[g(Y+Z)]\right)^2$ is convex. Applying the upper bound from (\[eq:8\]) completes the proof of (i).
We use a similar argument for (ii). We have that $$\mathrm{Var}[g(Y+Z)]\leq\mathbb{E}[(g(Y+Z)-\mathbb{E}[g(Y)])^2]\leq\mathbb{E}[(g(Y)-\mathbb{E}[g(Y)])^2]\,,$$ where the final inequality uses the convex ordering between $Y$ and $Y+Z$ (from which $\mathbb{E}[\phi(Y+Z)]\leq\mathbb{E}[\phi(Y)]$ for any concave function $\phi$) and the assumption that the mapping $x\mapsto (g(x)-\mathbb{E}[g(Y)])^2$ is concave. We now apply the lower bound from (\[eq:8\]) to complete the proof of (ii).
Example \[eg:Pearson\]: Stein kernel and further applications {#sec:more-examples-1}
=============================================================
We start by recalling a result taken from [@Stein1986 Equation (40), p.65], which was used in Example \[eg:Pearson\].
\[prop:perason\] A random variable with mean $\mu$ and variance $\sigma^2$ is of Pearson type if and only if there exist $\delta_1, \delta_2, \delta_3 \in {\mathbb R}$, not all equal to 0, such that $$\frac{p'(x)}{p(x) } = -\frac{(2\delta_1+1)(x-\mu) +
\delta_2}{\delta_1(x-\mu)^2+\delta_2(x-\mu)+\delta_3}.$$ In this case, its Stein kernel is $ \tau(x) = \delta_1 (x-\mu)^2 + \delta_2 (x-\mu) + \delta_3. $
To complement Example \[eg:Pearson\] and illustrate the scope of its application, we use the remainder of this appendix to present further examples along similar lines.
If $X \sim {NB}(r, \theta)$ has a negative binomial distribution with $\theta\in[0,1]$ and fixed $r\in{\mathbb N}$, and $\Theta_0 \sim{Beta}(\alpha, \beta)$ with $\alpha,\beta>0$, then $\Theta_2 \sim
{Beta}\left(\sum_{i=1}^nx_i+\alpha,nr+\beta\right)$. The Stein kernel for this Beta distribution is $\tau (\theta) = \frac{\theta (1 - \theta)}{\sum_{i=1}^nx_i+nr+
\alpha + \beta}$. Consequently, $$\frac{(\sum_{i=1}^nx_i+nr+\alpha +\beta+1)}{(\sum_{i=1}^nx_i+\alpha)(nr + \beta)} \mathbb{E} [\Theta_2 (1-\Theta_2) g'(\Theta_2)]^2 \leq \mathrm{Var}[g (\Theta_2)] \leq \frac{\mathbb{E}[\Theta_2 (1 - \Theta_2)g'(\Theta_2)^2]}{\sum_{i=1}^nx_i+nr+\alpha +\beta}.$$
If $X \sim {Wei}(k, \theta)$ has a Weibull distribution with $\theta>0$ and fixed $k>0$ (note that here we consider the Weibull density $x\mapsto \frac{k x^{k-1}}{\theta}\exp(-x^k/\theta), x>0$), and $\Theta_0 \sim{IG}(\alpha, \beta)$ with $\alpha,\beta>0$, then $\Theta_2 \sim
{IG}\left(n+\alpha,\sum_{i=1}^nx_i^k+\beta\right)$. The Stein kernel for this Inverse Gamma distribution is $\tau (\theta) = \frac{\theta^2}{n+\alpha-1}$. Consequently, $$\frac{n + \alpha -2}{(\sum_{i=1}^n x_i^k + \beta)^2} \mathbb{E}
[\Theta_2^2 g'(\Theta_2)]^2 \leq \mathrm{Var}[g(\Theta_2)] \leq \frac{\mathbb{E} [\Theta_2^2 g'(\Theta_2)^2]}{n + \alpha -1}.$$
If $X \sim {Gam}(k, \theta)$ has a Gamma distribution with $\theta, k>0$, and $\Theta_0 \sim{Gam}(\alpha, \beta)$ with $\alpha,\beta>0$, then $\Theta_2 \sim
{Gam}\left(nk+\alpha,\sum_{i=1}^nx_i+\beta\right)$. The Stein kernel for this Gamma distribution is $\tau (\theta) = \frac{\theta}{\sum_{i=1}^n x_i +
\beta}$. Consequently, $$\frac{\mathbb{E} [\Theta_2 g'(\Theta_2)]^2}{nk + \alpha} \leq \mathrm{Var}[g(\Theta_2)] \leq \frac{1}{\sum_{i=1}^n x_i + \beta} \mathbb{E} [\Theta_2 g'(\Theta_2)^2].$$
If $X \sim {Lap}(\mu, \theta)$ has a Laplace distribution with $\theta>0$ and fixed $\mu\in{\mathbb R}$, and $\Theta_0 \sim{IG}(\alpha, \beta)$ with $\alpha,\beta>0$, then $\Theta_2 \sim
{IG}\left(n+\alpha,\sum_{i=1}^n|x_i-\mu|+\beta\right)$. The Stein kernel can readily be deduced from previous examples, and we get $$\frac{n + \alpha -2}{(\sum_{i=1}^n |x_i - \mu| + \beta)^2} \mathbb{E} \left[ \Theta_2^2 g'(\Theta_2) \right]^2 \leq \mathrm{Var}[g(\Theta_2)] \leq \frac{1}{n + \alpha -1} \mathbb{E} \left[\Theta_2^2 g'(\Theta_2)^2 \right].$$
If $X \sim {Poi}(\theta)$ has a Poisson distribution with $\theta>0$, and $\Theta_0 \sim{Gam}(\alpha, \beta)$ with $\alpha,\beta>0$, then $\Theta_2 \sim {Gam}\left(\sum_{i=1}^nx_i+\alpha,n+\beta\right)$. The Stein kernel can readily be deduced from previous examples, and we get $$\frac{\mathbb{E} [\Theta_2 g'(\Theta_2)]^2}{\sum_{i=1}^nx_i+\alpha} \leq \mathrm{Var}[g(\Theta_2)] \leq \frac{1}{n+\beta} \mathbb{E}\left[ \Theta_2 g'(\Theta_2)^2 \right].$$
If $X \sim {U}(0,\theta)$ has a Uniform distribution with $\theta>0$, and $\Theta_0 \sim{Par}(\alpha, \beta)$ has a Pareto distribution with $\alpha,\beta>0$ (as a reminder, the density of such a Pareto distribution is $\theta\mapsto
\frac{\alpha\beta^\alpha}{\theta^{\alpha+1}}\mathbb{I}[\beta\leq\theta]$ where $\mathbb{I}[A]$ is the indicator function of the event $A$), then $\Theta_2 \sim {Par}\left(n+\alpha,\max(m(x),\beta)\right)$ with $m(x)=\max(x_1,\ldots,x_n)$. The Stein kernel for this Pareto distribution is $\tau (\theta) = \frac{\max(m(x) , \beta) - \theta}{n+\alpha - 1}
\theta$. Consequently, we get $$\begin{gathered}
\frac{(n + \alpha - 2)}{(n + \alpha)(\max(m(x) , \beta))^2}
\mathbb{E} \left[(\max(m(x) , \beta)-\Theta_2)\Theta_2 g'(\Theta_2)
\right]^2 \\
\leq \mathrm{Var}[g(\Theta_2)]
\leq \frac{1}{n+\alpha - 1}\mathbb{E}\left[ (\max(m(x) , \beta)-\Theta_2) \Theta_2 g'(\Theta_2)^2 \right].
\end{gathered}$$
[^1]: Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh EH14 4AS, UK. E-mail: f.daly@hw.ac.uk
[^2]: Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281, S9, Campus Sterre, 9000 Gent, Belgium. E-mail: fatemeh.ghaderinezhad@ugent.be
[^3]: Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281, S9, Campus Sterre, 9000 Gent, Belgium. E-mail: christophe.ley@ugent.be
[^4]: Department of Mathematics, Université libre de Bruxelles, Boulevard du Triomphe, CP210, B-1050 Bruxelles
|
---
abstract: 'This paper presents a novel control strategy for the coordination of a multi-agent system subject to high-level goals expressed as linear temporal logic formulas. In particular, each agent, which is modeled as a sphere with $2$nd order dynamics, has to satisfy a given local temporal logic specification subject to connectivity maintenance and inter-agent collision avoidance. We propose a novel continuous control protocol that guarantees navigation of one agent to a goal point, up to a set of collision-free initial configurations, while maintaining connectivity of the initial neighboring set and avoiding inter-agent collisions. Based on that, we develop a hybrid switching control strategy that ensures that each agent satisfies its temporal logic task. Simulation results depict the validity of the proposed scheme.'
author:
- 'Christos K. Verginis and Dimos V. Dimarogonas[^1]'
bibliography:
- 'references.bib'
title: '**Mode Switching Decentralized Multi-Agent Coordination under Local Temporal Logic Tasks** '
---
INTRODUCTION {#sec:Introduction}
============
The integration of temporal logic planning and multi-agent control systems has gained significant amount of attention during the last decade, since it provides planning capabilities that allow achievement of complex goals (see e.g., [@Chen2012; @Diaz2015; @Fainekos2009; @Filippidis2012; @Loizou2004; @Meng15; @Tumova2014; @Zhang2016; @verginis2017distributed; @verginis2017robust; @guo2016communication; @kloetzer2011multi; @nikou17TimedTemporal; @saha2014automated; @ulusoy2013optimality; @verginis18TASE]). Firstly, an abstracted discrete version (e.g., a transition system) of the multi-agent system is derived by appropriately discretizing the workspace and finding the control inputs that navigate the system among the discrete states. A task specification is then given as a temporal logic formula (e.g., linear temporal logic (LTL) or metric-interval temporal logic (MITL)) with respect to the discretized version of the system, and by employing formal verification techniques, a high-level discrete plan is found that satisfies the task. Finally, the control inputs associated with the transitions between the discrete states are applied to achieve the plan execution.
An appropriate abstraction of the continuous-time system to a transition system form necessitates the design of appropriate control inputs for the transition of the system among the discrete states. Most works in the related literature, when designing such discrete representations, either assume that there exist such control inputs or adopt simplified dynamics and employ optimization and input discretization techniques. Moreover, when deploying multi-robot teams, it is crucial to guarantee inter-agent collision avoidance. The latter is usually not taken into account in the related works, most of which unrealistically consider point-mass agents.
Collision avoidance properties during the multi-agent transitions are incorporated in [@Loizou2004] and [@Filippidis2012], where the authors adopt single-integrator models and appropriately constructed potential fields, namely navigation functions. These results, however, are not extendable to higher order dynamics in a straightforward way and are based on gain tuning, which might be problematic for real robot actuators. They also guarantee the multi-agent transitions from *almost* all (except for a set of measure zero) collision-free initial conditions, implying that there are initial configurations that drive the multi-agent system to local minima. Potential-based collision avoidance was also incorporated in our previous works [@verginis18TASE], where a centralized controller was employed, and [@verginis2017robust], where no explicit potential field was given. In the latter, the agents also start their transitions simultaneously, which induces a centralized feature to the scheme.
In this paper, we propose a novel hybrid control strategy for the coordination of a multi-agent system subject to complex specifications expressed as linear temporal logic (LTL) formulas over predefined points of interest in the workspace. We first use formal verification methodologies to derive a high-level navigation plan for each agent over these points that satisfies its LTL formula. Then, we design a continuous control protocol that guarantees the global navigation of an agent to a goal point while guaranteeing inter-agent collisions and connectivity maintenance of the initially connected agents. By “global", we mean up to a set of collision-free and connected (in the sense of a connected communication graph) initial configurations. The control scheme is decentralized, based on limited sensing capabilities of the agents, as well as robust to modeling uncertainties. Finally, by introducing certain priority variables for the agents, we develop a switching protocol that guarantees the sequential navigation of the agents to their goal points of interest and the satisfaction of their respective formulas.
This work can be considered as an extension of [@guo2016communication], where a similar strategy is followed. In [@guo2016communication], however, point-mass agents are considered and no inter-agent collision avoidance is taken into account. Moreover, the multi-agent transitions are not guaranteed globally; appropriate gain tuning achieves transitions from almost all initial conditions, except for a set of measure zero.
The rest of the paper is organized as follows. Section \[sec:Notation and Preliminaries\] introduces notation and preliminary background. Section \[sec:problem form\] provides the problem formulation and Section \[sec:main results\] discusses the proposed solution. Simulation results are given in Section \[sec:Simulation\] and Section \[sec:Conclusion\] concludes the paper.
Notation and Preliminaries {#sec:Notation and Preliminaries}
==========================
Notation {#subsec:Notation}
--------
The set of natural and real numbers is denoted by $\mathbb{N}$, and $\mathbb{R}$, respectively, and $\mathbb{R}_{\geq 0}$, $\mathbb{R}_{> 0}$ are the sets of nonnegative and positive real numbers, respectively. The notation $\|x\|$ implies the Euclidean norm of a vector $x\in\mathbb{R}^n$. The identity matrix is $I_n\in\mathbb{R}^{n \times n}$ and, given a sequence $s_1\dots s_n$ of elements in $S$, we denote by $(s_1\dots s_n)^\mathsf{\omega}$ the infinite sequence $s_1\dots s_n s_1\dots s_n\dots $ created by repeating $s_1\dots s_n$.
Task Specification in LTL {#subsec:LTL}
-------------------------
We focus on the task specification $\phi$ given as a Linear Temporal Logic (LTL) formula. The basic ingredients of a LTL formula are a set of atomic propositions $\Psi$ and several boolean and temporal operators. LTL formulas are formed according to the following grammar [@baier2008principles]: $\phi ::= \mathsf{true}\: |\:a\: |\: \phi_{1} \land \phi_{2}\: |\: \neg \phi\: |\:\bigcirc \phi\:|\:\phi_{1}\cup\phi_{2} $, where $a\in \Psi$, $\phi_1$ and $\phi_2$ are LTL formulas and $\bigcirc$, $\cup$ are the *next* and *until* operators, respectively. Definitions of other useful operators like $\square$ (*always), $\lozenge$ (*eventually) and $\Rightarrow$ (*implication) are omitted and can be found at [@baier2008principles]. The semantics of LTL are defined over infinite words over $2^{\Psi}$. Intuitively, an atomic proposition $\psi\in \Psi$ is satisfied on a word $w=w_1w_2\dots$, denoted by $w\models\psi$, if it holds at its first position $w_1$, i.e. $\psi\in w_1$. Formula $\bigcirc\phi$ holds true if $\phi$ is satisfied on the word suffix that begins in the next position $w_2$, whereas $\phi_1\cup\phi_2$ states that $\phi_1$ has to be true until $\phi_2$ becomes true. Finally, $\lozenge\phi$ and $\square\phi$ holds on $w$ eventually and always, respectively. For a full definition of the LTL semantics, the reader is referred to [@baier2008principles].***
Problem Formulation {#sec:problem form}
===================
Consider $N>1$ autonomous agents, with $\mathcal{N} \coloneqq \{1,\dots,N\}$, operating in $\mathbb{R}^n$ and described by the spheres $\mathcal{A}_i(x_i) \coloneqq \{y\in\mathbb{R}^n : \|x_i-y\| < r_i \}$, with $x_i \in\mathbb{R}^n$ being agent $i$’s center, and $r_i\in\mathbb{R}_{> 0}$ its bounding radius. We consider that there exist $K>1$ points of interest in the workspace, denoted by $c_k\in\mathbb{R}^n$, $\forall k\in\mathcal{K}\coloneqq \{1,\dots,K\}$, with $\Pi \coloneqq \{c_1,\dots,c_K\}$. Moreover, we introduce disjoint sets of atomic propositions $\Psi_i$, expressed as boolean variables, that represent services provided by agent $i\in\mathcal{N}$ in $\Pi$. The services provided at each point $c_k$ are given by the labeling functions $\mathcal{L}_i:\Pi\rightarrow2^{\Psi_i}$, which assign to each point $c_k$, $k\in\mathcal{K}$, the subset of services $\Psi_i$ that agent $i$ can provide in that region. Note that, upon the visit to $c_k$, agent $i$ chooses among $\mathcal{L}_i(c_k)$ the subset of atomic propositions to be evaluated as true, i.e., the subset of services it *provides* among the available ones. These services are abstractions of action primitives that can be executed in different regions, such as manipulation tasks or data gathering. In this work, we do not focus on how the service providing is executed by an agent; we only aim at controlling the agents’ motion to reach the regions where these services are available.
The agents’ motion is described by the following dynamics, inspired by rigid body motion:
\[eq:dynamics\] $$\begin{aligned}
&\dot{x}_i = v_i, \\
&B_i \dot{v}_i + f_i(x_i,v_i) + g_i = u_i,
\end{aligned}$$
where $v_i\in\mathbb{R}^n$ are the agents’ generalized velocities, $B_i\in\mathbb{R}^{n\times n}$ are positive definite matrices representing inertia, $g_i\in\mathbb{R}^n$ are gravity vectors, $u_i\in\mathbb{R}^n$ are the control inputs, and $f_i:\mathbb{R}^{2n}\to\mathbb{R}^n$ are terms representing modeling uncertainties, satisfying the following assumption.
\[ass:f\_i\] It holds that $\|f_i(x_i,v_i)\| \leq a_i\bar{f}_i(x_i)\|v_i\|$, $\forall (x_i,v_i)\in\mathbb{R}^{2n}$, $i\in\mathcal{N}$, where $a_i$ are *unknown* positive constants and $\bar{f}_i:\mathbb{R}^{2n}\to\mathbb{R}_{\geq 0}$ are known continuous functions.
Moreover, we consider that each agent has a certain priority $\mathsf{pr}_i\in\mathbb{N}$ in the multi-agent team, with higher $\mathsf{pr}_i$ denoting higher priority. Without loss of generality, we assume that these variables have been normalized so that $\exists i\in\mathcal{N} : \mathsf{pr}_i = 1$ and $|\mathsf{pr}_\ell - \mathsf{pr}_j| = 1$, $\forall \ell,j\in\mathcal{N}$, with $\ell \neq j$. The priority variables can be given off-line to the agents.
In addition, we consider that each agent has a limited sensing radius $d_{\text{con},i}\in\mathbb{R}_{>0}$, with $d_{\text{con},i} > \max_{j\in\mathcal{N}}\{r_i+r_j\}$, which implies that the agents can sense each other without colliding. Based on this, we model the topology of the multi-agent network through the undirected graph $\mathcal{G}(x) \coloneqq (\mathcal{N},\mathcal{E}(x))$, with $\mathcal{E}(x) \coloneqq \{(i,j)\in\mathcal{N}^2 : \|x_i - x_j \| \leq \min\{d_{\text{con},i}, d_{\text{con},j}\} \}$. We further denote $M(x)\coloneqq |\mathcal{E}(x)|$. Given the $m$ edge in the edge set $\mathcal{E}(x)$, we use the notation $(m_1,m_2)\in\mathcal{N}^2$ that gives the agent indices that form edge $m\in\mathcal{M}(x)$, where $\mathcal{M}(x)\coloneqq\{1,\dots,M(x)\}$ is an arbitrary numbering of the edges $\mathcal{E}(x)$. By also denoting $m_1$ as the tail and $m_2$ as the head of edge $m$, we define the $N\times M$ incidence matrix $D(\mathcal{G}(x)) \coloneqq [d_{im}]$, where $d_{im} = 1$ if $i$ is the head of edge $m$, $d_{im} = -1$ if $i$ is the tail of edge $m$, and $d_{im} = 0$, otherwise. Note that, for a connected graph $\mathcal{G}$, the sum of the rows of $D(\mathcal{G})$ equals zero. Next, we assume that the agents form initially a collision-free connected graph.
\[ass:initially connected\] The graph $\mathcal{G}(x(0))$ is nonempty, connected and $\mathcal{A}_i(x_i(0))\cap\mathcal{A}_j(x_j(0)) = \emptyset$, $\forall i,j\in\mathcal{N}$, with $i\neq j$.
As mentioned before, the agents, apart from satisfying their local LTL formulas, need to (a) preserve connectivity with their initial neighbors, and (b) guarantee inter-agent collision avoidance. More specifically, we will guarantee that the initial edge set $\mathcal{E}(x(0))$ will be preserved and that $\mathcal{A}_i(x_i(t))\cap\mathcal{A}_j(x_j(t)) = \emptyset$, $\forall i,j\in\mathcal{N}$, with $i\neq j$, $t\in\mathbb{R}_{>0}$.
In order to proceed, we need the following definitions:
\[def:agent in region\] An agent $i\in\mathcal{N}$, at configuration $x_i\in\mathbb{R}^n$, can provide a service at a point $c_k\in\mathbb{R}^n$, among the set $\mathcal{L}_i(\pi_k)$, if $c_k \in \mathcal{A}_i(x_i)$.
Let $x_i(t)\in\mathbb{R}^n$, $t\in\mathbb{R}_{\geq 0}$, be a trajectory of agent $i\in\mathcal{N}$. The *behavior* of agent $i$ is the tuple $\beta_i \coloneqq (c_{i1},\sigma_{i1}),(c_{i2},\sigma_{i2}),\dots$, with $c_{i\ell}\in \Pi$, $\forall \ell\in\mathbb{N},i\in\mathcal{N}$, and $c_{i\ell}\in\mathcal{A}_i(x_i(t))$, $\forall t\in \Delta t_{i\ell}\coloneqq [t_{i\ell},t'_{i\ell}] \subset \mathbb{R}_{\geq 0}$, $t_{i\ell} < t'_{i\ell} < t_{i(\ell+1)}$, $c_{k}\notin \mathcal{A}_i(x_i(t))$, $\forall k\in\mathcal{K}, t\in(t'_{i\ell},t_{i(\ell+1)})$, $\sigma_{i\ell}\in 2^{\Psi_i}$, $\sigma_{i\ell}\in(\mathcal{L}_i(c_{i\ell})\cup\emptyset)$.
Loosely speaking, a behavior consists of the sequence of points $c_{i1}c_{i2}\dots$ where agent $i$ can provide services at, at the time intervals $\Delta t_{i\ell},\ell\in\mathbb{N}$. In every point $c_{i\ell}$, agent $i$ chooses to provide the set $\sigma_{i\ell}$ of services among the $\mathcal{L}_i(c_{i\ell})$ available ones. Note that $\sigma_{i\ell}$ can be the empty set, implying that the agent may choose not to provide any services. Given the agent’s behavior $\beta_i$, the satisfaction of a task formula $\phi_i$ is defined as follows:
A behavior $\beta_i$ satisfies $\phi_i$ if there exists a subsequence $\widetilde{\sigma}_i$ $\coloneqq$ $\sigma_{k_{i1}}$ $\sigma_{k_{i2}}$ $\dots$ of $\sigma_{i1}$ $\sigma_{i2}$ $\dots$, with $k_{i1}$, $k_{i2}$, $\dots$ being a subsequence of $i1, i2,\dots$, such that $\widetilde{\sigma}_i \models \phi_i$.
The problem treated in this paper is the following:
Consider $N$ spherical autonomous agents with dynamics and $K$ points of interest in the workspace. Given the sets $\Psi_i$ and $N$ LTL formulas $\phi_i$ over $\Psi_i$, as well as Assumptions \[ass:f\_i\]-\[ass:initially connected\], develop a decentralized control strategy that achieves behaviors $\beta_i$, that yield the satisfaction of $\phi_i$, $\forall i\in\mathcal{N}$, while guaranteeing inter-agent collision avoidance and connectivity maintenance, i.e., $\mathcal{A}_i(x_i(t))\cap\mathcal{A}_j(x_j(t)) \neq \emptyset$, $\forall i,j\in\mathcal{N}$, with $i\neq j$, and $\|p_{m_1}(t) - p_{m_2}(t) \| \leq \min\{d_{\text{con},m_1},d_{\text{con},m_2}\}$, $\forall t\in\mathbb{R}_{\geq 0}, m\in\mathcal{M}(x(0))$.
Main Results {#sec:main results}
============
In this section we present the proposed solution, which consists of three layers: (i) an off-line plan synthesis for the discrete plan of each agent, i.e., the path of the goal points and the sequence of services to be provided; (ii) a distributed continuous control scheme that guarantees the navigation of one of the agents to a goal point of interest from *all* collision-free and connected (in the sense of $\mathcal{E}(x(0))$) initial configurations; (iii) a decentralized hybrid control layer that coordinates the discrete plan execution via continuous control law switching, to ensure the satisfaction of each agent’s local task.
Discrete Plan Synthesis {#subsec:discrete plan synthesis}
-----------------------
The discrete plan can be generated using standard techniques from automata-based formal synthesis. We first model the motion of each agent as a finite transition system $\mathcal{T}_i \coloneqq (\Pi',c_{i,0},\to_i,\Psi_i,\mathcal{L}_i)$, where $c_{i,0}$ represents the agent’s initial position $x_i(0)$, $\Pi'\coloneqq\Pi\cup\{c_{i,0}\}$ is the set of points of interest defined in Section \[sec:problem form\], expanded to include $c_{i,0}$, $\to_i\coloneqq \Pi\times\Pi$ is a transition relation, and $\Psi_i$, $\mathcal{L}_i$ are the sets of atomic propositions and labeling function, respectively, as defined in Section \[sec:problem form\]. Note that, by the definition of the transition relation, we consider that there can be transitions between any pair of points of interest. This is achieved in the continuous time motion by the proposed control scheme of the subsequent section. Next, each agent $i\in\mathcal{N}$ translates the LTL formula $\phi_i$ into a Büchi automaton $\mathcal{A}_{\phi_i}$ and builds the product $\widetilde{T}_i \coloneqq \mathcal{T}_i\otimes \mathcal{A}_{\phi_i}$. The accepting runs of $\widetilde{T}_i$ (that satisfy $\phi_i$) are projected onto $\mathcal{T}_i$ and provide for each agent a sequence of points to be visited and services to be provided in the prefix-suffix form: $\mathsf{plan}_i \coloneqq \ (c_{i1^\text{G}}, \sigma_{i1^\text{G}}) \ \dots \ (c_{il_i^\text{G}}, \sigma_{il_i^\text{G}}) \ ((c_{i(l_i+1)^\text{G}}, \sigma_{i(l_i+1)^\text{G}})\dots$ $(c_{iL_i^\text{G}}, \sigma_{iL_i^\text{G}}))^\mathsf{\omega}$, where $l_i,L_i \in \mathbb{N}$, with $l_i < L_i$, and $c_{i\ell^\text{G}}\in \Pi$, $\sigma_{i\ell^\text{G}}\in 2^{\Psi_i}, (\mathcal{L}_i(c_{i\ell^\text{G}})\cup\emptyset)$, $\forall \ell\in\{1,\dots,L_i\}$, $i\in\mathcal{N}$. More details regarding the followed technique are beyond the scope of this paper and can be found in [@baier2008principles]. Note that, in our work, LTL formulas are interpreted over the provided services along a trajectory, not the available ones. Hence, crossing of points of interest not included in $\mathsf{plan}_i$ (which might happen due to the collision and connectivity constraints, as explained in the next sections) does not influence the local LTL task satisfaction.
Continuous Control Design {#subsec:continuous control}
-------------------------
In this section we propose a decentralized control protocol for the transition of the agents to the points of interest, while guaranteeing inter-agent collision-avoidance and connectivity maintenance. More specifically, given a collision-free and connected (i.e., connected graph $\mathcal{G}(x(t_0))$) configuration of the agents at a time instant $t_0\in\mathbb{R}_{\geq 0}$, the proposed control scheme guarantees that exactly one agent $j\in\mathcal{N}$ navigates to a desired point, while preserving connectivity of the initial edge set and avoiding inter-agent collisions. Loosely speaking, connectivity maintenance forces the whole multi-agent team to navigate towards the desired point of agent $j$, while avoiding collisions. This is motivated by potential cooperative tasks of the agents at the points of interest (e.g. object transportation). Then, the hybrid coordination of the next section guarantees that all the agents will eventually reach their desired goals by an appropriate switching protocol based on the priority functions $\mathsf{pr}_i$.
Let the points $c_i\in\mathbb{R}^n$, $\forall i\in\mathcal{N}$, be some desired destinations of the agents. Consider the initial connected graph $\mathcal{G}_0=(\mathcal{N},\mathcal{E}_0)\coloneqq \mathcal{G}(x(t_0)) = (\mathcal{N},\mathcal{E}(x(t_0)))$, with $M_0 \coloneqq M(x(t_0))$ and edge numbering $\mathcal{M}_0 \coloneqq \mathcal{M}(x(t_0))$. Consider also the complete graph $\bar{\mathcal{G}} \coloneqq (\mathcal{N},\mathcal{E})$, with $\bar{\mathcal{E}}\coloneqq \{ (i,j), \forall i,j\in\mathcal{N} \text{ with } i < j\}$, $\bar{M}\coloneqq |\bar{\mathcal{E}}|$, and the edge numbering $\bar{\mathcal{M}}\coloneqq \{1,\dots,M_0,M_0+1,\dots,\bar{M}\}$, where $\{M_0+1,\dots,\bar{M}\}$ corresponds to the edges in $\bar{\mathcal{E}}\backslash \mathcal{E}_0$. In other words, we assume that the numbering of the extra edges $\bar{\mathcal{E}}\backslash\mathcal{E}_0$ starts from $M_0+1$.
Next, we construct the collision functions for all the edges $m\in\bar{\mathcal{M}}$. Let $\beta_{\text{col},m}:\mathbb{R}_{\geq 0}\to[0,\bar{\beta}_\text{col}]$, with $$\begin{aligned}
\beta_{\text{col},m}(x) \coloneqq \left\{ \begin{matrix}
\vartheta_{\text{col},m}(x) & 0 \leq x < \bar{d}_{\text{col},m}, \\
\bar{\beta}_\text{col} & \bar{d}_{\text{col},m} \leq x
\end{matrix} \right.,\end{aligned}$$ where $\vartheta_{\text{col},m}:\mathbb{R}_{\geq 0}\to[0,\bar{\beta}_\text{col}]$ is a continuously differentiable *strictly increasing* polynomial that renders $\beta_{\text{col},m}$ continuously differentiable, with $\vartheta_{\text{col},m}(0) = 0$, $\vartheta_{\text{col},m}(\bar{d}_{\text{col},m}) = \bar{\beta}_\text{col}$, $\forall m\in\bar{\mathcal{M}}$, and $\bar{\beta}_\text{col} $, $\bar{d}_{\text{col},m}$ are positive constants to be appropriately chosen. Then, for each edge $m\in\bar{\mathcal{M}}$, we can choose $\beta_{\text{col},m} \coloneqq$ $\beta_{\text{col},m}(\iota_m)$, where $\iota_m\coloneqq\|p_{m_1}-p_{m_2}\|^2 - (r_{m_1}+r_{m_2})^2$ and $\bar{d}_{\text{col},m}\coloneqq \underline{d}^2_{\text{con},m}- (r_{m_1}+r_{m_2})^2$, $\underline{d}_{\text{con},m} \coloneqq \min\{d_{\text{con},m_1},d_{\text{con},m_2}\}$, that vanishes when a collision between agents $m_1,m_2$ occurs. The term $\bar{\beta}_\text{col}$ can be any positive constant.
Next, we construct the connectivity functions for all the edges $m\in\mathcal{M}$. Let $\beta_{\text{con},m}:\mathbb{R}_{\geq 0}\to[0,\bar{\beta}_\text{con}]$, with $$\begin{aligned}
\beta_{\text{con},m}(x) \coloneqq \left\{ \begin{matrix}
\vartheta_{\text{con},m}(x) & 0 \leq x < \underline{d}^2_{\text{con},m} \\
\bar{\beta}_\text{con} & \underline{d}^2_{\text{con},m} \leq x \\
\end{matrix} \right.,\end{aligned}$$ where $\vartheta_{\text{con},m}:$ $\mathbb{R}_{\geq 0}$ $\to$ $[0,\bar{\beta}_\text{con}]$ is a cont. differentiable *strictly increasing* polynomial that renders $\beta_{\text{con},m}$ continuously differentiable, with $\vartheta_{\text{con},m}(0) = 0$, $\vartheta_{\text{con},m}(\underline{d}^2_{\text{con},m}) = \bar{\beta}_\text{con}$, $\forall m\in\mathcal{M}$. Then, for each edge $m\in\mathcal{M}$, we choose $\beta_{\text{con},m} \coloneqq \beta_{\text{con},m}(\eta_m)$, with $\eta_m\coloneqq \underline{d}^2_{\text{con},m} - \|p_{m_1}-p_{m_2}\|^2$, that vanishes at a connectivity break of edge $m$. The term $\bar{\beta}_\text{con}$ can be any positive constant. The aforementioned functions take into account the limited sensing capabilities of the agents, since the derivatives of $\beta_{\text{col},m}$ and $\beta_{\text{con},m}$ are zero when $\|p_{m_1}-p_{m_2}\| \geq \underline{d}_{\text{con},m}$, $\forall m\in\bar{\mathcal{M}}$. Note that all the necessary parameters for the construction of $\beta_{\text{col},m}$, $\beta_{\text{con},m}$ can be transmitted off-line to the agents $m_1,m_2$. Similarly to [@guo2016communication], we propose now the following decentralized control scheme, parameterized by the goal and mode of the agents: $$\begin{aligned}
&u_i(c_i,\mathsf{md}_i) \coloneqq \sum\limits_{m\in\bar{\mathcal{M}}}\alpha_{\text{col}}(i,m) \beta'_{\text{col},m}\frac{\partial \iota_m}{\partial x_{m_1}} + \notag \\
& +\sum\limits_{m\in\mathcal{M}_0}\alpha_\text{con}(i,m) \beta'_{\text{con},m}\frac{\partial \eta_m}{\partial x_{m_1}} - \mathsf{md}_i \gamma_i(c_i)+ g_i \notag \\
& -\Big(\hat{a}_i\bar{f}_i(x_i) + \mu_i\Big)v_i, \label{eq:control law}\end{aligned}$$ where $c_i\in\mathbb{R}^n$ is agent $i$’s desired destination, $\mathsf{md}_i\in\{0,1\}$ is the agent’s mode (active or passive); the functions $\alpha_{\text{col}}$, $\alpha_{\text{con}}$ are defined as $\alpha_\text{col}(i,m) = -\mu_{\text{col},m}$ if $i=m_1$ (agent $i$ is the tail of edge $m$), $\alpha_\text{col}(i,m) = \mu_{\text{col},m}$ if $i=m_2$ (agent $i$ is the head of edge $m$), and $\alpha_\text{col}(i,m) = 0$ otherwise, $\alpha_\text{con}(i,m) = -\mu_{\text{con},m}$ if $i=m_1$, $\alpha_\text{con}(i,m) = \mu_{\text{con},m}$ if $i=m_2$, and $\alpha_\text{con}(i,m) = 0$ otherwise, $i\in\mathcal{N}$; $\beta'_{\text{col},m} \coloneqq \frac{\partial }{\partial \iota_m}\left(\frac{1}{\beta_{\text{col},m}(\iota_m)}\right)$, $\beta'_{\text{con},m} \coloneqq \frac{\partial }{\partial \eta_m}\left(\frac{1}{\beta_{\text{con},m}(\eta_m)}\right)$, $\gamma_i(c_i)\coloneqq \mu_{c,i}(x_i - c_i)$; the constants $\mu_{\text{col},m}, \mu_{\text{con},m},\mu_{c,i}, \mu_i\in\mathbb{R}_{>0}$ are positive gains, $\forall m\in\bar{\mathcal{M}}$, $m\in\mathcal{M}_0$, $i\in\mathcal{N}$, and the terms $\hat{a}_i$ are adaptation signals that evolve according to $$\dot{\hat{a}}_i = \mu_{a,i}\bar{f}_i(x_i)\|v_i\|^2, \label{eq:adaptation laws}$$ with arbitrary bounded initial conditions $\hat{a}_i(t_0)$, and positive gains $\mu_{a,i}\in\mathbb{R}_{>0}$, $\forall i\in\mathcal{N}$. The intuition behind the parameters $\mathsf{md}_i$ is that only one of them can be true at time, meaning that only one agent navigates towards its desired point. After a successful navigation, the variable is activated for another agent, and so on. Section \[subsec:hybrid strategy\] describes the coordination strategy that decides about the activation of the variables $\mathsf{md}_i$. The navigation of the agent $j$ for which $\mathsf{md}_j = 1$ is guaranteed by the next theorem.
Consider a multi-agent team $\mathcal{N}$, described by the dynamics , at a collision-free and connected configuration at $t=t_0\in\mathbb{R}_{\geq 0}$, with desired destinations $c_i$, $\forall i\in\mathcal{N}$. Then, under Assumptions \[ass:f\_i\]-\[ass:initially connected\], the application of the control laws with $u_j = u_j(c_j,1)$ for a $j\in\mathcal{N}$ and $u_i=u_i(c_i,0)$, $\forall i\in\mathcal{N}\backslash\{j\}$ guarantees that $c_j\in\mathcal{A}_j(x_j(t_f))$ for a finite $t_f$, as well as $\mathcal{A}_i(x_i(t))\cap\mathcal{A}_n(x_n(t))=\emptyset$, $\forall i,n\in\mathcal{N}$, with $i\neq n$, and $\|p_{m_1}(t) - p_{m_2}(t) \| \leq \min\{d_{\text{con},m_1},d_{\text{con},m_2}\}$, $\forall t\geq t_0, m\in\mathcal{M}_0$, with bounded closed loop signals.
By taking into account that $\frac{\partial \iota_m}{\partial x_{m_1}} =- \frac{\partial \iota_m}{\partial x_{m_2}}$, $\forall m\in\bar{\mathcal{M}}$, $\frac{\partial \eta_m}{\partial x_{m_1}} = -\frac{\partial \eta_m}{\partial x_{m_2}}$, $\forall m\in\mathcal{M}_0$, we can write the control laws in vector form: $$\begin{aligned}
u =& (D(\mathcal{G}_0)\otimes I_n)\mu_{\text{con}}\beta_{\text{con}} + (D(\bar{\mathcal{G}})\otimes I_n)\mu_{\text{col}}\beta_{\text{col}} -\gamma_{\mathsf{md}}(x)\notag \\
& + g - h(x)v \label{eq:control law vector form}
\end{aligned}$$ where $g \coloneqq [g_1^\top,\dots,g_N^\top]^\top$, $x \coloneqq [x_1^\top,\dots,x_N^\top]^\top$, $v \coloneqq [v_1^\top,\dots,v_N^\top]^\top \in\mathbb{R}^{Nn}$, $h(x) = \text{diag}\{[\hat{a}_i\bar{f}_i(x_i)+\mu_i]_{i\in\mathcal{N}}\}\in\mathbb{R}^{Nn\times Nn}$, $\gamma_{\mathsf{md}}(x)\in\mathbb{R}^{Nn}$ is a vector of zeros except for the rows $nj,\dots, n(j+1)$, which are $\gamma_j(c_j)$; $\mu_{\text{con}} \coloneqq \text{diag}\{ [\mu_{\text{con},m} I_n]_{m\in\mathcal{M}_0} \}$, $\mu_{\text{col}} \coloneqq \text{diag}\{ [\mu_{\text{col},m} I_n]_{m\in\bar{\mathcal{M}}}\}\in\mathbb{R}^{Nn\times Nn}$, $D(\cdot)$ is the graph incidence matrix, as defined in Section \[sec:problem form\], and $\beta_\text{con} \coloneqq \left[\beta'_{\text{con},1}\left(\frac{\partial \eta_1}{\partial x_{1_1}}\right)^\top,\dots,\beta'_{\text{con},M_0}\left(\frac{\partial \eta_{M_0}}{\partial x_{(M_0)_1}}\right)^\top \right]^\top \in \mathbb{R}^{nM_0}$, $\beta_\text{col} \coloneqq \left[\beta'_{\text{col},1}\left(\frac{\partial \iota_1}{\partial x_{1_1}}\right)^\top,\dots,\beta'_{\text{col},\bar{M}}\left(\frac{\partial \iota_{\bar{M}}}{\partial x_{\bar{M}_1}}\right)^\top \right]^\top \in \mathbb{R}^{n\bar{M}}$.
Consider the positive definite Lyapunov candidate $V(x,v,\hat{a}) \coloneqq\frac{\mu_{c,j}}{2}\|x_j-c_j\|^2 + \frac{1}{2}\sum_{i\in\mathcal{N}}\Big(v_i^\top B_i v_i + \frac{1}{2\mu_{a,i}}\widetilde{a}_i^2 \Big)
+\sum_{m\in\bar{\mathcal{M}}} \frac{\mu_{\text{col},m}}{\beta_{\text{col},m}(\iota_m)} +
\sum_{m\in\mathcal{M}_0}\frac{\mu_{\text{con},m}}{\beta_{\text{con},m}(\eta_m)}$, where $\hat{a} = [\hat{a}_1,\dots,\hat{a}_N]^\top\in\mathbb{R}^N$, and $\widetilde{a}_i\coloneqq\hat{a}_i-a_i$, $\forall i\in\mathcal{N}$. The connectedness of $\mathcal{M}_0$ and collision-free initial conditions imply the existence of a finite constant $\bar{V}$ such that $V(t_0) \leq \bar{V}$. By taking the derivative of $V$ we obtain $\dot{V} = \gamma_j(c_j)^\top v_j + \sum_{i\in\mathcal{N}}\Big\{\widetilde{a}_i\bar{f}_i(x_i)\|v_i\|^2 + v_i^\top(u_i - g_i -
f_i(x_i,v_i)) \} - \Big(\widetilde{\beta}_\text{con}^\top(D(\mathcal{G}_0)\otimes I_n)^\top + \widetilde{\beta}_\text{col}^\top(D(\bar{\mathcal{G}})\otimes I_n)^\top\Big) v$, where $\widetilde{\beta}_\text{con}\coloneqq \mu_\text{con}\beta_\text{con}$, $\widetilde{\beta}_\text{col}\coloneqq \mu_\text{col}\beta_\text{col}$. By substituting the control and adaptation laws , and employing Assumption \[ass:f\_i\], we obtain $\dot{V} \leq\sum_{i\in\mathcal{N}}\{\|v_i \|\|f_i(x_i,v_i)\| -\hat{a}_i \bar{f}_i(x_i)\|v_i\|^2 + \widetilde{a}_i\bar{f}_i(x_i)\|v_i\|^2 - \mu_i\|v_i\|^2 \} \leq \sum_{i\in\mathcal{N}} \{a_i\bar{f}_i(x_i)\|v_i\|^2 -\hat{a}_i \bar{f}_i(x_i)\|v_i\|^2 + \widetilde{a}_i\bar{f}_i\|v_i\|^2 - \mu_i\|v_i\|^2\} = -\sum_{i\in\mathcal{N}}\mu_i\|v_i\|^2$. Hence, we conclude that $\dot{V} \leq 0$, which implies that $V(t) \leq V(t_0) \leq \bar{V}$. Therefore, we conclude that $\beta_{\text{col},m}(\iota_m) \geq \frac{\mu_{\text{col},m}}{\bar{V}}$ $\beta_{\text{con},m}(\eta_m) \geq \frac{\mu_{\text{con},m}}{\bar{V}}$, and $\|x_j - c_j\| \leq \frac{2\bar{V}}{\mu_{c,j}}$, i.e., the boundedness of $x_j$ (since $c_j$ is finite), the boundendess of $v_i, \hat{a}_i$, $\forall i\in\mathcal{N}$, as well as that the multi-agent trajectory is free of collisions and connectivity breaks, $\forall t\geq t_0$. Since the multi-agent system stays connected and $x_j$ is bounded, we conclude that the rest $x_i$, $i\in\mathcal{N}\backslash\{j\}$ are also bounded, $\forall t\geq t_0$. Moreover, by invoking LaSalle’s invariance principle, we conclude that the system will converge to the largest invariant set contained in $\mathbb{L} \coloneqq \{ (x,v,\hat{a})\in\mathbb{R}^{2Nn} : v_i = 0, \forall i\in\mathcal{N}\}$, which is the set $\widetilde{\mathbb{L}} \coloneqq \{ ((x,v,\hat{a}))\in\mathbb{R}^{2Nn} : \dot{v}_i = 0,v_i = 0, \forall i\in\mathcal{N}\}$. By considering the closed loop system - and taking into account the positive definiteness of $B_i$, we conclude that the system will converge to the configuration $$(D(\mathcal{G}_0)\otimes I_n)\widetilde{\beta}_\text{con}+ (D(\bar{\mathcal{G}})\otimes I_n)\widetilde{\beta}_\text{col} - \gamma_{\mathsf{md}}(x) =0. \label{eq:LaSalle}$$ Note that $\mathcal{G}_0$ and $\bar{\mathcal{G}}$ are connected graphs, and hence the sum of the rows of $D(\mathcal{G}_0)$ and $D(\bar{\mathcal{G}})$ is zero. In particular, let $D(\mathcal{G}_0) = [d_{0,1},\dots,d_{0,N}]^\top$, $D(\bar{\mathcal{G}})= [\bar{d}_1,\dots, \bar{d}_{N}]^\top$, where $d^\top_{0,i}\in\mathbb{R}^{M_0}$, $\bar{d}^\top_i\in\mathbb{R}^{\bar{M}}$, $i\in\mathcal{N}$, are the rows of $D(\mathcal{G}_0)$ and $D(\bar{\mathcal{G}})$, respectively. Then it holds that $\sum_{i\in\mathcal{N}}d_{0,i} = \sum_{i\in\mathcal{N}}\bar{d}_i = 0$. We can then write $D(\mathcal{G}_0)\otimes I_n = [d_{0,1}\otimes I_n,\dots,d_{0,N}\otimes I_n]^\top $, $D(\bar{\mathcal{G}})\otimes I_n= [\bar{d}_1\otimes I_n,\dots, \bar{d}_{N}\otimes I_n]^\top$ and hence becomes
\[eq:LaSalle element form\] $$\begin{aligned}
& [d_{0,i}\otimes I_n]^\top \widetilde{\beta}_\text{con} + [\bar{d}_i\otimes I_n]^\top \widetilde{\beta}_\text{col} = 0, \ \forall i \in \mathcal{N}\backslash\{j\}, \label{eq:LaSalle element form 1} \\
& \gamma_j(c_j) - [d_{0,j}\otimes I_n]^\top \widetilde{\beta}_\text{con} - [\bar{d}_j\otimes I_n]^\top \widetilde{\beta}_\text{col} = 0. \label{eq:LaSalle element form 2}
\end{aligned}$$
From we obtain that $\gamma_j(c_j) - [(-\sum_{i\in\mathcal{N}\backslash\{j\}} d_{0,i} )\otimes I_n]^\top\widetilde{\beta}_\text{con} - [(-\sum_{i\in\mathcal{N}\backslash\{j\}} \bar{d}_i )\otimes I_n]^\top\widetilde{\beta}_\text{col} = 0$, which implies $\gamma_j(c_j) + \sum_{i\in\mathcal{N}\backslash\{j\}}\{ [d_{0,i}\otimes I_n]^\top\widetilde{\beta}_\text{con} + [\bar{d}_i\otimes I_n]^\top\widetilde{\beta}_\text{col} \} = 0$ and in view of , $\gamma_j(c_j)= 0$. Therefore, it holds that $\lim_{t\to\infty}x_j(t) = c_j$, which implies that, for every $\varepsilon$, there exists a $t_f > t_0$ such that $\|x_j(t) - c_j\| < \varepsilon, \forall t \geq t_f$. Hence, since $x_j$ is the center of $\mathcal{A}_j(x_j)$, we conclude that there exists a finite $t_f$ such that $c_j\in\mathcal{A}_j(x_j(t_f))$, which leads to the conclusion of the proof.
Hybrid Control Strategy {#subsec:hybrid strategy}
-----------------------
In this section, we propose a decentralized switching strategy for each agent to decide on its own activity or passivity. Through this strategy, we integrate the discrete plan execution from Section \[subsec:discrete plan synthesis\] and the continuous control scheme from Section \[subsec:continuous control\] into a hybrid control scheme, which monitors the plan execution online. The desired plans for the agents, from Section \[subsec:discrete plan synthesis\], are $\mathsf{plan}_i \coloneqq \ (c_{i1^\text{G}}, \sigma_{i1^\text{G}}) \ \dots \ (c_{il_i^\text{G}}, \sigma_{il_i^\text{G}}) \ ((c_{i(l_i+1)^\text{G}}, \sigma_{i(l_i+1)^\text{G}})\dots$ $(c_{iL_i^\text{G}}, \sigma_{iL_i^\text{G}}))^\mathsf{\omega}$, i.e., agent $i\in\mathcal{N}$, has to pass through the points $c_{i1^\text{G}}, \dots, c_{iL_i^\text{G}}$ and provide the corresponding services $\sigma_{i1^\text{G}},\dots,\sigma_{iL_i^\text{G}}$, which satisfy formula $\phi_i$, i.e, $\sigma_{i1^\text{G}}\dots\sigma_{il_i^\text{G}}(\sigma_{i(l_1+1)^\text{G}} \sigma_{iL_i^\text{G}})^\mathsf{\omega}\models \phi_i$.
Let each agent have a counter variable $s_i$ initiated at $s_i = 1$, as well as a cycle counter $\kappa_i$, initiated at $\kappa_i = 1$, $\forall i\in\mathcal{N}$. Then, given the agent priority variables $\mathsf{pr}_i$, each agent executes $u_i = u_i(c_{is_i^\text{G}},1)$ if $\kappa_i = \mathsf{pr}_i$ and $u_i(c_{is_i^\text{G}},0)$ if $\kappa_i \neq \mathsf{pr}_i$. The agents update the cycle counter $\kappa_i$ every time the current active agent reaches its desired point, and the variable $s_i$ every time they reach their current desired point. Each agent provides the services $\sigma_{il^\text{G}}$ if $c_{il^\text{G}}\in\mathcal{A}_i(x_i)$ and $\kappa_i = \mathsf{pr}_i$, otherwise he does not provide any services. More specifically, we construct the following algorithm:
$\kappa_i \leftarrow 1, s_i \leftarrow 1$, $\forall i\in\mathcal{N}$\
Loosely speaking, agent $i$ provides the services $\sigma_{is^{\text{G}}}$ only if $c_{is_i^\text{G}}\in\mathcal{A}_i(x_i)$, i.e., if it is in the respective desired point of interest, and $\kappa_i = \mathsf{pr}_i$, i.e., it is its turn to be active. As soon as it reaches the point and provides the services, it updates its progressive goal index $s_i$, and everyone in the team updates the cycle counter $\kappa_i$, so that another agent becomes active. Note that the agents need to know when the current agent reaches its progressive goal and provides its services so that they update the counter variable $\kappa_i$. To that end, the current agent can simply communicate this information as soon as it provides its services. Since the communication graph is always connected, the information can propagate to all agents. Note that potential time delays in this inter-agent communication do not affect the overall strategy. A communication-free solution could be the use of state and input estimators along with the discontinuous change of the control law of the current agent [@guo2016communication].
In that way, all the agents eventually reach their goal points of interest and provide the corresponding services. More specifically, the resulting time trajectory of each agent yields the behavior $\beta_i = (c_{i1},\sigma_{i1})(c_{i2},\sigma_{i2})\dots$, and the desired behavior $\mathsf{plan}_i$ is a subsequence of $\beta_i$, with $\sigma_{i\ell} = \emptyset$, $\forall \ell : \sigma_{i\ell}\neq \sigma_{i\ell^\text{G}}$, i.e., agent $i$ does not provide any services in unplanned crossing of points of interest (while navigating to a desired point or being in passive mode), providing only the desired services at the corresponding desired points.
SIMULATION RESULTS {#sec:Simulation}
==================
We consider $N = 5$ holonomic spherical agents in $\mathbb{R}^3$, with $r_i=1\text{m}$, $d_{\text{con},i} = 4\text{m}$, priorities as $\mathsf{pr}_i = i$, $\forall i\in\mathcal{N}$, and initial positions $x_1 = [0,0,0]^\top\text{m}$, $x_2 = [-2.1,-2.3,2]^\top\text{m}$, $x_3 = [1.3,1.3,1.5]^\top\text{m}$, $x_4 = [-2,3.25,2.2]^\top\text{m}$, $x_5 = [2,2.4,-0.15]^\top\text{m}$, which give the edge set $\mathcal{E}_0 = \{(1,2),(1,3),(3,4),(3,5),(1,5)\}$. The complete edge set is $\bar{\mathcal{E}}$ $=$ $\{(1,2)$, $(1,3)$, $(3,4)$, $(3,5)$, $(1,5)$, $(1,4)$, $(2,3)$, $(2,4)$, $(2,5)$, $(4,5)\}$. We choose $B_i = b_{m_i} I_3$ and $f_i(x_i,v_i) = \alpha_i\|x_i\|\sin(w_{i,1}t + w_{i,2})v_i$, with $b_{m_i}, w_{i,1}$, $w_{i,2}$ randomly chosen in the interval $(1,2)$, $\forall i\in\mathcal{N}$. The points of interest are $c_1 = [10,10,10]^\top \text{m}$, $c_2 = [-5,0,5]^\top \text{m}$, $c_3 = [5,-2,-7]^\top \text{m}$, $c_4 = [0,-6,2]^\top \text{m}$. For simplicity, we consider that each agent can provide the services $\Psi_i = \{``\mathsf{r}_i",``\mathsf{b}_i",``\mathsf{g}_i",``\mathsf{m}_i"\}$, $\forall i\in\mathcal{N}$, and $\mathcal{L}_i(c_1) = \{``\mathsf{r}_i"\}$, $\mathcal{L}_i(c_2) = \{``\mathsf{b}_i"\}$, $\mathcal{L}_i(c_3) = \{``\mathsf{g}_i"\}$, $\mathcal{L}_i(c_4) = \{``\mathsf{m}_i"\}$ $\forall i\in\mathcal{N}$. The LTL formulas were taken as $\phi_1 = \square\lozenge(``\mathsf{r}_1"\land``\mathsf{r}_1"\bigcirc ``\mathsf{g}_1"\bigcirc \mathsf{m}_1 \bigcirc ``\mathsf{b}_1")$, $\phi_2 = \lozenge``\mathsf{m}_2"\land \square\lozenge(``\mathsf{r}_2"\land``\mathsf{b}_2")$, $\phi_3 = \lozenge``\mathsf{m}_3"\land \square\lozenge(``\mathsf{r}_3"\land``\mathsf{b}_3")$, $\phi_4 = \square\lozenge(``\mathsf{g}_4"\land``\mathsf{g}_4"\bigcirc ``\mathsf{b}_4"\bigcirc \mathsf{m}_4 \bigcirc ``\mathsf{g}_4")$, and $\phi_5 = ``\mathsf{r}_5"\land \square\lozenge(``\mathsf{b}_5"\land ``\mathsf{m}_5"\bigcirc ``\mathsf{g}_5")$. By following the procedure described in Section \[subsec:discrete plan synthesis\], we obtain the desired plans $\mathsf{plan}_1 = ((c_1,``\mathsf{r}_1")(c_3,``\mathsf{g}_1")(c_4,``\mathsf{m}_1")(c_2,``\mathsf{b}_1"))^\mathsf{\omega}$, $\mathsf{plan}_2 = (c_2,``\mathsf{b}_2")(c_4,``\mathsf{m}_2")((c_1,``\mathsf{r}_2")(c_2,``\mathsf{b}_2"))^\mathsf{\omega}$, $\mathsf{plan}_3 = (c_4,``\mathsf{m}_3")(c_3,``\mathsf{g}_3")((c_1,``\mathsf{r}_3")(c_2,``\mathsf{b}_3"))^\mathsf{\omega}$, $\mathsf{plan}_4 = ((c_3,``\mathsf{g}_4")(c_2,``\mathsf{b}_4")(c_4,``\mathsf{m}_4")(c_3,``\mathsf{g}_4"))^\mathsf{\omega}$, and $\mathsf{plan}_5 = (c_1,``\mathsf{r}_5")((c_4,``\mathsf{m}_5")(c_3,``\mathsf{g}_5")(c_2,``\mathsf{b}_5"))^\mathsf{\omega}$. We assume that the services are provided instantly by the agents. The control gains are chosen as $\mu_{c,i} = 3$, $\mu_i = 25$, $\mu_{\alpha,i} = 0.1$, $\forall i\in\mathcal{N}$, and $\mu_{\text{con},m} = \mu_{\text{col},m} = 0.1$, $\forall m\in\mathcal{M}_0$, $m\in\bar{\mathcal{M}}$. The simulation results are depicted in Fig. \[fig:gammas\]-\[fig:a\_hats\]. for $t\in[0,10^3]\text{sec}$. More specifically, Fig. \[fig:gammas\] shows the distance functions $\mathsf{md}_i\gamma_i$, $\forall i\in\mathcal{N}$. In the total time duration, all the agents execute their first goal of their respective plans, according to their assigned priorities, whereas agent $1$ executes its second goal as well; Fig. \[fig:beta\_col\] and \[fig:beta\_con\] illustrate the collision- and connectivity- associated terms $\beta_{\text{col}_m}(\iota_m)$, $\forall m\in\bar{\mathcal{M}}$, $\beta_{\text{con}_m}(\eta_{m})$, $\forall m\in\mathcal{M}_0$, which are always positive, verifying the collision avoidance and connectivity maintenance properties. Finally, Fig. \[fig:a\_hats\] depicts the adaptation variables $\hat{\alpha}_i$, $\forall i\in\mathcal{N}$, which are always kept bounded.
CONCLUSIONS AND FUTURE WORKS {#sec:Conclusion}
============================
This paper presented a hybrid coordination strategy for the motion planning of a multi-agent team under high level specifications expressed as LTL formulas. Inter-agent collision avoidance and connectivity maintenance is also guaranteed by the proposed continuous control protocol. Future efforts will be devoted towards addressing timed temporal tasks as well as including workspace obstacles.
[^1]: The authors are with the KTH Center of Autonomous Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden. Email: [{cverginis, dimos}@kth.se]{}. This work was supported by the H2020 ERC Starting Grant BUCOPHSYS, the European Union’s Horizon 2020 Research and Innovation Programme under the GA No. 731869 (Co4Robots), the Swedish Research Council (VR), the Knut och Alice Wallenberg Foundation (KAW) and the Swedish Foundation for Strategic Research (SSF).
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---
abstract: 'We study in detail, by experimental measurements, atomistic simulations and DFT transport calculations, the process of formation and the resulting electronic properties of atomic-sized contacts made of Au, Ag and Cu. Our novel approaches to the data analysis of both experimental results and simulations, lead to a precise relationship between geometry and electronic transmission – we reestablish the significant influence of the number of first neighbors on the electronic properties of atomic-sized contacts. Our results allow us also to interpret subtle differences between the metals during the process of contact formation as well as the characteristics of the resulting contacts.'
author:
- 'C. Sabater'
- 'W. Dednam'
- 'M. R. Calvo'
- 'M. A. Fernández'
- 'C. Untiedt'
- 'M. J. Caturla'
bibliography:
- 'j2c\_relativistic.bib'
title: The role of first neighbors geometry in the electronic and mechanical properties of atomic contacts
---
Introduction
============
Single atoms and molecules have been widely hailed as potential electronic devices over the last twenty years [@joachim_electronics_2000]. To make such devices a reality, metallic contact formation and the electrical characteristics of few-atom contacts, need to be understood in depth at the atomic level. The electrical conduction in single-atom contacts has been broadly studied both from an experimental and theoretical point of view [@agrait2003quantum], and single-atom contacts have been proposed as elementary circuit components, such as quantized resistors, capacitors [@wang1998capacitance] or switches. [@terabe2005quantized]
The conductance of few-atom contacts is given by the sum of contributions from quantized transport modes propagating at the contact junction and the number and transmission probabilities of those modes are determined by the size and chemical valence of the central part of the constriction [@agrait2003quantum]. For example, both a single-atom contact and a monoatomic chain of Au exhibit a resistance of around a quantum of conductance $G_{0}=2e^2/h$, which is in this case the signature of electronic transport through a single, fully open, quantum channel [@Otal13]. However, variations in the geometrical configuration of the leads[@sabater2013understanding], i.e., the number of neighboring atoms in the constriction, give rise to fluctuations of up to 20 percent in the conductance of a single atomic contact. Not only the electrical properties of single atom contacts are strongly influenced by their coordination to the leads, but also their mechanical properties. When two electrodes in the tunneling regime eventually come into contact, it is known, for certain materials and geometries, that the process of contact formation happens as a sudden jump. Nonetheless, jump to contact is not a generalized phenomenon and the process of formation may be smooth. [@Untiedt_2007] The probability of occurrence of jump to contact and the details of this process have already been suggested to strongly depend not only on the bulk mechanical properties of the material, such as its cohesive energy and Young’s modulus[@Trouwborst_2008; @Fern_ndez_2016], but also, for certain materials, e.g., Au or Cu, on the specific geometry of the contacting leads [@Kr_ger_2009; @Kroger16].
In this article, we focus on the influence of the first-neighbor configurations on the process of formation of single-atom contacts made of Au, Ag and Cu, as well as their associated conductance values. To this end, we combine atomistic simulations and quantum transport calculations [@pethica1988stability; @landman1990atomistic; @Bratkos; @brandbyge1995conductance; @sorensen1998mechanical; @dreher2005Aucontacts] with a detailed analysis of experimental results. We improve the statistical analysis carried out by Untiedt *et al.* [@Untiedt_2007] for Au, and compare our results with those obtained from the atomistic simulations we perform to determine the most likely first-neighbor structures at first contact, and corresponding conductance values we calculate from Density Functional Theory (DFT) methods. [@palacios2001fullerene; @palacios2002transport; @louis2003keldysh] From such a comparison between simulation and experimental results, we can relate the distribution of contact conductances to specific geometries. In agreement with the results published in Refs. [@Untiedt_2007; @sabater2013understanding] we find the most likely geometries to lie within four classes: monomers, dimers, Double Contacts (D.C) and Triple Contacts (T.C). Furthermore, we identify more specific structures within these classes and more interestingly, find the dispersion in conductance values for each of these classes to be a consequence of the variations in the number of first neighbors. Our analysis provides a precise assignment of the conductance values reported for these configurations, and remarkably, yields a broader distribution of conductance values for the monomer than in previous works on Au, ultimately explaining previous disagreements between experiments and theory. The reason for this can be traced to a higher dependence of the monomer’s conductance on the number of first neighbors. We complete our study by carrying out a similar analysis for Ag and Cu.
Methods
=======
Experimental methods
--------------------
Our atomic contacts are fabricated by performing several cycles of indentation and separation of two electrodes made of the same high purity (99.999%) metal, Au, Ag or Cu, under cryogenic vacuum at 4.2K. The electrical conductance of the junctions (obtained as the measured current divided by the applied voltage of 100 mV) is recorded while the two electrodes are carefully brought into contact in a scanning tunneling microscope (STM) setup, as described in previous works. [@Untiedt_2007; @sabater2013understanding] The traces of conductance as a function of electrode distance (Fig.\[autrhis\](a)) contain valuable information about the process of contact rupture and formation. When electrodes are close enough but not yet in contact, electrons may tunnel between them. In the tunneling regime, the conductance increases exponentially as the separation between leads decreases. The conductance increases smoothly until a sudden jump occurs, from the tunneling regime up to a clear plateau at around 1 $G_{0}$, indicating the formation of a monoatomic contact [@agrait2003quantum]. Examples of rupture and formation traces are displayed in Fig.1(a).
![a) Conductance trace for the formation and rupture of a gold atomic contact recorded in our STM-MCBJ setup at 4.2 K. b) Conductance histogram built from more than 1000 Au contact rupture traces.[]{data-label="autrhis"}](Figure1.pdf){width="50.00000%"}
Every realization of an atomic-size contact produces a slightly different conductance trace, which is suggestive of a variation in structural configurations. Therefore, a statistical analysis of the data is key to extracting information about the most probable configurations. An approach that is widely used in the literature [@agrait2003quantum] is the construction of a conductance histogram (such as the one in Fig. 1b for the case of rupture traces of Au), to determine the conductance values associated with the most probable configurations of the single-atom contact.
A more specific method for the study of contact formation was introduced by Untiedt et al. [@Untiedt_2007] As sketched in Fig. 1a, for each formation trace, the highest jump in conductance between two consecutive points is monitored. Two conductance values are then recorded, $G_{a}$, from which the jump occurs and $G_{b}$, the final value immediately after the jump. A density plot of the pairs $(G_{a},G_{b})$ (main panel in Fig. 2) displays the values of greatest probability from and to which the conductance jump occurs.
![Central panel: density plot constructed from the pairs $(G_a,G_b)$ obtained as described in the text from more than 2000 traces of formation of Au contacts. Right panel: Scatter plot showing the projection of the density plot on the $G_b$ axis. As shown in previous works [@Untiedt_2007], this projection can be fitted to the sum of three gaussian peaks (green line). The purple, orange and yellow lines represent the individual gaussian components. Bottom panel: Projection of density plot on the $G_a$ axis (scatter plot). The maximum above the dashed line has been left out of the analysis here in order to more clearly identify the components of the maximum below the line. The projection of the latter maximum can be fitted to the sum of two gaussian distributions (purple line). The individual components are shown as the yellow and orange lines.[]{data-label="Fig2"}](Figure2.pdf){width="45.00000%"}
As mentioned above, prior to contact formation, the tunneling conductance depends exponentially on the distance between electrodes as $G\simeq Ke^{-\frac{\sqrt{2m\phi}}{h}d}$, where $K$ is a proportionality constant which depends on the cross-sectional area and density of states at the Fermi level of the electrodes, $m$ corresponds to the electron mass and $\phi$ is the work function of the material. Since $G_a$ is the conductance in the tunneling regime immediately before jump to contact, its logarithm $log(G_a)$ [^1] is proportional to the distance between the electrodes from which the jump occurs. When the $G_a$ axis is plotted on a logarithmic scale, the density plot corresponding to formation of Au contacts, reveals shapes of the maxima that can be more easily interpreted than those previously reported in Ref. [@Untiedt_2007]
Data analysis
-------------
The projections of the density plot data on both $log(G_a)$ and $G_b$ axes (Fig. 2) can be fitted to a sum of gaussian peaks. This suggests that the density plot is formed by a number of maxima which are normally distributed in both variables. Therefore, we fit the data to the sum of three bivariate normal distributions, sketched as ellipses in Fig. 2 and labeled $D1$, $D2$ and $D3$, with different relative probabilities $p$. Each of these distributions is described by the expression: $$f(x,\mu,\Sigma)=\frac{1}{\sqrt{|\Sigma|}(2\pi)^2}e^{-\frac{1}{2}(x-\mu)'\Sigma^{-1}(x-\mu)}
\label{eq:xmusigma}$$
where $x=(log(G_a),G_b)$, $\bf{\mu}=(\mu_a, \mu_b)$ and $\Sigma=\left( \begin{array}{cc}
\sigma_a^2 &\rho\sigma_a\sigma_b \\
\rho\sigma_a\sigma_b & \sigma_b^2 \end{array} \right) $. $\mu_i$ and $\sigma_i$ represent the 2D equivalents of the unidimensional mean and standard deviation, respectively, and $\rho$ is the correlation parameter between variables $logG_a$ and $G_b$.
Experimental Results
====================
The experiments and analysis described in the previous section were repeated during the fabrication of over 2000 contacts made of Au, Ag and Cu. The output fitting parameters for the three materials are summarized in Table I. The characteristic parameters of the distributions can be graphically represented by an ellipse (for example, as the overlays in Fig. 2). The center of the ellipse ($\mu_a$,$\mu_b$) represents the (log$G_a$,$G_b$) position of the mean of the distribution, and the axes of the ellipse represent the standard deviations ($\sigma_a$,$\sigma _b$) in the respective conductance axis. The tilt of the ellipse is proportional to the correlation ($\rho$) between the two variables. The identification of three maxima is in good agreement with Ref. [@Untiedt_2007] for Au. This new analysis provides an opportunity to revise those results and carry out a more precise quantitative analysis of the data.
In analogy with Ref. [@Untiedt_2007], we find an isolated distribution with a low probability of occurrence, well above $G_0$ (labeled $D3$ in Fig. 2), while, at around 1 $G_0$, we find the sum of two distributions. Here we disentangle those two distributions and provide an estimate of their relative probabilities ($p$). Distribution $D1$ contains more than 50 percent of the data, while $D2$ contains around 30 percent. In this instance, the results for all three materials are similar.
[|p[1cm]{}||p[1cm]{}||p[1cm]{}|p[1cm]{}||p[1cm]{}|p[1cm]{}|p[1cm]{}|]{}\
& p (%) &$ \mu_{a} $&$\mu_{b} $& $\sigma_{a}$& $\sigma_{b}$& $\rho$\
**& 58 &-1.2 &0.9 &0.4 &0.2 &0.3\
**&32&-1.2 &1.0 &0.4 &0.05 &0.3\
**&10 &-1.1 &1.7 &0.4 &0.3 &0.2\
******
\
[|p[1cm]{}||p[1cm]{}||p[1cm]{}|p[1cm]{}||p[1cm]{}|p[1cm]{}|p[1cm]{}|]{}\
& p (%) &$ \mu_{a} $&$\mu_{b} $& $\sigma_{a}$& $\sigma_{b}$& $\rho$\
**&52 &-0.6 &1.1 &0.2 &0.2 &0.4\
**&30 &-0.9 &1.0 &0.2 &0.08 &0.5\
**&18 &-0.6 &1.9 &0.2 &0.2 & 0.4\
******
\
[|p[1cm]{}||p[1cm]{}||p[1cm]{}|p[1cm]{}||p[1cm]{}|p[1cm]{}|p[1cm]{}|]{}\
& p (%) &$ \mu_{a} $&$\mu_{b} $& $\sigma_{a}$& $\sigma_{b}$& $\rho$\
**&57 & -0.6 &1.0 &0.3 &0.2 &0.12\
**&29 &-0.8 &1.0 & 0.3 &0.08 &0.2\
**&14 &-0.6 &1.8 &0.3 &0.2 &0.2\
******
Moreover, on comparing the three materials, we discover a striking result: there is an important difference between the jump distance of Au versus Ag and Cu, represented by their mean values of log($G_a/G_0$) denoted for simplicity as $\mu_a$. This is the focus of a separate study [@PRL], in which we show that the origin of this phenomenon can be traced to the different strengths of relativistic effects in these materials. Besides the information given by the mean of each distribution, the standard deviation also provides a measure of the dispersion in each. Regarding the dispersion in the $G_a$ axis (*$\sigma_a$*), which may appear to be much larger in the case of Au, if scaled to the mean value, it is actually similar to the dispersion for Ag and Cu. This indicates, in all cases, that the variation in jump distance is a percentage of the average distance, which, in turn, supports the interpretation provided in Ref. [@PRL], that the dispersion in conductance originates from the large number of possible geometrical configurations.
While the dispersion in conductances before jump ($\sigma_a$) remain similar in all three distributions for each material, remarkably, the dispersions in $G_{b}$ (*$\sigma_b$*) exhibit significant differences. Distribution $D1$, in contact conductance $G_b$, is rather broad, while distribution $D2$, the second-most probable, exhibits a rather narrower dispersion in this parameter, as is evident from the widths of the ellipses in the $G_b$ axis $\sigma_b$. This point will be discussed further in light of atomistic simulations, but it already suggests that the conductance in contact of one of the distributions is considerably less sensitive to geometrical variations than the other.
Finally, we note that the correlation between $G_a$ and $G_b$, *$\rho$* (visible from the tilt of the ellipses) is very similar not only for the three distributions, but also for all three materials, indicating a slight tendency for contacts associated with shorter jump distances to exhibit higher conductances.
Besides the notable discrepancies in $\mu_a$, a comparison of the metals yields also a number of subtle differences that are connected to the longer jump distance in the case of Au. Firstly, the means $\mu_a$ of $D1$ and $D2$ for Au, occur at about the same distance, while $D3$’s mean has a slightly different value. However, for Ag and Cu, distributions $D1$ and $D3$ are centered at similar values of $log G_a$, while the contacts corresponding to $D2$ are established from a greater jump distance. Regarding the value of *$\mu_{Gb}$*, note the lower conductance value for $D1$ in the case of Au with respect to the other two distributions, as well as with respect to the corresponding values for Ag or Cu. Although differences between $D1$ and $D2$ are small and perhaps within error margins, this behavior is expected for the more “stretched out” structures formed in Au [@PRL].
Molecular Dynamic simulations and *Ab-Initio* calculations
==========================================================
Methodology
-----------
We have not found any analysis of experimental measurements of electronic transport in the literature, which can provide information about the geometry at or the instant just before contact is established. Therefore, in order to have an appreciation of the importance of the configuration of the atoms in the immediate vicinity of few-atom contacts, we simulate the experiments by means of classical molecular dynamics (CMD) and first-principles quantum transport calculations. An alternative approach is used in Refs. [@Hybertsen_2016; @Hybertsen_2017], in which a potential energy surface is calculated as an adiabatic trajectory by DFT. Metal junctions composed of small opposing fragments of Au, Ag or Cu are elongated/separated in small steps with a geometry optimization at each step
Molecular dynamics simulations are based on solving Newton’s second law for all the atoms, as they evolve from their initial positions. In such simulations, the potential used to model interactions between the atoms is semi-empirical. [@allen1989computer] The initial structure in the present work is independent of the metal and consists of 4736 atoms, oriented along the \[100\] crystallographic direction, as shown in panel a) of Fig. \[MDfigure\]. The result of solving Newton’s second law for this system is that we can obtain the classical trajectories of all the atoms in the structure, as it is ruptured and brought back into contact over many cycles. Extracting from these trajectories, then, the structure at first contact, as well as the one immediately before it, will, via DFT transport calculations [@palacios2001fullerene; @palacios2002transport; @louis2003keldysh], yield the conductance at the moment that contact is re-established.
![Snapshots of a gold nanocontact at different times during a molecular dynamics simulation. Panel: a) initial structure, arrows in a) and b) indicate the direction of elongation or compression. Panel c) shows a zoom-in of panel b), which is the step immediately before the contact shown in panel d) has formed.[]{data-label="MDfigure"}](Figure3.png){width="48.00000%"}
As mentioned above, all the simulations involving Au, Ag and Cu are based on the same initial seed structure. The simulations are run in a way that reproduces cyclic loading of the nanowire in analogy with a typical STM or mechanically controllable break junction (MCBJ) experiment. This is also an approach that was followed in our previous works. [@sabater2012mechanical; @sabater2013understanding] The interactions between the metal atoms are modeled by the semi-empirical, embedded atom method (EAM) potential. [@daw1983semiempirical] All the simulations have been realized by means of the Large-scale Atomic/Molecular Massively Parallel Simulator LAMMPS. [@plimpton1995fast; @lammps2] The potential parameters used for Au, Ag and Cu in this work, are taken from Ref. [@zhou2001atomic] The potential itself is derived in Ref. [@wadley2001potential]
Additionally, in order to mimic the conditions of the experiment as closely as possible, we simulate at the boiling point of liquid Helium, $4.2$ K. The Nose-Hoover thermostat [@nose1984molecular; @hoover1985canonical] serves to maintain the temperature constant during the cycles of retraction and approach of the nanoelectrodes in the simulations. Thermostatting is performed every 1000 simulation time steps, the time interval that is recommended by the developers of LAMMPS.[@lammps2]
The atoms that are located in the first three crystallographic planes from the top of the initial seed structure, as well as the corresponding three planes at the bottom, are pinned to their equilibrium bulk lattice positions so as to constrain their relative positions. The remaining atoms respond dynamically to the bulk motion of these “frozen" planes. Following, the entire structure is stretched lengthwise (vertically) by moving the frozen layers in opposite directions at a constant speed of $\sim$1 m/s. The arrows in Fig. \[MDfigure\], panels a) and b), illustrate the directions of the applied forces on both ends (top/bottom) during contact rupture and formation. A speed of $\sim$1 m/s may be many orders of magnitude greater than that employed in the experiments, but we argue that there is enough time for the structures to reach equilibrium, and not merely meta-stable states, because this speed is at least three orders of magnitude lower than that of sound in the bulk metals. [@sorensen1998mechanical] The low temperature used in our simulations also ensures that processes that would otherwise be important at microsecond time-scales, such as surface diffusion, remain negligible. In fact, at $4.2$ K, surface diffusion is inhibited by activation energies that are $3-4$ orders of magnitude higher than the thermal energy of the atoms. [@ibach2006physics]
To perform cyclic loading in CMD, the simulation domain is divided longitudinally into slices of equal height, corresponding to the interlayer spacing within the bulk crystal. In a face-centered cubic crystal, this spacing is half the lattice parameter along the \[100\] crystallographic axis. The slice containing the least number of atoms then corresponds to the minimum cross section of the nanocontact. Hence, the structure is stretched until the minimum-atom slice and either of the slices adjacent to it no longer contain any atoms as shown in Fig. \[MDfigure\] c). At this point, the motion is reversed and the two ruptured tips are brought back together at the same speed with which the structure was first broken. When the minimum-atom layer contains more than 15 atoms, the motion is once more reversed and the nanocontact is stretched until it breaks. This process is repeated at least 20 times. To clarify our terminology, we denote by one “cycle" a single rupturing and re-forming of the contact. It is crucial in our simulations to know at which time step during approach, first contact occurs. We detect this moment by monitoring the value of the minimum cross section, which happens when there are more than 0 atoms in the contact cross section. This means that contact has been (re-)established. Incidentally, the semi-empirical potentials describing the interactions between the atoms in the simulations, lead to first-contact distances ranging up to half-way between first and second neighbors in a bulk FCC lattice: $\sim3.5$ Åin the cases of Au and Ag, and $\sim3.0$ Åin the case of Cu. In past works, this has also been used as the criterion to identify the moment of first contact.[@Fern_ndez_2016; @Dednam2014contacts] Figure \[MDfigure\] c) and d) show the structure prior to and immediately after first contact, respectively.
Finally, to calculate the conductance of structures extracted from molecular dynamics simulation trajectories, we have used the electronic transport code ANT.G, [@ANTG] which depends on DFT parameters calculated by GAUSSIAN09. [@GAUSSIAN09] The structures obtained from CMD contain more than 4000 atoms. Therefore, in order to compute the conductance of these structures within a reasonable time via DFT calculations, it has been necessary to trim the region of interest down to around 500 atoms, keeping only those atoms that lie within a box smaller than the original simulation domain, and centered on the region of first contact, or minimum cross section. However, obtaining accurate conductance values required, in addition, that we had to assign a larger basis set of 11 valence electrons to 40 atoms in the contact region. The rest of the atoms were assigned a basis set of one valence electron.
Molecular Dynamics Results\[SecMD\]
-----------------------------------
For the analysis of the CMD results obtained after 20 cycles of contact rupture and formation, we have used a simple algorithm that counts the number of atoms in layers spaced vertically along the simulation domain. By keeping in mind that the three layers on opposite ends of the structures remain “frozen" internally during the simulations, i.e., that the lattice parameter of these layers stays fixed at the bulk value, we discretize the entire structure into a number of layers half the bulk lattice parameter in thickness. As lattice parameters, we used 4.08 Åfor Au and Ag, and 3.61 Åfor Cu. Consequently, during an approach (contact formation) phase, for example, we count, at every step, the number of atoms in each layer. Figure \[figexplacative\] a) shows how the layers are distributed along the length of the nanocontact. The plot in Fig. \[figexplacative\] c) was constructed by counting the number of atoms in each layer. Thus, in principle, a zoom-in of the atoms in the minimum cross section in a), located somewhere between layers 24 and 29, should lead us to conclude that the contact type is “4-1-1-4". Panel b) is such a zoom-in of panel a) and shows clearly what the contact type is. It therefore confirms, via visual inspection, the result inferred from panel c). The trace in Fig. \[figexplacative\] d) has been constructed by plotting the minimum of the parabola in c) against simulation time step. The resemblance to an experimental conductance trace is, at the very least, suggestive. Furthermore, we would like to point out that panel c) contains more information than is used for the purposes of the present article. In fact, such a plot can also give us an idea about the evolution of the sharpness of the contact. For example, blunt electrodes will give rise to broader parabolas than sharper tips. This tool could open the way to a novel analysis of the evolution of the contact in CMD, one that renders direct visualization unnecessary. In addition, a better counting algorithm could take advantage of it. All the results in Fig. \[figexplacative\] have been extracted from cycle 5 of the simulation involving Au, in which contact occurs at time step 85000.
The methodology followed to count atoms in the cross section is not unique. Other algorithms, such as the one developed by Bratkovsky *et al.* [@Bratkos] do not count an integer number of atoms and neighbors in the contact minimum cross section. In this work, we have modified the Bratkovsky algorithm to suit our purposes and count an integer number of atoms in the layers. We are well aware of the limitations of our method, therefore, to obtain complementary information, we calculate the conductance of the CMD structures via DFT and if, in the worst of cases, it differs very much from the expected value, we recheck the structure by visual inspection, and where necessary reassign an appropriate contact type.
![a) The atomic-sized gold contact simulated via MD, with the the layer positions indicated by dashed-dotted lines. b) A zoom-in of panel a), showing the type of contact identified by our algorithm, the results of which are shown in panel c). d) The number of atoms in the minimum cross-section as a function of simulation time step. The inset in c) is a zoom-in that shows when, during the simulation, exactly 1 atom remains in the minimum cross-section.[]{data-label="figexplacative"}](Figure4.jpg){width="51.00000%"}
Thus, we have employed the approach summarized in Fig. \[figexplacative\], to study the 3 metals and the 20 cycles of contact rupture-formation they undergo during the simulations. By following the criterion that is outlined in the next paragraph, we have been able to identify different types of contacts as well as their first neighbors, as detailed in Fig. \[Typeofcontact\].
![Illustration of the different types of contacts. Left column: low-coordination first-neighbor contacts. Right column: high-coordination first-neighbor contacts. Each of the single, double or triple contacts can also occur as monomers or dimers.[]{data-label="Typeofcontact"}](Figure5.jpg){width="49.00000%"}
Our criterion for identifying the contacts as single, double or triple involves counting the number of atoms in the minimum cross section between the leads, at the very moment when the corresponding layers become populated during the simulation. All three contact types can occur in a monomeric or dimeric configuration, as illustrated in Fig. \[Typeofcontact\]. The “low" and “high" coordination designations, irrespective of whether the contacts are monomeric or dimeric, depend on the number of first neighbors found by our algorithm, on either side of the minimum-atom layer. We have established the limit of first neighbors based on an exposed (001) FCC surface layer, which, as is known, is puckered by four-fold hollows, such that an adsorbed atom will have 4 first neighbors immediately beneath it. [@Fern_ndez_2016] Then, “low" coordination means equal to or less than 4 first neighbors, in both electrodes. As soon as the limit of 4 first neighbors is exceeded at one of the electrodes, that side is designated as “high" coordination. Figure \[Typeofcontact\] summarizes the typical contacts encountered in our simulations.
For some of the contacts that form, there is an indeterminate number of possible configurations, and therefore, to simplify the statistical analysis, we use an X to represent combinations with more than 4 first-neighbor atoms. Likewise, we use a Y in combinations where the number of first-neighbor atoms are in a similar range as or larger than X (See Fig. \[Typeofcontact\]).
Hence, we have simulated contact evolution over continuous loading cycles, and studied the electronic transport during contact formation by means of DFT calculations. After 20 cycles, some of the contact types are reproduced several times, while other contact types appear only once. Table \[MDandDFTtab\] records, for every cycle, the contact type and number of first-neighbor atoms according to the nomenclature outlined in Fig. \[Typeofcontact\]. In the same table, we have corrected the type of contact through visual inspection. Raw data about the type of contact, i.e., in the absence of visual inspection, is collected in table \[tablaraw\], in the appendix. Finally, the double and triple asterisks in table \[MDandDFTtab\] refer to those curious cases in which 2 or 3 atoms close to forming a contact, contribute to the conductance across the junction, but directly via tunneling.
DFT Calculations based on CMD simulations
-----------------------------------------
All the MD frames that have been analyzed from the point of view of the geometry in table \[tablaraw\], have also been analyzed via DFT conductance calculations. The results are shown in table \[MDandDFTtab\] and are, in addition, included in Fig. \[Projections\].
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The structures obtained from CMD simulations, which are limited in their ability to predict realistic structures, require interpretation via electronic transport calculations (if meaningful comparisons with the experimental results are to be made). Following this, upon comparing the calculated conductance and experimental density plots, we can extract information about the type of contact that is formed as well as the configuration of the first-neighbor atoms around it. The electronic transport across all the structures has been calculated by means of ANT.G [@ANTG], which interfaces with GAUSSIAN09. [@GAUSSIAN09] We have grouped the various contacts by type, and their mean conductance values and standard deviations are plotted in Fig. \[Projections\] as dots and vertical bars, respectively.
Discussion
==========
In this work, our aim is to find the origin of the subtle differences between materials, and identify the properties of types of contacts defined by their specific geometry. Elsewhere, we prove that relativistic effects are responsible for the large discrepancy between the jump-to-contact distances of Au and Ag [@PRL], represented by the respective means of their $G_a$ values.
To approach this problem, we use CMD as a tool to visualize the moment of first contact and identify the number and arrangement of the first neighbors. We cannot rely on CMD in the case of tunneling because the potentials only account indirectly for the effects of electrons, and hence, relativity, and, then, only very crudely. Furthermore, it is not possible, experimentally, to know the structure and geometry of the electrodes in the tunneling regime. In CMD, the structure before contact is, at times, preserved in contact, as illustrated in Figs. \[MDfigure\] c) and d). At other times, significant rearrangements occur and the before-contact structures are no longer preserved. For this reason, we confine our analysis to the first neighbors in the contact regime.
[|c|c|c|c|c|c|]{} **Type**& & **Coord** & **& **& **\
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![Projection of experimental $G_b$ values vs number of counts, for Au Ag and Cu. Data points and error bars: the conductance and standard deviation of the various simulated contacts. “Single" is denoted by Si, followed by Mo or Di, depending on whether “monomer" or “dimer". Hi and Lo represent high and low coordination[]{data-label="Projections"}](Figure6.jpg){width="50.00000%"}
The conductance values obtained via DFT from the CMD structures are summarized in table \[Tablesimply\]. The comparison of these results with the experimental distribution of values (Figure \[Projections\]) allows us to interpret our results in terms of the simulated geometry of the contacts. Double and triple contacts are simplified in Fig. \[Projections\], i.e., we don’t distinguish between high or low, or monomer or dimer. Thus, the blue dot and triangle represent mean values, and their error bars, the standard deviations obtained through grouping.
In spite of the reduced statistics (we have performed 20 loading cycles in CMD, on each metal), we observe how the distribution of conductance for the calculated geometries, classified as monomer, dimer and higher order contacts, mostly coincide with the three distributions obtained from the experimental data. We can therefore confirm the assignment by Untiedt *et al.* [@Untiedt_2007], of distributions $D1$ as monomer, $D2$ as dimer, and $D3$ as higher coordination. Our new simulations allow us to further classify the contacts into high- and low-coordination. This classification does not provide much additional interpretation of the experimental results due to the reduced statistics, but it does highlight the determining role of coordination on the conductance of atomic contacts. Our results for Au and Cu display a higher dispersion in conductance for monomers ($D1$) than dimers ($D2$) (Table \[Tablesimply\], Fig. \[Projections\]). The values listed in Table II exemplify how variations on the number of neighbors, for a dimer, have little repercussion on the value of the conductance. For a monomer, on the other hand, the number of neighbors result in large changes in conductance.
We also find that distribution $D3$ likely arises from a combination of double- and triple-contact structures. This leads to a wider distribution in conductance values, as can be seen also from the experimental data. Among these structures, we have identified, through conductance calculations, the triple contact, whose conductance values are in the $2-3$ $G_0$ range. In any event, there may be other structures that have not yet been identified, but that could be discovered by means of the new analysis methods introduced in this work.
Another important difference in the conductance values obtained from the simulations, is the small dispersion in $G_b$ of the monomeric and dimeric Cu structures as compared to Au. As alluded to earlier, in Cu, the dispersion in calculated $G_b$ of the monomer is twice that of the dimer, which is in agreement with the broader $D1$ profile relative to $D2$ in the experimental projections. In Au, the (experimental) $D2$ profile exhibits a very narrow distribution, similar to the narrow dispersion in values obtained for the low-coordinated single dimers from the simulations. This may suggest that these are actually the predominant structures occurring experimentally.
Finally, in the case of Au, we found a particularly good match between experimental and calculated means and standard deviations, particularly for the dimer, while, for the monomer, the calculated means are slightly over estimated. Since the simulations do not accurately capture the jump to contact, contact distances are probably shorter, leading to higher expected conductances.
SUMMARY
=======
By introducing a new statistical approach that permits identifying properties of atomic-sized contacts with greater precision, it has been possible to study, in detail, the process of formation of Au, Ag and Cu nanocontacts. This analysis allow us to identify with higher precision the distribution of values of conductance associated to different geometries, but also to extract information on the distance of contact formation for those geometries. Furthermore, we have used molecular dynamics to simulate the formation of atomic-sized contacts in STM/MCBJ experiments. These simulated contacts were, in turn, analyzed by means of a novel methodology that permits classifying their type and finding the number of first-neighbor atoms in their immediate vicinity. DFT transport calculations on the simulated structures provided a means of comparing theoretical results with the experimental data. We have demonstrated that the type of contact and the geometry of its first neighbors (shape, distance between first-neighbor atoms, and between them and the atomic contact itself) play decisive roles in electronic transport across the simulated contacts. Through a combination of the above three methods, we have found that the electronic transport across the atomic-sized contacts depends crucially on the number of first-neighbor atoms.
ACKNOWLEDGMENTS
===============
This work has been funded from the Spanish MEC through grants FIS2013-47328 and MAT2016-78625. C.S. gratefully acknowledges financial support from SEPE Servicio Público de Empleo Estatal. W.D. acknowledges funding from the National Research Foundation of South Africa through the Innovation Doctoral scholarship programme, Grant UID 102574. W.D. also thanks J. Fernandez-Rossier and J.J. Palacios for fruitful discussions.
APPENDIX
========
The methodology described in section \[SecMD\] and illustrated in Fig. \[figexplacative\] has been applied to the three materials during 20 MD rupture-formation cycles. Table \[tablaraw\] summarizes the obtained results. It records, for Au, Ag and Cu (in blue, red and green, respectively), the time step (in kilosteps, or, more precisely, picoseconds) when contact is established as well as the type of first contact that is formed during every cycle. Data marked with asterisks indicate that the algorithm has detected a contact when it has not really occurred. Through visual inspection we have selected the correct CMD timeframe in which contact actually occurred and also identified the type of contact.
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[^1]: log denotes here the common logarithm (base 10)
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---
abstract: '*Cook’s theorem* is commonly expressed such as any polynomial time-verifiable problem can be reduced to the *SAT* problem. The proof of *Cook’s theorem* consists in constructing a propositional formula $A(w)$ to simulate a computation of *TM*, and such $A(w)$ is claimed to be *CNF* to represent a polynomial time-verifiable problem $w$. In this paper, we investigate $A(w)$ through a very simple example and show that, $A(w)$ has just an appearance of *CNF*, but not a true logical form. This case study suggests that there exists the *begging the question* in *Cook’s theorem*.'
address: 'MIS, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80090 Amiens, France'
author:
- Yu LI
title: |
Case Study of the Proof of *Cook’s theorem*\
- Interpretation of $A(w)$
---
Introduction
============
*Cook’s theorem* [@cook1] is now expressed as any polynomial time-verifiable problem can be reduced to the *SAT (SATisfiability)* problem. The proof of *Cook’s theorem* consists in simulating a computation of *TM (Turing Machine)* by constructing a propositional formula $A(w)$ that is claimed to be *CNF (Conjonctive Normal Form)* to represent the polynomial time-verifiable problem \[1\].
In this paper we investigate whether this $A(w)$ is a true logical form to represent a problem through a very simple example.
Example
========
Polynomial time-verifiable problem and Turing Machine
-------------------------------------------------------
A polynomial time-verifiable problem refers to a problem $w$ for which there exists a *Turing Machine* $M$ to verify a certificat $u$ in polynomial time, that is, check whether $u$ is a solution to $w$.
Let us study a very simple polynomial time-verifiable problem :
Given a propositional formula $w=\neg x$ for which there exists a *Turing Machine* $M$ to verify whether a truth value $u$ of $x$ is a solution to $w$.
The transition function of $M$ can be represented as follows:
------- --- --------------- --- ----- -------
$q_0$ 0 $\rightarrow$ 1 $N$ $q_1$
$q_0$ 1 $\rightarrow$ 0 $N$ $q_1$
$q_1$ 1 $\rightarrow$ 1 $R$ $q_Y$
$q_1$ 0 $\rightarrow$ 0 $R$ $q_N$
------- --- --------------- --- ----- -------
where $N$ means that the tape head does not move, and $R$ means that the tape head moves to right; $q_Y$ refers to the state where $M$ stops and indicates that $u$ is a solution to $w$, and $q_N$ refers to the state where $M$ stops and indicates that $u$ is not a solution to $w$.
Computation of Turing Machine
------------------------------
A computation of $M$ consists of a sequence of configurations: $C(1), C(2), ..., C(T)$, where $T=Q(\mid w \mid)$ and $Q(n)$ is a polynomial function. A configuration $C(t)$ represents the situation of $M$ at time $t$ where $M$ is in a state, with some symbols on its tape, with its head scanning a square, and the next configuration is determined by the transition function of $M$.
Fig.1 and Fig. 2 illustrate two computations of $M$ on inputs : $x=0$ and $x=1$.
\[h\]
![The computation on input $x=0$.[]{data-label="fig1"}](figure1)
\[h\]
![The computation on input $x=1$.[]{data-label="fig2"}](figure2)
Form of $A(w)$
===============
According to the proof of *Cook’s theorem* [@cook1][@garey], the formula $A(w)$ is built by simulating a computation of $M$, such as $A(w)=B \land C \land D \land E \land F \land G \land H \land I $. $A(w)$ is claimed to represent a problem $w$.
We construct $A(w)$ for the above example.
Basic elements
--------------
The machine $M$ possesses:
- 4 states : $\{q_0, q_1, q_2=q_Y, q_3=q_N \}$, where $q_0$ is the initial state, and $q_2$, $q_3$ are two final states.
- 3 symbols : $\{ \sigma_1=b, \sigma_2=0, \sigma_3=1 \}$, where $\sigma_1$ is the blank symbol.
- 2 square numbers : $\{s=1, s=2\}$.
- 4 rules.
- $n$ is the input size, $n=2$; $p(n)$ is a polynomial function of $n$, and $p(2)=3$.
- 3 times ($t=1, t=2, t=3$) et 2 steps to verify a certificat $u$ of $w$, where $t=1$ corresponds to the time for the initial state of the machine.
Proposition symbols
-------------------
Three types of proposition symbols to represent a configuration of $M$ :
- $P_{s,t}^i$ for $1\leq i \leq 3$, $1\leq s \leq 2$, $1\leq t \leq 3$. $P_{s,t}^i$ is true iff at step $t$ the square number $s$ contains the symbol $\sigma_i$.
- $Q_{t}^i$ for $1 \leq i \leq 4$, $1\leq t \leq 3 $. $Q_{t}^i$ is true iff at step t the machine is in state $q_i$.
- $S_{s,t}$ for $1\leq s \leq 2$, $1\leq t \leq 3$ is true iff at step $t$ the tape head scans square number $s$.
Propositions
------------
1\. $E={E_1 \land E_2 \land E_3}$, where $E_t$ represents the truth values of $P_{s,t}^i$, $Q_{t}^i$ and $S_{s,t}$ at time $t$:
- $E_1 = Q_1^0 \land S_{1,1} \land P_{1,1}^{2} \land P_{2,1}^{1}$ ($x = 0 (\sigma_{2}$)); $E_1 = Q_1^0 \land S_{1,1} \land P_{1,1}^{3} \land P_{2,1}^{1}$ ($x = 1 (\sigma_{3}$))
- $E_2$ and $E_2$ are determined by the transition function of $M$
2\. $B={B_1 \land B_2 \land B_3}$, where $B_t$ asserts that at time $t$ one and only one square is scanned :
- $B_1 = (S_{1,1} \lor S_{2,1}) \land ( \neg S_{1,1} \lor \neg S_{2,1})$
- $B_2 = (S_{1,2} \lor S_{2,2}) \land ( \neg S_{1,2} \lor \neg S_{2,2})$
- $B_3 = (S_{1,3} \lor S_{2,3}) \land ( \neg S_{1,3} \lor \neg S_{2,3})$
3\. $C={C_1 \land C_2 \land C_3}$, where $C_t$ asserts that at time $t$ there is one and only one symbol at each square. $C_t$ is the conjunction of all the $C_{i,t}$.
$C_1 = C_{1,1} \land C_{2,1} $:
- $C_{1,1} = ( P_{1,1}^1 \lor P_{1,1}^2 \lor P_{1,1}^3) \land ( \neg P_{1,1}^1 \lor \neg P_{1,1}^2) \land ( \neg P_{1,1}^1 \lor \neg P_{1,1}^3) \land ( \neg P_{1,1}^2 \lor \neg P_{1,1}^3)$
- $C_{2,1} = ( P_{2,1}^1 \lor P_{2,1}^2 \lor P_{2,1}^3) \land ( \neg P_{2,1}^1 \lor \neg P_{2,1}^2) \land ( \neg P_{2,1}^1 \lor \neg P_{2,1}^3) \land ( \neg P_{2,1}^2 \lor \neg P_{2,1}^3)$
$C_2 = C_{1,2} \land C_{2,2} $:
- $C_{1,2} = ( P_{1,2}^1 \lor P_{1,2}^2 \lor P_{1,2}^3) \land ( \neg P_{1,2}^1 \lor \neg P_{1,2}^2) \land ( \neg P_{1,2}^1 \lor \neg P_{1,2}^3) \land ( \neg P_{1,2}^2 \lor \neg P_{1,2}^3)$
- $C_{2,2} = ( P_{2,2}^1 \lor P_{2,2}^2 \lor P_{2,2}^3) \land ( \neg P_{2,2}^1 \lor \neg P_{2,2}^2) \land ( \neg P_{2,2}^1 \lor \neg P_{2,2}^3) \land ( \neg P_{2,2}^2 \lor \neg P_{2,2}^3)$
$C_3 = C_{1,3} \land C_{2,3} $:
- $C_{1,3} = ( P_{1,3}^1 \lor P_{1,3}^2 \lor P_{1,3}^3) \land ( \neg P_{1,3}^1 \lor \neg P_{1,3}^2) \land ( \neg P_{1,3}^1 \lor \neg P_{1,3}^3) \land ( \neg P_{1,3}^2 \lor \neg P_{1,3}^3)$
- $C_{2,3} = ( P_{2,3}^1 \lor P_{2,3}^2 \lor P_{2,3}^3) \land ( \neg P_{2,3}^1 \lor \neg P_{2,3}^2) \land ( \neg P_{2,3}^1 \lor \neg P_{2,3}^3) \land ( \neg P_{2,3}^2 \lor \neg P_{2,3}^3)$
4\. $D={D_1 \land D_2 \land D_3}$, where $D_t$ asserts that at time $t$ the machine is in one and only one state.
- $D_1 = (Q_{1}^0 \lor Q_{1}^1 \lor Q_{1}^2 \lor Q_{1}^3) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^1) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^2) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^3) \land ( \neg Q_{1}^1 \lor \neg Q_{1}^2) \land ( \neg Q_{1}^1 \lor \neg Q_{1}^3) \land ( \neg Q_{1}^2 \lor \neg Q_{1}^3)$
- $D_2 = (Q_{2}^0 \lor Q_{2}^1 \lor Q_{2}^2 \lor Q_{2}^3) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^1) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^2) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^3) \land ( \neg Q_{2}^1 \lor \neg Q_{2}^2) \land ( \neg Q_{2}^1 \lor \neg Q_{2}^3) \land ( \neg Q_{2}^2 \lor \neg Q_{2}^3)$
- $D_3 = (Q_{3}^0 \lor Q_{3}^1 \lor Q_{3}^2 \lor Q_{3}^3) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^1) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^2) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^3) \land ( \neg Q_{3}^1 \lor \neg Q_{3}^2) \land ( \neg Q_{3}^1 \lor \neg Q_{3}^3) \land ( \neg Q_{3}^2 \lor \neg Q_{3}^3)$
5\. $F$, $G$, and $H$ assert that for each time $t$ the values of the $P_{s,t}^i$, $Q_{t}^i$ and $S_{s,t}$ are updated properly.
$F={F_1 \land F_2}$, where $F_t$ is the conjunction over all $i$ and $j$ of $F_{i,j}^t$, where $F_{i,j}^t$ asserts that at time $t$ the machine is in state $q_i$ scanning symbol $\sigma_j$, then at time $t+1$ $\sigma_j$ is changed into $\sigma_l$, where $\sigma_l$ is the symbol given by the transition function for $M$.
$F_1=F_{0,2}^1 \land F_{0,3}^1$ :
- $F_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor P_{1,2}^3)$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $F_{0,3}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^3 \lor P_{1,2}^2)$, with the rule $(q_0, 1 \rightarrow 0, N, q_1)$
$F_2=F_{1,2}^2 \land F_{1,3}^2$ :
- $F_{1,2}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,2}^2 \lor P_{1,2}^2)$, with the rule $(q_1, 0 \rightarrow 0, R, q_N)$
- $F_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,2}^3 \lor P_{1,3}^3)$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
$G={G_1 \land G_2}$, where $G_t$ is the conjunction over all $i$ and $j$ of $G_{i,j}^t$, where $G_{i,j}^t$ asserts that at time $t$ the machine is in state $q_i$ scanning symbol $\sigma_j$, then at time $t+1$ the machine is in state $q_k$, where $q_k$ is the state given by the transition function for $M$.
$G_1=G_{0,2}^1 \land G_{0,3}^1$ :
- $G_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor Q_{2}^1)$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $G_{0,3}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^3 \lor Q_{2}^1)$, with the rule $(q_0, 1 \rightarrow 0, N, q_1)$
$G_2=G_{1,2}^2 \land G_{1,3}^2$ :
- $G_{1,2}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,2}^2 \lor Q_{3}^3)$, with the rule $(q_1, 0 \rightarrow 0, R, q_N)$
- $G_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,3}^3 \lor Q_{3}^2)$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
$H={H_1 \land H_2}$, where $H_t$ is the conjunction over all $i$ and $j$ of $G_{i,j}^t$, where $H_{i,j}^t$ asserts that at time $t$ the machine is in state $q_i$ scanning symbol $\sigma_j$, then at time $t+1$ the tape head moves according to the transition function for $M$.
$H_1=H_{0,2}^1 \land H_{0,3}^1$ :
- $H_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor S_{1,2})$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $H_{0,3}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^3 \lor S_{1,2})$, with the rule $(q_0, 1 \rightarrow 0, N, q_1)$
$H_2=H_{1,2}^2 \land H_{1,3}^2$ :
- $H_{1,2}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,2}^2 \lor S_{2,3})$, with the rule $(q_1, 0 \rightarrow 0, R, q_N)$
- $H_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,3}^3 \lor S_{2,3})$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
6\. $I = (Q_{3}^2 \lor Q_{3}^3) \land (Q_{3}^2 \lor \neg Q_{3}^3) \land (\neg Q_{3}^2 \lor Q_{3}^3)$, asserts that the machine reaches the state $q_y$ or $q_N$ at time 3.\
Finally, $A(w)=B \land C \land D \land E \land F \land G \land H \land I $.
Conjunctive form of $A(w)$
===========================
We develop $A(w)$ as a computation of $M$ for $x=0$ as input (see Fig. 1) in order to clarify the real sense of $A(w)$.
Let us define the configuration and the transition of configurations of $M$ :
$C(t)$ : the truth values of $P_{s,t}^i$, $Q_{t}^i$, $S_{s,t}$ and their constraints.
$C(t) \rightarrow C(t+1)$ : $C(t)$ is changed to $C(t+1)$ according to the transition function of $M$.\
1. At $t=1$, $C(1) = E_1 \land B_1 \land C_1 \land D_1$ :
\[h\]
\[fig1.1\]
- $E_1 = Q_1^0 \land S_{1,1} \land P_{1,1}^{2} \land P_{2,1}^{1}$, representing the initial configuration where $M$ is in $q_0$, the tape head scans the square of number 1, and a string $0b$ is on the tape.
- $B_1 = (S_{1,1} \lor S_{2,1}) \land ( \neg S_{1,1} \lor \neg S_{2,1}) $.
- $C_1 = C_{1,1} \land C_{2,1} $:
- $C_{1,1} = ( P_{1,1}^1 \lor P_{1,1}^2 \lor P_{1,1}^3) \land ( \neg P_{1,1}^1 \lor \neg P_{1,1}^2) \land ( \neg P_{1,1}^1 \lor \neg P_{1,1}^3) \land ( \neg P_{1,1}^2 \lor \neg P_{1,1}^3) $
- $C_{2,1} = ( P_{2,1}^1 \lor P_{2,1}^2 \lor P_{2,1}^3) \land ( \neg P_{2,1}^1 \lor \neg P_{2,1}^2) \land ( \neg P_{2,1}^1 \lor \neg P_{2,1}^3) \land ( \neg P_{2,1}^2 \lor \neg P_{2,1}^3) $
- $D_1 = (Q_{1}^0 \lor Q_{1}^1 \lor Q_{1}^2 \lor Q_{1}^3) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^1) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^2) \land ( \neg Q_{1}^0 \lor \neg Q_{1}^3) \land ( \neg Q_{1}^1 \lor \neg Q_{1}^2) \land ( \neg Q_{1}^1 \lor \neg Q_{1}^3) \land ( \neg Q_{1}^2 \lor \neg Q_{1}^3)$
2\. At $t=2$, $C(2) = E_2 \land B_2 \land C_2 \land D_2$ is obtained from $C(1) \land (C(1) \rightarrow C(2))$.
\[h\]
\[fig1.1\]
$C(1) \rightarrow C(2)$ is represented by $F$, $G$ and $H$ at $t=1$ :
- $F_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor P_{1,2}^3)$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $G_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor Q_{2}^1)$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $H_{0,2}^1 = ( \neg Q_1^0 \lor \neg S_{1,1} \lor \neg P_{1,1}^2 \lor S_{1,2})$, with the rule $(q_0, 0 \rightarrow 1, N, q_1)$
- $E_2 = Q_2^1 \land S_{1,1} \land P_{1,2}^{3} \land P_{2,2}^{1}$, with $Q_2^1=1$, $S_{1,2}=1$, $P_{1,2}^{3}=1$, $P_{2,2}^{1}=1$, and other proposition symbols concerning $t=2$ are assigned with 0.
- $B_2 = (S_{1,2} \lor S_{2,2}) \land ( \neg S_{1,2} \lor \neg S_{2,2})$
- $C_2 = C_{1,2} \land C_{2,2} $:
- $C_{1,2} = ( P_{1,2}^1 \lor P_{1,2}^2 \lor P_{1,2}^3) \land ( \neg P_{1,2}^1 \lor \neg P_{1,2}^2) \land ( \neg P_{1,2}^1 \lor \neg P_{1,2}^3) \land ( \neg P_{1,2}^2 \lor \neg P_{1,2}^3) $
- $C_{2,2} = ( P_{2,2}^1 \lor P_{2,2}^2 \lor P_{2,2}^3) \land ( \neg P_{2,2}^1 \lor \neg P_{2,2}^2) \land ( \neg P_{2,2}^1 \lor \neg P_{2,2}^3) \land ( \neg P_{2,2}^2 \lor \neg P_{2,2}^3) $
- $D_2 = (Q_{2}^0 \lor Q_{2}^1 \lor Q_{2}^2 \lor Q_{2}^3) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^1) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^2) \land ( \neg Q_{2}^0 \lor \neg Q_{2}^3) \land ( \neg Q_{2}^1 \lor \neg Q_{2}^2) \land ( \neg Q_{2}^1 \lor \neg Q_{2}^3) \land ( \neg Q_{2}^2 \lor \neg Q_{2}^3)$
3\. At $t=3$, $C(3) = E_3 \land B_3 \land C_3 \land D_3$ is obtained from $C(2) \land (C(2) \rightarrow C(3))$.
\[h\]
\[fig1.1\]
$C(2) \rightarrow C(3)$ is represented by $F$, $G$ and $H$ at $t=2$ :
$F_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,2}^3 \lor P_{1,3}^3)$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
$G_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,3}^3 \lor Q_{3}^2)$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
$H_{1,3}^2 = ( \neg Q_2^1 \lor \neg S_{1,2} \lor \neg P_{1,3}^3 \lor S_{2,3})$, with the rule $(q_1, 1 \rightarrow 1, R, q_Y)$
- $E_3 = Q_3^2 \land S_{2,3} \land P_{1,3}^{3} \land P_{2,3}^{1}$, with $Q_3^2=1$, $S_{2,3}=1$, $P_{1,3}^{3}=1$, $P_{2,3}^{1}=1$ , and other proposition symbols concerning $t=3$ are assigned with 0.
- $B_3 = (S_{1,3} \lor S_{2,3}) \land ( \neg S_{1,3} \lor \neg S_{2,3})$
- $C_3 = C_{1,3} \land C_{2,3} $:
- $C_{1,3} = ( P_{1,3}^1 \lor P_{1,3}^2 \lor P_{1,3}^3) \land ( \neg P_{1,3}^1 \lor \neg P_{1,3}^2) \land ( \neg P_{1,3}^1 \lor \neg P_{1,3}^3) \land ( \neg P_{1,3}^2 \lor \neg P_{1,3}^3)$
- $C_{2,3} = ( P_{2,3}^1 \lor P_{2,3}^2 \lor P_{2,3}^3) \land ( \neg P_{2,3}^1 \lor \neg P_{2,3}^2) \land ( \neg P_{2,3}^1 \lor \neg P_{2,3}^3) \land ( \neg P_{2,3}^2 \lor \neg P_{2,3}^3)$
- $D_3 = (Q_{3}^0 \lor Q_{3}^1 \lor Q_{3}^2 \lor Q_{3}^3) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^1) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^2) \land ( \neg Q_{3}^0 \lor \neg Q_{3}^3) \land ( \neg Q_{3}^1 \lor \neg Q_{3}^2) \land ( \neg Q_{3}^1 \lor \neg Q_{3}^3) \land ( \neg Q_{3}^2 \lor \neg Q_{3}^3)$\
Therefore, the computation of $M$ for $x=0$ as input can be represented as :
$
C(1) \land (C(1) \rightarrow C(2)) \land (C(2) \rightarrow C(3)) \\
= C(1) \land C(2) \land C(3) \\
= (E_1 \land B_1 \land C_1 \land D_1) \land (E_2 \land B_2 \land C_2 \land D_2) \land (E_3 \land B_3 \land C_3 \land D_3)\\
= E \land B \land C \land D \\
= A(w)
$
It can be seen that $A(w)$ is just the conjonction of all configurations of $M$ to simulate a concret computation of $M$ for verifying a certificat $u$ of $w$. Given an input $u$ ($x=0$ or $x=1$ in this example), whether $M$ accepts it or not, $A(w)$ is always true. Obviously, $A(w)$ has just an appearance of conjunctive form, but not a true logical form.
Conclusion
==========
In fact, a true *CNF* formula is implied in the transition function of $M$ corresponding to $F$, $G$, $H$ as well as $ C(t) \rightarrow C(t+1)$, however the transition function of $M$ is based on the expressible logical structure of a problem.
Therefore, it is not that any polynomial time-verifiable problem can be reduced to the SAT problem, but any polynomial time-verifiable problem itself asserts that such problem is representable by a CNF formula. In other words, there exists the *begging the question* in *Cook’s theorem*.
Acknowledgements {#acknowledgements .unnumbered}
================
Thanks to Mr Chumin LI for his suggestion to use this simple example to study $A(w)$.
[00]{}\[ref:ref\]
Stephen Cook, The complexity of theorem proving procedures. Proceedings of the Third Annual ACM Symposium on Theory of Computing. p151-158 (1971)
Garey Michael R., David S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and company (1979)
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---
abstract: 'We present a Ginzburg-Landau theory of micro phase separation in a bidisperse chiral membrane consisting of rods of opposite handendness. This model system undergoes a phase transition from an equilibrium state where the two components are completely phase separated to a microphase separated state composed of domains of a finite size comparable to the twist penetration depth. Characterizing the phenomenology using linear stability analysis and numerical studies, we trace the origin of the discontinuous change in domain size that occurs during this to a competition between the cost of creating an interface and the gain in twist energy for small domains in which the twist penetrates deep into the center of the domain.'
author:
- Raunak Sakhardande
- Stefan Stanojeviea
- Arvind Baskaran
- Aparna Baskaran
- 'Michael F. Hagan'
- Bulbul Chakraborty
title: Theory of microphase separation in bidisperse chiral membranes
---
#### Introduction:
When two immiscible fluids are mixed, they typically undergo bulk phase separation. Applications ranging from food science, catalysis, and the function of cell membranes require the arrest of this phase separation to form microstructures. A common pathway to accomplish this is the introduction of a third component, such as a surfactant, that stabilizes interfaces between the two fluids [@safran2003statistical]. Here, we theoretically demonstrate a novel mechanism for microphase separation in fluid membranes that is mediated by the chirality of the constituent entities themselves, and hence does not require the introduction of a third component. In addition to identifying a design principle to engineer nano structured materials, this work could shed light on the role of chirality in compositional fluctuations and raft formation in biomembranes [@hyman2012beyond; @lingwood2010lipid; @weis1984two; @dietrich2001lipid; @simons2010revitalizing; @veatch2003separation].
Our theory is motivated by a recently developed colloidal-scale model system of fluid membranes, composed of fd-virus particles [@gibaud2012reconfigurable; @zakhary2013geometrical; @zakhary2014imprintable; @barry2009model; @sharma2014hierarchical; @barry2010entropy]. The system contains two species of virus particles that have opposite chirality and different lengths (Fig \[fig:phase\]). In the presence of a depletant, they self-assemble into a monolayer membrane that is one rod length thick. The competition between depletant entropy, mixing entropy of the two species, and molecular packing forces leads to a rich phase behavior within a membrane, including bulk phase separation of the two species, microdomain formation, and homogeneous mixing. In particular, the experiments find that in the regime where a single species forms a macroscopic membrane, limited only by the amount of material, a mixture of two species leads to the formation of circular monodisperse microdomains (rafts) of one species in a background of the other.
In this work, to understand the mechanisms controlling this raft formation, we develop a continuum Ginzburg-Landau theory that captures the physics of chirality and compositional fluctuation in a 2D binary mixture of rods with opposing chiralities. The primary physics that we incorporate into the theory is a coupling between the twist of the director field and the compositional fluctuations [@selinger1993chiral]. By using linear stability analysis and numerical solutions of the time-dependent Ginzburg-Landau equations, we show that the tendency of the molecules to twist arrests the phase separation of the two species, and stabilizes a droplet phase whose phenomenology closely mimics that observed in experiments. In particular, the theory shows a discontinuous jump in the droplet radius as the system transitions from a microphase separated state to bulk separation, a phenomenon observed in the experiments as well. In contrast, previously studied mechanisms of microphase separation lead to a droplet size that continuously diverges as the system approaches bulk phase separation [@elias1997macro; @bates2008block; @seul1995domain; @janssen2007aperiodic].
#### Model:
The Ginzburg-Landau (GL) model involves two fields: a director field $\hat{\mathbf{n}}\left( \mathbf{r}\right) $ that characterizes the orientation of the rods with respect to the membrane normal and a scalar field $\psi \left( \mathbf{r}\right) $, which characterizes the local composition of the membrane in terms of the two species. We choose a coordinate system in which the layer normal of the membrane lies along the $z$ axis and normalize the order parameter $\psi $ such that $\psi =\pm 1
$ correspond to the homogeneous one-component phases. The GL functional is taken to be of the form: $$\begin{gathered}
F = \int d^{2}\mathbf{r}\left[ \frac{1}{2}K_{1}( \nabla \cdot \mathbf{
\hat{n}}) ^{2}+\frac{1}{2}K_{2}\left( \mathbf{\hat{n}}\cdot \nabla
\times \mathbf{\hat{n}}-q\left( \psi \right) \right) ^{2} \right.\\ +\left . \frac{1}{2}
K_{3}\left( \mathbf{\hat{n}}\times \nabla \times \mathbf{\hat{n}}\right)
^{2} +\frac{C}{2}\sin ^{2}\theta-\frac{\psi ^{2}}{2}+\frac{\psi ^{4}}{4}+
\frac{\lambda_{\psi} }{2}\left( \nabla \psi \right) ^{2} \right]\end{gathered}$$
The physics incorporated in the GL functional can be summarized as follows : i) The first three terms arise from the Frank elasticity associated with director distortion, with $K_{1}$, $%
K_{2} $ and $K_{3}$ being the elastic constants associated with splay, twist and bend respectively [@DegenneBook]. The twist term involves a pitch $q(\psi \left( \mathbf{r}\right))$ that encodes the chirality and hence the associated tendency of the rods to develop a spontaneous non-zero twist. In a mixture of left and right handed rods, $q$ is naturally a function of the composition, which introduces a coupling between $\hat{\mathbf{n}}\left( \mathbf{r}\right) $ and $\psi \left( \mathbf{r}\right) $. ii) The term $\frac{C}{2}\sin ^{2}\theta $ encodes the fact that the rods in the membrane tend to align with the layer normal [@DegenneBook], and gives rise to the standard mechanism of twist expulsion seen in Smectic C systems. When $\psi =\pm 1$, the terms discussed in (i) and (ii) reduce to the theoretical description used successfully to describe single component chiral membranes in earlier works [@pelcovits2009twist; @kaplan2010theory; @kaplan2014colloidal]. iii) The compositional fluctuations encoded in the field $\psi $ are described by a standard $\psi ^{4}$ theory [*below*]{} the critical point that leads to bulk phase separation, with an energetic cost to forming interfaces controlled by the parameter $\lambda_{\psi} $. Thus, the difference in the length of the rods that leads to phase separation in the experimental system is represented as an effective interaction, and our 2D model does not include information about the spatial variation of the membrane in the third dimension.
In the following, we work in the single elastic constant approximation of the Frank elasticity: $K_{1}=K_{2}=K_{3}\equiv K$. We model the variation of $q$ with composition through a minimal linear coupling, $q(\psi)=q_{0}+a \psi$, which defines the coupling parameter $a$. We nondimensionalize the GL functional using the twist penetration depth $\lt\equiv \sqrt{\frac{K}{C}}$ as the characteristic length scale. Defining dimensionless parameters: $q_{0}^{\prime
}=\lt q_{0}$, $a^{\prime }=\lt a\left(
1-Ca^{2}\right) ^{1/2},$ $\lambda_{\psi} ^{\prime }=\frac{\lambda_{\psi} /\lambda _{t}^{2}%
}{\left( 1-Ca^{2}\right) }$ ,$\psi ^{\prime }=\frac{\psi }{\left(
1-Ca^{2}\right) ^{1/2}}$ and $C^{\prime }=\frac{C}{\left( 1-Ca^{2}\right)
^{2}}$ the GL functional becomes:$$\begin{aligned}
F&=\int d^{2}\mathbf{r}^{\prime }\left[f_\text{LC}+f_{\psi}+f_\text{Cross}\right] \nonumber \\
f_\text{LC}&=\frac{C^{\prime }}{2}\left[ \left( \nabla ^{\prime
}\cdot \mathbf{\hat{n}}\right) ^{2}+\left( q_{0}^{\prime 2}-q_{0}^{\prime }
\mathbf{\hat{n}}\cdot \nabla ^{\prime }\times \mathbf{\hat{n}}\right) \right] \nonumber \\
f_{\psi}&=\left[ -\frac{\psi ^{\prime 2}}{2}+\frac{
\psi ^{\prime 4}}{4}+\frac{\lambda_{\psi} ^{\prime }}{2}\left( \nabla ^{\prime
}\psi ^{\prime }\right) ^{2} \right] \nonumber \\
f_\text{Cross}&=\left[ -C^{\prime }a^{\prime }\psi ^{\prime }\left(
\mathbf{\hat{n}}\cdot \nabla ^{\prime }\times \mathbf{\hat{n}}\right)
+C^{\prime }a^{\prime }q_{0}^{\prime }\psi ^{\prime }\right]
\label{eq:GLequation}\end{aligned}$$ This nondimensionalized GL functional is used in all of our subsequent analysis and the $^{\prime }$’s are dropped for compactness of notation.
We model the dynamics by the time-dependent GL equations with a conserved composition field $\psi $: $\partial _{t}\psi =\nabla ^{2}\frac{\delta F}{\delta
\psi }$ (Model B dynamics), and a non-conserved director field $\partial _{t}
\mathbf{\hat{n}}=-\left( \mathbf{I}-\mathbf{\hat{n}\hat{n}}\right) \cdot
\frac{\delta F}{\delta \mathbf{\hat{n}}}$ (Model A dynamics)[@Hohenberg-Halperin]. The $\mathbf{\hat{n}}$ dynamics accounts explicitly for the fact that it is a unit vector. The time constants for the relaxation dynamics of $\psi$ and $\mathbf{\hat{n}}$ have been chosen to be same and set equal to 1. The resulting equations are : $$\partial _{t}\psi =\nabla ^{2}\left( -\psi +\psi ^{3}-\lambda_{\psi} \nabla
^{2}\psi -Ca\left( \mathbf{\hat{n}}\cdot \nabla \times \mathbf{\hat{n}}%
\right) +Caq_{0}\right) \label{dyn1}$$$$\begin{gathered}
\partial _{t} \mathbf{\hat{n}}=-\left( \mathbf{I}-\mathbf{\hat{n}\hat{n}}\right) \cdot\left(-C\nabla^2\mathbf{\hat{n}}-2Cq\nabla \times \mathbf{\hat{n}}+C\mathbf{\hat{n}} \times \nabla q \right. \\ \left.
+C(n_{x}\hat{x}+n_{y}\hat{y}) \right) \label{dyn2}\end{gathered}$$
![a) Schematic of the experimental system. b) The primary results of this work summarized in a phase diagram as a function of the smectic alignment parameter $C$ and the twist-composition coupling parameter $a$. The lines indicate the phase boundary between the microphase separated and bulk phase separated states as obtained from linear stability analysis (black/solid) and numerical integration of Eqs. \[dyn1\]-\[dyn2\] (green/dashed). The snapshots show configurations at steady state obtained from numerics at the indicated parameter values ([ o]{}). c) Illustration of the evolution to steady state for two parameter sets. The results shown here and in the rest of the paper are for a 60-40 mixture with $\lambda_\psi=0.1$, and $q_{0}=0.1$[]{data-label="fig:phase"}](phasediagram15.png){width="50.00000%"}
#### Linear Stability Analysis:
Eqs.(\[dyn1\]-\[dyn2\]) admit homogenous steady states of the form $\psi =\pm 1$ and $\mathbf{\hat{n}}=%
\mathbf{\hat{z}}$. As a first step in understanding the dynamics of phase separation, we analyze the instability of the homogeneous state to small fluctuations of the form $\psi=1+\delta \psi$ and $\mathbf{\hat{n}}=\mathbf{\hat{z}}+\mathbf{\delta n}$. We introduce Fourier transformed variables $\widetilde{X}\left( \mathbf{k},t\right)
=\int d^{2}\mathbf{r}e^{i\mathbf{k}\cdot \mathbf{r}}X\left( \mathbf{r}%
,t\right) $. Without loss of generality, we choose a coordinate system in the plane of the membrane such that the $x$ axis lies along the spatial gradient direction. We find that the longitudinal fluctuations in the director $\delta \widetilde{n}_{x}$ decouple from the other variables ([@supinfo]) and we obtain the linearized equations $$\begin{aligned}
\partial _{t}\left(
\begin{array}{c}
\delta \widetilde{{ \psi} } \\
\delta \widetilde{ {n}}_{y}
\end{array}
\right) =
\left(
\begin{array}{ccc}
-2k^{2}-\lambda_{\psi} k^{4} & ik^{3}Ca \\
-ikCa & -C-Ck^2
\end{array}%
\right) \left(
\begin{array}{c}
\delta \widetilde{ \psi } \\
\delta \widetilde{ {n}}_{y}
\end{array}%
\right) \label{LSM}\end{aligned}$$ The homogeneous state is found to be linearly unstable to modes $k$ that satisfy $$\lambda_{\psi} k^6-(Ca^2-2-\lambda_{\psi})k^4+2k^2 < 0 ~.
\label{eq:linear}$$
![(color online) a) The largest eigenvalue $\omega(k)$ of the linear stability matrix in Eq.(\[LSM\]) as a function of the wavevector $k$ for indicated values of the alignment strength $C$, with $a=0.8$ and $\lambda_{\psi}=0.1.$ b) Dependence of the optimal domain size on $Ca^2$ obtained by three different analysis methods: the steady-state mean radius of domains obtained by numerical integration, the radius which minimizes the GL free energy (calculated as described in the text), and wavelength corresponding to the fastest-growing mode calculated by linear stability analysis, for $\lambda_{\psi}=0.1$[]{data-label="fig:dispersion"}](figure2b.png){width="50.00000%"}
We see from Eq. \[eq:linear\] that the $k=0$ mode is always marginally stable, and that the linear instability is controlled only by the combination $Ca^2$ and does not depend individually on the strengths of the smectic alignment and the twist-composition coupling. At a critical value of $Ca^2$ determined by $(Ca^2 -2-\lambda_{\psi})^2 = 8\lambda_{\psi}$, the mode with $k_{\textrm{max}} = (2/\lambda_{\psi})^{1/4}$ becomes unstable[@supinfo]. Fig. \[fig:dispersion\] shows the largest eigenvalue $\omega(k)$ of the linear stability matrix in Eq.(\[LSM\]) for different parameters. For any non-zero value of $\lambda_{\psi}$, the instability thus occurs at a finite $k$, which demonstrates that the instability of the homogeneous phase is to microphase domains of a finite size. The transition from a macroscopically phase separated state (infinite domain size, $k=0$) to a microphase separated state should thus be accompanied by a discontinuity in the domain size [@supinfo].
Numerical analysis of Eqs.(\[dyn1\]-\[dyn2\]) verifies this discontinuous change in the domain size. We solve Eqs.(\[dyn1\]-\[dyn2\]) numerically by using an implicit convex splitting scheme to evolve the equation for $\psi$ and the forward Euler method to evolve the director field [@supinfo]. We initialize the system with random compositional fluctuations around a homogeneous mixture with $\psi=0.2$ and we explore the phase space spanned by $C$ and $a$. For most of the results shown here, we choose $\lambda_{\psi}=0.1$, as the interface width in the experiments is found to be much smaller than the twist penetration length [@sharma2014hierarchical]. Also, we set $q_{0}=0.1$ as the preferred chiral twists of the two species of rods in the experimental system are not equal. The phase diagram obtained from numerics are shown in Fig. \[fig:phase\]. It is evident that linear stability analysis captures all qualitative aspects of the numerically determined phase diagram. The steady state domain sizes obtained from numerics are shown in Fig. \[fig:dispersion\] and clearly demonstrate the discontinuous change accompanying the phase transition. The formation of finite size domains is controlled by a competition between chirality and interfacial tension. A similar competition exists even in a chiral membrane of a single species, where the interfacial tension exists between the membrane edge and the bulk polymer suspension. A theoretical analysis of this system [@pelcovits2009twist] showed a transition between membranes of finite size and unbounded macroscopic membranes. Within such a membrane, the twist is expelled to the edge, decaying over a length $\lt$, and the membrane size grows continuously as the transition is approached. Here we see that introducing a second species with opposite handedness into such a membrane provides a mechanism for the twist to penetrate the interior of the membrane. As shown in Fig. \[fig:phase\], the director twists at the edge of each domain, and then untwists (twists in the opposite direction) into the background. This twist is confined to within approximately $\lt$ of a domain edge. The ability of the interface to accommodate twist is the mechanism that leads to the formation of microdomains in the region of parameter space where each species by itself would form a macroscopic membrane.
To quantitatively unfold this mechanism, and to understand the discontinuous change in domain size that occurs at the transition to bulk phase separation, we examine how the spatial variations in $\psi$ and $\mathbf{\hat{n}}$ influence the free energy Eq. (\[eq:GLequation\]). To this end, we calculate the free energy of a domain of radius $R$ of one species in a background of the other. We do so by assuming profiles for $\psi$ and $\mathbf{\hat{n}}$ that are consistent with the results obtained from numerical integration \[[@supinfo] section 2\]. The optimal domain size is then determined by the value of $R$ at each $C_0$, $a$, $q_0$, $\lambda_\psi$, for which the free energy is minimized (Fig. \[fig:energy\]). The resulting domain sizes are consistent with those obtained from linear stability analysis and numerical integration (Fig. \[fig:dispersion\]).
The origin of discontinuity in domain size is revealed by examining the variations in different contributions to the free energy density ($f_{\textrm{LC}}$, $f_{\textrm{$\psi$}}$ and $f_{\textrm{Cross}}$) as the droplet size changes. Fig. \[fig:energy\] shows these variations for a parameter set in the microphase separation regime. Note that in an extensive system with clear scale separation between bulk and interface, the interfacial contribution to a free energy density decays with increasing domain size, while the bulk contribution remains constant. In contrast, we see that $f_{\textrm{LC}}$ and $f_\textrm{Cross}$ are super-extensive for small domain sizes, only becoming extensive asymptotically. This superextensivity is significant only for domain sizes of the order of the twist penetration length ($R\sim5 \lambda_\textrm{t}$). Thus, finite-sized domains appear only when the increase in $f_{\textrm{LC}}$ and $f_\textrm{Cross}$ with $R$ is sufficient to outcompete $f_\textrm{$\psi$}$ at these small domain sizes. As $Ca^2$ decreases, the super-extensive behavior diminishes, forcing the critical domain size (at which $f_{\textrm{LC}}$ and $f_\textrm{Cross}$ dominate over $f_\textrm{$\psi$}$) to larger $R$. At the threshold value of $Ca^2$, $f_{\textrm{LC}}$ and $f_\textrm{Cross}$ become extensive before dominating over the interfacial tension, and macrophase separation sets in.
The source of the super-extensive growth in $f_{\textrm{LC}}$ and $f_\textrm{Cross}$ can be understood from the dependence of twist profiles on $R$. For large $R$, twist decays exponentially from the domain edge (Fig. 2 in [@supinfo]); thus ensuring scale separation between the bulk and the interface. On the other hand, such a separation does not exist for small domains where the twist penetrates to the center of the domain.
In conclusion, we have presented a theory of microphase separation in membranes, which is driven by chirality of its constituent entities. The underlying mechanism of microphase separation can be traced to the the gain in twist energy in these structures, which can accommodate twist at the boundaries of domains. We have provided quantitative analysis that unfolds the precise factors leading to the appearance of microdomains. We have also shown that the microdomains have a natural length scale determined by the twist penetration depth, and therefore the domain size does not increase continuously as the system transitions to the macrophase separated state. Domains that are much larger than the twist penetration depth fail to gain enough free energy from the twisting at the interface to compensate for the free energy cost of creating an interface where the composition changes. By reducing $\lambda_{\psi}$, this limiting length can be made larger, however the transition is discontinuous for all finite values of $\lambda_{\psi}$. This feature of the microdomains is appealing from the perspective of creating nanostructures since the domain size can be tightly controlled.
#### Acknowledgment:
This work was supported by the Brandeis University NSF MRSEC, DMR-1420382. Computational resources were provided by the NSF through XSEDE computing resources (Stampede) and the Brandeis HPCC which is partially supported by DMR-1420382. We gratefully acknowledge Robert Meyer, Robert Pelcovits, Zvonimir Dogic, and Prerna Sharma for helpful discussions; we also thank Prerna Sharma for providing her experimental data.
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abstract: 'Feedback stabilization of magnetohydrodynamic (MHD) modes is studied in a cylindrical model for a tokamak with resistivity, viscosity and toroidal rotation. The control is based on a linear combination of the normal and tangential components of the magnetic field just inside the resistive wall. The feedback includes complex gain, for both the normal and for the tangential components, and the imaginary part of the feedback for the former is equivalent to plasma rotation. The work includes (1) analysis with a reduced resistive MHD model for a tokamak with finite $\beta$ and with stepfunction current density and pressure profiles, and (2) computations with full compressible visco-resistive MHD and smooth decreasing profiles of current density and pressure. The equilibria are stable for $\beta=0$ and the marginal stability values $\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$ (resistive plasma, resistive wall; resistive plasma, ideal wall; ideal plasma, resistive wall; ideal plasma, ideal wall) are computed for both cases. The main results are: (a) imaginary gain with normal sensors or plasma rotation stabilizes below $\beta_{rp,iw}$ because rotation supresses the diffusion of flux from the plasma out through the wall and, more surprisingly, (b) rotation or imaginary gain with normal sensors destabilizes above $\beta_{rp,iw}$ because it prevents the feedback flux from entering the plasma through the resistive wall to form a virtual wall. The effect of imaginary gain with tangential sensors is more complicated but essentially destabilizes above and below $\beta_{rp,iw}$. A method of using complex gain to optimize in the presence of rotation in the $\beta>\beta_{rp,iw}$ regime is presented.'
author:
- 'D. P. Brennan$^{*}$ and J. M. Finn$^{\dagger}$'
bibliography:
- 'BrennanFinn.bib'
title: Control of resistive wall modes in a cylindrical tokamak with plasma rotation and complex gain
---
$^{*}$Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544; $^{\dagger}$ Applied Mathematics and Plasma Physics, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
Introduction
============
Feedback stabilization of magnetohydrodynamic (MHD) modes with plasma resistivity and a resistive wall in tokamaks has received recent attention particularly because of the need to control disruptions[@bondeson-ward; @betti-freidberg; @finn_rw1; @finn_rw2; @finn_rw3; @finn_rw4; @Bhattacharyya; @Boozer-slow-rotation; @GA-res-wallI; @fitzpatrick-aydemir; @garofalo_2002; @Liu2006]. Studies have also been performed for reversed field pinches (RFPs)[@Bishop; @RichardsonFinnDelzanno; @Sassenberg]. Earlier studies in tokamak geometry[@Finn2004; @bondeson_pop; @Tangential-normalChuGlasser; @bondeson2001; @Chu2004; @mode-control; @Pustovitov2002] investigated sensing either the radial or the poloidal component of the magnetic field, concluding that it is better to sense the poloidal component, and that the latter measurement is of more use inside the wall[@Pustovitov2002; @Finn2004]. Results in Refs. [@Tangential-normalChuGlasser; @Finn2004] suggested that the advantages of tangential sensing are due to the fact that it is less sensitive to sensors that detect sidebands or feedback coils that excite sidebands.
In Ref. [@Finn2006] studies were performed with of *both* the radial and poloidal components (radial and *toroidal* components in the RFP context) but with idealized (single Fourier component) coils. The results showed that this approach has useful advantages over control based on either sensor alone. Whereas feedback based on sensing either field component alone is limited to the marginal stability point for resistive plasma modes with an ideal wall, feedback based on sensing both components can stabilize up to the ideal plasma - ideal wall limit. These results were presented in Ref. [@Finn2006], which used a very simple qualitative model based on reduced resistive MHD[@Strauss] for the plasma dynamics. More recent investigations in full visco-resistive MHD in a cylindrical model for RFPs[@RichardsonFinnDelzanno; @Sassenberg] have shown this ability to stabilize close to the ideal plasma - ideal wall limit, depending on the plasma viscosity and resistivity. In Ref. [@Sassenberg] the work in Ref. [@RichardsonFinnDelzanno] was extended to a model measuring the radial component and *two* tangential components of the magnetic field, again in RFP geometry. This work included the presence of two walls, the (inner) vacuum vessel and a better conducting external copper shell, with sensors between the two walls, as suggested by the configuration of the RFX-mod facility[@RFX-mod]. The results of this study also showed the possibility of stabilizing close to the ideal plasma - ideal wall limit, depending on the plasma viscosity and the placement of the sensors, and that the second tangential component (toroidal in tokamak geometry and poloidal in RFP geometry) is not important. The RFX-mod facility has the capability of sensing both the normal and toroidal components and applying a *pre-specified* linear combination of these[@RFXPiron; @RFXAuthors]. In the theoretical work in Refs. [@RichardsonFinnDelzanno; @Sassenberg], the normal and tangential components were considered independent. The RFP results with $\beta=0$ were parameterized in terms of the critical values of the equilibrium current density at the magnetic axis, i.e. $\lambda_{0}=(j_{||}/B)(r=0)$, namely $\lambda_{rp,rw}<\lambda_{rp,iw}<\lambda_{ip,rw}<\lambda_{ip,iw}$. These four values of $\lambda_{0}$ are, respectively the current limits for resistive plasma, resistive wall; resistive plasma, ideal wall; ideal plasma, resistive wall; and ideal plasma, ideal wall. The inner inequality $\lambda_{rp,iw}<\lambda_{ip,rw}$ was observed to hold[@RichardsonFinnDelzanno; @Sassenberg] for all RFP equilibria investigated. The other inequalities must always hold.
In this paper we investigate linear stability in a finite-$\beta$ cylindrical model with tokamak-like profiles, namely large toroidal aspect ratio $R/a$, large toroidal field $B_{z}\sim(R/a)B_{\theta}$ and decreasing profiles of current density $j_{z0}(r)$ and pressure $p_{0}(r)$. The decreasing $j_{z0}(r)$ profile leads to a monotonically increasing profile of the safety factor $q(r)=rB_{z0}/RB_{\theta0}(r)$ with $q\sim1$. We consider equilibria which are stable for zero pressure and characterize the stability properties without feedback in terms of the four marginal values of $\beta_{0}=2p_{0}(0)/B_{z0}(0)^{2}$, namely $\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$, analogous to the values of $\lambda=j_{||}/B$ at $r=0$ in the RFP studies. (As in the RFP studies, the middle inequality, which does not hold in general, has been observed to hold for all the equilibria we considered.) We again investigate the behavior with feedback proportional to the radial and poloidal magnetic field components, with gain factors $G$ and $K$, respectively. We also include toroidal plasma rotation and complex gain[@bondeson_prl] for both the normal component and the tangential component, i.e. $G$ and $K$. (Complex gain is attained by shifting the phase of the actuator coils relative to the sensor coils.) In Ref. [@FinnChacon1] it was argued that, in cylindrical geometry with a single $k_{z}$, the imaginary part $G_{i}=\text{Im}G$ is equivalent to rotation of the wall, which is in turn equivalent to rigid rotation of the plasma.
An aspect of our studies worth emphasizing is the inclusion of plasma resistivity as well as wall resistivity. This inclusion introduces two important marginal stability parameters, namely $\beta_{rp,rw}$ and $\beta_{rp,iw}$, that are absent in ideal MHD. Also, above the latter limit, modes are unstable but grow on the wall time $\tau_{w}$ and are therefore sensitive to plasma resistivity and react differently to plasma rotation.
As in the RFP control studies, the control is applied at a surface *external* to the resistive wall. This is in spite of the fact that in some current devices actuators are located inside the wall. Our focus on control applied outside the wall is motivated by the obvious potential problems of internal control coils, as well as the results shown here, indicating the possibility of stabilizing well above the resistive plasma-ideal wall threshold.
In Sec. 2 we describe the cylindrical MHD equilibria used in the analytic and the numerical studies. In the former case, the simplified equilibrium has large $B_{z0}$ and stepfunction models for $j_{z0}(r)$ and $p_{0}(r)$. In the latter the equilibrium is specified by smooth functions for $j_{z0}(r)$ and $p_{0}(r)$.
In Sec. 3 we describe the methods used to analyze the stability of the simplified model, as well as the full MHD model used to study the stability of the smooth profile equilibria. In the former we use reduced resistive MHD[@Strauss] in the viscoresistive (VR) regime, with a single resistive wall and control applied at a wall external to the resistive wall. We also formulate the problem with a layer in the resistive-inertial (RI) regime for comparison. The use of reduced MHD with plasma resistivity and stepfunction profiles enables us to obtain analytic results for which the various physical effects in the presence of plasma rotation and feedback with complex gains are transparent. The studies in full MHD enable us to determine how well the results of the simplified model represent those of the full model.
In Sec. 4 we show results using both models. We first present studies of the stability properties, in particular the four values $\beta_{rp,rw},\,\beta_{rp,iw},\,\beta_{ip,rw},\,\beta_{ip,iw}$, without rotation or gain. We then present results with real gains $G=G_{r}$ and $K=K_{r}$, with increasing $\beta_{0}\equiv2p_{0}(0)/B_{z0}(0)^{2}$.
In Sec. 5 we show results including rotation $\Omega$ and complex gain $G_{i}$, with $K_{i}=0$. The main result is that the behavior depends on the value of $\beta_{0}$ relative to $\beta_{rp,iw}$, the resistive plasma - ideal wall threshold. For $\beta_{0}<\beta_{rp,iw}$ plasma rotation $\Omega$ and $G_{i}$ (equivalent to wall rotation $\Omega_{w}$ and therefore equivalent to plasma rotation in the opposite direction) are stabilizing, leading to a larger region of stability in the $(K,G)$ space. This is because rotation of the plasma relative to the wall suppresses the resistive wall mode by preventing the flux from diffusing through the wall. For $\beta_{0}>\beta_{rp,iw}$, rotation relative to the wall is found to be *destabilizing*: in this regime, the resistive plasma mode is unstable even with an ideal wall, and for the feedback to succeed the flux needs to diffuse through the wall in order to form a virtual wall[@Bishop] inside the actual wall[@Bishop; @Finn2006; @RichardsonFinnDelzanno; @Sassenberg]. The stabilizing effect of rotation or $G_{i}$ for $\beta_{0}<\beta_{rp,iw}$ and the destabilizing effect for $\beta_{0}>\beta_{rp,iw}$ is similar to the dependence on the wall time observed in Ref. [@Finn2006]. For finite plasma rotation $\Omega\neq0$ the optimum value of $G_{i}$ when $\beta_{0}>\beta_{rp,iw}$ is that value which makes the equivalent wall rotation $\Omega_{w}$ equal to the plasma rotation $\Omega$, allowing the fastest penetration of the flux from the feedback coils.
In Sec. 6 we study the effects of $K_{i}$. It is also found that in this regime there is no simple equivalence between $K_{i}$ and plasma rotation, although $K_{i}$ affects the modes in a manner which has some similarity to rotation. Increasing $|K_{i}|$ is destabilizing for both $\beta_{0}<\beta_{rp,iw}$ and for $\beta_{0}>\beta_{rp,iw}$, so it is not possible to interpret $K_{i}$ in terms of equivalent wall rotation. There is also an optimal value of $K_{i}$ for $\Omega\neq0,\, G_{i}=0$, both above and below $\beta_{rp,iw}$. For $\beta_{0}>\beta_{rp,iw}$ this behavior is similar to that for $G_{i}$ for reduced MHD, but is more complicated for full MHD. For $\beta_{0}<\beta_{rp,iw}$ rotation is stabilizing and the optimal value of $K_{i}$ can generally expand the stable region.
The change in behavior across $\beta_{0}=\beta_{rp,iw}$ for all values of $\Omega,\,\, G_{i}$ and $K_{i}$ indicates the importance of plasma modeling including plasma resistivity.
In Sec. 7 we summarize and discuss the results presented, particularly the possibility of stabilization well above $\beta_{rp,iw}$ by optimization using complex gain. We also emphasize the fact that the simple analytic modeling predicts qualitatively most of the phenomena found by the more complete full MHD treatment, and that resistive MHD modeling is necessary to obtain these conclusions because the modes are resonant.
Equilibria
==========
The equilibrium for the simplified reduced MHD model is specified in terms of decreasing stepfunction profiles of current density and pressure, i.e.
$$\begin{array}{c}
B_{\theta0}(r)=r\,\,\,\,\text{for}\,\,\,\, r<a_{1}\\
=\frac{a_{1}^{2}}{r}\,\,\,\,\text{for}\,\,\,\, r>a_{1}\\
j_{z0}(r)=2\Theta(a_{1}-r)\\
B_{z0}(r)=B_{0}\,\,\,\,=\,\text{const}\\
p_{0}(r)=p_{0}(0)\Theta(a_{2}-r).
\end{array}\label{eq:Equilibrium}$$
Length scales are relative to $r_{w}$ and time scales to $r_{w}/v_{A}$, with $v_{A}$ based on the nominal equilibrium poloidal field $B_{\theta0}'(0)r_{w}$, so $B_{\theta}$ is normalized to have $B_{\theta0}'(0)=r_{w}=1$. The major radius $R$ satisfies $\epsilon\equiv r_{w}/R\ll1$. For equilbria in reduced MHD, we take $B_{\theta}\sim\epsilon B_{z}$ and $p\sim B_{\theta}^{2}\sim\epsilon^{2}B_{z}^{2}$. It follows that $B_{z}B_{z}'\sim\epsilon^{2}B_{z}^{2}$, so that at the steps at $r=a_{1}$ and $r=a_{2}$ we have $\Delta B_{z0}\sim\epsilon^{2}B_{z0}$. This means that it is consistent to treat $B_{z}$ as uniform and still have force balance in equilibrium. The $q$ profile is given by $$q(r)=q(0)\,\,\,\,\text{for}\,\,\,\, r<a_{1},\,\,\,\,\,\,=q(0)\frac{r^{2}}{a_{1}^{2}}\,\,\,\,\text{for}\,\,\,\, r>a_{1},$$ where $q(0)=B_{0}/R$ and $R$ is the major radius. The modes behave as $e^{im\theta+ikz}$ with $k=-n/R$ and $n=1$. We assume $q(0)<m/n$ but $q(a_{2})>m/n$, so that the four radii $a_{1},\, r_{t},\, a_{2},r_{w}$ satisfy $a_{1}<r_{t}<a_{2}<r_{w}$. Here, $r_{t}$ is the radius of the mode rational surface (tearing layer), which satisfies $q(r_{t})=m/n$, and $r_{w}$ is the radius of the resistive wall. See Fig. 1a. We also have a control surface at $r=r_{c}>r_{w}$. Plasma rotation is represented by a uniform equilibrium toroidal velocity $u_{z0}$.
The equilibrium used for the numerical studies in full MHD is specified by the toroidal current density $j_{z0}(r)$ and the pressure $p_{0}(r).$ The current density used is the ‘flattened model’ of Ref. [@FRS], with pressure $p_{0}(r)$ added having a profile similar to $j_{z0}(r)$. Specifically, we take $$B_{\theta0}(r)=\frac{r}{\left(1+(r/a_{1})^{2\nu}\right)^{1/\nu}}$$ with $\nu=4$, again normalized to have $B_{\theta0}'(0)=1$. Hence we have $$j_{z0}(r)=\frac{2}{\left(1+(r/a_{1})^{2\nu}\right)^{(\nu+1)/\nu}}.\label{eq:jz0-specification}$$ For the pressure we take a similar form with $\nu=6$, $$p_{0}(r)=\frac{p_{0}(0)}{\left(1+(r/a_{2})^{2\nu}\right)^{(\nu+1)/\nu}}.\label{eq:p0-specification}$$ Radial force balance $j_{\theta0}B_{z0}-j_{z0}B_{\theta0}=p_{0}'(r)$ gives the toroidal field by $$\frac{B_{z0}^{2}}{2}=\frac{B_{0}^{2}}{2}+p_{00}-p_{0}(r)-\int_{0}^{r}j_{z0}(r')B_{\theta0}(r')dr'.$$ We use the integration constant $B_{0}=B_{z0}(0)$ to specify $q(0)$, where $q(r)=rB_{z0}(r)/RB_{\theta0}(r)$, i.e. $q(0)=B_{0}/R$. Here, as above, the toroidal aspect ratio is $R/r_{w}$. These equilibrium quantities are shown in Fig. 1b. The equilibrium velocity $u_{z0}$ is again taken to be uniform.
Linear models
=============
In this section we describe the linear models used to compute the stability of the stepfunction and smooth equilibria introduced in the last section. In the first case, we do asymptotic matching with a viscoresistive (VR) or resistive inertial (RI) inner layer model, with outer regions derived from finite $\beta$ reduced ideal MHD without inertia. In the second case, we solve the complete resistive MHD equations with viscosity, compressional effects and parallel dynamics.
Simplified linearized MHD model
-------------------------------
The simplified model for treating resistive MHD modes in a large aspect ratio cylinder model for a tokamak with a resistive wall uses reduced MHD[@Strauss] with plasma resistivity, and with stepfunction profiles as described in Sec. 2. In this model the linear dynamics is described entirely in terms of the perturbed flux function $\tilde{\psi}=\tilde{A}_{z}$, with $\mathbf{\tilde{B}}=\nabla\tilde{\psi}(r,\theta,z)\times\hat{\mathbf{e}}_{z}$ – the toroidal field is not perturbed. The (perpendicular) velocity is given in terms of the perturbed streamfunction by $\tilde{\mathbf{v}}_{\perp}=\nabla\tilde{\phi}\times\hat{\mathbf{e}}_{z}$. The reduced MHD equations are given in the outer region (ideal MHD, zero inertia) by $$0=iF(r)\nabla_{\perp}^{2}\tilde{\psi}-\frac{im}{r}j_{0}'(r)\tilde{\psi}+\frac{2imB_{\theta0}^{2}(r)}{B_{0}^{2}r^{2}}\tilde{p},\label{eq:reduced-1}$$ $$\gamma_{d}\tilde{\psi}=iF(r)\tilde{\phi}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\gamma_{d}\tilde{p}=-\frac{im}{r}p_{0}'(r)\tilde{\phi},\label{eq:reduced-2}$$ where $F(r)=mB_{\theta0}(r)+kB_{0}$ and $\gamma_{d}$ is the Doppler shifted growth rate $\gamma+iku_{z0}=\gamma+i\Omega$. We obtain $$\nabla_{\perp}^{2}\tilde{\psi}=\frac{mj_{z0}'(r)}{rF(r)}\tilde{\psi}+\frac{2m^{2}B_{\theta0}^{2}(r)p_{0}'(r)}{B_{0}^{2}r^{3}F(r)^{2}}\tilde{\psi}\label{eq:OuterRegionEq}$$ $$=-A\delta(r-a_{1})\tilde{\psi}-B\delta(r-a_{2})\tilde{\psi},\label{eq:JumpConditions}$$ where $A=2m/a_{1}F(a_{1})$ and $B=m^{2}\beta_{0}a_{1}^{4}/a_{2}^{5}F(a_{2})^{2}$. Note that $B>0$ in general, and $A>0$ since $F(a_{1})=m-nq(a_{1})=m-nq(0)>0$. Also, $F(a_{2})=(a_{1}^{2}/a_{2}^{2})(m-nq(a_{2}))$ implies $B=m^{2}\beta_{0}/a_{2}(m-nq(a_{2}))^{2}$.
For this stepfunction modeling, the region outside $r=a_{2}$ satisfies $\nabla_{\perp}^{2}\tilde{\psi}=0$, by Eqs. (\[eq:OuterRegionEq\],\[eq:JumpConditions\]). Therefore, this model has the property that if it were modified by introducing a vacuum in the region $r_{p}<r<r_{w}$ with $r_{p}>a_{2}$, the equations would be unchanged. To the degree that $j_{z0}(r)$ and $p_{0}(r)$ in Eqs. (\[eq:jz0-specification\],\[eq:p0-specification\]) are very small near $r=r_{w}$, the same conclusions hold for the numerical full MHD model.
We write the flux $\tilde{\psi}$ as $$\tilde{\psi}(r)=\alpha_{1}\psi_{1}(r)+\alpha_{2}\psi_{2}(r)+\alpha_{3}\psi_{3}(r),\label{eq:CulhamExpansion}$$ where the basis functions $\psi_{1},\,\psi_{2},\,\psi_{3}$ are described and computed in the Appendix (see Fig. 10) for this stepfunction equilibrium. They have $\psi_{1}(0)=0$, $\psi_{1}(r_{t})=1$, $\psi_{1}(r_{w})=0$; $\psi_{2}(r_{t})=0$, $\psi_{2}(r_{w})=1$, $\psi_{2}(r_{c})=0$; and $\psi_{3}(r_{w})=0$, $\psi_{3}(r_{c})=1$. There are three conditions for the three unknowns $\alpha_{1},\,\alpha_{2},\,\alpha_{3}$. The first is the constant-$\psi$ VR tearing mode jump condition at the tearing layer at $r=r_{t}$. The second is the resistive thin-wall jump condition at $r=r_{w}$, and the third is the prescribed feedback control condition at the control surface $r=r_{c}$: $$\gamma_{d}\tau_{t}\tilde{\psi}(r_{t})=\left[\tilde{\psi}'\right]_{r_{t}},\label{eq:RPJump}$$ $$\gamma\tau_{w}\tilde{\psi}(r_{w})=\left[\tilde{\psi}'\right]_{r_{w}},\label{eq:RWJump}$$ $$\tilde{\psi}(r_{c})=-G\tilde{\psi}(r_{w})+K\tilde{\psi}'(r_{w}-).\label{eq:GainEquation}$$ Here again $\gamma_{d}$ is the Doppler shifted frequency $\gamma+iku_{z0}=\gamma+i\Omega$; the plasma velocity enters in only Eq. (\[eq:RPJump\]) and we assume that the velocity shear across the tearing layer is negligible. Also, $[\cdot]_{r_{t},r_{w}}$ represents the jump in radial derivatives at $r=r_{t}$ and $r=r_{w}$, respectively. Note that the gain $G$ multiplies the radial (normal) component $\tilde{B}_{r}=im\tilde{\psi}/r$ and $K$ multiplies the poloidal (tangential) normal component $\tilde{B}_{\theta}=-\tilde{\psi}'(r)$. For sensing of the normal component (for $G$ real), the measured field consists of the field due to the plasma perturbation as well as that due to the control coils. This point, which has been discussed as a reason for preferring tangential sensing[@okabayashi], has been discussed in Ref. [@FinnChacon1], where it was shown that, in cylindrical geometry with idealized coils (i.e. with a single poloidal Fourier component), the field due to the plasma alone has a simple proportionality to the total normal field. Although this issue is avoided for tangential sensing with $K$ real, it appears that for $\pi/2$ phase shift ($K$ imaginary) the same considerations apply.
The results ofRef. [@Sassenberg] show that, even in a model which contains a second tangential component (here $\tilde{B}_{z}$), this component is not very important and is zero if the measurements are made in a vacuum region between the plasma and the wall. The results in Ref. [@Sassenberg] also show that in the presence of an inner wall with a much shorter time constant, this inner wall can be treated as part of the vacuum for small $|\gamma|$, and that such a simple model with constant-$\psi$ matching and the thin-wall treatment is qualitatively accurate.
We obtain $$\gamma_{d}\tau_{t}\alpha_{1}=\Delta_{1}\alpha_{1}+l_{21}\alpha_{2},\label{eq:Basic-1}$$ $$\gamma\tau_{w}\alpha_{2}=l_{12}\alpha_{1}+\Delta_{2}\alpha_{2}+l_{32}\alpha_{3},\label{eq:Basic-2}$$ and $$\alpha_{3}=-G\alpha_{2}+K\left(-l_{12}\alpha_{1}+l_{22}^{(-)}\alpha_{2}\right).\label{eq:Basic-3}$$
The quantity $\Delta_{1}$ is the tearing mode matching condition at $r=r_{t}$ with an ideal wall at $r=r_{w}$, and the entry $\gamma_{d}\tau_{t}$ is based on the VR dispersion relation with the constant-$\psi$ approximation. (For the resistive-inertial or RI regime, $\gamma_{d}\tau_{t}$ is replaced by $(\gamma_{d}\tau_{t}')^{5/4}$, but in the presence of both viscosity and inertia the modes go over to the visco-resistive regime for $|\gamma_{d}|$ small.) The quantity $\Delta_{2}$ is the resistive wall matching condition at $r=r_{w}$ with ideal plasma conditions at $r_{t}$. The inductance coefficients $l_{12}=-\psi_{2}'(r_{w}-),\, l_{21}=\psi_{2}'(r_{t}-),\, l_{32}=\psi_{3}'(r_{w}+)$ as well as $\Delta_{1}=[\psi_{1}']_{r_{t}}$, $\Delta_{2}=[\psi_{2}']_{r_{w}}$, and $l_{22}^{(-)}=\psi_{2}'(r_{w}-)$ are computed in the Appendix. See Fig. 10. The pressure affects the values of $\Delta_{1}$, $\Delta_{2}$, and $l_{22}^{(-)}$ but, since there is no pressure gradient at $r=r_{t}$, is not included in the tearing layers, where otherwise it could have stabilizing or destabilizing effects[@GlasserGreenejohnson; @FinnManheimer].
Substituting Eq. (\[eq:Basic-3\]) into Eqs. (\[eq:Basic-1\],\[eq:Basic-2\]) we obtain $$\left(\begin{array}{cc}
\Delta_{1}-\gamma_{d}\tau_{t} & \,\,\,\,\,\,\,\,\,\, l_{21}\\
l_{12}-Kl_{32}l_{12} & \,\,\,\,\,\,\,\,\,\,\Delta_{2}-\gamma\tau_{w}-Gl_{32}+Kl_{32}l_{22}^{(-)}
\end{array}\right)\left(\begin{array}{c}
\alpha_{1}\\
\alpha_{2}
\end{array}\right)=0\label{eq:2X2-eigenvalue-primitive}$$ or $$\left(\begin{array}{cc}
\frac{\Delta_{1}}{\tau_{t}}-i\Omega-\gamma & \,\,\,\,\,\,\,\,\,\,\frac{l_{21}}{\tau_{t}}\\
\frac{l_{12}-Kl_{32}l_{12}}{\tau_{w}} & \,\,\,\,\,\,\,\,\,\,\frac{\Delta_{2}-Gl_{32}+Kl_{32}l_{22}^{(-)}}{\tau_{w}}-\gamma
\end{array}\right)\left(\begin{array}{c}
\alpha_{1}\\
\alpha_{2}
\end{array}\right)=0\,\,\,\,\,\,\,\,\,\text{-- or}\,\,\,\,\,\,\,\,\,(\mathsf{A}-\gamma\mathsf{I})\vec{\boldsymbol{\boldsymbol{\alpha}}}=0.\label{eq:2X2-eigenvalueEq}$$ The off-diagonal terms couple the resistive plasma ideal wall (rp,iw) mode and the ideal plasma resistive wall (ip,rw) mode. This leads to a dispersion relation from $\text{det}(\mathsf{A}-\gamma\mathsf{I})=0$, or $\gamma^{2}-T\gamma+D=0$, where $T=\text{trace}\mathsf{A}$ and $D=\text{det}\mathsf{A}$, giving $\gamma=T/2\pm\sqrt{(T/2)^{2}-D}$. For RI tearing modes rather than VR modes, $\gamma_{d}\tau_{t}$ is replaced by $(\gamma_{d}\tau_{t}')^{5/4}$. Notice that for $\tau_{t}\rightarrow\alpha\tau_{t}$, $\tau_{w}\rightarrow\alpha\tau_{w}$ and $\Omega\rightarrow\Omega/\alpha$, $\gamma$ is replaced by $\gamma/\alpha$, for either the VR or RI versions. This shows that for $\tau_{t}/\tau_{w}$ fixed marginal stability is unaffected by changes to $\Omega\tau$, where $\tau=\sqrt{\tau_{t}\tau_{w}}$.
Linearized full MHD model
-------------------------
In this subsection we discuss the full, compressional MHD model used with smooth current density and pressure profiles. Denoting perturbed quantities by a tilde, the visco-resistive MHD model reduces to the following coupled equations: $$\gamma_{d}\mathbf{\tilde{v}}=\left(\boldsymbol{\nabla}\times\mathbf{\tilde{B}}\right)\times\mathbf{B}_{0}+\mathbf{j}_{0}\times\tilde{\mathbf{B}}-\nabla\tilde{p}+\nu\mathbf{\nabla^{2}}\mathbf{\tilde{v},}\label{eq:2a}$$ $$\gamma_{d}\mathbf{\tilde{B}}=\boldsymbol{\nabla}\times\left[\mathbf{\tilde{v}}\times\mathbf{B}_{0}-\eta\boldsymbol{\nabla}\times\mathbf{\tilde{B}}\right],\label{eq:2b}$$
$$\gamma_{d}\tilde{p}=-\tilde{\mathbf{v}}\cdot\nabla p_{0}-\Gamma p_{0}\nabla\cdot\tilde{\mathbf{v}},\label{eq:adiabatic}$$
where again $\gamma_{d}=\gamma+iku_{z0}=\gamma+i\Omega$, only contributing a constant Doppler shift due to the uniform toroidal equilibrium flow. The normalization is such that the (assumed uniform) equilibrium density is unity; $\Gamma=5/3$ is the adiabatic index. As in Sec. 2, all perturbations are of the form $e^{i(m\theta+kz-\omega t)}$, where $\omega=i\gamma$ is the complex frequency and the toroidal mode number is given by $n=-kR$. For current density and pressure profiles that are smoothed forms of the stepfunction profiles of Sec. 2, and for large aspect ratio $R/r_{w}$ so that reduced MHD is fairly accurate, we obtain results that are in good agreement with those obtained with the model of Sec. 3.1. These equations are put in dimensionless form as before, with time in Alfvén units using the nominal poloidal field $B_{\theta0}'(0)r_{w}$ and lengths scaled to $r_{w}$, so that $B_{\theta0}'(0)r_{w}=r_{w}=1$. The results are reported in terms of $\beta_{0}=2p_{0}(0)/B_{z0}(0)^{2}$, the Lundquist number $S=\tau_{r}/\tau_{A}$ and the magnetic Prandtl number $Pr=\nu/\eta$. The aspect ratio used is $R/r_{w}=5$.
In ideal MHD modeling, the modes can be influenced by continuum damping. We include plasma resistivity, and therefore the continuum is replaced be discrete damped modes, and collisional transport (represented by plasma resistivity and viscosity) causes damping in place of the continuum damping of ideal MHD.
$${\color{black}\gamma_{d}\tilde{B}_{r}(r_{w})=i\mathbf{k}\cdot\mathbf{B}_{0}\tilde{v}_{r}}\label{eq:BC_IdealOhm}$$
$$\gamma\tau_{w}\tilde{B}_{r}(r_{w})=[\tilde{B}_{r}^{\prime}]_{r_{w}}\label{eq:RWCondition}$$
$$im\tilde{v_{r}}/r+r\partial_{r}(\tilde{v}_{\theta}/r)=0\label{eq:Zero-stress}$$
$$ik\tilde{v_{r}}+\partial_{r}\tilde{v_{z}}=0\label{eq:Another-Zero-stress}$$
$$\partial_{r}(r\tilde{B}_{\theta})-im\tilde{B_{r}}=0\label{eq:ZeroCurrentDensity}$$
$$\partial_{r}\tilde{B_{z}}-ik\tilde{B_{r}}=0\label{eq:AnotherZeroCurrentDensity}$$
$$\gamma_{d}\tilde{p}=-\tilde{v}_{r}\partial_{r}p_{0}(r_{w})-\Gamma p_{0}(r_{w})(\nabla\cdot\tilde{\mathbf{v}})_{r_{w}}\label{eq:Adiabatic}$$
$$\tilde{B_{r}}(r_{c})=[-(Gr_{w}-K)\tilde{B_{r}}(r_{w})+Kr_{w}\tilde{B_{r}}'(r_{w}-)]/r_{c}.\label{eq:ControlInNumericalBCs}$$
The comments made after Eq. (\[eq:GainEquation\]) apply as well to the essentially identical control scheme of Eq. (\[eq:ControlInNumericalBCs\]).
Notice that **$\tilde{v}_{r}$** at the wall is allowed, consistent with ideal MHD (Eq. (\[eq:BC\_IdealOhm\])) and the finite $\tilde{B}_{r}$ due to the wall resistivity. Equations (\[eq:ZeroCurrentDensity\]) and (\[eq:AnotherZeroCurrentDensity\]) represent the tangential components of the plasma current being set to zero, consistent with Eq. (\[eq:BC\_IdealOhm\]), and preventing an artificial resistive boundary layer near the wall. (Skin currents in the wall irrelevant.) no-stress boundary condition on $\tilde{\mathbf{v}}$, reasonable since we are modeling plasmas for which the region near the wall consists of either cold plasma or vacuum. In general with the thin wall boundary condition, $\tilde{B}_{r}$ is continuous across the wall, while the jump in the gradient of $\tilde{B}_{r}$ represents the current induced in the wall. These boundary conditions are idealized, and to be sure a more complete treatment of the interaction with the wall is possible. However, results which we show in the next section indicate that these boundary conditions do not allow artificial boundary layers near the walls, producing results that are very similar (in the numerical modeling) or identical (for the analytic treatment) to results that would be obtained with a vacuum region just inside the resistive wall.
Results with zero rotation and gain parameters
==============================================
In this section we present results obtained with both the simplified model, handled analytically as described in Sec. 3.1, and the full MHD model of Sec. 3.2.
Let us first consider $G=K=0$ and $\Omega=0$ with the simplified model. Because the $q(r)$ profile is increasing, the negative step $\Delta j_{z0}$ at $a_{1}<r_{t}$ contributes a destabilizing influence. The diffuse current density profile in Sec. 2 has a destabilizing influence for $r<r_{t}$ and a stabilizing influence for $r_{t}<r<r_{w}$. The negative step $\Delta p_{0}$ is stabilizing for $a_{2}<r_{t}$ or for $a_{2}>r_{t}$ but we assume the latter. In fact, for $\beta_{0}=0$ (or for $a_{2}<r_{t}$) the mode is an *internal* mode, concentrated in the region $0<r<r_{t}$ and therefore insensitive to the resistive wall. For $\beta_{0}>0$ the mode is also driven at $r=a_{2}>r_{t}$ and is therefore no longer localized to $r<r_{t}$ and is sensitive to the resistive wall. In RFPs, i.e. for decreasing $q(r)$ profiles, the current density contribution is stabilizing for $r<r_{t}$ and destabilizing for $r_{t}<r<r_{w}$, so that the mode is not internal, i.e. is sensitive to the resistive wall. Thus, the four stability thresholds in $\lambda=j_{||}/B$ at $r=0$ are distinct and can occur at zero $\beta$. In a toroidal rather than a cylindrical model for a tokamak, the modes are more sensitive to the resistive wall because of poloidal mode coupling. The results in the Appendix show that $\Delta_{1}$ has a destabilizing term due to the current step at $r=a_{1}$, $\sim1/(A+\delta_{1})$, and one due to the pressure step at $r=a_{2}$, $\sim1/(B+\delta_{2})$; those results also show that $\Delta_{2}$ has a destabilizing term from the pressure step only $\sim1/(B+\delta_{2})$. In the Appendix, we discuss why $\Delta_{1}>\Delta_{2}$, and hence $\beta_{rp,iw}<\beta_{ip,rw}$, for typical parameters for this model.
For $\Omega=G=K=0$, we have $T=\text{trace}\mathsf{A}=\Delta_{1}/\tau_{t}+\Delta_{2}/\tau_{w}$ and $D\equiv\text{det}\mathsf{A}=(\Delta_{1}\Delta_{2}-l_{12}l_{21})/\tau_{t}\tau_{w}$. Also, note that $(T/2)^{2}-D=\left[(\Delta_{1}/\tau_{t}-\Delta_{2}/\tau_{w})^{2}+l_{12}l_{21}\tau_{t}\tau_{w}\right]/4$, which is nonnegative. (See the Appendix.) As $\beta_{0}$ is increased from zero, we reach marginal stability $\gamma=0$ at $D\equiv\text{det}\mathsf{A}=0$ or $$\Delta_{1}=\frac{l_{12}l_{21}}{\Delta_{2}}.\label{eq:beta1}$$ This is the resistive plasma-resistive wall limit $\beta=\beta_{rp,rw}$.[^1] Next, we set $\tau_{w}=\infty$ to find, from Eq. (\[eq:2X2-eigenvalueEq\]), $\gamma(\gamma-\Delta_{1}/\tau_{t})=0,$ so that the resistive plasma-ideal wall stability limit $\beta=\beta_{rp,iw}$ has $\Delta_{1}=0$. Similarly, setting $\tau_{t}=\infty$ we find that the resistive wall-ideal plasma limit $\beta=\beta_{ip,rw}$ is at $\Delta_{2}=0$. The condition $\Delta_{2}<\Delta_{1}$ guarantees that $\beta_{rp,iw}<\beta_{ip,rw}$. (In Ref. [@finn_rw2] it was concluded that resistive wall modes could be stabilized by slow rotation for $\beta_{ip,rw}<\beta_{0}<\beta_{rp,iw}$, the area called Region III in Ref. [@finn_rw2]. This range of $\beta_{0}$ is empty for the case we consider, with $\beta_{rp,iw}<\beta_{ip,rw}$.) The analogous ordering for zero-beta reversed field pinches, i.e. $\lambda_{rp,iw}<\lambda_{ip,rw}$, holds for all reasonable RFP profiles[@RichardsonFinnDelzanno; @Sassenberg]. The ideal wall-ideal plasma limit $\beta_{ip,iw}$ occurs when $\Delta_{1}=\Delta_{2}=\infty$, both occurring where $B+\delta_{2}\rightarrow0-$, as discussed in the Appendix. The growth rates for very large $\Delta_{1}$ andare not accurate because the constant-$\psi$ approximation and the thin-wall approximation are not accurate there, but the qualitative behavior is correct and the marginal stability points are still valid. We choose parameters $a_{1}=0.5,\, a_{2}=0.8,\, r_{w}=1,\, r_{c}=1.5,\, q(0)=0.9$, and find $r_{t}=0.745$. We summarize in Table I. Notice that, although $\beta_{ip,rw}<\beta_{ip,iw}$ must hold, the extrapolation process to $S=\infty$ makes it difficult (as well as unimportant) to obtain more than two-place accuracy.
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Model$\downarrow$ $\beta_{0}\rightarrow$ $\beta_{rp,rw}$ $\beta_{rp,iw}$ $\beta_{ip,rw}$ $\beta_{ip,iw}$
---------------------------------------------- ---------------------------------------- ------------------ ------------------ ----------------------------------------------
Analytic $0.045$ $0.101$ $0.383$ $0.440$
(Analytic) ($\Delta_{1}=l_{12}l_{21}/\Delta_{2}$) ($\Delta_{1}=0$) ($\Delta_{2}=0$) ($\Delta_{1},\,\Delta_{2}\rightarrow\infty$)
Numerical $S=10^{5}$ $0.06$ $0.12$ $\sim1.5^{*}$ $\sim1.5^{*}$
Table 1. Marginally stable $\beta$ values, for the simplified and numerical models with parameters as in Figs. (1) and (2). Note ([\*]{}) that the two ideal plasma limits are estimated from extrapolations of the growth rate curves of $S>10^{8}$ to the marginal point in the ideal MHD regime $S\rightarrow\infty$.
------------------------------------------------------------------------
In Fig. 2a we show the growth rate $\gamma$ in poloidal Alfvén units for $\tau_{w}=10^{3},\,\tau_{t}=10^{4}$, showing the marginal stability points as in Table 1. We also show $\Delta_{1}$ and $\Delta_{2}$ as functions of $\beta_{0}$. In Fig. 2b we show $\gamma\tau_{A}$ vs $\beta_{0}$ for the full MHD model. The value $\beta_{rp,iw}$ is found by setting $\tau_{w}$ very large; $\beta_{ip,rw}$ is found by a convergence study for large Lundquist number $S$.
In Fig. 3 we include feedback (real $G$, $K$) but with $\Omega=G_{i}=K_{i}=0$, and show the stability diagram for four values of $\beta_{0}$, both for the simplified model and the full MHD model. Note that $G=K=0$ is in the stable region for the lowest value of $\beta_{0}$ in Fig. 3a, consistent with ${\normalcolor {\color{red}{\normalcolor \beta_{0}<\beta_{rp,rw}}}}$. Also, the results are consistent with the top line becoming vertical for $\beta_{0}=\beta_{rp,iw}=0.101$, and that the slope of the top line approaches that of the bottom line as $\beta\rightarrow\beta_{ip,iw}$, where $\Delta_{1},\,\Delta_{2}\rightarrow\infty$. The results in Fig. 3b and c, with the full MHD model, show similar results. It is thus possible to stabilize the tearing mode above $\beta_{rp,iw}$ and, for the simplified model, technically up to $\beta_{ip,iw}$. In Ref. [@Finn2006] a stable window was shown to exist up to $\beta_{ip,iw}$, as in the present results; in Refs. [@RichardsonFinnDelzanno; @Sassenberg], with finite viscosity, the limit was slightly below $\beta_{ip,iw}$. Indeed, stability ($\text{Re}(\gamma)<0$) is guaranteed for the simplified model if the trace in Eq. (\[eq:2X2-eigenvalueEq\]) is negative and the determinant is positive. The trace condition for stability with feedback, $T<0$, gives $$G>Kl_{22}^{(-)}+\frac{\Delta_{2}}{l_{32}}+\frac{\Delta_{1}}{l_{32}}\frac{\tau_{w}}{\tau_{t}},\label{eq:TraceCondition}$$ and depends on $\tau_{w}/\tau_{t}$[@Finn2006]. The determinant condition $D>0$ is independent of $\tau_{w}/\tau_{t}$ and gives $$\Delta_{1}G<\Delta_{1}Kl_{22}^{(-)}+\frac{\Delta_{1}\Delta_{2}}{l_{32}}+\frac{l_{12}l_{21}}{l_{32}}\left(Kl_{32}-1\right).\label{eq:DeterminantCondition}$$ (Recall that $\Delta_{1}$ can have either sign; this inequality is valid for either sign of $\Delta_{1}$.) The upper and lower straight lines correspond to the determinant condition in Eq. (\[eq:DeterminantCondition\]) and the trace condition in Eq. (\[eq:TraceCondition\]), respectively. The upper line (independent of $\tau_{w}/\tau_{t}$) is the marginal stability curve for the purely growing tearing mode. The lower line (with intercept depending on $\tau_{w}/\tau_{t}$) corresponds to a complex root driven unstable by the feedback below the line. See Refs. [@Finn2006; @RichardsonFinnDelzanno; @Sassenberg]. Notice that the determinant condition indeed gives a vertical line at $\beta_{rp,iw}$, where $\Delta_{1}=0$, and it is also clear that the slopes of the lines become equal for $\Delta_{1}$ large, so that the stable region disappears as $\beta\rightarrow\beta_{ip,iw}$.
If we look for the intersection of the $T=0$ line and the $D=0$ line we find $$\Delta_{1}^{2}=\frac{\tau_{t}}{\tau_{w}}l_{12}l_{21}\left(Kl_{32}-1\right).\label{eq:IntersectionPoint}$$ That is, at this intersection we must have $l_{32}K>1$. Note that this implies that the coupling coefficient $\sim a_{21}$ in Eq. (\[eq:2X2-eigenvalueEq\]) is negative in the stable region. Also, this holds regardless of the sign of $\Delta_{1}$, i.e. with $\beta_{0}$ on either side of $\beta_{rp,iw}$ and for $\beta_{0}=\beta_{rp,iw}$ ($\Delta_{1}=0$) this intersection has $K=1/l_{32}$. More importantly, this intersection occurs for rapidly increasing $K$ (and $G$) as $\Delta_{1}$ increases. The slopes in $(G,K)$ of the marginal stability lines, from Eqs. (\[eq:TraceCondition\],\[eq:DeterminantCondition\]), approach each other rapidly as $\Delta_{1}$ increases, so that, although the theoretical limit for feedback stabilization is $\beta_{ip,iw}$, the practical limit is a few times $\beta_{rp,iw}$. This practical limit can be below or above $\beta_{ip,rw}$.
Another point is that, whereas the two straight lines intersect at a point for the simplified model, the lower stability boundary in Fig. 3(c) develops curvature for the full MHD model. This curvature is reproduced qualitatively by using the RI version of the simplified model, with $\gamma_{d}\tau_{t}\rightarrow(\gamma_{d}\tau_{t}')^{5/4}$
The major conclusions of this section are that feedback stabilization appears to be practically possible well above $\beta_{rp,iw}$ and possibly above $\beta_{ip,rw}$. Further, the simplified model captures well the qualitative behavior of the full MHD model. We also note that in a toroidal configuration at moderate aspect ratio the ideal plasma limits will be much lower, as the toroidicity affects the stability at the same order as pressure and current, and thus the stable regions will more easily approach $\beta_{ip,iw}$.
Results with plasma rotation and complex gain $G_{i}$
=====================================================
In this section we show analytic and numerical results for the appropriate equilibria, with plasma rotation and complex gain $G_{i}$, both for the simplified model and the full MHD model.
Figure 4, for $\beta_{0}<\beta_{rp,iw}$, shows that the stable region increases in size with $\Omega$ ($G_{i}=K_{i}=0$) in this range for the simplified model. ($\Omega\rightarrow-\Omega$ gives identical results for the growth rate, with $\gamma\rightarrow\gamma^{*}$, both for the simplified model and for the full MHD model.) This stabilization is expected because the mode in this regime is a resistive wall tearing mode, and plasma rotation relative to the wall stabilizes by supressing flux from penetrating the wall. A close look at the numerical results in Fig. 4b shows that for low rotation, for $0<\Omega<0.001$, the stable region actually shrinks along some sections of the marginal stability curve. This is related to the fact that in the RI regime low rotation initially destabilizes resistive wall modes, followed by stabilization for higher rotation. This behavior is explained by the mode-coupling picture of Ref. [@finn_rw4] and is even more noticeable for ideal plasma resistive wall modes[@betti-freidberg; @finn_rw4]. Larger $\Omega$ is stabilizing for the full MHD model and the expanding stable region develops a tail toward negative $G$ and $K$. This general behavior of stabilization as $\Omega$ increases is consistent with the observation in Ref. [@Finn2006] that increasing $\tau_{w}/\tau_{t}$ is stabilizing in this regime.
The curvature seen in Fig. 4b with $\Omega=0$ at the tip is seen all along the curve to the right. This is consistent with the fact that the RI model is reasonable for this curve because marginal stability there has real frequency; and in the RI regime the $(\gamma_{d}\tau_{t}')^{5/4}$ term does indeed cause curvature (not shown.) On the upper (left) curve, marginal stability has $\gamma=0$, so that the VR dispersion relation is correct there, giving a linear marginal stability curve.
Figure 5 shows a case with $\beta_{0}>\beta_{rp,iw}$, in which the stable area is observed to *decrease* as the plasma rotation $\Omega$ increases, for both the simplified model and full MHD. The explanation is as follows: In this regime the tearing mode is unstable even with an ideal wall, so *lower* $\Omega$ allows the feedback flux to penetrate the resistive wall faster. These results can be interpreted in terms of a virtual wall inside $r=r_{w}$ for the upper curve, but not the lower curve, which has complex frequency even for $\Omega=0$. As discussed in Ref. [@FinnChacon1], there is an equivalence between $G_{i}$ and wall rotation in the presence of a single value of $k_{z}=k$. (So this equivalence is not exact in nonlinear theory.) This is evident in Eq. (\[eq:2X2-eigenvalueEq\]): the effective wall rotation rate $\Omega_{w}=ku_{zw}$ is given by $$\Omega_{w}=\frac{l_{32}G_{i}}{\tau_{w}}.\label{eq:RotationGiEquivalence}$$ Results (not shown) with $G_{i}$ such that the equivalent wall rotation $\Omega_{w}$ is equal to the values of the plasma rotation in Fig. 5 give identical results.
Figure 6 shows a case, again with $\beta_{0}>\beta_{rp,iw}$ and the same parameters but with $\Omega=0.005,\, K_{i}=0$ and four values of $G_{i}$. Note however, that in Fig. 6 both the analytic and numerical models have $\tau_{w}=2\times10^{4}$. In the configuration of Fig. 1(a) we calculate $l_{32}=2.2$. The value of $G_{i}$ corresponding to $\Omega_{w}=\Omega$ is $G_{iw}=\Omega_{w}\tau_{w}/l_{32}$ in the analytic model, showing that the optimal value of $G_{i}$ is where the relative rotation rate vanishes, $\Omega-\Omega_{w}=0$, at $G_{i}=45$. Also, the stability regions are symmetric about $G_{i}=G_{iw}$: In the plasma frame $G_{i}$ enters in Eq. (\[eq:2X2-eigenvalue-primitive\]) asand $\gamma\rightarrow\gamma^{*}$ shows that $\gamma_{real}$ is an even function of $G_{iw}-G_{i}$. Similar behavior is seen for the full MHD model, with optimal $G_{i}\approx40$. Indeed, the boundary conditions related to the resistive wall and feedback (Sec. 3.2) also show this equivalence between rotation and $G_{i}$.
As expected, it is clear from this discussion that there is some advantage in having two resistive walls, with complex gain to give effective rotation to the outer wall[@gimblettRotatingSecondWall; @Fitzpatrick-Jensen; @FinnChacon1], in the regime $\beta_{0}<\beta_{rp,iw}$. However, there is no such advantage in the regime $\beta_{0}>\beta_{rp,iw}$, since optimal control in this latter regime allows the flux from the outside to penetrate the wall to get into the plasma.
As in the previous section, we conclude that the simplified model captures the essential physics of the full MHD model.
Studies with plasma rotation and complex gain $K_{i}$
=====================================================
In this section we show results with $G_{i}=0$ but with imaginary gain $K_{i}$, both for the simplified model and the full MHD model.
The effect of $K_{i}$, the imaginary part of the tangential gain $K$, on the results is not as transparent as that of $G_{i}$ because $K$ occurs in two matrix elements in Eq. (\[eq:2X2-eigenvalueEq\]). In Fig. 7a we show results using the simplified model with parameters as in Fig. 2, with $\beta<\beta_{rp,iw}$, $\Omega=0$ and four values of $K_{i}$. In Fig. 7b we show corresponding results with $\beta_{0}>\beta_{rp,iw}$. Symmetry about $K_{i}=0$ is apparent in both and is easily proved by arguments like those in the previous section. As with $\Omega$ and $G_{i}$, increasing $|K_{i}|$ shrinks the stable region for $\beta_{0}>\beta_{rp,iw}$. However, we observe that the stable region also shrinks with increasing $|K_{i}|$ for $\beta_{0}<\beta_{rp,iw}$, indicating that the behavior with respect to $K_{i}$ differs significantly from the behavior with varying $\Omega$.
Results with finite $\Omega$ and $K_{i}$, for $\beta_{0}<\beta_{rp,iw}$ are shown in Fig. 8 for the simplified and full MHD models. Here we see a difference between the simplified and reduced MHD models. In this range of $\beta_{0}$, for the simplified model the stable region is largest near ${\color{magenta}{\normalcolor K_{i}=0}}$, and returns the result to near that of $\Omega=0$ in Fig. 4(a) for $K_{i}=-3$, but decreases the stable region outside of these values. However, no symmetry about the optimal value is observed. In Fig. 8(b) the numerical results with finite $\Omega$ and $K_{i}$ are shown, where the optimal $K_{i,opt}\approx1$ and the stable region decreases in size more slowly for $K_{i}<0$ than for $K_{i}>0$, in qualitative agreement with the simplified model. The optimal $K_{i}$ can appear on either side of $K_{i}=0$ here, as the effects of wall time and plasma response compete, but the stable regions tend to be more prominent for $K_{i}<0$ for $\Omega>0$.[^2]
Results with $\beta_{0}>\beta_{rp,iw}$ and finite $\Omega$ and $K_{i}$ are shown in Fig. 9(a) for the simplified model. In the results shown in Fig. 9a, the stable region is largest for $K_{i}=-1$ but shrinks as $K_{i}$ changes away from this optimal value. The result is not symmetric as can be seen by the similarity between the results for $K_{i}=-4$ and $K_{i}=0$. Results in Fig. 9(b) for the full MHD model have some similar aspects, but differ in that the width of the stable region increases with $K$ as the boundary becomes curved with increasing $K_{i}$. Again, no symmetry about any value of $K_{i}$ is observed. These results show that large $K_{i}$ (positive or negative) destabilize, but for moderate $K_{i}$ with rotation, the full MHD results vary significantly from those of reduced MHD. It is in general true that with rotation the stable region has some optimal $K_{i}$, but in full MHD it is not the same shape as $\Omega=K_{i}=0$. It can in fact be larger in some cases. In contrast to previous sections, we observe that the results using the full MHD model are captured by the simplified model in a broad sense, but some differences are observed in detail.
Though not shown here, in highly limited regions of parameter space as the stable regions approach marginality, weakly growing modes can appear within and distort the stable regions in the full MHD description. Likewise, isolated regions of stability can appear in the unstable region near marginality, rapidly moving to negative $G$ and $K$ as the original stable region moves to positive $G$ and $K$. These behaviors in marginally stable regions of parameter space are beyond the scope of this paper, but will be considered in context as we next look to investigate analogous systems in toroidal geometry.
Summary and conclusions
=======================
In this paper we have used a cylindrical linear model for a tokamak to make initial investigations in tokamak geometry into feedback control using complex gains $G$ and $K$, multiplying the measured radial and poloidal magnetic field components, respectively, in the presence of plasma resistivity and rotation. This model has four stability thresholds in the following order: $\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$, where $rp$ and $ip$ represent resistive plasma and ideal plasma, respectively, and $rw$ and $iw$ stand for resistive wall and ideal wall. We have determined the region of stability as a function of the real parts of the gains $G$ and $K$. For $\beta_{0}<\beta_{rp,iw}$, rotation $\Omega$ or imaginary gain $G_{i}$, which is equivalent to rotation of the resistive wall[@FinnChacon1], stabilizes. This is because in this regime, the tearing mode is unstable with a resistive wall but not with an ideal wall, and rotation can easily stabilize resistive wall tearing modes[@finn_rw1]. In this regime, $K_{i}$ is actually destabilizing and is therefore not equivalent to rotation. For $\beta_{0}>\beta_{rp,iw}$, on the other hand, plasma rotation $\Omega$ and $G_{i}$ are both destabilizing, while results for $K_{i}$ are more complex. Above $\beta_{rp,iw}$ and for nonzero plasma rotation $\Omega$, the optimal value for $G_{i}$ is the value for which the equivalent wall rotation equals the plasma rotation, and for this value of $G_{i}$ stability is possible well above $\beta_{rp,iw}$, as for $\Omega=G_{i}=0$. There is also an optimum value of $K_{i}$ in both ranges of $\beta_{0}$, but its value and shape in $G,K$ space cannot easily be determined by a simple equivalence with rotation, indeed the situation for $\beta_{0}>\beta_{rp,iw}$, shown in Fig. 9(b), is more complex than for $\beta_{0}<\beta_{rp,iw}$. These results have been found by both analysis on a reduced resistive MHD model with simple stepfunction current density and pressure profiles and a general MHD model with smooth profiles; the results and conclusions from both models are very similar.
The fact that rotation or $G_{i}$ is stabilizing for $\beta_{0}<\beta_{rp,iw}$ and destabilizing for $\beta_{0}>\beta_{rp,iw}$ suggests the importance of modeling resistive wall modes and their control including plasma resistivity, at least for resonant modes. The use of ideal MHD modeling with a resistive wall tacitly assumes that $\beta_{rp,rw}\lessapprox\beta_{ip,rw}$ and $\beta_{rp,iw}\lessapprox\beta_{ip,iw}$, which is not consistent with the results from our cylindrical model, namely $\beta_{rp,rw}<\beta_{rp,iw}<\beta_{ip,rw}<\beta_{ip,iw}$. If the latter ordering holds in toroidal geometry, then modeling using non-ideal MHD for resistive wall modes in toroidal geometry is also important.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of D. P. Brennan was supported by the DOE Office of Science, Fusion Energy Sciences under Contract No DE-SC0004125. The work of J. M. Finn was supported by the DOE Office of Science, Fusion Energy Sciences and performed under the auspices of the NNSA of the U.S. DOE by LANL, operated by LANS LLC under Contract No DEAC52- 06NA25396.
Appendix. Calculations for stepfunction model {#appendix.-calculations-for-stepfunction-model .unnumbered}
=============================================
In this appendix we show the steps necessary to compute $\psi_{1},\,\psi_{2},$ and $\psi_{3}$, i.e. the quantities $l_{12},\, l_{21},\, l_{32},\,\Delta_{1},\,\Delta_{2},$ and $l_{22}^{(-)}$. We first define auxiliary functions $\phi_{1},\,\phi_{t},\,\phi_{2},\,\phi_{w},$ and $\phi_{c}$ with $\phi_{1}(0)=0,\,\phi_{1}(a_{1})=1,\,\phi_{1}(r_{t})=0$. The four radii are $a_{1}$, where the current density step is; $r_{t}$, the tearing layer; $a_{2}$, where the pressure step is; $r_{w}$, the radius of the resistive wall; and $r_{c}$, the position of the control surface. The other three functions $\phi_{t},\,\phi_{2},\,\phi_{w}$ are defined similarly. We have $$\phi_{1}(r)=(r/a_{1})^{m}\,\,\,\text{for}\,\, r<a_{1}\,\,\,\text{and}\,\,\,$$ $$\phi_{1}(r)=\frac{(r_{t}/r)^{m}-(r/r_{t})^{m}}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}}\,\,\text{for}\,\,\, a_{1}<r<r_{t}.$$ Similar expressions hold for $\phi_{t},\dots,\phi_{c}$. See Fig. 10. We find $$\phi_{1}'(a_{1}-)=\frac{m}{a_{1}};\,\,\,\phi_{1}'(a_{1}+)=-\frac{m}{a_{1}}\frac{(r_{t}/a_{1})^{m}+(a_{1}/r_{t})^{m}}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}}.$$ This leads to $$\delta_{1}\equiv[\phi_{1}']_{a_{1}}=-\frac{2m}{a_{1}}\frac{(r_{t}/a_{1})^{m}}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}}.$$ All other quantities are computed in the same manner: $$k_{t1}=\phi_{t}'(a_{1}+)=\frac{2m}{a_{1}}\frac{1}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}},$$ $$k_{1t}=-\phi_{1}'(r_{t}-)=\frac{2m}{r_{t}}\frac{1}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}},$$ $$k_{2t}=\phi_{2}'(r_{t}+)=\frac{2m}{r_{t}}\frac{1}{(a_{2}/r_{t})^{m}-(r_{t}/a_{2})^{m}},$$ $$k_{t2}=-\phi_{t}(a_{2}-)=\frac{2m}{a_{2}}\frac{1}{(a_{2}/r_{t})^{m}-(r_{t}/a_{2})^{m}},$$ $$k_{w2}=\phi_{w}'(a_{2}+)=\frac{2m}{a_{2}}\frac{1}{(r_{w}/a_{2})^{m}-(a_{2}/r_{w})^{m}},$$ $$k_{2w}=-\phi_{2}'(r_{w}-)=\frac{2m}{r_{w}}\frac{1}{(r_{w}/a_{2})^{m}-(a_{2}/r_{w})^{m}},$$ $$k_{ww}^{(-)}=-\phi_{w}'(r_{w}-)=\frac{m}{r_{w}}\frac{(r_{w}/a_{2})^{m}+(a_{2}/r_{w})^{m}}{(r_{w}/a_{2})^{m}-(a_{2}/r_{w})^{m}},$$ $$k_{cw}=\phi_{c}'(r_{w}+)=\frac{2m}{r_{w}}\frac{1}{(r_{c}/r_{w})^{m}-(r_{w}/r_{c})^{m}},$$ $$\delta_{t}=[\phi_{t}']_{r_{t}}=-\frac{m}{r_{t}}\left[\frac{(r_{t}/a_{1})^{m}+(a_{1}/r_{t})^{m}}{(r_{t}/a_{1})^{m}-(a_{1}/r_{t})^{m}}+\frac{(a_{2}/r_{t})^{m}+(r_{t}/a_{2})^{m}}{(a_{2}/r_{t})^{m}-(r_{t}/a_{2})^{m}}\right],$$ $$\delta_{2}=[\phi_{2}]_{a_{2}}=-\frac{m}{a_{2}}\left[\frac{(a_{2}/r_{t})^{m}+(r_{t}/a_{2})^{m}}{(a_{2}/r_{t})^{m}-(r_{t}/a_{2})^{m}}+\frac{(r_{w}/a_{2})^{m}+(a_{2}/r_{w})^{m}}{(r_{w}/a_{2})^{m}-(a_{2}/r_{w})^{m}}\right],$$ and $$\delta_{w}=[\phi_{w}']_{r_{w}}=-\frac{m}{r_{w}}\left[\frac{(r_{w}/a_{2})^{m}+(a_{2}/r_{w})^{m}}{(r_{w}/a_{2})^{m}-(a_{2}/r_{w})^{m}}+\frac{(r_{c}/r_{w})^{m}+(r_{w}/r_{c})^{m}}{(r_{c}/r_{w})^{m}-(r_{w}/r_{c})^{m}}\right].$$
We set $\psi(r)=a_{1}\phi_{1}(r)+a_{t}\phi_{t}(r)+a_{2}\phi_{2}(r)$. The condition $\psi_{1}(a_{1})=1$ implies $\alpha_{t}=1$, and Eq. (\[eq:JumpConditions\]) implies $[\psi_{1}']_{a_{1}}=-A$ and $[\psi_{2}']_{a_{2}}=-B$. From these we find $\delta_{1}a_{1}+k_{t1}a_{t}=-Aa_{1}$ and $\delta_{2}a_{2}+k_{t2}a_{t}=-Ba_{2}$ or $$a_{1}=-\frac{k_{t1}}{A+\delta_{1}},\,\,\,\,\, a_{2}=-\frac{k_{t2}}{B+\delta_{2}}.$$ We conclude $$\Delta_{1}=\delta_{t}-\frac{k_{1t}k_{t1}}{A+\delta_{1}}-\frac{k_{2t}k_{t2}}{B+\delta_{2}}.$$ Similar calculations, plus the fact that $\phi_{c}=\psi_{3}$ show $$l_{12}=-\frac{k_{t2}k_{2w}}{B+\delta_{2}},$$ $$\Delta_{2}=\delta_{w}-\frac{k_{w2}k_{2w}}{B+\delta_{2}},$$ $$l_{22}^{(-)}=\frac{k_{w2}k_{2w}}{B+\delta_{2}}+k_{ww}^{(-)},$$ $$l_{21}=-\frac{k_{w2}k_{2t}}{B+\delta_{2}},$$ and $$l_{32}=k_{cw}.$$ A sketch of $\psi_{1}-\psi_{3}$, as well as $\phi_{1},\phi_{t},\phi_{2},\phi_{w}$, and $\phi_{c}$ is shown in Fig. 10.
The terms proportional to $1/(A+\delta_{1})$ are due to the destabilizing influence of the current density gradient at $a_{1}$. Those proportional to $1/(B+\delta_{2})$ are due to the destabilizing influence of the pressure gradient at $a_{2}$. The condition $\Delta_{1}>\Delta_{2}$ gives $$\Delta_{1}-\Delta_{2}=\delta_{t}-\delta_{w}-\frac{k_{1t}k_{t1}}{A+\delta_{1}}.$$ The term $\delta_{t}-\delta_{w}$ depends only one the geometry, i.e. on $a_{1},\, r_{t},\, a_{2},\, r_{w}$, and $r_{c}$. It is positive if $r_{w}-a_{2}$ or $r_{c}-r_{w}$ is small enough, which we assume. The term $-k_{1t}k_{t1}/(A+\delta_{1})$, from the drive by the current gradient inside $r_{t}$, is positive for $A\sim\Delta j_{z0}$ small and goes to infinity as $A+\delta_{1}\rightarrow0-$. We consider cases in which the drive due to the current, while not sufficient to drive the instability for zero pressure, is fairly large, so that $\beta_{rp,rw}$ and $\beta_{rp,iw}$ are small. In the simplified model, the values $\beta_{ip,rw}$ and $\beta_{ip,iw}$ are fairly large (and those values for the numerical model are large) because for an ideal plasma $\tilde{\psi}(r_{t})$ is zero, and therefore any unstable mode must be driven solely by the pressure gradient in the region $r_{t}<r<r_{w}$. Summarizing, for the geometry and and profiles we consider, $\Delta_{1}-\Delta_{2}$ should be positive, which implies $\beta_{rp,iw}<\beta_{ip,rw}$. Poloidal mode coupling in a torus, $m\rightarrow m\pm1$, prevent the shielding of the mode inside $r=r_{t}$ from the region for $r>r_{t}$. This will be the subject of a future publication.
Notice that $l_{12},\, l_{21}$ are positive for $B\rightarrow0$ (pressure $p_{0}\rightarrow0$) and go to infinity as $B+\delta_{2}\rightarrow0-$. Also, $\Delta_{1},\,\Delta_{2}\rightarrow+\infty$ as $B+\delta_{2}\rightarrow0-$, and $l_{22}^{(-)}\rightarrow-\infty$ in this limit. As we shall discuss in Sec. III, the limit $B+\delta_{2}\rightarrow0-$, where $\Delta_{1},\,\Delta_{2}\rightarrow+\infty$, is the ideal plasma-ideal wall limit $\beta_{ip,iw}$.
These quantities are used in the dispersion relation in Eq. (\[eq:2X2-eigenvalueEq\]) to obtain the results in Sec. 3-6.
with the same $\beta_{0}$ values
having
In (a) the optimal value of $K_{i}$ is -1. In (b) the stability regions are more complex, but optimal for $K_{i}$ for small $K_{i}$.
Sketch of basis functions $\phi_{1},\,\phi_{t},\,\phi_{2},\,\phi_{w}$ used to derive functions $\psi_{1},\,\psi_{2}$, and $\psi_{3}$ of the Appendix, also shown.
[^1]: We find stability if $1-(a_{1}/r_{c})^{2m}<m-nq(0)$, so if we take $a_{1}=0.5,\, r_{c}=1.5$, $m=2,\, n=1$ we get $q(0)<1+(a_{1}/r_{c})^{4}=1.01$.
[^2]: Indeed, an argument along the lines of that in the previous section shows that $\gamma_{real}$ is a symmetric function of $\Omega\tau_{w}+l_{32}l_{22}^{(-)}K_{i}$ and $K_{i}$, the first dependence coming through the $a_{22}$ matrix element and the second dependence from the $a_{21}$ matrix element. But such a function is not symmetric in $K_{i}$ with $\Omega$ held fixed.
|
---
abstract: |
Tropical manifolds are polyhedral complexes enhanced with certain kind of affine structure. This structure manifests itself through a particular cohomology class which we call the eigenwave of a tropical manifold. Other wave classes of similar type are responsible for deformations of the tropical structure.
If a tropical manifold is approximable by a 1-parametric family of complex manifolds then the eigenwave records the monodromy of the family around the tropical limit. With the help of tropical homology and the eigenwave we define tropical intermediate Jacobians which can be viewed as tropical analogs of classical intermediate Jacobians.
address:
- 'Université de Genève, Mathématiques, Villa Battelle, 1227 Carouge, Suisse'
- 'Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506 USA'
author:
- Grigory Mikhalkin
- Ilia Zharkov
title: Tropical eigenwave and intermediate Jacobians
---
[^1]
Tropical spaces and tropical manifolds
======================================
In this section we briefly recall basic concepts of tropical spaces relevant for our paper. For more details we refer to [@Mik06] and [@MR]. The main assumption we make is that our the tropical space is regular at infinity.
Tropical spaces
---------------
A tropical affine $n$-space ${\mathbb T}^n$ is the topological space $[-\infty,\infty)^n$ (homeomorphic to the $n$th power of a half-open interval) enhanced with a collection of functions ${\mathcal O}_{\operatorname{pre}}=\{f\}$, $f:U\to{\mathbb T}=[-\infty,\infty)$. Here $U\subset{\mathbb T}^n$ is an open set and $f$ is a function that can be expressed as $$\label{fx}
f(x)=\max_{j\in A} (jx + a_j)$$ for a finite set $A\subset{\mathbb Z}^n$ and a collection of numbers $a_j\in{\mathbb T}$, such that the scalar product $jx$ is well-defined as a number in ${\mathbb T}$ (i.e. is finite or $-\infty$) for any $x\in U$.
The collection of functions ${\mathcal O}_{\operatorname{pre}}$ is a presheaf which gives rise to a sheaf ${\mathcal O}$ of [*regular functions*]{} on ${\mathbb T}^n$ (which we will also denote ${\mathcal O}_{{\mathbb T}^n}$ indicating the space where it is defined to avoid ambiguity). ${\mathcal O}$ is called the [*structure sheaf*]{} on ${\mathbb T}^n$.
It is convenient to stratify the space ${\mathbb T}^n$ by $${\mathbb T}^\circ_I:= \{y\in {\mathbb T}^n \ :\ y_i=-\infty, i\in I \ \text{ and } \ y_i>-\infty, i\notin I \},$$ where $I\subset \{1,\dots,n\}$. Each $T^\circ_I$ is isomorphic to ${\mathbb R}^{n-|I|}$ and we set ${\mathbb T}_I$ to be its closure in ${\mathbb T}^n$.
To write down a regular function on ${\mathbb R}^n$ all we need is the [*integral affine structure*]{} on ${\mathbb R}^n$. This allows us to distinguish functions ${\mathbb R}^n\to{\mathbb R}$ which are affine with linear parts defined over ${\mathbb Z}$. Thus the tropical structure on ${\mathbb T}^n$ can be thought of as an extension of the integral affine structure in ${\mathbb R}^n$ where the overlapping maps are compositions of linear transformations in ${\mathbb R}^n$ defined over ${\mathbb Z}$ with arbitrary translations in ${\mathbb R}^n$.
Given a subset $U\subset{\mathbb T}^N$ we say that a [*continuous*]{} map $U\to{\mathbb T}^M$ is [*integral affine*]{} if it restricts to an affine map ${\mathbb R}^N\to {\mathbb R}^M$ with integral linear part. We say that a partially defined map $h:{\mathbb T}^{N} \dashrightarrow {\mathbb T}^{M}$ is [integral affine]{} if it is defined on a subset $U\supset{\mathbb R}^N$ and is integral affine there. Extending $h$ whenever we can by continuity we see that for each $I\subset\{1,\dots,N\}$ $h$ is defined everywhere or nowhere on ${\mathbb T}^\circ_I$.
The automorphisms of a subset $U\subset{\mathbb T}^N$ are invertible integral affine maps $U\to U$. For example, the automorphisms ${\operatorname{Aut}}({\mathbb R}^N) \cong {\operatorname{GL}}_N ({\mathbb Z}) \ltimes {\mathbb R}^N$ form a group of all integral affine transformations of ${\mathbb R}^N$ while ${\operatorname{Aut}}({\mathbb T}^{N})\cong {\mathbb R}^N$ only consists of translations. We also note that automorphisms of ${\mathbb T}^{s} \times {\mathbb R}^{N-s}$ translate an $s$-dimensional affine subspace of ${\mathbb R}^N$ parallel to the ${\mathbb T}^{s}$ factor to another one with the same property.
A [*convex polyhedral domain*]{} $D$ in ${\mathbb T}^{N}$ is defined as the intersection of a finite collection of half-spaces $H_k$ of the form $$\label{H_k}
H_k=\{x\in{\mathbb T}^N\ |\ jx\le a\}\subset{\mathbb T}^{N}$$ for some $j\in{\mathbb Z}^N$ and $a\in{\mathbb R}$. The boundary ${\partial}H_k$ is given by the equation $jx=a$. A [*mobile face*]{} $E$ of $D$ is the intersection of $D$ with the boundaries of some of its defining half-spaces given by . The adjective [*mobile*]{} stands here to distinguish such faces among more general faces of $X$ which we will define later and which are allowed to have support in ${\mathbb T}^N{\smallsetminus}{\mathbb R}^N$, i.e. be disjoint from ${\mathbb R}^N\subset {\mathbb T}^{N}$. (They have reduced mobility and are called [*sedentary*]{}).
The [*dimension*]{} of a convex polyhedral domain $D$ is its topological dimension. Observe that for each mobile face $E$ of $D$ the intersection $$E^\circ=E\cap{\mathbb R}^N$$ is non-empty. The intersection $E^\circ$ is called the non-infinite part of a mobile face. Each mobile face of $D$ is a convex polyhedral domain itself (although perhaps of smaller dimension).
We say two domains $D\subset {\mathbb T}^{N}$ and $D'\subset {\mathbb T}^{M}$ are isomorphic if there is an integral affine map ${\mathbb T}^{N} \dashrightarrow {\mathbb T}^{M}$ which restricts to a homeomorphism $D\to D'$ (in particular, it has to be defined everywhere on $D$).
We say that a convex polyhedral domain $D\subset {\mathbb T}^{N}$ is [*regular at infinity*]{} if for every $I\subset \{1,\dots, N\}$ the intersection $D\cap ({\mathbb T}^{\circ}_I)$ is either empty or is a $(\dim D-|I|)$-dimensional polyhedral domain in ${\mathbb T}^\circ_I\cong {\mathbb R}^{N-|I|}$.
\[def:poly\_complex\] An $n$-dimensional polyhedral complex $Y=\bigcup D\subset {\mathbb T}^{N}$ is the union of a finite collection of convex $n$-dimensional polyhedral domains $D$, called the [*facets*]{} of $Y$ subject to the following property. For any collection $\{D_j\}$ of facets, their intersection $\bigcap D_j$ is a face of each $D_j$. Such intersections are called the (mobile) faces of $Y$. Clearly they are themselves polyhedral domains in ${\mathbb T}^{N}$.
We say that $Y$ is regular at infinity if all its faces are regular at infinity.
In this paper we assume that all polyhedral complexes are regular at infinity.
Let $E$ be an $(n-1)$-dimensional mobile face in $Y$ and $D_1,\dots,D_l\subset {\mathbb T}^{N}$ be the facets adjacent to $E$. Take the quotient of ${\mathbb R}^{N}$ by the linear subspace parallel to $E^\circ$, the non-infinite part of $E$. The balancing condition requires that $$\label{bal-cond}
\sum\limits_{k=1}^l\epsilon_k=0,$$ where the $\epsilon_k$ are the outward primitive integer vectors parallel to the images of $D_k$ in this quotient.
A polyhedral complex $Y\subset{\mathbb T}^N$ is called [*balanced*]{} if all of its $(n-1)$-dimensional faces satisfy the balancing condition.
More generally we can consider spaces that locally look like balanced polyhedral complexes, i.e. admit a covering by open sets $U_\alpha$ enhanced with open embeddings (charts) $$\phi_\alpha:U_\alpha\to Y_\alpha\subset{\mathbb T}^{N_\alpha}$$ where each $Y_\alpha\subset{\mathbb T}^{N_\alpha}$ is a balanced polyhedral complex. In this paper we assume in addition that each $Y_\alpha$ is regular at infinity.
We may express compatibility of different charts by requiring that the corresponding overlapping maps are induced by integral affine maps ${\mathbb T}^{N_\alpha}\dashrightarrow{\mathbb T}^{N_\beta}$. Or, equivalently, we may use the structure sheaf and enhance each $Y_\alpha\subset{\mathbb T}^{N_\alpha}$ with the sheaf ${\mathcal O}_{Y_\alpha}$ induced from ${\mathcal O}_{{\mathbb T}^{N_\alpha}}$. Its pull-back under $\phi_\alpha$ is a sheaf on $U_\alpha$. Two charts $\phi_\alpha$ and $\phi_\beta$ are compatible if the corresponding restrictions to $U_\alpha\cap U_\beta$ agree.
We arrive to the following definition of a tropical space.
\[def-tropspace\] A tropical space is a topological space $X$ enhanced with a cover of compatible charts $\phi_\alpha:U_\alpha\to Y_{\alpha}\subset{\mathbb T}^{N_\alpha}$ to balanced polyhedral complexes as above and which satisfies the finite type condition below.
The tropical space $X$ is regular at infinity if it admits charts to polyhedral complexes regular at infinity.
The charts induce a sheaf ${\mathcal O}_X$ on $X$ which we call the structure sheaf of $X$.
The number of charts $\phi_\alpha$ covering $X$ is finite while each chart is subject to the following property. If $\{x_j\in U_\alpha\}_{j=1}^\infty$ is a sequence such that $\phi_\alpha(x_j)$ converges to a point $y\in{\mathbb T}^{N_\alpha}$ then either the sequence $\{x_j\}$ converges inside the topological space $X$ or there exists a coordinate in ${\mathbb T}^{N_\alpha}$ such that its value on $y$ is $-\infty$ while its value on any point in $\phi_\alpha(U_\alpha)$ is finite.
It is easy to see that this finite type condition is a reformulation of the one from [@MR].
Sedentary points and faces
--------------------------
Let $D \subset {\mathbb T}^{N}$ be a polyhedral domain. It is convenient to treat the intersections $D \cap{\mathbb T}_I$ for $I\subset\{1,\dots,N\}$ also as its faces (at infinity). If we need to distinguish such faces from the mobile ones we have defined before we call these new faces [*sedentary*]{}.
\[def:sedentarity\] We say that $$E_I:=E \cap{\mathbb T}_I$$ is a face of $D$ if $E$ is a mobile face of $D$. The [*sedentarity*]{} of the face $E_I$ is $s=|I|$, while its [*refined sedentarity*]{} is $I$.
Clearly, the [mobile]{} faces (defined previously) are the faces of sedentarity $0$. If $Y\subset {\mathbb T}^{M}$ is a polyhedral complex then we define a (possibly sedentary) face of $Y$ as a face of a facet in $Y$.
We will use the notation $F\prec^s_j E$ when $F$ is a face of $E$ of codimension $j$ and sedentarity $s$ higher. It is also convenient to introduce the following terminology.
A face $E$ of $Y$ is called [*infinite*]{} if either it is not compact or it contains a higher sedentary subface. Otherwise $E$ is called finite (even if the sedentarity of $E$ itself is positive).
Note that even though a face $F\subset Y$ of sedentarity $I$ may be adjacent to several facets, it is always presented as $$F=E\cap{\mathbb T}^I$$ for a unique mobile face $E\subset Y$ which we call the [*parent*]{} of $F$ (as long as $Y$ is regular at infinity). The set of faces of $Y$ with the same parent $E$ is called the [*family*]{} of $E$. In case $E$ is compact the regularity at infinity forces its family to have a very simple combinatorial structure.
\[prop:family\] Let $E\subset Y$ be a compact mobile face containing a face of a maximal sedentarity $s$. Then its family $\Pi(E)$ forms a lattice poset (under $\prec_j^j$), isomorphic to the face poset of a simplicial cone of dimension $s$. The maximal sedentary face in the poset is finite.
Note also that a face $F$ of sedentarity $I$ completely determines the integral affine structure of its parent face $E$ in the neighborhood of ${\mathbb T}_I$. Namely, we have the following proposition.
Let $\pi_I:{\mathbb T}^N\to{\mathbb T}^I$ be the projection taking a point $(x_1,\dots,x_N)$ to the point whose $j$-th coordinate is $x_j$ if $j\notin I$ and $-\infty$ otherwise. The parent face $E$ of $F$ is contained in $\pi_I^{-1}(F)$. Furthermore, for a small open neighborhood $U\supset {\mathbb T}_I$ we have $$E\cap U= \pi_I^{-1}(F)\cap U.$$
In other words for a sufficiently small $\epsilon>-\infty$ we have $(x_1,\dots,x_N)\in E$ whenever $\pi_I(x_1,\dots,x_N)\in F$ and $x_j<\epsilon$ for any $j\in I$. Thus the directions parallel to the $j$-th coordinate in ${\mathbb T}^N$ for $j\in I$ are quite special for $E$. We orient them toward the $-\infty$-value of the coordinate and call them [*divisorial directions*]{}, see Figure \[mobile\]. Their positive linear combinations span the [*divisorial cone*]{} while all linear combination span the [*divisorial subspace*]{} in ${\mathbb R}^N$. The primitive integral vector along a divisorial direction (pointing towards $-\infty$ as the direction itself) is called a [*divisorial vector*]{}.
![Mobile and sedentary faces of a polyhedral domain in ${\mathbb T}^N$.[]{data-label="mobile"}](faces.pdf){height="45mm"}
One important observation is that the divisorial vectors are invariant with respect to any integral affine automorphism of ${\mathbb T}^{|I|}\times{\mathbb T}_I^\circ$. Thus they are intrinsically defined for $F$ and so is the divisorial subspace which we denote by $W^{div}$.
Tangent spaces {#sec:tangent}
--------------
Let $y$ be a point in the relative interior of a face $F$ of sedentarity $I$ in a balanced polyhedral complex $Y\subset{\mathbb T}^N$. Let $\Sigma(y)$ be the cone in ${\mathbb T}^\circ_I\cong{\mathbb R}^{N-|I|}$ consisting of vectors $u\in{\mathbb T}^\circ_I$ such that $y+\epsilon u\in Y\cap {\mathbb T}^\circ_I$ for a sufficiently small $\epsilon>0$ (depending on $u$). We denote the intersection of all maximal linear subspaces contained in $\Sigma(y)$ by $W'(y)$.
Clearly, the cones $\Sigma(y)$ can be canonically identified for all points $y$ in the relative interior of the same face, and so can be the vector spaces $W'(y)$. We say that $y_m\in Y$ is a [*nearby mobile point*]{} to $y$ if $y_m$ belongs to the relative interior of the parent face to $F$.
For a point $y\in Y$ we define $W(y)$, the [*wave tangent space*]{} at $y$, as $W'(y_m)$ for a nearby mobile point $y_m$. The (conventional) tangent space $T(y)$ at $y$ is defined as the linear span of $\Sigma(y)$ in ${\mathbb T}^\circ_I\cong{\mathbb R}^{N-|I|}$, where $I$ is the refined sedentarity of $y$.
Note that there are two essential distinctions in defining $T(y)$ and $W(y)$. To define $W(y)$ we always move to a nearby mobile point $y_m$. The space $W'(y)$ itself, is naturally a quotient of $W(y)$ by the divisorial subspace $W^{div}(y)$.
On the other hand, for $T(y)$ we work in a vector space ${\mathbb T}^\circ_I$, which is naturally the quotient ${\mathbb R}^N/W^{div}(y)$, but we take the linear span of the cone instead of the vector space contained in it.
If we need to specify the space $Y$ for the tangent space $T(y)$ we write $T_Y(y)$, and similarly for $W(y)$. The following proposition is straightforward.
\[prop:differentials\] An integral affine map $h:{\mathbb T}^N\dashrightarrow{\mathbb T}^M$ induces linear maps $dh^W: W_Y(y)\to W_{h(Y)}(h(y))$ and $dh^T: T_Y(y)\to T_{h(Y)}(h(y))$ whenever $h$ is well-defined on $y$. We call these maps differentials of $h$.
The differentials are natural in the following sense. If $g :{\mathbb T}^M\dashrightarrow{\mathbb T}^L$ is another integral affine map defined on $h(y)$, then the induced differentials satisfy $d(g\circ h)=(dg)\circ (dh)$.
Let $x\in X$ be now a point in a tropical space.
The tangent spaces $W_{Y_\alpha}(\phi_\alpha(x))$ (resp. $T_{Y_\alpha}(\phi_\alpha(x))$) for different charts $\phi_\alpha$ are identified by the differentials of the overlapping maps. The resulting spaces $W(x)$ and $T(x)$ are called the wave tangent space and the (conventional) tangent space to the tropical space $X$ at its point $x$.
The tangent spaces $T(x)$ and $W(x)$ carry natural integral structure. We denote the corresponding lattices by $T_{\mathbb Z}(x)$ and $W_{\mathbb Z}(x)$.
Polyhedral structures {#sec:polyhedral}
---------------------
Sometimes a tropical space $X$ comes with a structure of an (abstract) polyhedral complex, which is not always the case.
We say that a tropical space $X$ is [*polyhedral*]{} if there are finitely many closed subsets $\Delta_j\in X$ (called [*facets*]{}) with the following properties.
- For each $\Delta_j$ there exists a chart such that $\Delta_j\subset U_\alpha$ and $\phi_\alpha(\Delta_j)$ is a facet of the balanced polyhedral complex $Y_\alpha\subset{\mathbb T}^{N_\alpha}$.
- For any collection $\{\Delta_j\}$ of facets of $X$ and any face $\Delta_j$ in this collection the intersection $\bigcap \Delta_j$ is a face of $\Delta_j$.
Note that we may work with tropical polyhedral spaces in the same way as we work with balanced polyhedral complexes in ${\mathbb T}^N$. In particular, we can define in the same way their faces (which will denote by $\Delta$), both mobile and sedentary, parent faces with their families, divisorial directions, and any other notion which is intrinsically defined, that is stable under allowed integral affine maps. For instance, Proposition \[prop:family\] will read:
\[prop:subface\] Let $X$ be a compact polyhedral tropical space. For every face $\Delta$ of sedentarity $s$ there is a unique (parent) face $\Delta_0$ of sedentarity 0 such that $\Delta\prec^s_s\Delta_0$. The cells of $X$ with the same $\Delta_0$, the family of $\Delta_0$, form a lattice poset $\Pi(\Delta_0)$ isomorphic to the face poset of a simplicial cone. Every face of $X$ belongs to exactly one family poset $\Pi$. The maximal sedentary face $\Delta_{\min}$ in a poset is finite.
We will denote the $k$-skeleton of a polyhedral tropical space X (that is the union of $(\le k)$-dimensional faces) by ${\operatorname{Sk}}_k(X)$. It is often convenient to take the covering $\{U_\alpha\}$ by open stars of vertices. That is, each $U_v$ is the union of relative interiors of faces of $X$ adjacent to the vertex $v$. Then the relative interior of a face $\Delta$ is contained in every $U_v$ if $v$ is a vertex of $\Delta$.
Another useful feature of a compact polyhedral tropical space is that we can define its first baricentric subdivision. For a finite cell we take an arbitrary point in its interior for its baricenter.
![\[fig:bar\] Baricentric subdivision of an infinite cell. The dotted faces have higher sedentarity.](bar.pdf){height="30mm"}
For an infinite cell we take for its baricenter the baricenter of its unique most sedentary (necessarily finite, cf. Proposition \[prop:subface\]) subface (see Figure \[fig:bar\]). That is, we first choose baricenters of maximal sedentary faces and then name them also as baricenters of any adjacent faces of lower sedentarity. The subdivision of each face of $X$ into simplices is constructed as usual by the flags of its subfaces of [*minimal*]{} sedentarity.
The baricentric subdivision of $X$ is [*not*]{} a polyhedral tropical space as we defined it. It violates the regularity at infinity property. Nevertheless, it is very convenient to have a triangulation of $X$. This enables us to define simplicial versions of the (co)homology theories which are very useful for carrying out explicit calculations.
Combinatorial stratification {#sec:stratification}
----------------------------
Notice that a polyhedral structure on a tropical space (if it exists) is in no way unique. In this subsection we define a combinatorial stratification which is not always polyhedral, but is naturally defined on any tropical space $X$.
We say that two points $x,x'\in X$ are [*combinatorially equivalent*]{} if there exists a path connecting $x$ to $x'$ along which both the dimension of the wave tangent space $W$ and the sedentarity remain constant. [*A combinatorial stratum*]{} of the tropical space $X$ is a class of combinatorial equivalence.
We will denote combinatorial strata of $X$ by ${\mathcal E}$ and use the notation ${\mathcal E}\prec {\mathcal E}'$ if the stratum ${\mathcal E}$ lies on the boundary of ${\mathcal E}'$.
\[eg:elliptic\] Consider the circle $E_l$ of length $l$, otherwise called a [*tropical elliptic curve*]{}. $E_l$ is a tropical space: we can present it as a tropical polyhedral space by choosing, e.g., three distinct points so that they split $E_l$ into three facets. This subdivision is not unique as we can move these points around or consider a subdivision into a larger number of facets. The combinatorial stratification for $E_l$ is trivial: it consists of a single stratum $E_l$.
Let two points $x,y \in U_\alpha\subset X$ belong to one chart $\phi: U_\alpha \to Y$ of $X$ and they sit in some strata $x\in {\mathcal E}_x$ and $y\in {\mathcal E}_y$. If ${\mathcal E}_x={\mathcal E}_y$, that is if they belong to the same stratum, one can canonically identify the tangent spaces $T (y) = T (x)$ and $W(x) = W(y)$. The identification is natural in the following sense. If the points also belong to another common covering open subset $U_\beta$ it commutes with the differentials induced by the overlapping map.
In other words, we get flat connections on the bundles $T$ and $W$ over each combinatorial stratum of $X$.
Furthermore, if ${\mathcal E}_x\prec {\mathcal E}_y$ then one has two natural maps $$\label{eq:strata_maps}
\iota: T_{\mathbb Z}(y) \to T_{\mathbb Z}(x) \text{ and } \pi : W_{\mathbb Z}(x) \to W_{\mathbb Z}(y),$$ (note the different directions) defined as follows. If $I(\phi(y))=I(\phi(x))$ then any face adjacent to $\phi(x)$ is contained in some face adjacent to $\phi(y)$ and $\iota$ is given by inclusion. If $I(\phi(y))\neq I(\phi(x))$ (note that we must have $I(\phi(y))\subset I(\phi(x))$) then $\iota$ is the projection along the divisorial directions indexed by $I(\phi(x)){\smallsetminus}I(\phi(y))$. The map $\pi$ is given by inclusion of the linear spaces spanned by the corresponding parent faces.
Again the maps $\iota$ and $\pi$ are natural in the sense that they commute with the overlapping differentials.
Tropical manifolds
------------------
First we recall a construction of a balanced polyhedral fan associated to a matroid ([@AK], see also e.g. [@Shaw], [@MR]).
A matroid $M=(M,r)$ is a finite set $M$ together with a rank function $r:2^M\to{\mathbb Z}_{\ge0}$ such that we have the inequalities $r(A\cup B)+r(A\cap B) \le r(A) + r(B)$ and $r(A)\le |A|$, where $|A|$ is the number of elements in $A$, for any subsets $A,B\subset M$ as well as the inequality $r(A)\le r(B)$ whenever $A\subset B$. Subsets $F\subset M$ such that $r(A)>r(F)$ for any $A\supset F$ are called [*flats*]{} of $M$ of rank $r(F)$. Matroid $M$ is [*loopless*]{} if $r(A)=0$ implies $A=\emptyset$.
The so-called [*Bergman fan*]{} of a loopless matroid $M$ is a polyhedral fan $\Sigma_M\subset {\mathbb R}^{|M|-1}$ constructed as follows. Choose $|M|$ integer vectors $e_j\subset{\mathbb Z}^{|M|-1}\subset{\mathbb R}^{|M|-1}$, $j\in M$ such that $\sum\limits_{j\in M} e_j=0$ and any $|M|-1$ of these vectors form a basis of ${\mathbb Z}^{|M|-1}$. To any flat $F\subset M$ we associate a vector $$e_F:=\sum\limits_{j\in F} e_j\in{\mathbb R}^{|M|-1}.$$ E.g, $e_M=e_\emptyset=0$, but $e_F\neq 0$ for any other (proper) flat $F$. To any flag of flats $F_{i_1}\subset\dots\subset F_{i_k}$ we associate a convex cone generated by $e_{F_{i_j}}$. We define $\Sigma_M$ to be the union of such cones, which is, clearly, an $(r(M)-1)$-dimensional integral simplicial fan. It is easy to check (cf. [@AK]) that it satisfies the balancing condition, so that $\Sigma_M$ is a tropical space, called the [*Bergman fan*]{} of $M$.
The matroid $M$ is called [*uniform*]{} if $r(A)=|A|$ for any $A\subset M$. Note that the Bergman fan of a uniform matroid is a complete unimodular fan in ${\mathbb R}^{|M|-1}$ with $|M|$ maximal cones.
A tropical space $X$ is called [*smooth*]{}, or a [*tropical manifold*]{}, if all its charts $\phi_\alpha$ are open embeddings to $Y_\alpha = \Sigma _M \times {\mathbb T}^{s}\subset{\mathbb T}^{|M|-1}\times{\mathbb T}^s$ for some loopless matroid $M$ and a number $s\ge 0$. (Here $s$ is the maximal sedentarity in this chart and $n=r(M)-1+s$ is the dimension of our tropical manifold $X$.)
Tropical manifolds can be thought of as tropical spaces without points of multiplicity greater than 1, see [@MR], thus we use the term [*smooth*]{}. Note that smoothness is a property of the tropical space $(X,{\mathcal O}_X)$ alone, it does not involve presentation of $X$ as a polyhedral complex.
Homology groups
===============
Singular tropical homology
--------------------------
Let $x\in X$ be a point in a tropical space. Choose a sufficiently small open set $U\ni x$ and an embedding $\phi :U \to Y \subset {\mathbb T}^N$. Then for points $y$ such that $\phi(y)$ lies in an adjacent face to $\phi(x)$ we have a natural map between lattices in the tangent spaces $\iota: T_{\mathbb Z}(y) \to T_{\mathbb Z}(x)$, cf. .
\[def-fk\] The group ${\mathcal F}_k(x)$ is defined as the subgroup of the $k$th exterior power $\Lambda^k(T_{\mathbb Z}(x))$ generated by the products $\iota(v_1)\wedge\dots\wedge \iota(v_k)$ with $v_1,\dots,v_k\in T_{\mathbb Z}(y)$ for a point $y$ such that $\phi(y)$ lies in an adjacent face to $\phi(x)$ of the same sedentarity. It is important that all $k$ elements $v_j$ come from a single adjacent face. The group ${\mathcal F}^k(x)$ is defined as ${\operatorname{Hom}}({\mathcal F}_k(x),{\mathbb Z})$.
The discussion at the end of Section \[sec:stratification\] tells us that the groups ${\mathcal F}_k(x)$ and ${\mathcal F}_k(y)$ are canonically identified if $x$ and $y$ belong a single chart $U_\alpha$ and lie in a single stratum ${\mathcal E}$ of $X$. Furthermore, if for two points $x,y$, still in the same chart, we have ${\mathcal E}_x\succ{\mathcal E}_y$, then there are natural homomorphisms $$\label{cosheafmap}
\iota:{\mathcal F}_k(x)\to{\mathcal F}_k(y).$$
If three points $x,y,z\in U$ lie in the strata with incidence ${\mathcal E}_x\succ {\mathcal E}_y \succ {\mathcal E}_z$ then the three corresponding maps form a commutative diagram. In other words, if we consider the set of strata in the $U_\alpha \subset X$ as a category (under inclusions) then ${\mathcal F}_k$ forms a contravariant functor from strata of $U_\alpha$ to abelian groups (cf. Proposition \[functor-sheaf\]).
We may interpret our data as a system of coefficients suitable to define singular homology groups on $X$. Namely, we consider the finite formal sums $$\sum \beta_\sigma \sigma,$$ where each $\sigma:\Delta\to X$ is a singular $q$-simplex which has image in a single chart $U_\sigma$ and is such that for each relatively open face $\Delta'$ of $\Delta$ the image $\sigma(\Delta')$ is contained in a single combinatorial stratum ${\mathcal E}_{\Delta'}$ of $X$. Slightly abusing the notations we’ll identify the source and the image of $\sigma$ with the singular simplex $\sigma$ itself and say that $\tau=\sigma|_{\Delta'}$ is a face of $\sigma$. Here $\beta_\sigma\in {\mathcal F}_k({\mathcal E}_\Delta \cap U_\sigma)$.
These chains form a complex $C_\bullet(X; {\mathcal F}_k)$ with the differential ${\partial}$ given by the standard singular differential followed by the maps \[cosheafmap\]. We call such compatible singular chains with coefficients in ${\mathcal F}_k$ [*tropical chains*]{}. The groups $$H_{p,q}(X)=H_q(C_\bullet(X; {\mathcal F}_p), {\partial})$$ are called the [*tropical homology*]{} groups.
These homology groups is a version of singular homology groups of a topological space $X$ (after imposing the condition of compatibility of singular chains with the charts and combinatorial strata).
A priori the groups $H_{p,q}(X)$ depend on the covering. Indeed, if we refine the covering the tropical chains will be more restrictive. However the usual chain homotopy arguments apply and show that the resulting homology groups are canonically isomorphic. Thus we can conclude that the tropical homology groups are independent of the covering $\{U_\alpha\}$.
In case $X$ has a polyhedral structure one can require the singular chains to be compatible with the polyhedral face structure on $X$, rather than with its combinatorial structure. Clearly, the homology groups defines by the two complexes are canonically isomorphic. For polyhedral $X$ there are other equivalent ways for constructing tropical homology groups: simplicial, cellular. This is what we are going to consider next.
Cellular and simplicial tropical homology
-----------------------------------------
We assume $X$ is polyhedral and compact throughout this subsection. The main advantage of dealing with cellular and simplicial chain groups is that they are finitely generated. This will give an effective way to calculate the tropical homology.
Recall that $X$ comes with a subdivision into convex polyhedral domains. We define the cellular chain complex $$C^{cell}_q(X; {\mathcal F}_p)=\oplus {\mathcal F}_p(\Delta)=\oplus H_q(\Delta,{\partial}\Delta;{\mathcal F}_p(\Delta)).$$ Here the direct sum is taken over all $q$-dimensional faces $\Delta$ of the subdivision. The homology $H_q(\Delta,{\partial}\Delta;{\mathcal F}_p(\Delta))$ of the pair with constant coefficients equals ${\mathcal F}_p(\Delta)$ since each $q$-dimensional face $\Delta$ in $X$ is topologically a closed $q$-disk (recall that $X$ is compact).
Our next step is to define the boundary homomorphism ${\partial}: C^{cell}_q(X; {\mathcal F}_p)\to C^{cell}_{q-1}(X; {\mathcal F}_p)$. The ${\partial}$ is the composition of the maps $$\label{homcelld1}
H_q(\Delta,{\partial}\Delta; {\mathcal F}_p(\Delta))\to H_{q-1}({\partial}\Delta; {\mathcal F}_p(\Delta))\to
H_{q-1}({\partial}\Delta,{\partial}\Delta\cap{\operatorname{Sk}}_{q-2}(X);{\mathcal F}_p(\Delta)),$$ the isomorphism $$\label{isocelld}
H_{q-1}({\partial}\Delta,{\partial}\Delta\cap{\operatorname{Sk}}_{q-2}(X);{\mathcal F}_p(\Delta))
\to
\oplusH_{q-1}(\Delta',{\partial}\Delta';{\mathcal F}_p(\Delta)),$$ where the direct sum is taken over all $(q-1)$-dimensional subfaces $\Delta'\prec\Delta$, and $$\label{homcelld2}
\oplus H_{q-1}(\Delta',{\partial}\Delta';{\mathcal F}_p(\Delta)) \to
\oplus H_{q-1}(\Delta',{\partial}\Delta';{\mathcal F}_p(\Delta')).$$ In the first homomorphism is the boundary homomorphism of the pair $(\Delta,{\partial}\Delta)$ and the second one is induced by the inclusion of the pairs $(\Delta,\emptyset)\subset (\Delta,{\partial}\Delta)$. The isomorphism comes from the excision as the quotient space ${\partial}\Delta/({\partial}\Delta\cap{\operatorname{Sk}}_{q-2}(X))$ is homeomorphic to a bouquet of $(q-1)$-dimensional spheres, one sphere for each $(q-1)$-dimensional subface $\Delta'\prec\Delta$. Finally, the homomorphism is induced by .
The homology groups of the cellular chain complex $(C^{cell}_\bullet(X; {\mathcal F}_p), \partial)$ are called the cellular tropical homology groups $H^{cell}_\bullet(X; {\mathcal F}_p)$. If one has $X$ covered by the open stars of vertices we have the following identification.
The cellular tropical homology groups $H^{cell}_\bullet(X; {\mathcal F}_p)$ are canonically isomorphic to the (singular) tropical homology groups $H_\bullet(X; {\mathcal F}_p)$.
As in algebraic topology with constant coefficients to prove this isomorphism we need to use cellular homotopy. Let us recall that by the cellular homotopy argument the inclusion ${\operatorname{Sk}}_q(X)\to X$ induces an epimorphism $$\label{skq}
H_j({\operatorname{Sk}}_q(X);{\mathcal F}_p)\to H_j(X;{\mathcal F}_p)$$ for $j\le q$ (which is an isomorphism for $j< q$). Note that even though ${\mathcal F}_p$ is not a constant coefficient system, all cellular homotopy takes place within a single cell, so the classical argument also holds here.
Consider the homomorphism (in singular homology groups) induced by the inclusion of pairs $({\operatorname{Sk}}_q(X),\emptyset)\subset ({\operatorname{Sk}}(X),{\operatorname{Sk}}_{q-1}(X))$ $$H_q (X; {\mathcal F}_p)\to H_q ({\operatorname{Sk}}_q(X),{\operatorname{Sk}}_{q-1}(X);{\mathcal F}_p)=C^{cell}_q(X; {\mathcal F}_p).$$ Its image consists of cycles by the construction of the boundary map in the short exact sequence of the pair and thus it gives us a homomorphism $$\label{skq-cell}
H_q ({\operatorname{Sk}}_q(X); {\mathcal F}_p)\to H^{cell}_q (X; {\mathcal F}_p).$$
Note that by cellular homotopy the kernel of coincides with the kernel of for $j=q$. To see surjectivity of we consider an element $c\in H^{cell}_q(X; {\mathcal F}_p)$. Subdividing the faces of $X$ into simplices if needed we may represent $c$ by a singular chain in $C_\bullet({\operatorname{Sk}}_q(X);{\mathcal F}_p)$, whose boundary ${\partial}c$ is null-homologous in $$C^{cell}_{q-1}({\operatorname{Sk}}_{q-1}(X), {\operatorname{Sk}}_{q-2}(X);{\mathcal F}_p).$$ But $H_{q-1}({\operatorname{Sk}}_{q-2}(X); {\mathcal F}_p)=0$ by the dimensional reason and thus ${\partial}c$ must also vanish in $H^{cell}_{q-1}({\operatorname{Sk}}_{q-1}(X); {\mathcal F}_p)$. Thus we may correct $c$ (by adding to it a singular chain in ${\operatorname{Sk}}_{q-1}(X)$ whose boundary coincides with ${\partial}c$) to make it a cycle in $C_\bullet(X;{\mathcal F}_p)$.
Next observation will be very useful when we define the cap product action by the wave class.
\[lem:divisible\] Let $\gamma= \sum \beta_\Delta \Delta$ be a cellular cycle in a compact tropical polyhedral space $X$. Then each $\beta_\Delta$ is divisible by the divisorial vectors of $\Delta$.
We only have to check this for infinite cells $\Delta$. Since $X$ is compact, $\Delta$ must have a boundary face $\Delta_q$ (of sedentarity one higher) for every divisorial direction $q$. But the coefficient of $\partial \gamma$ at $\Delta_q$ comes only from the projection of $\beta_\Delta$ along $q$.
There is a simplicial variant of the tropical homology arising from the first baricentric simplicial chains on $X$. (The baricentric subdivision of $X$ was described at the end of Section \[sec:polyhedral\]). Then we can consider the baricentric simplicial chain complex with coefficients in ${\mathcal F}_p$ as a subcomplex $C^{bar}_\bullet(X; {\mathcal F}_p)$ of $C_\bullet(X; {\mathcal F}_p)$.
Note that the cellular chain complex $C^{cell}_\bullet(X; {\mathcal F}_p)$ can be viewed as a subcomplex of $C^{bar}_\bullet(X; {\mathcal F}_p)$, where all coefficients on simplices of the same cell are taken equal. Applying the standard chain homotopy arguments for constant coefficients one can show that this inclusion $$C^{cell}_\bullet(X; {\mathcal F}_p) \hookrightarrow C^{bar}_\bullet(X; {\mathcal F}_p)$$ is again a quasi-isomorphism. This allows us to identify both baricentric simplicial and cellular homology with the tropical homology.
In [@IKMZ] it is shown that in the case when $X$ is a smooth projective tropical manifold that comes as the limit of a complex 1-parametric family the groups $H_{p,q}(X)$ can be obtained from the limiting mixed Hodge structure of the approximating family. In particular, we have the equality $$h^{p,q}(X_t)=\operatorname{rk}H_{p,q}(X),$$ for the Hodge numbers $h^{p,q}(X_t)$ of a generic fiber $X_t$ from the approximating family.
In Section \[section:konstruktor\] we will show that there is a fairly small subcomplex of $C^{bar}_\bullet(X; {\mathcal F}_p)$, called konstruktor, which suffices to calculate the homology groups $H_{p,q}(X)$ in the smooth projective realizable case.
Tropical cohomology groups
--------------------------
Finally we define tropical [*cochains*]{} $C^\bullet(X; {\mathcal F}^p)$ to be certain linear functionals on charts/strata compatible ${\mathbb Z}$-singular chains with values in $\bigoplus_{\alpha, {\mathcal E}} {\mathcal F}^p({\mathcal E}\cap U_\alpha)$. Namely, if a simplex $\sigma$ lies in ${\mathcal E}\cap U_\alpha$ then we require the value of the cochain to lie in ${\mathcal F}^p({\mathcal E}\cap U_\alpha)$. If $\sigma$ also lies in $U_\beta$, then its value in ${\mathcal F}^p({\mathcal E}\cap U_\beta)$ should coincide with its value in ${\mathcal F}^p({\mathcal E}\cap U_\alpha)$ via the differential of the overlapping map.
Then one can define the differential as the usual coboundary followed by the maps dual to $$\delta \alpha (\sigma) = \alpha ({\partial}\sigma) \in \bigoplus_{\tau\subset \sigma} {\mathcal F}^p(\Delta_\tau) \rightarrow {\mathcal F}^p(\Delta_\sigma).$$ We can define the [*tropical cohomology*]{} groups $$H^{p,q}(X)=H^q(C^\bullet(X; {\mathcal F}^p), \delta).$$
Sheaf/cosheaf (co)homology
--------------------------
To make connections with sheaf (co)homology theories we use the coefficient systems ${\mathcal F}_p$ to define a constructible cosheaf with respect to the combinatorial stratification of $X$. With a slight abuse of notations we denote this cosheaf also by ${\mathcal F}_p$. A cosheaf is a suitable notion to take homology, just like sheaf for cohomology.
First we construct the pre-cosheaf in each open chart $U_\alpha$. Given an open set $U\subset U_\alpha$ we consider the poset formed by the connected components of intersections of the strata of $U_\alpha$ with $U$. The order is given by adjacency. This poset can be represented by a quiver (oriented graph) $\Gamma(U)$. Each vertex $v\in\Gamma(U)$ corresponds to a connected component of the intersection $U\cap {\mathcal E}$ of the open set $U$ and a stratum ${\mathcal E}$ of $U_\alpha$. A single stratum can produce several vertices in $\Gamma(U)$, see Figure \[cosheaf-explanation\].
To each vertex $v$ we associate the coefficient group ${\mathcal F}_p(v)={\mathcal F}_p({\mathcal E})$. To an arrow from $v$ to $w$ we associate the relevant homomorphism $i_{vw}: {\mathcal F}_p(v)\to {\mathcal F}_p(w)$ from . The groups ${\mathcal F}_p(v)$ with maps $i_{vw}$ thus form a representation of the quiver $\Gamma(U)$.
![\[cosheaf-explanation\] An open set in a polyhedral complex and the corresponding quiver. Here ${\mathcal F}_1(U) \cong {\mathbb Z}^4$.](quiver2.pdf){height="50mm"}
\[def-fu\] ${\mathcal F}_p(U)$ is the quotient of the direct sum $\bigoplus_{v\in\Gamma(U)} {\mathcal F}_p(v)$ by the subgroup generated by the elements $a - i_{vw}(a)$ for all pairs of connected vertices $(v,w)$, and all $a\in {\mathcal F}_p(v)$.
Note that an inclusion $U\subset V\subset U_\alpha$ induces a morphism between the corresponding quivers $\Gamma(U)\to\Gamma(V)$ with isomorphisms at the corresponding vertices. This map clearly preserves the equivalence relation, and hence descends to the map ${\mathcal F}_p(U) \to {\mathcal F}_p(V)$. Thus, we get a covariant functor from the open sets $U\subset U_\alpha$ (with morphism given by inclusions) to free abelian groups $U\mapsto {\mathcal F}_p(U)$. It is easy to check that all sequences $$\label{exact:cosheaf}
\bigoplus_{i,j} {\mathcal F}_p(U_i \cap U_j) \to \bigoplus_i {\mathcal F}_p(U_i) \to {\mathcal F}_p(U) \to 0,$$ where $U=\bigcup U_i$, are exact. Thus the functor $U\mapsto{\mathcal F}_p(U)$ is a cosheaf (cf., e.g. [@Bredon]) on the open set $U_\alpha$.
To define the sheaf ${\mathcal F}^p$ we need a contravariant functor $U\mapsto{\mathcal F}^p(U)$. Let $\Gamma(U)$ to be the directed graph as before with all arrows reversed. We set ${\mathcal F}^p(U)$ to be the subgroups of $\bigoplus_{v\in\Gamma(U)} {\mathcal F}^p(v)$, where the collections of elements $\{a_v\in {\mathcal F}^p(v)\}$ are compatible with all the morphisms dual to . Note that these collections are precisely the ones annihilated by the elements $a- i_{vw}(a)$ from the Definition \[def-fu\], and thus $F^p(U)={\operatorname{Hom}}({\mathcal F}_p(U), {\mathbb Z})$. Dualizing the exact sequences we see that the functor $U\mapsto{\mathcal F}^p(U)$ is a (constructible) sheaf on $U_\alpha$.
Finally we can glue together the sheaves and cosheaves defined on all open charts $U_\alpha$ (see, e.g., [@hartshorne], Ch. II, Exer. 1.22, for the sheaf version). We get a well defined cosheaf ${\mathcal F}_k$ and sheaf ${\mathcal F}^k$ on $X$ as long as we have the isomorphisms $\psi_{\alpha\beta}$ between the charts which satisfy $\psi_{\alpha\beta}\circ \psi_{\beta\gamma} = \psi_{\alpha\gamma}$ (see Proposition \[prop:differentials\]).
The combinatorial strata of $X$ form a category. Its objects are the strata themselves. There is a unique morphism from ${\mathcal E}$ to ${\mathcal E}'$ if ${\mathcal E}\prec {\mathcal E}$, and no morphisms otherwise. Our reasoning above can be formalized into the following general statement.
\[functor-sheaf\] Suppose $X$ has an open covering $\{U_\alpha\}$ and covariant functors ${\mathcal F}_\alpha$ for each ${\mathcal U}_\alpha$ from the combinatorial strata of ${\mathcal U}_\alpha$ to abelian groups which are compatible on the overlaps in the sense of Proposition \[prop:differentials\]. Then gluing gives rise to a constructible sheaf on $X$. Contravariant functors yields a constructible cosheaf on $X$.
If the functors ${\mathcal F}_\alpha$ behave naturally with respect to refinements of the covering $\{U_\alpha\}$ the resulting (co)sheaf ${\mathcal F}$ does not depend on the covering.
Finally we can use the sheaf-theoretic or Čech homology and cohomology for cosheaves ${\mathcal F}_p$ and sheaves ${\mathcal F}^p$. The standard algebraic topology techniques identify all these homology theories with the tropical (co)homology.
There are natural isomorphisms $$H_{p,q}\cong H_q(X, {\mathcal F}_p) \quad \text{and} \quad H^{p,q}\cong H^q(X, {\mathcal F}^p),$$ where on the right hand side are the sheaf-theoretic (co)homology groups.
Tropical waves
==============
Waves and cowaves
-----------------
There is also another collection of sheaves and cosheaves that can be associated to a tropical space $X$. Recall that for every point $x\in X$ we defined the wave tangent spaces $W(x)$ in Section \[sec:tangent\].
\[def-wk\] We define $W_k(x)$ as the exterior power $\Lambda^k W(x)$. We also consider the dual vector space $W^k(x)$.
In any given chart $U_\alpha$ the $W(x)$ can be canonically identified for points in a single stratum ${\mathcal E}$. Thus we may write $W_k({\mathcal E}\cap U_\alpha)=W_k(x)$ for any point $x\in {\mathcal E}\cap U_\alpha$. For a pair ${\mathcal E}\prec{\mathcal E}'$ of two adjacent strata the map induces the natural homomorphisms $$\label{Warrow}
\pi: W_k({\mathcal E})\to W_k({\mathcal E}') \ \ \ \text{and}\ \ \ \hat\pi: W^k({\mathcal E}')\to W^k({\mathcal E}).$$ By Proposition \[functor-sheaf\] the coefficient system $W_k$ defines a constructible sheaf ${\mathcal W}_k$ on $X$, whereas the $W^k$ defines a cosheaf ${\mathcal W}^k$ for every integer $k\ge 0$.
Tropical [*wave*]{} and [*cowave*]{} groups, respectively, are $$\label{tropwave-defn}
H^q(X; {\mathcal W}_k)
\ \ \ \text{and}\ \ \
H_q(X; {\mathcal W}^k).$$
Again, we can think of these groups from the sheaf-theoretic point of view or as stratum-compatible singular (co)homology with coefficients in the systems $W_k$ and $W^k$.
\[eg:nodal\] Let us consider a tropical genus 2 curve $C$ with a simple double point. The underlying topological space of $C$ is a wedge of two circle, i.e. it is a graph with a single vertex $v$ and two edges that are glued to $v$, see Figure \[fig:genus2singular\].
The tropical structure in the interior of each edge is isomorphic to an open interval of finite length in ${\mathbb R}$ (treated as the tropical torus ${\mathbb T}^\times={\mathbb T}{\smallsetminus}\{-\infty\}$).
![Nodal genus 2 curve.[]{data-label="fig:genus2singular"}](genus2singular){width="2in"}
The tropical structure at the vertex $v$ is such that the four primitive vectors divide into 2 pairs of opposite vectors. This means that the chart at $v$ is given by a map to ${\mathbb R}^2$ such that a neighborhood of $v$ in $C$ goes to the union of coordinate axes and the four primitive vectors near $v$ go to the unit tangent vectors to those axes.
Thus, ${\mathcal F}_1(v)={\mathbb Z}^2$ and $W_1(v)=0$. On the other hand every point $x$ in the interior of either edge has the groups ${\mathcal F}_1(x)={\mathbb Z}$ and $W_1(x)={\mathbb R}$. The group ${\mathcal F}_0(x)$ is always ${\mathbb Z}$ and $W_0(x)={\mathbb R}$ for any point $x$. From the two term cell complex one can easily calculate $$H_0(C; {\mathcal F}_0)\cong {\mathbb Z}, \quad H_0(C; {\mathcal F}_1)\cong {\mathbb Z}, \quad H_1(C; {\mathcal F}_0)\cong {\mathbb Z}^2 ,\quad H_1(C;{\mathcal F}_1)\cong {\mathbb Z},$$ and $$H^0(C; {\mathcal W}_0)\cong {\mathbb R}, \quad H^0 (C; {\mathcal W}_1)=0, \quad H^1(C; {\mathcal W}_0)\cong {\mathbb R}^2, \quad H^1 (C; {\mathcal W}_1) \cong {\mathbb R}^2.$$
In general, ${\mathcal F}_0$ and ${\mathcal W}_0$ are constants, thus for $p=0$ we recover the ordinary topological homology and cohomology groups.
We have $H_{0,q} =H_q(X;{\mathbb Z})$, $H_{q}(X;{\mathcal W}^0)= H_q(X;{\mathbb R})$, $H^{0,q}=H^q(X;{\mathbb Z})$, $ H^{q}(X;{\mathcal W}_0)= H^q(X;{\mathbb R})$.
Pairing of ${\mathcal F}$ and ${\mathcal W}$
--------------------------------------------
The importance of the wave classes stems from their action on the tropical homology via a natural bilinear map $$\cap: H^r (X; {\mathcal W}_k) \otimes H_{q}(X; {\mathcal F}_p \otimes {\mathbb R}) \to H_{q-r}(X; {\mathcal F}_{p+k} \otimes {\mathbb R})$$ which we are going to define now. On the chain level this map is just the standard cap product between singular chains and cochains coupled with the wedge product on the coefficients $\wedge: W_k \otimes {\mathcal F}_p \to {\mathcal F}_{p+k} \otimes {\mathbb R}$.
Let us clarify the meaning of the wedge multiplication. For a mobile point $x\in X$ the wave tangent space $W(x)$ is naturally a subspace in $T(x)$ hence the product makes sense on the nose. For any sedentary point $x$ the wave space $W(x)$ naturally projects (along the divisorial directions) to $W'(x)$, which is a subspace of $T(x)$. When taking the wedge product we first apply this projection.
In details, let $\alpha$ be a compatible $r$-cochain with coefficients in $W_k$ and $\gamma=\sum \beta\sigma$ be a tropical $q$-chain with coefficients in ${\mathcal F}_p$. For each singular simplex $\sigma$ we denote by $\sigma_{0\dots r}$ its first $r$-face (spanned by the first $r+1$ vertices of $\sigma$) and by $\sigma_{r\dots q}$ its last $(q-r)$-face. Then we set $$\label{eq:cap_product}
\alpha\cap \gamma = \sum (\alpha(\sigma_{0\dots r})\wedge \beta) \sigma_{r\dots q} .$$ Here we push the value of $\alpha$ at the face $\sigma_{0\dots r}$ to the simplex $\sigma$ with the sheaf map and then push the value of the result from $\sigma$ to $\sigma_{r\dots q}$ using the cosheaf map.
We will need the following local observation. Let us assume we live in a single chart $U_\alpha$.
\[lemma:wedge\] Let ${\mathcal E}'\prec {\mathcal E}$ be a pair of adjacent strata in $U_\alpha$. Then the diagram $$\xymatrix{
W_k({\mathcal E}) \otimes {\mathcal F}_p({\mathcal E}) \ar@<2pc>[d]^{\iota} \ar[r]^--{\wedge} & {\mathcal F}_{p+k}({\mathcal E}) \otimes {\mathbb R}\ar[d]^{\iota}\\
W_k({\mathcal E}') \ar@<2pc>[u]^{\pi} \otimes {\mathcal F}_p({\mathcal E}') \ar[r]^--{\wedge} & {\mathcal F}_{p+k}({\mathcal E}') \otimes {\mathbb R}}$$ is commutative in the sense that for any $\alpha\in W_k({\mathcal E}')$ and $\beta \in {\mathcal F}_p({\mathcal E})$ one has $\iota( \pi (\alpha) \wedge \beta)= \alpha \wedge \iota (\beta)$.
The wedge product is bilinear with respect to inclusion and quotient (in fact, all) homomorphisms between free abelian groups.
For each $r\le q$ the cap product descends to a natural bilinear map in homology $$\cap: H^r (X; {\mathcal W}_k) \otimes H_{q}(X; {\mathcal F}_p \otimes {\mathbb R}) \to H_{q-r}(X; {\mathcal F}_{p+k} \otimes {\mathbb R}).$$
The statement follows at once from the usual Leibnitz formula $$(-1)^r {\partial}(\alpha \cap \gamma)= (\delta \alpha) \cap \gamma + \alpha \cap {\partial}\gamma.$$ Note that the wedge products in $\delta(\alpha \cap \gamma)$ and $(\partial \alpha) \cap \gamma$ are taken in $\sigma$ and then pushed to ${\mathcal F}_{p+k} (\sigma_{r\dots \hat i \dots q})$. On the other hand the wedge products in $\alpha \cap \delta \gamma$ are taken in $\sigma_{0\dots \hat i \dots q}$ and then pushed to ${\mathcal F}_{p+k} (\sigma_{r\dots \hat i \dots q})$. But Lemma \[lemma:wedge\] allows us to identify the results.
The group $H^1(X;{\mathcal W}_1\otimes{\mathbb R})$ and deformations of the tropical structure of $X$
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In this section we assume that $X$ is compact. Recall that $X$ has a covering by charts $\phi_\alpha: U_\alpha \to Y_\alpha \subset {\mathbb T}^{N_\alpha}$. The transition maps on the overlaps are given by integral affine maps $\psi_{\alpha\beta}: {\mathbb T}^{N_\alpha} \dashrightarrow {\mathbb T}^{N_\beta}$.
As a topological space $X$ can be presented as the quotient of the disjoint union of its covering sets $\bigsqcup_\alpha (U_\alpha)$ by the following equivalence relation. We say two points $x\in U_\alpha$ and $y\in U_\beta$ are equivalent if $\psi_{\alpha\beta} \circ \phi_\alpha(x)=\phi_\beta(y)$. Reflexivity of equivalence says that $\psi_{\alpha\beta} = \psi_{\beta\alpha}^{-1}$ (as partially defined maps). Transitivity translates as the cocycle condition, or as the composition rule, $\psi_{\beta\gamma}\circ \psi_{\alpha\beta} = \psi_{\alpha\gamma}$.
Conversely, given open subsets $\phi_\alpha(U_\alpha)\subset Y_\alpha \subset {\mathbb T}^{N_\alpha}$ and a collection of integral affine maps $\psi_{\alpha\beta}$ satisfying $\psi_{\alpha\beta} = \psi_{\beta\alpha}^{-1}$ and $\psi_{\beta\gamma}\circ \psi_{\alpha\beta} = \psi_{\alpha\gamma}$ we can define a topological space $X$ as the quotient of $\bigsqcup_\alpha \phi_\alpha(U_\alpha)$ by the equivalence given by the $\psi$’s.
$X$ will be a tropical space provided all subsets $\phi_\alpha(U_\alpha)$ remain open in the quotient and $X$ satisfies the finite type condition. Moreover we will get an isomorphic tropical space if the $\psi_{\alpha\beta}$ are changed by a “coboundary” (twisted by automorphisms $\psi_\alpha: {\mathbb T}^{N_\alpha} \dashrightarrow {\mathbb T}^{N_\alpha}$ for some $\alpha$).
Let $\tau$ be a class in $H^1(X;{\mathcal W}_1)$. We can assume that the covering $\{U_\alpha\}$ is fine enough so that $\tau$ can be represented by a Čech 1-cocycle $\tau_{\alpha\beta}\in W (U_\alpha \cap U_\beta)$. We can also assume that all $U_\alpha$ and $U_\alpha\cap U_\beta$ are connected. Then $W (U_\alpha \cap U_\beta)$ consists of vectors parallel to all mobile strata in $U_\alpha \cap U_\beta$. We can think of $W (U_\alpha \cap U_\beta)$ as a subspace in ${\mathbb R}^{N_\alpha} \subset {\mathbb T}^{N_\alpha} $ via the map $\phi_\alpha$, or in ${\mathbb R}^{N_\beta} \subset {\mathbb T}^{N_\beta}$ via $\phi_\beta$.
By shrinking the $U_\alpha$ if necessary it will also be convenient to assume that slightly larger open subsets $V_\alpha\supset \overline{U_\alpha}$ do not contain any new strata other than those already in the $U_\alpha$. For instance if $X$ is polyhedral we can take $U_\alpha$ to be the open stars of vertices, “shrunk” a little bit.
Now for $\epsilon >0$ we modify the overlapping maps $\psi_{\alpha\beta}: {\mathbb T}^{N_\alpha} \dashrightarrow {\mathbb T}^{N_\beta}$ by precomposing them with the translation by $\epsilon\tau_{\alpha\beta}$. Since $\tau_{\alpha\beta}=-\tau_{\beta\alpha}$ the new relation is reflexive. Also since $\tau$ is a cocycle the new maps $\psi^{\epsilon\tau}_{\alpha\beta}$ satisfy the composition rule. Thus they define a new equivalence relation and we call the corresponding quotient space $X_{\epsilon\tau}$ the deformation of $X$.
For $\epsilon >0$, small enough, $X_{\epsilon\tau}$ is a tropical space.
We only need to show that $X_{\epsilon\tau}$ is of finite type and each of the $U_\alpha$ is still an open subset in $X_{\epsilon\tau}$. For the latter it is enough to show that each $U_\alpha\cap U_\beta$ is open. But this is clear since by condition that slight enlargements of $U_\alpha$ contain no new strata, no new strata can appear in $U_\alpha\cap U_\beta$ for small enough $\epsilon$. The argument for the finite type condition is similar.
The deformed tropical space is especially easy to visualize in the polyhedral case. Namely, $X_{\epsilon\tau}$ has the same combinatorial face structure, but the faces of $X_{\epsilon\alpha}$ themselves may have different shapes and sizes. E.g. if $X$ is 1-dimensional, the lengths of the edges of $X$ and $X_{\epsilon\alpha}$ may be different.
Straight classes
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Straight cycles in tropical homology
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We start with a natural generalization of balanced polyhedral complexes in ${\mathbb T}^N$ to a situation where a facet can have a weight. A weighted balanced polyhedral complex $Y\subset{\mathbb T}^N$ is a union of a finite number of facets $D$ as before, but now each $D$ is enhanced with an integer weight $w(D)\in{\mathbb Z}$ subject to the following weighted balancing condition for every $(n-1)$-dimensional mobile face $E\subset Y$. As in we consider all facets $D_1,\dots,D_l\subset {\mathbb T}^{N}$ adjacent to $E$ and take the quotient of ${\mathbb R}^{N}$ by the linear subspace parallel to $E^\circ$, the non-infinite part of $E$. The weighted balanced condition is $$\label{wbal-cond}
\sum\limits_{k=1}^l w(D_k)\epsilon_k=0.$$ We say that the weighted balanced complex $Y$ is [*effective*]{} if the weights of all its facets are positive.
Just as balanced polyhedral complexes form local models for tropical spaces, effective weighted balanced polyhedral complexes form models for [*weighted tropical spaces*]{}.
A weighted tropical space is a topological space $X$ enhanced with a weight function $w:X\dashrightarrow{\mathbb N}$ defined on an open dense set $A\subset X$ and a sheaf ${\mathcal O}_X$ of functions to ${\mathbb T}$ such that there exists a finite covering of compatible charts $\phi_\alpha: U_\alpha\to Y_\alpha \subset {\mathbb T}^{N_\alpha}$ with the following properties.
- $Y_\alpha$ is an effective weighted balanced polyhedral complex in ${\mathbb T}^{N_\alpha}$.
- For the relative interior $D^\circ$ of any facet $D\subset Y_\alpha$ we have $\phi_\alpha^{-1} (D^\circ)\subset A$ while the weight function $w$ is constant on $\phi_\alpha^{-1}(D^\circ)$ and equal to the weight of $D$.
- For each facet $D\subset Y_\alpha$ there exists a ${\mathbb Z}$-linear transformation $\Phi_D:{\mathbb Z}^{N_\alpha}\to {\mathbb Z}^{N_\alpha}$ of determinant $w(D)$ such that ${\mathcal O}_X|_{D^\circ\cap U_\alpha}$ is induced by $\Phi_D^{-1}\circ\phi_\alpha$.
We may reformulate the last condition of this definition by saying that each facet $D$ comes with a sublattice of index $w(D)$ of the tangent lattice $T_{{\mathbb Z}}(x)$, $x\in D$. This sublattice is locally constant and does not depend on the choice of charts. Note that not every weighted balanced polyhedral complex in ${\mathbb T}^N$ is a weighted tropical space in this sense as it is not always possible to consistently choose such a sublattice. However no such sublattice for the facets of $Z$ is needed for the following definition.
Let $X$ be a tropical space. A subspace $Z\subset X$ enhanced with a weight function $$w:Z\dashrightarrow{\mathbb Z}$$ defined on an open dense set $A\subset Z$ is called a [*straight tropical $p$-cycle*]{} if for every chart $\phi_\alpha:U_\alpha\to Y_\alpha\subset{\mathbb T}^{N_\alpha}$ of $X$ there exists a weighted $p$-dimensional balanced polyhedral complex $Z_\alpha\subset{\mathbb T}^{N_\alpha}$ such that $\phi_\alpha: Z\cap U_\alpha \to Z_\alpha$ is an open embedding, and for the relative interior $D^\circ$ of any facet $D\subset Z_\alpha$ we have $\phi_\alpha^{-1} (D^\circ)\subset A$. The weight function $w$ is constant on $\phi_\alpha^{-1}(D^\circ)$ and equal to the weight of $D$.
\[specialfundclass\] Each straight tropical $p$-cycle $Z\subset X$ gives rise to a canonical element $[Z]\in H_{p,p}(X)$ in the tropical homology group of $X$.
We choose a sufficiently fine (topological) triangulation of $Z=\bigcup \sigma$ so that each $p$-simplex $\sigma$ from the triangulation lies in a single chart $U_\alpha$ and in a single combinatorial stratum ${\mathcal E}$ of $X$. In particular each $\sigma$ carries the weight $w(\sigma)$ induced from $Z$. An orientation of $\sigma$ defines the canonical volume element ${\operatorname{Vol}}_\sigma\in{\mathcal F}^X_p(\sigma)$ given by the generator of $\Lambda^p(W^Z_{{\mathbb Z}}(\sigma))\cong{\mathbb Z}$. Inverting the orientation of $\sigma$ will simultaneously invert the sign of ${\operatorname{Vol}}_\sigma$. Thus the product ${\operatorname{Vol}}_\sigma \sigma$ is a well-defined tropical chain in $C_p (X;{\mathcal F}_p)$. Then the weighted balancing condition for $Z$ ensures that $$\gamma_Z= \sum_{\sigma\subset Z} w(\sigma) {\operatorname{Vol}}_\sigma \sigma$$ is a cycle in $C_p (X;{\mathcal F}_p)$. Its class is clearly independent of the triangulation and gives the desired element $[Z]\in H_p(X;{\mathcal F}_p)=H_{p,p}(X)$.
Elements of $H_{p,p}$ realised by straight tropical cycles as in Proposition \[specialfundclass\] are called straight homology classes (or, in other existing terminology, [*special*]{} or [*algebraic*]{}). They form a subgroup $$H_{p,p}^{straight}(X) \subset H_{p,p}(X).$$
Recall that the tropical $N$-dimensional projective space ${{\mathbb T}{\mathbb P}}^N$ may be obtained by gluing $N+1$ affine charts ${\mathbb T}^N$ with the help of integral affine maps, cf. e.g. [@MR]. A topological subspace $X\subset{{\mathbb T}{\mathbb P}}^N$ is called a [*projective tropical space*]{} if the intersection of $X$ with any such chart is a balanced polyhedral complex. A projective tropical space has a non-trivial straight homology class $$[H^X_p]\in H_{p,p}^{straight}(X)$$ (called the [*hyperplane section*]{}) in any dimension $p=0,\dots,n=\dim X$.
To see this we start from the case $X={{\mathbb T}{\mathbb P}}^N$. Consider the equations $x_{j}=c_j$, $j=p+1,\dots,n$, $c_j\in{\mathbb R}$, in a chart ${\mathbb T}^N\subset{{\mathbb T}{\mathbb P}}^{N}$. They define a $p$-dimensional linear space parallel to a coordinate plane. We may take for $H_p$ the topological closure of this linear space in ${{\mathbb T}{\mathbb P}}^N$. Clearly, the homology class $[H_p]$ does not depend on the choice of the ${\mathbb T}^N$-chart or on permutation of coordinates in this chart. Furthermore, $H_0$ is a point and thus $[H_0]\neq 0$ in $H_{0,0}({{\mathbb T}{\mathbb P}}^N)\cong {\mathbb Z}$. Note that this also implies that $[H_p]\neq 0$ in $H_{p,p}({{\mathbb T}{\mathbb P}}^N)$ as we may choose the transverse representatives $H_p$ and $H_{p'}$ so that $H_p\cap H_{p'}=H_{p+p'-N}$, cf. [@Shaw]. It is easy to show that any element of $H_{*,*}({{\mathbb T}{\mathbb P}}^N)$ is generated by $[H_p]$, $p=0,\dots,N$.
A similar construction can be made for general projective tropical spaces $X\subset {{\mathbb T}{\mathbb P}}^N$. We take $H^X_p=H_{N+p-n}\cap X$ where $H_{N+p-n}$ is chosen to be transverse to $X$ with the help of translations in ${\mathbb R}^N$. But in addition to those hyperplane sections and their powers $H_{*,*}(X)$ may have additional, more interesting, straight classes.
Straight cowaves
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A notion of straight classes exists also for cowaves. Once again, let $Z\subset X$ be a subspace such that each chart $\phi_\alpha$ takes $Z$ to a $q$-dimensional polyhedral complex in ${\mathbb T}^{N_\alpha}$ (which we no longer assume balanced). We refer to such subspace of $X$ as a [*straight subspace*]{}.
In this subsection we assume that $\dim W(x)=m$ for some $m$ almost everywhere on $Z$. In other words we assume that each open facet of $Z$ sits in the $m$-skeleton of $X$, but outside of the $(m-1)$-skeleton of $X$. We call such straight subspaces $Z$ [*purely $m$-skeletal*]{}. E.g. $Z$ is $n$-skeletal if no open facets of $Z$ intersect $\operatorname{Sk}_{n-1}(X)$.
A [*coweight function*]{} on $Z$ is a function $$x \mapsto cow(x)\in W^m(x)$$ defined on an open dense set $A\subset Z$. Here we assume that $\dim W(x)=m$ whenever $x\in A$, so we have $W^m(x)\approx{\mathbb Z}$.
This is a dual notion to the weight function. But while the weight function was integer-valued, here we do not have a canonical isomorphism between $W^m(x)$ and ${\mathbb Z}$, it is only canonical up to sign.
Let $x\in A$ be inside of a facet of $Z$ parallel to a $q$-dimensional affine space $L$ (in a chart $\phi_\alpha$). As in the previous subsection, we may consider the volume element ${\operatorname{Vol}}_L\in W_q(x)$ which is well-defined by the integer lattice in $L$ and a choice of orientation of $L$. Given this choice we have a well-defined map $$\lambda\mapsto cow(x)(\lambda\wedge{\operatorname{Vol}}_L),$$ $\lambda\in W_{m-q}(x)$ and thus an element in $W^{m-q}(x)$, a group that depends only on the open facet of $Z$ containing $x$. Thus any $q$-simplex $\sigma$ embedded to the same facet and parallel to $L$ defines a canonical chain with coefficients in $W^{m-q}(x)$. In particular, a triangulation of a coweighted purely $m$-skeletal $q$-dimensional polyhedral pseudocomplex $Y$ gives rise to a cowave chain in $C_q(X;{\mathcal W}^{m-q})$. Such cowave chains are called [*straight*]{}.
As in Proposition \[specialfundclass\] we may associate a singular chain with the coefficients in $W^{m-q}$ to $Z$ by using a combinatorial stratification of $Z$.
A coweighted straight subspace $Z\subset X$ is called [*cobalanced*]{} if the resulting chain is a cycle. We may refine this into a local notion by saying that $Z$ is cobalanced at $x\in Z$ if $x$ is disjoint from the support of the boundary of the resulting special cowave cochain.
Note that once an orientation of $W(x)$ is chosen we may identify coweight and weight at $x$.
Suppose that a $q$-dimensional coweighted straight subspace $Z$ is purely $m$-skeletal and that $x\in Z$ belongs to a relative interior of a $(q-1)$-dimensional face (in a chart) with $\dim W(x)=m$. Then $Z$ is cobalanced at $x$ if and only if $Z$ is balanced at $x$.
Note that $x$ must belong to the same combinatorial stratum of $X$ as its small open neighbourhood in $Z$, since $Z$ is purely $m$-skeletal and $\dim W(x)=m$. Thus ${\mathcal W}^m|_Z$ is locally trivial near $x$ and we may translate coweights into weights simultaneously for the whole neighbourhood with the help of an arbitrary orientation of $W(x)$.
At the same time if $\dim W(x)<m$ then the cobalancing condition is different from the balancing condition. We believe that study of straight cowaves might be useful, particularly in the context of mirror symmetry.
The eigenwave
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The eigenwave $\phi$
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There is a canonical element $\phi\in H^1(X;{\mathcal W}_1)$ for every compact tropical space $X$. Unfortunately, it does not have a preferred representative as a singular wave cocycle in the case when $X$ has points of positive sedentarity. Rather we shall represent it in the quotient space $C^1(X;{\mathcal W}_1)/B_{div}^1(X;{\mathcal W}_1)$ where the subspace $B_{div}^1(X;{\mathcal W}_1) \subset C^1(X;{\mathcal W}_1)$ will consist of certain coboundaries. Note that this ambiguity will not cause us any problems with the cap product of $\phi$ and the homology cycles because, as we will see, taking product with elements in $B_{div}^1(X;{\mathcal W}_1)$ annihilates any singular cycle.
Let $C_{div}^0(X;{\mathcal W}_1) \subset C^0(X;{\mathcal W}_1)$ be the subspace of 0-wave cochains whose values on points $x\in X$ are in $W^{div}(x)$. We let $B_{div}^1(X;{\mathcal W}_1) \subset C^1(X;{\mathcal W}_1)$ consist of the coboundaries of the cochains from $C_{div}^0(X;{\mathcal W}_1)$. The elements $\gamma \in B_{div}^1(X;{\mathcal W}_1)$ are characterized by the property that on any singular 1-simplex $\tau$ the values $\gamma(\tau)$ belong to the subspace $W^{div}(\tau) \subset W(\tau)$ spanned by the divisorial subspaces at the boundary points of $\tau$.
We are ready to define the eigenwave class $\phi \in C^1(X;{\mathcal W}_1)/B_{div}^1(X;{\mathcal W}_1)$. Let us first consider the case when all points of $X$ have zero sedentarity, in particular $B_{div}^1(X;{\mathcal W}_1)=0$. In such case we define the value of $\phi$ on a singular 1-simplex $\tau:[0,1]\to X$ as $\tau(1)-\tau(0)$. Recall that our singular chains are assumed to be compatible with the combinatorial stratification of $X$ so that $\tau((0,1))$ is contained in a single combinatorial stratum and a single tropical chart. This means that the difference $\tau(1)-\tau(0)$ can be interpreted as a vector in the tangent space to this stratum and therefore in $W(\tau)$.
Returning to the general case, if $x\in X$ is of positive sedentarity we choose a nearby mobile point $y_x$ which maps to $x$ under the projection along divisorial directions. If $x\in X$ is mobile we set $y_x=x$.
The element $\phi \in C^1(X;{\mathcal W}_1)/B_{div}^1(X;{\mathcal W}_1)$ is defined on a 1-simplex $\tau:[0,1]\to X$ as the vector $w_\tau:= y_{\tau(1)}-y_{\tau(0)} \in W(\tau)$.
Clearly the ambiguity in $w_\tau$ resulting from different choices of $y_x$ is confined to $B_{div}^1(X;{\mathcal W}_1)$. The next proposition asserts that $\phi$ defines a class in $ H^1(X;{\mathcal W}_1)$, which we call the [*eigenwave*]{} of $X$. We denote this class also by $\phi$, this should not cause any confusion.
$\delta \phi =0$.
By definition the value of $\delta\phi$ on a 2-simplex $\sigma$ is the sum of the values of $\phi$ on the three edges $\tau_1,\tau_2,\tau_3$ of $\sigma$. This is clearly zero (perhaps after applying the maps $\pi: W(\tau) \to W(\sigma)$ in case some of the $\tau_i$ land in different strata).
Action of the eigenwave $\phi$ and its powers on tropical homology {#subsection:wave_action}
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The $k$-th cup powers of $\phi$ are also (higher degree) wave classes $\phi^k\in H^k(X;{\mathcal W}_k)$. One can define the value of $\phi^k$ on a $k$-simplex $\sigma$ modulo the ideal in $W_k(\sigma)$ generated by the $W^{div}$ for all vertices in $\sigma$. Namely, for an edge $\tau\prec \sigma$ let $w_\tau \in W(\sigma)$ stand for the vector $y_{\tau(1)}-y_{\tau(0)}\in W(\tau)$ pushed to $W(\sigma)$. Then $$\label{eq:phi^k}
\phi^k (\sigma)=w_{\sigma_{01}}\wedge\dots \wedge w_{\sigma_{k-1, k}} =: w_\sigma \in W_k (\sigma).$$ Taking the cap product with $\phi^k=[\phi^k_{sing}]$ gives us the homomorphism: $$\label{eq:wave_action}
\phi^k \cap: H_q(X;{\mathcal F}_p \otimes {\mathbb R}) \to H_{q-k}(X;{\mathcal F}_{p+k} \otimes {\mathbb R}).$$
In case $X$ is compact and polyhedral we consider its baricentric subdivision and think of the $H_q(X; {\mathcal F}_p)$ as simplicial or cellular homology groups. The advantage is that we can define the cap product with $\phi^k$ on the cycle level $$\label{eq:wave_action_cycle}
\phi^k: C^{cell}_q({\mathcal F}_p) \to C^{bar}_{q-k}({\mathcal F}_{p+k}\otimes {\mathbb R}).$$ Below we give two different descriptions of the map depending on the choice of vertex ordering. The first result is a cycle in $C^{bar}_{q-k}({\mathcal F}_{p+k}\otimes {\mathbb R})$ while the second one is still in $C^{cell}_{q-k}({\mathcal F}_{p+k}\otimes {\mathbb R}) \subset C^{bar}_{q-k}({\mathcal F}_{p+k}\otimes {\mathbb R})$.
We recall the notion of the dual cells in the first baricentric subdivision of a polyhedral complex. Let $\Delta\in X$ be a $q$-cell. For any [*finite*]{} $j$-dimensional face $\Delta' \prec\Delta$ of the sedentarity $s(\Delta')=s(\Delta)$ its [*dual cell*]{} $\hat\Delta'_\Delta$ in the baricentric subdivision of $\Delta$ is defined as the union of all $(q-j)$-simplices in $bar(\Delta)$ containing the baricenters of $\Delta$ and $\Delta'$. We can think of $\hat \Delta'_\Delta$ as a simplicial $(q-j)$-chain. The orientations of the pair $\Delta'$ and $\hat \Delta'_\Delta$ are taken to agree with the original orientation of $\Delta$.
Let $\gamma= \sum \beta_\Delta \Delta$ be a cycle in $C^{cell}_q(X;{\mathcal F}_p)$. Then according to Lemma \[lem:divisible\] the coefficients $\beta_\Delta$ for all $\Delta \subset X$ have to be divisible by the divisorial directions of $\Delta$. In particular, the wedge product of $\beta_\Delta$ with any element in $\wedge^k (W(\Delta)/W^{div}(\Delta))$ gives a well-defined element in ${\mathcal F}_{p+k}(\Delta)\otimes {\mathbb R}$. We can also think of $\gamma=\sum_{\sigma\in bar (\Delta)} \beta_\Delta \sigma$ as an element in $C^{bar}_q(X;{\mathcal F}_p)$.
[**Description 1:**]{} We label the vertices of each $q$-simplex $\sigma$ in $bar(\Delta)$ according to the dimension of the largest cells whose baricenters they represent (recall that several faces of $\Delta$ of different sedentarity may have the same baricenter). In this case the cycle $\phi^k \cap \gamma \in C^{bar}_{q-k}(X;{\mathcal F}_{p+k})$ is supported on the dual subdivision inside the $q$-skeleton of $X$.
Precisely, for every $k$-face $\Delta'$ of $\Delta$ let $w_{\Delta'}\in W_k(\Delta)$ denote the volume element associated to $\Delta'$ as in . Clearly, $w_{\Delta'}$ equals the sum of all $w_{\sigma_{0\dots k}}$ (taken with appropriate signs) for the $k$-simplices $\sigma_{0\dots k}$ forming the baricentric triangulation of $\Delta'$. Then one can easily calculate from the definition of the cap product: $$\label{eq:wave_action1}
\phi^k \cap (\sum_{\sigma\in bar (\Delta)} \beta_\Delta \sigma) = \sum_{\Delta' \prec\Delta} (w_{\Delta'}\wedge\beta_\Delta) \hat \Delta'_\Delta ,$$ where the sum is taken over all $k$-dimensional faces of $\Delta$. Note that higher sedentary $k$-faces don’t appear in the sum because $\beta_\Delta$ vanishes when pushed to these higher sedentary faces.
![The two descriptions of the wave action on a 2-cell $\sigma$. The support of $\phi_{sing} \cap \sigma$ is red and the framing is blue.[]{data-label="fig:wave_action"}](wave_action.pdf){height="28mm"}
[**Description 2:**]{} Here we label the vertices of each $\sigma$ in the opposite order to the description 1. That is the baricenters with the smaller numbers correspond to the larger faces. Now the cycle $\phi^k \cap \gamma\in C^{bar}_{q-k}(X;{\mathcal F}_{p+k})$ is supported on the $(q-k)$-skeleton of $X$.
Precisely, for every $(q-k)$-face $\Delta'$ of $\Delta$ let $\hat w_{\Delta'}\in W_k(\Delta)$ denote the polyvector corresponding to the integration along the chain $\hat \Delta'_\Delta$. Note that the faces $\sigma_{k\dots q}$ lie in the $(q-k)$-faces of $\Delta$. The polyvectors $w_{\sigma_{0\dots k}}$ sum to $\hat w_{\Delta'}$ for those simplices $\sigma\in bar (\Delta)$ whose faces $\sigma_{k\dots q}$ give the same simplex in $bar(\Delta')$. Then again from the definition of the cap product we can write: $$\label{eq:wave_action2}
\phi^k \cap (\sum_{\sigma\in bar(\Delta)} \beta_\Delta \sigma) = \sum_{\Delta' \prec\Delta} (\hat w_{\Delta'}\wedge\beta_\Delta) \Delta',$$ where the sum is taken now over all $(q-k)$-dimensional faces of $\Delta$.
It is straight forward to check that in both cases the resulting chain $$\phi^k \cap (\sum_\Delta\sum_{\sigma\in bar(\Delta)} \beta_\Delta \sigma)$$ is a cycle.
\[conj:isomorphism\] Let $X$ be a smooth compact tropical variety. Then for $q\ge p$ $$\phi^{q-p} \cap: H_q(X;{\mathcal F}_p \otimes {\mathbb R}) \to H_{p}(X;{\mathcal F}_{q} \otimes {\mathbb R})$$ is an isomorphism.
We will prove the conjecture in the realizable case in Section \[section:konstruktor\] though we believe that realizability assumption is not necessary. Certain amount of smoothness, on the other hand, is essential. In the non-smooth case even the ranks of $H_q(X; {\mathcal F}_p)$ and $H_{p}(X; {\mathcal F}_{q})$ may not agree. A simple example is provided by the nodal genus 2 curve (see Example \[eg:nodal\]).
The action of the eigenwave $\phi$ is trivial on straight tropical $(p,p)$-classes.
If $\gamma\in H_{p,p}^{straight}(X)$ then $\phi\cap\gamma=0$.
Any vector parallel to a simplex $\sigma$ of a special tropical cycle turns to zero after the wedge product with the volume element of $\sigma$.
Intermediate Jacobians
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Tropical tori
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Let $V$ be a $g$-dimensional real vector space containing two lattices $\Gamma_1,\Gamma_2$ of maximal rank, that is $V \cong \Gamma_{1,2}\otimes {\mathbb R}$. Suppose we are given an isomorphism $Q:\Gamma_1\to \Gamma_2^*$, which is symmetric if thought of as a bilinear form on $V$.
The torus $J=V/\Gamma_1$ is the [*principally polarized tropical torus*]{} with $Q$ being its polarization. The tropical structure on $J$ is given by the lattice $\Gamma_2$. If, in addition $Q$ is positive definite, we say that $J$ is an [*abelian variety*]{}.
The map $Q:\Gamma_1\to \Gamma_2^*$ provides an isomorphism of $J=V/\Gamma_1$ with the tropical torus $V^*/\Gamma_2^*$. The tropical structure on the latter is provided by the lattice $\Gamma_1^*$.
The above data $(V, \Gamma_1, \Gamma_2, Q)$ is equivalent to a non-degenerate real-valued quadratic form $Q$ on a free abelian group $\Gamma_1\cong {\mathbb Z}^g$. The other lattice $\Gamma_2 \subset V := \Gamma_1\otimes {\mathbb R}$ is defined as the dual lattice to the image of $\Gamma_1$ under the isomorphism $V\to (V)^*$ given by $Q$.
Let us take the free abelian group $\Gamma_1=H_q(X; {\mathcal F}_p)\cong{\mathbb Z}^g$ with $p+q=\dim X$, and $p\le q$. We define the tropical intermediate Jacobian as the torus above together with a symmetric bilinear form $Q$ on $H_q(X; {\mathcal F}_p)$.
The form $Q$ is a certain intersection product on tropical cycles which we define in two ways. The first definition is manifestly symmetric while the second definition descends to homology. And then we show that the two definitions are equivalent.
Unfortunately we are not able to show in this paper that the form is non-degenerate, though we believe that in the smooth and compact case this should be true (cf. Conjecture \[conj:non\_degenerate\]).
Intersection product
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Let $X$ be a compact tropical space of dimension $n$. For a singular simplex $\sigma$ we denote its relative interior by $int(\sigma)$. We abuse the notation $int(\sigma)$ to denote also its image in $X$.
We say that a tropical chain $\sum\beta_\sigma \sigma\in C_{q}(X; {\mathcal F}_{p})$ is [*transversal to the combinatorial stratification of $X$*]{} (or, simply, transversal) if for any simplex $\sigma$ and any face $\tau\prec_k\sigma$ we have
- $int(\tau)$ meets strata of X only of dimension $(n-k)$ and higher;
- if $\tau$ lies in a sedentary stratum of $X$ then $\beta_\sigma$ is divisible by all corresponding divisorial directions.
We say that two transversal tropical chains $\sum\beta_{\sigma'} \sigma'\in C_{q'}(X; {\mathcal F}_{p'})$ and $\sum\beta_{\sigma''} \sigma'' \in C_{q''}(X; {\mathcal F}_{p''})$ form a [*transversal pair*]{} if the following holds. For every pair of simplices $\sigma', \sigma''$ from these chains and any choice of their faces $\tau'\prec\sigma', \tau''\prec \sigma''$, if the interiors $int(\tau'), int(\tau'')$ lie in the same stratum ${\mathcal E}$ then $int(\tau'), int(\tau'')$ are transversal in the usual sense as smooth maps to ${\mathcal E}$.
If a pair of simplices $\sigma', \sigma''$ from the transversal pair have non-empty intersection then all three submanifolds $\sigma' , \sigma'', \sigma'\cap \sigma''$ are supported on the same maximal stratum ${{\mathcal E}_{\sigma'\cap \sigma''}}$ of $X$ (and on no smaller strata). The oriented triple $\sigma' , \sigma'', \sigma'\cap \sigma''$ determines an integral volume element ${\operatorname{Vol}}_{{\mathcal E}_{\sigma'\cap \sigma''}}$ as well as its dual volume form $\Omega_{{\mathcal E}_{\sigma'\cap \sigma''}}$. By transversality, $\sigma'\cap \sigma''$ has dimension $q'+q''-n$. We can choose a singular chain $\sum \tau$ representing its relative fundamental class agreeing with the orientation of $\sigma'\cap \sigma''$.
Let $\gamma'=\sum\beta_{\sigma'} \sigma'\in C_{q'}(X; {\mathcal F}_{p'})$ and $\gamma''=\sum\beta_{\sigma''} \sigma'' \in C_{q''}(X; {\mathcal F}_{p''})$ be a transversal pair of tropical chains. We define the following bilinear product with values in the cowave chains: $$\label{eq:dot_product}
\gamma'\cdot\gamma''=\sum_
{\tau\subset\sigma'\cap \sigma''}
\Omega_{{\mathcal E}_\tau} (\beta_{\sigma'}\wedge \beta_{\sigma''}) \cdot \tau \in C_{q'+q''-n}(X; {\mathcal W}^{n-p'-p''}).$$
Note that $\gamma'\cdot\gamma''$ has no support on infinite simplices $\tau\subset\sigma'\cap \sigma''$ since the divisorial directions in $W^{div}(\tau)$ divide both $\beta_{\sigma'}$ and $\beta_{\sigma''}$.
If $q'+q'' < n$ or $p'+p''>n$ then $\gamma' \cdot \gamma''=0$ for dimensional reasons. In what follows we will tacitly assume this is not the case.
From now on we assume that $X$ is a compact smooth tropical space. Our goal will be to show that in this case the above product descends to homology.
First we show that we can deform all cycles to a transverse position. Since the question is local we can work in a chart $\phi_\alpha: U_\alpha \to Y \subset {\mathbb T}^N$. The next lemma says that we can move a tropical cycle $\gamma$ off a face $E$ of $Y$, if it intersects it in higher than expected dimension, not changing it outside the open star ${\operatorname{St}}(E)$.
\[transversal-E\] Let $\gamma \in C_q(X,{\mathcal F}_p)$ be a (singular) tropical cycle in a tropical $n$-dimensional manifold $X$ and let $E$ be an $l$-face of $Y$ in a chart $\phi_\alpha: U_\alpha \to Y \subset {\mathbb T}^N$. Then there exists a cycle $\gamma'=\sum \beta_\sigma \sigma\in C_q(X;{\mathcal F}_p)$ homologous to $\gamma$ and such that for any $(q-k)$-face $\tau$ of a simplex $\sigma$ we have $int(\tau)\cap E=\emptyset$, i.e. $\tau$ is not supported on $E$ whenever $k+l<n$. In addition, $\gamma'$ satisfies to the following properties:
- $\gamma \cap (X{\smallsetminus}(U_\alpha \cap {\operatorname{St}}(E))) = \gamma'\cap (X{\smallsetminus}(U_\alpha \cap {\operatorname{St}}(E)))$,
- the chain $\gamma-\gamma'$ is the boundary of a tropical $(q+1)$-chain supported in $U_\alpha \cap{\operatorname{St}}(E)$,
- if $E$ has positive sedentarity then any simplex $\sigma$ such that $\sigma \cap E \ne \emptyset$ has its coefficient $\beta_\sigma$ divisible by all divisorial vectors corresponding to $E$.
First let us consider the case when $E$ is mobile. Working in a chart we can assume $Y$ is the Bergman fan for some loopless matroid $M$. Clearly, any matroid $M$ contains a uniform submatroid $M_0\subset M$ of the same rank $r(M)$ (by submatroid we mean a subset with the restriction of the rank function). Thus, we have a sequence $M_0 \subset \dots \subset M_{|M|-r(M)}=M$ of submatroids of $M$ such that $M_{j+1}$ is obtained from $M_{j}$ by adding one element $\epsilon_{j+1}$. We may form a matroid $H_j$ of rank $r(M_j)-1$ by setting a new rank function $r_{H_j}$ on $M_j$, $r_{H_j}(A)=r_M(A\cup \epsilon_{j+1})-1$ for $A\subset M_j$.
The fan $Y_{M_{j+1}}\subset {\mathbb R}^{|M_j|}$ maps to the fan $Y_{M_j}\subset {\mathbb R}^{|M_j|-1}$ by projection along the coordinate corresponding to the element $\epsilon_{j+1}$. If the matroid $H_j$ has loops this map $$\tau_j:Y_{M_{j+1}}\to Y_{M_j}$$ is an isomorphism.
![The matroids $M_{j+1}, M_j$ and $H_j$ and their corresponding fans. The fan $Y'_{M_{j+1}}$ is the unshaded part of $Y_{M_{j+1}}$. The shaded part of $Y_{M_{j+1}}$ is ${\operatorname{St}}(e_{\epsilon_{j+1}})$.[]{data-label="fig:lemma59"}](lemma59.pdf){height="60mm"}
Otherwise note that the Bergman fan $Y_{H_j}$ is a subfan of $Y_{M_j}$. Also we denote by $Y'_{M_{j+1}}$ the subfan of $Y_{M_{j+1}}$ containing only those cones whose corresponding flags do [*not*]{} have two flats differing just by $\epsilon_{j+1}$, see Fig. \[fig:lemma59\]. Then $\tau_j: Y'_{M_{j+1}}\to Y_{M_j}$ is a one-to-one map linear on the cones, cf. [@Shaw]. Indeed, $\tau_j$ contracts precisely those cones of $Y_{M_{j+1}}$ which are parallel to $e_{\epsilon_{j+1}}$.
$Y_{M_0}$ is a complete fan in ${\mathbb R}^{r(M)-1}$ and the lemma is trivial since the coefficients ${\mathcal F}_p = \Lambda^p {\mathbb Z}^{r(M)-1}$ are constant on all strata and we may deform $\gamma$ into a general position (subdividing simplices in $\gamma$ if needed to keep the chain strata-compatible). Inductively we suppose that the lemma holds for $Y_{M_j}$ and the matroid $H_j$ is loopless and then prove that the lemma holds for $Y_{M_{j+1}}$.
We denote by ${\operatorname{St}}(e_{\epsilon_{j+1}})$ the complement of $Y'_{M_{j+1}}$ in $Y_{M_{j+1}}$. It really is the open star of $e_{\epsilon_{j+1}}$ (in the coarsest face structure of $Y_{M_{j+1}}$). Note that ${\operatorname{St}}(e_{\epsilon_{j+1}})\cong Y_{H_j}\times{\mathbb R}$.
If $E\subset{\operatorname{St}}(e_{\epsilon_{j+1}})$ we may use the inductive assumption for projections to $Y_{H_j}$ (it has smaller dimension) together with a deformation along a generic vector field parallel to $e_{\epsilon_{j+1}}$.
If $E\not\subset{\operatorname{St}}(e_{\epsilon_{j+1}})$, that is $E$ is contained in $Y'_{M_{j+1}}$ we have $\dim(\tau_j(E))=\dim(E)=l$. Consider singular $q$-simplices from $\gamma$ with the interiors mapped to ${\operatorname{St}}(E)$ and such that their closures intersect $E$. These simplices form a chain $\gamma_E$ which can be considered as a relative cycle modulo its boundary ${\partial}\gamma_E$. We have ${\partial}\gamma_E\cap E=\emptyset$. Furthermore, $\tau_j({\partial}\gamma_E)$ is a $(q-1)$-cycle in the $(n-1)$-dimensional tropical manifold $Y_{H_j}$. By induction on dimension we may assume that $\tau_j({\partial}\gamma_E)\cap {\operatorname{St}}(e_{\epsilon_{j+1}})$ can be deformed in $Y_{H_j}$ to a cycle with simplices without faces of dimension larger than $q-n+l$ whose relative interiors are contained in $E$. As ${\operatorname{St}}(e_{\epsilon_{j+1}})\cong Y_{H_j}\times{\mathbb R}$ such deformation lifts to $Y_{M_{j+1}}$ and can be extended to a deformation of $\gamma$ in $Y_{M_{j+1}}$.
By induction on $j$ there exists a tropical chain $b_j\in C_{q+1}(Y_{M_j};{\mathcal F}_p)$ such that the relative interiors of $k$-faces of singular simplices of $\gamma'_j={\partial}B_j - \tau_j (\gamma)$ are disjoint from $E$. This assumption holds for any face structure on $Y_{M_j}$, in particular for the one compatible with $Y_{H_j}$. Then the relative interiors of all $q$-dimensional simplices are disjoint from $Y_{H_j}$ and we can form $\tilde b_j\in C_{q+1}(Y_{M_{j+1}};{\mathcal F}_p)$ and $\tilde \gamma'_j\in C_{q+1}(Y_{M_{j+1}};{\mathcal F}_p)$ by applying $\tau_j^{-1}|_{Y_{M_j}{\smallsetminus}Y_{H_j}}$ to $b_j$ and $\gamma'_j$. Note that ${\partial}\tilde b_j - \gamma - \tilde \gamma'_j$ must have the coefficients vanishing under $\tau_j$, even though generated from the facets of $Y'_{M_{j+1}}$. Such coefficients must be supported on ${\operatorname{St}}(e_{\epsilon_{j+1}})$ and thus we may apply the same reasoning as in the case of $E\subset{\operatorname{St}}(e_{\epsilon_{j+1}})$.
Finally, let us now consider the sedentary case, that is let $E$ be a sedentarity $s$ face of $Y_M\times{\mathbb T}^s$ with $s=|I|>0$. Let $\xi_j$ be the divisorial vectors, and let $V_J:=\wedge_{j\in J} \xi_j$ denote the divisorial $|J|$-polyvector for each $J\subset I$. We will need to deform $\gamma$ to $\gamma'$ so that no $(q-s)$, or smaller, -dimensional face of a simplex $\sigma$ in $\gamma'$ meets $Y_M\times\{-\infty\}$ (here $\{-\infty\}\in{\mathbb T}^s$ is the point of sedentarity $s$). In $Y_M\times{\mathbb R}^I$ the groups ${\mathcal F}_p$ split into the direct sum $\oplus_{J\subset I} {\mathcal F}_p^J$, where ${\mathcal F}_p^J$ consists of elements divisible by the polyvector $V_J$, and no larger $V_{J'}$. (The splitting is not canonical, it depends on a chart). Accordingly, we have a decomposition $\gamma=\sum_{J\subset I} \gamma_J$ into cycles.
If $J\ne I$, that is there exists $j\notin J$, we may push $\gamma_J$ from $E$ with the help of a vector field parallel to $x_j$. Note that $\gamma_J$ remains a cycle after such deformation as $\xi_{j}$ is not present in the coefficients of $\gamma_J$. Thus by induction on sedentarity we may assume $J=I$.
The cycle $\gamma_I$ has coefficients in ${\mathcal F}_{p-s}^{Y_M}\otimes V_I$, and hence can be interpreted as a relative cycle modulo ${\partial}{\mathbb T}^I={\mathbb T}^I{\smallsetminus}{\mathbb R}^I$ with coefficients in ${\mathcal F}_{p-s}^{Y^M}$ (as $V_I$ vanishes on ${\partial}{\mathbb T}^I$ and constant otherwise) and $(T^I,{\partial}T_I)$ is homeomorphic to the pair ${\mathbb R}^{s-1}\times ({\mathbb R}_{\ge 0},\{0\})$ of a half-space and its boundary. Thus $\gamma_I$ may be deformed to a product (after simplicial subdivision) of the relative fundamental cycle in the $s$-dimensional half-space with some $(q-s)$-dimensional singular cycle. In particular, $E$ will not meet any codimension $<s$ face of a $q$-simplex in a deformed cycle.
Let $\Sigma = \bigcup \sigma$ be an integral polyhedral fan (with its cones $\sigma$ oriented). Then using the inclusion homomorphisms we can form the complex $C^{(p)}_k:= \oplus_{\dim \sigma =k} {\mathcal F}_p(\sigma)$. In case $\Sigma$ is a matroidal fan the statement of Lemma \[transversal-E\] is equivalent to that the complex $C^{(p)}_\bullet$ has only the highest homology.
When $X$ is not smooth the statement of the Lemma is not true. For example let $X$ be a union of two 2-planes in ${\mathbb R}^4$ intersecting in a point. Consider an unframed path (that is cycle in $C_1(X;{\mathcal F}_0)$ through the vertex which starts in one plane and ends in the other plane. Any deformation of this path will still have to go through the vertex.
\[lemma:transversal\] Let $X$ be a tropical manifold. Then
1. Every class in $H_q(X; {\mathcal F}_p)$ is represented by a transversal cycle.
2. Every pair of classes in $H_{q'}(X; {\mathcal F}_{p'})$ and $H_{q''}(X; {\mathcal F}_{p''})$ is represented by a transversal pair of cycles.
3. If $\gamma'_1, \gamma'_2$ are two cycles which represent the same class in $H_{q'}(X; {\mathcal F}_{p'})$ and both form transversal pairs with a cycle $\gamma''\in C_{q''}(X; {\mathcal F}_{p''})$, then there is $b\in C_{q'+1}(X; {\mathcal F}_{p'})$ which form a transversal pair with $\gamma''$, and such that $\partial b=\gamma'_1-\gamma'_2$.
We may start from any tropical cycle and deform it to a transversal position by applying Lemma \[transversal-E\] stratum by stratum starting from $0$-dimensional faces and then higher-dimensional strata. (Note that in a chart the open star of any face can intersect only faces of higher dimension).
Suppose that we have two transversal cycles. Since any stratum ${\mathcal E}$ is a manifold we can make interiors of faces of the simplices from these cycles transversal in ${\mathcal E}$ by a small deformation with the help of the usual Sard’s theorem. In any chart this deformation extends to a small deformation in ${\operatorname{St}}({\mathcal E})$. Making this procedure stratum by stratum in the order of non-decreasing dimension we make any pair of cycles transversal. A similar argument applies to the relative cycle in the last statement of the corollary.
If $p'+p''+q'+q''=2n$ we can give a numerical value to the product $\gamma'\cdot\gamma''$ by integrating the $(n-p'-p'')$-form $\Omega_{{\mathcal E}_\tau} (\beta_{\sigma'}\wedge \beta_{\sigma''})$ over the $(q'+q''-n)$-simplex $\tau$. Indeed, since $\beta_{\sigma'}\wedge \beta_{\sigma''}$ is divisible by all divisorial directions corresponding to sedentary faces of $\tau=\sigma'\cap \sigma''$, the integration can be carried over in the quotient space to those (infinite) coordinates, thus giving a finite answer. Thus we define $$\label{eq:pairing}
\int \gamma' \cdot \gamma'':=\sum_
{\tau \subset \sigma'\cap \sigma''}
\int_\tau\Omega_{{\mathcal E}_\tau}(\beta_{\sigma'}\wedge \beta_{\sigma''}) \in {\mathbb R}.$$
The most interesting case to us is when $p'+q'=p''+q''=n$. Assuming $q'+q''\ge n$ we can use the eigenwave action on one of the cycles in the pair to make them of complementary dimensions, after which the integration becomes just summing over the intersection points $${\langle}\gamma',\gamma''{\rangle}:=\int \gamma' \cdot \gamma''= \sum_{x\in |\gamma'|\cap|\gamma''|} \Omega_x (\beta'_x\wedge\beta''_x),$$ where $\beta'_x, \beta''_x$ are the coefficients at $\sigma',\sigma''$ for their intersection points $x\in\sigma'\cap \sigma''$.
\[prop:wave\_commute\] Let $\gamma'=\sum\beta_{\sigma'} \sigma'\in C_{q'}(X; {\mathcal F}_{p'})$ and $\gamma''=\sum\beta_{\sigma''} \sigma'' \in C_{q''}(X; {\mathcal F}_{p''})$ be a transversal pair of tropical cycles with $p'+q'=p''+q''=n$ and $q'+q''\ge n$. Let $k:=q'-p''= q''-p' \ge0$. Then $${\langle}\phi^k \cap \gamma', \gamma''{\rangle}=\int \gamma'\cdot\gamma''.$$
First we need a representative of the cycle $\phi^k \cap \gamma'$ such that it still forms a transversal pair with $\gamma''$. We fix first and second baricentric subdivisions of the simplices $\sigma'$ in $\gamma'$. Then by transversality of $\gamma''$ we can assume that the intersection of each $\sigma'$ with $\gamma''$ is supported on the star skeleton of $\sigma'$. That is $\sigma'\cap |\gamma''|$ consists of the $k$-simplices of the first baricentric subdivision of $\sigma'$ spanned by the baricenters of the $q'-k, \dots, q'$-dimensional faces $\tau$ of $\sigma'$. We label the $k$-simplices in the first baricentric subdivision of $\sigma'$ by the flags of its faces $(\tau_0\prec\dots \prec \tau_k)$.
![Intersection in $\sigma'$: $ |\gamma''|$ (in red), $|\phi^k \cap \gamma'|$ (in blue).[]{data-label="fig:intersection"}](intersection.pdf){height="30mm"}
Then the result of the wave action from Description 1 on $\beta_{\sigma'}\sigma'$ gives the following chain (see Fig. \[fig:intersection\]) $$\sum_{\tau_0\prec\dots \prec \tau_k} (w_{\tau_0\prec\dots \prec \tau_k}\wedge\beta_{\sigma'}) \widehat{(\tau_0\prec\dots \prec \tau_k)},$$ where $w_{\tau_0\prec\dots \prec \tau_k}\in W_k (\Delta_{\sigma'})$ is the polyvector associated to the simplex $(\tau_0\prec\dots \prec \tau_k)$, and $\widehat{(\tau_0\prec\dots \prec \tau_k)}$ is its star dual in the second baricentric subdivision (cf. definition in Section \[subsection:wave\_action\]). When intersected with $\gamma''$ only the simplices $(\tau_0\prec\dots \prec \tau_k)$ with maximal dimensional flags enter and we see that the result coincides with the definition of $\int \gamma' \cdot \gamma''$.
\[prop:product\_homology\] Let $X$ be smooth. Then the intersection product ${\langle}\ , \ {\rangle}$ on cycles descends to a pairing on homology $H_q(X;{\mathcal F}_p) \otimes H_{p}(X; {\mathcal F}_{q}) \to {\mathbb R}$.
Suppose that we have two homologous cycles $\gamma'_1\in C_q(X;{\mathcal F}_p)$ and $\gamma'_2 \in C_q(X;{\mathcal F}_p)$. Let $b\in C_{q+1}(X;{\mathcal F}_p)$ be the connecting chain, i.e. ${\partial}b=\gamma'_1-\gamma'_2$. According to Corollary \[lemma:transversal\] we can assume that each of the three $\gamma_1', \gamma_2', b$ forms a transversal pair with a cycle $\gamma'' \in C_p(X; {\mathcal F}_q)$.
It is clear that ${\partial}(b \cdot \gamma'')$ coincides with the $\gamma_1' \cdot \gamma'' - \gamma_2' \cdot \gamma''$ on the interiors of the maximal strata of $X$. Thus it is enough to show that $b \cdot \gamma''$ has no boundary on codimension 1 mobile strata of $X$ (according to Lemma \[transversal-E\] the intersection has no support on infinite simplices). This is local so we can work in a chart $\phi_\alpha: U_\alpha \to Y \subset {\mathbb T}^N$.
Let $E$ be a codimension 1 face of $Y$ and let $D_1,\dots,D_k$ be the adjacent facets at $E$. We choose $v_1,\dots,v_k$, the corresponding primitive vectors such that $\sum_{i=1}^k v_i=0$ (not just modulo the span of $E$). Let $x$ be a point in the relative interior of $E$ where $b$ intersects $\gamma''$, and let $\tau_1, \dots, \tau_k$ be the intervals in the support of $b \cdot \gamma''$ adjacent to $x$. Each $\tau_i$ lies in $D_i$. Let $\beta_i'\in {\mathcal F}_p(D_i)$ and $\beta_i''\in {\mathcal F}_q(D_i)$ be the coefficients of the simplices of $b$ and of $\gamma''$, respectively, which intersect at the $\tau_i$.
Since $\gamma''$ is a cycle, we have $\sum_i \beta_i''=0$. We can write each $$\beta_i''=v_i\wedge \bar \alpha_i'' + \alpha_i'',$$ where $\bar\alpha_i'' \in W_{q-1}(E)$ and $\alpha_i'' \in W_q (E)$.
Recall that our tropical space $X$ is smooth. In particular, this means that the fan at $E$ modulo linear span of $E$ is matroidal. That is, $\sum_{i=1}^k v_i=0$ is the [*only*]{} linear relation among the $v_i$’s. This together with $\sum_i \beta_i''=0$ implies that $$\sum_{i=1}^k \alpha_i''=0 \quad \text{ and } \quad \bar\alpha_1''=\dots=\bar\alpha_k'' =: \bar\alpha''.$$
Similarly, $\sum_i \beta_i'=0$ since ${\partial}b$ cannot have support at $x$. Hence we can write $$\beta_i'=v_i\wedge \bar \alpha' + \alpha_i',$$ with $\sum \alpha_i'=0$, $\alpha_i' \in W_p (E)$ and $\bar\alpha' \in W_{p-1}(E)$. Note that in the product $$\begin{gathered}
\beta_i'\wedge\beta_i''=(v_i\wedge \bar \alpha' + \alpha_i')\wedge (v_i\wedge \bar \alpha'' + \alpha_i'')
=v_i\wedge(\bar\alpha'\wedge \alpha_i'' + \alpha_i'\wedge \bar \alpha'')\end{gathered}$$ only the cross terms survive. Now we are ready to evaluate ${\partial}(b\cdot \gamma'')$ at $x$: $$\begin{gathered}
\sum_i \Omega_{\Delta_i} [ v_i\wedge(\bar\alpha'\wedge \alpha_i'' + \alpha_i'\wedge \bar \alpha'') ]
= \sum_i \Omega_{\Delta} (\bar\alpha'\wedge \alpha_i'' + \alpha_i'\wedge \bar \alpha'') \\
= \Omega_{\Delta} (\bar\alpha'\wedge \sum_i \alpha_i'' +\sum_i \alpha_i'\wedge \bar \alpha'') = 0.\end{gathered}$$
Finally we restrict to the case when both $\gamma', \gamma''$ are cycles in $C_{q}(X; {\mathcal F}_{p})$ with $p+q=n$. Then $\gamma'\cdot \gamma'' = \gamma'' \cdot \gamma'$. Indeed, assuming the orientation of $\tau$ is chosen, taking the product in the opposite order will result in the change of sign of the volume form $\Omega_{{\mathcal E}_\tau}$ according to the parity of $p$. On the other hand this parity will also affect the coefficients product: $\beta'\wedge\beta''= (-1)^p \beta''\wedge\beta'$, both effects cancel in $\Omega_{{\mathcal E}_\tau}(\beta'\wedge\beta'')$. This observation combined with Propositions \[prop:wave\_commute\] and \[prop:product\_homology\] lead to the final statement.
\[theorem:intersection\] Let $X$ be compact and smooth. The product on cycles descends to a symmetric bilinear form on $H_q(X; {\mathcal F}_p)$ for any $p+q=n$.
\[conj:non\_degenerate\] This form is non-degenerate.
Appendix: Konstruktor and the eigenwave action in the realizable case {#section:konstruktor}
=====================================================================
Tropical limit and the Steenbrink-Illusie spectral sequence
-----------------------------------------------------------
Suppose $X$ is the tropical limit of a complex projective one-parameter degeneration $\mathcal X \to \Delta^*$. Then $X$ is naturally polyhedral. We assume also that $X$ is smooth. In this case the refined stable reduction theorem [@Mumf] allows us assume the following (see details in [@IKMZ]).
- $X$ is unimodularly triangulated. This means that the finite cells are unimodular simplices and the infinite cells are products of unimodular simplices and unimodular cones spanned by the divisorial vectors.
- The finite part of $X$ is identified with the dual Clemens complex of the degeneration with simple normal crossing central fiber $Z=\cup Z_{\alpha}$. This means that the components of $Z$ are labelled by vertices of zero sedentarity and their intersections $Z_{\alpha_0} \cap\dots \cap Z_{\alpha_k}=:Z_\Delta$ are labelled by (finite) simplices $\Delta=\{\alpha_0\dots\alpha_k\}$ of $X$ of zero sedentarity.
\[theorem:isomorphism\] Let $X$ be a realizable smooth projective tropical variety. Then for $q\ge p$ $$\phi^{q-p}: H_q(X; {\mathcal F}_p) \otimes{\mathbb Q}\to H_{p}(X; {\mathcal F}_{q}) \otimes{\mathbb Q}$$ is an isomorphism.
In this algebraic setting the eigenwave itself is an integral class in $H^1(X;{\mathcal W}_1)$ (recall that $W$ carries a natural lattice). Hence in the statement we can avoid tensoring the tropical homology groups with ${\mathbb R}$. However its proof relies on the isomorphism in Theorem \[theorem:main\] which we can assert only over ${\mathbb Q}$. Although we believe that the theorem remains true over ${\mathbb Z}$ its proof may be more delicate.
We will prove the theorem by comparing the eigenwave action with the classical monodromy action $T: H_k(X_t, {\mathbb Q}) \to H_k(X_t, {\mathbb Q})$, where $X_t$ is a general fiber in $\mathcal X$. The idea that the monodromy can be represented by a cap product with certain cohomology class appeared before in the Calabi-Yau case. The second author [@Zh00] proved a related conjecture of Gross [@Gr98] that for toric hypersurfaces the monodromy can be described as the fiber-wise rotation by a natural section of the SYZ fibration. Later Gross and Siebert ([@GS10], Section 5.1) explored the relation between the monodromy and the cap product in the logarithmic setting.
Notations:
- $\Delta$ or $\Delta'$ will always denote a finite face of $X$ of sedentarity 0, in particular, a simplex.
- $ H_{2l}(\Delta)[-r] = H_{2l}(Z_{\Delta},{\mathbb Q})$, Tate twisted by $[-r,-r]$.
- $H_{2l}(k)[-r] = \oplus H_{2l}(\Delta)[-r]$, where $\Delta$ runs over all $k$-simplices in $X$ as above.
First we recall the classical spectral sequence which calculates the limiting mixed Hodge structure of the family $\mathcal X$ (see, e.g. [@Steen], Chapter 11). This spectral sequence (from now on referred to as the Steenbrink-Illusie’s, or SI for short) has the first term $${E}^1_{r, k-r}=\bigoplus_{i\ge \max\{0, r\}} H_{k+r-2i}(2i-r)[r-i],$$ and it degenerates at $E_2$ abutting to homology of the smooth fiber $X_t$ of $\mathcal Z$ with the monodromy weight filtration.
Since all strata in $Z$ are blow ups of projective spaces, the odd rows in Steenbrink-Illusie’s $E^1$ vanish. Removing those and making shifts in the even rows we relabel the terms by $$\tilde{E}^1_{q,p}:={E}^1_{q-p, 2p}=\bigoplus_{i\ge \max\{0, q-p\}} H_{2q-2i}({2i+p-q})[q-p-i].$$ The first differential $d=d'+d''$ consists of the map $d'$ induced by strata inclusion and the Gysin map $d''$: $$\begin{split}
& d' :H_{2l}({k})[-r] \to H_{2l}({k-1})[-r] \\
& d'' : H_{2l}({k})[-r] \to H_{2l-2}({k+1})[-r-1].
\end{split}$$
For reader’s convenience we write the beginning of the $\tilde{E}^1$ term: $$\xymatrix{
H_0(4)[-4] & & & &
\\
H_0(3)[-3] &
\txt{$H_0(4) [-3]$ \\ $\oplus H_2(2)[-2]$} \ar[l]_{d} \ar[ul]_\nu & & &
\\
H_0(2)[-2] &
\txt{$H_0(3)[-2] $ \\ $\oplus H_2(1)[-1]$} \ar[l]_{d} \ar[ul]_\nu &
\txt{$H_0(4)[-2] $ \\ $\oplus H_2(2) [-1]$ \\ $\oplus H_4(0) $} \ar[l]_{d} \ar[ul]_\nu & &
\\
H_0(1)[-1] &
\txt{$H_0(2) [-1]$ \\ $\oplus H_2(0)$} \ar[l]_{d} \ar[ul]_\nu &
\txt{$H_0(3) [-1]$ \\ $\oplus H_2(1)$} \ar[l]_{d} \ar[ul]_\nu &
\txt{$H_0(4) [-1]$ \\ $\oplus H_2(2)$} \ar[l]_{d} \ar[ul]_\nu &
\\
H_0(0) & H_0(1) \ar[l]_{d} \ar[ul]_\nu & H_0(2) \ar[l]_{d} \ar[ul]_\nu & H_0(3) \ar[l]_{d} \ar[ul]_\nu & H_0(4) \ar[l]_{d} \ar[ul]_\nu
}$$ The monodromy operator $\nu=\frac1{2\pi i} \log T$ acts along the diagonals by the Tate twist isomorphism $H_{2l}(k)[-r] \to H_{2l}(k)[-r-1]$ or by 0 if the corresponding group is missing (cf. [@Steen], Chapter 11).
Propellers
----------
Next we will give a combinatorial description of the SI groups and the differential in terms of [*propellers*]{} - the “local tropical cycles” in $X$.
Some more notations:
- Recall that $\Delta, \Delta', \Delta''$ always denote finite faces of $X$ of sedentarity 0.
- We write $\Delta\prec_k\Delta'$ or $\Delta'\succ_k\Delta$, if $\Delta$ is a face of $\Delta'$ of codimension $k$.
- ${\operatorname{Link}}_l(\Delta)$ consists of sets $\bar q=\{q_1,\dots,q_l\}$ where each $q_i$ is either a vertex or a divisorial vector, such that the vertices of $\Delta$ together with elements of $\bar q$ span a face (infinite, in case $\bar q$ contains divisorial vectors) adjacent to $\Delta$ of dimension $l$ higher. We denote the corresponding face by $\{\Delta \bar q\}$ and often drop the brackets from the notation (e.g., as below) when they become cumbersome.
- ${\operatorname{Link}}_l^0(\Delta) \subset {\operatorname{Link}}_l(\Delta)$ consists of those sets $\bar q=\{q_1,\dots,q_l\}$ where $q_i$ are allowed to be only vertices (not the divisorial vectors). In this case $\{\Delta \bar q\}$ is finite.
- ${\operatorname{Vol}}_{\Delta\bar q}$ is the integral volume element in the (oriented) face $\{\Delta \bar q\}$.
Let $\Delta$ be an oriented finite cell of sedentarity 0. One can naturally identify (see [@IKMZ] for details) the homology groups $H_{2l}(\Delta)$ with the space of local tropical relative $l$-cycles around $\Delta$. That is, we consider formal ${\mathbb Q}$-linear combinations $$\sum_{\bar q \in {\operatorname{Link}}_l(\Delta)} \rho_{\bar q} \{\Delta {\bar q}\}$$ of (possibly infinite) cells $\{\Delta {\bar q}\}\succ_l\Delta$ which are balanced along $\Delta$. We call these local cycles [*propellers*]{} and abusing the notation we continue denoting this group by $H_{2l}(\Delta)$ (there is no Tate twist however).
Then one can identify the Gysin map $d'': H_{2l}(\Delta) \to H_{2l-2}(\Delta')$ with the restriction of the propeller to a consistently oriented finite simplex $\Delta'\succ_1 \Delta$. Put together $$\label{eq:d''}
d'' ( \sum_{\bar q \in {\operatorname{Link}}_l(\Delta)} \rho_{\bar q} \{\Delta {\bar q}\} )= \sum_{q \in {\operatorname{Link}}_1^0(\Delta)} ( \sum_{\bar r \in {\operatorname{Link}}_{l-1} (\Delta q)} \rho_{q \bar r} \{\Delta {q \bar r}\} ).$$
The inclusion map $d': H_{2l}(\Delta) \to H_{2l}(\Delta')$, where $\Delta'= \Delta {\smallsetminus}v$ is consistently oriented facet of $\Delta$, is somewhat more tricky. Let $c=\sum_{\bar q \in {\operatorname{Link}}_l(\Delta)} \rho_{\bar q}\{\Delta{\bar q}\}
$ be an element in $H_{2l}(\Delta)$. For any ${\bar q \in {\operatorname{Link}}_l(\Delta)}$ let $\{\Delta'{\bar q}\}=\{\Delta{\bar q}{\smallsetminus}v\}$ be the corresponding cell containing $\Delta'$. Then the image of $ d' c$ in $H_{2l}(\Delta')$ will be $$\label{eq:d'}
\sum_{\bar q \in {\operatorname{Link}}_l(\Delta)} \rho_{\bar q} \{\Delta'{\bar q}\} + \sum_{\bar r \in {\operatorname{Link}}_{l-1}(\Delta)} \rho_{v \bar r} \{\Delta{\bar r}\},$$ where the coefficients $\rho_{v \bar r}\in {\mathbb Q}$ are chosen to make the result balanced along $\Delta'$. There is always a unique such choice (cf. [@IKMZ]), namely, the $\rho_{v \bar r}$ can be read off from the balancing condition for $c$ along $\{\Delta{\bar r}\}$: $$\label{eq:descent}
\sum_q \rho_{q \bar r} \overrightarrow{(\Delta' q)} + \rho_{v \bar r} \overrightarrow{(\Delta' v)} =0 \quad \mod \{\Delta' \bar r\},$$ where $\overrightarrow{(\Delta' q)}$ means the divisorial vector $q$, or the vector from any vertex of $\Delta'$ to $q$ (well defined mod $\Delta'$) if $q$ is a vertex, and same for $\overrightarrow{(\Delta' v)}$.
From now on we will not distinguish between the classical geometric Steenbrink-Illusie $E_1$ complex and its interpretation via complex of propellers. One of the main results in [@IKMZ] is the following statement.
\[theorem:main\] $\tilde{E}^2_{q,p}\cong H_{q}(X;{\mathcal F}_p) \otimes {\mathbb Q}$.
Konstruktor
-----------
Now we provide another realization of the Steenbrink-Illusie’s $E_1$ complex in terms of specific tropical simplicial chains. The collection of these chains which we call [*konstruktor*]{} forms a subcomplex of $C_\bullet^{bar} (X, {\mathcal F}_\bullet)$, and we can refer to Theorem \[theorem:main\] to see that the inclusion is a quasi-isomorphism. A wonderful feature of the konstruktor is that the eigenwave acts on its elements precisely as the monodromy operator $\nu$ acts on the terms in the Steenbrink-Illusie’s $E_1$.
Let us fix the first baricentric subdivision of $X$. We elaborate a little bit on already used notation of the dual cell.
- For a pair $\Delta\succ \Delta'$ of finite simplices of sedentarity 0 in $X$, and $\bar q \in {\operatorname{Link}}_l(\Delta)$ we let $ \hat \Delta'_{\Delta \bar q}$ denote the dual cell to $\Delta'$ in the face $\{\Delta \bar q\}$ of $X$, that is the union of all simplices in the baricentric subdivision containing baricenters of both $\Delta'$ and $\{\Delta{\bar q}\}$.
- In the summation formulae to follow we assume the terms with $ \hat \Delta'_{\Delta \bar q}$ are not present if $\Delta'$ is not a zero sedentarity finite face of $\{\Delta \bar q\}$.
Let $\Delta$ be a finite $k$-simplex of sedentarity 0 in $X$, and $r\le k$ a non-negative integer. To any propeller, that is a local tropical $l$-cycle $$c=\sum_{\bar q \in {\operatorname{Link}}_l(\Delta)} \rho_{\bar q} \{\Delta{\bar q}\} \in H_{2l}(\Delta)$$ we associate a simplicial chain $c[-r] \in C^{bar}_{k+l-r}(X,{\mathcal F}_{l+r})$ as follows (note that $c[0]$ now has other meaning than just $c$): $$c[-r] =\sum_{\bar q \in {\operatorname{Link}}_{l}(\Delta)}
\sum_{
\begin{subarray}{c}
\Delta'\prec \Delta\\
\dim \Delta'= r
\end{subarray}
}
(\rho_{\bar q} {\operatorname{Vol}}_{\Delta'\bar q}) \hat \Delta'_{\Delta \bar q} .$$ The orientation of $\hat \Delta'_{\Delta \bar q}$ is consistent with the original orientation of $\Delta$ and the choice of the volume element ${\operatorname{Vol}}_{\Delta'\bar q}$. Clearly for each $r$ between 0 and $k$ the map $$(\cdot) [-r]: H_{2l}(k) \to C^{bar}_{k+l-r}(X,{\mathcal F}_{l+r})$$ is an injective group homomorphism. We denote its image in $C^{bar}_{k+l-r}(X,{\mathcal F}_{l+r})$ by $K_l(k)[-r]$.
The [*konstruktor*]{} is the subgroup of $C^{bar}_{\bullet}(X,{\mathcal F}_{\bullet})$ generated by the $K_l(k)[-r]$ for all $k$, $l$ and $r$ . Note that $K_l(k)[-r]$ intersect trivially for different triples $k,l,r$.
Next we want to show that for each $p$ the $\oplus_r K_{p-r}(\bullet -p+2r)[-r]$ is indeed a subcomplex of $C^{bar}_\bullet(X,{\mathcal F}_p)$ isomorphic to the SI complex $\tilde{E}_1^{\bullet,p}$. This follows at once from comparing the SI differentials $d=d'+d''$ with the simplicial boundary ${\partial}$.
\[prop:konstruktor\] $ {\partial}(c[-r]) = (d'c)[-r] + (d''c)[-r-1]$.
For the proof we need two linear algebra identities. Let $\sigma',\sigma''$ be two opposite faces in a unimodular simplex $\sigma=\{\sigma' \sigma''\}$. Then one has $$\sum_{\tau''\prec_1\sigma''} {\operatorname{Vol}}_{\sigma'\tau''}={\operatorname{Vol}}_{\sigma'} \wedge {\operatorname{Vol}}_{\sigma''}=\sum_{\tau'\prec_1\sigma'} {\operatorname{Vol}}_{\tau'\sigma''},$$ where, say, the left equality easily follows from the case when $\sigma'$ is a vertex. Here all $\tau'$ are oriented consistently with $\sigma'$, and all $\tau''$ with $\sigma''$. We will need this identity in the form $$\label{eq:simplex}
\sum_{\Delta'\prec_1\Delta} {\operatorname{Vol}}_{\Delta'\bar q}= \sum_{q\in {\operatorname{Link}}_1^0(\Delta)} {\operatorname{Vol}}_{\Delta \bar q{\smallsetminus}q},$$ where $\Delta$ is a finite simplex and $\bar q\in {\operatorname{Link}}_l(\Delta)$. Note that the divisorial vectors (if any) in $\bar q$ just multiply both sides of the identity for finite simplices.
The second identity involves a relation among the balancing coefficients $\rho_{v \bar r}$ from for $c=\sum \rho_{\bar q} \{\Delta{\bar q}\}$. One can show (cf. [@IKMZ]) that they satisfy a refined version of . Namely, for $\Delta'\prec \Delta \prec \{\Delta \bar q\}$ we have $$\sum_q \rho_{q \bar r} \overrightarrow{(\Delta' q)} +\sum_{v\in \Delta{\smallsetminus}\Delta'} \rho_{v \bar r} \overrightarrow{(\Delta' v)} = 0 \quad \mod \{\Delta' \bar r\}$$ for faces $\Delta'\prec\Delta$ of codimension possibly higher than 1. Multiplying the above by ${\operatorname{Vol}}_{\Delta'\bar r}$ we arrive at $$\label{eq:vol_balance}
\sum_{q\in {\operatorname{Link}}_1(\Delta)} \rho_{q \bar r} {\operatorname{Vol}}_{\Delta' q \bar r} = - \sum_{v \in \Delta{\smallsetminus}\Delta'} \rho_{v \bar r} {\operatorname{Vol}}_{\Delta' v \bar r}.$$
Now we are ready to proof the proposition. Let $c=\sum \rho_{\bar q} \{\Delta{\bar q}\}$, then we can write $$c[-r] =\sum_{
\begin{subarray}{c}
\Delta'\prec_{k-r} \Delta\\
\bar q \in {\operatorname{Link}}_{l}(\Delta)
\end{subarray}
}
(\rho_{\bar q} {\operatorname{Vol}}_{\Delta'\bar q}) \hat \Delta'_{\Delta{\bar q}} .$$ The topological boundary of each cell $\Delta'_{\Delta{\bar q}}$ consists of two types:
- Type 1: cells in the form $\Delta''_{\Delta{\bar q}}$ for faces $\Delta''\succ_1\Delta'$ of $\{\Delta \bar q\}$. If the cell $\Delta'_{\Delta{\bar q}}$ includes divisorial directions then its coefficient ${\operatorname{Vol}}_{\Delta'\bar q}$ in $c[-r]$ is divisible by all divisorial vectors. Hence the type 1 part of the boundary ${\partial}(c[-r])$ is, in fact, supported on the faces $\Delta''_{\Delta{\bar q}}$ for finite $\Delta''$. Thus $\Delta''_{\Delta{\bar q}}$ in the formulae below make sense.
- Type 2: cells in the form $\Delta'_{\Delta{\bar q}{\smallsetminus}v}$ where $v$ is a vertex or a divisorial vector in $\{\Delta \bar q\}$ which is not in $\Delta'$.
Next we show that these two boundary types endowed with the framing correspond to the $d''$ and $d'$ differentials in the SI complex, respectively, see Figure \[fig:h\_2(1)\].
![$d=d'+d'': H_2(1)\to H_2(0)\oplus H_0(2)[-1]$. (Framing coefficient vectors are not to scale).[]{data-label="fig:h_2(1)"}](h_2_1){width="4.5in"}
Boundary of type 1: $$\begin{gathered}
\sum_{\Delta', \bar q} \ \sum_{q\in \bar q} (\rho_{\bar q} {\operatorname{Vol}}_{\Delta' \bar q}) \hat{\{\Delta' q\}}_{\Delta{\bar q}}
+ \sum_{\bar q}\sum_{\Delta'\prec_1 \Delta''\prec \Delta} (\rho_{\bar q} {\operatorname{Vol}}_{\Delta'\bar q}) \hat\Delta''_{\Delta{\bar q}}\\
= \sum_{q\in {\operatorname{Link}}_1^0(\Delta)}
\left( \sum_
{
\begin{subarray}{c}
\Delta''\prec \Delta q,\ \Delta''\not\prec\Delta\\
\bar r \in {\operatorname{Link}}_{l-1}(\Delta q)
\end{subarray}
}
(\rho_{q \bar r} {\operatorname{Vol}}_{\Delta''\bar r}) \hat\Delta''_{\Delta{q \bar r}}
+ \sum_
{
\begin{subarray}{c}
\Delta''\prec \Delta\\
\bar r \in {\operatorname{Link}}_{l-1}(\Delta q)
\end{subarray}
}
(\rho_{q \bar r} {\operatorname{Vol}}_{\Delta''\bar r})\right)
\hat\Delta''_{\Delta{q \bar r}}\\
=\sum_{q \in {\operatorname{Link}}_1^0(\Delta)} \sum_
{
\begin{subarray}{c}
\Delta''\prec \Delta q\\
\bar r \in {\operatorname{Link}}_{l-1}(\Delta q)
\end{subarray}
}
(\rho_{q \bar r} {\operatorname{Vol}}_{\Delta''\bar r} ) \hat\Delta''_{\Delta{q \bar r}} .\end{gathered}$$ Here in the second summand we used the identity for the pair $\Delta' \prec \Delta''\bar q$. From one can easily see that this coincides with $(d''c)[-r-1]$.
Boundary of type 2: $$\begin{gathered}
\sum_{q\in {\operatorname{Link}}_1(\Delta)} \sum_{
\begin{subarray}{c}
\Delta'\prec \Delta\\
\bar r \in {\operatorname{Link}}_{l-1}(\Delta)
\end{subarray}
}
( \rho_{q \bar r} {\operatorname{Vol}}_{\tau q \bar r}) \hat \Delta'_{\Delta{\bar r}}
+ \sum_{v\in\Delta } \sum_
{
\begin{subarray}{c}
\Delta'\prec \Delta{\smallsetminus}v\\
\bar q \in {\operatorname{Link}}_{l}(\Delta)
\end{subarray}
}
(\rho_{\bar q} {\operatorname{Vol}}_{\Delta' \bar q}) \hat\Delta' _{\Delta{\bar q}{\smallsetminus}v} \\
= \sum_
{
\begin{subarray}{c}
v\in\Delta\\
\Delta' \prec \Delta{\smallsetminus}v \\
\end{subarray}
}
\left( \sum_{\bar r \in {\operatorname{Link}}_{l-1}(\Delta)}
(\rho_{v \bar r} {\operatorname{Vol}}_{\tau v \bar r}) \hat \Delta'_{\Delta{\bar r}}
+ \sum_{\bar q \in {\operatorname{Link}}_{l}(\Delta)}
(\rho_{\bar q} {\operatorname{Vol}}_{\tau\bar q}) \hat \Delta'_{\Delta{\bar q}{\smallsetminus}v} \right).\end{gathered}$$ Here in the first summand we used the identity for each $\Delta', \bar r$ with the sign compensated by the orientation of $\hat\Delta'_{\Delta{\bar r}}$ and the choice of $ {\operatorname{Vol}}_{\tau v \bar r}$. Taking the sum of over all vertices $v\in \Delta$ we easily identify the last expression with $(d'c)[-r]$.
Combining the above proposition with Theorem \[theorem:main\] we can conclude that the konstruktor complex can be used to calculate the tropical homology groups $H_q(X; {\mathcal F}_p)$:
The inclusion of the konstruktor $\oplus_r K_{p-r}(\bullet -p+2r)[-r]$ into the complex $C^{bar}_\bullet(X;{\mathcal F}_p)$ is a quasi-isomorphism for each $p$.
Finally, since all infinite cells in the konstruktor chains have coefficients divisible by the divisorial directions we can use the explicit description of the eigenwave action on it. Then unveiling the konstruktor definition we arrive at the following.
\[konstruktor\_action\] For any $c\in H_{2l}(\Delta)$ one has $\phi \cap (c[-r])=c[-r-1]$.
Now we can combine all above observations to prove the claimed isomorphism $$\phi^{q-p}: H_q(X; {\mathcal F}_p) \to H_{p}(X; {\mathcal F}_{q}).$$
The cap product action of the eigenwave $\phi^{q-p}$ on the homology $H_q(X, {\mathcal F}_p)$ can be induced from its action on the konstruktor, which is a simplicial chain subcomplex. But it agrees there with the classical action of the monodromy $\nu^{q-p}$ on the $E_1$ term of the SI spectral sequence. On the other hand it is well known that the $\nu^{q-p}$ induces an isomorphism on the associated graded pieces with respect to the monodromy weight filtration on $H_{p+q}(X_t)$, which are calculated on the $E_2$ term of the SI spectral sequence.
We are grateful to Ilia Itenberg for numerous useful discussions. We also wish to thank the referee for pointing out several mistakes and suggesting many exposition improvements. Finally we would like to thank the Max-Planck-Institut-für-Mathematik for its hospitality during the special program “Tropical Geometry and Topology".
[KKMS73]{}
F. Ardila and C. Klivans. . J. Comb. Theory, Ser. B 96(1): 38-49 (2006).
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M. Gross and B. Siebert. . J. Algebraic Geom. 19 (2010), no. 4, 679-780.
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G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat. . Lect. Notes in Math. 339, Springer 1973.
G. Mikhalkin. . Proceedings of the International. Congress of Mathematicians, Madrid 2006, 827-852.
G. Mikhalkin and J. Rau. Book in preparation.
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[^1]: Research is supported in part by the NSF FRG grant DMS-0854989 (G.M. and I.Z.), the TROPGEO project of the European Research Council (G.M.) and the Swiss National Science Foundation grants 140666 and 141329 (G.M.).
|
---
abstract: 'Observational evidence for the radial alignment of satellites with their dark matter host has been accumulating steadily in the past few years. The effect is seen over a wide range of scales, from massive clusters of galaxies down to galaxy-sized systems, yet the underlying physical mechanism has still not been established. To this end, we have carried out a detailed analysis of the shapes and orientations of dark matter substructures in high-resolution N-body cosmological simulations. We find a strong tendency for radial alignment of the substructure with its host halo: the distribution of halo major axes is very anisotropic, with the majority pointing towards the center of mass of the host. The alignment peaks once the sub-halo has passed the virial radius of the host for the first time, but is not subsequently diluted, even after the halos have gone through as many as four pericentric passages. This evidence points to the existence of a very rapid dynamical mechanism acting on these systems and we argue that tidal torquing throughout their orbits is the most likely candidate.'
author:
- 'Maria J. Pereira, Greg L. Bryan and Stuart P. D. Gill'
title: Radial Alignment in Simulated Clusters
---
Introduction
============
Anisotropy in galaxy orientations has been a matter of debate for several decades, and many conflicting reports have been published. Past studies have found evidence for three different types of alignment: alignment between clusters [@bin82; @pli02], between the brightest cluster galaxy (BCG) and the satellite distribution [e.g., @yan06] and between the orientation of satellites and their host [@haw75; @per05; @agu06]. This last type of alignment, which we will refer to as *radial* alignment and is the focus of this paper, has been the hardest to confirm [@tre92; @tor07], since it requires high quality data on small scales. In recent years, this field has seen a resurgence, largely due to the arrival of the Sloan Digital Sky Survey (SDSS) [@aba05]. The SDSS provides accurate measurements of isophotal shapes for millions of galaxies, and this has finally allowed large statistical studies of galaxy alignments to be performed. @per05 targeted galaxies in massive X-ray selected clusters and found a significant tendency for their radial alignment. This result has since been confirmed by @fal07a for a larger sample of groups optically selected from the SDSS. On smaller scales, @agu06 found a tendency for satellite galaxies in the SDSS to be radially aligned with their host galaxy, whereas on large scales, @man06 found a very significant correlation between the orientations of galaxies and the surrounding density field traced by galaxy overdensities.
Initially motivated by the prospect of using galaxy orientations to probe their formation histories, these studies are now also driven by the need to calibrate weak lensing and cosmic shear measurements. A key assumption for lensing techniques is that the population of galaxies being lensed is randomly oriented. Some intrinsic alignments between galaxies can be dealt with easily: *e.g.* downweighting close pairs readily removes contamination by alignments induced in interacting systems. However, as @hir03 pointed out, if galaxy orientations are affected by their surrounding density field (*e.g.* a galaxy cluster), then they will also be correlated with the orientations of the background population of galaxies that is being lensed by that field. This correlation between widely separated redshift bins cannot trivially be removed.
Given the growing body of evidence suggesting that galaxy orientations are anisotropic, and the pressing need for an accurate quantification of intrinsic alignments for weak lensing, it seems crucial that we try and find the physical cause behind these anisotropies. There are two commonly proposed explanations. The first, initially developed by @pee69 in his tidal torque theory (TTT), explains the anisotropy as a left-over primordial effect. TTT ascribes the orientation and rotation of galaxies to torquing during their formation. It therefore follows that the signal should be stronger on the outskirts of the cluster, and that it wanes with time, such that older, more relaxed clusters should exhibit less tendency for alignment. The other alternative, proposed by @per05, is a dynamical mechanism, i.e. an interaction with the tidal field of the host cluster that gets progressively stronger during infall and is not erased by subsequent orbital motions. Observational studies have so far been unable to distinguish between the two, due to difficulties in constraining galaxy orbits and in measuring galaxy shapes accurately out to large redshifts.
A different approach is needed, and a few numerical studies have recently been published on this subject. Studies of simulated halo shapes and orientations have been performed around voids [@bru07], along filaments and sheets [@ara07; @alt06; @hah07] and in a Milky-Way type halo [@kuh07]. Anisotropies are reported in all three environments. The advantages of working with simulated clusters are obvious - 3D spatial information means we do not suffer dilution from projection effects. Also, with enough temporal and spatial resolution, we can follow the galaxies as they fall into the cluster along filaments, and beyond, as they orbit inside the cluster. By tracking the effect’s evolution with time, we will be able to more precisely determine its source.
We start (§2.1) by introducing the simulations used for this analysis and describing the properties of the eight host halos. Our methods for finding the substructure halos (§2.2) and determining their shapes (§2.3) follow, along with a study of the reliability of our shape measurements. With this information, we then show in section 3.1 that cosmological dark-matter simulations do indeed produce radial alignment in clusters at z $\approx 0$, and we study the correlation of this effect with various parameters, such as host halo mass and distance from the cluster center (§3.2). Having established the importance of the alignment effect in our simulations, we use the high temporal resolution to study its evolution with time in section 3.3, and its dependence on orbital phase (§3.4). We argue in section 4.1 that tidal torquing by the host halo tidal field is responsible for the alignment of substructure, and compare our results with previous observations in §4.2. We end (§4.3) by briefly speculating on the possible consequences of such a mechanism for the morphological and orbital evolution of galaxies in clusters.
Simulations and Analysis
========================
The Data
--------
The $N$-body simulations used in this work are presented in detail in @gil04, and we describe them here only briefly. Using the open source adaptive mesh refinement code `MLAPM` [@kne01], a set of four initial conditions at redshift $z=45$ in a standard $\Lambda$CDM cosmology ($\Omega_0 = 0.3,\Omega_\lambda =
0.7, \Omega_b h^2 = 0.04, h = 0.7, \sigma_8 = 0.9$) were created. From an initial distribution of $512^{3}$ particles in a box 64[$h^{-1}{\ }{\rm Mpc}$]{} wide and with a mass resolution of $m_p = 1.6 \times 10^{8}$[$h^{-1}{\ }{\rm M_{\odot}}$]{}, the closest eight particles were iteratively collapsed, reducing the particle number to 128$^3$ particles. These low resolution initial conditions were then evolved until $z=0$, at which point eight clusters were selected in the mass range 1–3$\times 10^{14}$[$h^{-1}{\ }{\rm M_{\odot}}$]{}. All particles within two times the virial radius were then tracked back to their initial positions at $z=45$, where they were regenerated to their original mass resolution and positions. These high resolution pockets are surrounded by a “buffer" zone with eight times the original mass resolution, which itself is nested in particles that are 64 times more massive than the particles at the center of the cluster. These initial conditions were then re-simulated to $z=0$, recording 63 outputs from $z=1.5$ to $z=0$ so that $\Delta t \approx 0.17$ Gyrs. A summary of the eight host halos is presented in Table \[HaloDetails\], and on quick inspection it should be immediately apparent that the eight hosts have widely varying masses and assembly histories. We calculate these quantities as follows: the virial radius is defined as the distance at which the average halo density drops below $\rho_{halo}(r_{vir}) = \Delta_{vir}\rho_b$, where $\Delta_{vir} = 340$ and $\rho_b$ is the cosmological background density. The virial mass is defined to be the mass inside this radius. We calculate each host’s age as the time elapsed since their formation, which is defined, following @lac93, at the redshift where the halo first contains half of its present-day mass.
Halo [$R_{\rm vir}$]{} [$M_{\rm vir}$]{} [$z_{\rm form}$]{} age $N_{\rm sat}(<r_{\rm vir})$
------ ------------------- ------------------- -------------------- ------ -----------------------------
\# 1 1.34 2.82 1.18 8.37 166
\# 2 0.97 1.05 0.87 7.17 49
\# 3 1.06 1.38 0.84 7.01 98
\# 4 1.06 1.38 0.75 6.57 71
\# 5 1.34 2.80 0.59 5.65 168
\# 6 1.06 1.39 0.50 5.06 97
\# 7 1.00 1.16 0.43 4.52 54
\# 8 1.37 3.00 0.30 3.42 152
: \[HaloDetails\]Summary of the eight host dark matter halos at $z=0$. Distance is measured in [$h^{-1}{\ }{\rm Mpc}$]{}, mass in 10$^{14}$[$h^{-1}{\ }{\rm M_{\odot}}$]{}, and age in Gyrs. Only satellites with more than 200 particles are tallied in the last column.
Identifying Substructure
------------------------
Simulation outputs merely tell us what the particle spatial and kinetic distribution is at each redshift. They give us no information about particle assignment or halo identity - which particle belongs to which halo? There is no unique answer to this question, mainly because there are many different ways in which a halo can be defined.
A number of sophisticated algorithms have been developed to locate halos within simulations [@dav85; @fre88; @ber91; @sut92; @wei97; @kly97]. They face many challenges: the dynamic environment of cosmological simulations blurs halo boundaries, and halos are continually undergoing mergers or being stripped within a host potential, making it impossible to clearly define a halo edge. Furthermore, most of these do a poor job at finding substructure in very dense background regions, and although nearly all algorithms now use kinetic information to remove gravitationally unbound particles, they are generally not too concerned with background contamination, which can be safely ignored for most applications.
Unfortunately, these issues are especially problematic for our analysis: once the substructure halos cross the virial radius of the host cluster, contrast is lost, and the particle background from the host becomes very significant. If we were to mistakenly assign background particles from the cluster to our substructure halos, this could mimic the radial alignment effect we are looking for, since cluster particles themselves are radially distributed. We solve this problem by finding the substructure halos early on, at the formation redshift ($z_{form}$) for each host. At these early times, the hosts are still starting to assemble and halos are not as clustered, and therefore much easier to identify. Once the halos have been found, their individual particle distributions can then be tracked forward in time through any environment, even into the densest cores of clusters, without suffering from background contamination.
A more detailed description of our halo finding and tracking methods can be found in @gil04, so we provide here only a brief summary. We find and truncate all the halos in our simulation volume at $z_{form}$ using the AMIGA Halo Finder (AHF), the successor of the MLAPM Halo Finder (MHF) [@gil04]. AHF uses the adaptive grids of AMIGA to locate halos within the simulation. AMIGA’s adaptive refinement meshes follow the density distribution *by construction. The grid structure naturally “surrounds” the halos, as the halos are simply manifestations of overdensities. As AMIGA’s grids are adaptive it constructs a series of embedded grids, the higher refinement grids being subsets of grids on lower refinement levels. AHF takes this hierarchy of nested isolated grids and constructs a “grid tree”. Within that tree, each branch represents a halo, thus identifying halos, sub-halos, sub-sub-halos and so on.*
Once we have found all the halos and sub-halos in our simulations at this redshift, we can start tracking their particle distributions through time. The main disadvantage of this method is that any subsequent accretion (after $z_{form}$) onto the halos will, by design, be ignored. This seems a reasonable compromise - halos within halos travel through their environment too quickly to accrete significant amounts of particles and we assume that any particles acquired before the halo enters the host settle into the potential well isotropically, so that we are still obtaining a fair sample of the shape of the halos by only including the particles that were present at $z_{form}$.
At each time step we look at the distribution of particles for each halo and, after recalculating their center of mass, we check if each particle is still bound to the halo. This is an iterative process: starting at the center of the halo and moving outwards, we calculate each particle’s kinetic and potential energy in the halo’s reference frame and remove all particles that have velocities, $v > b v_{esc}$, where $b = 1.5$ is the bound factor, and the only free parameter in our algorithm. We repeat the process until no further particles are removed or a minimum number ($N_p = 200$, *c.f.* §2.3) of particles has been reached. Particles that are determined to be unbound are subsequently ignored. This is a completely effective way of removing the cluster background, as particles that do not belong to the substructure halo will be quickly left behind. It also allows us to track debris being stripped off the subhalos as they orbit inside the cluster.
When all unbound particles have been removed, we fit an NFW distribution to the radial profile of the remaining particles. We define the halo’s radius as the distance at which the average halo density drops below $\rho_{halo}(r_{vir}) = \Delta_{vir}(z)\rho_b(z)$, where $\Delta_{vir}(z)$ is the virial overdensity at that redshift, and discard any particles that lie outside this limit. However, this radius is almost never reached in the case of substructure, in which case the radius of the halo is defined as the distance to the furthest bound particle. Once we have determined which particles belong to which halo at each timestep, we are ready to measure their shapes.
Shape Measurements
------------------
How can we condense a three dimensional particle distribution into a few simple parameters describing its shape? With no prior knowledge of how the particles are distributed this is a difficult task. However, halos produced in dark matter cosmological simulations seem to follow a universal density profile [@nfw96], and are generally well fit by triaxial ellipsoids [@fre88; @all06]. The simplest way to do this is to calculate the inertia tensor of the distribution, $I_{jk} = \sum_i m_i r_{i,j} r_{i,k}$, which is then diagonalized to find the principal axes of the halo. However, this procedure is not ideal, since it weights particles by $r^2$, and therefore results in a shape measurement that is overly biased by the outlying particle distribution.
A better measure [@ger83] is the *reduced* inertia tensor: $$\label{eq inertia tensor}
\tilde{I}_{jk} = \sum_i m_i \frac{ r_{i,j} r_{i,k} }{r_i^2}.$$ which weights particles equally regardless of their distance to the center of the halo, using only the directional information of halo particles to calculate their shapes. The eigenvectors and eigenvalues of this reduced form of the inertia tensor give us the principal axes of the halo and a measure of their relative lengths ($b/a, c/a$), although the latter are substantially overestimated, as we shall see.
The main source of uncertainty in determining the shapes of our halos is the small number of particles we sample their potentials with. We want to characterize halo alignments over as wide a mass range as possible, so we want to know what the minimum number of particles is that will still give us a reliable measure of a halo’s shape. The ability to determine the orientation of a halo’s major axis also depends strongly on the value of $b/a$: An oblate halo with $b \approx a$, will be almost degenerate in its major/intermediate axis orientations.
In order to address these questions, we generated a set of fake triaxial NFW halos with varying numbers of particles ($N_p$) and intermediate-to-major axis ratios ($b/a$) and fed them through our pipeline. For each value of $N_p$ and $b/a$ we performed 100 random realizations of an NFW halo and calculated the angle between the major axis direction measured and that which was input. The dispersion in these values, $\theta_{acc}$, is then a good estimate of the accuracy of our measurement. The minor-to-major axis ratio ($c/a$) does not appear to affect the determination of the major axis direction, and the results presented in figure \[err1\] are therefore only for prolate halos with $b = c$.
As expected, our accuracy depends very strongly on the number of particles sampled - the points on the left panel are well fit by a relation of the form: $\theta_{acc} \propto N^{-0.54}$. We want a compromise between individual halo accuracy and sample size - at values of $N_p<200$, $\theta_{acc}$ increases rapidly, and we pick this, somewhat arbitrarily, for our lower limit on $N_p$. If $b/a = 0.8$, our measurements of these halos would be accurate by $\theta_{acc} \approx {\ensuremath{10^\circ}} $. However, $\theta_{acc}$ also depends strongly on $b/a$. When $N_p = 200$, $b/a < 0.8$ is required to maintain the [$10^\circ$]{} error, with an increase in $b/a$ leading to a rapid decrease in accuracy. Figure \[err1\] refers to the input values of $b/a$. In fact, the measured ellipticities are much higher, although the two are tightly correlated: $(b/a)_{input}=(b/a)^{0.45}_{measured}$. We place an upper limit of 0.8 on the intrinsic axis ratios, which translates to a limit on the measured values of $b/a < 0.9$.
With our limits in place for the minimum number of particles and maximum axis ratios, we are ready to start analyzing our results. It is worth noting, however, that both these error sources would bias our shapes randomly: there is no preferred direction that will be selected if the halos are under-sampled or too spherical. This in turn implies that the results on alignment presented in the next section are, if anything, conservative.
Results {#res}
=======
Alignment at $z=0$ {#z0sec}
------------------
The quantity we will focus on is the angle, $\phi$, between the major axis of each halo and the vector connecting the halo to the center of the host. If halos are oriented randomly in space, the cosine of $ \phi$ will be uniformly distributed between 0 and 1, with a mean value, $\left< \cos{\phi} \right>$, of 0.5. When $\cos \phi \approx 1$ the halo is pointing toward the host center, whereas when $\cos \phi \approx 0$ it is aligned tangentially to it, so that a value of $\left< \cos{\phi} \right> > 0.5$ implies an overall tendency for radial alignment. The standard error on $\left< \cos{\phi} \right>$ is $\sigma_{\left< \cos{\phi} \right>} = \sigma/\sqrt{N}$, where $N$ is the sample size and $\sigma$ is its standard deviation. We show in Figure \[z0\] a histogram of $\cos{\phi}$ for all halos within $2 r_{vir}$ of each of the eight hosts. It is immediately apparent that our distribution is inconsistent with isotropy at a very high significance level: $\left< \cos{\phi} \right> = 0.66 \pm 0.01$, with most halos pointing toward the center of the host halo.
While it is clear that the results of figure \[z0\] confirm previous observational reports of radial alignment, a precise quantitative comparison is rather difficult, and we defer this discussion to §\[obs\]. Nevertheless, much can be learned from a qualitative study of the effect’s behavior and correlation with individual (and host) halo properties. Figure \[z0mass\] shows the same histogram as in Figure \[z0\] but now for two separate halo populations, segregated by mass. There does not appear to be a significant distinction between the two populations. This tells us not only that the alignment effect is mass independent, but also confirms the experiments in §2.3 that show that resolution effects are unimportant in the lowest mass halos considered in our analysis ($N_p>200$).
We can also study how the effect depends on extrinsic characteristics of the halo, *e.g.* the distance to the center of the host, or the host mass. We searched for correlations with different global host properties such as mass and age, and found none. The alignment mechanism appears to be universal, in that it is present with approximately the same strength in hosts with widely varying mass, formation times and assembly histories. This surprising result is also seen in observational studies: @per05 found no correlation of the alignment strength with the dynamical state of the clusters inferred from their x-ray morphologies.
Dependence on Distance to Cluster Center
----------------------------------------
Figure \[z0dist\] shows the dependence of the effect on the distance from the cluster center. All halos at redshifts $z<z_{form}$ are included in this analysis, in order to enhance the overall signal. The behavior appears very smooth: $\left< \cos{\phi} \right>$ rises gradually as the host is approached, peaks slightly past its virial radius, and then decreases again toward the center. It is striking that already at a distance of three virial radii there is a small, consistent, tendency for radial alignment. At this distance, how can the halo already “feel" the presence of the host? This is easily understood once we consider that clusters form at the intersection of filaments, and hence that most filaments will be radially aligned with respect to their nearest clusters. If there is a primordial alignment of halos with respect to the filaments in which they form, then even at large distances this will appear as a radial alignment in our analysis. This type of primordial alignment at large radii was seen by @ara07 in their study of filamentary structures, where they found similar values for $\left< \cos{\phi} \right>$ (their figure 2e).
The main focus of this paper, however, is what happens closer to the host. As the halo falls in, the amplitude of the alignment increases dramatically, reaching a peak of $\left< \cos{\phi} \right> = 0.72$ at about one-half of the virial radius of the cluster, before decreasing again gradually inside the core. The increase of the alignment with decreasing distance matches the behaviour found by @fal07a in their study of SDSS groups, and is to be expected if the effect is caused by the tidal field of the host, but the dip at small radii, $r < 0.3r_{vir}$, has not yet been observed. This is most likely due to the severe projection effects that dominate the cores of observed clusters.
What causes this behavior? It appears that the alignment that is set-up in the infall regions is being disrupted in the inner regions of the cluster. What causes the disruption? Is this primarily a spatial effect caused by the environment of the cluster core, or a temporal one, given that galaxies closest to the center have been in the cluster environment for longer? And what produced the alignment in the first place? The best way to answer these questions is to take advantage of the extra dimension provided by simulations, and explore the evolution of this effect with time.
\[distsec\]
Evolution with Redshift
-----------------------
The evolution of the alignment with redshift is plotted in figure \[zall8\] for each of the eight clusters independently. Perhaps the most striking feature of this plot is the self-similarity of the different curves. Every cluster appears to go through exactly the same evolution, regardless of size or formation time, such that at $z=0$ they are practically indistinguishable, as described in the previous section. Figure \[zall8\] also reveals that the clusters evolve monotonically, with the strength of the effect increasing steadily since the formation time of each cluster to the present day.
Whatever the source of the disruption at the cluster cores, it is seemingly not strong enough to dilute the overall alignment signal. There are two possible explanations: It could be that, even though alignment is disrupted once the halo reaches the core of the host, the constant infall of pristinely aligned halos results in an overall increase of the average alignment per host. Alternatively, the misalignment seen at the cores could be short-lived - a feature of each halo’s orbital motion through the potential of the host.
Distinguishing between these two alternatives requires a different approach: we need to track halos on their way toward the cluster, and then trace their orbits inside the virial radius of the host.
Evolution with Orbital Phase
----------------------------
Figure \[orbit\] shows the alignment evolution stacked for all halos throughout their orbits. Initially halos are tracked relative to the amount of time (in Gyrs) remaining until they cross the virial radius of the host for the first time. Once they cross this threshold, halo orbital times are normalized at each passage through pericenter and apocenter.
We again detect a small alignment at large distances from the cluster , which we believe is evidence for a primordial alignment along filaments as discussed in §\[distsec\]. As the host is approached the signal increases significantly, peaking just before the first pericentric passage, and then a periodic oscillation ensues, which follows the halo’s orbital period closely. On average, the tendency for alignment is much larger within the host than before, although the alignment tendency changes dramatically with orbital phase. It follows that the dip observed near the cluster cores in figure \[z0dist\] is in fact a result of the misalignment observed at pericenter, and, most importantly, that it is not disruptive, since the alignment tendency is restored well before the next apocenter is reached. In fact, the alignment is quite constant throughout the rest of the orbit and seems to increase slightly at each passage. This evidence points to a stable dynamical effect that is set-up as the halo orbits around the cluster.
Further insights can be obtained by exploring the orientations of the halos with respect to their orbits. We define a new angle, $\beta$, as the angle between each halo’s major axis and the halo’s velocity, and plot the mean value of its cosine for all halos vs. orbital phase in Figure \[orbal\]. The similarities in behavior between radial and orbital alignment at large distances are simply a consequence of the radial nature of the orbits themselves - halos form and travel along filaments toward the intersecting nodes where clusters reside. In fact, even inside the hosts, orbits are quite eccentric, with an average apocentric to pericentric distance ratio of $4:1$.
Could it be, then, that the radial alignment we observe within the virial radius is just a tendency for halos to be aligned along their orbits coupled with the fact that orbits are, on average, quite radial? Figure \[orbal\] tells us that this is not the case: once inside the cluster, we find that the orbital alignment is also correlated with orbital phase, but whereas the radial alignment is almost instantly recovered after pericenter, the orbital alignment increases again much more slowly, and only after reaching the next apocenter. This asymmetry around pericenter seems, at first, surprising, but, as will be shown in the following section, follows as a natural consequence of tidal torquing by the cluster potential throughout the halo’s orbit.
Discussion
==========
Tidal Torquing as a Mechanism for Alignment {#tor}
-------------------------------------------
Once radial and orbital alignment information is combined, a clearer picture emerges of what is going on inside these clusters. As the halo approaches pericenter along an eccentric orbit, it is continually torqued along the direction of the potential gradient, i.e. halos tend to point toward the host center, and, because their orbits are fairly eccentric, also along their orbital direction. At pericenter, the halo is moving too fast for the torquing to be completely effective, which causes the dip in radial alignment. It is nevertheless enough to torque the halo away from its orbital direction and back toward the cluster center, in a figure rotation that is co-planar with its orbital rotation and in the same direction. The radial alignment is quickly reinstated, but orbital alignment is lost as the halo progresses towards apocenter. Steady torquing throughout the orbit keeps halos oriented toward the cluster center and away from the direction of their orbits until after the apocentric passage, where orbital alignment increases steadily towards pericenter, and a new cycle begins. Figure \[sketch\] illustrates this behaviour with a sketch of a halo’s rotation as it orbits around the cluster.
If halo orbits were circular, halos would quickly become tidally locked and maintain radial alignment throughout their orbits. In reality, their orbits are quite eccentric, and their orbital speed varies significantly. Halos do not react to the tidal torquing quickly enough through the pericentric passage, and the narrow dips observed are the result. In fact, idealized numerical experiments involving a single halo in a circular orbit around a static host invariably lead to tidal locking of the halo, although the time required for locking varies significantly with the original orientation of the halo (C. M. Simpson & K. V. Johnston, private communication). Interestingly, for halos that start out already pointing toward the host center, the time required is rather short, of the order of an orbital period or less. Further support for this tidal torquing hypothesis is shown in figure \[orbitb\]. Although we believe our shape measurements to be robust to random outliers, it is possible that strongly distorted outer shells, caused, e.g., by tidal stripping, could significantly bias the result. As a test, we apply four different particle cuts to each of our halos by varying the boundedness criteria on the particle velocities: instead of throwing out all particles for which $v>b v_{esc}$, where $b = 1.5$, we exclude alternately particles that have velocities greater than 1, 0.75 and 0.5 times the escape velocity. For the most conservative criteria, which only retains particles that have velocities $v < 0.5 v_{esc}$, more than $70\%$ of the particles are discarded, and we are only probing the very bound cores of the halos. Figure \[orbitb\] makes clear that stripping cannot possibly be the sole cause of the alignment effect, since even the most conservative cut shows significant alignment.
Nevertheless, a trend is observed, in that the most bound particles show slightly less tendency for alignment overall. This could be the result of tidal stripping in the outer layers, but more likely it is a simple statistical effect: Because we only consider halos with $N_p>200$ in this analysis, as we progressively exclude more particles from the halo with decreasing $b$, some halos fall below this limit and are consequently ignored. Hence a decrease in $b$ implies a smaller number of halos in each sampled bin, which reduces the signal-to-noise, and brings $\left< \cos{\phi} \right>$ closer to $0.5$. Despite this trend, the conclusion remains that halo shapes are not significantly warped by tidal stripping, and that tidal torquing of the entire halo is a better explanation for the effect.
A number of early numerical and analytical studies support the importance of tidal torques within clusters . @mil82 performed a set of numerical experiments on a rotating bar in an external force field and observed tidal braking of the rotation, with a rate that was inversely related to the square of the cluster crossing time. More recently, numerical $N$-body experiments by @cio94 showed that the time required for the alignment of a prolate galaxy with the tidal field of a cluster is much shorter than the Hubble time, and on the order of a few times the galaxy’s intrinsic dynamical time. Using a different approach, @usa97 studied tidal effects on gaseous ellipsoids orbiting in a central potential analytically, predicting that galaxies in eccentric orbits should have their long-axis trapped toward the direction of the radius vector of the cluster.
While this paper was being written, two studies were published on halo alignments that describe similar results. @kuh07 studied the alignment of substructure around a Milky Way type halo using the Via Lactea simulation, and observed a radial alignment tendency that is preserved throughout the halos’ orbits. @fal07b looked at several different types of alignment in a set of dark matter hosts at $z=0$, finding similar levels of radial alignment that increase with decreasing distance to the host.
A Comparison with Observations {#obs}
------------------------------
The results of §\[res\] certainly seem to substantiate the observational evidence for radial alignment of cluster galaxies. A quantitative comparison, however, is not easily made. In order to properly “observe" these simulations, semi-analytic models of galaxy formation are required to extrapolate from the dark matter halos to the luminous components embedded within. These then need to be projected, interlopers and survey limits accounted for, and the resulting image fed through traditional source extraction and isophotal analysis pipelines. This is a laborious procedure and it cannot yet provide accurate results, since galaxy formation models are still largely unconstrained in a crucial parameter: the alignment between luminous and dark matter.
Observationally, studies of the alignment and relative ellipticity of the two components are currently only possible for gravitational lens galaxies, a very rare class of objects. @kee98 analysed a sample of 17 lenses, mostly isolated early-types, and found that the luminous component of the lens generally aligns with its inner halo to $\le$ [$10^\circ$]{}. In order to probe the shapes of the halos to larger radii, stacked galaxy-galaxy weak lensing studies are needed. These are just now becoming feasible, and preliminary results appear somewhat contradictory [@hoe04; @man06b].
Most theoretical studies have concentrated on the formation of disk galaxies and their angular momentum, where some misalignment between baryonic and dark matter spin is commonly seen (e.g. @bos03). On the other hand, @bai05 find that the orientations of simulated halos and their embedded disks are largely uncorrelated at large radii, and almost perfectly aligned at small ($r< 0.1r_{vir}$).
The current uncertainty in this parameter makes it impossible to accurately predict the orientation of the galaxies that would populate our halos. However, the results presented in this paper suggest a gravitational origin for the alignment mechanism, and it is therefore reasonable to expect that the two components should react similarly to it. Furthermore, the tidal torquing within clusters is so effective that the halos appear to “forget" their original orientations before a single orbit is completed, which renders the original alignment between light and dark matter relatively unimportant.
We therefore compare the dark matter alignment directly with the galaxy observations of @per05. We project each halo’s dark matter particles along the three spatial dimensions in our simulation and compute the 2D inertia tensor of their projected distribution. The angle between the halo’s 2D major axis and its projected separation from the cluster center can then be measured. We include in this sample all galaxies within 2 virial radii of the cluster center - interlopers are not accounted for, since the SDSS galaxies we are comparing our results to are all spectroscopically confirmed cluster members. Figure \[2d\] shows the results of this 2D analysis and compares them with the SDSS observations. We plot all three independent projections and note that the dispersion in their values should give us a fair estimate of the error introduced by the projection procedure itself.
The dark matter alignment is much stronger than that observed: $\left< \theta \right> _{halo} = {\ensuremath{34^\circ}}.5 \pm {\ensuremath{0^\circ}}.9$ whereas $\left< \theta \right> _{gal} = {\ensuremath{42^\circ}}.79 \pm {\ensuremath{0^\circ}}.55$. This must be in some part a reflection of how much harder it is to measure accurate galaxy position angles on an survey image than for a well-resolved halo in a cosmological simulation, where dynamical information allows for a much cleaner background removal. Nonetheless, the dilution caused by this measurement noise cannot wholly account for the significant difference in radial alignment between the two components. Given the nature of the alignment mechanism established in §\[tor\], it is perhaps not too surprising that dark matter halos are more strongly aligned. One would naively expect the dark matter halos to be more easily torqued, given that they are much more extended (providing a longer lever) and have generally lower spins ( and therefore less gyroscopic resistance) than their luminous counterparts.
Possible Consequences of Tidal Torquing in Clusters
---------------------------------------------------
Halo alignments have traditionally been studied either as a probe of their formation history, or as a contaminant to weak lensing studies. Now that we have established that the leading mechanism behind halo alignments within clusters is a dynamical effect present throughout their lifetime, it is interesting to speculate on what possible evolutionary consequences this mechanism might have for the halos affected.
Figure \[sketch\] shows us that at each point in the orbit, the torque acts to rotate the halo away from its orbital direction, which necessarily results in a deceleration of the halo’s orbital motion, inducing orbital decay. The halos analysed in this study do indeed show a tendency for orbital circularization: @gil04b showed that halos with more pericentric passages have smaller orbital eccentricites. They argued that dynamical friction was not a likely cause, and tentatively ascribed the effect to the growth of the host halo instead. While the velocity dispersion of the satellites is seen to depend on the host halo mass, it seems possible that at least part of the orbital decay observed is a natural result of the constant torquing throughout the halo’s orbit. This is an interesting prospect, since an extra source of orbital decay could potentially help solve the outstanding problem of cD formation in massive clusters, as well as alleviate some unresolved discrepancies between observed satellite populations and the generally low efficacy of dynamical friction predicted by numerical studies (*e.g.* @has03 [@taf03]). We are currently investigating the importance of this induced orbital decay, and this will be the subject of a future paper.
Another possible consequence of the strong torquing of dark matter halos within hosts is the possibility of disk warping. Because of their high angular momentum, disks will naturally resist tidal torquing more effectively than the surrounding dark matter halo, which will introduce a misalignment between the halo and the disk. Even though recent studies of (isolated) disk-halo alignments show that their orientations are largely uncorrelated at large radii [@bai05], the same is not true for the inner halos ($r<0.1r_{vir}$), where the rotational axis of the disk is seen to lie very close to the minor axis of the inner halo. We have shown that tidal torquing affects all particles in the halos, even the most bound, so it is not unreasonable to expect that the inner shells should also feel these torques. The question then remains whether the disk within will align itself accordingly, or whether the misalignment could be a possible cause of warping of the disk, but this will also require further study.
Conclusions
===========
There is growing observational evidence that a satellite’s major axis is preferentially aligned with the radial vector linking the satellite to its host. This tendency for satellites to point at their hosts has been seen on both cluster and group scales [@per05; @agu06]. Motivated by this result, we have used a suite of cosmological N-body simulations to investigate the alignment between satellite and host dark matter halos.
We take particular care to separate satellite and cluster particles using a combined halo finder plus tracker. In this method, an adaptive halo finder [@gil04] is used to initially identify a set of satellite sub-halos that we subsequently track as they enter and orbit the cluster, removing particles as they become unbound. The advantage of this approach is that we can be sure to use only genuine sub-halo particles and exclude “background" cluster particles that might bias our shape measurements. We then use the reduced inertia tensor to measure the shapes and orientations of all sub-halos which end up inside the virial radii of a set of eight simulated clusters. We highlight here the main results obtained from this analysis:
- The satellites in the simulations show a strong tendency to point toward the cluster center. The mean cosine of the angle between the major axis of each halo and the cluster center is $\left< \cos{\phi} \right> = 0.66 \pm 0.01$, where an isotropic distribution would have $\left< \cos{\phi} \right> = 0.5$. This tendency for alignment is found for all clusters at all redshifts analyzed, and does not appear to depend on the mass of the cluster or the satellite.
- The amplitude of the alignment is a strong function of radius, with a small but significant effect extending out to many virial radii from the cluster. This signal, which has been seen in previous work [@ara07], is most likely left over from the primordial imprint of the surrounding large-scale structure and can be ascribed to tidal torques exerted at early times, when the cluster-size perturbations were just turning around [e.g., @pee69].
- Closer to the cluster center, within 1-2 virial radii, the amplitude of the alignment increases dramatically to a peak of $\left< \cos{\phi} \right> = 0.72$ at about one-half of the virial radius, and then falls slowly closer to the cluster center.
- When examined as a function of orbital phase for a given satellite, we find that the alignment increases rapidly as the satellite falls into the cluster for the first time and remains high after that, except for a short period during pericenter passage, when it dips precipitously. It is this short-lived dip which gives rise to the decrease in $\left< \cos{\phi} \right>$ close to the cluster center.
Based on these results, we conclude that the strong alignment seen at small radius — within two virial radii — is due to [*tidal torquing*]{} by the cluster halo. The idea is very simple – the galaxy is only in a stable equilibrium if it is pointing at the cluster center; otherwise there is a net torque which acts to rotate the galaxy towards this equilibrium point. We demonstrate that the alignment is seen both in the outer and inner parts of the satellite, indicating that it is not due to some process (such as tidal stripping) which impacts only the outer, poorly bound, part of the sub-halo. We also briefly review previous literature which has investigated the impact of tidal torques on collisionless systems using analytic approximations or idealized simulations, and find that the expected amplitude and timescale is sufficient to produce the alignments we see.
Although we study only dark-matter halos, we expect this effect to extend to the luminous part of galaxies, as observations seem to indicate. This will have an observational impact on weak lensing studies and may also modify the distribution of stars in a satellite, as well as the satellite’s orbital properties. We will investigate these possibilities in future work.
The simulations presented in this paper were carried out on the Beowulf cluster at the Centre for Astrophysics & Supercomputing, Swinburne University. We would like to thank Jeff Kuhn, Kathryn Johnston and Christine Simpson for helpful discussions. Greg Bryan acknowledges support from NSF grants AST-05-07161, AST-05-47823, and AST-06-06959, as well as the National Center for Supercomputing Applications.
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abstract: 'How much information about an unknown quantum state can be obtained by a measurement? We propose a model independent answer: the information obtained is equal to the minimum entropy of the outputs of the measurement, where the minimum is taken over all measurements which measure the same “property” of the state. This minimization is necessary because the measurement outcomes can be redundant, and this redundancy must be eliminated. We show that this minimum entropy is less or equal than the von Neumann entropy of the unknown states. That is a measurement can extract at most one meaningful bit from every qubit carried by the unknown states.'
address: |
$^a$ Service de Physique Théorique, Université Libre de Bruxelles,\
CP 225, Bvd. du Triomphe, B1050 Bruxelles, Belgium.\
$^b$ Isaac Newton Institute, University of Cambridge, Cambridge CB3 0EH, U.K.\
$^c$ BRIMS, Hewlett-Packard Labs., Bristol BS12 SQZ, U.K.
author:
- 'S. Massar$^a$ and S. Popescu$^{bc}$'
title: 'How Much Information can be Obtained by a Quantum Measurement ?'
---
Introduction
============
Quantum mechanics has at its core a fundamental statistical aspect. Suppose you are given a single quantum particle in a state $|\Psi\rangle$ unknown to you. There is no way to find what $|\Psi\rangle$ is - to find it out you need an infinite ensemble of quantum particles, all prepared in the same state. Indeed, the different properties which characterize the state are, in general, complementary to one another; measuring one disturbes the rest. Only if an infinite ensemble is given can one find out the state. But infinite ensembles don’t exist in practice. Given a finite ensemble of identically prepared particles, how well can one estimate the state? The problem is a fundamental one for understanding the very basis of quantum mechanics. It has been investigated by many authors, see for instance [@Hel][@H2], and it constitutes probably the oldest problem in what is at present called “quantum information". Here we approach this problem from a new point of view which, we think, leads to a deeper understanting.
What is the optimal way to estimate the quantum state given a finite ensemble? As such the question is not well posed. Indeed, since we cannot completely determine the state , i.e. completely determine all its properties, we must decide which particular property we want to determine. For an ensemble of spins, for example, estimating as well as possible the mean value of the $z$ spin component is, obviously, a different question than estimating as well as possible the mean value of the $x$ spin component.
But things are in fact even more complicated. The apparent benign words “as well as possible" in the previous paragraph are not well defined. Indeed, “as well as possible" actually means “as well as possible given a specific measure of what “well“ means”. Obviously, one can imagine many different measures. For example, suppose that a source emits states $|\psi_i\rangle$ with probability $p_i$. The problem is to design a measurement at the end of which we must guess which state was emitted. Let the guess be $|\phi_j^{guess}\rangle $, and let the measure of success (fidelity) be $$F_{ij}=|\langle \phi_j^{guess}|\psi_i\rangle |^2\ ,\label{1}$$ i.e. the absolute value square of the scalar product in between the true state $|\psi_i\rangle$ and the guess $|\phi_j^{guess}\rangle$. The goal is to optimize the measurement such that it yields the highest average fidelity $$F=\sum_{i,j} p_i F_{ij}p(j|i)\ .\label{2}$$ where $p(j|i)$ is the probability to make guess $j$ if the state is $|\psi_i\rangle $. On the other hand, one can imagine another fidelity function, such as $$F'_{ij}=|\langle \phi_j^{guess}|\psi_i\rangle |^4\ .
\label{3}$$ Or one could try to optimize the mutual information $$I=
- \sum_i p_i \ln p_i + \sum_j p_j \sum_i p(i|j)\ln p(i|j)
\label{4}$$ or any other measure.
The important point to notice about the above different problems is that the different fidelities (2-4) not only define different scales according to which we measure the degree of success in estimating the state, but also, implicitly, define which property of the state we are actually estimating. If all the different fidelities where to lead to the same optimal measurements, we could say that we learn the same property about the state but just expressed in a different way. However the different fidelities will in general lead to different optimal measurements which means that in each case we learn a different property about the system.
To summarize, in general each particular estimation problem is completely different from the other, they measure different properties and their degree of success is measured on different scales, with the scales also defining implicitly what exactly is the property we estimate.
That one can learn different properties is a fact of life inherent to quantum mechanics. But there is no reason not to use the same scale to gauge how successful we have been in learning the property we decided to measure. The aim of this paper is to propose such a universal scale, and in the process to introduce a novel approach to quantum state estimation.
Main idea
=========
The central point of our approach starts from a simple but fundamental question: what do we actually learn from a measurement on a state? Let us illustrate this question by an example. We shall contrast two situations. Consider a source which emits spin 1/2 particles. In the first case the particles are polarized with equal probability along either the $+z$ ($|\uparrow_z\rangle $) or $-z$ ($|\downarrow_z\rangle $) directions. In the second case the states are polarized along random directions uniformly distributed on the sphere. Suppose we want to identify the states as well as possible according to the fidelity eq. (\[2\]). In the first case it is obvious that a measurement along $\sigma_z$ perfectly identifies the state, hence the fidelity is $F=1$. In the second case, it has been shown [@MP] that the measurement along $\sigma_z$ is also optimal. But in this case the states cannot be identified perfectly, and the fidelity is only $F=2/3$.
Nevertheless the two situations seem extremely similar. In both cases we perform the same measurement. And in both cases before we perform the measurement we know that the outcomes of the measurement are either $+1$ or $-1$, and the a priori probabilities of the two outcomes are equal. When we perform the measurement this uncertainty is resolved. Hence in both cases the measurement yields 1 bit of information. Our main idea is to interpret this quantity as the information we extract from the state. Incidentally we note that in both cases this information (the Shannon information of the outcomes) equals the von Newmann entropy of the unknown states (both are equal to 1).
This idea might seem paradoxical at first sight because in one case we completely recognize the state whereas in the other case we recognize it badly. To understand let us introduce a classical source that decides which quantum state is emitted from the quantum source (see figure 1). In the first case the classical source must only specify one bit (either $+z$ or $-z$) to determine which state is emitted. In the second case it must provide a direction $\underline{n}_{in}$ (ie. an infinite number of bits) in order to specify the state $|\uparrow_{\underline{n}_{in}}\rangle $. In both cases one extracts one bit of information. In the first case this means that the classical information supplied by the source is completely recovered. In the second case information is lost. However it is now clear that the loss does not occur during the measurement, but during the first step, where classical information is converted into quantum.
(300,200) (10,100)
To summarize, the quantum state estimation problem as presented in figure 1 consists of a chain of events which starts with a classical source that tells the quantum source what state to emit, and ends with the measurement. The fidelity measures the overall performance of the chain since it is proportional to the scalar product $\underline{n}_{in} . \underline{n}_{guess}$. On the other hand the number of bits in the output characterizes how much information is extracted by the measurement. Therefore in this article we shall focus on the latter quantity.
Main Result
===========
The preceding discussion suggests that the Shannon information of the outcomes $$I_{output}^S = - \sum_j p_j \ln p_j\ ,$$ where $p_j= \sum_i p(j|i)$ is the probability of outcome $j$, measures how much information is extracted from the state. This idea however has to be refined.
The main problem is that there may be redundancies in the outputs of the measurement. As a trivial example, a measurement could be accompanied by the flip of a coin, and the outcomes of the measurement would consist of both the outcomes of the measurement proper and the outcomes of the coin flip. This adds one bit to the entropy of the outputs without telling anything about the system. In less trivial examples involving POVM’s and ancillas, redundancies can arise in a less obvious way, and it is not immediate how they can be identified and eliminated.
Our main result is that no matter what property of the system one wants to measure, when the redundancy is eliminated, the remaining Shannon information of the outputs has a universal upper bound which is the von Neumann entropy of the quantum source: $$I_{output}^S (no\ redundancy) \leq I^{VN}_{input}\ ,
\label{S}$$ where $I^{VN}_{input} = -Tr \rho \ln \rho$ is the Shannon information of the quantum source and $\rho$ is the density matrix of the quantum source $\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|$.
One does not always attain equality in eq. (\[S\]). Indeed some questions are more informative about the system then others. Less informative questions can be answered by measurements whose output entropy is smaller. More informative questions require measurements with more entropy. But the most detailed questions can always be answered in $I^{VN}_{input}$ bits.
Strategy
========
The main problem we face in deriving eq. (\[S\]) is to eliminate the redundancy. In order to do this we shall proceed in several steps.
1. The first step is to decide which property we are interested in. We may fix the property directly (for instance decide to measure the average of $\sigma_z$) or implicitly by choosing a fidelity. In the rest of the this paper we shall adopt the second approach.
2. We then look at optimal measurements, that is measurements which maximize the fidelity. In general there is an entire class of such measurements.
3. We perform a second optimization. Namely among the optimal measurements we look for the measurements which minimize $I_{output}^S$.
This double optimization strategy has already been considered for some particular cases in [@DBE][@LPT][@LPT2].
One expects that this strategy yields measurements which have no spurious redundancy. However as we will find out later through some examples, redundancies cannot be completely eliminated by the above procedure and we will have to further modify it.
These further modifications are motivated by the classical and quantum theory of information[@Sh][@S] which suggest the idea of performing measurements on blocks of quantum states, rather than on individual particles. Thus we shall allow the measuring device to accumulate a large number $L$ of input states before making a collective measurement on the $L$ states simultaneously. It is in the context of these collective measurements that we make the two optimizations (points 2 and 3 above) and thereby eliminate the spurious redundancies.
We want to emphasize that this procedure cannot increase the fidelity since the subsequent particles are completely uncorrelated. However by considering measurements on large blocks we can hope to reduce the redundancy of the measurement, ie. the entropy of the outcomes, by making “better use” of each outcome.
Two technicalities have to be taken into account. First of all we must take care not to modify the definition of fidelity as we go from measurements on single particles to block measurements. That is the fidelity must still be the fidelity of each state individually, rather than the fidelity for the whole block. Second we should not require the measurement to absolutely maximize the fidelity, since then using block measurements does not help to reduce the entropy (this follows once more from the fact that the subsequent states are completely uncorrelated). However, following the ideas of information theory, we shall only require that the measurement has a fidelity approaching arbitrarily closely the optimum. In this framework we shall prove eq. (\[S\]).
To summarize, there is no best way of estimating an unknown quantum state. Different measurements will learn about different properties of the state, and it is up to us to choose which property we want to learn about. However once we fix the property we want to learn about, we show that quantitatively one cannot learn more than $I_{input}^{VN} = -
Tr \rho \ln \rho$ bits about this property. That is a measurement can extract at most one meaningful bit from each qubit coming from the source.
Examples {#example}
========
Before embarking on a proof of our result, we give two examples which illustrate the main points that must be taken into account in the proof.
In the first example there are two possible input states $|\psi_1\rangle = \alpha
|\uparrow\rangle + \beta |\downarrow\rangle $ and $|\psi_2\rangle = \alpha
|\uparrow\rangle - \beta |\downarrow\rangle $ which occur with equal probability. The density matrix of the source is $\rho = \alpha^2
|\uparrow\rangle \langle \uparrow| + \beta^2 |\downarrow\rangle
\langle \downarrow|$ which is different from the identity for $\alpha \neq \beta$ Therefore the von Newmann entropy of the input states $I_{input}^{VN} < 1$ qubit.
In this example we use a fidelity defined as follows: after each measurement one must guess whether the state is $|\psi_1\rangle $ or $|\psi_2\rangle $. In case of a correct guess one receives a score of $+1$, and for an incorrect guess one receives a score of $-1$. The aim is to maximize the average score. The techniques of section \[fidelity\] can be used to show that the optimal measurement is a von Neumann measurement of $\sigma_x$, see figure 2. The two outcomes of this measurement occur with equal probability, and hence $I_{output}^S=1 > I_{input}^{VN}$.
(110,60) (60,20)[(3,2)[42]{}]{} (105,45)[$\psi_1$]{} (60,20)[(-3,2)[42]{}]{} (5,45)[$\psi_2$]{} (60,20)[(1,0)[50]{}]{} (105,25)[$\uparrow_x$]{} (60,20)[(-1,0)[50]{}]{} (5,25)[$\downarrow_x$]{}
In this example, a natural first step in eliminating the redundancy is to project blocks of input states onto their probable subspace[@S][@JS]. This projection succeeds with arbitrarily high probability, and affects the input states arbitrarily little. But it reduces the dimensionality of the Hilbert space of the input states from $2^N$ to $2^{N I_{input}^{VN}}$. Hence if we can prove that there is a von-Newman measurement restricted to the probable subspace that is optimal, we will have proved our claim. However the construction of such a von-Newmann measurement is non trivial, as is illustrated in the next example.
In our second example there is no “most probable” subspace because the density matrix of the inputs is completely random. In this example there are three input states $|\psi_1\rangle = |\uparrow\rangle $, $| \psi_2\rangle ={1 \over 2}
|\uparrow\rangle
+{\sqrt{3} \over 2}|\downarrow\rangle $, $\psi_3\rangle ={1 \over 2}
|\uparrow\rangle
-{\sqrt{3} \over 2}|\downarrow\rangle $, each occurring with equal probability $p_i = 1/3$. The density matrix of these states is $\rho = I/2$ and their entropy is $I_{input}^{VN} = 1$ qubit. The fidelity is defined as above: after the measurement one must guess which was the input state. If the guess is correct one scores $+1$ point, if the guess is incorrect, one scores $-1$ points. The aim is to maximize the average score (fidelity).
(110,110) (50,50)[(0,1)[50]{}]{} (55,95)[$\psi_1$]{} (50,50)[(3,-2)[42]{}]{} (87,30)[$\psi_2$]{} (50,50)[(-3,-2)[42]{}]{} (0,30)[$\psi_3$]{}
Using the techniques of section \[fidelity\], one can show that the elements of an optimal POVM are necessarily proportional to the three projectors $|\psi_1\rangle \langle \psi_1| ,
|\psi_2\rangle
\langle \psi_2| , |\psi_3\rangle \langle \psi_3|$, see figure 3. Therefore the optimal POVM whose output entropy is minimum is $\{
{2 \over 3} |\psi_1\rangle \langle \psi_1| , {2 \over 3}
|\psi_2\rangle
\langle \psi_2| , {2 \over 3} |\psi_3\rangle \langle \psi_3|\}$. In this case $I_{output}^S = \ln 3 > 1$ bits. The other optimal measurements have larger $I_{output}^S > \ln 3$ bits. One can also show that there is no measurement on blocks of $L$ input states whose fidelity is strictly equal to the optimum and whose output entropy is less then $L\ln
3$ bits. However if one only requires that the fidelity is arbitrarily close to the maximum, then in the asymptotic limit ($L\to \infty$) the output entropy can be made arbitrarily close to $L$ bits, thereby attaining the bound eq. (\[S\]). The main difficulty of the proof will be to construct such a measurement on large blocks whose output entropy is equal to $L$ bits and whose fidelity is arbitrarily close to the optimal fidelity.
Plan of the Proof
=================
The main part of this paper is devoted to proving the bound eq. (\[S\]). In section \[fidelity\] we introduce a large class of fidelities, and derive some properties of the optimal measurements. In section \[Fid\] we show how to generalize these fidelities to measurements on large blocks of input states. At the end of section \[Fid\] we are in a position to state with precision a first version of our main result, eq. (\[S\]). In section \[other\] we extend the notion of fidelity and state a slightly more general version of our result. In section \[comp\] we show how to construct a measurement on large blocks which has little redundancy. In section \[optimal\] we derive an intermediate result concerning the fidelity of the measurement constructed in section \[comp\]. If the states are uniformly distributed in Hilbert space (ie. the density matrix is proportional to the identity, $\rho =
I/d$) , then this intermediate result already proves our main claim eq. (\[S\]). When the states are not uniformly distributed in Hilbert space, we must first project blocks of states onto the probable subspace before using the intermediate result of section \[optimal\]. This is done in section \[prob\] and completes the proof of eq. (\[S\]).
Fidelity
========
Let us consider the general setup described in figure 1. The states emitted by the quantum source $|\psi_i\rangle$ belong to a Hilbert space of dimension $d$. They occur with probability $p_i$. Their density matrix is $\rho = \sum_i p_i |\psi_i\rangle \langle \psi_i|$ with $Tr \rho =1$. The most general measurement on the input states is a POVM with $M$ element: $a_j \geq 0$, $\sum_{j=1}^M a_j = I_d$.
We introduce the fidelity in the following way. To each outcome $j$ of the measurement we associate a state $|\phi_j^{guess}\rangle $ which is our “guess” as to what the input state was. The correctness of this guess is measured by a function, the fidelity, which depends on the input state and the guessed state $f(|\phi_j^{guess}\rangle ,|\psi_i\rangle )$. For instance $f$ could have the form eq. (\[1\]) or (\[3\]). The mean fidelity is then: $$F = \sum_i p_i \sum_j p(j|i)
f(|\phi_j^{guess}\rangle ,|\psi_i\rangle )\ .
\label{F}$$ where the probability to obtain outcome $j$ if the state is $ |
\psi_i\rangle $ is $$p(j|i) = \langle \psi_i| a_j | \psi_i\rangle$$ An optimal measurement is one which maximizes the mean fidelity $F$.
This is a rather general formulation of the state estimation problem. However the fidelity is not the most general one could consider. To see this let us consider the optimization of $F$. When we make the optimization, we must compare the value of $F$ for different POVM’s, [*however the guessed states $|\phi_j^{guess}\rangle $ are kept fixed.*]{} That is the guessing strategy is fixed once and for all, and we try to optimize the measurement for fixed guessing strategy. The advantage of formulating the fidelity in this way is technical: it ensures that the fidelity depends linearly on the POVM elements. We shall show in section \[other\] how to extend our result to more general fidelities for which the guessed states $|\phi_j^{guess}\rangle $ are not kept fixed.
We summarize here the main properties of optimal measurements for the fidelity eq. (\[F\]), see also [@H2][@MP].
First of all note that we can always take the optimal POVM to consist of one dimensional projectors $b_j = |b_j\rangle \langle b_j|$ (The $b_j$ are not normalized). Indeed refining a POVM can only increase the fidelity. This can be seen formally in the following way: suppose the $a_j$ are an optimal POVM, but not necessarily made out of one dimensional projectors. Then each $a_j$ can always be decomposed as $a_j = \sum_k |b_{jk}\rangle \langle b_{jk}|$ since it is a positive operator. Inserting this into the expression for $F$ one sees that the $b_{jk}$ (to which we associate the guessed state $\phi_j^{guess}$) are also optimal.
Thus we can optimize $F$ in the class of POVM’s whose elements are one dimensional projectors $|b_j\rangle \langle b_j|$. These projectors are subject to the unitarity condition $\sum_j |b_j\rangle \langle b_j| =I_d$. This can be implemented by introducing $d^2$ Lagrange multipliers $\lambda_{\mu\nu}$ which we group into one operator $\hat \lambda$: $$\begin{aligned}
F & =& \sum_i p_i \sum_j\langle \psi_i | b_j\rangle \langle b_j |
\psi_i\rangle f(\psi_i, \phi_j^{guess})
- Tr [ \hat \lambda (\sum_j| b_j\rangle \langle b_j | - I_d)]\nonumber\\
&=& \sum_j Tr [ (\hat F_j - \hat \lambda) | b_j\rangle \langle b_j | ]
+
Tr \hat \lambda\ ,
\label{FFF}\end{aligned}$$ where $\hat F_j = \sum_i p_i | \psi_i\rangle \langle \psi_i |
f(\psi_i, \phi_j^{guess})$. If we vary this with respect to $\langle b_j|$, we obtain the equations $$(\hat F_j - \hat \lambda) | b_j\rangle =0\ .
\label{flb}$$ Inserting this into eq. (\[FFF\]) shows that $F= Tr \hat \lambda$.
Eq. (\[flb\]) is the essential equation to find optimal measurements explicitly. For instance consider the first example of section \[example\]. There are two input states $\psi_1$ and $\psi_2$ and two guessed states $\phi^{guess}_1 = |\psi_1\rangle $ and $\phi^{guess}_2 = |\psi_2\rangle $. If the input state is $|\psi_1\rangle $ and one guesses $\phi^{guess}_1$, then $f=+1$, whereas if the input state is $|\psi_2\rangle $ and one guesses $\phi^{guess}_1$, then $f=-1$, hence $\hat F_1 = {1\over 2} (|\psi_1\rangle \langle \psi_1| -
|\psi_2\rangle \langle \psi_2|)= +\alpha
\beta \sigma_x$. Similarly $\hat F_2 = {1\over 2} (|\psi_2\rangle \langle \psi_2| -
|\psi_1\rangle \langle \psi_1|)= -\alpha
\beta \sigma_x$. The task is then to find an operator $\hat \lambda$ such that null eigenvectors of $\hat F_{1,2} - \hat \lambda$ can satisfy the completeness relation. The only possibility is $\hat \lambda = \alpha \beta
I$. Therefore the optimal measurement is along the $x$ axis, and $F_{max} = 2 \alpha \beta$. The second example of section \[example\] can be treated along similar lines.
An important consequence of eq. (\[flb\]) is an explicit expression for the value of $F$ if the measurement is not optimal. Consider a measurement $a'_j$ which is not optimal, but each positive operator $a'_j$ is “close” to the corresponding operator $b_j$ of the optimal measurement. We then decompose the operator $a'_j$ in terms of its components along $|b_j\rangle $: $a'_j = X_j |b_j\rangle \langle b_j| + Y_j|b_j\rangle \langle
b_j^\perp| +
Y^*_j |b_j^\perp\rangle \langle b_j| +
z_j$ where the state $|b_j^\perp\rangle $ is orthogonal to $|b_j\rangle $ and the operator $z_j$ obeys $z_j |b_j\rangle =0$, $\langle b_j| z_j =0$. Inserting this decomposition into the expression for $F$, we obtain $$\begin{aligned}
F(a') &=& Tr \hat \lambda + \sum_j
Tr [ (\hat F_j - \hat \lambda) a'_j]
\nonumber\\
&=& F_{max} + \sum_j Tr [ (\hat F_j - \hat \lambda) z_j]
\nonumber\\
&\geq& F_{max} - C \sum_j Tr z_j \ ,\label{boundF}\end{aligned}$$ where we have used eq. (\[flb\]) and $C$ is some positive constant independent of $j$. This expresses in a simple way how much the fidelity differs from its maximal value in terms of how much the measurement differs from the optimal measurement.
Fidelity for measurements on large blocks {#Fid}
=========================================
As discussed above it is necessary to also consider measurements on large blocks of $L$ input states $| \psi_{i_1}
... \psi_{i_L}\rangle $. The fidelity for measurements on large blocks is $$F_L = \sum_{i1,...,i_L} p_{i_1}...p_{i_L}
\sum_{j=1}^N \langle \psi_{i_1} ... \psi_{i_L}| A_j | \psi_{i_1}
... \psi_{i_L}\rangle {1 \over L} \sum_{k=1}^L f(\psi_{i_k} ,
\phi^{guess}_{j_k}) \ ,
\label{FL}$$ where $A_j$ is the measurement on the $L$ input states. The guessed state is the product $|\Phi_{j}^{guess}\rangle =
|\phi_{1_j}^{guess}
... \phi_{L_j}^{guess}\rangle $. The fidelity is taken to be the average of the fidelities for each state $| \psi_{i_1}\rangle ,...,|\psi_{i_L}\rangle $. This ensures that eq. (\[FL\]) is just the average of the fidelities eq. (\[F\]), as can be seen by rewriting $F_L$ as $$\begin{aligned}
F_L &=& {1 \over L} \sum_{k=1}^L \sum_{j=1}^N \sum_{i_k} p_{i_k}
\langle \psi_{i_k}| A_j^{(k)} | \psi_{i_k}\rangle f(\psi_{i_k} ,
\phi^{guess}_{j_k}) \ ,\end{aligned}$$ where the operators $A_j^{(k)}$ are the operators $A_j$ restricted to the space of particle $k$: $$\begin{aligned}
A_j^{(k)}&=& Tr_{l \neq k} \left( \prod_{l' \neq k}\rho_{l'}\right)
A_j \ .
\label{FLtwo}\end{aligned}$$
Note that a possible measurement that maximizes $F_L$ is built out of the measurement $\{ a_i \}$ which maximize eq. (\[F\]): $$A_j = a_{j_1}\otimes ...\otimes a_{j_L} \ .
\label{optt}$$ This measurement has $M^L$ outcomes. And in general $M$ will be larger than $2^{I_{input}^{VN}}$.
Our main result is that one can always construct optimal measurements with $2^{I_{input}^{VN}}$ outcomes per input state which also maximize $F$. Stated with precision we shall prove the following result:
Consider a state estimation problem in which the unknown state $|\psi_i\rangle $ have density matrix $\rho = \sum_i p_i
|\psi_i\rangle
\langle \psi_i|$ and von Neumann entropy $I_{input}^{VN} = - Tr \rho \ln \rho$. The quality of the state estimation is measured by a fidelity of the form eq. (\[F\]). Given any $\epsilon > 0$ and $\eta > 0$, then there exists $L_0$ such that for any $L \geq L_0$, and any $N$ larger than $2^{L (I_{input}^{VN} +\eta)}$, there exists a measurement on sequences of $L$ input states which has $N$ outcomes and attains a fidelity $F_L \geq F_{max} -
\epsilon$. The Shannon entropy of the outputs per input state, $I_{output}^S$, can therefore be made equal or less then $ I_{input}^{VN} + \eta $.
It is this result that will be proven in sections \[comp\] to \[prob\].
Other Fidelities {#other}
================
Our main result, as stated with precision at the end of the preceding section, applies only to fidelities of the form eq. (\[F\]) with fixed guessed states. In this section we enquire whether it can be generalized to other fidelities?
As a first generalization, we consider fidelities of the form eq. (\[F\]), but for which both the POVM elements $\{ a_j\}$, and the guessed states are undetermined and must be varied to find the optimum estimation strategy. That is whereas in section \[fidelity\] the specification of an estimation strategy consisted only of the POVM elements $\{ a_j\}$, it now consists of the set $\{ a_j ,
\phi_j^{guess}\}$ which comprises both the POVM elements and the guessed states. An example of such more general fidelities was considered in [@MP]. The unknown states $|\psi_i\rangle$ where taken to be $n$ spin 1/2 particles all polarized along the same direction $\Omega$ and the fidelity was taken to be the scalar product of one spin polarized along $\Omega$ with one spin polarized along the guessed direction $f=|\langle \uparrow_\Omega |
\uparrow_{\Omega_{guess}}\rangle|^2$.
It is easy to show that our main result eq. (\[S\]) also applies to such more general fidelities for which both the POVM elements and the guessed states can be varied. First note that one can always find an optimal estimation strategy with only a finite number $M$ of outcomes [@DBE; @LPT]. Associated to each outcome is a guessed state $\phi_j^{guess (OPT)}$, $j=1,...,M$. Let us now consider the subclass of estimation strategies $\{ a_j ,
\phi_j^{guess(OPT)}\}$ for which the guessed states are fixed to be an optimal set and only the POVM elements can vary. Note that the optimal fidelity for this subclass is equal to the optimal fidelity for the more general estimation strategy since the guessed states are taken to be optimal. Since for this subclass only the POVM elements can vary, we are in the conditions of section \[fidelity\] and \[Fid\]. The result stated at the end of section \[Fid\] therefore applies. Hence there exists a measurement on large blocks whose output entropy is less or equal to the von Newmann entropy of the input states and whose fidelity is greater then the optimal fidelity minus $\epsilon$. This shows that our main result also holds for these more general fidelities.
One can however construct even more general fidelities (for instance by taking the fidelity to be non linear in the POVM elements). For such more general fidelities it is an open question whether our claim also applies. One example of such more general fidelities is the mutual information eq. (\[4\]). For this particular example our claim also holds. This is discussed in the next section.
Relation to the classical capacity of a quantum channel {#conc}
=======================================================
In the state estimation problem as presented in figure 1, the classical source specifies in a completely random manner which quantum state is emitted. The task of the measurement is to recognize as well as possible which state was emitted by the quantum source. It is instructive to compare this to the problem of classical communication through a quantum channel[@H][@HJSWW]. In this case the classical source chooses a controlled subset of all possible sequences (called code words) in such a way that they can be recognized (almost) perfectly by the receiver. He can then communicate classical information reliably through the quantum channel. The relation between the two problems is that in the communication problem the receiver must recognize the code words, so he is confronted with a state estimation problem, although it is a particular one.
For this reason the two problems are related both conceptually and formally. On the conceptual side, a corollary of our main result is an alternative proof of Holevo’s upper bound on the classical capacity of a quantum channel[@H] in the case where the quantum channel consists of pure states. Indeed if the message is to be transmitted faithfully, Bob must recognize the code words with high fidelity. We can now view the code words as the states $|\psi_i\rangle$ that are emitted by the quantum source in figure 1. The von Newmann entropy of the words is less than $n I^{VN} (\rho)$ where $n$ is the number of letters in a word and $I^{VN} (\rho)$ is the von Newmann entropy of the letters. Recall now that the question answered in this paper is to find, among all the measurements which recognize the input words with high fidelity, those whose output has the minimum entropy. Clearly this minimum entropy is an upper bound to the capacity of the channel. We have shown that it is less or equal to the von Neumann entropy of the channel. Thus the quantum channel has a classical capacity less than $I^{VN} (\rho)$ bits per word, confirming Holevo’s result.
On the formal side, the techniques we have used to construct a measurement which minimizes the entropy of the outputs are closely related and inspired by the techniques used to construct a decoding measurement which maximizes the capacity of the channel [@HJSWW]. There is however a very important difference with the communication problem. Indeed in that case one can easily build a measurement with a small number of outcomes (corresponding to a few code words, ie. to a small capacity), and the task is to try to [*maximize*]{} the number of outcomes of the measurement while continuing to recognize the code words faithfully. In this paper we can easily build a measurement with a high fidelity (ie. which is optimal), but with a large redundancy in the output. The difficulty is to [*minimize*]{} the number of outcomes (the redundancy) while keeping the measurement optimal. Nevertheless the mathematical technique that we use in section \[comp\] to decrease the number of outcomes without substantially modifying the measurement is related to the techniques used in [@HJSWW].
Eliminating redundancy {#comp}
======================
Our aim in this section is to construct a measurement with less outcomes than the optimal measurement eq. (\[optt\]). The next two sections will be devoted to prove that this measurement does not diminish the fidelity. This measurement is very similar to the measurement used in [@HJSWW] to decode a classical message sent through a quantum communication channel.
We start from the optimal POVM acting on one input state and decomposed into one dimensional projectors $b_i = |b_i\rangle \langle b_i|$. We express it in terms of the normalized operator $\tilde b_i = |\tilde b_i\rangle \langle \tilde
b_i| = b_i / Tr (b_i)$ as $b_i = \beta_i \tilde b_i$. (Throughout the text we shall denote normalized operators by ${\tilde{ }}\ $). The $\beta_i$ sum to $\sum_i
\beta_i = d$ obtained by taking the trace of the completeness relation.
We now construct $N$ operators acting on the space of $L$ input states: $$\tilde B_j= |\tilde B_j\rangle \langle \tilde B_j| = \tilde b_{j_1}
\otimes \ldots \otimes \tilde b_{j_L}
\label{Bj}$$ where each $ \tilde b_{j_k}$ is chosen randomly and independently from the set $\tilde b_1, \ldots , \tilde b_M$ with probabilities $p_1= \beta_1/d ,..., p_M = \beta_M /d$.
The $|\tilde B_j\rangle $ span a subspace $H_B$ of the Hilbert space of the $L$ input states. In this subspace the operator $B=\sum_j \tilde B_j$ is strictly positive, hence we can construct the operators $$C_j = |C_j\rangle \langle C_j| = B^{-1/2} \tilde B_j B^{-1/2}\ .
\label{C}$$ The $C_j$ are positive operators, which sum up to the identity in $H_B$: $\sum_{j=1}^N C_j = \Pi _{B}$ where $\Pi_B$ is the projector onto $H_B$. The POVM we shall use consists of the $C_j$ and the projector onto the complementary subspace $C_0=I_{d^L} - \Pi_{B}$ ($I_{d^L}$ is the identity on the Hilbert space of the $L$ input states).
Our strategy in the next sections will be to compute the average fidelity $\overline {F_L }$, where the average is taken over possible choices of $B_j$ in eq. (\[Bj\]). We shall show that the average of $F_L$ satisfies our main result stated at the end of section \[Fid\]. Therefore there necessarily are some choices of $B_j$ which also satisfy our main result.
But first we derive some important properties of the $C_j$. We shall obtain mean properties, where the mean is the average over choices of $B_j$ in eq. (\[Bj\]).
- The mean of $\tilde B_j$ is $\overline{ \tilde B_j} = I_{d^L} / d^L$.
- The mean of $B$ is: $$\begin{aligned}
\overline{ B }
&=& \sum_{j=1}^N \overline{\tilde B_j } =
{N \over d^L} I_{d^L} \ .\end{aligned}$$ This motivates our writing $$B = {N \over d^L}\left ( I_{d^L} +
\Delta \right)$$ and subsequently making expansions in $\Delta$.
- The dimension of $H_B$ is $$\begin{aligned}
dim_{H_B} &=& \sum_j Tr C_j =\sum_j Tr B^{-1} \tilde B_j
= {d^L \over N} \sum_j Tr {1 \over I_{d^L} +
\Delta } \tilde B_j \nonumber\\
&\geq& {d^L \over N}
\sum_j Tr (I_{d^L} - \Delta ) \tilde B_j \ .\end{aligned}$$ Furthermore $$\begin{aligned}
Tr \Delta \tilde B_j &=&
Tr ({d^L \over N} B - I_{d^L} ) \tilde
B_j \nonumber\\
&=& Tr \left[ {d^L \over N} ( \tilde B_j +\sum_{k \neq j}
\tilde B_k \tilde B_j ) - \tilde B_j \right]\end{aligned}$$ where we have used the fact that $ \tilde B_j^2 = \tilde B_j$. We now take the average of this expression. Using the fact that for $k \neq j$, $ \tilde B_k$ and $ \tilde B_j$ are independent, the average of $ \tilde B_k
\tilde
B_j (k \neq j)$ is the product of the averages$ \overline{ \tilde B_k \tilde B_j } =
\overline{
\tilde
B_j } \ \overline{
\tilde B_k } =I_{d^L} / d^{2L}
$. And hence $\overline {\sum_{k \neq j}
\tilde B_k \tilde B_j } = (N-1) I_{d^L} / d^{2L}$. Putting all together, we find $\overline{ {Tr \Delta \tilde B_j}}
= {d^L -1 \over N} $ and $$d^L \geq \overline{ {dim H_B} }
\geq d^L (1 - {d^L -1 \over
N}) \ .
\label{dim}$$ This shows that if $N$ is slightly larger than the dimension of the Hilbert space $d^L$, then the $C_j$ ($j \neq 0$) fill the Hilbert space.
- Finally we need to know how much the $C_j$ differ from the $\tilde B_j$. We write $|C_j \rangle = \alpha_j |\tilde B_j\rangle +
|B_j^\perp\rangle $ and compute $\alpha^2_j$: $$\begin{aligned}
\alpha^2_j &=& Tr C_j\tilde B_j \nonumber\\
&=& Tr\tilde B_j B^{-1/2}\tilde B_j B^{-1/2} \nonumber\\
&=& \left( Tr \tilde B_j B^{-1/2} \right)^2 \nonumber\\
&\geq&{d^L \over N} \left( 1 - {1\over 2}Tr\tilde B_j \Delta
\right)^2
\ .\end{aligned}$$ Hence $$\begin{aligned}
\overline{\alpha^2_j }
&\geq& {d^L \over N} ( 1 - \overline{ Tr\tilde B_j \Delta })\nonumber\\
&=& {d^L \over N} (1 - {d^L -1 \over N})\ .\end{aligned}$$ This is then used to compute the average of $\langle
B_j^\perp|B_j^\perp
\rangle $: $$\overline{\langle B_j^\perp|B_j^\perp\rangle }= \overline{ Tr C_j } -
\overline{ Tr C_j B_j }
\leq {d^L \over N} {d^L -1 \over N}\ ,
\label{betaj}$$ which shows that the $C_j$ are arbitrarily close to the $\tilde B_j$ when $N > d^L$.
An intermediate result {#optimal}
======================
In this section we shall prove the following intermediate result:
Suppose that the input states $|\psi_i\rangle $ belong to a Hilbert space of dimension $d$ and have a density matrix $\rho = \sum_i p_i |\psi_i\rangle
\langle
\psi_i| $. Denote by $\rho_{max}$ the largest eigenvalue of $\rho$. Consider measurements on blocks of $L$ input states. Give yourself any positive number $\eta >0$. Let $N$ be any integer larger than $2^{L (2\ln d + \ln \rho_{max} +\eta)}$. Then there exist measurements with $N$ outcomes with a fidelity $F_L \geq F_{max} - R 2^{- L \eta}$ where $R$ is a positive constant.
In the next section we shall combine this intermediate result with the concept of probable subspace of a long sequence of states to prove our claim in full generality.
To prove this intermediate result, we proceed as follows:
Let $\{ b_j=|bj\rangle \langle b_j| \}$ be a POVM that maximizes the fidelity $F$ eq. (\[F\]). Using the algorithm of eq. (\[Bj\]) to (\[C\]) we construct a measurement $C_j$ , $j=0,...,N$ acting on the space of $L$ copies of the input states.
Let us consider the fidelity for the measurement $C_j$: $$\begin{aligned}
F_L &=& \sum_{j=0}^N {1 \over L} \sum_{k=1}^L \sum_{i_k} p_{i_k}
\langle \psi_{i_k}| C_j^{(k)} | \psi_{i_k}\rangle
f(\psi_{i_k} , \phi^{guess}_{j_k})
\label{Smult2}\end{aligned}$$ where the $C_j^{(k)} = Tr_{l \neq k} \left( \prod_{l' \neq
k}\rho_{l'}\right) C_j$ are defined as in eq. (\[FLtwo\]).
We can decompose $C_j^{(k)}$ (for $j \neq
0$) according to its components along $|\tilde b_{jk}\rangle $: $C_J^{(k)}= X_{jk} |\tilde b_{jk}\rangle \langle \tilde b_{jk}|
+ Y_{jk} | \tilde b_{jk}\rangle \langle b_{jk}^\perp| +
Y_{jk}^* | b_{jk}^\perp\rangle \langle \tilde b_{jk}|
+ z_{jk}$ where $z_{jk} |\tilde b_{jk}\rangle = 0$, $\langle \tilde b_{jk}|z_{jk} =0$. Inserting this expression in eq. (\[Smult2\]), and using eq. (\[boundF\]), yields $$\begin{aligned}
F_L
&\geq& {1 \over L} \sum_{k=1}^L \left (
F_{max} - C \sum_{j=1}^N Tr z_{jk}
- C Tr C_0^{(k)}\right ) \end{aligned}$$ where the last term comes from the $C_0 = I_{d^L} - \Pi_B$ outcome.
It remains to calculate $Tr C_0^{(k)}$ and $Tr z_{jk}$. We start with the former $$\begin{aligned}
C_0^{(k)} &=& Tr_{l \neq k} (\prod_{l' \neq k} \rho_{l'}) (I_{d^L} -
\Pi_B)\nonumber\\
&\leq & (\rho_{max})^{L-1} Tr (I_{d^L} -
\Pi_B) = (\rho_{max})^{L-1} (d^L - dim \ H_B)\label{C0}\end{aligned}$$ where $\rho_{max}$ is the largest eigenvalue of $\rho$.
To estimate $Tr z_{jk}$ we recall the decomposition of $|C_j\rangle = \alpha_j |\tilde B_j\rangle + |B_j^\perp\rangle $. We can further decompose $|B_j^\perp\rangle $ according to whether when restricted to the space of the $k$’th particle, it is equal to $|b_{j_k}\rangle $ or not: $|B_j^\perp\rangle $ = $|\tilde b_{j_k}\rangle |\phi\rangle +
|\tilde b_{j_k}^\perp\rangle |\chi\rangle $. Inserting this into the trace which yields $C_j^{(k)}$, we obtain $$\begin{aligned}
C_j^{(k)}& =& Tr_{l \neq k}(\prod_{l' \neq k} \rho_{l'})
\left (\alpha_j| \tilde B_j\rangle
+ | \tilde b_{j_k}\rangle |\phi\rangle + | \tilde
b_{j_k}^\perp\rangle
|\chi\rangle \right)
\left ( \alpha_j^*
\langle \tilde B_j | +...
..\right)\nonumber\\
&=& | \tilde b_{j_k}\rangle \langle \tilde b_{j_k}| X_{jk}
+ | \tilde b_{j_k}\rangle \langle \tilde b_{j_k}^\perp| Y_{jk} + | \tilde
b_{j_k}^\perp\rangle \langle \tilde b_{j_k}|Y_{jk}^*
+ | \tilde b_{j_k}^\perp\rangle \langle \tilde b_{j_k}^\perp| Z_{jk}
\ .\end{aligned}$$ The coefficients $X_{jk}$, $Y_{jk}$, $Z_{jk}$ are easily calculated. The one of interest is $Z_{jk} = Tr z_{jk}$: $$\begin{aligned}
Z_{jk} &=& Tr \prod_{l' \neq k} \rho_{l'}
|\chi\rangle \langle \chi|
\nonumber\\
&\leq & (\rho_{max})^{(L-1)}
\langle \chi|\chi\rangle \nonumber\\
&\leq & (\rho_{max})^{(L-1)} \langle B_j^\perp|B_j^\perp\rangle
\ .\end{aligned}$$
Inserting these bounds into the expression for $F_L$ we obtain $$\begin{aligned}
F_L
&\geq& {1 \over L} \sum_{k=1}^L \left ( F_{max} - {C (\rho_{max})^{L-1}}
\langle B_j^\perp |B_j^\perp\rangle - {C (\rho_{max})^{L-1}}
(d^L - dim H_B) \right)\ .
\label{bound}\end{aligned}$$ We now take the average of this expression over all possible choices of $b_{jk}$ operators in eq. (\[Bj\]). Inserting eq. (\[dim\]) and (\[betaj\]) yields $$\overline{F_L} \geq F_{max}
-2 C (\rho_{max} )^{L-1} d^L{d^L - 1 \over N} \ .$$ Therefore if $N\geq 2^{L(2\ln d + \ln \rho_{max} + \eta)}$, then $\overline{F_L} \geq F_{max} - R 2^{-L \eta}$ where $R=2 C /\rho_{max}$. This proves the intermediate result.
Note that if the input states are uniformly distributed in Hilbert space, ie. $\rho = I/d$, then this intermediate result directly implies our main claim. Indeed when $\rho = I/d$, $\rho_{max} = 1/d$, then $\overline{F_L} \geq F_{max} - R 2^{-L\eta}$ if $N\geq 2^{L(\ln d + \eta)}
= 2^{L(I_{input}^{VN} + \eta)}$. When the input states are not uniformly distributed in Hilbert space, we must use the notion of probable Hilbert space of a long sequence to prove our main result. This is done in the next section.
Measurements on probable subspaces {#prob}
==================================
We now combine the result of the previous section with the notion of probable subspace of large blocks of states.
We first recall the properties of the probable subspace[@S][@JS]. Consider a long sequence of $L'$ input states $|\psi_{i_1}...\psi_{i_{L'}}
\rangle $. The density matrix of these states is $\rho= \prod_{k=1}^{L'}\rho_k$. The projector $\Pi$ onto the probable subspace has the properties that given $\epsilon'>0$, $\eta' >0$, and for $L'$ sufficiently large,
1. $Tr \Pi \rho \geq 1 - \epsilon'$, ie. the probability to be in the probable subspace is arbitrarily close to $1$.
2. $\Pi$ and $\rho$ commute, ie. the eigenvectors of $\rho$ are either eigenvectors of $\Pi$ or of $1-\Pi$. And furthermore the eigenvectors which are common to $\Pi$ and $\rho$ have eigenvalues comprised between $2^{L'(-H - \eta')}\leq (\rho_{L'})_i \leq 2^{L'(-H + \eta')}$
3. From these two properties it follows that the dimension of the probable Hilbert space is bounded by $( 1 - \epsilon' ) 2^{L'(H - \eta')}\leq Tr \Pi \leq 2^{L'(H +
\eta')}$
Let us now show that measurements restricted to the probable subspace are arbitrarily close to optimal. Suppose that $A_j$ is a measurement that optimizes the state determination problem eq. (\[FL\]) for sequences of $L' $ input states (for instance the measurement eq. (\[optt\]). Consider the POVM consisting of the operators $A'_j = \Pi A_j \Pi$ (to which we associate the unmodified guessed states $\phi_{j_k}^{guess}$) and the operator $I-\Pi$ (to which we associate the minimal value of the fidelity $f_{min}$). The fidelity for this measurement is
$$\begin{aligned}
F_{L'} &=&
\sum_{i_1 ... i_{L'}} p_{i_1}... p_{i_{L'}}
\sum_{j=1}^N \langle \psi_{i_1} ... \psi_{i_{L'}}|
\Pi A_j \Pi | \psi_{i_1}
... \psi_{i_{L'}}\rangle {1 \over L'}
\sum_{k=1}^{L'} f(\psi_{i_k} , \phi_{j_k})
\nonumber\\
& &+ \sum_{i_1 ... i_{L'}} p_{i_1}... p_{i_{L'}}
\langle \psi_{i_1} ... \psi_{i_{L'}}| 1- \Pi
| \psi_{i_1}
... \psi_{i_{L'}}\rangle
f_{min}
\nonumber\\
&\geq& F_{max} \nonumber\\ & &
-
\sum_{i_1 ... i_{L'}} p_{i_1}... p_{i_{L'}}
\sum_{j=1}^N \langle \psi_{i_1} ... \psi_{i_{L'}}|
A_j - \Pi A_j \Pi | \psi_{i_1}
... \psi_{i_{L'}}\rangle
{1 \over{L'} } \sum_{k=1}^ {L'}f(\psi_{i_k} , \phi_{jk})
\nonumber\\
& &
+ f_{min} Tr \rho (1 - \Pi) \ . \end{aligned}$$
We bound the second term by $$\begin{aligned}
&& | \sum_{i_1 ... i_{L'}} p_{i_1}... p_{i_{L'}}
\sum_{j=1}^N \langle \psi_{i_1} ... \psi_{i_{L'}}|
A_j - \Pi A_j \Pi | \psi_{i_1}
... \psi_{i_{L'}}\rangle
{1 \over{L'} } \sum_{k=1}^{L'} f(\psi_{i_k} , \phi_{jk}) |
\nonumber\\
&\leq&f_{max} \sum_{j=1}^N |
\sum_{i_1 ... i_{L'}} p_{i_1}... p_{i_{L'}}
\langle \psi_{i_1} ... \psi_{i_{L'}}| ( A_j - \Pi A_j \Pi)
| \psi_{i_1}
... \psi_{i_{L'}}\rangle
|
\nonumber\\
&=&f_{max} \sum_{j=1}^N | Tr [ \rho (A_j - \Pi A_j \Pi) ]| \nonumber\\
&=& f_{max} Tr[ ( \rho - \Pi \rho \Pi) \sum_{j=1}^N A_j]
= f_{max} Tr \rho (I - \Pi
)\nonumber\\
&\leq& \epsilon ' f_{max}\end{aligned}$$ where $f_{max}$ is the maximum value of the fidelity and we have used the fact that $\rho - \Pi \rho \Pi$ is a positive operator, and therefore that $ Tr [ \rho (A_j - \Pi A_j \Pi) ] \geq 0$ which allows us to remove the absolute value sign and put the sum over $j$ inside the trace.
Putting everything together we have $$F_{L'} \geq F_{max} - \epsilon' (f_{max} - f_{min} )\ .
\label{fmax}$$ This shows that the restriction of the measurement to the probable Hilbert space diminishes the fidelity by an arbitrarily small amount $\epsilon'(f_{max} - f_{min} )$.
We can now build a measurement which satisfies our main result as stated at the end of section \[Fid\]. We decompose the input states into blocks of $L'$ states. On each of these blocks we first carry out a partial measurement $\Pi$ and $I - \Pi$ to know whether it is in the probable subspace or not. If the result is $I- \Pi$ the sequence is discarded. The sequences which pass the test are kept.
We now take the sequences which have passed the test as the input states in the intermediate result. These sequences belong to a Hilbert space of dimension $dim\
H_{probable} \leq 2^{L' (I_{input}^{VN} + \eta')}$ and the largest eigenvalue of their density matrix is $\rho_{max} \leq 2^{L' (-I_{input}^{VN} + \eta')}$. To apply the intermediate result, we take an integer $L$ and an $\eta>0$. Then there exists a measurement on blocks of $L$ sequences which has a number of possible outcomes equal to any integer $N$ larger than $2^{L ( L' (I_{input}^{VN} + 3 \eta') + \eta)}
= 2^{L L' ( I_{input}^{VN} + 3 \eta' + \eta / L')}$ and which has a fidelity larger than $F_{LL'} \geq F_{max} - \epsilon' (f_{max} - f_{min})
- R 2^{- L \eta}$ where $R$ is a positive constant.
Let us calculate the entropy $I_{outputs}^S$ of the outputs of this measurement. We need less than $I_{\epsilon'} = -\epsilon' \ln \epsilon'
- (1- \epsilon') \ln (1- \epsilon')$ bits to describe whether or not the input state passes the first test of belonging to the probable Hilbert space or not. If it does then we need less than $\ln N$ bits to encode the output of the measurement on the $L$ blocks of probable sequences. Therefore the total number of bits we need to describe the outcome of this measurement on $L L' $ elementary input states is $I_{output}^S \leq \ln N + L I_{\epsilon'}$. Replacing $N$ by its bound, we have $I_{output}^S \leq L L'
( I_{input}^{VN} + ( 3 \eta' + \eta / L' + I_{\epsilon'}/ L' )$. Since $\epsilon'$, $\eta'$ and $\eta$ can be chosen arbitrarily small, and $L'$ arbitrarily large, our claim is proven.
Conclusion
==========
In this paper we have obtained a quantitative estimate of how much information can be obtained by a quantum measurement. We considered optimal measurements, that is measurements which maximize a fidelity function. We then enlarged the set of optimal measurements in two ways. First we considered optimal measurements that act collectively on large blocks of input states rather than measurements restricted to act on each state separately. Secondly we did not require the fidelity of the measurements to be exactly equal to the optimal fidelity, but only that it be arbitrarily close to the optimal fidelity. In this context we showed that whatever property of a quantum system one wants to learn about, one can learn at most one bit of information about every qubit of quantum information carried by the unknown quantum system. That is, the Shannon entropy of the outcomes of optimal measurements can always be made equal or less than the von Newmann entropy of the unknown quantum states.
[**Acknowledgments :**]{} S.M. would like to thank Utrecht University where most of this work was carried out. He is a “chercheur qualifié” of the Belgian National Research Fund.
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|
---
abstract: 'We introduce a set of minimal simplified models for dark matter interactions with the Standard Model, connecting the two sectors via either a scalar or pseudoscalar particle. These models have a wider regime of validity for dark matter searches at the LHC than the effective field theory approach, while still allowing straightforward comparison to results from non-collider dark matter detection experiments. Such models also motivate dark matter searches in multiple correlated channels. In this paper, we constrain scalar and pseudoscalar simplified models with direct and indirect detection experiments, as well as from existing LHC searches with missing energy plus tops, bottoms, or jets, using the exact loop-induced coupling with gluons. This calculation significantly affects key differential cross sections at the LHC, and must be properly included. We make connections with the Higgs sector, and conclude with a discussion of future searches at the LHC.'
author:
- 'Matthew R. Buckley'
- David Feld
- Dorival Gonçalves
title: Scalar Simplified Models for Dark Matter
---
Introduction \[sec:intro\]
==========================
The case for the existence of dark matter is strong. Decades of evidence from multiple independent lines [@Zwicky:1933gu; @Rubin:1980zd; @Olive:2003iq; @Ade:2013zuv] reveal that this form of matter has a significant role in the composition and evolution of our Universe (for a review, see [*e.g.*]{}, Ref. [@Bertone:2004pz]). No particle in the Standard Model is a suitable candidate for dark matter and so we need new physics to explain it. Though we lack evidence of the nature of the dark sector, if particle dark matter has a mass at the TeV scale or lower and was ever in thermal equilibrium in the early Universe, we have good reason to expect interactions with the visible sector to be within reach of our present experiments. However, this is of course not guaranteed.
Perhaps the best known example of such dark matter is a weakly-interacting massive particle which becomes a thermal relic with the appropriate energy density after freeze-out. This type of dark matter is realized in many extensions of the Standard Model introduced to solve other problems of a theoretical nature ([*e.g.*]{} Naturalness and Hierarchy). However, looking beyond this class of dark matter, even models of non-thermal dark matter often require significant annihilation cross sections into either the Standard Model or some hidden sector, so as not to overclose the Universe [@Buckley:2011kk]. It is therefore well-motivated to search for dark sector particles in a range of experiments, including the Large Hadron Collider (LHC).
When looking for dark matter, we can cast the experimental reach in terms of specific models of dark matter which are UV-complete. These models usually have a number of additional new particles with more significant interactions with the Standard Model than the dark matter itself. The canonical example of this sort is the supersymmetric neutralino, which is accompanied by a host of new charged and colored superpartners. Despite the advantage of UV-complete models, interpreting results in this way has some drawbacks: $i$) the results may be difficult to recast for new models; $ii$) correlating results with non-collider experiments may be very dependent on UV-complete parameters; $iii$) focusing on a specific high-energy model runs the risk of overlooking other experimentally interesting channels; and $iv$) tuning the experimental selection criteria could reduce the sensitivity to other types of dark matter.
In order to approach the problem in a somewhat model-independent way while still allowing for comparison between different classes of experiments, it has been useful to present the results of experimental searches in an effective field theory (EFT) framework [@Cao:2009uw; @Goodman:2010yf; @Goodman:2010ku]. The EFT approach assumes contact term interactions between dark matter and SM particles with the particle(s) connecting the two sectors integrated out of the low-energy spectrum. The validity of the EFT approach diminishes in the regime where the momentum transfer cannot be neglected relative to the (unknown) mass of the heavy particles. For direct detection this condition is usually satisfied, as long as mediators are not extremely light, as the momentum scale is on the order of 10 keV. Indirect detection and thermal freeze-out involve the annihilation of non-relativistic dark matter and so the EFT is applicable as long as the mediator is significantly heavier than twice the dark matter mass, assuming no additional new particles in the theory [@Abdallah:2014hon].
However, when considering the production of dark matter at particle colliders through high $p_T$ visible particles recoiling against invisible dark matter [@Birkedal:2004xn; @Feng:2005gj; @Beltran:2008xg; @Konar:2009ae; @Beltran:2010ww; @Bai:2010hh; @Rajaraman:2011wf; @Fox:2011pm; @Bai:2012xg; @Fox:2012ee; @Carpenter:2012rg], the momentum transfer in dark matter pair production events is large enough to render the EFT assumption invalid for a significant range of dark matter masses, couplings, and mediator masses [@Bai:2010hh; @Fox:2011fx; @Fox:2011pm; @Shoemaker:2011vi; @Fox:2012ee; @Weiner:2012cb; @Busoni:2013lha; @Buchmueller:2013dya; @Buchmueller:2014yoa; @Busoni:2014sya; @Busoni:2014haa]. As the momentum flowing through the production diagram is proportional to both the transverse momentum of the dark matter particles ([*i.e.*]{} the missing transverse momentum, or MET) and the transverse momentum of recoiling visible particles required for the trigger, this issue will be even more pressing at the LHC Run-II, as the trigger requirements on MET and jet $p_T$ will be higher than those used in Run-I. Rather than viewing the invalidity of the EFT formalism as a drawback, it should be seen as an optimistic statement: if dark matter is being produced at colliders, it is generally the case that new mediating particles are being produced as well. As we look to interpret results from dark matter experiments and design new search strategies at the LHC, a balance should be struck between the very general (but often inapplicable) EFT approach and a full theory like supersymmetry. One solution has been found in [*Simplified Models*]{} [@Alwall:2008ag; @Alves:2011wf; @Goodman:2011jq], which resolve the contact interaction into a single exchange particle, without adding in the full complexity of a UV-complete model. By specifying the spin and gauge quantum numbers of the dark matter and the mediators, the parameter space can be made relatively small, allowing an easy conversion of bounds between experiments and theories. Previous papers have discussed colored mediators [@An:2013xka; @DiFranzo:2013vra; @Papucci:2014iwa], which result in $t$-channel production of dark matter in a manner very similar to squarks in supersymmetry. Other works have considered vector and axial vector $Z'$ models [@An:2012va; @Frandsen:2012rk; @Busoni:2014haa], which cause $s$-channel dark matter production at colliders.
In this paper we consider a class of simplified models with a spin-0 scalar or pseudoscalar mediator, which allows $s$-channel production of dark matter from Standard Model partons at the LHC. These models are attractive in their simplicity, requiring only a minimal extension of the Standard Model’s particle content. New scalars or pseudoscalars can also be easily accommodated in extended Higgs sectors, and it is not unreasonable to expect the Higgs to have contact with the dark sector. As with other simplified models, scalar mediators predict LHC signatures in a number of correlated channels; this can be used to our advantage when designing new searches.
As previous works [@Haisch:2012kf; @Haisch:2013fla; @Haisch:2013ata; @Busoni:2014sya; @Ghorbani:2014qpa; @Crivellin:2014qxa] have pointed out, scalar and pseudoscalar mediator models and EFTs face unique simulation issues at colliders. Making the well-motivated assumption that the mediator couplings to Standard Model fermions proportional to the Higgs Yukawas, the mediator is primarily produced at the LHC through a loop-induced interaction with gluons. As was noted in the context of scalar EFTs, this loop-induced coupling must be calculated assuming large momentum transfer, as the trigger requirements at the LHC for most dark matter searches require significant transverse momentum in the event. Just as large momenta requires the expansion of a point-like dark matter-Standard Model EFT interaction to include a mediator, the mediator-gluon interaction must also be resolved as the momentum transfer increases $p_{T\phi}=\mathcal{O}( 2m_t)$. A sketch of the successive levels of effective theories is shown in Figure \[fig:momentum\_cartoon\]. As we will show, the large momentum transfer at the LHC forces us to fully resolve the top-loop induced coupling, just as it forces us to resolve the mediator in the EFT.
![A heuristic diagram presenting the successive levels of effective theories that must be expanded as the momentum flow (proportional to the MET) through the interaction increases. On the left we have the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$. In the center two effective theories with either $(m_\phi \rightarrow \infty, \mbox{finite}~m_t)$ (top) or $(\mbox{finite}~m_\phi,m_t \rightarrow \infty)$ (bottom). On the right the Full Theory with finite $(m_\phi, m_t)$. \[fig:momentum\_cartoon\]](./pictorial_graphs){width="0.73\columnwidth"}
In this paper, we provide two benchmark models for scalar and pseudoscalar mediated simplified models, with a five-dimensional parameter space. We demonstrate the non-negligible effects of resolving the mediator loop-induced coupling to gluons in collider simulations, compared to the effective interactions. We derive bounds on these parameters using data from direct and indirect detection, as well as predictions assuming that the dark matter is a thermal relic. We then show the existing constraints on these benchmarks from a number of Run-I LHC searches, including – but not limited to – the MET plus jets searches that have been of primary interest previously. This comprehensive set of bounds on scalar mediators has not been previously collected, and underlines the necessity of multiple complimentary channels when searching for dark matter at the LHC [@Lin:2013sca].
In Section \[sec:models\] we set up our two benchmark models for scalar and pseudoscalar mediators. We introduce a set of parameters which describe the relevant phenomenology for current and future experimental results. In this section we also show the effects of the resolved top-loop on the distribution of transverse momentum at colliders. In Section \[sec:noncollider\] we show constraints on these models from non-collider physics: direct and indirect detection, as well relic abundance cross section. Constraints from existing LHC Run-I missing energy searches are discussed in Section \[sec:collider\_bounds\] in three channels: missing transverse energy with associated jets, with associated top quark pairs, and with associated bottom quarks. We apply our constraints to the special case of the 125 GeV Higgs as the scalar mediator in Section \[sec:higgs\]. We then conclude by outlining additional searches and improvements that could be made for future analyses.
Simplified Models {#sec:models}
=================
In this paper we consider interactions between Dirac fermion dark matter $\chi$ and Standard Model fermions mediated by either a new scalar $\phi$ or a new pseudoscalar $A$. Our choice of fermionic dark matter is somewhat arbitrary; our results would translate to the scalar dark matter case with minor modifications, though this assumption would introduce additional parameters. Our two benchmark models take the form
$$\begin{aligned}
{\cal L}_S & = & {\cal L}_\text{SM}+ \frac{1}{2} (\partial_\mu \phi)^2 - \frac{1}{2} m_\phi^2 \phi^2
+i \bar{\chi} \slashed{\partial} \chi - m_\chi \bar{\chi} \chi - g_\chi \phi \bar{\chi}\chi
- \sum_{\text{fermions}} g_v \frac{y_f}{\sqrt{2}} \phi \bar{f}f \;, \label{eq:Lphi} \\
{\cal L}_A & = & {\cal L}_\text{SM}+\frac{1}{2} (\partial_\mu A)^2 - \frac{1}{2} m_A^2 A^2
+i \bar{\chi} \slashed{\partial} \chi - m_\chi \bar{\chi} \chi - i g_\chi A \bar{\chi}\gamma^5 \chi
- \sum_{\text{fermions}} i g_v \frac{y_f}{\sqrt{2}} A \bar{f} \gamma^5f . \label{eq:LA}\end{aligned}$$
Here, ${\cal L}_\text{SM}$ is the Lagrangian of the Standard Model. Such models introduce five free parameters: dark matter mass $m_\chi$, mediator mass $m_{\phi}$ or $m_A$, the dark matter-mediator coupling $g_\chi$, the flavor-universal Standard Model-mediator coupling $g_v$, and the mediator width $\Gamma_{\phi}$ or $\Gamma_A$.[^1] Keeping the width as a free parameter leaves open the possibility that the mediator has other couplings to additional particles, perhaps in an expanded dark sector. Furthermore, as the cross section for dark matter production, annihilation, and scattering to nucleons is proportional to product of the couplings $\left( g_\chi g_v \right)^2$ and the width depends on the sum of terms proportional to $g_\chi^2$ and $g_v^2$ separately, by keeping the width as a free parameter, we can set limits on the combination $g_\chi g_v$ as a function of the width without specifying the individual couplings $g_v$ and $g_\chi$. This is how we will present our bounds in Sections \[sec:noncollider\] and \[sec:collider\_bounds\].
We set the fermion couplings proportional to the SM Yukawa couplings, using the Minimal Flavor Violating (MFV) assumption [@D'Ambrosio:2002ex]. This avoids introducing precision constraints from flavor measurements. Additionally, note that the left-handed Standard Model fermions are $SU(2)_L$ doublets and the right-handed fermions are singlets, while the dark matter cannot be primarily an $SU(2)_L$ multiplet with $Y\neq 0$, due to direct detection bounds. If $\chi$ is a complete Standard Model gauge singlet, then the mediator $\phi$ or $A$ must have some mixing with the Higgs sector to interact with both the doublet fermions and the dark matter, justifying the Yukawa-proportional coupling assumption. Another possibility is that dark matter is a doublet-singlet mixture, as in the case of a neutralino, allowing the mediator to be an $SU(2)_L$ doublet while still avoiding direct detection constraints. This again involves mass terms in the dark sector proportional to the electroweak symmetry breaking scale, which suggests (though does not require) couplings proportional to Yukawa terms.
We assume that the coupling $g_v$ is universal across all the families of quarks and leptons. One could loosen this requirement without introducing large flavor violation. Taking a cue from two-Higgs doublet models for example, the up-type and down-type couplings could be varied independently. We will not explore this possibility in detail here, but we note such deviations from the baseline model would change the ratios of expected signals in the various collider channels we consider. This again motivates a broad set of experimental searches.
As we have seen, this set of simplified models has some obvious connections with the Higgs sector [@Fox:2011pm; @Djouadi:2012zc]. As a gauge-singlet scalar, the mediator $\phi$ will generically mix with the neutral Higgs. If the SM Higgs is part of an extended Higgs sector, then the pseudoscalar $A$ would fit easily into the model (for example, as the pseudoscalar in a two-Higgs doublet model). If the models are so intimately related to Higgs physics, one might expect some coupling to $W$ and $Z$ bosons, which we do not allow in our baseline models. We justify this omission by noting that even for scenarios where the scalar and/or pseudoscalar are part of a Higgs sector, deviations from alignment in supersymmetry are constrained to be small [@Craig:2013hca; @Carena:2013ooa], which in turn implies that the coupling to $W/Z$ bosons of new scalars and pseudoscalars in the Higgs sector would likely be small compared to the $125$ GeV Higgs.
Similarly, we would expect explicit dimension-4 $\phi-h$ or $A-h$ couplings in our Lagrangians Eqs. and . In a full UV-complete theory, into which the simplified model presumably fits, these couplings would be set by some unspecified dynamics. In this work, we set them to zero for simplicity, as we did for the $W$ and $Z$ couplings.
Analogously to the production of the Higgs, the dominant form of dark matter production at the LHC would be through gluon fusion, as the tree-level couplings to the light quarks are Yukawa-suppressed. This production mode is dominantly through the loop induced $g-g-\phi (A)$ coupling. Representative diagrams for the leading-jet process are shown in Figure \[fig:loop\_feyndiagram\]. Note that in the production of the mediators in channels with associated $b$ or $t$ quarks is largely dominated by the tree-level terms, though as in Higgs production, loop effects can be important in the $\phi(A)+$ heavy flavor channels. If the external particles in the loop induced $g-g-\phi (A)$ interaction are on-shell, then it can be exactly calculated in a single coupling value, as in Higgs physics. At leading-order, the on-shell Lagrangians for our two benchmark models gain the additional terms [@toploop; @pseudoloop1; @pseudoloop2; @pseudoloop3; @Harlander:2005if] $$\begin{aligned}
{\cal L}_{S,\text{loop}} = \frac{\alpha_s}{8 \pi} \frac{g_v}{v} \tau [1+(1-\tau) f\left(\tau \right)] G^{\mu\nu}{G}_{\mu\nu}\phi
\;, \qquad \qquad
{\cal L}_{A,\text{loop}} = \frac{\alpha_s}{4 \pi} \frac{g_v}{v} \tau f\left(\tau \right) G^{\mu\nu}\tilde{G}_{\mu\nu}A
\;,
\label{eq:lag_loop}\end{aligned}$$ where $\tau = 4 m_t^2/m_{\phi (A)}^2$, $y_t$ is the top Yukawa, $v$ is the Higgs vacuum expectation value, and the function $f(\tau)$ is defined as $$\begin{aligned}
f(\tau) =
\begin{cases}
\arcsin^2 \frac{1}{\sqrt{\tau}} \;, & \tau \ge 1 \;, \\
-\frac{1}{4} \left( \log \frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}} - i \pi \right)^2\;, & \tau < 1\;.
\end{cases} \end{aligned}$$ We should emphasize that the effective coupling approximation can be accurately calculated for arbitrary top and mediator masses. However, for associated production of $\phi$ or $A$ plus jets at collider, with momenta and energy scales where the loop induced top contributions start to be resolved, that is $p_{T,\phi} = \mathcal{O}(2m_t)$, this effective operator breaks down and the one-loop dynamics should be taken into account. Also note that the scalar coupling to gluons is suppressed relative to the pseudoscalar by $\gtrsim30\%$ for mediator masses below $\sim 400$ GeV. This will result in slightly weaker bounds on the scalar model relative to pseudoscalars in channels where the gluon coupling dominates ([*i.e.*]{}, LHC monojets).
![Sample of the leading-order Feynman diagrams, in the Full Theory with finite top mass effects, contributing to the scalar plus jet production at the LHC.[]{data-label="fig:loop_feyndiagram"}](./feynman.pdf){width="\columnwidth"}
![Missing energy distribution for the process $pp\rightarrow \bar{\chi}\chi + j$ in the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$ (equivalent to the left panel of Fig. \[fig:momentum\_cartoon\]), for a finite mediator mass with an effective coupling to gluons $m_t \rightarrow \infty$ (lower center panel of Fig. \[fig:momentum\_cartoon\]) and the Full Theory including the top mass effects (right panel of Fig. \[fig:momentum\_cartoon\]). On the left panel we display the results for a light mediator and on the right for a very heavy one (equivalent to the upper center panel of Figure \[fig:momentum\_cartoon\]). These distributions were generated at the parton level with [MCFM]{} and LHC at 8 TeV.[]{data-label="fig:etmiss_intro"}](./etmiss_intro_mh100 "fig:"){width="0.4\columnwidth"} ![Missing energy distribution for the process $pp\rightarrow \bar{\chi}\chi + j$ in the EFT $\mathcal{O}_G=\alpha_s/\Lambda^{3}\,\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}$ (equivalent to the left panel of Fig. \[fig:momentum\_cartoon\]), for a finite mediator mass with an effective coupling to gluons $m_t \rightarrow \infty$ (lower center panel of Fig. \[fig:momentum\_cartoon\]) and the Full Theory including the top mass effects (right panel of Fig. \[fig:momentum\_cartoon\]). On the left panel we display the results for a light mediator and on the right for a very heavy one (equivalent to the upper center panel of Figure \[fig:momentum\_cartoon\]). These distributions were generated at the parton level with [MCFM]{} and LHC at 8 TeV.[]{data-label="fig:etmiss_intro"}](./etmiss_intro_mh1200 "fig:"){width="0.4\columnwidth"}
In Section \[sec:collider\_bounds\] we will discuss further details of the missing transverse energy searches with associated jets used the LHC experiments. For this section, it is sufficient to state that significant transverse missing momentum is required (that is, large transverse momentum of the $\phi$ or $A$), along with large momentum of at least one jet, in order to pass the trigger and selection criteria. In events without additional heavy flavor tagging, the primary production vertex for the $\phi$ or $A$ will be through the top-loop coupling to gluons, in association with a hard emission of initial state radiation, see Figure \[fig:loop\_feyndiagram\].
In Figure \[fig:etmiss\_intro\], we show the missing transverse momentum distribution (MET or $\slashed{E}_T$) for $p p \to \bar{\chi} \chi+ j$ at the 8 TeV LHC, setting $m_\chi = 10$ GeV. Following our sketch (in Figure \[fig:momentum\_cartoon\]) of the inclusion of integrated-out particles as we resolve effective operators, we present the differential MET distribution from dark matter production for three different interaction hypothesis:
1. for the direct production through an EFT interaction with gluons, $\alpha_s/\Lambda^{3}\,\left[\bar{\chi}\chi G_{\mu\nu}G^{\mu\nu}\right]$;
2. for the production via a scalar mediator with an effective $g-g-\phi$ interaction vertex, as in Eq. . For comparison purposes, we show both a light (100 GeV) on-shell mediator and very heavy (1200 GeV) mediator which gives dark matter through off-shell production; and
3. for the production via a scalar mediator where the top-loop has been taken into account via the exact one-loop computation. We show once more a very light ($m_\phi=100$ GeV) and a very heavy ($m_\phi \rightarrow \infty$) mediator scenarios.
All these distributions were generated using [MCFMv6.8]{} [@Campbell:2010ff; @hj], where we have extended the process implementation ${pp\rightarrow H(A)+j \rightarrow \tau^+ \tau^-+j}$ in [MCFM]{} to accommodate the off-shell mediator production and decay to a dark matter pair. The hard scales are defined as $\mu_F^2=\mu_R^2=m_{\phi(A)}^2+p_{Tj}^2$. For further details on the event generation see Section \[sec:collider\_bounds\].
From Figure \[fig:etmiss\_intro\], we observe that for heavy mediators above $\mathcal{O}(1~\mbox{TeV})$ and $m_t \rightarrow \infty$ the [*Simplified Model*]{} can be well described by the EFT. However, for light mediators ($m_\phi=100$ GeV) or finite top mass we see that this approximation breaks down. Moreover, if accurate conclusions about such models are to be drawn from LHC data, it is clearly necessary to include the mediator-gluon interaction (induced by the heavy-quark loops) when the characteristic energies are above $\mathcal{O}(2m_t)$.
At every stage of returning the integrated particles to the spectrum (as pictorially presented in Figure \[fig:momentum\_cartoon\]), we see significant changes in the differential cross sections. There is a large decrease in the tail of the MET distributions as first the mediator and then the top-loop are correctly taken into account. Ignoring these effects in the simplified scalar model will lead to an over-prediction of the cross section at the LHC for a given set of parameters, and thus overly strong limits. Furthermore, when using search techniques that rely on detailed knowledge of the kinematic shape ([*e.g.*]{} razor variables [@Fox:2012ee; @Rogan:2010kb; @Chatrchyan:2011ek]), it is of course necessary to fully and correctly understand the shape of the signal distributions.
Before moving on to the bounds on the benchmark models, it is useful to consider the widths and branching ratios we might expect in our models of interest. In Figure \[fig:widths\], we show the partial widths for $\phi$ and $A$ decaying into Standard Model particles and dark matter as a function of mass $m_{\phi(A)}$, assuming $m_\chi = 10$ GeV and $g_v = g_\chi = 1$. It is straightforward to rescale the relevant widths if these assumptions are loosened. As can be seen, if $g_v \sim g_\chi$ and $m_\chi \ll m_\phi/2$, the decay of the mediator into dark matter is expected to dominate, unless the mediator is heavy enough for the top channel to open. This is a result of the small Yukawa couplings for the lighter fermions. It is also worth pointing out that differences in rate between the scalar and pseudoscalar partial decays are given by a distinct scaling pattern with the particle velocity $\beta_\chi=\sqrt{1-4m_\chi^2/m_\phi}$. Namely, the scalar presents a stronger suppression $\Gamma_{\phi\rightarrow\chi\chi} \propto \beta_\chi^3$ when compared to the pseudoscalar, $\Gamma_{A\rightarrow\chi\chi} \propto \beta_\chi$. As a result, when the dark matter mass is close to the kinematic limit $2m_\chi \sim m_{\phi(A)}$, we should expect constraints on the couplings of scalars to be weaker than those placed on the couplings to pseudoscalars. When the dark matter is much lighter than the mediator, the coupling constraints on the two models should be equivalent, as in this regime $\beta^{3} \sim \beta \sim 1$.
![The width $\Gamma$ of the scalar $\phi$ (left) and pseudoscalar $A$ (right) decaying into pairs of 10 GeV dark matter (black dotted), top quarks (green), bottom quarks (red), tau leptons (blue), $\gamma\gamma$ (black dashed), and the total width (black solid), as a function of the parent mass $m_\phi$ or $m_A$. Widths are calculated assuming $g_v = g_\chi =1$. \[fig:widths\]](./phi_width_scaled.pdf "fig:"){width="0.5\columnwidth"}![The width $\Gamma$ of the scalar $\phi$ (left) and pseudoscalar $A$ (right) decaying into pairs of 10 GeV dark matter (black dotted), top quarks (green), bottom quarks (red), tau leptons (blue), $\gamma\gamma$ (black dashed), and the total width (black solid), as a function of the parent mass $m_\phi$ or $m_A$. Widths are calculated assuming $g_v = g_\chi =1$. \[fig:widths\]](./A_width_scaled.pdf "fig:"){width="0.5\columnwidth"}
Non-Collider bounds {#sec:noncollider}
===================
In this section, we derive bounds on our benchmark model parameters, using direct and indirect detection experimental results, as well as the thermal relic abundance calculation. These bounds are complimentary to those set by colliders, which we will consider in Section \[sec:collider\_bounds\]. However, we caution that care must be taken in extrapolating bounds between different classes of experiments, as there are both particle physics and astrophysical assumptions that must be kept in mind. For example, the direct detection limits rely on an assumption about the local dark matter density and velocity distributions, the latter of which is expected to vary from the standard assumptions used in the experimental results [@Kuhlen:2009vh; @Lisanti:2010qx; @Mao:2012hf; @Mao:2013nda; @Kuhlen:2013tra; @Lee:2013wza; @Bozorgnia:2013pua]. While it is possible to some degree to disentangle the astrophysical uncertainties to place limits on the fundamental parameters [@Fox:2010bu; @Fox:2010bz; @Fairbairn:2012zs; @Pato:2012fw; @DelNobile:2013cta; @Feldstein:2014gza], we cannot lose sight of the assumptions that went into the analysis. Similarly, the parameters that are required to obtain a thermal relic abundance can be changed significantly if additional particles (beyond the minimal set in our benchmark simplified models) are present in the spectrum, or if the flavor-universal assumption for the coupling $g_v$ is lifted. Furthermore, we have no direct knowledge that the dark matter is a thermal relic. Thus, we wish to emphasize that no single result presented here should be taken as the final word on the limits for our models, since these searches – along with those of the colliders – are complimentary and approach the problem from different angles. Despite the caveats, these limits are useful in that they provide a sense of the size of the parameters which might be necessary to obtain a viable model of dark matter, and allow us to focus on regions where particular classes of experiments may dominate.
Direct Detection
----------------
Direct detection experiments measure the recoil energy from WIMP-nucleus scattering, placing an upper limit on the dark matter-nucleon elastic scattering cross section. This, like all the bounds we discuss in this paper, requires coupling the dark and visible sectors, and so limits on the scattering cross section provide a constraint on the combination of couplings $g_\chi g_v$. The pseudoscalar model has no velocity or momentum independent scattering cross section with protons and neutrons, and so has no significant limits from direct detection. However, assuming Dirac dark matter, the scalar mediator induces a spin-independent cross section and so the model parameters are constrained by a number of experiments. The strongest bounds at present come from LUX [@Akerib:2013tjd] for $m_\chi \gtrsim 6$ GeV and, at lower dark matter masses, by CDMS-lite [@Agnese:2013jaa].
The fundamental Lagrangian parameters are translated into dark matter-nucleon scattering cross sections using $$\begin{aligned}
\sigma_{\chi-p,n} & = & \frac{\mu^2}{\pi} f_{p,n}^2, \\
f_{p,n} & = & \sum_{q=u,d,s} f_q^{p,n}\frac{m_{p,n}}{m_q} \left(\frac{g_\chi g_v y_q}{\sqrt{2}m_\phi^2} \right) + \frac{2}{27} f_\text{TG}^{p,n} \sum_{q=c,b,t} \frac{m_{p,n}}{m_q} \left(\frac{g_\chi g_v y_q}{\sqrt{2}m_\phi^2} \right),\end{aligned}$$ where $\mu$ is the dark matter-nucleon reduced mass, and the parameters $f^{p,n}_q$ and $f^{p,n}_\text{TG}$ are proportional to the quark expectation operators in the nucleon. These must be extracted from lattice QCD simulations [@Belanger:2008sj; @Young:2009zb; @Toussaint:2009pz; @Giedt:2009mr; @Fitzpatrick:2010em], and we adopt the values from Ref. [@Fitzpatrick:2010em]. For the purposes of this paper, there is no significant difference between the proton and neutron $f_{p,n}$, and so our dark matter scattering is essentially isospin-conserving.
The finite width is not relevant to these constraints (barring widths of order $m_\phi$), so the bound is placed on the combination $g_\chi g_v$ as a function of dark matter and mediator masses, independent of width. In Figure \[fig:direct\_bounds\], we show the upper limits placed by LUX and CDMS-lite at the 95% confidence level (CL) on the coupling combination $g_\chi g_v$, as a function of the scalar mediator and dark matter masses. The discontinuity visible at $m_\chi \sim 6$ GeV is a result of the sharply weakening LUX bounds being overtaken by the CDMS-lite constraint. As we will continue to do throughout this paper, we include limits on the combination of couplings well above the perturbativity bound $g_\chi g_v \gtrsim4\pi$. Clearly, such enormous couplings are not part of a sensible perturbative quantum field theory. We include them for completeness, and to allow some comparison of the sensitivity of the different classes of experiments.
![Contour plot of 95% CL upper bounds on the coupling combination $g_\chi g_v$ from LUX [@Akerib:2013tjd] and CDMS-lite [@Agnese:2013jaa] direct detection searches on the scalar mediator benchmark model as a function of the mediator mass $m_\phi$ and dark matter mass $m_\chi$. \[fig:direct\_bounds\]](./dd_bounds_scaled.pdf){width="0.6\columnwidth"}
Indirect Detection
------------------
Indirect detection searches look for dark matter annihilating to Standard Model particles in the Universe today. Such processes could be seen by finding an otherwise unexplained excess of gamma rays or positrons coming from an area of expected high dark matter density. While direct detection searches place non-trivial limits on scalar mediator models, such models result in thermally averaged cross sections $\langle \sigma v\rangle$ which are proportional to $v^2$. The velocity $v$ of dark matter today is very small $\lesssim 10^{-2}c$, and so scalar mediators do not result in significant signals in indirect searches. The velocity-averaged annihilation cross section into Standard Model fermion final states for our two benchmark models are [@Buckley:2013jwa] $$\begin{aligned}
\langle\sigma v\rangle(\chi\bar{\chi} \to \phi^* \to f\bar{f}) & = & \sum_{f} N_f \frac{3g_\chi^2 g_v^2 y_f^2 (m_\chi^2-m_f^2)^{3/2}}{8\pi m_\chi^2\left[ (m_\phi^2 - 4m_\chi^2)^2 + m_\phi^2 \Gamma_\phi^2 \right]} T \\
\langle\sigma v\rangle(\chi\bar{\chi} \to A^* \to f\bar{f}) & = & \sum_{f} N_f \frac{g_\chi^2 g_v^2 y_f^2}{4\pi \left[ (m_A^2 - 4m_\chi^2)^2 + m_A^2 \Gamma_A^2 \right]} \left [m_\chi^2 \sqrt{1-\frac{m_f^2}{m_\chi^2}} +\frac{3m_f^2}{4 m_\chi \sqrt{1-\frac{m_f^2}{m_\chi^2}} } T \right]\end{aligned}$$ Here, $N_f$ is the number of colors of the fermion $f$, and $T$ is the temperature of the dark matter. As $T\propto v^2$, of our two simplified models, only the pseudoscalars have a thermal annihilation cross section with a velocity-independent term. Thus, only the pseudoscalar mediator gives significant annihilation in the Universe today with non-trivial bounds set by indirect detection.
Of particular interest, due to their sensitivity to multiple decay channels, are indirect searches for gamma-ray annihilation, either from direct annihilation (resulting in gamma rays with a characteristic energy of $E_\gamma = m_\chi$), or from a cascade of Standard Model decays after annihilation into heavy, charged, and unstable Standard Model particles, which provide a continuum of gamma rays. For gamma-ray energies (and thus dark matter masses) below approximately a TeV, the Fermi Gamma-Ray Space Telescope (FGST) provides the best bounds at present [@Abdo:2010ex; @GeringerSameth:2011iw; @Ackermann:2011wa; @Geringer-Sameth:2014qqa]. In particular in this paper we will use the bounds set by the FGST in Ref. [@Ackermann:2011wa], searching for dark matter annihilation in dwarf spheroidal galaxies orbiting the Milky Way (see also Ref. [@GeringerSameth:2011iw] for an independent analysis). At the moment these are the most constraining.
We comment that there is an excess of gamma rays from the Galactic Center reported in the FGST data-set [@Goodenough:2009gk; @Hooper:2010mq; @Hooper:2011ti; @Boyarsky:2010dr; @Abazajian:2012pn; @Hooper:2012sr; @Hooper:2013rwa; @Gordon:2013vta; @Huang:2013pda; @Abazajian:2014fta; @Daylan:2014rsa]. Though the source of these gamma rays is still uncertain [@Abazajian:2010zy; @Wharton:2011dv; @Hooper:2013nhl; @Carlson:2014cwa], if interpreted in terms of dark matter, it could be be accommodated by annihilation through a pseudoscalar mediators with Standard Model couplings proportional to Yukawas [@Abdullah:2014lla; @Basak:2014sza; @Berlin:2014pya; @Arina:2014yna; @Cheung:2014lqa; @Balazs:2014jla], as in our benchmark simplified model.
![Contour plot of 95% CL upper bounds on $g_\chi g_v$ derived from indirect detection constraints set by the FGST dwarf spheroidal analysis [@Ackermann:2011wa] in the $b\bar{b}$ channel, as a function of the pseudoscalar mediator mass $m_A$ and the dark matter mass $m_\chi$. The width is set assuming $g_v = g_\chi$, which is relevant only near resonance. \[fig:indirect\_bounds\]](./id_bounds_scaled.pdf){width="0.59\columnwidth"}
In this paper, we use only the 95% CL upper limits on the indirect annihilation cross section into pairs of $b$-quarks from the FGST dwarf analysis [@Ackermann:2011wa], converted to limits on our model parameters by calculating the velocity averaged cross section $\langle \sigma v\rangle$ (see Ref. [@Buckley:2013jwa] for details) evaluated at $v \to 0$. Constraints on $g_\chi g_v$ are shown in Figure \[fig:indirect\_bounds\]. The width $\Gamma_A$ can play an important role here near resonance, so to reduce the parameter space we choose a width under the assumption that the two couplings are equal. This has only a minor effect on the majority of the parameter space. We further assume that no other annihilation channels are present.
Thermal Relic Abundance
-----------------------
By measuring CMB anisotropies, surveys such as the Planck mission have measured the dark matter contribution to the Universe’s energy budget to be $\Omega_{\chi}h^2 = 0.1187 \pm 0.0017$ [@Ade:2013zuv]. From standard Boltzmann relic density calculations [@Kolb], this implies a thermal annihilation cross section of $\langle \sigma v\rangle \sim 3 \times 10^{-26}$ cm$^3$/s. If we assume that $\phi$ is the only connection between the dark and visible sectors, and we further assume that the dark matter is a thermal relic, we can calculate the couplings $g_\chi$ and $g_v$ necessary for the production of the observed density of dark matter.
As with indirect detection, near resonance ($m_\phi \sim 2 m_\chi$) we must assume knowledge of the mediator width $\Gamma_{\phi(A)}$. We make the same assumption as before: that the width is calculated as if $g_v = g_\chi$. Annihilation near resonance can have significant effects on the cross section during thermal freeze-out, which we take into account using the methods outlined in Ref. [@3Excpns]. Away from resonance, the thermally averaged cross section becomes identical to that calculated for the indirect detection constraints, evaluated at the freeze-out temperature $T_f = m_\chi/x_f \sim m_\chi/ 25$.
Additionally, when $m_\phi < m_\chi$, dark matter can annihilate in the process $\bar{\chi}\chi \to \phi \phi$, followed by decay of the $\phi$. Thus a thermal relic can be obtained even when $g_v \sim 0$, as long as the $\phi$ is not sufficiently long-lived as to decay after Big Bang Nucleosynthesis. Such detector-stable particles are completely consistent as a dark matter mediator, but may require searches targeted towards displaced vertices. For the purposes of this paper, will not consider these models in more detail here, though the possibility should not be ignored. The required combinations of couplings $g_\chi g_v$ in order to obtain a thermal abundance are shown in Figure \[fig:thermal\_bounds\], assuming the only open channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$. We again emphasize that the regions of mass and coupling parameter space that do not yield a correct thermal relic under our specific set of assumptions are still of great interest, and so these constraints should not be taken as the final word on dark matter physics. Recall that we are discussing a simplified scenario, which presumably fits into a larger model of the dark sector. If the couplings are too small to give the correct relic abundance, then the simplified model predicts an over-abundance of dark matter from thermal processes. However, entropy dilution could reduce the dark matter density, if the physics in the Early Universe is non-standard [@Hooper:2013nia]. Somewhat more prosaically, the full theory of the dark sector could contain additional mediating particles that increase the annihilation cross section [@ArkaniHamed:2008qn]. If the couplings under consideration are larger than required for thermal annihilation, then non-thermal models of dark matter (such as asymmetric dark matter) are an attractive possibility [@Kaplan:2009ag; @Cohen:2009fz; @Belyaev:2010kp; @Davoudiasl:2010am; @Buckley:2010ui; @Buckley:2011kk].
![Required values of $g_\chi g_v$ as a function of mediator mass $m_{\phi(A)}$ and dark matter mass $m_\chi$ assuming that dark matter is a thermal relic and the only annihilation channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$, for the scalar (left) and pseudoscalar (right) simplified models. \[fig:thermal\_bounds\]](./thermal_scalar.pdf "fig:"){width="0.42\columnwidth"} ![Required values of $g_\chi g_v$ as a function of mediator mass $m_{\phi(A)}$ and dark matter mass $m_\chi$ assuming that dark matter is a thermal relic and the only annihilation channel is $\bar{\chi}\chi \to \phi(A) \to \bar{f}f$, for the scalar (left) and pseudoscalar (right) simplified models. \[fig:thermal\_bounds\]](./thermal_pseudo.pdf "fig:"){width="0.42\columnwidth"}
Collider bounds {#sec:collider_bounds}
===============
Having placed bounds on our simplified models from direct detection, indirect detection, and under the assumption that the dark matter obtains the thermal relic abundance, we now turn to bounds from the LHC. The most obvious signature for dark matter at colliders is missing transverse momentum (more colloquially, missing transverse energy). When dark matter is produced it escapes the detector unseen, leaving an imbalance of momentum which can be measured in the transverse plane. This missing transverse momentum is a powerful signature for new physics models. MET signatures must be accompanied by some associated production of visible particles, both for momentum conservation and triggering. We consider three signatures in this paper: MET with associated untagged jets, MET with two associated dileptonic tops, and MET plus one or two $b$-tagged jets.
In all these searches, we follow our previous policy of setting upper bounds on the combination $g_v g_\chi $. However, unlike the previous examples, the branching ratios of the mediators $\phi$ or $A$ are integral to the bounds set. By setting the limit on the combination of couplings, the mediator width $\Gamma_{\phi(A)}$, which depends on $g_\chi^2$ and $g_v^2$ separately, must be specified as an independent parameter.
Both the simplified models and EFTs can consider scenarios where the mass hierarchy is inverted ($2m_\chi > m_{\phi(A)}$). For EFTs, this makes no difference (other than bringing into question the applicability of the effective operator approach). However, in our simplified models, if the mediator is light enough to be produced at a collider, but the dark matter is heavy enough so that it cannot be the product of on-shell decay of the mediator, then it is likely that better search strategies would be those based around the decays of the mediator into visible final states. For heavy mediators ([*i.e.*]{} $m_{\phi(A)} \gtrsim 1$ TeV at the LHC) the searches for dark matter with masses satisfying $2m_\chi < m_{\phi(A)}$ would be reliant on the off-shell mediator production. For scalars and pseudoscalar mediators, however, the current constraints in this regime from the LHC turn out to be extremely weak. As a result, in this paper, we will concentrate on the $m_{\phi(A)} > 2m_\chi$ regime, and leave the remainder of the mass plane for future work.
Considering the importance of the width on the collider constraints for much of the accessible parameter space, we chose to parametrize the derived limits on $g_\chi g_v$ at fixed dark matter and mediator masses, varying the width $\Gamma_{\phi(A)}$. We choose two mediator masses: ${m_{\phi(A)} = 100}$ GeV, and 375 GeV, and $m_\chi = 40$ GeV. For on-shell mediator production, the bounds could be easily extrapolated to other dark matter masses (up to the kinematic limit $2m_\chi = m_{\phi(A)}$) by rescaling the overall branching ratio into dark matter at a new mass point. Recall that the kinematic suppression for scalars ($\beta^3$) will be more significant than that of pseudoscalars ($\beta$) for the 100 GeV benchmark, as a 40 GeV dark matter particle is near the kinematic threshold.
Mono-jet Search
---------------
At a hadron collider, unless the mediator has large couplings to $W/Z/\gamma$ compared its coupling to the colored partons, we would expect the strongest constraints to come from the production of dark matter in association with an initial state jet radiation [@Goodman:2010ku; @Beltran:2010ww; @Fox:2011pm; @Goodman:2010yf; @Rajaraman:2011wf; @Fox:2012ee]. Both ATLAS [@ATLASmonojet] and CMS [@CMSmonojet] have performed dedicated “monojet” searches using Run-I LHC data at $\sqrt{s} = 8$ TeV. We note that the “monojet” moniker is something of a misnomer, as these analyses do allow a second high-$p_T$ jet in the sample.
We use results from CMS [@CMSmonojet] to derive bounds on couplings for our benchmark models. The CMS search used a data sample corresponding to an integrated luminosity of $19.5$ fb$^{-1}$. Events are required to have one jet with $p_{Tj} > 110$ GeV. A second jet is allowed, but no more than two jets with $p_{Tj}> 30$ GeV. Signal events are grouped into seven MET bins: $\slashed{E}_T > 250$, 300, 350, 400, 450, 500, and 550 GeV. The CMS Collaboration has provided the number of events in each bin that can be accommodated as signal at the 95% CL, which we use to place bounds on $g_\chi g_v$ as a function of $m_{\phi(A)}$, $m_\chi$, and $\Gamma_{\phi(A)}$, using the most constraining limit from any of the seven MET signal bins.
As we showed in Section \[sec:models\], the treatment of the $g$–$g$–$\phi(A)$ interaction as an effective operator would introduce significant errors in the extrapolated bounds on the model parameters. Hence, accurate distributions of MET and jet $p_T$ require simulation of $\phi$ or $A$ plus a hard parton including the exact heavy-quark loop effects. We implement this in [MCFMv6.8]{} [@Campbell:2010ff; @hj], modifying the process ${pp\rightarrow H(A)+j \rightarrow \tau^+ \tau^-+j}$ in [MCFM]{} to produce events files which can be subsequently showered and hadronized by [Pythia8]{} [@Sjostrand:2006za; @Sjostrand:2007gs], then fed into a detector simulator. Note that, while the CMS analysis allows a second jet, our [MCFM]{} simulation is limited to one hard parton, though additional jets are generated through the [Pythia8]{} parton shower. See Refs. [@hjets; @Campanario:2010mi; @Buschmann:2014twa] for issues pertaining the simulation of the second jet including the top mass effects. In addition, we generalized the [MCFM]{} implementation including the possibility of off-shell mediator production. As there are no full Next-to-Leading order (NLO) predictions including the top mass effects for this process in the literature, we include these effects via a flat correction factor $K\sim 1.6$ obtained using the infinite top mass limit [@kfactor] . Our hard scales are defined as $\mu_F^2=\mu_R^2=m_{\phi(A)}^2+p_{Tj}^2$, and we used the [CTEQ6L1]{} parton distribution functions [@cteq].
![Missing transverse momentum differential cross sections for the scalar (left panel) and pseudoscalar (right panel) mediators. The leading order effective gluon couplings are shown as dashed lines, and the exact loop-induced calculations are solid. We assume the LHC at 8 TeV. \[fig:pt\_distributions\]](./etmiss_scalar.pdf "fig:"){width="0.4\columnwidth"} ![Missing transverse momentum differential cross sections for the scalar (left panel) and pseudoscalar (right panel) mediators. The leading order effective gluon couplings are shown as dashed lines, and the exact loop-induced calculations are solid. We assume the LHC at 8 TeV. \[fig:pt\_distributions\]](./etmiss_pseudo.pdf "fig:"){width="0.4\columnwidth"}
![Lower limit on the coupling $g_\chi g_v$ set by the CMS monojet search as a function of dark matter mass $m_\chi$, assuming mediators of 100 GeV, $\Gamma_{\phi(A)}/m_{\phi(A)} = 10^{-3}$, and exclusively on-shell production of dark matter. The constraint on the scalar mediator is shown in red and pseudoscalars in blue. \[fig:dm\_dependence\]](./dm_dependence.pdf){width="0.48\columnwidth"}
Whereas the primary effect on the bounds placed on the combination $g_\chi g_v$ from varying the width $\Gamma_{\phi(A)}$ is just a rescaling of the branching ratio to dark matter, there can be small secondary effects when the width is significant compared to the mediator mass. To investigate these effects, as well as demonstrate the importance of the full simulation on the bounds, we also generate dark matter events in our two simplified models using [MadGraph5]{} [@Alwall:2011uj; @Alwall:2014hca]. This implementation starts with the inclusion of our [*Simplified Model*]{}, presented in Eq. , into [Feynrules]{} [@Alloul:2013bka] which generates a model file that is subsequently used by [MadGraph]{}. In [MadGraph]{} we produce $\phi(A)$ events matched up to two jets via the MLM scheme [@Mangano:2006rw]. We also include the detector simulation through [Delphes3]{} [@deFavereau:2013fsa]. In Figure \[fig:pt\_distributions\], we compare the distributions for the leading jet $p_T$ and the MET in the narrow width approximation generated by both [MCFM]{} and [MadGraph5]{}, after the CMS event selection criteria. As in Figure \[fig:etmiss\_intro\], the effective gluon operator overestimates the distribution tails, which would lead to an overly aggressive bound on the couplings. Notice that these differential distributions do not differ from the exact result by just a flat factor, but have different shapes. While these effects are important here, they will be even more critical in future LHC runs, where the energies will be higher and the MET cuts will be harsher. To confirm the consistency of our implementation, we have produced results in the EFT limit $(m_t\rightarrow \infty,m_\phi\rightarrow \infty)$ and validated it against the CMS EFT bounds [@CMSmonojet].
Using these simulations, we place 95% CL bounds on $g_\chi g_v$ as a function of $\Gamma_{\phi(A)}/m_{\phi(A)}$, for 100 and 375 GeV mediators and $m_\chi =40$ GeV. Our results are shown in Figure \[fig:scalar\_bounds\] for the scalar mediator and Figure \[fig:pseudo\_bounds\] for the pseudoscalar. Two points from these results should be addressed in detail.
1. The different dependence on the scalar and pseudoscalar widths on $\beta$ have an important effect on the results. For the light mediator, the scalar partial width into dark matter ($\propto\beta^{3}$) significantly reduces the total cross section when compared to the pseudoscalar ($\propto\beta$). As a result, the couplings to the scalar must be larger than the pseudoscalar for the $100$ GeV mediators. For the heavy mediator, neither scenario has a significant kinematic suppression. This is dependent on our choice of dark matter mass; as the dark matter mass increases, we expect to see the scalar bounds weaker faster than the pseudoscalar. This is explicitly an effect due to on-shell production of the mediator; if the dark matter mass was heavier than $m_{\phi(A)}/2$, then the monojet channel would only be sensitive to production of dark matter via an off-shell mediator, which does not scale with the kinematic suppression factor. In Figure \[fig:dm\_dependence\], we show the scaling of the monojet bound as a function of dark matter mass, assuming a 100 GeV scalar or pseudoscalar (and $\Gamma_{\phi(A)}/m_{\phi(A)} = 10^{-3}$).
2. In the case of the [MCFM]{} results, the changing width only causes a rescaling of the total rate of mediator production times decay into dark matter through the changing branching ratios. While this is the dominate effect for the finite width calculation, there is a subleading effect at $\Gamma_{\phi(A)}/m_{\phi(A)} \gtrsim 0.1$, where the tail of the mediator $p_T$ distribution (and thus the MET) can be increased relative to the narrow width approximation. This is a result of the mediator being able to be produced with $q^2$ very far away from the expected mass, convolved with the proton parton distribution functions. For the 100 GeV mediators, as the width is increased this secondary effect causes the bound on $g_\chi g_v$ to weaken less rapidly than one would expect from the branching ratio alone. The effect is negligible for the 375 GeV mediators.
Heavy Flavor Searches
---------------------
One would expect that the strongest constraint that the LHC can place on the dark matter decay channels of our benchmark scalar and pseudoscalar mediators comes from the general jets plus missing transverse energy search discussed previously, as the production cross section here is highest. However, channels with missing energy associated with particles other than untagged jets can have significantly lower backgrounds (and different systematics) than the monojets. Therefore, we can and should consider searches in additional channels. Though we will often find that limits placed on the couplings will be weaker than those placed by the monojet search, this approach is still critical as the LHC continues to ramp up to higher energies and luminosities. Recall that we are working with a simplified model, purposefully constructed to minimize the number of free parameters. Therefore, under these assumptions we can predict the exact ratio of signal strength in multiple channels, as the cross section for each is set by the same masses and couplings. However, we must be open to deviations from the simplified model. For example, if the couplings to up- and down-type couplings are not set by a universal coupling $g_v$, or if the loop-induced gluon coupling does not depend solely on the couplings to top and bottom quarks, then it is quite possible that the signal in the monojet channel could be suppressed relative to other production mechanisms. Discovery in more than one channel would also allow better understanding of the theoretical underpinnings of any new physics.
With that motivation in mind, it is clearly important to look for new physics in many associated channels. Even when considering modifications to the baseline models, it is still reasonable to assume that the interactions with fermions are largely MFV, and therefore that the mediator is most strongly coupled to the heaviest fermions. Therefore, we show here limits on production of the $\phi$ or $A$ in association with top and bottom quarks, followed by the invisible decay of the mediator into dark matter. Some of the main production diagrams for such processes are shown in Figure \[fig:feyn\_heavy\].
![Representative Feynman diagrams contributing to heavy quark flavor plus dark matter production at the LHC in our [*Simplified Models*]{}. \[fig:feyn\_heavy\]](./feyn_heavy.pdf){width="0.5\columnwidth"}
![95% CL upper limits on $g_\chi g_v$ for scalar mediators from collider searches as a function of $\Gamma_\phi/m_{\phi}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) scalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. The grayed-out region indicates where the minimum width consistent with $g_\chi g_v$ is greater than the assumed width. \[fig:scalar\_bounds\]](./scalar_combo_bounds_100_40_scaled.pdf "fig:"){width="0.5\columnwidth"}![95% CL upper limits on $g_\chi g_v$ for scalar mediators from collider searches as a function of $\Gamma_\phi/m_{\phi}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) scalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. The grayed-out region indicates where the minimum width consistent with $g_\chi g_v$ is greater than the assumed width. \[fig:scalar\_bounds\]](./scalar_combo_bounds_375_40_scaled.pdf "fig:"){width="0.5\columnwidth"}
We use the CMS dedicated search for dark matter produced in events with dileptonic tops [@CMSttbar], performed on 19.7 fb$^{-1}$ of integrated luminosity at the 8 TeV LHC. The analysis requires exactly two isolated leptons with individual $p_T > 20$ GeV and $\sum{p_t} > 120$ GeV, and at least two jets with $p_T > 30$ GeV. The invariant mass of the leptons must be greater than $20$ GeV, and if they are the same flavor, a $Z$-mass veto of $|m_{\ell\ell} - 91~\mbox{GeV}|>15$ GeV is applied. The two jets are required to have invariant mass of less than $400$ GeV. The signal region is $\cancel{E}_T > 320$ GeV. As with the monojet analysis described previously, we can straightforwardly recast the CMS limits to apply to our benchmark models, based on the number of events seen in their signal region. Signal was generated using [MadGraph5]{}, passed through the [Pythia6]{} and [Delphes3]{} pipeline described earlier. As in the monojet case, we validate our results using the dark matter EFT to compare with the CMS results. We show the bounds from this channel on $g_\chi g_v$ for our benchmark mediator models (for mediators of 100 and 375 GeV, and 40 GeV dark matter) as a function of mediator width in Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\].
Finally, we can consider the associated production of the mediator $\phi$ or $A$ with $b$-quarks. Until recently, no dedicated dark matter search similar to the monojet or dileptonic top plus MET analyses has been performed for the process $p p \rightarrow \chi \bar{\chi} +b \bar{b}$, and constraints could only be extracted using the sbottom searches $p p \rightarrow \tilde{b}^{*} \tilde{b} \rightarrow \chi \bar{\chi} + b \bar{b}$ from CMS [@CMSbbar] and ATLAS [@Aad:2013ija]. These searches have selection criteria which are far from ideal for the kinematics of the simplified models, but they do place relevant constraints directly on the tree-level interaction between $b$-quarks and the mediator.
Recently however, ATLAS has published a dedicated search for dark matter produced in associated with $b$-tagged jets in 20.3 fb$^{-1}$ of 8 TeV data [@Aad:2014vea]. Two signal categories in this search are relevant for our analysis here. In both, the analysis vetoes events with leptons that have $p_T > 20$ GeV and requires $\cancel{E}_T > 300$ GeV. The azimuthal angle between all jets and the MET must be $\Delta \phi >1$. Signal Region SR1 requires one or two jets, at least one of which must be $b$-tagged (at a $60\%$ efficiency) and have $p_T > 100$ GeV. Signal region SR2 requires three or four jets in the event, again requiring at least one to be $b$-tagged with $p_T > 100$ GeV. If a second $b$-tagged jet exists, it must have $p_T> 60$ GeV, and the second highest $p_T$ jet must have $p_T > 100$ GeV. ATLAS provides the 95% CL upper limit on the number of events in each signal region which can be accommodated by new physics, and we validate our simulation using the EFT results.
We again generate our signal events using [MadGraph5]{}, through the tree-level coupling of the mediator and the $b$-quarks. As with the monojet search, for each of our benchmark models, we use the strongest limit on $g_\chi g_v$ set by either of these signal regions.
![95% CL upper limits on $g_\chi g_v$ for pseudoscalar mediators from collider searches as a function of $\Gamma_A/m_{A}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) pseudoscalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the indirect detection limit in the $b\bar{b}$ channel from FGST. \[fig:pseudo\_bounds\]](./pseudoscalar_combo_bounds_100_40_scaled.pdf "fig:"){width="0.5\columnwidth"}![95% CL upper limits on $g_\chi g_v$ for pseudoscalar mediators from collider searches as a function of $\Gamma_A/m_{A}$, assuming 40 GeV dark matter and 100 GeV (left) and 375 GeV (right) pseudoscalar mediators. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the indirect detection limit in the $b\bar{b}$ channel from FGST. \[fig:pseudo\_bounds\]](./pseudoscalar_combo_bounds_375_40_scaled.pdf "fig:"){width="0.5\columnwidth"}
The results from this analysis are shown along with our previous limits as a function of mediator width in Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\]. Along with the bounds derived from colliders, we include the direct and indirect constraints (for scalar and pseudoscalar models, respectively) and the required value of $g_\chi g_v$ to obtain the thermal relic abundance. While it is a very useful benchmark to compare the experimental sensitivity, note that coupling values that diverge from that required for a thermal relic are still experimentally and theoretically interesting: as we consider only a [*Simplified Model*]{}, we do not attempt to specify the full theory. Further, we do not even know that dark matter is in fact a thermal relic. If dark matter was generated through some asymmetric process (like baryons), then one would not expect the low-energy annihilation channels to obtain a thermal abundance.
In Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], we also show the exclusion region of coupling-width parameter space that is theoretically inconsistent. While we cannot specify a width only from the coupling combination $g_\chi g_v$, we can calculate the minimum possible width (assuming only decays into the dark matter and the Standard Model fermions) that is consistent with a given value of $g_\chi g_v$. That is, for a given width $\Gamma_{\phi(A)}$, we find the minimum value of the product $g_\chi g_v$ which would allow $$\Gamma_{\phi(A)} > \frac{g_\chi^2 m_{\phi(A)} }{8\pi} \left(1-\frac{4m_\chi^2}{m_{\phi(A)}^2} \right)^{n/2}+\sum_f \frac{g_v^2 y_f^2 m_{\phi(A)}}{16\pi} \left(1-\frac{4m_f^2}{m_{\phi(A)}^2} \right)^{n/2},$$ for any values of that $g_\chi$ and $g_v$ which satisfy the product constraint (here $n= 1$ for pseudoscalars and 3 for scalars). We gray-out the regions of $g_\chi g_v$ parameter space where minimum width possible for any $g_\chi$ and $g_v$ is larger than the assumed $\Gamma_{\phi(A)}$.
Examining Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], it is interesting to note that the top constraints on the scalar mediator are competitive (within the accuracy of our simulated search) with those of the monojet channel at low mediator masses. This is due to the relative suppression of the scalar coupling to gluons compared to the coupling to the fermions Eq. . The pseudoscalar gluon coupling does not have the same level of suppression, leading to a larger production cross section in the monojet channel, and thus better bounds when compared to the heavy flavor channel. As the mediator mass increases, the production of a heavy particle in association with the two massive tops is suppressed, and the monojet constraint regains its preeminence for the scalar model.
The $b$-tagged channel places significantly weaker constraints on these models than the monojet or the top channels. However, as this probes directly the coupling to the down-sector, it would be sensitive to deviations the universal coupling assumption in a way that the top channel is not, as the top channel relies on the same coupling as the loop-induced monojet search, unless new colored particles coupling to the mediator exist in the spectrum. The direct detection constraints are also very powerful compared to the collider reach (though for dark matter masses less than $\sim 6$ GeV, the colliders are more constraining) for scalar mediators, while the pseudoscalars are much less constrained by the indirect searches, are comparable with the current LHC constraints. However, as we argued previously, multiple probes in multiple channels are still necessary, as simple modifications of the basic model or experiment-specific backgrounds and uncertainties could increase the sensitivity of one mode while decreasing another. In our search for new physics, we must exhaust all reasonable search strategies.
Higgs Mediators \[sec:higgs\]
=============================
As we have often mentioned throughout this work, there are obvious connections between our scalar and pseudoscalar simplified models and Higgs physics. In addition to the possible embedding of the simplified models into extended Higgs sectors, the couplings (both tree-level and loop-induced) even in the general scenarios have many similarities with Higgs physics (due in part to the MFV assumption). The correct technique for generation of high $p_T$ events through the gluon-mediator coupling was also inherited from Higgs physics. With these considerations, it is reasonable to ask what bounds can be set on the 125 GeV Higgs itself, assuming that it is the scalar mediator between the visible and the dark sector. This is the well-known “Higgs Portal” scenario for dark matter [@Burgess:2000yq; @Davoudiasl:2004be; @Patt:2006fw; @Andreas:2008xy; @Barger:2008jx; @Lerner:2009xg; @He:2009yd; @Barger:2010mc; @Djouadi:2011aa; @Kanemura:2010sh; @Mambrini:2011ik; @He:2011de; @Han:2013gba; @Greljo:2013wja; @Okada:2013bna; @Chacko:2013lna; @deSimone:2014pda; @Endo:2014cca; @Drozd:2014yla; @Englert:2013gz] (similarly, one could consider the “dilaton” portal [@Bai:2009ms; @Agashe:2009ja; @Blum:2014jca; @Efrati:2014aea]). Collider bounds on the 125 GeV Higgs decaying to dark matter can be placed in two ways. First, just as we have done previously, we can place limits on the total cross section from the monojet and heavy flavor channels, which can be translated into limits on the coupling of the Higgs to dark matter. Secondly, we can use the experimental measurements of the Higgs width to constrain the addition of new channels to Higgs decay.
![95% CL upper limits on $g_\chi g_v$ for the 125 GeV Higgs from collider searches as a function of the width $\Gamma$, assuming 40 GeV dark matter. The limit from the CMS monojet search is shown as the solid colored (red or blue) line for the Full Theory including heavy quark mass effects [MCFM]{} calculation. The [MadGraph]{} effective operator CMS monojet constraint is shown in dashed color. The shaded region indicates an extrapolation of the finite width effects to the [MCFM]{} results. The constraint from the top pair plus missing energy search is the dashed black line, and the $b$-jet plus missing energy search limit is the dotted black line. The horizontal solid black line shows the direct detection limit from LUX and CDMS-lite. Three vertical lines show experimental limits on the 125 GeV Higgs’ width assuming Standard Model couplings and an invisible branching ratio of 54% [@Chatrchyan:2014tja] (dotted purple), the upper limit on the width from interference with the $Z$ [@Khachatryan:2014iha] (dashed purple), and the maximum possible width from the $4\ell$ lineshape (solid purple) [@Chatrchyan:2013mxa]. \[fig:higgs\_bounds\]](./higgs_combo_bounds_40_scaled.pdf){width="0.5\columnwidth"}
We can extract constraints on the total width of the Higgs in three different ways. First, if we require that the coupling to the Standard Model is exactly that of [*the*]{} Standard Model Higgs, then by requiring the visible production and decay channels are consistent with observations, the total invisible branching ratio must be less than $0.54$ at 95% CL [@Chatrchyan:2014tja] (see also Ref. [@Aad:2014iia]). Given the Standard Model Higgs width of 4.1 MeV [@Heinemeyer:2013tqa], the addition of a decay to dark matter saturating this bound gives a total width of at most 8.9 MeV. Furthermore, as this assumes that $g_v = 1$, in this restricted subset of the model space, the dark matter coupling can be constrained to be less than $$g_\chi^2 \leq \frac{8\pi}{m_h} \left(1-\frac{4m_\chi^2}{m_h^2} \right)^{-3/2} \times \left( 8.9~\mbox{MeV} \times 0.54 \right).$$
This chain of logic does require that the Higgs couplings be exactly the Standard Model values. Somewhat weaker constraints can be placed on the invisible branching ratio once this assumption has been lifted. This does not extend to statements about the total width. Though perhaps unlikely from a theoretical standpoint, it is possible that a larger branching ratio to dark matter could be compensated by larger couplings for the production of the Higgs, leaving the rates for the observed channels unchanged [@Belanger:2013kya].
The second method of measuring the Higgs width relaxes the requirement that the couplings to the fermions and gauge bosons are as in the Standard Model, and places a bound on the width via the measured interference of the Higgs and the $Z$. This constrains the Higgs width to be $\Gamma_h < 17.4$ MeV [@CMS-PAS-HIG-14-002; @Khachatryan:2014iha]. However, as with the invisible Higgs decay measurement, this interference effect does make some assumptions about the production mechanism of the Higgs [@Englert:2014aca; @hjets]. The third method remains fully agnostic as to the Higgs couplings. This is the most robust, but least constraining measurement: the direct measurement from the $h \to ZZ^* \to 4\ell$ channel, which has measured $\Gamma_h < 3.4$ GeV [@Chatrchyan:2013mxa].
In Figure \[fig:higgs\_bounds\], we show the collider and direct detection constraints on the 125 GeV Higgs boson as a function of total width, assuming a coupling to dark matter $g_\chi$ (unlike Figures \[fig:scalar\_bounds\] and \[fig:pseudo\_bounds\], note that the horizontal axis is $\Gamma_h$, not $\Gamma_{\phi(A)}/m_{\phi(A)}$). As before, we parametrize the coupling to the Standard Model fermions as $g_v$. Given the present concordance between experiment and theory, the primary model-building focus for Higgs physics appears to be concentrating on scenarios with $g_v\sim 1$, and it appears to be difficult to find models where large deviations from the Standard Model prediction is consistent with all Higgs data in a realistic extension of the Standard Model [@sfitter]. As we saw in the general scalar mediator, the collider bounds are much less constraining than those set by direct detection experiments. While the collider constraints are relatively insensitive to dark matter masses below $m_h/2$, the direct detection bounds weaken significantly significantly if the dark matter is below $\sim 6$ GeV.
It is surprising to see that the associated top channel is comparable here to the monojets, given the experimental difficulties in probing Standard Model $tth$ production. However, recall that the Standard Model search in this channel is forced to rely on $h\to b\bar{b}$ decay. If we assume a significant branching ratio of the 125 GeV Higgs into invisible dark matter, the much lower backgrounds in the dileptonic top plus MET channel allow the experiments to set a bound comparable to that of the monojets.
Conclusions
===========
The next few years of data from the LHC Run-II has the potential to shed new light on the nature of dark matter. The EFT formalism has been very useful in the analysis of Tevatron and LHC data, allowing straightforward comparisons to direct and indirect searches, and moving dark matter searches in a more model-independent direction. However, the powerful bounds set by the LHC push the theory into a regime where the EFT often does not generally apply. This should be a cause for optimism: the break-down of the consistency of the EFT implies that, for much of the parameter space, if the LHC can produce dark matter then it can also produce associated particles that mediate the interaction between the dark sector and our own.
Previous works have introduced various simplified models which bridge the theoretical divide between the EFT and complete models such as supersymmetry. We add to this work by constructing two benchmark models of spin-0 mediators coupling to dark matter consisting of Dirac fermions. While such attractive models have been considered in the past, we – for the first time – provide a comprehensive set of constraints from direct detection, indirect searches, and three collider channels associated with missing transverse energy.
As previous works have noted, care must be taken when simulating scalar mediated missing energy searches at the LHC, as these are primarily produced through a top-loop induced coupling to gluons. As the transverse momentum flowing through this loop is large compared to $2m_t$ (and may be large compared to the mediator mass), it has been demonstrated that working in approximations of infinite top mass and/or on-shell gluons can incorrectly predict the MET and $p_T$ distributions. In this paper, we clearly show the impact of these effects on the distribution of jet $p_T$ and MET, which are critical to missing energy searches at the LHC, and outline appropriate techniques for simulating these models. These issues will become even more important in future LHC runs, where higher energies will force harsher MET and $p_T$ cuts, further increasing the deviation between the distributions predicted by an effective operator treatment of the loop-coupling, and the correct one.
For our benchmark models, the monojet channel remains the most constraining out of all the collider bounds. However, associated heavy flavor searches are important; associated production with tops can rival the monojet channel in the low mediator mass region. As such these additional searches should be pursued as complimentary to the monojet bounds, sensitive to different combination of couplings. Similarly, the direct detection bounds place much more powerful limits on the couplings for scalar models, assuming the dark matter mass is heavier than $\sim 6$ GeV. However, there are astrophysical uncertainties inherent to direct detection limits, and the LHC searches provide an complimentary testing ground, one that independent of the uncertainties on our local dark matter density and velocity distribution. Similar astrophysical uncertainties also relate to bounds placed by indirect detection, and further collider searches may be a key factor in resolving the active debate about claimed signals from the Galactic Center. As can be seen from Figure \[fig:pseudo\_bounds\], the current constraints already touch on the relevant parameter space for $m_\chi \sim 40-50$ GeV, and can indeed rule out simplified models with mediators much heavier than 100 GeV as the source of the anomaly. Though modifications of the benchmark simplified model can explain the Galactic Center excess with particles that have vanishing LHC cross sections [@Abdullah:2014lla; @Cheung:2014lqa], it is interesting that one of the simplest scenarios is not yet ruled out, yet lies within realistic reach of the LHC in the near future.
The searches we extracted bounds from in this paper were pre-existing and easily adapted to our simplified models. However, as should be clear, many other possible channels exist, which would place complimentary bounds on the couplings of our benchmark models. In addition to further missing energy searches in association with heavy flavor – in particular, searches with $\tau$ leptons, which would probe the mediator-lepton coupling – we suggest that future work should also consider the constraints from decays of mediators back into Standard Model particles.
Given couplings $g_\chi$ and $g_v$ which are of the same order of magnitude, one would expect decays to dark matter to dominate. However, it is possible that $g_v \gg g_\chi$, or that the dark matter itself is kinematically inaccessible as a decay product of the mediator. In this second case, though some missing energy constraints can be placed from dark matter production via off-shell mediators, the collider production cross section of the mediator itself would be far higher. Channels with decays to $b\bar{b}$, $\tau \tau$, top pairs, or the experimentally clean $\gamma\gamma$ signatures are all likely candidates for dark matter simplified models, particularly with the spin-0 mediators considered here. If the width is small, than long-lived mediators are possible, and searches for displaced decays back to visible particles could place important limits on models with small $g_v$ and $g_\chi$ which would be otherwise inaccessible. CMS, ATLAS, and Tevatron have performed searches in some of the prompt channels, though their results are typically presented in terms of two-Higgs doublet models. As we have described in this paper, such signatures can be relevant to a large range of models, and could be an important part of our search for the new physics of dark matter.
Acknowledgements {#acknowledgements .unnumbered}
================
MRB thanks Tim Tait, Dan Hooper, and Maria Spiropulu for helpful comments and suggestions.
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[^1]: If referring to both the scalar and pseudoscalar models simultaneously, we will use mediator mass $m_{\phi(A)}$ and mediator width $\Gamma_{\phi(A)}$.
|
A. R. Mirotin\
**FREDHOLM AND SPECTRAL PROPERTIES OF TOEPLITZ OPERATORS ON $H^p$ SPACES OVER ORDERED GROUPS**\
amirotin@yandex.ru
Toeplitz operators on spaces $H^p(G)\ (1< p<\infty)$ associated with compact connected Abelian group $G$ with ordered dual are considered and the generalization of the classical Gohberg-Krein theorem on the Fredholm index of such operators with continuous symbols is proved. Applications to spectral theory of Toeplitz operators are given and examples of evident computation of index have been considered.
**А. Р. Миротин\
ФРЕДГОЛЬМОВЫ И СПЕКТРАЛЬНЫЕ СВОЙСТВА ТЁПЛИЦЕВЫХ ОПЕРАТОРОВ В ПРОСТРАНСТВАХ $H^p$ НАД УПОРЯДОЧЕННЫМИ ГРУППАМИ**
[Рассматриваются тёплицевы операторы в пространствах $H^p(G)\ (1< p<\infty)$, ассоциированных с компактной связной абелевой группой $G$, группа характеров которой упорядочена, и в случае линейного порядка доказывается теорема об индексе Фредгольма для таких операторов с непрерывным символом, обобщающая классическую теорему Гохберга-Крейна. Указываются приложения полученных результатов к спектральной теории тёплицевых операторов и рассматриваются примеры явного вычисления индекса. ]{}
**§1. Введение**
Классическая теория тёплицевых операторов, возникшая первоначально как теория операторов в гильбертовом пространстве $H^2$ на группе $\mathbb{T}$ вращений окружности, была в дальнейшем перенесена на банаховы пространства $H^p\ (1<p<\infty)$ на $\mathbb{T}$ (см., например, [@BS]). Данная работа посвящена некоторым вопросам фредгольмовой и спектральной теории тёплицевых операторов в пространствах $H^p\ (1<p<\infty)$ на компактных абелевых группах.
Всюду ниже $G$ есть нетривиальная связная компактная абелева группа с нормированной мерой Хаара $m$ и (вообще говоря, частично) упорядоченной группой характеров $X$, $T$ – положительный конус в $X$, порождающий $X$. Другими словами, в группе $X$ выделена подполугруппа $T$, содержащая единичный характер ${\bf 1}$ и такая, что $T\cap T^{-1}=\{{\bf 1}\}$ и $X=T^{-1}T$. При этом полугруппа $T$ индуцирует в $X$ частичный порядок, согласованный со структурой группы, по правилу $\xi\leq\chi:=\chi\xi^{-1}\in
T$. Очевидно, что $T$ – конфинальная часть $X$ и направленное множество относительно рассматриваемого порядка (если $\lambda_1,\lambda_2\in T$, то $\lambda_1\leq \lambda_1\lambda_2,\lambda_2\leq \lambda_1\lambda_2$). Если дополнительно $T\cup T^{-1}=X$, то этот порядок линеен (пишем $X$ л. у.). Хорошо известно, что (дискретная) абелева группа $X$ может быть линейно упорядочена тогда и только тогда, когда она не имеет кручения (см., например, [@Rud]), что, в свою очередь, равносильно тому, что её группа характеров $G$ связна [@Pont] (при этом линейный порядок в $X$, вообще говоря, не единственен).
В описанной выше ситуации тёплицевы операторы $T_\varphi$ с символом $\varphi\in L^\infty(G)$ в обобщённых пространствах Харди $H^2(G)$ были определены Дж. Мёрфи в [@Mur87] и интенсивно изучались (см., например, [@Mur87] – [@XuChen]). В частности, было показано, что соответствующие обобщённые тёплицевы алгебры обладают рядом интересных свойств, если порядок линеен. Более общее определение тёплицева оператора над группами дано в [@IJPAM], [@Corr]. Рассматривались также обобщения и дискретных тёплицевых операторов, в том числе и с матричным символом, см. статью [@EMRS] и библиографию там (автор признателен рецензенту, обратившему его внимание на эту работу).
Ниже мы рассматриваем тёплицевы операторы в пространствах $H^p(G)\ (1< p<\infty)$ и в случае линейного порядка доказываем теорему об индексе Фредгольма для таких операторов с непрерывным символом, обобщающую классическую теорему Гохберга-Крейна. Попутно установлены обобщения теорем Брауна-Халмоша и Хартмана-Винтнера. Указаны приложения полученных результатов к спектральной теории тёплицевых операторов и рассмотрены примеры явного вычисления индекса. При этом, хотя ряд результатов для общих групп $G$ оказался аналогичен классическому случаю $G=\mathbb{T}$, обнаружились и существенные отличия. Например, если $G\not=\mathbb{T}$, то не каждый тёплицев оператор с неаннулирующимся непрерывным символом фредгольмов, существуют группы, для которых фредгольмовость соответствующих им тёплицевых операторов равносильна обратимости, существенный спектр тёплицева оператора, вообще говоря, не совпадает с множеством значений его символа и т. д. (см. теорему 4 и следствия 1, 2, 3, 5, 6, 7 и 11, а также примечание рецензента к примеру 2). Таким образом, в определенном смысле группа $\mathbb{T}$ в этом круге вопросов оказалась уникальной.
Отметим, что в работах [@MurIrish91] — [@Mur93] изучались индексы и обобщённые индексы тёплицевых операторов в пространствах $H^2(G)$ (случай архимедова порядка рассматривался ранее в [@CDSS]). В частности, в [@Mur93] получено обобщение теоремы об индексе при условии, что группа $X$ линейно упорядочена и содержит наименьший положительный элемент. При этом использовались методы теории $C^\ast$-алгебр, не переносящиеся на случай банаховых пространств $H^p(G)$.
**§2. Тёплицевы операторы над упорядоченными группами**
Ниже через $Pol(G)$ ($Pol_T(G)$) будет обозначаться пространство тригонометрических полиномов (соответственно, тригонометрических полиномов аналитического типа) на группе $G$, т. е. линейная оболочка множества $X$ (соответственно, $T$) в $L^p(G)$. При $1\leq p\leq\infty$ через $H^p(G)$ обозначим подпространство таких функций $f \in L^p(G)$, преобразование Фурье $\widehat f$ которых сосредоточено на $T$, с нормой, индуцированной из $L^p(G)$ [@HL]. Легко видеть, что $Pol_T(G)$ содержится в $H^p(G)$ (кроме того, $A_T(G)\subset H^\infty(G)$, где алгебра $A_T(G)$ есть равномерное замыкание множества $Pol_T(G)$, см. лемму 2).
**Лемма 1.** *При $1\leq p<\infty$ пространство $H^p(G)$ инвариантно относительно преобразования $g\mapsto g^\ast$, где $g^\ast(y)=\overline {g(y^{-1})}$, и совпадает с замыканием в $L^p(G)$ множества $Pol_T(G)$.* Доказательство. Инвариантность $H^p(G)$ относительно преобразования $g\mapsto g^\ast$ проверяется непосредственно с учётом инвариантности меры Хаара группы $G$ относительно отражения. Обозначим замыкание в $L^p(G)$ множества $Pol_T(G)$ через $H^p_T(G)$. Включение $H^p_T(G)\subseteq H^p(G)$ следует из того, что $H^p(G)$ замкнуто в $L^p(G)$ и содержит $Pol_T(G)$.
Пусть $F$ есть такой линейный непрерывный функционал на $L^p(G)$, что сужение $F|H^p(G)=0$. Тогда найдётся функция $f\in L^q(G)$ (здесь и ниже $p^{-1}+q^{-1}=1$), для которой $$F(h)=\int\limits_G h(x)\overline{f(x)}dm(x)=0$$ при всех $h\in H^p(G)$. В частности, преобразование Фурье $\widehat f(\chi)=0$ при всех $\chi\in T$. Если теперь $g\in H^p(G)$, то $\widehat f\widehat g=0$, а тогда и свёртка $f\ast g=0$, т. е.
$$\int\limits_G f(xy^{-1})g(y)dm(y)=0.$$ при почти всех $x\in G$. Отсюда следует, что $F(g^\ast)=0$. Следовательно, по теореме Хана-Банаха $g^\ast\in H^p_T(G)$, а потому и $g\in H^p_T(G)$, поскольку последнее пространство также инвариантно относительно преобразования $g\mapsto g^\ast$.
**Лемма 2.** [1)]{} *Для любого $p\in [1,\infty]$ имеет место включение $H^\infty(G)\cdot H^p(G)\subseteq H^p(G)$. Более того, для каждого фиксированного $p\in [1,\infty)$* $$H^\infty(G)=\{h\in L^\infty(G)| hH^p(G)\subseteq H^p(G)\}.\eqno(2.1)$$
[2)]{} *Пространство $H^\infty(G)$ есть банахова алгебра относительно поточечных операций и нормы, индуцированной из $L^\infty(G)$*. Доказательство. 1) Пусть $h\in H^\infty(G),\ f\in H^p(G), p\in [1,\infty)$. Тогда $hf\in L^p(G)$, и при $\chi\in X\setminus T,\ \xi\in T$ получаем $\widehat{h\xi}(\chi)=\widehat{h}(\xi^{-1}\chi)=0$, так как $\xi^{-1}\chi\in X\setminus T$. Следовательно, $\widehat{hg}(\chi)=0$ для любого аналитического полинома $g\in Pol_T(G)$. Выберем теперь последовательность $g_n\in Pol_T(G)$, сходящуюся к $f$ в $L^p(G)$ (лемма 1). Если $\chi\in X\setminus T$, то по доказанному выше с учётом неравенства Гёльдера ($h\overline{\chi}\in L^q(G)$) $$0=\widehat{hg_n}(\chi)=\int\limits_G h\overline{\chi}g_ndm\to\widehat{hf}(\chi)\ (n\to\infty),$$ а потому $hf\in H^p(G)$.
Если же $h\in L^\infty(G)$ и $hH^p(G)\subseteq H^p(G)$, то $h\cdot 1\in H^p(G)$, что доказывает (2.1). В свою очередь, (2.1) влечёт включение $H^\infty(G)\cdot H^\infty(G)\subseteq H^\infty(G)$.
2\) Это следует из 1) и замкнутости $H^\infty(G)$ в $L^\infty(G)$.
**Определение 1.** [*Проектор Рисса*]{} $P_T: Pol(G)\to Pol_T(G)$ определяется равенством $$P_T(\sum\limits_{\chi\in M}c_\chi \chi)=\sum\limits_{\chi\in M\cap
T}c_\chi \chi.$$
Известно [@B], [@Hel] (см. также [@Rud Глава 8]), что $P_T$ $L^p$-ограничен при $1<p<\infty$, а потому продолжается до ограниченного проектора $P_T:L^p(G)\to H^p(G)$ (теорема Бохнера-Хелсона).
**Определение 2.** Пусть $1< p<\infty$. Тёплицев оператор $T_\varphi$ в $H^p(G)$ с символом $\varphi\in L^\infty(G)$ определяется следующим образом: $$T_\varphi f=P_T(\varphi f),\ f\in H^p(G).$$
Далее через $M_\varphi$ будет обозначаться оператор умножения на измеримую функцию $\varphi$, действующий в пространстве $L^p(G)$ (или его подпространствах).
Нижеследующее обобщение теоремы Брауна-Халмоша (относительно классического случая см., например, [@BS]) справедливо для произвольной компактной абелевой группы $G$ с линейно упорядоченной группой характеров. Для его доказательства нам понадобится
**Лемма 3.** *Если для измеримой функции $\varphi$ на $G$ оператор умножения $M_\varphi$ отображает $Pol(G)$ в $L^p(G)$ и $L^p$-ограничен ($1<p<\infty$), то $\varphi\in L^\infty (G)$, и $\|M_\varphi\|=\|\varphi\|_\infty$.* Доказательство. Интерес представляет случай $M_\varphi\ne O$. Допустим, что $\varphi\notin L^\infty (G)$. Тогда найдётся такое компактное $K\subset G$, что $m(K)>0$ и $|\varphi(x)|>2\|M_\varphi\|$ при $x\in K$. Для любого $\varepsilon>0$ выберем открытое $U\supset K$, удовлетворяющее условию $m(U)<m(K)+\varepsilon$, и пусть непрерывная функция $f:G\to [0;1]$ такова, что $f|K=1,\ f|(G\setminus U)=0$ (лемма Урысона). По теореме Вейерштрасса-Стоуна существует полином $q\in Pol(G)$, такой, что $\|f-q\|_\infty<\varepsilon$. Отсюда следует, что $|q(x)|\geq 1-\varepsilon$ при $x\in K$, а потому $$\|M_\varphi q\|_p\geq \left(\int\limits_K|\varphi|^p|q|^p dm\right)^{1/p}\geq
(1-\varepsilon)\left(\int\limits_K|\varphi|^p dm\right)^{1/p}>$$ $$>(1-\varepsilon)2\|M_\varphi\|(m(K))^{1/p}.$$
С другой стороны, $\|f-q\|_p<\varepsilon$, и значит $\|q\|_p<\|f\|_p+\varepsilon$. А поскольку к тому же $\|f\|_p<m(U)^{1/p}$, то $$\|M_\varphi q\|_p\leq \|M_\varphi\|(\|f\|_p+\varepsilon)\leq \|M_\varphi\|(m(U)^{1/p}+\varepsilon)<$$ $$<\|M_\varphi\|((m(K)+\varepsilon)^{1/p}+\varepsilon).$$
Следовательно, для любого $\varepsilon>0$ справедливо неравенство $$(1-\varepsilon)2\|M_\varphi\|m(K)^{1/p}<\|M_\varphi\|((m(K)+\varepsilon)^{1/p}+\varepsilon),$$ что невозможно. Последнее утверждение леммы хорошо известно.
Далее мы полагаем для $f\in L^p(G), g\in L^q(G)
(p\in(1,\infty), p^{-1}+q^{-1}=1)$ $$\langle f,g \rangle:= \int\limits_G f\overline g dm.$$
**Теорема 1.** *Пусть $X$ л. у., $1< p<\infty$. Если ограниченный оператор $A:Pol_T(G)\to
H^p(G)$ таков, что для некоторой функции $a:X\to \mathbb{C}$ при любых $\chi_1, \chi_2\in T$ справедливо равенство $\langle A\chi_1, \chi_2\rangle=a(\chi_1{ \chi_2}^{-1})$, то существует такая функция $\varphi\in L^\infty(G)$, что $A=T_\varphi$, причём её преобразование Фурье $\widehat\varphi=a$. Более того, $ \|\varphi\|_\infty\leq \|T_\varphi\|\leq
c_p\|\varphi\|_\infty$, где $c_p=\|P_T\|$. В частности, $\|T_\varphi\|
=\|\varphi\|_\infty$, если $p=2$.* Доказательство. Рассмотрим направленность $b_\chi=\overline{\chi}A\chi\in L^p(G)\ (\chi\in T)$. Так как она ограничена ($\|b_\chi\|_p\leq \|A\|$ для всех $\chi\in T$), то, переходя, если нужно, к поднаправленности, можно считать, что $b_\chi$ слабо сходится в $L^p(G)$ к элементу $\varphi\in L^p(G)$. В частности, $\lim_{\chi\in T}\langle b_\chi,\xi\rangle=\langle \varphi,\xi\rangle$ при всех $\xi\in X$. Заметим, что $\langle b_\chi,\xi\rangle=\langle A\chi,\chi\xi\rangle=a(\xi^{-1})$, если $\chi\xi\in T$. Поскольку $T$ конфинально в $X$, в пределе получаем, что $\langle \varphi,\xi\rangle=a(\xi^{-1})$ при всех $\xi\in X$, т. е. $\widehat\varphi=a$.
Оператор $M_\varphi$, очевидно, отображает $Pol(G)$ в $L^p(G)$; покажем, что он $L^p$-ограничен. В силу последнего равенства имеем при всех $f,g\in Pol(G)$ $$\langle M_\varphi f,g\rangle=\langle M_{\bar{\chi}}AM_\chi f,g \rangle,$$ так как обе части совпадают при $f,g\in X$. Поэтому с учётом неравенства Гёльдера $$|\langle M_\varphi f,g\rangle|\leq\| M_{\bar{\chi}}AM_\chi f\|_p\|g\|_q\leq \|A\|\|f\|_p\|g\|_q,$$ и следовательно $$\|M_\varphi f\|_p=\sup\{|\langle M_\varphi f,g\rangle|| g\in Pol(G), \|g\|_q\leq 1\}\leq \|A\|\|f\|_p.$$ Таким образом, оператор $M_\varphi$ $L^p$-ограничен, а потому $\varphi\in L^\infty (G)$ по лемме 3. Одновременно мы показали, что $\|M_\varphi\|\leq \|A\|$.
Далее, так как при $\chi_1,\chi_2\in T$ $$\langle T_\varphi \chi_1,\chi_2\rangle=\widehat\varphi(\chi_1\chi_2^{-1})=a(\chi_1\chi_2^{-1})
=\langle A\chi_1,\chi_2\rangle,$$ то $A=T_\varphi$.
Наконец, $$\|\varphi\|_\infty=\|M_\varphi\|\leq\|A\|=\|T_\varphi\|=\|P_T M_\varphi\|\leq c_p\|\varphi\|_\infty,$$ и доказательство теоремы полностью завершено.
**§3. Фредгольмовость и индекс тёплицевых операторов**
**с непрерывным символом**
Для формулировки и доказательства теоремы об индексе требуется определенная подготовка.
Прежде всего, мы определим индекс вращения для функций из некоторой подгруппы группы $C(G)^{-1}$ обратимых элементов алгебры $C(G)$. Начнём с определения индекса вращения характера группы $G$ (ниже $\#F$ обозначает число элементов конечного множества $F$).
**Определение 3.** В каждом из следующих случаев определим индекс вращения характера $\chi\in X$ следующим образом:
[1)]{} ${\rm ind}\chi = \#(T\setminus \chi T)$, если $\chi\in T$ и множество $T\setminus \chi T$ конечно;
[2)]{} ${\rm ind}\chi ={\rm ind}\chi_1-{\rm ind}\chi_2$, если $\chi=\chi_1\chi_2^{-1}$, где $\chi_j\in T$, причём оба множества $T\setminus \chi T_j$ конечны $(j=1,2)$.
В остальных случаях считаем, что характер не имеет индекса.
Множество характеров, имеющих индекс, обозначим $X^i$.
Далее нам понадобится результат Г. Бора и E. ван Кампена, согласно которому любая функция $\varphi\in C(G)^{-1}$ представима в виде $\chi e^g$, где $g\in C(G),\ \chi\in X$ [@vK]. Более того, как следует из одного результата Е. А. Горина (см. замечание после доказательства теоремы 2 в [@Gor]), характер $\chi$ в этом разложении определяется по $\varphi$ однозначно (позднее этот факт был переоткрыт в [@MurIrish91]).
**Определение 4.** Рассмотрим функцию $\varphi\in C(G)^{-1}$ с разложением Бора-ван Кампена $\varphi=\chi e^g\ (g\in C(G),\ \chi\in X)$. Если $\chi\in
X^i$, то положим $${\rm ind}\varphi={\rm ind}\chi.$$ В противном случае будем считать, что функция $\varphi$ не имеет индекса. Множество функций из $C(G)^{-1}$, имеющих индекс, обозначим $\Phi(G)$. Таким образом, $\Phi(G)=X^i\cdot \exp(C(G))$, причём по причине отмеченной выше единственности характера в разложении Бора-ван Кампена, $X\cap \exp(C(G))=\{{\bf 1}\}$.
В следующей теореме перечислены основные свойства множеств $X^i$, $\Phi(G)$ и отображения ${\rm ind}\ (H^\bot$ обозначает аннулятор подмножества $H$ группы $G$, а знак $\sqcup$ — дизъюнктное объединение множеств; напомним, что подгруппа $\Xi$ упорядоченной группы $X$ называется *выпуклой*, если отрезок $[\xi_1,\xi_2]$ содержится в $\Xi$, лишь только $\xi_1,\xi_2\in\Xi$).
**Теорема 2.** [1)]{} *Множество $X^i$ есть выпуклая подгруппа группы* $X$;
[2)]{} *отображение ${\rm ind}:X^i\to \Bbb{Z}$ определено корректно и есть сохраняющий порядок гомоморфизм на группу вида $a\Bbb{Z}
\ (a\in \Bbb{Z}_+)$, который, если $X$ л. у., а $X^i$ нетривиальна, является порядковым изоморфизмом на* $\Bbb{Z}$;
[3)]{} *множество $\Phi(G)$ есть открытая подгруппа группы $C(G)^{-1}$ с открытыми компонентами связности $\chi\exp(C(G))$, где $\chi$ пробегает $X^i$;* [4)]{} *отображение ${\rm ind}:\Phi(G)\to \Bbb{Z}$ есть гомоморфизм групп и принимает постоянное значение ${\rm ind}\chi$ на компоненте связности $\chi\exp(C(G))$ ($\chi\in X^i$);* [5)]{} *если $X$ л. у., то для любого непрерывного гомоморфизма $\gamma:\mathbb{T}\to G$, удовлетворяющего условию $X^i\nsubseteq\gamma(\mathbb{T})^\bot$, найдётся такое целое $k_\gamma\ne 0$, что при всех $\varphi\in \Phi(G)$ справедливо равенство* $${\rm ind}\varphi=k_\gamma^{-1}{\rm wn}(\varphi\circ\gamma),$$ *где ${\rm wn}$ обозначает классический индекс вращения функций из $C(\mathbb{T})^{-1}$.* Доказательство. 1), 2) Сначала заметим, что $$T\cap X^i=\{\chi\in T| \#(T\setminus \chi T)<\infty\}.$$ Действительно, если $\chi\in T\cap X^i$, то $\chi=\chi_1^{-1}\chi_2$, причём $\#(T\setminus \chi_j T)<\infty\ (j=1,2)$. Тогда $\chi_2=\chi_1\chi$ (т. е. ${\bf 1}\leq\chi\leq\chi_2$), а потому $T\setminus \chi T\subseteq T\setminus \chi_2
T$. Отсюда следует, что $\#(T\setminus \chi T)<\infty$, и ${\rm
ind}\chi\leq {\rm ind}\chi_2$ (заодно мы показали, что $\chi_2\in T\cap X^i$ влечёт $[{\bf 1},\chi_2]\subset X^i$, и что отображение ${\rm ind}:T\to \Bbb{Z}_+$ сохраняет порядок). Значит, $T\cap X^i\subseteq\{\chi\in T| \#(T\setminus \chi T)<\infty\}.$ Обратное включение очевидно.
Установим корректность определения ${\rm ind}\chi$. При $\chi_1,
\chi_2 \in T$ справедливо равенство $$T\setminus \chi_1\chi_2 T=(T\setminus \chi_1
T)\sqcup\chi_1(T\setminus \chi_2 T).$$ Поэтому, если $\chi_1, \chi_2$ имеют индексы, то $\chi_1 \chi_2$ тоже его имеет (т. е. $T\cap X^i$ есть полугруппа) и ${\rm
ind}(\chi_ 1\chi_2)={\rm ind}\chi_1+{\rm ind}\chi_2$. Из последнего равенства следуют корректность определения ${\rm
ind}\chi$ (включая совпадение двух определений ${\rm
ind}\chi$ в случае $\chi\in T$) и гомоморфность индекса.
Далее, $X^i=(T\cap X^i)^{-1}(T\cap X^i)$ есть подгруппа группы $X$, порождённая полугруппой $T\cap X^i$. Докажем её выпуклость. Пусть $\xi_1\leq\xi\leq\xi_2$, где $\xi_1=\chi_1^{-1}\chi_2,\
\xi_2=\chi_3^{-1}\chi_4,\ (\chi_j\in T\cap X^i, j=1,\ldots,4)$. Тогда $\chi_2\chi_3\leq\chi_1\chi_3\xi\leq\chi_1\chi_4$, причём из первого неравенства следует, что $\chi_5:=\chi_1\chi_3\xi\in T$, а из второго, – что $\chi_5\in X^i$ (см. начало доказательства теоремы). Поэтому $\xi=(\chi_2\chi_3)^{-1}\chi_5\in X^i$.
Покажем, что отображение ${\rm ind}$ сохраняет порядок. Пусть, как выше, $\xi_1\leq\xi\leq\xi_2$, где $\xi_1=\chi_1^{-1}\chi_2,\
\xi_2=\chi_3^{-1}\chi_4,\ (\chi_j\in T\cap X^i, j=1,\ldots,4)$. Тогда $\chi:=\xi_2\xi_1^{-1}\in T\cap X^i$. Поэтому $\chi_2\chi_3\chi=\chi_1\chi_4$, откуда следует, что ${\rm ind}\chi_2+{\rm ind}\chi_3\leq {\rm ind}\chi_1+{\rm ind}\chi_4$, т. е. ${\rm ind}\xi_1\leq {\rm ind}\xi_2$. Кроме того, ${\rm ind}(X^i)$, будучи подгруппой группы $\Bbb{Z}$, имеет вид $a\Bbb{Z}\ (a\in\Bbb{Z}_+)$.
Предположим теперь, что $X$ л. у., а $X^i$ нетривиальна, и докажем инъективность гомоморфизма ${\rm ind}:X^i\to \Bbb{Z}$, т. е. равенство ${\rm
Ker}({\rm ind})=\{{\bf 1}\}$. Пусть $\xi=\chi_1^{-1}\chi_2,\ \chi_j\in T\cap X^i, j=1,2$ и ${\rm ind}\xi=0$, т. е. ${\rm ind}\chi_1={\rm ind}\chi_2$. Если, для определённости, $\chi_1\leq\chi_2$, то $T\setminus \chi_1 T
\subseteq T\setminus \chi_2 T$, а потому здесь имеет место равенство (это конечные множества с одинаковым числом элементов), откуда $\chi_1 T=\chi_2 T$. Последнее равенство влечёт $\chi_1\leq\chi_2$ и $\chi_2\leq\chi_1$, т. е. $\chi_1=\chi_2$. Таким образом, группа $X^i$ порядково изоморфна $a\Bbb{Z}$, а потому ${\rm ind}(X^i)={\rm ind}(a\Bbb{Z})=\Bbb{Z}$, поскольку $a\not=0$.
3\) Равенство $\Phi(G)=X^i\cdot \exp(C(G))$ показывает, что $\Phi(G)$ есть подгруппа группы $C(G)^{-1}$ (причём из равенства $X^i\cap \exp(C(G))=\{{\bf 1}\}$ следует, что она изоморфна прямому произведению $X^i\times \exp(C(G))$). Далее, хорошо известно, что множество $\exp(C(G))$ открыто и является компонентой единицы в $C(G)^{-1}$. Следовательно, подмножества $\chi\exp(C(G))\subset C(G)^{-1}$ открыты и связны, и осталось заметить, что при различных $\chi$ они попарно не пересекаются по причине единственности характера в разложении Бора-ван Кампена.
4\) Это следует из 2) и того факта, что ${\rm ind}(\exp(C(G)))=\{ 1\}$.
5\) Так как ${\rm ind}(e^g)={\rm wn}(e^{g\circ\gamma})=0$ при $g\in C(G)$, интерес представляет случай $\varphi\in X^i$. С учётом утверждения 2) можно, исключая тривиальный случай, считать также, что группа $X^i$ изоморфна $\mathbb{Z}$. Поскольку у таких групп любые два гомоморфизма в $\mathbb{Z}$ пропорциональны, найдётся число $c_\gamma$, для которого ${\rm ind}\chi=c_\gamma{\rm wn}(\chi\circ\gamma)$ при всех $\chi\in X^i$. Если в последнем равенстве положить $\chi=\chi_1$, где $\chi_1\in X^i,\ {\rm ind}\chi_1=1$, то получим $c_\gamma=k_\gamma^{-1}$, где $k_\gamma={\rm wn}(\chi_1\circ\gamma)$, что и требовалось доказать.
ЗАМЕЧАНИЕ 1. Утверждение [2)]{} этой теоремы показывает, что в случае линейного порядка группа $X^i$ либо порядково изоморфна $\Bbb{Z}$, либо тривиальна.
ЗАМЕЧАНИЕ 2. Из утверждений [3)]{} и [4)]{} следует, что определённый выше индекс вращения функций из $\Phi(G)$ обладает свойствами классического индекса вращения и, в частности, является гомотопическим инвариантом.
ЗАМЕЧАНИЕ 3. Характер $\chi\in T$ называется положительным конечным элементом группы $X$, если для любого $\xi\in T\setminus\{\bf{1}\}$ найдётся такое натуральное $n$, что $\xi^n\geq\chi$. Группа $F(X)$, порождённая всеми положительными конечными элементами, называется группой конечных элементов группы $X$ (эта группа играет важную роль в [@Mur93]). Из теоремы 2 легко следует, что $X^i=F(X)$, если $X^i$ нетривиальна. В то же время рассмотрение подгрупп $X\subseteq \mathbb{R}$ (наделённых естественным порядком и дискретной топологией) показывает, что $F(X)$ может быть нетривиальной при тривиальной $X^i$.
**Следствие 1.** *Если группа $X$ л. у. и совпадает с $X^i$, то $G$ изоморфна одномерному тору $\mathbb{T}$*. В самом деле, так как группа $X$ предполагается нетривиальной, это сразу следует из замечания 1.
Для случая одномерного тора известно, что все [*полукоммутаторы*]{} $[T_\varphi, T_\psi):=T_\varphi T_\psi-T_{\varphi \psi}\ (\varphi,
\psi\in C(\mathbb{T}))$ компактны в $H^p(\mathbb{T})$. Для групп, отличных от $\mathbb{T}$, это уже не так.
**Следствие 2.** *Пусть группа $X$ л. у., $\chi\in T$. Полукоммутатор $[T_\chi,
T_{\bar{\chi}})$ компактен в $H^p(G)$ тогда и только тогда, когда $\chi\in X^i$. Следовательно, если операторы $[T_\chi,
T_{\bar{\chi}})$ компактны в $H^p(G)$ при всех $\chi\in T$, то $G$ изоморфна одномерному тору.* Доказательство. Пусть $I$ — единичный оператор в $H^p(G)$. Поскольку $(I-T_\chi T_{\bar{\chi}})\zeta=0$ при $\zeta\in \chi T$ и $(I-T_\chi T_{\bar{\chi}})\zeta=\zeta$ при $\zeta\in T\setminus\chi T$, а $T$ — линейно независимая система, порождающая пространство $H^p(G)$ (лемма 1), оператор $-[T_\chi,
T_{\bar{\chi}})=I-T_\chi T_{\bar{\chi}}$ является ограниченным проектором на подпространство $L \subset H^p(G)$, порождённое системой $T\setminus\chi T$. Компактность этого проектора равносильна конечномерности $L$, т. е. конечности множества $T\setminus\chi T$, что доказывает первое утверждение. Если теперь операторы $[T_\chi,
T_{\bar{\chi}})$ компактны в $H^p(G)$ при всех $\chi\in T$, то $T\subset X^i$, а потому $X=X^i$ и осталось воспользоваться следствием 1.
**Следствие 3.** *Если группа $X\not=\mathbb{Z}$ линейно и архимедово упорядочена, то $X^i$ тривиальна.* Действительно, известно, что линейно и архимедово упорядоченная группа не содержит собственных нетривиальных выпуклых подгрупп, а равенство $X=X^i$ влечёт $X=\mathbb{Z}$ в силу утверждения 2) теоремы 2.
**Предложение 1.** *Пусть $\varphi, \psi\in L^\infty(G)$. Если $\bar{\varphi}\in
H^\infty(G)$ или $\psi\in H^\infty(G)$, то* $T_{\varphi\psi}=T_{\varphi}T_\psi$. Доказательство. 1) Если $\psi\in H^\infty(G)$, то для любого $f\in H^p(G)$ $$T_{\varphi}T_\psi f=T_{\varphi}(\psi f)=P_T(\varphi\psi
f)=T_{\varphi\psi}f.$$
2\) Пусть теперь $\bar{\varphi}\in H^\infty(G)$. В силу доказанного выше при любых $\chi_1, \chi_2\in T$ справедливо равенство $$\langle T_{\bar{\varphi}\psi}\chi_1, \chi_2\rangle=
\langle T_\psi T_{\bar{\varphi}}\chi_1, \chi_2\rangle.
\eqno(3.1)$$ Как и в классическом случае $G=\mathbb{T}$, легко проверить, что при $f, g \in H^2(G),\ \varphi\in L^\infty (G)$ $$\langle T_{\varphi}f,g\rangle = \langle f,T_{\bar{\varphi}}g\rangle.\eqno(3.2)$$ С учётом последнего равенства левая часть (3.1) приобретает вид $\langle \chi_1, T_{\varphi\bar{\psi}}\chi_2\rangle$.
С другой стороны, дважды применяя равенство (3.2), для правой части (3.1) получаем выражение $\langle \chi_1,
T_{\varphi}T_{\bar{\psi}}\chi_2\rangle$. Следовательно, полагая $h:=T_{\varphi\bar{\psi}}\chi_2-T_{\varphi}T_{\bar{\psi}}\chi_2$, выводим, что $0=\langle \chi_1,h \rangle=\overline{\widehat h(\chi_1)}$, т. е. ${\widehat h}|T=0$. Но $h\in H^p(G)$, а потому ${\widehat h}|(X\setminus T)=0$. Таким образом, $T_{\varphi\bar{\psi}}\chi_2=T_{\varphi}T_{\bar{\psi}}\chi_2$ при всех $\chi_2\in T$, т. е. $T_{\varphi\bar{\psi}}|Pol_T(G)=T_{\varphi}T_{\bar{\psi}}|Pol_T(G)$, откуда и следует утверждение предложения.
**Предложение 2.** *Пусть группа $X$ л. у. Для любой функции $\varphi\in C(G)$ оператор $T_{e^\varphi}$ обратим в $H^p(G)\ (1<p<\infty)$* . Доказательство. Воспользуемся методом доказательства теоремы 7.1 из [@MurIEOT92]. По теореме Вейерштрасса-Стоуна найдётся такой тригонометрический полином $q\in Pol(G)$, что $\|{\bf 1}-e^{\varphi-q}\|_\infty<1/c_p\ (c_p=\|P_T\|_p)$. Пусть $q_1=P_Tq,\ q_2=q-q_1$. Тогда $q_1, \overline{q_2}\in Pol_T(G)$ (у нас $X=T\cup T^{-1}$). Лемма 2 показывает теперь, что $e^{\pm
q_1}, e^{\pm\overline{q_2}}\in H^\infty(G)$, а потому в силу предложения 1 существуют обратные операторы $(T_{e^{q_i}})^{-1}=T_{e^{-q_i}}\ (i=1,2)$. Повторное применение предложения 1 даёт $T_{e^\varphi}=T_{e^{q_2}}T_{e^{\varphi
-q}}T_{e^{q_1}}$. Осталось заметить, что оператор $T_{e^{\varphi
-q}}$ обратим, поскольку $$\|I-T_{e^{\varphi -q}}\|=\|T_{{\bf 1}-e^{\varphi -q}}\|\leq
c_p\|{\bf 1}-e^{\varphi -q}\|_\infty<1.$$
Для обобщения теоремы Хартмана-Винтнера (относительно классического случая см., например, [@BS]) нам потребуется следующий простой факт. Ниже через $w$-$\lim_{\chi\in T}h_\chi$ ($s$-$\lim_{\chi\in T}h_\chi$) обозначается слабый (соответственно, сильный) предел направленности $h:T\to
LB(L^p(G))$ ($LB(L^p(G))$ обозначает алгебру линейных ограниченных операторов в $L^p(G)$).
**Лемма 4.** $w$-$\lim\limits_{\chi\in T}M_\chi=O$. Доказательство. Пусть $f\in L^p(G),\ g\in L^q(G)$ ($p$ и $q$ – сопряжённые показатели, $1< p<\infty$). Тогда функция на $X$ $$\chi\mapsto \langle M_\chi f,g \rangle= \overline{\int\limits_G
\overline\chi \overline f g dm}$$ стремится к нулю на бесконечности вместе с преобразованием Фурье функции $\overline fg\in L^1(G)$. Следовательно, для любого $\varepsilon >0$ найдётся такое конечное множество $F=\{\xi_1,\ldots,\xi_n\}\subset X$, что $|\langle M_\chi f,g
\rangle|<\varepsilon$ при $\chi\notin F$. Выберем характеры $\chi_i,\eta_i\in T$ таким образом, что $\xi_i^{-1}=\chi_i^{-1}\eta_i\ (i=1,\ldots,n)$, и положим $\chi_\varepsilon=\chi_1 \ldots \chi_n$. Тогда $\chi_\varepsilon\xi_i^{-1}\in T$, т. е. $\chi_\varepsilon\geq \xi_i$ при всех $\xi_i\in F$. Поэтому из неравенства $\chi>\chi_\varepsilon$ следует, что $\chi\notin F$, а потому и $|\langle M_\chi f,g
\rangle|<\varepsilon$, что и завершает доказательство леммы.
Теперь обобщение теоремы Хартмана-Винтнера доказывается по существу так же, как и в классическом случае. Приведём доказательство для полноты изложения (напомним, что для ограниченного оператора $T$ в банаховом пространстве $Y$ запись $T\in\Phi_+(Y)$ означает, что его образ ${\rm Im}T$ замкнут, а ядро ${\rm Ker}T$ конечномерно, а запись $T\in\Phi_-(Y)$ означает конечномерность факторпространства $Y/{\rm Im T}$; операторы из $\Phi_-(Y)\cup\Phi_+(Y)$ называются полуфредгольмовыми, а операторы из $\Phi_-(Y)\cap\Phi_+(Y)$ — фредгольмовыми в $Y$).
**Теорема 3.** *Если оператор $T_\varphi$ полуфредгольмов в $H^p(G) (1<p<\infty)$, то его символ $\varphi$ обратим в алгебре $L^\infty (G)$.* Доказательство. Если $T_\varphi\in\Phi_+(H^p(G))$, то обозначим через $K$ проектор $H^p(G)\to {\rm Ker}T_\varphi$. Тогда при некотором $\delta>0$ и всех $f\in H^p(G)$ справедливо неравенство (см., например, [@BS утверждение 1.12 (g)]) $$\|T_\varphi f\|+\|Kf\|\geq\delta\|f\|.$$ Полагая здесь $f=P_Tg$, имеем $$\|P_T M_\varphi P_Tg\|+\|KP_Tg\|\geq\delta\|P_Tg\|.$$ Далее, $g=P_Tg+Qg$, где $Q=I-P_T$. Поэтому $\|P_Tg\|\geq\|g\|-\|Qg\|$. С учётом этого, предыдущее неравенство влечёт $$\|P_T M_\varphi P_Tg\|+\|P_TKP_Tg\|\geq\delta (\|g\|-\|Qg\|),$$ то есть $$\|P_T M_\varphi P_Tg\|+\|P_TKP_Tg\|+\delta\|Qg\|\geq\delta \|g\|.$$ Заменяя тут $g$ на $M_\chi g$, получаем в силу изометричности $M_\chi$, что $$\|M_{\chi^{-1}}P_T M_\varphi P_TM_\chi g\|+\|P_TKP_TM_\chi g\|+\delta\|M_{\chi^{-1}}QM_\chi g\|
\geq\delta \|g\|.\eqno(3.3)$$
Заметим теперь, что семейство $E_\chi:=M_{\chi^{-1}}P_TM_\chi\ (\chi\in T)$ равномерно ограничено. Кроме того, $E_\chi\xi=\xi (\xi\in X)$ при $\chi\xi\in T$ (т. е. при $\chi\geq \xi^{-1}$), а потому $\lim_{\chi\in T}E_\chi q=q$ при $q\in Pol(G)$. Следовательно, $s$-$\lim_{\chi\in T}E_\chi=I$ и $s$-$\lim_{\chi\in T}M_{\chi^{-1}}QM_\chi =O$.
В свою очередь отсюда следует, что $$s\mbox{-}\lim\limits_{\chi\in T}M_{\chi^{-1}}P_T M_\varphi P_TM_\chi=
s\mbox{-}\lim\limits_{\chi\in T}E_\chi M_\varphi E_\chi=M_\varphi.$$ В самом деле, так как $E_\chi\xi=\xi$ при $\chi\geq \xi^{-1}$, то направленность $E_\chi M_\varphi E_\chi\xi \to M_\varphi \xi$ по $\chi\in T$. По линейности это же верно при замене характера $\xi$ любым тригонометрическим полиномом, и осталось заметить, что семейство $E_\chi M_\varphi E_\chi\ (\chi\in T)$ равномерно ограничено.
Поскольку оператор $P_TKP_T$ компактен, $s$-$\lim_{\chi\in T}P_TKP_TM_\chi =O$ в силу леммы 4. Теперь (3.3) влечёт $\|M_\varphi g\|\geq\delta \|g\|$, откуда $\|\varphi\|_\infty\geq\delta$.
Пусть теперь $T_\varphi\in\Phi_-(H^p(G))$. Положим ${\stackrel{\circ}{H^p_-}}(G)={\rm Ker} P_T$. Тогда $L^p(G)=H^p(G)\dot + {\stackrel{\circ}{H^p_-}}(G),\ L^q(G)=H^q(G)\dot +
{\stackrel{\circ}{H^q_-}}(G)$ ($p$ и $q$ – сопряжённые показатели), а потому $$T_\varphi\in\Phi_-(H^p(G))\Longleftrightarrow T_\varphi P_T+Q\in\Phi_-(L^p(G)),$$ $$T_{\bar{\varphi}}\in\Phi_+(H^q(G))\Longleftrightarrow T_{\bar{\varphi}} P_T+Q\in\Phi_+(L^q(G)).$$ Поскольку пространство $L^q(G)$ сопряжено $L^p(G)$, а оператор $T_{\bar{\varphi}} P_T+Q$ сопряжён оператору $T_\varphi P_T+Q$ (см. (3.2)), то $T_{\bar{\varphi}}\in\Phi_+(H^q(G))$, и всё свелось к случаю, рассмотренному выше.
Сейчас мы в состоянии доказать теорему об индексе.
**Теорема 4.** *Пусть группа $X$ л. у., $\varphi\in C(G)$. Оператор $T_\varphi$ в пространстве $H^p(G)\ (1<p<\infty)$ фредгольмов тогда и только тогда, когда $\varphi\in \Phi(G)$. При этом* $${\rm Ind} T_{\varphi}=-{\rm ind}\varphi.$$ Доказательство. Необходимость. Если оператор $T_\varphi$ фредгольмов, то $\varphi\in C(G)^{-1}$ по предыдущей теореме. Пусть $\varphi=\chi e^g\ (g\in C(G),\ \chi\in X)$ – разложение Бора-ван Кампена. По предложению 1 $T_\varphi=T_{e^g}T_\chi$, если $\chi\in T$, и $T_\varphi=T_\chi T_{e^g}$, если $\chi\in T^{-1}$. При этом оператор $T_{e^g}$ обратим (предложение 2), а потому фредгольмов нулевого индекса. Следовательно, оператор $T_\chi$ тоже фредгольмов, причём ${\rm Ind} T_{\varphi}={\rm Ind}
T_{\chi}$.
Рассмотрим вопрос о фредгольмовости и индексе оператора $T_\chi$. Возможны два случая.
1\) $\chi\in T$. Поскольку ${\rm Ker} T_\chi=\{0\}$, оператор $T_\chi$ фредгольмов тогда и только тогда, когда пространство ${\rm Coker}
T_\chi=H^p(G)/\chi H^p(G)$ конечномерно, и при этом ${\rm Ind}
T_{\chi}=-\dim (H^p(G)/\chi H^p(G))$. Заметим, что соотношения ортогональности для характеров влекут равенство $\langle\xi,\eta
\rangle=0$ при $\xi\in T\setminus \chi T, \eta\in\chi T$. По непрерывности $\langle\xi,f \rangle=0$ при $\xi\in T\setminus \chi
T, f\in\chi H^p(G)$. Поэтому характеры из $T\setminus \chi T$ попарно не эквивалентны ${\rm mod}(\chi H^p(G))$. Тогда из сказанного выше следует, что если $\chi\notin X^i$, т. е. множество $T\setminus \chi T$ бесконечно, то оператор $T_\chi$ не фредгольмов. Пусть теперь $\chi\in X^i$, т. е. $T\setminus \chi T$ конечно. Тогда подпространство ${\rm
span}(T\setminus \chi T)\dot +\chi H^p(G)$ замкнуто в $H^p(G)$ (см., например, [@Shef Глава I, п. 3.3]). Поскольку оно содержит $Pol_T(G)$, справедливо равенство ${\rm span}(T\setminus
\chi T)\dot +\chi H^p(G)=H^p(G)$ (лемма 1), откуда $H^p(G)/\chi H^p(G)={\rm
span}(T\setminus \chi T)$. Следовательно, в рассматриваемом случае $T_\chi$ фредгольмов и
$${\rm Ind} T_{\chi}=-\dim (H^p(G)/\chi H^p(G))=-{\rm ind} \chi=-{\rm
ind}\varphi.$$
2\) Пусть теперь $\chi\in T^{-1}$. Далее мы неоднократно будем пользоваться равенством $I=T_\chi T_{\bar{\chi}}$, вытекающим из предложения 1. Если оператор $T_\chi$ фредгольмов, то это равенство показывает, что оператор $T_{\bar{\chi}}$ тоже фредгольмов. Тогда $\chi^{-1}\in X^i$ в силу 1), а потому и $\chi\in X^i$. Более того, из этого же равенства следует, что $${\rm Ind} T_{\chi}=-{\rm Ind} T_{\chi^{-1}}=-{\rm ind} \chi=-{\rm
ind}\varphi.$$
Обратно, если $\chi\in X^i$, то и $\chi^{-1}\in X^i$, а потому оператор $T_{\bar{\chi}}$ фредгольмов. Третий раз воспользовавшись нашим равенством, получаем фредгольмовость оператора $T_\chi$ и в этом случае.
Итак, в любом случае оператор $T_\chi$ фредгольмов тогда и только тогда, когда $\chi\in X^i$. Из этого включения, в частности, следует, что $\varphi\in \Phi(G)$, и доказательство необходимости завершено. Попутно мы установили и формулу для ${\rm Ind} T_{\varphi}$.
Достаточность. Если $\varphi\in \Phi(G)$, т. е. $\varphi=\chi
e^g$, где $g\in C(G),\ \chi\in X^i$, то оператор $T_\chi$ фредгольмов по доказанному выше. Тогда оператор $T_{\varphi}$ тоже фредгольмов (он равен $T_{e^g}T_\chi$, если $\chi\in T$, и $T_\chi T_{e^g}$, если $\chi\in T^{-1}$), что и завершает доказательство.
ЗАМЕЧАНИЕ 4. В работе [@Mur93] рассматривалась обобщённая фредгольмовость и обобщённый индекс (Бройора) тёплицевых операторов в $H^2(G)$ с непрерывным символом для случая, когда $X$ счётна и группа её конечных элементов $F(X)$ нетривиальна. В частности, при этих условиях в [@Mur93 теорема 4.2] показано, что оператор $T_{\varphi}$ фредгольмов в этом обобщённом смысле тогда и только тогда, когда характер $\chi$ из разложения Бора-ван Кампена $\varphi=\chi e^g$ принадлежит группе $F(X)$. Сравнение этого результата с теоремой 4 и замечанием 3 показывает, что в случае, когда группа $X^i$ нетривиальна, обобщённая фредгольмовость оператора $T_{\varphi}$ равносильна обычной.
Всюду далее предполагается, что $X$ л. у.
**Следствие 4.** *Пусть $\varphi\in C(G)$. Следующие утверждения равносильны:*
[1)]{} $T_\varphi$ *фредгольмов нулевого индекса*;
[2)]{} $\varphi\in \exp (C(G))$;
[3)]{} $T_\varphi$ *обратим*.
**Следствие 5.** *Пусть $\varphi\in C(G)$, а группа $X\ne
\mathbb{Z}$ линейно и архимедово упорядочена. Если оператор $T_\varphi$ фредгольмов, то он обратим.* Доказательство. В силу следствия 3 в этой ситуации имеет место равенство $\Phi(G)=\exp(C(G))$.
Справедливость следствия 5 была отмечена в [@Mur91 c. 356] (там рассматривался случай $p=2$).
**Следствие 6.** *Если любой оператор $T_\chi, \ \chi\in T$ фредгольмов, то $G$ изоморфна $\mathbb{T}$*. Доказательство. Если $T_\chi$ фредгольмов, то $\chi\in
\Phi(G)\cap T=X^i\cap T$. Поэтому из условия следует, что $T\subset X^i$, откуда $X=X^i$ и осталось применить следствие 1.
Рассмотрим примеры вычисления индекса, первый из которых обобщает теоремы 3.1 и 3.2 из работы [@Mur91], где рассматривались пространства $H^2(\Bbb{T}^d)$ и символы специального вида.
[**Пример 1**]{}. Пусть $G=\Bbb{T}^d,\ d>1$, $X=\Bbb{Z}^d_{{\rm lex}}$ (нижний индекс указывает, что порядок лексикографический), и $\varphi\in C(\Bbb{T}^d)$. Тогда оператор $T_{\varphi}$ фредгольмов в $H^p(G)\ (1<p<\infty)$, если и только если его символ имеет вид
$$\varphi(t_1,\ldots,t_d)= t_d^{n_d}e^{g(t_1,\ldots,t_d)}$$ для некоторых $n_d\in\Bbb{Z},\ g\in C(\Bbb{T}^d)$.
При этом $${\rm Ind}T_{\varphi}=-n_d.$$
В самом деле, по определению лексикографического порядка положительный конус в этом случае суть $$T=\{n\in \mathbb{Z}^d| n_1>0\}\sqcup\{n\in \mathbb{Z}^d| n_1=0, n_2>0\}
\sqcup$$ $$\cdots\sqcup \{n\in \mathbb{Z}^d| n_1= n_2=\ldots=n_{d-1}=0,
n_d>0\}\sqcup\{0\}.$$ Отсюда следует, что $X^i= \{n\in \mathbb{Z}^d| n_1=
n_2=\ldots=n_{d-1}=0\}$ (точка $(0,\ldots,0,1)$ принадлежит $X^i$ и осталось воспользоваться замечанием 1 после теоремы 2). Это значит, если отождествить точку $n\in \Bbb{Z}^d$ с характером $\chi_n(t)=t^n$ группы $\Bbb{T}^d$, что $$\Phi(G)=\{t_d^{n_d}e^{g(t)}| g\in C(\Bbb{T}^d)\},$$ и утверждение следует из теоремы 4.
[**Пример 2**]{}. Пусть $G=\Bbb{T}^\infty,$ и $X=\Bbb{Z}^\infty_{{\rm lex}}$ — аддитивная группа всех финитных последовательностей целых чисел с положительным конусом $$T=\{0\}\sqcup\{n\in \mathbb{Z}^\infty| n_1>0\}\sqcup\{n\in \mathbb{Z}^\infty| n_1=0, n_2>0\}
\sqcup\cdots.$$
В этом примере группа $X^i$ тривиальна, а потому оператор $T_{\varphi}$ с непрерывным символом фредгольмов в $H^p(G)\ (1<p<\infty)$, если и только если он обратим, т. е. его символ имеет вид $\varphi=e^g$, где $g\in C(\Bbb{T}^\infty)$.
[**Пример 3**]{}. Пусть $G=\Bbb{T}^\infty,$ $X=\Bbb{Z}^\infty$, причём линейный порядок в $X$ задаётся положительным конусом $T$, состоящим из нуля и тех точек группы $\Bbb{Z}^\infty$, последняя ненулевая координата у которых положительна. Поскольку $(1,0,0,\ldots)\in X^i$, группа $X^i$ состоит из точек вида $(n_1,0,0,\ldots)$, где $n_1\in \mathbb{Z}$. Рассуждая как в примере 1, получаем, что оператор $T_{\varphi}$, где $\varphi\in C(G)$, фредгольмов в $H^p(G)\ (1<p<\infty)$ тогда и только тогда, когда его символ имеет вид
$$\varphi(t_1,t_2,\ldots)= t_1^{n_1}e^{g(t_1,t_2,\ldots)}$$ для некоторых $n_1\in\Bbb{Z},\ g\in C(\Bbb{T}^\infty)$. Кроме того,
$${\rm Ind}T_{\varphi}=-n_1.$$
**§4. Спектры и существенные спектры тёплицевых операторов**
Применим полученные результаты к исследованию структуры спектров тёплицевых операторов. Всюду ниже, если не оговорено противное, группа $X$ считается линейно упорядоченной, а символ $\varphi$ – непрерывным.
В случае символа $\varphi\in C(\mathbb{T})$ известно, что существенный спектр Фредгольма $\sigma_e(T_\varphi)=\varphi(\mathbb{T})$. Для других групп ситуация иная.
**Следствие 7.** *Если $\varphi\in L^\infty(G)$, то $\sigma_e(T_\varphi)\supseteq \sigma(\varphi)$, где $\sigma(\varphi)$ — множество существенных значений функции $\varphi$. Причём, если $G$ не изоморфна $\mathbb{T}$, для $\varphi\in
C(G)^{-1}\setminus\Phi(G)$ включение строгое.* Доказательство. Первое утверждение сразу следует из теоремы 3. Пусть теперь $\varphi\in C(G)^{-1}\setminus\Phi(G)$ (последнее множество не пусто по следствию 1, если $G$ не изоморфна $\mathbb{T}$). Тогда оператор $T_\varphi$ не фредгольмов (теорема 4), а потому $0
\in\sigma_e(T_\varphi)$.
Далее под [*дырами*]{} компактного связного множества $\varphi(G)$ подразумеваются ограниченные компоненты его дополнения $\mathbb{C}\setminus\varphi(G)$.
**Теорема 5.** *Пусть $\varphi\in C(G)$. Тогда* [1)]{} *существенный спектр Фредгольма $\sigma_e(T_\varphi)$ получается из множества $\varphi(G)$ присоединением тех его дыр $\Lambda$ (если они существуют), для которых $\varphi-\lambda\notin \Phi(G)$ при всех (при одном)* $\lambda\in
\Lambda$;
[2)]{} *спектр $\sigma_(T_\varphi)$ получается из $\sigma_e(T_\varphi)$ присоединением тех дыр $\Lambda$ множества $\varphi(G)$ (если они существуют), для которых $\varphi-\lambda\in \Phi(G)\setminus \exp(C(G))$ при всех (при одном)* $\lambda\in
\Lambda$;
[3)]{} *существенный спектр Вейля* $\sigma_{w}(T_\varphi)=\sigma(T_\varphi)$;
[4)]{} *спектры $\sigma_e(T_\varphi)$ и $\sigma(T_\varphi)$ связны.* Доказательство. 1) Теорема Бора-ван Кампена вкупе с единственностью разложения означает, что факторгруппа $C(G)^{-1}/\exp(C(G))$ изоморфна $X$, поэтому далее мы будем эти группы отождествлять (значительно более общий факт установлен в [@Gor]). Обозначим через $\pi$ каноническое отображение $$C(G)^{-1}\to C(G)^{-1}/\exp(C(G))=X$$ и рассмотрим отображение $$\alpha_\varphi: \mathbb{C}\setminus\varphi(G)\to
X, \lambda\mapsto\pi(\varphi-\lambda).$$ Оно непрерывно, а значит постоянно на каждой компоненте множества $\mathbb{C}\setminus\varphi(G)$ в силу дискретности $X$. В соответствии с теоремой 4 $$\sigma_e(T_\varphi)=\{\lambda\in\mathbb{C}| T_{\varphi-\lambda}\notin \Phi(H^p((G))\}=
\{\lambda\in\mathbb{C}| \varphi-\lambda\notin \Phi(G)\}=$$ $$=\varphi(G)\sqcup\{\lambda\in\mathbb{C}
\setminus\varphi(G)| \alpha_\varphi(\lambda)\notin X^i\}. \eqno(4.1)$$
Поскольку оператор $T_{\varphi-\lambda}$ обратим при больших $\lambda$, и $\alpha_\varphi$ постоянно на компонентах множества $\mathbb{C}\setminus\varphi(G)$, неограниченная компонента этого множества не пересекается с $\sigma_e(T_\varphi)$, а потому формула (4.1) равносильна первому утверждению теоремы.
2\) Воспользуемся равенством $$\sigma(T_\varphi)=\varphi(G)\sqcup\{\lambda\in\mathbb{C}\setminus\varphi(G)|
\alpha_\varphi(\lambda)\ne
{\bf 1}\},\eqno(4.2)$$ которое следует из того, что в силу следствия 4 $\lambda\notin
\sigma(T_\varphi)$ тогда и только тогда, когда $\varphi-\lambda\in
\exp(C(G))$, т. е. когда $\lambda\notin \varphi(G)$ и $\alpha_\varphi(\lambda)= {\bf 1}$. Осталось заметить, что с учётом формулы (4.1) правая часть (4.2) есть $$\sigma_e(T_\varphi)\sqcup\{\lambda\in\mathbb{C}\setminus\varphi(G)|
\alpha_\varphi(\lambda)\in X^i\setminus\{{\bf 1}\}\}.$$
3\) В соответствии с одним утверждением М. Шехтера [@Shech] (см. также [@Conw теорема XI.6.12]; [@Er Глава 3]) $\sigma_{w}(T_\varphi)$ получается из $\sigma_e(T_\varphi)$ присоединением тех точек $\lambda\in \mathbb{C}$, для которых оператор $T_{\varphi-\lambda}$ полуфредгольмов и ${\rm Ind}T_{\varphi-\lambda}\ne 0$. С учётом теоремы 4 и утверждения 2) это означает, что $\sigma(T_\varphi)\subseteq\sigma_{w}(T_\varphi)$, что и требовалось доказать.
4\) Это следует из того, что $\sigma(T_\varphi)$ и $\sigma_e(T_\varphi)$ представимы в виде объединения связного множества $\varphi(G)$ и (возможно) некоторых его дыр.
ЗАМЕЧАНИЕ 5. При $p=2$ связность $\sigma(T_\varphi)$ была доказана в работе [@MurIrish91], а связность $\sigma_e(T_\varphi)$ — в [@MurIEOT92] даже для случая пространств $H^2$, порождённых алгебрами Дирихле.
**Следствие 8.** *Спектральные радиусы* $|T_\varphi|_{ess}=|T_\varphi|=\|\varphi\|_\infty$. Доказательство. Поскольку $\sigma(T_\varphi)$ получается из множества $\varphi(G)$ присоединением некоторых его дыр, то $|T_\varphi|=\sup|\varphi(G)|$. Первое равенство доказывается аналогично.
**Следствие 9.** *Для любого компактного оператора $K$ в пространстве $H^p(G)\ (1<p<\infty)$ справедливо неравенство $\|T_\varphi+K\|\geq (1/c_p)\|T_\varphi\|$, где $c_p=\|P_T\|$. В частности, если оператор $T_\varphi$ компактен в $H^p(G)$, то он нулевой.* Доказательство. Поскольку существенный спектр Вейля инвариантен относительно компактных возмущений, имеем с учётом теоремы 5, следствия 8 и теоремы 1 $$\|T_\varphi+K\|\geq |T_\varphi+K|\geq|T_\varphi|=\|\varphi\|_\infty\geq (1/c_p)\|T_\varphi\|.$$
**Следствие 10.** *Пусть $p=2$. Если $A$ есть замкнутая симметричная подалгебра алгебры $C(G)$, обладающая тем свойством, что при любых $\varphi, \psi\in A$ полукоммутаторы $[T_\varphi, T_\psi)$ компактны в $H^2(G)$, то $\sigma_e(T_\varphi)
=\varphi(G)$ при $\varphi\in A$.* Доказательство. Отображение $j$ из $A$ в алгебру Калкина ${\cal C}(H^2(G))$, ставящее в соответствие каждой функции $\varphi\in A$ класс смежности $T_\varphi +K(H^2(G))$, является \*-гомоморфизмом алгебр, который изометричен в силу теоремы 1 и следствия 9. Остаётся заметить, что алгебра $j(A)$, будучи $C^*$-подалгеброй алгебры Калкина, наполнена в ней.
ЗАМЕЧАНИЕ 6. Примером алгебры $A$, удовлетворяющей условиям следствия 10, может служить $C^\ast$-подалгебра алгебры $C(G)$, порождённая характером $\chi\in X^i$, поскольку компактность полукоммутаторов $[T_{\chi^k},T_{\chi^j}),\ k,j\in\mathbb{Z}$ легко следует из предложения 1 и следствия 2. Следовательно, в этом случае $\sigma_e(T_\chi)=\chi(G)$. На самом деле, как показывает следствие 11, последнее равенство верно для любого $p\in (1,\infty)$ (ниже $\mathbb{D}$ обозначает открытый единичный круг в $\mathbb{C}$).
**Следствие 11.** *Пусть $\chi\in X,\ \chi\ne \bf{1}$. Тогда*
[1)]{} *спектр $\sigma(T_\chi)$ совпадает с замкнутым единичным кругом $\overline{\mathbb{D}}$*;
[2)]{} *при $\chi\in X^i$ существенный спектр Фредгольма $\sigma_e(T_\chi)
=\chi(G)=\Bbb{T}$;*
[3)]{} *при $\chi\notin X^i$ существенный спектр Фредгольма $\sigma_e
(T_\chi)= \overline{\mathbb{D}}$.* Доказательство. 1) Равенство $\chi(G)=\mathbb{T}$ следует из того, что $\chi(G)$ — нетривиальная связная подгруппа окружности. При этом для единственной ограниченной компоненты $\mathbb{D}$ дополнения $\mathbb{C}\setminus \chi(G)$ имеем, используя обозначения из доказательства предыдущей теоремы, $\alpha_\chi|\mathbb{D}=\alpha_\chi(0)\ne {\bf 1}$, поскольку $\chi\notin\exp(C(G))$ в силу единственности разложения Бора-ван Кампена. Теперь равенство $\sigma(T_\chi)=\overline{\mathbb{D}}$ следует из формулы (4.2).
2\) Это вытекает из теоремы 5 и того факта, что в рассматриваемом случае для единственной дыры $\mathbb{D}$ множества $\chi(G)$ имеем $\mathbb{D}\cap
\sigma_e(T_\chi)=\emptyset$, так как $\chi-0\in \Phi(G)$.
3\) Поскольку теперь $\chi-0\notin \Phi(G)$, то $\sigma_e(T_\chi)=\mathbb{T}\cup\mathbb{D}$ по теореме 5.
Статья опубликована в [@SbMath].
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abstract: 'Numerous important problems can be framed as learning from graph data. We propose a framework for learning convolutional neural networks for arbitrary graphs. These graphs may be undirected, directed, and with both discrete and continuous node and edge attributes. Analogous to image-based convolutional networks that operate on locally connected regions of the input, we present a general approach to extracting locally connected regions from graphs. Using established benchmark data sets, we demonstrate that the learned feature representations are competitive with state of the art graph kernels and that their computation is highly efficient.'
bibliography:
- 'dlnf.bib'
---
Introduction
============
With this paper we aim to bring convolutional neural networks to bear on a large class of graph-based learning problems. We consider the following two problems.
1. Given a collection of graphs, learn a function that can be used for classification and regression problems on unseen graphs. The nodes of any two graphs are *not* necessarily in correspondence. For instance, each graph of the collection could model a chemical compound and the output could be a function mapping unseen compounds to their level of activity against cancer cells.
2. Given a large graph, learn graph representations that can be used to infer unseen graph properties such as node types and missing edges.
We propose a framework for learning representations for classes of directed and undirected graphs. The graphs may have nodes and edges with multiple discrete and continuous attributes and may have multiple types of edges. Similar to convolutional neural network for images, we construct locally connected neighborhoods from the input graphs. These neighborhoods are generated efficiently and serve as the receptive fields of a convolutional architecture, allowing the framework to learn effective graph representations.
The proposed approach builds on concepts from convolutional neural networks (CNNs) [@Kunihiko:1980; @atlas:1987; @lecun:1998; @lecun:2015] for images and extends them to arbitrary graphs. Figure \[fig-grid\] illustrates the locally connected receptive fields of a CNN for images. An image can be represented as a square grid graph whose nodes represent pixels. Now, a CNN can be seen as traversing a node sequence (nodes $1$-$4$ in Figure \[fig-grid\](a)) and generating fixed-size neighborhood graphs (the $3$x$3$ grids in Figure \[fig-grid\](b)) for each of the nodes. The neighborhood graphs serve as the receptive fields to read feature values from the pixel nodes. Due to the implicit spatial order of the pixels, the sequence of nodes for which neighborhood graphs are created, from left to right and top to bottom, is uniquely determined. The same holds for NLP problems where each sentence (and its parse-tree) determines a sequence of words. However, for numerous graph collections a problem-specific ordering (spatial, temporal, or otherwise) is missing and the nodes of the graphs are not in correspondence. In these instances, one has to solve two problems: (i) Determining the node sequences for which neighborhood graphs are created and (ii) computing a normalization of neighborhood graphs, that is, a unique mapping from a graph representation into a vector space representation. The proposed approach, termed [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}, addresses these two problems for arbitrary graphs. For each input graph, it first determines nodes (and their order) for which neighborhood graphs are created. For each of these nodes, a neighborhood consisting of exactly $k$ nodes is extracted and normalized, that is, it is uniquely mapped to a space with a fixed linear order. The normalized neighborhood serves as the receptive field for a node under consideration. Finally, feature learning components such as convolutional and dense layers are combined with the normalized neighborhood graphs as the CNN’s receptive fields.
Figure \[fig-architecture\] illustrates the [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}architecture which has several advantages over existing approaches: First, it is highly efficient, naively parallelizable, and applicable to large graphs. Second, for a number of applications, ranging from computational biology to social network analysis, it is important to visualize learned network motifs [@milo:2002]. [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}supports feature visualizations providing insights into the structural properties of graphs. Third, instead of crafting yet another graph kernel, [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}learns application dependent features without the need to feature engineering. Our theoretical contributions are the definition of the normalization problem on graphs and its complexity; a method for comparing graph labeling approaches for a collection of graphs; and a result that shows that [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}generalizes CNNs on images. Using standard benchmark data sets, we demonstrate that the learned CNNs for graphs are both efficient and effective compared to state of the art graph kernels.
Related Work {#sec:related}
============
Graph kernels allow kernel-based learning approaches such as SVMs to work directly on graphs [@Vishwanathan:2010]. Kernels on graphs were originally defined as similarity functions on the nodes of a single graph [@Kondor:2002]. Two representative classes of kernels are the skew spectrum kernel [@Kondor:2008] and kernels based on graphlets [@Kondor:2009; @Shervashidze:2009]. The latter is related to our work, as it builds kernels based on fixed-sized subgraphs. These subgraphs, which are often called motifs or graphlets, reflect functional network properties [@milo:2002; @alon:2007]. However, due to the combinatorial complexity of subgraph enumeration, graphlet kernels are restricted to subgraphs with few nodes. An effective class of graph kernels are the Weisfeiler-Lehman (WL) kernels [@Shervashidze:2011]. WL kernels, however, only support discrete features and use memory linear in the number of training examples at test time. [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}uses WL as one possible labeling procedure to compute receptive fields. Deep graph kernels [@Yanardag:2015] and graph invariant kernels [@Orsini:2015] compare graphs based on the existence or count of small substructures such as shortest paths [@Borgwardt:2005], graphlets, subtrees, and other graph invariants [@haussler:1999; @Orsini:2015]. In contrast, [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}learns substructures from graph data and is not limited to a predefined set of motifs. Moreover, while all graph kernels have a training complexity at least *quadratic* in the number of graphs [@Shervashidze:2011], which is prohibitive for large-scale problems, [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}scales *linearly* with the number of graphs.
Graph neural networks (GNNs) [@Scarselli:2009] are a recurrent neural network architecture defined on graphs. GNNs apply recurrent neural networks for walks on the graph structure, propagating node representations until a fixed point is reached. The resulting node representations are then used as features in classification and regression problems. GNNs support only discrete labels and perform as many backpropagation operations as there are edges and nodes in the graph *per learning iteration*. Gated Graph Sequence Neural Networks modify GNNs to use gated recurrent units and to output sequences [@li:2015].
Recent work extended CNNs to topologies that differ from the low-dimensional grid structure [@bruna:2013; @henaff:2015]. All of these methods, however, assume one global graph structure, that is, a correspondence of the vertices across input examples. [@duvenaud:2015] perform convolutional type operations on graphs, developing a differentiable variant of one specific graph feature.
Background
==========
We provide a brief introduction to the required background in convolutional networks and graph theory.
Convolutional Neural Networks
-----------------------------
CNNs were inspired by earlier work that showed that the visual cortex in animals contains complex arrangements of cells, responsible for detecting light in small local regions of the visual field [@hubel:1968]. CNNs were developed in the $1980$s and have been applied to image, speech, text, and drug discovery problems [@atlas:1987; @LeCun:1989; @lecun:1998; @lecun:2015; @WallachDH:2015]. A predecessor to CNNs was the Neocognitron [@Kunihiko:1980]. A typical CNN is composed of convolutional and dense layers. The purpose of the first convolutional layer is the extraction of common patterns found within local regions of the input images. CNNs convolve learned filters over the input image, computing the inner product at every image location in the image and outputting the result as tensors whose depth is the number of filters.
Graphs
------
A graph $G$ is a pair $(V, E)$ with $V = \{v_1, ..., v_n\}$ the set of vertices and $E \subseteq V \times V$ the set of edges. Let $n$ be the number of vertices and $m$ the number of edges. Each graph can be represented by an adjacency matrix $\mathbf{A}$ of size $n \times n$, where $\mathbf{A}_{i,j} = 1$ if there is an edge from vertex $v_i$ to vertex $v_j$, and $\mathbf{A}_{i,j} = 0$ otherwise. In this case, we say that vertex $v_i$ has *position* $i$ in $\mathbf{A}$. Moreover, if $\mathbf{A}_{i,j} = 1$ we say $v_i$ and $v_j$ are *adjacent*. Node and edge attributes are features that attain one value for each node and edge of a graph. We use the term attribute value instead of label to avoid confusion with the graph-theoretical concept of a labeling. A walk is a sequence of nodes in a graph, in which consecutive nodes are connected by an edge. A path is a walk with distinct nodes. We write $\mathbf{d}(u, v)$ to denote the distance between $u$ and $v$, that is, the length of the shortest path between $u$ and $v$. $N_1(v)$ is the $1$-neighborhood of a node, that is, all nodes that are adjacent to $v$.
**Labeling and Node Partitions.** <span style="font-variant:small-caps;">Patchy-san</span> utilizes graph labelings to impose an order on nodes. A graph labeling $\ell$ is a function $\ell: V \rightarrow S$ from the set of vertices $V$ to an ordered set $S$ such as the real numbers and integers. A graph labeling procedure computes a graph labeling for an input graph. When it is clear from the context, we use *labeling* to refer to both, the graph labeling and the procedure to compute it. A ranking (or coloring) is a function $\mathbf{r}:V \rightarrow \{1, ..., |V|\}$. Every labeling induces a ranking $\mathbf{r}$ with $\mathbf{r}(u) < \mathbf{r}(v)$ if and only if $\ell(u)>\ell(v)$. If the labeling $\ell$ of graph $G$ is injective, it determines a total order of $G$’s vertices and a unique adjacency matrix $\mathbf{A}^{\ell}(G)$ of $G$ where vertex $v$ has position $\mathbf{r}(v)$ in $\mathbf{A}^{\ell}(G)$. Moreover, every graph labeling induces a partition $\{V_1, ..., V_n\}$ on $V$ with $u, v \in V_i$ if and only if $\ell(u)=\ell(v)$. Examples of graph labeling procedures are node degree and other measures of centrality commonly used in the analysis of networks. For instance, the *betweeness centrality* of a vertex $v$ computes the fractions of shortest paths that pass through $v$. The Weisfeiler-Lehman algorithm [@weisfeiler:1968; @douglas2011weisfeiler] is a procedure for partitioning the vertices of a graph. It is also known as color refinement and naive vertex classification. Color refinement has attracted considerable interest in the ML community since it can be applied to speed-up inference in graphical models [@Kersting:2009; @Kersting:2014] and as a method to compute graph kernels [@Shervashidze:2011]. [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}applies these labeling procedures, among others (degree, page-rank, eigenvector centrality, etc.), to impose an order on the nodes of graphs, replacing application-dependent orders (temporal, spatial, etc.) where missing.
**Isomorphism and Canonicalization.** The computational problem of deciding whether two graphs are isomorphic surfaces in several application domains. The graph isomorphism (GI) problem is in NP but not known to be in P or NP-hard. Under several mild restrictions, GI is known to be in P. For instance, GI is in P for graphs of bounded degree [@luks:1982]. A canonicalization of a graph $G$ is a graph $G'$ with a fixed vertex order which is isomorphic to $G$ and which represents its entire isomorphism class. In practice, the graph canonicalization tool [[<span style="font-variant:small-caps;">Nauty</span>]{}]{}has shown remarkable performance [@McKay:2014].
Learning CNNs for Arbitrary Graphs
==================================
When CNNs are applied to images, a receptive field (a square grid) is moved over each image with a particular step size. The receptive field reads the pixels’ feature values, for each channel once, and a patch of values is created for each channel. Since the pixels of an image have an implicit arrangement – a spatial order – the receptive fields are always moved from left to right and top to bottom. Moreover, the spatial order uniquely determines the nodes of each receptive field and the way these nodes are mapped to a vector space representation (see Figure \[fig-grid\](b)). Consequently, the values read from two pixels using two different locations of the receptive field are assigned to the same relative position if and only if the pixels’ structural roles (their spatial position within the receptive field) are identical.
To show the connection between CNNs and [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}, we frame CNNs on images as identifying a sequence of nodes in the square grid graph representing the image and building a normalized neighborhood graph – a receptive field – for each node in the identified sequence. For graph collections where an application-dependent node order is missing and where the nodes of any two graphs are not yet aligned, we need to determine for each graph (i) the sequences of nodes for which we create neighborhoods, and (ii) a unique mapping from the graph representation to a vector representation such that nodes with similar structural roles in the neighborhood graphs are positioned similarly in the vector representation.
We address these problems by leveraging graph labeling procedures that assigns nodes from two different graphs to a similar relative position in their respective adjacency matrices if their structural roles within the graphs are similar. Given a collection of graphs, [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}(<span style="font-variant:small-caps;">Select</span>-<span style="font-variant:small-caps;">Assemble</span>-<span style="font-variant:small-caps;">Normalize</span>) applies the following steps to each graph: (1) Select a fixed-length sequence of nodes from the graph; (2) assemble a fixed-size neighborhood for each node in the selected sequence; (3) normalize the extracted neighborhood graph; and (4) learn neighborhood representations with convolutional neural networks from the resulting sequence of patches. In the following, we describe methods that address the above-mentioned challenges.
graph labeling procedure $\ell$, graph $G=(V, E)$, stride $s$, width $w$, receptive field size $k$ $V_{\mathtt{sort}}$ = top $w$ elements of $V$ according to $\ell$ $i=1, j=1$ $\mathsf{f} = \textsc{ReceptiveField}(V_{\mathtt{sort}}[i])$ $\mathsf{f} = \textsc{ZeroReceptiveField}()$ apply $\mathsf{f}$ to each input channel $i = i + s$, $j = j + 1$
Node Sequence Selection
-----------------------
Node sequence selection is the process of identifying, for each input graph, a sequence of nodes for which receptive fields are created. Algorithm \[alg:sequence\] lists one such procedure. First, the vertices of the input graph are sorted with respect to a given graph labeling. Second, the resulting node sequence is traversed using a given stride $s$ and for each visited node, Algorithm \[alg:example\] is executed to construct a receptive field, until exactly $w$ receptive fields have been created. The stride $s$ determines the distance, relative to the selected node sequence, between two consecutive nodes for which a receptive field is created. If the number of nodes is smaller than $w$, the algorithm creates all-zero receptive fields for padding purposes. Several alternative methods for vertex sequence selection are possible. For instance, a depth-first traversal of the input graph guided by the values of the graph labeling. We leave these ideas to future work.
vertex $v$, receptive field size $k$ set of neighborhood nodes $N$ for $v$ $N = [v]$ $L = [v]$ $L = \bigcup_{v \in L} N_1(v)$ $N = N \cup L$ the set of vertices $N$
Neighborhood Assembly
---------------------
For each of the nodes identified in the previous step, a receptive field has to be constructed. Algorithm \[alg:example\] first calls Algorithm \[alg:assembly\] to assembles a local neighborhood for the input node. The nodes of the neighborhood are the candidates for the receptive field. Algorithm \[alg:assembly\] lists the neighborhood assembly steps. Given as inputs a node $v$ and the size of the receptive field $k$, the procedure performs a breadth-first search, exploring vertices with an increasing distance from $v$, and adds these vertices to a set $N$. If the number of collected nodes is smaller than $k$, the $1$-neighborhood of the vertices most recently added to $N$ are collected, and so on, until at least $k$ vertices are in $N$, or until there are no more neighbors to add. Note that at this time, the size of $N$ is possibly different to $k$.
Graph Normalization
-------------------
The receptive field for a node is constructed by *normalizing* the neighborhood assembled in the previous step. Illustrated in Figure \[fig-normalize\], the normalization imposes an order on the nodes of the neighborhood graph so as to map from the unordered graph space to a vector space with a linear order. The basic idea is to leverage graph labeling procedures that assigns nodes of two different graphs to a similar relative position in the respective adjacency matrices if and only if their structural roles within the graphs are similar.
To formalize this intuition, we define the optimal graph normalization problem which aims to find a labeling that is optimal relative to a given collection of graphs.
Let $\mathcal{G}$ be a collection of unlabeled graphs with $k$ nodes, let $\ell$ be an injective graph labeling procedure, let $\mathbf{d}_{\mathbf{G}}$ be a distance measure on graphs with $k$ nodes, and let $\mathbf{d}_{\mathbf{A}}$ be a distance measure on $k\times k$ matrices. Find $\hat{\ell}$ such that $$\label{equation-graph-normalization}
\hat{\ell} = \operatorname*{arg\,min}_{\ell} \mathbb{E}_{\mathcal{G}}\left[\left|\mathbf{d}_{\mathbf{A}}\left(\mathbf{A}^{\ell}(G), \mathbf{A}^{\ell}(G')\right) - \mathbf{d}_{\mathbf{G}}(G, G')\right|\right].$$
The problem amounts to finding a graph labeling procedure $\ell$, such that, for any two graphs drawn uniformly at random from $\mathcal{G}$, the expected difference between the distance of the graphs in vector space (with respect to the adjacency matrices based on $\ell$) and the distance of the graphs in graph space is minimized. The optimal graph normalization problem is a generalization of the classical graph canonicalization problem. A canonical labeling algorithm, however, is optimal only for isomorphic graphs and might perform poorly for graphs that are similar but not isomorphic. In contrast, the smaller the expectation of the optimal normalization problem, the better the labeling aligns nodes with similar structural roles. Note that the similarity is determined by $\mathbf{d}_{\mathbf{G}}$.
vertex $v$, graph labeling $\ell$, receptive field size $k$ $N = \textsc{NeighAssemb}(v, k)$ $G_{\mathtt{norm}} = \textsc{NormalizeGraph}(N, v, \ell, k)$ $G_{\mathtt{norm}}$
We have the following result concerning the complexity of the optimal normalization problem.
\[thm:normalization\_NP\] Optimal graph normalization is NP-hard.
By reduction from subgraph isomorphism.
[[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}does *not* solve the above optimization problem. Instead, it may compare different graph labeling methods and choose the one that performs best relative to a given collection of graphs.
subset of vertices $U$ from original graph $G$, vertex $v$, graph labeling $\ell$, receptive field size $k$ receptive field for $v$ compute ranking $\mathbf{r}$ of $U$ using $\ell$, subject to\
$\forall u,w \in U: \mathbf{d}(u,v) < \mathbf{d}(w,v) \Rightarrow \mathbf{r}(u)<\mathbf{r}(w)$ $N = $ top $k$ vertices in $U$ according to $\mathbf{r}$ compute ranking $\mathbf{r}$ of $N$ using $\ell$, subject to\
$\forall u,w \in N: \mathbf{d}(u,v) < \mathbf{d}(w,v) \Rightarrow \mathbf{r}(u)<\mathbf{r}(w)$ $N = U$ and $k-|U|$ dummy nodes $N = U$ construct the subgraph $G[N]$ for the vertices $N$ canonicalize $G[N]$, respecting the prior coloring $\mathbf{r}$ $G[N]$
\[thrm:graph\_norm\_expectation\] Let $\mathcal{G}$ be a collection of graphs and let $(G_1,G_1'), ..., (G_N,G_N')$ be a sequence of pairs of graphs sampled independently and uniformly at random from $\mathcal{G}$. Let $\hat{\theta}_{\ell} := \sum_{i=1}^{N} \mathbf{d}_{\mathbf{A}}\left( \mathbf{A}^{\ell}(G_i), \mathbf{A}^{\ell}(G_i')\right) / N$ and $\theta_{\ell} := \mathbb{E}_{\mathcal{G}}\left[\left| \mathbf{d}_{\mathbf{A}}\left( \mathbf{A}^{\ell}(G), \mathbf{A}^{\ell}(G')\right) - \mathbf{d}_{\mathbf{G}}(G, G')\right|\right]$. If $\mathbf{d}_{\mathbf{A}} \geq \mathbf{d}_{\mathbf{G}}$, then $\mathbb{E}_{\mathcal{G}}[\hat{\theta}_{\ell_1}] < \mathbb{E}_{\mathcal{G}}[\hat{\theta}_{\ell_2}]$ if and only if $\theta_{\ell_1} < \theta_{\ell_2}$.
Theorem \[thrm:graph\_norm\_expectation\] enables us to compare different labeling procedures in an unsupervised manner via a comparison of the corresponding estimators. Under the assumption $\mathbf{d}_{\mathbf{A}} \geq \mathbf{d}_{\mathbf{G}}$, the smaller the estimate $\hat{\theta}_{\ell}$ the smaller the absolute difference. Therefore, we can simply choose the labeling $\ell$ for which $\hat{\theta}_{\ell}$ is minimal. The assumption $\mathbf{d}_{\mathbf{A}} \geq \mathbf{d}_{\mathbf{G}}$ holds, for instance, for the edit distance on graphs and the Hamming distance on adjacency matrices. Finally, note that all of the above results can be extended to directed graphs.
The graph normalization problem and the application of appropriate graph labeling procedures for the normalization of local graph structures is at the core of the proposed approach. Within the [[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}framework, we normalize the neighborhood graphs of a vertex $v$. The labeling of the vertices is therefore constrained by the graph distance to $v$: for any two vertices $u, w$, if $u$ is closer to $v$ than $w$, then $v$ is always ranked higher than $w$. This definition ensures that $v$ has always rank $1$, and that the closer a vertex is to $v$ in $G$, the higher it is ranked in the vector space representation.
Since most labeling methods are not injective, it is necessary to break ties between same-label nodes. To do so, we use [[<span style="font-variant:small-caps;">Nauty</span>]{}]{} [@McKay:2014]. [[<span style="font-variant:small-caps;">Nauty</span>]{}]{}accepts prior node partitions as input and breaks remaining ties by choosing the lexicographically maximal adjacency matrix. It is known that graph isomorphism is in PTIME for graphs of bounded degree [@luks:1982]. Due to the constant size $k$ of the neighborhood graphs, the algorithm runs in time polynomial in the size of the original graph and, on average, in time linear in $k$ [@Babai:1980]. Our experiments verify that computing a canonical labeling of the graph neigborhoods adds a negligible overhead.
Algorithm \[alg:normalization\] lists the normalization procedure. If the size of the input set $U$ is larger than $k$, it first applies the ranking based on $\ell$ to select the top $k$ nodes and recomputes a ranking on the smaller set of nodes. If the size of $U$ is smaller than $k$, it adds disconnected dummy nodes. Finally, it induces the subgraph on the vertices $N$ and canonicalizes the graph taking the ranking $\mathbf{r}$ as prior coloring.
We can relate <span style="font-variant:small-caps;">Patchy-san</span> to CNNs for images as follows.
Given a sequence of pixels taken from an image. Applying <span style="font-variant:small-caps;">Patchy-san</span> with receptive field size $(2m-1)^2$, stride $s$, no zero padding, and $1$-WL normalization to the sequence is identical (up to a fixed permutation of the receptive field) to the first layer of a <span style="font-variant:small-caps;">CNN</span> with receptive field size $2m-1$, stride $s$, and no zero padding.
It is possible to show that if an input graph is a square grid, then the $1$-WL normalized receptive field constructed for a vertex is always a square grid graph with a unique vertex order.
Convolutional Architecture
--------------------------
[[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}is able to process both vertex and edge attributes (discrete and continuous). Let $\mathtt{a}_v$ be the number of vertex attributes and let $\mathtt{a}_e$ be the number of edge attributes. For each input graph $G$, it applies normalized receptive fields for vertices and edges which results in one $(w, k, \mathtt{a}_v)$ and one $(w, k, k, \mathtt{a}_e)$ tensor. These can be reshaped to a $(wk, \mathtt{a}_v)$ and a $(wk^2, \mathtt{a}_e)$ tensors. Note that $\mathtt{a}_v$ and $\mathtt{a}_e$ are the number of input channels. We can now apply a $1$-dimensional convolutional layer with stride and receptive field size $k$ to the first and $k^2$ to the second tensor. The rest of the architecture can be chosen arbitrarily. We may use merge layers to combine convolutional layers representing nodes and edges, respectively.
Complexity and Implementation
=============================
[[<span style="font-variant:small-caps;">Patchy-san</span>]{}]{}’s algorithm for creating receptive fields is highly efficient and naively parallelizable because the fields are generated independently. We can show the following asymptotic worst-case result.
Let $N$ be the number of graphs, let $k$ be the receptive field size, $w$ the width, and $O(f(n,m))$ the complexity of computing a given labeling $\ell$ for a graph with $n$ vertices and $m$ edges. <span style="font-variant:small-caps;">Patchy-san</span> has a worst-case complexity of $O(N w (f(n,m)+ n\log(n) + \exp(k)))$ for computing the receptive fields for $N$ graphs.
Node sequence selection requires the labeling of each input graph and the retrieval of the $k$ highest ranked nodes. For the creation of normalized graph patches, most computational effort is spent applying the labeling procedure $\ell$ to a neighborhood whose size may be larger than $k$. Let $\overline{d}$ be the maximum degree of the input graph $G$, and $U$ the neighborhood returned by Algorithm \[alg:assembly\]. We have $|U| \leq (k-2) \overline{d} \leq n$. The term $\exp(k)$ comes from the worst-case complexity of the graph canonicalization algorithm [[<span style="font-variant:small-caps;">Nauty</span>]{}]{}on a $k$ node graph [@miyazaki:1997].
For instance, for the Weisfeiler-Lehman algorithm, which has a complexity of $O((n+m) \log(n))$ [@Berkholz:2013], and constants $w \ll n$ and $k \ll n$, the complexity of <span style="font-variant:small-caps;">Patchy-san</span> is linear in $N$ and quasi-linear in $m$ and $n$.
Experiments
===========
We conduct three types of experiments: a runtime analysis, a qualitative analysis of the learned features, and a comparison to graph kernels on benchmark data sets.
![\[fig-runtime\] Receptive fields per second rates on different graphs.](runtimes.pdf){width="47.00000%"}
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Runtime Analysis
----------------
We assess the efficiency of <span style="font-variant:small-caps;">Patchy-san</span> by applying it to real-world graphs. The objective is to compare the rates at which receptive fields are generated to the rate at which state of the art CNNs perform learning. All input graphs are part of the collection of the Python module <span style="font-variant:small-caps;">graph-tool</span>[^1]. For a given graph, we used <span style="font-variant:small-caps;">Patchy-san</span> to compute a receptive field for *all* nodes using the $1$-dimensional Weisfeiler-Lehman [@douglas2011weisfeiler] (1-WL) algorithm for the normalization. **torus** is a periodic lattice with $10,000$ nodes; **random** is a random undirected graph with $10,000$ nodes and a degree distribution $P(k) \propto 1/k$ and $k_{\max} = 3$; **power** is a network representing the topology of a power grid in the US; **polbooks** is a co-purchasing network of books about US politics published during the $2004$ presidential election; **preferential** is a preferential attachment network model where newly added vertices have degree $3$; **astro-ph** is a coauthorship network between authors of preprints posted on the astrophysics arxiv [@newman:2001]; **email-enron** is a communication network generated from about half a million sent emails [@leskovec:2009]. All experiments were run on commodity hardware with 64G RAM and a single 2.8 GHz CPU.
Figure \[fig-runtime\] depicts the receptive fields per second rates for each input graph. For receptive field size $k=5$ and $k=10$ <span style="font-variant:small-caps;">Patchy-san</span> creates fields at a rate of more than $1000/s$ except for **email-enron** with a rate of $600/s$ and $320/s$, respectively. For $k=50$, the largest tested size, fields are created at a rate of at least $100/s$. A CNN with $2$ convolutional and $2$ dense layers learns at a rate of about $200$-$400$ training examples per second on the same machine. Hence, the speed at which receptive fields are generated is sufficient to saturate a downstream CNN.
Feature Visualization
---------------------
The visualization experiments’ aim is to qualitatively investigate whether popular models such as the restricted Boltzman machine (RBM) [@freund:1992] can be combined with <span style="font-variant:small-caps;">Patchy-san</span> for unsupervised feature learning. For every input graph, we have generated receptive fields for all nodes and used these as input to an RBM. The RBM had $100$ hidden nodes and was trained for $30$ epochs with contrastive divergence and a learning rate of $0.01$. We visualize the features learned by a single-layer RBM for $1$-dimensional Weisfeiler-Lehman (1-WL) normalized receptive fields of size $9$. Note that the features learned by the RBM correspond to reoccurring receptive field patterns. Figure \[fig-features\] depicts some of the features and samples drawn from it for four different graphs.
[**Data set**]{} MUTAG PCT NCI1 PROTEIN D & D
---------------------------- ------------------------- -------------------------- --------------------------- --------------------------- --------------------------
Max 28 109 111 620 5748
Avg 17.93 25.56 29.87 39.06 284.32
Graphs 188 344 4110 1113 1178
SP \[\] $85.79 \pm 2.51$ $58.53 \pm 2.55$ $73.00 \pm 0.51$ $75.07 \pm 0.54$ $> 3$ days
RW \[\] $83.68 \pm 1.66$ $57.26 \pm 1.30$ $> 3$ days $74.22 \pm 0.42$ $> 3$ days
GK \[\] $81.58 \pm 2.11$ $57.32 \pm 1.13$ $62.28 \pm 0.29$ $71.67 \pm 0.55$ $78.45 \pm 0.26$
WL \[\] $80.72 \pm 3.00\ (5s)$ $56.97 \pm 2.01\ (30s)$ $80.22 \pm 0.51\ (375s)$ $72.92 \pm 0.56\ (143s)$ $77.95 \pm 0.70\ (609s)$
PSCN $k$=$5$ $91.58 \pm 5.86\ (2s)$ $59.43 \pm 3.14\ \ (4s)$ $72.80 \pm 2.06\ \ (59s)$ $74.10 \pm 1.72\ \ (22s)$ $74.58 \pm 2.85\ (121s)$
PSCN $k$=$10$ $88.95 \pm 4.37 \ (3s)$ $62.29 \pm 5.68\ \ (6s)$ $76.34 \pm 1.68\ \ (76s)$ $75.00 \pm 2.51\ \ (30s)$ $76.27 \pm 2.64\ (154s)$
PSCN $k$=$10^{\mathtt{E}}$ $92.63 \pm 4.21 \ (3s)$ $60.00 \pm 4.82\ \ (6s)$ $78.59 \pm 1.89\ \ (76s)$ $75.89 \pm 2.76\ \ (30s)$ $77.12 \pm 2.41\ (154s)$
PSLR $k$=$10$ $87.37 \pm 7.88$ $58.57 \pm 5.46$ $70.00 \pm 1.98$ $71.79 \pm 3.71$ $68.39 \pm 5.56$
Graph Classification
--------------------
Graph classification is the problem of assigning graphs to one of several categories. [**Data Sets.**]{} We use $6$ standard benchmark data sets to compare run-time and classification accuracy with state of the art graph kernels: MUTAG, PCT, NCI1, NCI109, PROTEIN, and D&D. MUTAG [@debnath:1991] is a data set of $188$ nitro compounds where classes indicate whether the compound has a mutagenic effect on a bacterium. PTC consists of $344$ chemical compounds where classes indicate carcinogenicity for male and female rats [@toivonen:2003]. NCI1 and NCI109 are chemical compounds screened for activity against non-small cell lung cancer and ovarian cancer cell lines [@wale:2006]. PROTEINS is a graph collection where nodes are secondary structure elements and edges indicate neighborhood in the amino-acid sequence or in 3D space. Graphs are classified as enzyme or non-enzyme. D&D is a data set of $1178$ protein structures [@Dobson:2003] classified into enzymes and non-enzymes.
[**Experimental Set-up.**]{} We compared <span style="font-variant:small-caps;">Patchy-san</span> with the shortest-path kernel (SP) [@Borgwardt:2005], the random walk kernel (RW) [@Gaertner:2003], the graphlet count kernel (GK) [@Shervashidze:2009], and the Weisfeiler-Lehman subtree kernel (WL) [@Shervashidze:2011]. Similar to previous work [@Yanardag:2015], we set the height parameter of WL to $2$, the size of the graphlets for GK to $7$, and chose the decay factor for RW from $\{10^{-6}, 10^{-5}, ..., 10^{-1}\}$. We performed $10$-fold cross-validation with <span style="font-variant:small-caps;">LIB-SVM</span> [@Chang:2011], using $9$ folds for training and $1$ for testing, and repeated the experiments $10$ times. We report average prediction accuracies and standard deviations.
For <span style="font-variant:small-caps;">Patchy-san</span> (referred to as PSCN), we used $1$-dimensional WL normalization, a width $w$ equal to the average number of nodes (see Table \[fig-classification\]), and receptive field sizes of $k=5$ and $k=10$. For the experiments we only used node attributes. In addition, we ran experiments for $k=10$ where we combined receptive fields for nodes and edges using a merge layer ($k=10^{\mathtt{E}}$). To make a fair comparison, we used a single network architecture with two convolutional layers, one dense hidden layer, and a softmax layer for all experiments. The first convolutional layer had $16$ output channels (feature maps). The second conv layer has $8$ output channels, a stride of $s=1$, and a field size of $10$. The convolutional layers have rectified linear units. The dense layer has $128$ rectified linear units with a dropout rate of $0.5$. Dropout and the relatively small number of neurons are needed to avoid overfitting on the smaller data sets. The only hyperparameter we optimized is the number of epochs and the batch size for the mini-batch gradient decent algorithm <span style="font-variant:small-caps;">rmsprop</span>. All of the above was implemented with the <span style="font-variant:small-caps;">Theano</span> [@bergstra:2010] wrapper <span style="font-variant:small-caps;">Keras</span> [@chollet:2015]. We also applied a logistic regression (PSLR) classifier on the patches for $k=10$.
Moreover, we ran experiments with the same set-up[^2] on larger social graph data sets (up to $12000$ graphs each, with an average of $400$ nodes), and compared <span style="font-variant:small-caps;">Patchy-san</span> with previously reported results for the graphlet count (GK) and the deep graphlet count kernel (DGK) [@Yanardag:2015]. We used the normalized node degree as attribute for <span style="font-variant:small-caps;">Patchy-san</span>, highlighting one of its advantages: it can easily incorporate continuous features.
[**Data set**]{} GK \[\] DGK \[\] PSCN $k$=$10$
------------------ ------------------ ------------------ ------------------
COLLAB $72.84 \pm 0.28$ $73.09 \pm 0.25$ $72.60 \pm 2.15$
IMDB-B $65.87 \pm 0.98$ $66.96 \pm 0.56$ $71.00 \pm 2.29$
IMDB-M $43.89 \pm 0.38$ $44.55 \pm 0.52$ $45.23 \pm 2.84$
RE-B $77.34 \pm 0.18$ $78.04 \pm 0.39$ $86.30 \pm 1.58$
RE-M5k $41.01 \pm 0.17$ $41.27 \pm 0.18$ $49.10 \pm 0.70$
RE-M10k $31.82 \pm 0.08$ $32.22 \pm 0.10$ $41.32 \pm 0.42$
: \[table-social-graphs\] Comparison of accuracy results on social graphs \[\].
[**Results.**]{} Table \[fig-classification\] lists the results of the experiments. We omit the results for NCI109 as they are almost identical to NCI1. Despite using a one-fits-all CNN architecture, the CNNs accuracy is highly competitive with existing graph kernels. In most cases, a receptive field size of $10$ results in the best classification accuracy. The relatively high variance can be explained with the small size of the benchmark data sets and the fact that the CNNs hyperparameters (with the exception of epochs and batch size) were not tuned to individual data sets. Similar to the experience on image and text data, we expect <span style="font-variant:small-caps;">Patchy-san</span> to perform even better for large data sets. Moreover, <span style="font-variant:small-caps;">Patchy-san</span> is between $2$ and $8$ times more efficient than the most efficient graph kernel (WL). We expect the performance advantage to be much more pronounced for data sets with a large number of graphs. Results for betweeness centrality normalization are similar with the exception of the runtime which increases by about $10$%. Logistic regression applied to <span style="font-variant:small-caps;">Patchy-san</span>’s receptive fields performs worse, indicating that <span style="font-variant:small-caps;">Patchy-san</span> works especially well in conjunction with CNNs which learn non-linear feature combinations and which share weights across receptive fields.
<span style="font-variant:small-caps;">Patchy-san</span> is also highly competitive on the social graph data. It significantly outperforms the other two kernels on four of the six data sets and achieves ties on the rest. Table \[table-social-graphs\] lists the results of the experiments.
Conclusion and Future Work
==========================
We proposed a framework for learning graph representations that are especially beneficial in conjunction with CNNs. It combines two complementary procedures: (a) selecting a sequence of nodes that covers large parts of the graph and (b) generating local normalized neighborhood representations for each of the nodes in the sequence. Experiments show that the approach is competitive with state of the art graph kernels.
Directions for future work include the use of alternative neural network architectures such as RNNs; combining different receptive field sizes; pretraining with RBMs and autoencoders; and statistical relational models based on the ideas of the approach.
Acknowledgments {#acknowledgments .unnumbered}
===============
Many thanks to the anonymous ICML reviewers who provided tremendously helpful comments. The research leading to these results has received funding from the European Union’s Horizon 2020 innovation action program under grant agreement No 653449-TYPES.
[^1]: https://graph-tool.skewed.de/
[^2]: Due to the larger size of the data sets, we removed dropout.
|
---
abstract: 'We show the rather counterintuitive result that entangled input states can strictly enhance the distinguishability of two entanglement-breaking channels.'
author:
- 'Massimiliano F. Sacchi'
title: 'Entanglement can enhance the distinguishability of entanglement-breaking channels'
---
The class of entanglement-breaking channels—trace-preserving completely positive maps for which the output state is always separable—has been extensively studied [@eb1; @eb2; @Hv; @King1; @Shor; @King2; @VDC; @shir2]. More precisely, a quantum channel ${\cal E}$ is called entanglement breaking if $({\cal
E} \otimes I)(\Gamma )$ is always separable, i.e., any entangled density matrix $\Gamma $ is mapped to a separable one. The convex structure of entanglement-breaking channels has been thoroughly analyzed in Refs. [@eb1; @eb2]. Moreover, the properties of such a kind of channels have allowed to obtain a number of results for the hard problem of additivity of capacity in quantum information theory [@Hv; @King1; @Shor; @King2; @VDC; @shir2; @tcv; @hw; @shir1].
Channels which break entanglement are particularly noisy in some sense. In order to check if a channel is entanglement-breaking it is sufficient to look at the separability of the output state corresponding just to an input maximally entangled state [@eb1], namely ${\cal E}$ is entanglement-breaking iff $({\cal E} \otimes
I)(|\beta \rangle \langle \beta |)$ is separable for $|\beta \rangle =
d^{-1/2} \sum _{j=0}^{d-1} |j \rangle \otimes |j \rangle $, $d$ being the dimension of the Hilbert space. Another equivalent condition [@eb1] is that the channel ${\cal E} $ can be written as $$\begin{aligned}
{\cal E}
(\rho )=\sum _k \langle \phi _k |\rho |\phi _k \rangle |\psi _k
\rangle \langle \psi _k |
\;,\end{aligned}$$ where $\{ |\phi _k \rangle \langle \phi _k |\}$ gives a positive operator-valued measure (POVM), namely $\sum _k |\phi _k \rangle
\langle \phi _k |=I $ [@fin]. The last formulation has an immediate physical interpretation: an entanglement-breaking channel can be simulated by a classical channel, in the sense that the sender can make a measurement on the input state $\rho $ by means of a POVM $\{ |\phi _k \rangle \langle \phi _k |\}$, and send the outcome $k$ via a classical channel to the receiver who then prepares an agreed-upon pure state $|\psi _k \rangle $. For the above reason one could think that entanglement—the peculiar trait of quantum mechanics—may not be useful when one deals with entanglement-breaking channels. In fact, entanglement breaking channels have zero quantum capacity [@hw].
In this report, however, we will show a situation in which the use of entanglement can be relevant also for entanglement-breaking channels, namely when one is asked to optimally discriminate two entanglement-breaking channels, as in the quantum hypothesis testing scenario [@hel]. What we mean is that an entangled input state can [*strictly*]{} enhance the distinguishability of two given entanglement-breaking channels. We will make use of some recent results [@discr] on the optimal discrimination of two given quantum operations. In particular, a complete characterization of the optimal input states to achieve the minimum-error probability has been given for Pauli channels [@discr], along with a necessary and sufficient condition for which entanglement strictly improves the discrimination. Such a condition is the following.
Given with a priori probability $p_1$ and $p_2=1-p_1$ two Pauli channels $$\begin{aligned}
{\cal E}_i (\rho )= \sum_{\alpha =0}^3 q_i^{(\alpha )}\, \sigma
_\alpha \,\rho \, \sigma _\alpha \;,\qquad {i=1,2,} \end{aligned}$$ where $\{\sigma _1\,,\sigma _2\,,\sigma _3 \}= \{ \sigma _x\,,\sigma
_y\,,\sigma _z\}$ denote the customary spin Pauli matrices, $\sigma _0 = I$, and $\sum _{\alpha =0}^3 q_i^ {(\alpha)}
= 1$, the use of entanglement strictly improves the discrimination iff [@discr] $$\begin{aligned}
\Pi _{\alpha =0}^3 \, r_\alpha < 0 \;,\label{}\end{aligned}$$ with $$\begin{aligned}
r_\alpha =p_1 \, q_1^{(\alpha )} - p_2 \, q_2^{(\alpha )}
\;.\label{ral}\end{aligned}$$ Moreover, the optimal input state can always be chosen as a maximally entangled state.
In the following we explicitly show the case of two entanglement-breaking channels that are strictly better discriminated by means of a maximally entangled input state. Let us consider for simplicity two different depolarizing channels $$\begin{aligned}
{\cal E}_i^ D(\rho )= q_i \, \rho + \frac {1-q_i}{3}\,
\sum_{\alpha =1}^3 \sigma
_\alpha \, \rho \, \sigma _\alpha \;, \qquad q_1 \neq q_2\;,\end{aligned}$$ The two channels are supposed to be given with a priori probability $p_1=p$ and $p_2=1-p$, respectively. The coefficients $r_\alpha $ of Eq. (\[ral\]) are given in this case by $$\begin{aligned}
&& r_0=p\,q_1 -(1-p)\,q_2 \;,
\nonumber \\& &
r_1=r_2=r_3=p \, \frac {1-q_1}{3}- (1-p)\,\frac
{1-q_2}{3}\;.\end{aligned}$$ Hence, entanglement strictly enhances the distinguishability of the two channels ${\cal E}_1^D$ and ${\cal E}_2^D$ iff $$\begin{aligned}
&&[p\,q_1 -(1-p)\,q_2]
\left [ p \,\frac {1-q_1}{3}- (1-p)\,\frac
{1-q_2}{3}\right ]
< 0\;,\label{cond1}\end{aligned}$$ or equivalently $$\begin{aligned}
&&(q_1+q_2)(2-q_1-q_2) p^2 -(q_1-2q_1q_2+3q_2-2q_2^2)p \nonumber \\& &
+ q_2(1-q_2)<0
%(q_1+q_2)(2-q_1-q_2) \, p^2 -(q_1-2q_1q_2+3q_2-2q_2^2)\, p + q_2(1-q_2)<0
\;.\label{cond2}\end{aligned}$$ The solution of Eq. (\[cond2\]) for the prior probability $p$ versus $q_1$ and $q_2$ is given by $$\begin{aligned}
&&\frac{1-q_2}{2-q_1-q_2} < p < \frac{q_2}{q_1+q_2}\qquad \hbox{for
}\quad q_1 < q_2 \;, \nonumber \\& &
\frac{q_2}{q_1+q_2}< p < \frac{1-q_2}{2-q_1-q_2}
\qquad \hbox{for
}\quad q_1 > q_2
\;.\label{sol}\end{aligned}$$ A depolarizing channel is entanglement breaking iff $q \leq 1/2$, where $q$ is the probability pertaining to the identity transformation. This fact can be easily checked by applying the PPT condition [@ppt1; @ppt2] to the Werner state [@ws] $({\cal
E}\otimes I)(|\beta \rangle \langle \beta |)$, where $|\beta \rangle $ denotes the maximally entangled state $|\beta \rangle =\frac{1}{\sqrt
2}(|00 \rangle +|11 \rangle )$. It follows that the solution in Eq. (\[sol\]) for $q_1,q_2 \leq 1/2$ gives examples of situations where a maximally entangled input state strictly improves the distinguishability of two entanglement-breaking channels.
![The grey region represents the value of the a priori probability $p$ for which the discrimination between a depolarizing channel with $q\leq 1/2$ (an entanglement-breaking channel) and a completely depolarizing channel is strictly enhanced by using a maximally entangled input state.[]{data-label="f:fig1"}](entbreak)
In Fig. 1 we plot such a set of solutions for the a priori probability $p$ in the case of discrimination between an entanglement-breaking depolarizing channel with $q_1=q\leq 1/2$ and a completely depolarizing channel $q_2=1/4$.
In conclusion, in the problem of discriminating two quantum operations the relevant object is the map corresponding to the their difference, which is not a completely positive map. Using entangled states at the input of entanglement-breaking channels give output separable states that, however, can be better discriminated since they live in a higher dimensional Hilbert space. Curiously, we note that, on the other hand, when we are asked to optimally discriminate two arbitrary unitary transformations—which are of course entanglement-preserving operations—entanglement never enhances the distinguishability [@1; @2; @3].
*Acknowledgments.* This work has been sponsored by INFM through the project PRA-2002-CLON, and by EC and MIUR through the cosponsored ATESIT project IST-2000-29681 and Cofinanziamento 2003.
[99]{} M. Horodecki, P. W. Shor, and M. B. Ruskai, Rev. Math. Phys. [**15**]{}, 629 (2003). M. B. Ruskai, Rev. Math. Phys. 15, 643 (2003). A. S. Holevo, Russian Math. Surveys [**53**]{}, 1295 (1999). C. King, J. Math. Phys. [**43**]{}, 1247 (2002). P. W. Shor, J. Math. Phys. [**43**]{}, 4334 (2002). C. King, Quant. Information and Computation [**3**]{}, 186 (2003). G. Vidal, W. Dür, J.I. Cirac, Phys. Rev. Lett. [ **89**]{}, 027901 (2002). A. S. Holevo, M. E. Shirokov, and R. F. Werner, quant-ph/0504204. D. Kretschmann and R. F. Werner, New J. Phys. [**6**]{}, 26 (2004). A. S. Holevo and R. F. Werner, Phys. Rev. A [**63**]{}, 032312 (2001). M. E. Shirokov, quant-ph/0408009. For simplicity, we are considering finite dimensional Hilbert space, for which the POVM has rank-one elements. This may not be the case in infinite dimension [@shir2]. C. W. Helstrom, [*Quantum Detection and Estimation Theory*]{} (Academic Press, New York, 1976). M. F. Sacchi, quant-ph/0505183. A. Peres, Phys. Rev. Lett. [**77**]{}, 1413 (1996). M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A [**223**]{}, 1 (1996). R. F. Werner, Phys. Rev. A [**40**]{}, 4277 (1989). A. M. Childs, J. Preskill, and J. Renes, J. Mod. Opt. [**47**]{}, 155 (2000). A. Acín, Phys. Rev. Lett. [ **87**]{}, 177901 (2001). G. M. D’Ariano, P. Lo Presti, and M. G. A. Paris, Phys. Rev. Lett. [**87**]{}, 270404 (2001).
|
---
abstract: 'We describe a new algebraic technique, utilising transfer matrices, for enumerating self-avoiding lattice trails on the square lattice. We have enumerated trails to 31 steps, and find increased evidence that trails are in the self-avoiding walk universality class. Assuming that trails behave like $A \lambda ^n n^{11 \over 32}$, we find $\lambda = 2.72062 \pm 0.000006$ and $A = 1.272 \pm 0.002$.'
author:
- A R Conway
- A J Guttmann
bibliography:
- 'comb.bib'
---
= 10000
Enumeration of self avoiding trails on a square lattice using a transfer matrix technique.
Department of Mathematics,\
The University of Melbourne,\
Parkville 3052,\
Australia
Department of Theoretical Physics,\
Oxford University,\
1 Keble Road,\
Oxford OX1 3NP\
U.K.
v\#1 \#1 \#1[\#1]{} \#1
\#1[\#1]{} \#1
\#1[ ]{}
\#1\#2\#3
\[\#3\]
History
=======
Over the years, the study of the trails problem has provided an interesting counterpoint to the corresponding SAW problem. While self avoiding walks are connected open non-intersecting paths on a lattice, and hence no site or bond may be visited more than once, lattice trails are open paths on a lattice which may re-visit sites, but not bonds. Thus SAWs are a proper subset of trails. First seriously studied by Malakis [@trails:first], a number of exact and numerical results were obtained by Guttmann [@trails:exact; @trails:num].
It has been shown by Hammersley [@saw:ham] that SAWs have a connective constant: that is some value $\mu$ such that if there are $c_n$ SAWs of length $n$, then $\log \mu= lim_{n\to\infty} \log(c_n) / n $ exists and is finite and non-zero. Later, Hammersley and Welsh proved that $c_n=\mu^n \exp(O(\sqrt{n}))$.
These results were carried over to trails by Guttmann[@trails:exact]. If $t_n$ is the number of trails of length $n$, and $\lambda$ denotes the connective constant for trails then $c_n\leq t_n$ and $\mu \leq \lambda$. It was also shown[@trails:exact] that the critical behaviour is in the saw universality class for trails on the honeycomb lattice, no such proof has been found for the square lattice case. The earlier series were found to be rather poorly converged compared to saw series of similar length, with the exponent of the trails generating function being $\gamma \approx 1.40$, compared to the saw result $\gamma = {{43}\over{32}}
= 1.34375$. The connective constant was estimated as $\lambda = 2.7215 \pm 0.002$. This poor convergence prompted Guttmann and Osborn [@trails:montecarlo] to carry out a Monte-Carlo study, using the Berretti-Sokal [@saw:montecarlo] algorithm, using walks up to 200 steps. They found $\gamma = 2.7205 \pm 0.0016$ and $\gamma = 1.348 \pm 0.11$. A biased estimate of the connective constant, assuming $\gamma = 1.34375$, gave the critical point estimate $\lambda =
2.72059 \pm 0.0008$. Recently Lim and Meirovitch [@trails:scan] used an entirely different Monte Carlo algorithm, the scanning simulation method. They obtained the estimates $\lambda = 2.72058 \pm 0.00020$ and $\gamma = 1.350 \pm 0.012$.
In this work we report a substantial extension of the series expansion of the generating function for square lattice trails. The finite-lattice method plus transfer matrices described here allows 31 terms to be obtained on a work station (an IBM 6000/530 with 256MB of memory). The method is described below. The complexity of our algorithm is in fact worse than exponential, compared to $\lambda^{n}$ required by a conventional enumeration algorithm, where n is the maximum number of steps. However for intermediate values of length, say 50-100 steps, it is in fact substantially faster. The detailed performance is discussed in Section \[algcomplexity\]
Algorithm
=========
The algorithm for enumerating self avoiding trails is very similar to the algorithm for enumerating self avoiding walks described in [@saw:saw39]. We will summarise this process, expanding and pointing out the differences between walks and trails where appropriate. Where there is no difference, the word [*paths*]{} is used to denote either walks or trails.
Introduction
------------
The basis of this method is the transfer matrix technique on a finite lattice. This enables one to count the total number of paths on a square lattice (or other type of lattice, with the appropriate modifications). We shall firstly discuss paths that can fit into a finite lattice.
\[algintro\]
The fundamental problem with enumerating self avoiding paths that the self avoiding constraint is non-local. One can’t just say, for instance “There are $x$ number of ways of getting from $(0,0)$ to $(a,b)$ and $y$ number of ways to get from $(a,b)$ to $(c,d)$, so there must then be $xy$ ways to get from $(0,0)$ to $(c,d)$.” However, if we [*could*]{} do something like this, it could save a great deal of time, since $xy$ is typically much larger than $x+y$...that is, it is faster to count $x$ steps followed by $y$ steps than it is to count $xy$ steps.
If one draws a boundary line through the (finite) lattice, one notices that the self avoiding constraint works independently on both sides. This means that it would be possible to work out, for all the possible boundary conditions, how many patterns to the left and right of the boundary there are that give those boundary conditions. A boundary condition is the set of bonds cut by the boundary line, plus a description of their interconnectedness. Thus, one can consider the two partial paths given in figure \[exampbound\] as behaving in exactly the same manner, if all further growth takes place on the right of the boundary. The number of partial paths to the left and right with a given boundary can then be multiplied and summed over all boundaries to give the required number of paths.
Note that we usually do not want the total number of paths on a certain sized lattice, but rather the number of paths of a certain length on that lattice.
To cope with this, instead of just counting the number of partial paths to either side of the boundary, one can count the number of partial paths of $n$ steps, $g_n$, and then make a generating function $G(x)=\sum_{n=0}^\infty g_n x^n$. Then, when one multiplies the generating functions for either side of the boundary, one ends up with a total generating function, from which the number of paths of the appropriate length can be easily extracted.
However, this leaves the tasks of actually counting those paths, and of matching them up. This task can be simplified by noticing that we could add in a second boundary (figure \[twobound\]) separating just a single site.
Now, we have three independent areas. The matching process is however not much more complicated. First, one works out the generating functions for all possible boundary conditions on the left. Secondly, for each of the boundary conditions on the left, one works out which new paths can be created by adding this new point. This will create zero or more possible new paths to the left of the second line, with perhaps different generating functions. When two or more different “first boundary” conditions create the same “second boundary” condition, the generating functions should be added. An example of the new boundary conditions created is shown in figure \[exampnew\]
This process can be represented as matrix multiplication. We start off with a column vector of generating functions, where each element corresponds to a particular (first) boundary condition. We then perform a linear transformation of this column vector to another column vector containing a generating function for each (second) boundary condition. This is the same as multiplying on the left by a rectangular matrix. This is the reason for the name “transfer matrix technique”. Note that the column matrices are very long, so this rectangular matrix will be exceedingly large. However, there is no need to store this matrix as the matrix is very sparse, and elements can be calculated on demand.
This process can add one extra point. But there is no reason that this process can not be continued to add a second point... or a third... or the rest of the lattice, one site at a time. Once the end of the lattice is reached, the boundary conditions are very simple to match: nothing is allowed to cross the end of the lattice. Similarly, there is no reason why one cannot start at the very beginning of the lattice. One then begins with only one initial boundary condition — no connections, and the trivial generating function $1$. Thus one can generate all the possible paths using only the transfer matrix techniques. This means one never has to explicitly count paths. This process is potentially significantly faster than counting paths individually.
Defining Boundary conditions
----------------------------
The boundary specification is called a “signature” and is based on a series of numbers, one for each of the bonds crossed by the boundary line. For a lattice of width $W$ bonds, there will be $W+1$ (vertical line) or $W+2$ (vertical line with a kink) of these bonds.
\[algbound\]
Each bond crossed may be characterised by one of three possibilities. Firstly, it may be unoccupied, in which case it is easy to specify. Assign it the number ‘0’. Secondly, the bond may be occupied, and lead to a dangling end. That is, the pathlet connected to the bond ends somewhere to the left of the boundary. This is also easy to specify — assign it the number ‘1’. Note that these assignments are arbitrary, there is no inherent meaning in this code. The third possibility is that the bond is occupied, and is connected (via some route to the left of the boundary) to some other bond on the boundary. In order to fully specify this, one must somehow uniquely define which bond it is connected to.
The arrangement of possible interconnections is severely limited in the self avoiding walk case, as the pathlets cannot cross. This enables a very efficient encoding: If one labels the ‘top’ of a pathlet with a ‘2’ and the bottom with a ‘3’, then this will uniquely specify the way the bonds are connected. For, if there is a ‘2’ at some point in the signature, one can find the corresponding ‘3’ by moving down the signature until the next ‘3’ is found, subject to the condition that every time a ‘2’ is crossed, one must ignore an extra ‘3’. So for instance, in the signature “232233”, the first ‘2’ matches the first ‘3’, whilst the second ‘2’ matches the last ‘3’. A computer can then store each code number for each bond as two binary bits, so for $W\leq 14$ the whole signature fits nicely into a 32 bit binary word, which is very convenient for current computers.
For self avoiding trails, we do not have this nice restriction. Self avoiding trails can cross themselves at a site, as it is only the bonds that have to avoid one another. This means that one cannot get away with a clever encoding of just two symbols. Indeed, no finite number of symbols will do for all values of $W$, as will be presently shown. This leaves the explicit option, where the code number for each bond specifies the index of the bond to which it is attached (plus 1). The index of a bond is the bond’s position on the boundary — $1$ up to $W+2$ inclusive. The addition of one is to prevent mistaking a dangling end with a connection to the first bond. This means each bond will have a number from 0 to $W+3$ associated with it. If we restrict $W\leq 12$ in a computer program, then each bond fits into 4 bits, and the total signature is 56 bits, which is a little more clumsy to deal with, but not difficult.
Note that the specific numbers mentioned above for restrictions on $W$ are in no way restrictions on the algorithm, just on a particular programming implementation. We only used $W=7$ anyway, due to finite computer resources.
Paradoxically, the “implicit” coding of the self avoiding walk boundaries is significantly easier for a computer program to deal with than the “explicit” encoding for the self avoiding trails. This is due to the fact that most operations deal with local changes, when adding a site. These local changes are easy to implement with the implicit coding.
Note that in [@saw:saw39] we used a permutation of the numbering system described above for SAWs: we used ‘3’ as the dangling end marker, and ‘1’ and ‘2’ for the loop ends. We have changed notation here for consistency.
The coding of the signatures are actually of vital importance, as the total time and memory requirements of the algorithm are polynomials in $W$ times the number of different signatures ($W^2$ for space, and $W^4$ for time), as discussed in section \[algcomplexity\]
For self avoiding walks, an upper bound on the number of different signatures is obvious: $4^{W+2}$, as there are $4$ different possibilities for each bond, and $W+2$ possible bonds. Actually, there are significantly fewer than this since not all combinations are possible: one can’t have more than two floating ends (as it would be impossible to make them into a connected walk), and one can’t end a loop with a $3$ before starting it with a $2$. It turns out that the number of possible signatures grows like a polynomial in $W$ times $3^W$.
For self avoiding trails, the situation is again worse. The number of possible boundary conditions can be evaluated exactly in a straight forward manner. Let $B_n$ be the number of boundary conditions for a strip of width $n$, and $L_n$ the number of boundary conditions [*excluding “dangling” paths*]{}. That is, all of the $n+2$ bonds on a boundary condition counted in $L_n$ must either be unused bonds, or may be connected to another bond on the boundary. In [@saw:saw39] these were marked by a ‘3’. A formula for $B_n$ in terms of $L_n$ is easy to obtain: any boundary counted in $B_n$ may have no dangling ends (giving a term $L_n$), or it may have one dangling end in any of $n+2$ places (giving a term of $(n+2)L_{n-1}$), or it may have two dangling ends giving a term of $(n+2)(n+1)L_{n-2}$. Thus we have $$B_n=L_n+(n+2)L_{n-1}+(n+2)(n+1)L_{n-2}
\label{defB}$$ Now to work on an equation for $L_n$: The first bond may be unoccupied, giving a term of $L_{n-1}$, or it may be connected to one of $n+1$ other bonds, giving a term of $(n+1)L_{n-2}$. Thus $$L_n=L_{n-1}+(n+1)L_{n-2}
\label{defL}$$ Initial conditions are $L_{-1}=1$, $L_{-2}=1$ and $L_{-3}=0$.
These are worrying equations as they grow faster than exponentially due to the $(n+1)L_{n-2}$ term in equation (\[defL\]). This faster than exponential growth is the reason why no finite set of symbols could cope with encoding the connections for all values of $W$.
Actual values are given in table \[BLtable\], along with the number $s_n$ (taken from [@saw:saw39]) of possible boundary conditions for self avoiding walks.
Irreducible Components
----------------------
Directly using the transfer matrix method is not as much of a saving as could be expected, due to the very large number of vectors. If we want to count all paths up to a maximum length of $2n+1$, then at first it looks as though a square $2n+1$ wide is needed to cope with a perfectly vertical path. However, it is possible to use the symmetry relation between the horizontal and vertical axes, so that only paths up to a width of $n$ need to be calculated: for paths of width $n+1$ or greater, we can say that they must have height $n$ or less, and thus their mirror images will have already been counted. More formally, if $G_{ij}$ is the number of paths with $i$ horizontal and $j$ vertical steps, then $G_{ij}=G_{ji}$ and thus if we know $G_{ij}$ for $i\leq 2n+1$ and $j\leq n$, we really know $G_{ij}$ for all $i+j\leq 2n+1$. That is, we know the total number of paths of length up to $2n+1$ steps. This means that we could work with strips of width $n$, length $2n+1$ and obtain coefficients up to and including $2n+1$.
There is still a further improvement. Suppose that we break up all the paths (of vertical steps $\leq n=2M+1$) into two classes:
[**Irreducible**]{} paths have no place where a horizontal line could be drawn across the lattice intersecting exactly one vertical bond. As there are a maximum of $2M+1$ vertical bonds in the path, and there must be at least two vertical bonds for each unit of width (to satisfy the irreducibility definition), these paths must all fit into a strip of width $M$ bonds.
[**Reducible**]{} paths have at least one place where the horizontal line can be drawn, intersecting just one bond of the path. These paths have the nice property that the self avoiding constraint will act independently both above and below this line. All that is needed is to calculate the number of self avoiding paths above and below independently. This is a smaller problem, and indeed, can be further split up, until the entire path can be considered to be made up of an irreducible “top” section, then one or more sections composed of a vertical bond and an irreducible “middle” section, then finally a vertical bond and an irreducible “bottom” section. As all of these irreducible subsections will have fewer than $n$ steps, they will fit into a strip of width $M$.
All in all, these two optimisations allow calculations on a strip of width $M$ bonds to provide the number of paths with widths up to $n=2M+1$ and thus paths with total number of bonds up to $2n+1=4M+3$. Since the number of partial generating functions rises exponentially with strip width, these two optimisations reduce the complexity of the problem enormously.
However, it makes the counting task a little more difficult: we have to extract these “top”, “middle” and “bottom” sections individually. To facilitate this, the irreducible paths can be named as described below, based upon their starting and end points. Note that a distinction is made here between paths and routes. A [**path**]{} has a specific starting point: a [**route**]{} does not. This means that there are exactly half as many routes as paths.
\#1
$$
Name
-- -- ------ ------------
P n/a
Q $w^{2M}$
R $w^{2M}$
S $w^{2M+1}$
T $w^{3M}$
$$
Note that routes with two bottom ends are not included, as they are the same (in number and shape) as $R$, and similarly routes with one bottom end and one middle end are not given a name as they are covered by $S$. Note that all the routes above are irreducible.
The name in this table is the name of the generating functions associated with that variable in this paper. There are six generating functions associated with each letter in this paper, as per the following pattern:
- $Q(u,w)$ is the generating function for irreducible routes of the required shape with the power of $u$ giving the number of horizontal bonds, and $w$ representing vertical bonds.
- $Q_W(u,w)$ is the same, except only for those irreducible routes of width exactly $W$.
- $Q^*(u,w)$ is the generating function for [*all*]{} (i.e. both reducible and irreducible) routes of the required shape.
- $Q^*_W(u,w)$ is the same, except for all routes with width exactly $W$.
- $Q(u,w,z)$ is the generating function for irreducible routes of the required shape with the power of $u$ giving the number of horizontal bonds, $w$ representing vertical bonds, and $z$ the total width.
- $Q^*(u,w,z)$ is the generating function for all routes of the required shape with the power of $u$ giving the number of horizontal bonds, $w$ representing vertical bonds, and $z$ the total width.
Note that the same terminology applies to variables other than $Q$, with [*routes*]{} changed to [paths]{} where appropriate. The three variable generating function is the most general: the width $W$ generating functions can be extracted from the appropriate power of $z$, and the generating functions in two variables can be produced from the functions in three variables by setting $z=1$. That is, $$Q(u,w)=Q(u,w,1)$$ $$Q(u,w,z)=\sum_{n=0}^{\infty} z^n Q_n(u,w)$$
Note that if a path is on a strip the width of which is too small for the definition to make sense, then the corresponding generating function is zero: i.e. $Q_0$, $Q_1$, $R_0$, $S_0$, $S_1$, and $T_0$ are all zero.
Of these five functions, $P$ is easy to determine. There is one horizontal path of length zero, and two paths of every other length (one in each direction). Thus $$P(u,w,z)=1+2u+2u^2+2u^3+....={{1+u}\over{1-u}}$$
Now define another variable, $X$. This will represent the total number of irreducible middle sections. That is, the number of ways of going from a point at the bottom of an irreducible section to a point on the top. Note that every element of $T$ can be considered as a path [*restricted so as to not go below the starting point*]{}. Thus, $T$ copes with all the parts of $X$ of width at least one. For the zero width case, we just want paths from one point on a line to another point $P$. Thus $$X=T+P.$$ This is the reason for defining $P$ to be paths, whilst $Q$, $R$, $S$, and $T$ are routes.
This is a typical $X$: , and this is the corresponding $wzX$:
$X$ refers to just a single irreducible middle section. This can be extended to an arbitrary middle section by noting that a “middle section” can be formed from either a vertical bond ($wz$), or two vertical bonds with an $X$ in between, ($wzXwx$), or any number of extra $wzX$ terms. Define a new variable $V$ to be a total (reducible) “middle section”, then $$V=wz\left(1+wzX+(wzX)^2+(wzX)^3+...\right)={{wz}\over{1-wzX}}.
\label{eqnV}$$
Note that the top and bottom of a $V$ are [*always*]{} vertical bonds, so a $V$ can attach to [*any*]{} irreducible component which has an end at its top or bottom. This can be a $P$, an $R$, an $S$ or a $T$. Note that the $R$ has two ends to which connections can be made, so we must count it twice. $P$ is not counted twice since it is a path, not a route. Define the generating function of end components, $E$ as $$E=P+2R+S+T.$$
Now all the reducible routes can be calculated. Each consists of one end piece, $E$, a joint $V$ and another end piece $E$. Thus reducible routes are $EVE$. Irreducible routes (with some vertical component, i.e. not $P$) are $Q+2R+2S+T$. $R$ and $S$ are counted twice to allow for routes with two bottom ends or one bottom and one middle end respectively. To get the total number of paths then, we take the number of paths with no vertical component, $P$, and add in twice the number of routes with vertical components. This gives $$C=P+2(Q+2R+2S+T+EVE)$$ as the total number of paths.
Obtaining the irreducible components
------------------------------------
So far only $P(u,w,z)$ is known. In order to calculate the number of self avoiding paths up to length $4M+3$, $Q(u,w)$, $R(u,w)$, $S(u,w)$ and $T(u,w)$ must be known accurate to $u^{4M+3}$ and to $w^{2M+1}$.
Suppose that it were possible to obtain the starred polynomials $Q^*$, $R^*$, $S^*$ and $T^*$ as functions of three variables. Then $R=R^*$, as all paths starting from the top and ending at the top are irreducible.
Calculating the others is a little more difficult. Consider the generalisation of $X$ to $X^*$. $X^*$ will be equal to the sum of the irreducible parts $X$, plus reducible paths starting at the bottom and ending at the top. These are expressible as $XVX$, so we have $X^*=X+XVX$. Using equation \[eqnV\], this can be inverted to give $$X={{X^*}\over{1+wzX^*}}
\label{eqnXXstar}$$ which can be expanded in a formal binomial series to give $$X=X^* \left( 1 - wzX^* + w^2 z^2 X^{*2} - ... \right)$$ If $X^*$ is known to some order in $u$ and $w$ for powers up to $z^M$, then $X$ can be determined to the same order. Since $X$ is made up of $P$ (which is zero for widths other than 0), and $T$, which has the lowest power of $w$ being three times the power of $z$, order is preserved up to $w^{3M+2}$ and to the original order in $u$. Thus, if $X^*$ is known to $u^{4M+3}$ and $v^{2M+1}$, this is preserved in the calculation of $X$. So, by using the third variable, one can go from $X^*$ to $X$, and thence $T$. Without using the third variable $z$, the generating function $X^*$ would only be correct to terms of order $w^M$ rather than $w^{2M+1}$.
Similarly, if we define $Y=2R+S$ (connections at the bottom, but not the top), then $Y^*=2R^*+S^*=2R+S+XVY=Y(1+XV)$, so there is an expression for $Y$ similar to equation (\[eqnXXstar\]), $$Y={{Y^*}\over{1+XV}}$$ One can then obtain $Y$ and thence $S$ from $Y^*$ and hence $S^*$, in a manner similar to that used to obtain $T$ from $T^*$ via $X$ and $X^*$.
Lastly, $Q^*=Q+YVY$ so $$Q=Q^*-YVY$$ and $Q$ can also be obtained in a similar manner.
\[usered\]
This means that all the irreducible components can be obtained from reducible components given the full three variable information, and accuracy to
- $M$ in $z$ (i.e. to width $M$)
- $2M+1$ in $w$
- $4M+3$ in $u$.
Obtaining reducible components
------------------------------
Suppose that we could count all the paths on a certain finite lattice with constraints upon where the paths can start or end. Define the generating function in variables $u$ to order $4M+3$ and $w$ to order $2M+1$ for paths on a strip of width $K$ as $G_K(a,b,c)$, where $a$, $b$, and $c$ are $+$ or $-$ depending upon whether one can start or end paths on the top of the strip, the bottom of the strip, and/or the middle of the strip respectively. Ensure that all paths included in these generating functions start flush at the left of the lattice so that we do not need to worry about uniqueness in the horizontal direction.
\[getred\]
Now, by considering how the walks that fit into the strip can be made up of the reducible functions defined above, the latter can be defined as an invertible linear combination of the former. One inverts this relation and gets the reducible components needed in section \[usered\] from the $G_K(+,-,-)$, $G_K(+,+,-)$, $G_K(-,-,+)$ and $G_K(+,-,+)$, for $K$ from $0$ to $M$.
These relations are (as taken from [@saw:saw39]) $$\begin{aligned}
R_{m}& =& G_{m}(+,-,-) - G_{m-1}(+,-,-) \\
Q^{*}_{m}& =& G_{m}(-,-,+) - G_{m-1}(+,-,+) - \sum_{n=1}^{m-1}
\left(Q_n^*+R_n+S_n^*\right) \\
S^{*}_{m} &=& G_{m}(+,-,+) - G_{m-1}(+,-,+) - G_{m}(+,-,-) -
G_{m-1}(+,-,-) \\
&& \quad - {{P-1}\over 2} -Q_m^* - \sum_{n=1}^{m-1}
\left(Q_n^*+2*S_n^*+T_m^*\right) \\
T^{*}_{m}& =& G_{m}(+,+,-) - 2G_{m}(+,-,-) \\\end{aligned}$$
Counting paths on strips
------------------------
The transfer matrix technique can be used to obtain the generating functions $G_K$ that are needed in section \[getred\].
Suppose we are working on a lattice of width $W$ and length $4W+3$. As mentioned before, one starts with one partial generating function (boundary to the left of the entire lattice, no bonds used, generating function 1). Then add on sites as described in the next paragraph one at a time, working along the matrix column by column. At each site one stores for each valid signature the partial generating function. After processing the first column, one can remove the signature with no bonds occupied, as any animal based upon this signature will not lie flush against the left of the lattice, and by removing it we satisfy the horizontal uniqueness criterion.
To process a site, one cycles through all the stored signatures, processing each individually, creating a new set of signatures. Note that two or more signatures may produce the same signature after processing. In this case the partial generating functions for these two signatures should be added.
All that is left is to describe exactly what to do when each site is added for a particular signature. The site that is being added will have two bonds coming in (to the left of the new boundary), and another two bonds leaving (to the right).
One must firstly see if the walk can be finished at this point, and if so, add in the partial generating function to a total generating function which will give the final $G_K$ once all sites have been processed. In order to be able to accumulate a partial walk, two conditions must be satisfied. Firstly, there must be no occupied bonds in the signature other than those coming into the bond being processed. Secondly, one of the three following conditions must hold:
- There must be a single dangling end coming in to the site being processed (type ‘1’ in the signature coding), and it is valid to start or stop a path at this point (determined by the $+$ or $-$ parameters in the particular $G_K$ being computed.
- Or there may be two dangling ends that connect at this site.
- Or (only in the case of trails) there may be a loop completed at this site and it is valid to start or stop a path at this point.
We will first discuss the possibilities for the new signatures if one cannot start or stop a path at the site being processed.
If one is counting walks, and there is only one bond going into the site, then that one bond must emerge from either of the two bonds coming out of the site. This gives two new signatures, one with the old generating function multiplied by $w$ (emerging vertically), and one with the old generating function multiplied by $u$ (emerging horizontally). In future we will not mention these multiplications.
Again for walks, one may have both bonds entering the site occupied. In this case neither output bond may be occupied, as one can’t have more than two occupied bonds touching a site for self avoiding walks. What happens depends upon the specific case. If the two bonds are attached together, then a loop has been formed which is illegal, so no signatures are generated. If the two bonds are dangling ends, then attaching them would make an entire dangling path, which is not allowed. In the remaining cases, one does produce a new valid signature, and one must adjust the coding for the bond(s) in the signature to which the just processed bonds were attached.
Again for walks, if there are no bonds coming in, then there are two possibilities: no bonds coming out, or a new path being started at this point – that is two bonds coming out and connected to each other.
Further possibilities exist if one can stop or start from the site being processed.
For walks with no bonds coming in, one can now have one dangling end coming out either of the two outgoing bonds. With one bond coming in which is not a dangling bond, the pathlet it belongs to can be terminated at this site, and the bond to which it is attached elsewhere in the signature becomes a dangling bond. Note that each of these steps increases the number of dangling bonds in the signature, and one must check that the total number of dangling bonds does not exceed two, as this would mean that any path one tries to construct must have at least three ends!
These are summarised in table 2 of [@saw:saw39].
For trails, the situation is significantly more complicated, as bond loops and crossings are allowed, but the basic idea remains the same.
First, consider what can be done without starting or stopping.
The same possibilities as in the walks case (without stopping or starting at the site being processed) hold, with some extra possibilities when there are two bonds coming in. Firstly, both bonds could “bounce” and come out as two bonds with the same connections. Secondly, they could cross, and come out as two bonds with interchanged connections. Thirdly, if the two bonds coming in meet, and in the walks case would have produced no bonds coming out, one may also have two new connected bonds coming out, as occurred in the walks case when no bonds went in.
If one is allowed to start or stop at the site being processed, things get much more complicated. The actions can best be described by two stages.
In the first stage, associated with terminating incoming pathlets, one forms all the possibilities already described, and adds in the following possibilities:
- For one bond entering which is not an dangling end, the pathlet may be terminated at this site, and the other end of the pathlet converted to a dangling end (as was done for walks). No occupied bonds emerge.
- For two bonds entering, one a dangling end, and the other a pathlet, the pathlet may terminate (making the other end of the pathlet a dangling bond) and the dangling end can continue from either of the two new bonds.
- For two ends of the same pathlet entering, one end may terminate at the current site, and the other end (now a dangling end) may take either of the two new bonds. As either end of the pathlet may terminate, there are four new signatures produced.
- For two ends of different pathlets entering, there are the same four possibilities as above, except that this time it is a pathlet leaving, not a dangling end, and some other bond in the signature will become a dangling end. A fifth possibility is for both incoming pathlets to terminate, producing two dangling ends elsewhere in the signature and no bonds coming out.
In the first and last case above, there is the possibility of no bonds coming out. Again, one can add a new two bond loop in both cases as in the walks case when no bonds went in.
The second stage is associated with adding dangling ends at the leaving stage. If any of the signatures formed from the first stage have either or both of the outgoing bonds unoccupied, either or both may be filled with dangling ends.
\[calcG\]
Of course, when forming new dangling ends, one must remember the constraint that the total number of dangling ends in the signature may not exceed two.
Algorithm complexity
--------------------
One now has all the ingredients for the algorithm. One uses the transfer matrix technique to get all the $G_K$ terms for $K$ going up to some value $W$ (\[calcG\]), then obtain the reducible generating functions (\[getred\]) and thus obtain the irreducible generating functions and final answer (\[usered\]).
\[algcomplexity\]
Of these three stages, the first (\[calcG\]) is exceedingly time and memory consuming, whilst the second (\[getred\] and \[usered\]) is fast (polynomial in $W$ time) and uses little memory.
Since the first stage is the bottleneck, we shall discuss it exclusively in terms of complexity.
The total memory required will be bounded by the number of possible boundary conditions, multiplied by the total space per generating function (proportional to $W^2$), multiplied by two, since one may need to store both the incoming and outgoing partial generating function. In practice, this last factor is nowhere near as high as two, since as soon as a signature has been fully processed, the data associated with it may be discarded.
The total time required is proportional to the total amount of memory that needs to be processed (as above) times the number of sites that have to be processed (proportional to $W^2$), times the average number of new signatures per old signature. This last factor is pretty much independent of $W$. For trails it is significantly larger than walks.
The basic result is that the time and memory requirements are a small polynomial times the number of boundary conditions. The number of boundary conditions is therefore the most significant factor in the complexity of this algorithm.
For self avoiding walks, the number of boundary conditions grows like a polynomial in $W$ times $3^W$. Thus the dominant complexity of this method for self avoiding walks is $3^{n\over 4}$, where $n$ is the number of steps required. This comes from the fact that $n=4W+3$. The alternative, direct enumeration, grows like $\lambda^n$, where $\lambda$ is the connective constant for self avoiding walks. Note that $\lambda$ is significantly greater than $3^{1\over 4}$ (approximately twice $3^{1\over 4}$ in fact), so this algorithm is exponentially faster than direct enumeration.
For trails, the situation is not as good. The analysis in section \[algbound\] shows that the number of boundary conditions grows faster than exponentially. Thus, for very long trails, direct enumeration will be a more efficient algorithm! However, consulting table \[BLtable\] shows that trails are not all that much worse than walks for small values of $W$. So for small values of $W$, this transfer matrix method is actually more efficient than direct enumeration. Fortunately, the values of $W$ for which this algorithm is faster than directed enumeration are such that this algorithm is faster for $n$ at least 50, which is far beyond the capacity of current computers.
This algorithm is also amenable to parallelisation in the same manner as the self avoiding walk algorithm described in [@saw:saw39].
This algorithm was implemented in a C program using modular arithmetic, and was used to obtain trails of up to 31 steps. They are given in table \[tabenum\].
Analysis of series
==================
The method of analysis used is based on first and second order differential approximants. It was used in previous papers [@alg:critwalk; @saw:saw29; @saw:saw39] in which the related saw problem was studied, and is described in detail in [@alg:diff]. In summary, we construct near-diagonal inhomogeneous differential approximants, with the degree of the inhomogeneous polynomial increasing from 1 to 8 in steps of 1. For first order approximants (K=1), 12 approximants are constructed that utilise a given number of series coefficients, N. Rejecting occasional defective approximants, we form the mean of the estimates of the critical point and critical exponent for fixed order of the series, N. The error is assumed to be two standard deviations. A simple statistical procedure combines the estimates for different values of N by weighting them according to the error, with the estimate with the smallest error having the greatest weight. As the error tends to decrease with the number of terms used in the approximant, this procedure effectively weights approximants derived from a larger number of terms more heavily.
For second order approximants (K=2), 8 distinct approximants are constructed for each value of N. We find that as the number of series terms increases, the estimate of the critical exponent decreases. We show below that this is due to rather strong correction-to-scaling terms, much stronger than for the saw case. Because of this, the estimates we quote below should be treated as over estimates of the exponent and critical point. $$x_c = 0.367597 \pm 0.00002
\quad \gamma = 1.352 \pm 0.01 \quad (K=1)$$ $$x_c = 0.3676 \pm 0.0001
\quad \gamma = 1.348 \pm 0.008 \quad (K=2)$$
These results provide some support for the view that the trails are in the saw universality class. The critical point estimate can be refined if we assume that $\gamma = 1.34375$ exactly, which is the saw value. To refine the estimate of the critical point, linear regression is used. There is a strong correlation between estimates of the critical point and critical exponent. This is quantified by linear regression, and in this way the biased estimates (biased at $\gamma = 43/32$) are obtained.
We find $$x_c = 0.367564 \pm 0.000008 \quad (K=1)$$ $$x_c = 0.367562 \pm 0.000007\quad (K=2)$$
These are combined to give our best estimate for the connective constant $\lambda = 1/x_c = 2.72062 \pm 0.00006$, which is in agreement with previous estimates, but rather more accurate than any previous estimate.
The much slower rate of convergence of the trails series critical point estimates compared to the corresponding saw estimates is presumably due to stronger “correction-to-scaling” terms. We have investigated this possibility using three different methods. Firstly, we used the method of Baker and Hunter [@saw:transform] which transforms the series so that poles of the Padé approximants to the transformed series furnish estimates of the reciprocals of the exponents. However we found that the singularity on the negative real axis at $-x_c$ masked the presence of any confluent singularity at $x_c$. Accordingly, we split the series in two, treating the odd and even subsequences as independent series. In this way, we found exponents with the values $\approx 1.35$ and $\approx 1.0$ from the even sub-sequence. The smaller exponent was not well identified however. This implies a correction-to-scaling exponent of $\approx 0.35$. The odd subsequence gave no evidence of any exponent apart from the leading one.
The next method we used was the method of Adler et al. [@saw:scaling], in which a correction-to-scaling exponent is assumed, and then a transformation is applied which maps this non-analytic correction term to an analytic correction term. Padé analysis of the transformed series should then give the correct leading exponent. We tried various values of the correction to scaling exponent, and found that a value around $0.75$ resulted in a series which gave the correct critical exponent of $\gamma = 1.34375$.
The third method is the same as that used in our recent study of saws [@saw:saw39]. In that method we [*assume*]{} the correction to scaling exponent, and fit the series coefficients to the assumed form. The fit is judged reasonable if the sequences of amplitude estimates appear to converge well. This is not a particularly sensitive method, but is useful in that it does provide amplitude estimates as well as. From the two values of the correction-to-scaling exponent found above, we tried an intermediate value of $0.5$. Given that the s.a.w. exponent appears to be $1.5$, this seemed a reasonable thing to try. As well as the correction-to-scaling term, there is another singularity on the negative real axis. For saws, Guttmann and Whittington [@saw:negroot] showed that this was at $x = -x_c$. That proof applies [*mutatis mutandis*]{} to trails. We assume that universality of exponents applies to non-physical singularities also - a result supported by our series analysis. Then the singularity on the negative real axis will also have the same exponent as the energy at the physical singularity - as for saws - and so we expect the generating function for trails to behave like $$\begin{aligned}
T(x) &=& \Sigma t_nx^n \sim A(x)(1-\lambda x)^{-43/32}[1 +
B(x)(1-\lambda x)^{\Delta} ... ] \\
&& \quad + D(x)(1+\lambda x)^{-1/2}.\end{aligned}$$
The exponent for the singularity on the negative real axis reflects the fact that, as noted above, that term is expected to behave as the energy, and hence to have exponent $\alpha - 1$, where $\alpha = {{1}\over{2}}$. From the above, it follows that the asymptotic form of the coefficients, $c_n$, behaves like: $$t_n \sim \lambda^n[a_1n^{11/32} + b_1n^{11/32-\Delta }
+ (-1)^nd_1n^{-3/2}] \label{ampest}$$
The three amplitudes, $a_1, b_1, d_1$ come from the leading singularity , the correction-to-scaling term and the term on the negative real axis respectively. A small program written in Mathematica was used to fit successive triples of coefficients, $ c_{n-2}, c_{n-1}$ and $c_n$ for $n$ = 6,7,8,...,31. The results (with $\Delta = {1 \over 2}$) are shown in Table \[amp1\]
At first sight, these appear to be converging rather well. Closer inspection reveals that the sequences have a turning point at around n=29. We next tried a higher value of $\Delta$, choosing $\Delta = 0.75$ in agreement with the prediction of the transformation method of Adler et al. cited above. The results are shown in Table \[amp2\]
These sequences of amplitudes appear to be converging reasonably well, and support the earlier finding that the correction-to-scaling exponent is around $0.75$. If this is correct, we can extrapolate the above sequences and find $a_1 = 1.272 \pm 0.002$, $b_1 = -0.32 \pm 0.02$ and $d_1 = 0.035 \pm 0.004$. Even if the correction to scaling exponent were not as estimated, the leading amplitude is still likely to be within the quoted range.
We would like to thank Ian G Enting for introducing us to the finite lattice method. One of us (A.R.C.) would like to thank the A.O. Capell, Wyselaskie and Daniel Curdie scholarships. The other (A.J.G.) would like to thank the ARC for financial support.
----- -------- --------- --------
$n$ $L_n$ $B_n$ $s_n$
-1 1 2 2
0 2 5 5
1 4 13 13
2 10 38 37
3 26 116 106
4 76 382 312
5 232 1310 925
6 764 4748 2767
7 2620 17848 8314
8 9496 70076 25073
9 35696 284252 75791
10 140152 1195240 229495
----- -------- --------- --------
: Number of boundary conditions for trails ($B_n$) and for SAWs ($s_n$). Values for $s_n$ come from[@saw:saw39].
\[BLtable\]
----- -----------------
$n$ $t_n$
0 1
1 4
2 12
3 36
4 108
5 316
6 916
7 2628
8 7500
9 21268
10 60092
11 169092
12 474924
13 1329188
14 3715244
15 10359636
16 28856252
17 80220244
18 222847804
19 618083972
20 1713283628
21 4742946484
22 13123882524
23 36274940740
24 100226653420
25 276669062116
26 763482430316
27 2105208491748
28 5803285527724
29 15986580203460
30 44028855864492
31 121187822490084
----- -----------------
: Numbers of trails $t_n$ of $n$ steps
\[tabenum\]
----- -------- --------- --------
$n$ $d_1$ $b_1$ $a_1$
21 0.0289 -0.1805 1.2795
22 0.0309 -0.1833 1.2801
23 0.0296 -0.1849 1.2805
24 0.0311 -0.1868 1.2809
25 0.0306 -0.1874 1.2810
26 0.0313 -0.1885 1.2812
27 0.0310 -0.1889 1.2813
28 0.0314 -0.1894 1.2814
29 0.0315 -0.1893 1.2814
30 0.0316 -0.1894 1.2814
31 0.0319 -0.1890 1.2813
----- -------- --------- --------
: Sequences of amplitude estimates assuming $\Delta = {1 \over 2}$ Refer equation (\[ampest\])
\[amp1\]
----- -------- --------- --------
$n$ $d_1$ $b_1$ $a_1$
21 0.0281 -0.2544 1.2661
22 0.0316 -0.2615 1.2668
23 0.0289 -0.2670 1.2673
24 0.0318 -0.2727 1.2679
25 0.0298 -0.2765 1.2683
26 0.0321 -0.2809 1.2687
27 0.0303 -0.2878 1.2690
28 0.0321 -0.2902 1.2692
29 0.0309 -0.2902 1.2695
30 0.0323 -0.2930 1.2697
31 0.0313 -0.2949 1.2698
----- -------- --------- --------
: Sequences of amplitude estimates assuming $\Delta = {3 \over 4}$ Refer equation (\[ampest\])
\[amp2\]
|
---
abstract: |
We show that the first law of the black hole thermodynamics can lead to the tunneling probability through the quantum horizon by calculating the change of entropy with the quantum gravity correction and the change of surface gravity is presented clearly in the calculation. The method is also applicable to the general situation which is independent on the form of black hole entropy and this verifies the connection of black hole tunneling with thermodynamics further. In the end we discuss the crucial role of the relation between the radiation temperature and surface gravity in this derivation.** **
PACS classification codes: 04.70.Dy, 04.60.-m
Keywords: Black hole; Tunneling; Quantum horizon; Radiation temperature
author:
- 'Baocheng Zhang$^{a,b}$'
- 'Qing-yu Cai$^{a}$'
- 'Ming-sheng Zhan$^{a}$'
title: '**Hawking radiation as tunneling derived from Black Hole Thermodynamics through the quantum horizon**'
---
**Introduction**
================
About 30 years ago, Hawking discovered [@swh75] that when considering quantum effect black holes could radiate particles as if they were hot bodies with the temperature $\kappa/2\pi$ where $\kappa$ was the surface gravity of the black hole and explained [@hh76] the particles of radiation as stemming from vacuum fluctuations tunneling through the horizon of the black hole with Hartle together. But the semiclassical derivation of Hawking based on the Bogoliubov transformation didn’t have the directly connection with the view of tunneling. Parikh and Wilczek [@pw00] calculated directly the particle flux from the tunneling picture and made the tunneling physical explanation holds firm basis. In their consideration the energy conservation played a fundamental role and the outgoing particle itself created the barrier [@pw04]. After this, there have been some works which have extended the Parikh-Wilczek tunneling framework to different cases [@rzg05; @jwc06] and the question of information loss has been discussed in this framework [@mv05; @mv052]. Recently the general approach has been suggested [@sk08] for the tunneling of matter from the horizon by using the first law of thermodynamics or the conservation of energy. On the other side the tunneling probability has also been calculated [@tp08] directly through the change of the entropy that is proportional to area by the first law of thermodynamics, which verifies the connection of black hole radiation with thermodynamics [@swh76] further.
We have noticed that when the quantum gravity effect is considered the tunneling formula can also be obtained by Parikh-Wilczek method and the Hawking temperature relation [@amv05; @ma060; @ma06]. In this paper we will proceed this kind of consideration by using the same method as in Ref. [@tp08] but for the entropy which is modified by the logarithmic term caused by quantum gravity effect as in Ref. [@amv05]. In the new method we show clearly the necessary change of the surface gravity when considering the quantum gravity effect and the crucial role which the Hawking temperature relation plays. We note that the method could be extended to general situation where the tunneling probability is obtained by calculating the change of entropy, independent on the form of the entropy, from the first law of black hole thermodynamics. The generalization verifies the connection of black hole tunneling with thermodynamics further.
In this paper we take the unit convention $k=\hbar=c=G=1$.
The first law of black hole thermodynamics and entropy
======================================================
The first law of black hole thermodynamics [@ht99] states:
If one throws a small amount of mass into a static non-charged and non-rotated black hole, it will settle down to a new static black hole [@ep]. This change can be described as $dM=\frac{\kappa}{8\pi}dA$, which is analogue to the usual first law of thermodynamics $dM=TdS$. The case is the same for radiating a small amount of mass from black hole [@swh76].
According to Hawking, the temperature of black hole is taken as $T=\frac
{\kappa}{2\pi}$, so the entropy can be obtained as $S=\frac{1}{4}A$. It has been shown [@tp08] that the tunneling formulas for static, spherically symmetric black hole radiation are obtained by the first law of thermodynamics and the area-entropy relation, even if the radiation temperature is different from the Hawking temperature. From the first law of black hole thermodynamics, we can see that if the black hole temperature is changed, the area-entropy relation will also be changed. Note that in Ref. [@tp08] the author calculated the tunneling probability by using the entropy being proportional to horizon area and so the temperature was also proportional to the surface gravity. But when considering the entropy which is modified by the logarithmic term due to quantum gravity effect [@amv05], it looks as if the black hole temperature were not proportional to the black hole surface gravity. Then in such situation, could the tunneling probability be obtained by calculating the change of entropy with log-area term modification when considering the quantum gravity effect in the same way as in Ref. [@tp08]? The answer is positive! Before discussing this problem, we will first present the method proposed by Pilling.
Thermodynamics and tunneling
============================
In this section we will review the method, presented in Ref. [@tp08], which is used to obtain the tunneling probability directly from black hole thermodynamics. Let us start by writing the metric for a general spherically symmetric system in ADM form [@kw95], $$ds^{2}=-N_{t}(t,r)^{2}dt^{2}+L(t,r)^{2}[dr+N_{r}(t,r)dt]^{2}+R(t,r)^{2}d\Omega^{2}.$$ The metric is used for the situation where the geometry is spherically symmetric and has a Killing vector which is timelike outside the horizon. Specially one can consider the case of a massless particle and fix the gauge appropriately ($L=1,R=r$) which is particularly useful to study across horizon phenomena. So the metric becomes $$ds^{2}=-N_{t}(r)^{2}dt^{2}+[dr+N_{r}(r)dt]^{2}+r^{2}d\Omega^{2}, \label{me}$$ The metric is well behaved on the horizon and for a four dimensional spherically Schwarzschild solution, $N_{t}=1,N_{r}=\sqrt{\frac
{2M}{r}}$ ($M$ is the mass of the black hole), for a four dimensional Reissner-Nordstrom solution, $N_{t}=1,N_{r}=\sqrt{\frac{2M}{r}-\frac{Q^{2}}{r^{2}}}$ ($M$ is the mass and $Q$ is the charge of the black hole). And we also note that for $N_{t}=\sqrt{\frac{f(r)}{g(r)}},N_{r}=f(r)\sqrt
{\frac{1-g(r)}{f(r)g(r)}}$, the metric (\[me\]) becomes the same as that in Ref. [@tp08].
Now let us consider the Parikh-Wilczek tunneling [@pw00]. Supposed the mass of the black hole is fixed and the mass is allowed to fluctuate, then the shell of energy $E$ travels on the geodesics given by the line element (\[me\]). Taking into account self-gravitation effects, the outgoing radial null geodesics near the horizon are given approximately by $$\overset{.}{r}=N_{t}(r)-N_{r}(r)\simeq(N_{t}^{^{\prime}}(R)-N_{r}^{^{\prime}}(R))(r-R)+O((r-R)^{2}), \label{rng}$$ where the horizon, $r=R$, is determined from the condition $N_{t}(R)-N_{r}(R)=0$ and the last formula is the expansion of the radial geodesics in power of $r-R$.
According to the definition of a time-like Killing vector the surface gravity of the black hole near the horizon is obtained as $$\kappa\simeq N_{t}^{^{\prime}}(R)-N_{r}^{^{\prime}}(R). \label{sg}$$
So the radiation temperature is $$T=\frac{\kappa}{2\pi}=\frac{N_{t}^{^{\prime}}(R)-N_{r}^{^{\prime}}(R)}{2\pi}.
\label{rt}$$
Now we consider the black hole thermodynamics in the region near the horizon. The change of the Bekenstein-Hawking entropy, if the mass of black hole changes from $M_{i}$ to $M_{f}$, is given as $$\Delta S=\int_{M_{i}}^{M_{f}}\frac{dS}{dM}dM=\int_{M_{i}}^{M_{f}}2\pi
R\frac{dR}{dM}dM. \label{bh}$$
Considering the small path near $R$, we can insert the mathematical identity $\operatorname{Im}\int_{r_{i}}^{r_{f}}\frac{1}{r-R}dr=-\pi$ in the formula (\[bh\]). Thus we obtain $$\Delta S=-2\operatorname{Im}\int_{M_{i}}^{M_{f}}\int_{r_{i}}^{r_{f}}\frac
{R}{r-R}\frac{dR}{dM}dM. \label{bhe}$$
Using (\[rt\]) and the expression of the temperature in thermodynamics $\frac{1}{T}=\frac{\partial S}{\partial E}$, we attain $$R\frac{dR}{dM}=\frac{1}{N_{t}^{^{\prime}}(R)-N_{r}^{^{\prime}}(R)}. \label{ts}$$
Then the equations (\[rng\]) and (\[ts\]) give the final form of the change of entropy (\[bhe\]) as $$\Delta S=-2\operatorname{Im}\int_{M_{i}}^{M_{f}}\int_{r_{i}}^{r_{f}}\frac
{dR}{\overset{.}{r}}dM=-2\operatorname{Im}I,$$ where $I$ is the action for an s-wave outgoing positive particle in WKB approximation.
So the tunneling probability is given as $$\Gamma\thicksim e^{\Delta S}=e^{-2\operatorname{Im}I}$$
Thus we obtain the tunneling probability from the change of entropy as a direct consequence of the first law of black hole thermodynamics by using the same method as that in Ref. [@tp08]. Let us emphasize that in the original method the author uses the general radiation temperature different from the Hawking temperature in order to discuss the factor of 2 problem. However the new temperature is still proportional to the surface gravity like the Hawking temperature and only the proportional relation is crucial for the discussed problem in this paper. So we take the Hawking temperature as the black hole temperature without loss of generality.
The tunneling through the quantum horizon
=========================================
We note that for spherically symmetric black holes a generalized treatment [@sk08] has been suggested, in which the tunneling probability is gotten directly from the principle of conservation of energy by calculating the imaginary part of the action in WKB approximation and the method is independent on the form of black hole entropy. For the entropy which is proportional to area [@pw00] or contains the logarithmic modification caused by the presence of quantum gravity [@amv05], we know that the tunneling probability has been obtained by calculating the imaginary part of the action in WKB approximation. Recently Pilling has suggested that the tunneling probability is obtained directly from the first law of thermodynamics by calculating the change of the entropy being proportional to area, even if the radiation temperature is different from the Hawking temperature [@tp08]. Then could the Pilling method be applied to the situation where the entropy is modified by logarithmic term when considering quantum gravity effect? In the following we will discuss the problem.
First we take into account the modification of entropy caused by the presence of quantum gravity effect which gives a leading order correction with a logarithmic dependence on the area besides reproducing the familiar Bekenstein-Hawking linear relation [@dvf95; @km00; @dnp04] $$S_{QG}=\frac{A}{4L_{P}^{2}}+\alpha\ln\frac{A}{L_{p}^{2}}+O(\frac{L_{p}^{2}}{A}),$$ where $A$ is the area of black hole horizon and $L_{p}$ is the Planck length. The relation exists in string theory and loop quantum gravity. The difference is that $\alpha$ is negative in the case of loop quantum gravity [@gm05], but in String Theory the sigh of $\alpha$ depends on the number of field species appearing in the low energy approximation [@sns98]. It is noted that there is an interesting phenomenon that this log-area correction is closely related to black hole remnant when the coefficient $\alpha$ is negative [@lx07].
Along Pilling’s line we calculate the tunneling probability by using the entropy modified by quantum gravity effect. For briefness, we write $$S_{QG}=\frac{1}{4}A+\alpha\ln A=\pi R^{2}+\alpha\ln(4\pi R^{2}).$$ where the logarithmic correction can also be obtained by considering the one-loop effects of quantum matter fields near a black hole [@dvf95; @dnp04]. Whatever consideration we take, the spacetime will change. If we continue to use the spacetime represented by (\[me\]), the wrong result will be gotten. We can see this point clearly from the following calculation.
When the mass of the black hole changes from $M_{i}$ to $M_{f}$, we have $$\Delta S=\int_{M_{i}}^{M_{f}}\frac{dS}{dM}dM=\int_{M_{i}}^{M_{f}}\left( 2\pi
R+\frac{2\alpha}{R}\right) \frac{dR}{dM}dM=\Delta S_{1}+\Delta S_{2},
\label{qe}$$ where $\Delta S_{1}=\int_{M_{i}}^{M_{f}}2\pi R\frac{dR}{dM}dM$, $\Delta
S_{2}=\int_{M_{i}}^{M_{f}}\frac{2\alpha}{R}\frac{dR}{dM}dM$. It is noted that if we continue to calculate according to the same method presented in the section above, it will be found that $\Delta S_{1}=-2\operatorname{Im}\int_{M_{i}}^{M_{f}}\int_{r_{i}}^{r_{f}}\frac{dR}{\overset{.}{r}}dM$ by using the surface gravity (\[sg\]). It seems that $\Delta S_{2}$ is not related to the action of the black hole and so is not related to the tunneling probability. This is inconsistent with the result obtained in Ref. [@amv05] where $$\Gamma(E)\thicksim e^{-2\operatorname{Im}I}=\left( 1-\frac{E}{M}\right)
^{2\alpha}e^{\left( -8\pi ME(1-\frac{E}{2M})\right) }.$$ Consequently, we see that the imaginary part of the action is expressed as the change of the whole entropy but not that of partial entropy. The calculation above shows that when considering the entropy with logarithmic correction, the spacetime will change and carry the quantum gravity effect. On the other side we note that in Ref. [@tp08] the author uses the entropy being proportional to area, so the radiation temperature is obviously proportional to the surface gravity. But here we take the entropy $S_{QG}=\frac{1}{4}A+\alpha\ln A$, it looks as if formally the temperature were not proportional to the surface gravity according to the first law of black hole thermodynamics. A straight way to contain the quantum gravity effect is to use the thermodynamic relation to get the surface gravity afresh. From thermodynamics, the temperature can be given as $$\frac{1}{T}=\frac{dS}{dM}=\left( 2\pi R+\frac{2\alpha}{R}\right) \frac
{dR}{dM}\equiv\frac{2\pi}{\kappa}.$$
Thus we can get the surface gravity in the entropy with logarithmic modification as $$\kappa\equiv2\pi/\frac{dS_{QM}}{dM}=\frac{2\pi}{\left( 2\pi R+\frac{2\alpha
}{R}\right) \frac{dR}{dM}}, \label{qsg}$$ where the surface gravity is not only dependent on the mass of the black hole but also dependent on the coefficient $\alpha$ which accords with the consideration that the surface gravity carries the quantum gravity correction.
We note that if we want to obtain the relation $\Delta S=-2\operatorname{Im}I$ as that in the section above when considering the entropy with logarithmic correction, we have to find the method to calculate the radial null geodesic trajectory which is difficult to be calculated because we don’t know the property of such spacetime clearly. At the same time it has been pointed out that the quantum entropy comes from counting states in a quantum theory, whereas geodesics make sense in a classical spacetime. So the concept of geodesics has to be managed carefully when the logarithmic correction of entropy is explained as quantum gravity effect [@amv05; @ma060; @ma06]. However, the attained result is consistent with the explanation of the tunneling probability of quantum mechanics. Thus the feasibility of using the concept of geodesics means that there maybe exist the physical reason to explain the mathematical consistency. We note that the logarithmic correction of the black hole entropy can be obtained from the purely quantum gravity effect and can also be obtained from the one-loop effects of quantum matter fields near a black hole [@dnp04]. The difference lies in the value of the parameter $\alpha$, but the problem here is not concerned about it. So we can calculate the geodesics by considering the one-loop effects of quantum matter fields near a black hole. It is noted that in such consideration the expression of spacetime presented in (\[me\]) could still be used [@dvf95; @jwy85], but some quantities, such as the mass, the temperature, the surface gravity and so on, has to be changed. On the other hand we also note that here the modification of surface gravity (\[qsg\]) is consistent with the result obtain by considering the one-loop correction as in Ref. [@jwy85; @ls88]. For example, for Schwarzschild spacetime, the classical surface gravity is expressed as $\kappa_{0}=\frac{1}{4M}$ and the radius is $R=2M$, so by Eq. (\[qsg\]) the modified surface gravity can be gotten as $\kappa\simeq$ $\kappa_{0}(1-\frac{\alpha}{4\pi M^{2}})$ which accords with the modified surface gravity due to one loop back reaction effects [@jwy85; @ls88].
In the following we will show that the tunneling probability can be recovered by calculating the change of the entropy with logarithmic modification. By (\[qsg\]), we can write the change of entropy as $$\Delta S=\int_{M_{i}}^{M_{f}}\left( 2\pi R+\frac{2\alpha}{R}\right)
\frac{dR}{dM}dM=\int_{M_{i}}^{M_{f}}\frac{2\pi dM}{\kappa}. \label{ec}$$
Because the spacetime (\[me\]) can still be used, so the radial null geodesic trajectory is written as [@sp99; @jw06], $$\overset{.}{r}\simeq\kappa(r-R), \label{rngt}$$ where the formula can be gotten by using Eq. (\[rng\]) and Eq. (\[sg\]) and it must be stressed that here the surface gravity and the event horizon have been changed and are different from that appeared in the section above. Thus we replace the surface gravity in Eq. (\[ec\]) by Eq. (\[rngt\]), insert the mathematical identity $\operatorname{Im}\int_{r_{i}}^{r_{f}}\frac{1}{r-R}dr=-\pi$ and have $$\Delta S=-2\operatorname{Im}\int_{M_{i}}^{M_{f}}\int_{r_{i}}^{r_{f}}\frac
{1}{\overset{.}{r}}drdM=-2\operatorname{Im}I.$$
So the tunneling probability is gotten as $$\Gamma\thicksim e^{\Delta S}=e^{-2\operatorname{Im}\int_{r_{i}}^{r_{f}}p_{r}dr}. \label{tpc}$$ In this way we have finished the calculation of black hole tunneling probability from the first law of thermodynamics when considering the quantum gravity effect. In the derivation the introduction of the new surface gravity is a crucial step because this maintains the general expression of the relation between radiation temperature and surface gravity. We can see that the Hawking temperature relation $T=\frac{\kappa}{2\pi}$ or the proportional relation between the radiation temperature and surface gravity is key in the calculation here as that in [@sk08; @tp08; @amv05]. In general, when we discuss the connection of the black hole radiation as tunneling with thermodynamics, only if we accept the Hawking temperature relation or the proportional relation between the temperature and surface gravity, can we obtain the tunneling probability directly from the first law of the black hole thermodynamics, not dependent on the form of entropy of the black hole, which is seen by writing the change of the entropy as $$\Delta S=\int_{M_{i}}^{M_{f}}\frac{dS}{dM}dM=\int_{M_{i}}^{M_{f}}\frac{1}{T}dM=\int_{M_{i}}^{M_{f}}\frac{2\pi dM}{\kappa}.$$ Thus we can conclude that it is the relation between black hole temperature and surface gravity that plays the crucial role that relates the black hole thermodynamics with the tunneling picture of the black hole. At the same time the concept of geodesics has to be managed carefully.
After the Hawking temperature was discovered, there have also been several other methods [@hh76; @sp99] to derive the same result as that obtained by Hawking [@swh75]. Recently, however, it has been pointed out [@aas06] that the tunneling approach produces a temperature that is double the original Hawking temperature, which is used to question either the tunneling methods or the value of Hawking temperature. This problem is discussed again in [@pm07] where the authors consider the incoming solution besides the outgoing solution and uses the ratio of the outgoing and incoming probabilities to recover the Hawking temperature. The factor of 2 problem about black hole temperature is also discussed generally in Ref. [@tp08].
**Conclusion**
==============
We have showed that the tunneling probability can be obtained from the first law of thermodynamics by using the entropy with logarithmic modification which contains the quantum gravity effect and the change of the surface gravity has been presented clearly in the calculation. We have also showed the important role that the relation between the radiation temperature and the surface gravity plays. One can note that our derivation can be generalized only by starting from the first law of thermodynamics $dM=TdS$ and the relation $T=\frac{\kappa}{2\pi}$ instead of considering the form of the black hole entropy. The generalization verifies the connection of black hole radiation with thermodynamics further.
Acknowledgement
===============
We are grateful to the anonymous referee for his/her critical comments and helpful advice. This work is fund by National Natural Science Foundation of China (Grant. No. 10504039 and 10747164).
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University of Adelaide preprint:\
ADP-95-60/T205\
(to appear: Mod. Phys. Lett. [**1996**]{})
[**A New Hypothesis on the Origin of the Three Generations\
**]{}
Introduction
============
The Standard Model has proven very successful in every area of particle physics, including recent high-energy collider experiments – e.g. see Ref.[@1]. However, it has three features which are not well understood: the origin of mass, the three fermion generations and the phenomenon of CP violation. The question of mass is usually framed in terms of (fundamental) Higgs fields [@Higgs] and why the corresponding Yukawa couplings take particular values – see however Ref.[@2]. Instead, we believe that one should ask whether a formulation of the Standard Model with massless fermions makes sense. For example, it is well known that QED with massless electrons is not well defined at the quantum level [@Muta; @8].
In this paper we consider the pure Standard Model with gauge symmetry $SU(3) \otimes SU(2)_L \otimes U(1)$ and no additional interaction; that is, assuming no additional unification. We examine the physical theory that corresponds to the bare Standard Model Lagrangian with no elementary Higgs and just one generation of fermions and gauge bosons which all have zero bare mass. At asymptotic scales, where the U(1) coupling is significantly greater than the asymptotically free SU(3) and SU(2)$_L$ couplings, the left and right handed states of any given charged fermion couple to the U(1) gauge boson with different charges. At the Landau scale there will be three separate phase transitions corresponding to each of the right-right, right-left and left-left interactions becoming supercritical. These transitions correspond to three generations of fermions. As one passes through each transition from a higher scale (shorter distance) the corresponding scalar condensate “melts”, releasing a dynamical fermion into the Dirac phase studied in the laboratory. In this picture the three generations emerge as quasi-particle states built on a “fundamental fermion” interacting self-consistently with the condensates.
Clearly this proposal differs in a fundamental manner from the conventional approaches to the Standard Model. While the conceptual framework is extremely simple and elegant, the techniques for dealing with non-perturbative physics at the Landau scale are not well developed. In particular, at the present stage we are not able to present a rigorous, quantitative derivation of all of the features of the Standard Model. Nevertheless, we believe that the potential for understanding so many phenomena, including mass, CP-violation and the generations, is so compelling that the ideas should be presented at this stage.
The structure of the paper is the following. In order to introduce the ideas we first review the change in the vacuum of pure QED near a point-like nucleus with charge greater than $137$, a problem that has received enormous effort [@Grein1; @Grein2; @Orsay]. We then consider the analogous case of QED at the Landau scale, from which we conclude that in pure QED the electron would self-consistently generate its own mass. Having thus introduced the basic notions we turn to the full Standard Model, considering in turn the generations of charged fermions and the origin of CP violation, the neutrinos and the mass of the vector bosons.
The Supercritical Phase of Non-Asymptotically Free Gauge Theory
================================================================
Several decades of work have revealed that non-asymptotically free (NAF) U(1) gauge theories with zero bare fermion mass are capable of generating their own renormalised mass. The prime example of such a theory is, of course, QED with zero bare mass, where both analytic [@mas; @fk; @Miran] and numerical (lattice) calculations [@Latt] have shown that one finds a finite renormalised mass and a non-trivial, ultraviolet (UV), stable fixed point. Without this the theory is trivial; that is, the charge is completely screened by the interactions so that the theory is equivalent to a free field theory.
The possibility that QED may be trivial was originally suggested by Landau and co-workers [@Land1; @Land2] and Fradkin [@Frad]. Consider the runnning coupling in perturbative (weakly coupled) QED. The one loop vacuum polarisation implies $$\alpha (m^2) = { \alpha (\lambda^2) \over 1 + {\alpha (\lambda^2) \over 3 \pi}
\ln { \lambda^2 \over m^2 } }.$$ If we take $\lambda^2 \rightarrow \infty$ (the continuum limit of QED with a finite cut-off) then $\alpha (m^2)$ vanishes for all $m^2$. The same result applies in the limit of zero mass gap (i.e. $m^2 \rightarrow 0$), namely the coupling $\alpha (m^2)$ vanishes again. Recent work by Kocić et al. [@Kocic] has shown that this “zero charge problem” persists when the magnetic interaction of the electron is also included, despite the fact that it tends to screen the vacuum polarisation.
Another indication of the problem of massless QED is the fact that one cannot renormalise it (perturbatively) on mass shell ( which is a necessary condition for the electron to be a physical particle). The only alternative, which was once again suggested by Landau et al. [@Land1] (see also Dirac [@Dirac]), is that non-perturbative effects near the Landau scale mean that QED has a non-trivial, UV, stable fixed point. A number of groups [@mas; @fk; @Miran; @Miran2] have shown that QED, in quenched, ladder approximation, has a non-trivial, UV, stable fixed point at $\alpha_c = {\pi \over 3}$, which separates the weakly and strongly interacting phases. The theory is trivial for bare coupling $\alpha < \alpha_c$, whereas for $\alpha > \alpha_c$ the chiral symmetry of the massless bare theory is spontaneously broken by the interactions leading to the formation of tightly bound states – much like the $Z > 137$ point nucleus problem in QED. Kogut et al. [@Latt] have found that this UV, stable fixed point survives in unquenched lattice QED. Estimates of the value of $\alpha_c$ (the critical bare coupling) from Schwinger-Dyson and lattice calculations range between 0.8 and 2 \[18-21\].
Given that a NAF U(1) gauge theory has a non-trivial, UV, stable fixed point, $\alpha_c$, it follows that the theory has a two phase structure. We let $\lambda_c$ denote the scale at which $\alpha$ reaches the fixed point $\alpha_c$ and call the phases at scales above and below the critical scale $\lambda_c$ the Landau and Dirac phases respectively. Perturbative QED (and the Standard Model) is formulated entirely in the Dirac phase of theory ($ \mu < \lambda_c $). The theory seems to behave as a gauged Nambu-Jona-Lasinio model [@NJL] in the Landau phase [@Latt; @Bard; @Miran3].
The connection with supercritical phenomena (in particular, the large-$Z$, point-nucleus problem) suggests a simple physical interpretation of this theory. Since massless, perturbative QED is not a consistent theory because of the “zero charge problem”, we consider perturbative QED with a finite renormalised mass and sketch how this mass could be recovered self-consistently in a complete formulation of QED.
The coupling $\alpha$ increases until we reach the critical scale $\lambda_c$ where the interaction of the fermions with the gauge field becomes supercritical. To understand what happens at this transition it is helpful to consider the analogous problem of a static, large-$Z$, point nucleus in QED [@Grein1; @Grein2; @Orsay]. There the $1s$ bound state level for the electron falls into the negative energy continuum at $Z=137$. If we attempt to increase $Z$ beyond 137 the point nucleus becomes a resonance: an electron moves from the Dirac vacuum to screen the supercritical charge which then decays to $Z-1$ with the emission of a positron.
If the electron itself were to acquire a supercritical charge at very large scales, [O]{}$(\lambda_c)$, it would not be able to decay into a positive energy bound state together with another electron because of energy momentum conservation. In this case, the Dirac vacuum itself would decay to a new supercritical vacuum state. Since the vacuum is a scalar, this transition necessarily involves the formation of a scalar condensate which spontaneously breaks the (near perfect) chiral symmetry of perturbative QED at large momenta. The Dirac vacuum is a highly excited state at scales $\mu \geq \lambda_c$ and one must re-quantise the fields with respect to the new ground state vacuum in the Landau phase of the theory. The Dirac electron of perturbative QED would freeze out of the theory as a dynamical degree of freedom and the running coupling would freeze at $\alpha (\lambda_c)$. Perturbative QED, which requires a finite electron mass, is formulated entirely in the Dirac phase of the theory. The normal ordering mismatch between the zero point energies of the scalar vacua in the Dirac and Landau phases of QED means that the electron in the Dirac phase always feels a uniform, local, scalar potential. This potential must be included in the Hamiltonian for perturbative QED. The minimal gauge invariant, local, scalar operator that we can construct is the scalar mass term $m_e [ {\overline e} e ]$. A self-consistent treatment of QED appears to generate its own mass.
Generations in the Standard Model
=================================
We now discuss how the considerations of the previous section carry over to the Standard Model. The Standard Model differs from QED at very large momentum in that the U(1) gauge boson coupling to a fermion depends on its chirality. The right-right, right-left and left-left fermion interactions have different strengths for the Dirac leptons ($e, \mu$ and $\tau$) and the quarks. As we now explain, this important difference means that a non-perturbative solution of the Standard Model requires three generations of fermions. The fermion gauge boson interaction in the electroweak sector is described by the Standard Model Lagrangian with symmetry $SU(3) \otimes SU(2)_L \otimes U(1)$: $${\overline \Psi}_L \Biggl( \hat{\partial} - i g_1 \hat{B} - i g_2
\hat{\underline{W}}.{1 \over 2}
{\underline \tau} \Biggr) \Psi_L
+ {\overline \Psi}_R \Biggl( \hat{\partial} - i g_1 \hat{B} \Biggr) \Psi_R.$$ Here $\Psi_L$ and $\Psi_R$ include the left and right handed fermions according to the Standard Model. We use $\alpha_1 = {g_1^2 \over 4 \pi}$, $\alpha_2 = {g_2^2 \over 4 \pi}$ and $\alpha_s$ to denote the U(1), SU(2) and colour SU(3) couplings respectively.
Consider the Standard Model evolved to some very large scale, much greater than the “unification scales" where the U(1) coupling $\alpha_1 = \alpha_2$ and $\alpha_1 = \alpha_s$. As we approach the Landau scale the $Z^0$ evolves to become the U(1) gauge boson as $\sin^2 \theta_W \rightarrow 1$. Since the SU(2) and SU(3) sectors of the Standard Model are asymptotically free [@GG] the $W^{\pm}$, the photon and the gluon have effectively disappeared at these scales. The $Z^0$ mass increases logarithmically with increasing $\mu^2$ and can be treated as negligible at the Landau scale so that the theory behaves as a U(1) gauge field coupling to left and right handed fermions with different charges. The $Z^0$ coupling to the fermions is\
$$-i g_1 \gamma_{\mu}
\Biggl( c_L {1 - \gamma_5 \over 2} + c_R {1 + \gamma_5 \over 2} \Biggr)$$ where the left and right handed charges $c_L g_1$ and $c_R g_1$ are given in Table 1. (Here $l$ denotes the charged leptons and $\nu_l$ the corresponding neutrinos. We use $q^{*}$ and $q_{*}$ to denote the upper and lower components of the electroweak quark doublet.)
[ccc]{}\
\
& $c_L$ & $c_R$\
\
\
$l$ & $ - {1 \over 2} + \sin^2 \theta_W $ & $ \sin^2 \theta_W$\
\
$\nu_l$ & $ + {1 \over 2} $ & 0\
\
$q^{*}$ & $ + {1 \over 2} - {2 \over 3} \sin^2 \theta_W $ & $ - {2 \over 3} \sin^2 \theta_W$\
\
$q_{*}$ & $ - {1 \over 2} + {1 \over 3} \sin^2 \theta_W $ & $ + {1 \over 3} \sin^2 \theta_W$\
\
The idea that we wish to develop is the following. Consider a “fundamental fermion", which is defined in the pure Landau phase of the Standard Model. Since the left and right handed charges have different values, it follows that the left-left, left-right and right-right fermion interactions will, in general, become sub-critical at different scales as we evolve the theory through the supercritical transitions to lower $\mu^ 2$. The first interaction to become sub-critical as we decrease $\mu^2$ is the left-left interaction, followed by the left-right and then the right-right interactions. Each transition is associated with the melting of a scalar condensate which releases a dynamical fermion into the Dirac phase of the Standard Model. These Dirac fermions interact self-consistently with the condensates in the Landau phase of the theory. In this picture the three fermion generations emerge as three quasi-particle states in the Dirac phase which correspond to the “fundamental fermion" in the Landau phase and which couple to the gauge field with identical charge.
Let us now outline how this structure should be manifest from the opposite direction, as we evolve the Standard Model upwards from the “low scale" of the laboratory towards the Landau scale. For simplicity, we first consider the charged leptons. In the absence of any other physics the Standard Model should undergo a rich series of phase transitions near the Landau scale as each of the right-right, right-left and left-left fermion interactions become supercritical with increasing $\mu^2$. These transitions can be classified into one of two types: “static" transitions and “vacuum" transitions. “Static" transitions involve the decay of the left or right handed component of a “heavy" fermion $\Psi_h$ into a “light" fermion $\Psi_l$ together with the formation of a $(\Psi_h {\overline \Psi}_l)$ bound state (like the decay of a large-$Z$ point nucleus). The supercritical component of $\Psi_h$ becomes a resonance between the critical scales for the static transition and the vacuum transition at which the $\Psi_h$ freezes into the Landau phase. “Vacuum" transitions involve the decay of the fermionic vacuum from the Dirac into the Landau phase and the formation of a scalar condensate. Static transitions do not affect the symmetry or generation structure (which is given by the vacuum transitions). They do affect the scale at which the vacuum transition involving $\Psi_h$ takes place. The charge of the “resonance fermion" increases more slowly with increasing $\mu^2$ than we would predict using perturbative arguments alone so that vacuum transitions which involve the resonance fermion are pushed to a higher scale.
At very large scales where $\sin^2 \theta_W \rightarrow 1$, the charges of the left and right handed charged leptons become $c_L \rightarrow {1 \over 2}$ and $c_R \rightarrow 1$ respectively. The interaction between two right handed fermion fields (eg. $e^-_R$, $e^+_L$) is the first to go supercritical. One finds the static decays of the right handed muon $\mu^-_R$ and tau $\tau^-_R$, viz. $\tau^-_R \rightarrow (\tau^-_R e^+_L) \ e^-_R$, and also the vacuum transition involving the right handed electron at a critical scale $\lambda_c^{RR}$. Since the vacuum is a scalar, this vacuum transition must be associated with the formation of a scalar condensate. It is important to consider what has happened to the left handed electron at this point. The Dirac vacuum for the left-handed electrons collapses at $\lambda_c^{RR}$ because of the axial anomaly [@ABJ], whereby the chirality of a charged lepton in the Dirac phase is not conserved in the presence of a background gauge field. The anomaly has a simple interpretation in a two phase NAF gauge theory [@Bud]. Consider the gauge-invariant axial-vector current in perturbative QED with an explicit UV cut-off, which we shall take to be equal to $\lambda_c^{RR}$. The anomaly appears as a flux of chirality (or spin) over the cut-off – and into the Landau phase of the theory. If one turns off the anomaly, the Dirac vacuum for the left handed electrons is highly excited with respect to the Landau vacuum for the right handed electrons at $\mu \geq \lambda_c^{RR}$.Via the axial anomaly, the left-handed electrons condense with the right-handed electrons to form the Landau vacuum which is created at $\lambda_c^{RR}$ and the electron completely freezes out of the theory.
At scales $\mu \geq \lambda_c^{RR}$ the remaining charged lepton degrees of freedom are the $\mu$ and the $\tau$. Here the right handed muons and taus are supercritical resonances while the left handed muons and taus are still perturbative fermions. The left handed charge evolves significantly faster than the right handed charge with increasing $\mu^2$ and the left handed fermions drive the dynamics. The muon freezes into the Landau phase at the critical scale $\lambda_c^{LR}$ for the left-right vacuum transition and the tau freezes out at the left-left vacuum transition. The latter is catalysed by the axial anomaly in the same way as the right-right vacuum transition. The three self-supercritical transitions (right-right, left-right and left-left) yield three condensates in the Landau phase of the Standard Model.
The same arguments hold in the quark sector but there is one important new point to note. The upper and lower components of the electroweak quark doublet $q^{*}$ and $q_{*}$ become self-supercritical at different scales because of the different coupling of the U(1) gauge boson to each of the $q^{*}$ and $q_{*}$ quarks. This means that the eigenstates of the $W^{\pm}$-quark interaction in the Standard Model (which define the components of the quark doublet) and the quark mass eigenstates are not identical. The three generations of quarks mix according to a unitary (Kobayashi Maskawa) matrix which, in general, gives CP violation in the quark sector. To see that we have a CP violating interaction at large scales, consider the vector, vector, axial-vector triangle diagram. This is anomaly free in the pure Dirac phase of the Standard Model when we sum over $l$, $\nu_l$, $q^{*}$ and $q_{*}$ propagating in the triangle loop. At intermediate momentum scales, where one component of the quark doublet has frozen into the Landau phase and the other component remains in the Dirac phase, there is a nett three-gauge-boson contact interaction in the Dirac phase of theory which carries the CP-odd quantum numbers of the axial anomaly. This corresponds to regularising the UV behaviour of the triangle amplitude with a slightly different cut-off for each component of the electroweak doublet in perturbation theory. Since this cut-off is so much greater than any mass scales that are currently amenable to experiment this contact interaction does not harm either anomaly cancellation or the renormalisability of the Standard Model.
As the right-handed (Dirac) neutrino is non-interacting in the Standard Model we cannot see how to form a scalar, neutrino condensate at the supercritical transitions. Of course, each type of neutrino will sense the corresponding charged lepton transition (through the coupling $\nu_l \rightarrow W l \rightarrow \nu_l$). While it may be that this coupling gives rise to a mass, $\nu_l$, of the order of $G_F m_l$ in the Dirac phase, it seems most likely that the neutrinos are massless. (The Standard Model with massive gauge bosons and massless neutrinos can be renormalised on mass shell [@aoki].) If this is the case it is trivial that there should be no Kobayashi-Maskawa matrix in the lepton sector: one can simultaneously diagonalise the eigenstates of mass and the $W^{\pm}$-lepton interaction.
The resonance structure offers a possible reason why the top quark is so much heavier than the bottom quark. The relative separation of the left-right and left-left transitions is greater for the $q^*$ (top quark) than the $q_*$ (bottom quark). This means that the top quark has further to evolve than the bottom quark to get from $\lambda_c^{RL}$ to $\lambda_c^{LL}$ with a slowly increasing left handed charge; the top quark freezes out at a much higher scale than the bottom quark and has a much higher mass. Similarly, the charm quark has a lot further to go than the strange quark between the right-right and left-right transitions.
The dynamical chiral symmetry breaking which gives us the fermion masses also gives mass to the gauge bosons. The gauge fixing in the “fundamental” bare Lagrangian (with zero mass) does not involve the $0^{-+}$ Goldstone bosons, which are generated with the dynamical chiral symmetry breaking when we turn on the vacuum polarisation. The propagators for the gauge bosons are transverse in covariant (eg. Landau) gauge: $$\Pi_{\mu \nu} = f^2 \Biggl( g_{\mu \nu} - {p_{\mu} p_{\nu} \over p^2} \Biggr)$$ When we evaluate the gauge boson self-energies using the Schwinger-Dyson equations (eg. in leading logarithm approximation [@top]) we find a non-transverse mass term in $\Pi_{\mu \nu}$ which is proportional to $g_{\mu \nu}$. The transversity of $\Pi_{\mu \nu}$ is restored by the mixing of the gauge bosons with the $0^{-+}$ Goldstone bosons. The fermion, gauge-current vertex becomes: $$\Biggl( \gamma_{\mu} {1 \over 2} (1 - \gamma_5) {\tau^a \over 2} \Biggr)_{{\rm
bare}}
\rightarrow
\Biggl( \gamma_{\mu} {1 \over 2} (1 - \gamma_5) {\tau^a \over 2} -
f_{ab} g_b {p_{\mu} \over p^2} \Biggr)_{({\rm Standard \ Model})}$$ where $f_{ab}$ is the current-Goldstone transition amplitude and $g_b$ denotes the Goldstone-fermion coupling. The Higgs mass and the Goldstone parameters $f_{ab}$ and $g_b$ are determined by the mass of the top quark $m_t$ and the running QCD coupling $\alpha_s(m_t^2)$. As Gribov has emphasised [@top], the Schwinger-Dyson equations for the Higgs and Goldstone self-energies involve all the fermions on an equal footing. The top quark becomes important only because of its large mass; it has no special interaction.
Conclusions
===========
We have argued, on quite general grounds, that in the absence of elementary Higgs (or other, additional, physics) the Standard Model may generate its own mass and three generations of fermions as a result of super-critical phenomena at the Landau scale. We have not considered gravity and one may worry that, at least in a perturbative treatment, the Landau scale is larger than the Planck mass. However, we believe that the scenario presented in this paper is compelling and certainly merits further investigation. One could speculate that in a non-perturbative treatment the physics of the Planck scale and the Landau scale may in fact be coupled.
It is clearly important to explore the physics of the Landau scale in the laboratory. This is difficult in the U(1) sector because of the large momentum scales involved. On the other hand, in QCD the Landau scale is in the infra-red and it might be that one can learn a little about the mechanism proposed here through the study of phenomena such as quark confinement and hadronisation \[30-33\].
[**Acknowledgements\
**]{}
We would like to thank T. Goldman, V. N. Gribov, P. A. M. Guichon, C. A. Hurst, R. G. Roberts, D. Schütte, R. Volkas and A. G. Williams for helpful discussions. We would also like to thank J. Speth for his hospitality at the KFA Jülich where this work began. This work was supported in part by the Australian Research Council and the Alexander von Humboldt Foundation.
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|
---
abstract: 'In a finite temperature Thomas-Fermi theory with realistic nuclear interactions, we construct caloric curves for finite nuclei enclosed in a sphere of about $4 - 8$ times the normal nuclear volume. The specific heat capacity $C_v$ shows a peaked structure that is possibly indicative of a liquid-gas phase transition in finite nuclear systems.'
address:
- '$^{1)}$ Cyclotron Institute, Texas A&M University, College Station, TX 77843-3366, USA'
- '$^{2)}$ Department of Physics, McGill University, 3600 University St., Montreal, PQ, H3A 2T8 Canada'
- '$^{3)}$ Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Calcutta - 700064, India'
author:
- 'J. N. De$^{1*)}$, S. Das Gupta$^{2)}$, S. Shlomo$^{1)}$ and S. K. Samaddar$^{3)}$'
title: 'CALORIC CURVE FOR FINITE NUCLEI IN THOMAS-FERMI THEORY'
---
1 cm
[*Keywords:*]{} Caloric curve; Specific heat; Thomas-Fermi; Phase transition 1 cm
The equation of state (E.O.S.) of nuclear matter with realistic effective interactions shows a typical Van der Waals type behavior and a critical temperature of $\approx 15 - 20$ MeV [@kwh; @jmz; @bssd]. Supported by the experimental observation of a power law behavior in the mass or charge distribution in proton [@fi; @gkj] and heavy ion induced reactions[@chit; @lynen], the idea of liquid-gas phase transition in nuclear matter or finite nuclear systems [@jmz; @siem; @pan; @gkm] has gotten considerable interest in the literature. Theoretical speculations and possible experimental indications of a limiting temperature [@lb; @blv; @drss; @gue; @nat] in finite nuclei at $\approx 5 - 7$ MeV, above which the nucleus becomes unstable and breaks up into many fragments, also calls for a possible connection between the limiting temperature and the phase transition. Phase transitions are normally signalled by peaks in the specific heat at constant volume, $C_v$ as temperature increases. Fragmentation calculations in the microcanonical algorithm of Gross [@gross] and in the Copenhagen canonical description [@bdms; @bbims] show such peaks. Recent calculations by Das Gupta et. al. [@gupta] in the lattice gas model for fragmentation also show such a structure. Renewed interest in this subject was further fueled by the recent experimental observation [@poch] in the caloric curve of a near constancy of temperature in the excitation energy range of $\approx 4 - 10$ MeV/nucleon in Au + Au collisions. This prompted us to find out whether the trends in the caloric curve as seen in the experiment or in fragmentation calculations are reproduced in a finite temperature Thomas-Fermi (TF) theory. To our knowledge this is the first calculation of its kind with a realistic effective interaction. In the context of an exactly solvable Fermion model, Rossignoli et al [@rpm] have earlier calculated the specific heat of a finite nucleus in the grand canonical mean field theory with Lipkin’s model hamiltonian, but found no structure in it as a function of temperature. The structure appeared in the canonical calculation, with inclusion of correlations.
In our refined Thomas-Fermi (TF) model, the interaction density is calculated with a Seyler-Blanchard type [@sb] momentum and density dependent finite range two-body effective interaction [@drss]. The interaction is given by
$$v_{{\rm eff}}(r,p,\rho)=C_{l,u}[v_{1}(r,p)+v_{2}(r,\rho)]
\eqno (1)$$ $$v_{1}=-(1-p^{2}/b^{2})f({\bf r_{1}},{\bf r_{2}})$$ $$v_{2}=d^{2}[\rho_{1}(r_{1})+\rho_{2}(r_{2})]^{n}
f({\bf r_{1}},{\bf r_{2}})
\eqno (2)$$ with $$f({\bf r_{1}},{\bf r_{2}})=\frac{e^{- \mid {\bf r_{1}}-{\bf r_{2}} \mid } /
a}{ \mid {\bf r_{1}}-{\bf r_{2}} \mid /a} .
\eqno (3)$$ Here $a$ is the spatial range and $b$ the strength of repulsion in the momentum dependence of the interaction, $r=\mid {\bf r_{1}}-{\bf r_{2}}
\mid $ and $p= \mid {\bf p_{1}}-{\bf p_{2}} \mid $ are the relative distance and relative momenta of the two interacting nucleons. The subscripts $l$ and $u$ in the strength $C$ refer to like pair (n-n or p-p) or unlike pair (n-p) interaction respectively, $d$ and $n$ are measures of the strength of the density dependence of the interaction and $\rho_1$ and $\rho_2$ are the densities at the sites of the two nucleons.
The potential parameters are determined for a fixed value of $n$ from a fit of the well-established bulk nuclear properties and the value of $n$ is determined [@drss] from a fit of the Giant Monopole Resonance energies over a broad mass spectrum.
The Coulomb interaction energy density is given by the sum of the direct and exchange terms. They are given by
$$\varepsilon_{D}(r)=e^{2} \pi \rho_{p}(r) \:
\int \: dr^{\prime} \, r^{\prime \, 2} \rho_{p}(r^{\prime})
{\rm g}(r,r^{\prime}),
\eqno (4)$$ and $$\varepsilon_{{\rm ex}}(r)=-\frac{3e^{2}}{4\pi}(3\pi^{2})^{1/3}
\rho_{p}^{4/3}(r) .
\eqno (5)$$ Here $\rho_p(r)$ is the proton density and $${\rm g}(r,r^{\prime})=\frac{(r+r^{\prime})-\mid r-r^{\prime} \mid}
{rr^{\prime}} .
\eqno (6)$$ With the potential chosen, the total energy density at a temperature $T$ is then written as $$\varepsilon(r)=\sum_{\tau} \, \rho_{\tau}(r) [T \, J_{3/2}(\eta_{\tau}(r))/
J_{1/2}(\eta_{\tau}(r)) {(1- m_{\tau}^{\ast}(r) V_{\tau}^{1}(r))}
+\frac{1}{2}V_{\tau}^{0}(r)]
\eqno (7)$$ Here $\tau$ refers to neutron or proton, the $J$’s are the usual Fermi integrals, $V_{\tau}^{0}$ is the single particle potential ( for protons, it includes the Coulomb term), $V_{\tau}^{1}$ is the potential term that comes with momentum dependence and is associated with the effective mass $m_{\tau}^{\ast}$. The fugacity $\eta_{\tau}(r)$ is defined as $$\eta_{\tau}(r)=[\mu_{\tau}-V_{\tau}^{0}(r)-V_{\tau}^{2}(r)]/T
\eqno (8)$$ where $\mu_{\tau}$ is the chemical potential and $V_{\tau}^{2}$ is the rearrangement potential that appears for a density-dependent interaction. The total energy per particle at any temperature is then given by $$E(T)= \int \, \varepsilon(r) \, {\rm d}^{3} r /A .
\eqno (9)$$ Once the interaction energy density is known, the nuclear density can be obtained self-consistently and other observables of physical interest calculated. For details on the finite temperature TF theory, we refer to Ref. [@drss].
Since the continuum states of a nucleus at nonzero temperature are occupied with a finite probability given by a Fermi factor [@balian], the particle density does not vanish at large distances. The observables then depend on the size of the box in which the calculations are performed. Guided by the practice that many calculations for heavy ion collisions are done by imposing that thermalisation occurs in a freeze-out volume, we fix a volume and find out the excitation energy as a function of temperature which allows for the determination of the specific heat at constant volume.
We choose two systems, namely $^{150}$Sm and $^{85}$Kr. In the context of very heavy ion collisions at intermediate or higher energies, this mass range is of experimental interest. The calculations have been done for two confinement volumes, one at $V =4.0V_0$ and the other at $V = 8.0V_0$, where $V_0$ is the normal volume of the nucleus at zero temperature. The calculations at zero temperature are independent of the volumes taken; at low temperature of $\approx 1 - 2$ MeV, the observables are nearly independent of the volume. As the temperature increases, the central density is depleted. In Figure 1, the proton densities for $^{150}$Sm calculated in the volume $V = 8.0V_0$ are displayed for four temperatures, $T = 5$ MeV (dashed curve), $T = 9$ MeV (dotted curve), $T = 9.5$MeV (dash-dotted curve) and $T = 10$ MeV (full curve). At $T = 5$ MeV, the central density is depleted by $\approx 4\%$ compared to zero temperature density, but has a long thin tail spread to the boundary. The behaviours at $T = 9$ and 9.5 MeV are qualitatively the same, but with further depletion in the central density and a thicker tail. Beyond $T = 9.5$MeV, the change in the density starts being abrupt and the whole system looks like a uniform distribution of matter inside the volume. This is shown by a representative density distribution at $T = 10$MeV. The slight bump seen in the outer edge of the density is due to the Coulomb force. In Figure 2, the proton density at $T = 10$ MeV for the system at $V = 8.0V_0$ (dashed curve) is compared with that calculated at $V = 4.0V_0$ (full curve). The density calculated in smaller volume still shows a structure and the central density is depleted by only about $20\%$ even at this high temperature.
The excitation energy per particle $E^{*}$ is defined as $E^{*} = E(T) - E(T=0)$. In Figure 3, we display the caloric curve for the system $^{150}$Sm. The upper dashed curve corresponds to $V = 4.0V_0$ while the lower full curve corresponds to $V = 8.0V_0$. At lower density, the excitation energy rises faster. For both volumes, initially the temperature rises faster with excitation energy, then its rise is slower. For the lower density, a kink is observed in the caloric curve at $T \approx 10$ MeV, after which the excitation energy rises almost linearly with temperature. For the higher density, the kink is much smaller and appears at a somewhat higher temperature. In Figure 4, the corresponding specific heats $C_v$ defined as $$C_{v}=\left( {\rm d}E^{*}/{\rm d} T \right)_{v}
\eqno (10)$$ are displayed. Since we use units of MeV for both energy and temperature, the calculated $C_v$ is dimensionless. For both volumes, the specific heat shows a peak, the peak being much sharper for the case of a larger volume. For the smaller volume, the peak is at $T \approx 10.5$ MeV while for the large volume the peak is shifted down by $\approx 1$ MeV. We believe that the kink in the caloric curve or the peak in the specific heat are related to a phase transition in finite nuclei. From our calculations, we find that this transition temperature is weakly dependent on the confinement volume beyond $V = 8V_0$, e.g., for V as high as $20V_0$, the transition temperature is shifted down further by only $\approx 1$MeV. The classical value of $C_v = 3/2$ is reached at $T \approx 11$ MeV for the case with $V = 8.0V_0$ while for the smaller volume, it is reached at $T \approx 13$ MeV. This is expected as the interaction becomes weaker either with increased volume or with increased temperature.
In Figure 5, the caloric curve for the lower mass system $^{85}$Kr is shown. The trends are nearly the same as in Figure 3. Figure 6 displays the specific heat for this system. In the calculation with $V = 4V_0$, a broad bump in the specific heat at $T \approx 11$ MeV is seen. In calculations with expanded volume ($8V_0$), the system shows a sharp peak at $T \approx 10.5$ MeV. This peak is, however, not as sharp as the one for the heavier system. In calculations on limiting temperature in the model of liquid-gas phase equilibrium, the influence of Coulomb forces has often been emphasized [@bssd; @jaq] in the instability of the system. In the present calculation, we see a relatively small effect on the transition temperature. With the Coulomb force switched off, the transition temperature is shifted up by$\approx 1$ MeV for both the confinement volumes $4V_0$ and $8V_0$ and the matter density becomes more uniform. This transition temperature is somewhat lower compared to the critical temperature for asymmetric nuclear matter [@bssd] with isospin asymmetry equal to that of the nucleus.
To summarize, we have calculated the caloric curve and the specific heat for two systems in a self-consistent Thomas-Fermi theory at two volumes, namely at 4 and 8 times the normal nuclear volume. The specific heat $C_v$ shows a peaked structure possibly signalling a liquid-gas phase transition at a temperature of $\approx 10$ MeV which is lower than the calculated critical temperature for infinite nuclear matter but larger compared to the calculated limiting temperature for finite real nuclei [@drss]. In simplistic model calculations [@rpm], it has been shown that the inclusion of correlations brings in features reminiscent of a phase transition in a system when no phase transition is evident in the usual mean field calculation; it would therefore be interesting to see whether fluctuations with two-body correlations bring down the phase transition temperature obtained in our TF calculation.
The authors acknowledge fruitful discussions with Dr. E. Ramakrishnan. One of the authors (J. N. D.) gratefully acknowledges the hospitality of the Cyclotron Institute, Texas A&M University where this work was completed. This work is supported by the U.S. Department of Energy under grant DE-FE05-86ER40256, by the Natural Sciences and Engineering Research Council of Canada and by the U.S. National Science Foundation under grant PHY-9413872.
\* On leave of absence from the Variable Energy Cyclotron Centre, 1/AF, Bidhannagar, Calcutta- 700 064, India
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[**FIGURE CAPTIONS**]{}
[**Fig. 1**]{} The proton density profile for the system $^{150}$Sm calculated at four temperatures in the volume $V = 8.0V_0$. The dashed, dotted, dash-dot and full lines correspond to temperatures $T = 5,9,9.5$ and 10 MeV respectively.
[**Fig. 2**]{} The proton density profile for the system $^{150}$Sm calculated at temperature $T = 10$ MeV in two different volumes. The full and dashed lines correspond to calculations at $V = 4.0V_0$ and $V = 8.0V_0$, respectively.
[**Fig. 3**]{} The temperature plotted as a function of excitation energy per particle (caloric curve) for the system $^{150}$Sm. The dashed curve corresponds to calculations with volume $V = 4.0V_0$ while the full curve corresponds to $V = 8.0V_0$.
[**Fig. 4**]{} The specific heat per particle plotted as a function of temperature for the system $^{150}$Sm. The dashed curve corresponds to calculations with volume $V = 4.0V_0$ while the full curve corresponds to $V = 8.0V_0$.
[**Fig. 5**]{} Same as Figure 3 for the system $^{85}$Kr.
[**Fig. 6**]{} Same as Figure 4 for the system $^{85}$Kr.
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abstract: 'The influence of hydroxyectoine on the properties of the aqueous solution in presence of DPPC lipid bilayers is studied via semi-isotropic constant pressure (NPT) Molecular Dynamics simulations. We investigate the solvent-co-solute behavior in terms of Kirkwood-Buff integrals as well as hydrogen bond life times for an increasing hydroxyectoine concentration up to 0.148 mol/L. The observed preferential exclusion mechanism identifies hydroxyectoine as a kosmotropic osmolyte. Our findings in regards to the DPPC lipid bilayer indicate an increase of the surface pressure as well as the solvent accessible surface area in presence of higher hydroxyectoine concentrations. The results are in agreement to the outcome of recent experiments. With this study, we are able to validate the visibility of co-solute-solute-solvent effects for low and physiologically relevant osmolyte concentrations.'
author:
- 'Jens Smiatek$^{1,2}$'
- Rakesh Kumar Harishchandra$^3$
- 'Hans-Joachim Galla$^3$'
- Andreas Heuer$^2$
title: 'Low concentrated hydroxyectoine solutions in presence of DPPC lipid bilayers: a computer simulation study'
---
[Osmolytes, Molecular Dynamics simulations, DPPC lipid bilayers, Kosmotropes, Preferential Exclusion, Kirkwood-Buff theory]{}
Introduction
============
Osmolytes allow extremophilic microorganisms to resist harsh living conditions [@Lentzen06; @Driller08]. Typical examples for these species are ectoine and hydroxyectoine which are zwitterionic, strong water binding and low-molecular weight organic molecules. The functionalities of these molecules in living organisms among others are given by the protection of protein conformations [@Lentzen06; @Driller08; @Yu04; @Yu07; @Smiatek12] and the fluidization of lipid membranes [@Galla10; @Galla11]. The protective properties become mainly important under environmental stress conditions, e. g. high temperature, extreme dryness and salinity [@Knapp99]. Several studies have identified that several combinations of osmolytes can be found in biological cells. The concentration of a single osmolyte in these mixtures varies between 0.1 to 1.0 mol/L ([@Yancey2005] and references therein). Due to the fact that a lot of osmolytes are not affecting the cell metabolism, specific molecules like the ectoines are also commonly called compatible solutes.\
Recent studies were focusing on the molecular functionality of the ectoines as well as the analysis of the protective behavior [@Yu04; @Yu07]. The theoretical framework for the explanation of co-solute-solute effects has been mainly established in terms of the preferential exclusion [@Timasheff02] and the transfer free energy model [@Rose2008].\
These models focus on the strong ordering of the local water shell around the osmolytes and the exclusion from the immediate hydration shell of the solute which results in a preferential hydration behavior and a stabilization of the solutes native structure [@Timasheff02; @Rose2008]. Among these, more refined versions of theories have been in addition published which explicitly rely on the properties of chaotropic and kosmotropic behavior [@Collins97; @Collins04; @Collins07; @Ninham12] and the corresponding interaction with macromolecular surfaces. Co-solutes which strengthen the water hydrogen network are called kosmotropes (structure makers) while osmolytes which weaken the water network structure are called chaotropes (structure breaker). The separation of co-solutes into these two species is not unique and straightforward [@Ninham12]. Interactions and binding properties between kosmotropes and chaotropes can be predicted by the ’law of matching water affinities’ [@Collins04; @Ninham12]. A main point of this theory is the investigation of the corresponding hydration free energies which loosely depend among other factors on the molecular charge [@Ninham12]. One of the major achievements of the ’law of matching water affinities’ is the molecular description of a repulsive behavior for kosmotropic osmolytes from polar surfaces and vice versa, the attraction of chaotropic agents like urea. The preferential binding of urea has been validated in recent computer simulations [@Horinek11] while additional studies have observed a kosmotropic behavior for ectoine in terms of a preferential exclusion mechanism around Chymotrypsin Inhibitor II [@Yu07].\
It has been stated that kosmotropic co-solutes typically accumulate in the second or third hydration shell of the solvated macromolecule. In regards to their high charging and affinity for water molecules, it is assumed that this appearance strongly influences the first and the second hydration shell of the polar solute. The consequence of this behavior is given by a diminished number of solute-water hydrogen bonds which is compensated by a significant shrinkage of the solute surface to maintain a constant hydrogen bond surface density. It is commonly believed that this shrinkage in size is the molecular reason for the preservation of native protein conformations in presence of kosmotropic co-solutes [@Collins04; @Ninham12].\
Although the stabilizing effects on proteins in presence of specific osmolytes have been studied extensively before, less is known about compatible solutes and their interactions with bilayers. A small number of theoretical studies have focused on sugars like trehalose and their interactions with lipid membranes and monolayers [@dePablo03; @Hunenberger04; @Pastor05; @Hunenberger06; @Sum06; @Hunenberger08; @Hunenberger10]. For high molar concentrations of trehalose, it has been found that replacement of water molecules by the formation of additional sugar-membrane hydrogen bonds plays a major role [@Hunenberger04]. Despite this interpretation, it has been also discussed that the effects observed in sugar-DPPC mixtures can be only systematically explained by an interplay of several mechanisms [@Hunenberger10].\
In addition to theoretical studies, experimental findings have indicated a significant broadening of the liquid expanded (LE) - liquid condensed (LC) phase transition of monolayers in presence of ectoine and hydroxyectoine [@Galla10; @Galla11]. This was mainly indicated by the study of the corresponding surface pressure area isotherms. A main result of these studies was the observation of a surface pressure increase for higher hydroxyectoine concentrations. In addition it was supposed that the domain sizes of the liquid condensed regions significantly shrink in presence of hydroxyectoine which corresponds to a variation of the line tension [@Galla10; @Schwille07; @Baumgart08; @Ruckenstein97]. In regrds to the biological function, the above mentioned effects are of particular important for signaling processes and cell repair [@Driller08; @Galla10].\
In regards to the preferential exclusion/binding behavior of osmolyte-solute-solvent mixtures, computer simulations allow a detailed study of the corresponding molecular mechanisms. A theoretical framework which allows to distinguish between exclusion and binding behavior has been established in terms of the Kirkwood-Buff theory of solutions [@Kirkwood51; @Ben-Naim92; @Trout03]. The corresponding analysis has been therefore successfully applied to the study of urea and polyglycine interactions [@Horinek11]. It has been shown that the calculation of the Kirkwood-Buff integrals allows the effective determination of transfer free energies in addition to the detection of kosmotropic as well as chaotropic behavior [@Yu07; @Horinek11].\
In this paper, we study the properties of an aqueous hydroxyectoine solution in presence of DPPC lipid bilayers via semi-isotropic constant pressure (NPT) all-atom Molecular Dynamics simulations. The concentration of hydroxyectoine is low but physiologically relevant [@Yancey2005] with a maximum value of 0.148 mol/L. We have been inspired to use these small concentrations due to recent experimental findings for aqueous hydroxyectoine-DPPC monolayer mixtures [@Galla10; @Galla11]. Most of the simulation studies usually employ high co-solute concentrations which are often above one mole per liter to study pronounced behavior at unphysiological conditions [@Yu04; @Yu07; @Horinek11]. With this study, we are able to validate the observation of effects at smaller concentrations in agreement to experimental findings.\
Our main results include the characterization of hydroxyectoine as a kosmotropic osmolyte which strengthens the water hydrogen bond network. We are further able to validate a weakening of DPPC-water hydrogen bond interactions in presence of hydroxyectoine. In regards to the DPPC lipid bilayer properties, our results validate an increase of the surface pressure and the solvent-accessible surface area in agreement to the experimental results [@Galla10; @Galla11; @Smiatek12]. We emphasize the importance of electrostatic interactions between co-solutes and solutes for the understanding of the observed effects by the calculation of the bilayer electrostatic potential.\
The paper is organized as follows. In the next section we shortly introduce the theoretical background. In the third section we illustrate the simulation details and the methodology. The results for the solvent properties and the DPPC lipid bilayer are presented and discussed in the fourth section. We briefly conclude and summarize in the last section.
Theoretical Background
======================
Kirkwood-Buff integrals and preferential binding parameter
----------------------------------------------------------
The evaluation of statistical mechanics methods on the co-solvent and solvent distribution function allows important insights into the preferential exclusion as well as binding behavior in terms of the corresponding Kirkwood-Buff theory which has been introduced in the early 1950’s [@Kirkwood51; @Ben-Naim92]. The radial distribution function of molecules or atoms $\beta$ around solutes $\alpha$ can be expressed by $$\label{eq:rdf}
g_{\alpha\beta}(r) = \frac{\rho_{\beta}(r)}{\rho_{\beta,\infty}}$$ where $\rho_{\beta}(r)$ denotes the local density of $\beta$ at a distance $r$ around the solute and $\rho_{\beta,\infty}$ the global density in the bulk phase [@Leach01]. The Kirkwood-Buff integral is given by the integration of Eqn. \[eq:rdf\] $$\label{eq:KBI}
G_{\alpha\beta} = \lim_{R\rightarrow\infty} G_{\alpha\beta}(R) = \lim_{R\rightarrow\infty}\int_{r=0}^{r=R}4\pi r^2(g_{\alpha\beta}(r)-1)dr$$ where the above relation is valid in the limit of $R=\infty$ [@Yu07; @Horinek11; @Kirkwood51; @Ben-Naim92; @Trout03]. Eqn. \[eq:KBI\] can be used to calculate the excess coordination number of molecules or atoms of $\beta$ (hydroxyectoine) around $\alpha$ (DPPC) via [@Yu07; @Kirkwood51; @Ben-Naim92] $$N_{\beta}^{xs} = \rho_{\beta,\infty}G_{\alpha\beta}= \rho_{\beta,\infty}\lim_{R\rightarrow\infty} G_{\alpha\beta}(R)$$ which allows to evaluate the preferential binding coefficient $\nu_{\beta\gamma}(R)$ under the assumption of finite distances $R$ with $$\label{eq:bind}
\nu_{\beta\gamma}(R) = \rho_{\beta,\infty}(G_{\alpha\beta}(R)-G_{\alpha\gamma}(R)) = N_{\beta}^{xs}(R) - \frac{\rho_{\beta,\infty}}{\rho_{\gamma,\infty}}N_{\gamma}^{xs}(R)$$ where the indices $\gamma$ represent solvent molecules while $\alpha$ and $\beta$ are the solute DPPC and the co-solute hydroxyectoine. A negative value for the preferential binding coefficient of Eqn. \[eq:bind\] implies a preferential exclusion of hydroxyectoine from the lipid bilayer surface while a positive value indicates a preferential binding [@Yu07; @Trout03].
Hydrogen bonds and water relaxation times
-----------------------------------------
The formation of lipid bilayers in aqueous solutions is mainly driven by the hydrophobic effect [@Ball08]. The detailed molecular mechanism for the hydrophobic effect is still heavily under debate but it is consensus that the formation and cleavage of hydrogen bonds is of main importance [@Ball08]. For a detailed investigation of the co-solute-solvent properties, we have analyzed the water hydrogen bond characteristics in presence of varying hydroxyectoine concentrations.\
We apply the Luzar-Chandler definition of hydrogen bonds [@Luzar96; @Luzar2000], which restricts a maximum length of 0.35 nm between the interacting oxygen and hydrogen atom and an angle of not more than 30 degrees. The value for the life time allows an estimate of the relative strength for the corresponding hydrogen bond. This argument is inspired by results of transition state theory which connects the rate constant $k_F$ to the activation free energy $\Delta G^{*}$ via $$\label{eq:rate}
k_F = \frac{1}{\tau_F}= \frac{k_BT}{h} \exp\left(-\frac{\Delta G^{*}}{k_BT}\right)$$ with the thermal energy $k_BT$, the forward life time $\tau_F$ and the Planck constant $h$ [@vanderspoel06]. In terms of the underlying statistical analysis of hydrogen bonds [@vanderspoel06], Eqn. \[eq:rate\] reveals that longer life times correspond to larger free activation energies which indicates a strengthening of the hydrogen bond network.\
In addition to the hydrogen bond analysis, we have also calculated the dipolar reorientation time of water molecules in presence of hydroxyectoine. The evaluation of this quantity gives access to an estimation of the translational solvent entropy. It has been discussed that solvent entropies may play a significant role in contributions to the solvation free energies and therefore the exposure of hydrophilic and hydrophobic surfaces [@Ball08; @Finkelstein; @Ninham12]. To study this property, we have investigated the autocorrelation time for the dipolar orientation $\vec{\mu}$ between two water molecules in presence of DPPC lipid bilayers and hydroxyectoine which is given by $$\label{eq:mu}
<\vec{\mu}(t)\vec{\mu}(t_0)>\sim\exp(-t/\tau)^\beta.$$ It can be seen that the autocorrelation function follows a stretched exponential behavior which is dependent on the exponent $\beta$ [@Stanley05]. The evaluation of the corresponding times $\tau$ allows a quantitative determination of the influence of hydroxyectoine on the water dynamics.
Surface pressure
----------------
As it has been found out in recent experiments for DPPC monolayers in presence of hydroxyectoine [@Smiatek12; @Galla10; @Galla11], a significant variation of the surface tension can be observed for increasing physiological hydroxyectoine concentrations. The surface tension can be calculated by $$\gamma = \int (P_N-P_T(z)) dz$$ which can be also expressed for the ease of computation by $$\label{eq:gamma}
\gamma = \frac{1}{2}\left<L_z \left(P_{zz}-\frac{1}{2}(P_{xx}+P_{yy})\right)\right>$$ where $P_N$ denotes the normal pressure with $P_{zz}$ as the z-component of the pressure tensor and $P_T$ the transversal pressure given by $1/2(P_{xx}+P_{yy})$ as defined by the x- and the y-components. The factor $1/2$ accounts for two interfaces in contact with water [@Pastor95; @Kindt04; @Tieleman07; @Tieleman10]. The box length in z-direction is denoted by $L_{z}$.\
The surface pressure is given by $$\label{eq:pi}
\Pi = \gamma_0-\gamma$$ where $\gamma_0$ expresses the experimental surface tension of water at 300 K (71.6 mN/m) [@CPC] which allows a direct comparison with experiments as discussed in Ref. [@Pastor05].\
Simulation Details
==================
We have performed Molecular Dynamics simulations in explicit SPC/E water [@Straatsma87] with the software package GROMACS [@Berendsen95; @Hess08; @Spoel05].
![Structure of neutral hydroxyectoine.[]{data-label="fig0"}](fig0.eps)
The chemical structure of hydroxyectoine ((4S,5S)-2-methyl-5-hydroxy-1,4,5,6-tetrahydropyrimidine-4-carboxylic acid) in its neutral form is presented in Fig. \[fig0\]. The derivation of the force field and the topology of hydroxyectoine is in detail described in Ref. [@Smiatek12] where it has been also found that the zwitterionic form in aqueous solution is more stable than the neutral counterpart. We follow this finding by using purely zwitterionic molecules in our MD simulations. The force field for the lipids and the starting structure with 64 DPPC molecules [@Tieleman] were modeled with the parameters presented in Ref. [@Berger97].\
The Molecular Dynamics simulations have been carried out with periodic boundary conditions. The simulation box has initial dimensions of $(4.72450\times 4.23190\times 9.95050)$ nm$^3$. We performed simulations with $2,4,6$ and 8 hydroxyectoine molecules which correspond to effective concentrations of $0.037, 0.074, 0.111$ and $0.148$ mol/L. Electrostatic interactions have been calculated by the Particle Mesh Ewald sum [@Pedersen95]. The time step was $\delta t=2$ fs and the temperature was kept constant by a Nose-Hoover thermostat [@Frenkel96] at 300 K. All bonds have been constrained by the LINCS algorithm [@Fraaije97].\
After energy minimization and a 10 ns constant volume simulation to ensure the conformational equilibration of the lipid molecules [@Hunenberger10], we conducted a 20 ns equilibration run followed by a 30 ns semi-isotropic constant pressure (NPT) data production simulation. A Parrinello-Raman barostat has been used with a rescaling time step of 2 ps. The reference pressure in the x/y- and the z-direction was 1 bar and a compressibility of $4.5\times 10^{-5}$ bar$^{-1}$ was used. The solvent accessible surface area $\Sigma_{tot}$ was calculated by the sum of spheres centered at the atoms of the studied molecule, such that a spherical solvent molecule can be placed in closest distance and in agreement to van-der-Waals interactions by following the constraint that other atoms are not penetrated [@Scharf95].\
Hydrogen bonds have been defined as present if the distance between the interacting atoms is less than 0.35 nm and the interaction angle is not larger than 30 degrees. The hydrogen bond density which is calculated for the DPPC lipid bilayers is given by the number of hydrogen bonds divided by the hydrophilic solvent accessible surface area $\rho_{HB} = <N_{HB}/\sigma_{HPL}>$.
Results
=======
Solvent properties
------------------
The average position and the distribution of hydroxyectoine in front of the DPPC lipid bilayer can be easily determined by the evaluation of the pair radial distribution function (rdf). Due to the fact, that the nitrogen of DPPC is slightly positively charged ($+0.55 e$) and the oxygens in the carboxy group of hydroxyectoine are negatively charged ($-0.87 e$), we have decided to evaluate the rdf between these two atoms due to favorable electrostatic interactions. Furthermore we have calculated for a comparisoon the rdf for nitrogen in DPPC and oxygen in water molecules. The results for a hydroxyectoine concentration of 0.148 mol/L are shown in Fig. \[fig1\]. It can be seen that the pair radial distribution function $g(r)$ between DPPC and hydroxyectoine reveals a significant appearance of the compatible solute at a distance of 0.35 to 0.85 nm which is in good agreement to previous assumptions [@Smiatek12].
![Pair radial distribution function $g(r)$ for nitrogen in DPPC ($N_{DPPC}$) and oxygen in hydroxyectoine (red line), respectively oxygen in water (blue line) for a hydroxyectoine concentration of 0.148 mol/L. \[fig1\]](fig1.eps)
Comparing the pair radial distribution function for the nitrogen and water oxygen to determine the position of the first hydration shell reveals that a large amount of compatible solutes are accumulated at the second hydration shell of the lipid bilayer. This can be validated by the smaller occurrence of hydroxyectoine at 0.3 nm which roughly corresponds to the peak of the first hydration shell. Thus it can be concluded that direct interactions between hydroxyectoine and DPPC in terms of hydrogen bonds are less important for these concentrations as it was also stated in [@Galla11].\
To further investigate the hydration behavior of DPPC, we have calculated the water hydrogen bond density at the DPPC hydrophilic solvent accessible surface area $\sigma_{HL}$. We have found a nearly constant value of $\rho_{HB}=7.43 \pm 0.01$ nm$^{-2}$ for the hydrogen bond density averaged over all hydroxyectoine concentrations. The results show no significant deviations concerning higher hydroxyectoine concentrations such that it can be assumed, that the concentration of the co-solutes does not affect the overall water hydrogen bond density at the DPPC surface. In addition, we have evaluated the number of direct contacts between hydroxyectoine and DPPC in terms of hydrogen bonds where we have found a value of $<N_{{HB}}>=0.6\pm 0.5$ for the highest hydroxyectoine concentration. All other values for lower concentrations are vanishing or nearly identical. Compared to the number of hydrogen bonds between water and DPPC and their contributions to the total energy, it can be concluded that the influence of the DPPC-hydroxyectoine hydrogen bonds is nearly negligible. Regarding the water-replacement theory and compared to the results of trehalose-DPPC systems [@Hunenberger04], it can be concluded that the reduction of the number of water molecules in front of the DPPC lipid bilayer by hydroxyectoine as it has been assumed for proteins [@Collins04] is a minor effect which has not been detected in our simulations. Despite the additionally proposed diminished solvent accessible surface area [@Collins04], instead we have observed an increasing surface area for the lipid bilayer. We will discuss this point in more detail in the next section.\
For a detailed investigation of the preferential exclusion behavior, we have calculated the corresponding Kirkwood-Buff integrals [@Horinek11; @Kirkwood51; @Ben-Naim92; @Trout03]. The results for the preferential binding parameter (Eqn. \[eq:bind\]) are shown in Fig. \[fig2\].
![Preferential binding coefficient $\nu_{\beta\gamma}(R)$ for distances $r$ up to 0.55 nm for all four concentrations in presence of hydroxyectoine.[]{data-label="fig2"}](fig2.eps)
It can be clearly seen that the preferential binding coefficient $\nu_{\beta\gamma}(r)$ at lipid bilayer distances up to 0.55 nm is negative for all hydroxyectoine concentrations. The amount of exclusion increases with the concentration of the compatible solutes. This also clearly states that hydroxyectoine is preferentially excluded from the DPPC bilayer surface due to energetic reasons [@Trout03]. Hence, the observed behavior is in good agreement to the predicted behavior for kosmotropic osmolytes [@Collins04].\
In the framework of recent theories, it has been stated that highly charged ions can be interpreted as kosmotropes (structure maker) [@Collins04]. These molecules tend to strengthen the water-water structure by increasing hydrogen bond lifetimes. The influence of chaotropic solutes like urea on the dynamics of water-water hydrogen bonds has been investigated in Ref. [@Horinek11]. It was found that urea in contrast to kosmotropes decrease the strength of the water hydrogen bond network. We have analyzed the corresponding characteristics for hydroxyectoine in terms of the hydrogen bond transition state theory and the orresponding statistical analysis [@vanderspoel06]. The corresponding forward life times of hydrogen bonds for water-water interactions and water-DPPC hydrogen bond interactions are shown in Fig \[fig3\].
![Water-water hydrogen bond forward lifetimes $\tau_F$ (top) and DPPC-water hydrogen bond life times (bottom) for increasing hydroxyectoine concentrations.[]{data-label="fig3"}](fig3.eps)
It can be clearly seen that hydroxyectoine leads to an increase of the hydrogen bond life times in presence of higher concentrations. Due to the fact that the forward life times are proportional to $\log(\tau_F)\sim \Delta G^*$, an increase of the life times also indicates a higher activation free energy barrier $\Delta G^{*}$. Hence the kosmotropic properties of hydroxyectoine are obvious and can be identified by a strengthening of the water hydrogen bond network. In addition, we have observed a saturation plateau of life times for hydroxyectoine concentrations $c\geq 0.074$ mol/L. The reverse behavior can be also observed for DPPC-water hydrogen bonds which means decreasing life times for increasing hydroxyectoine concentrations until a saturation plateau is reached. Thus, it can be concluded that the hydration properties of DPPC bilayers in terms of energetic contributions are slightly disturbed in presence of low molar hydroxyectoine solutions. These findings are in good agreement to the ’law of matching water affinities’ which states that kosmotropic agents weaken the hydration behavior of polar solute surfaces [@Collins04]. Nevertheless, the pronounced influence of the DPPC lipid bilayer on the water dynamics can be also observed in terms of long lifetimes as it was also discussed in a recent publication [@Debnath2010]. It was mentioned that in close vicinity to the bilayer, a significant decrease of water diffusion coefficients can be validated.\
Finally we have calculated the he hydrogen bond life times for hydroxyectoine and water where we have found a nearly constant values of $<\tau_F> = 4.27 \pm 0.22$ ps for all concentrations.
![Increase for the dipolar autocorrelation time in presence of hydroxyectoine.[]{data-label="fig4"}](fig4.eps)
To illustrate the importance of hydroxyectoine on the water dynamics and therefore the entropic contributions in terms of hydration behavior, we have also analyzed the water dipole reorientation times (Eqn. \[eq:mu\]). It has been stated that the general entropy of the water is significantly influenced by the number of intermolecular hydrogen bonds and the values for the reorientation times [@Ball08; @Finkelstein]. By the evaluation of the water dipole autocorrelation function for each concentration, we have identified a value for $\beta$ in Eqn. \[eq:mu\] of $0.86\pm 0.05$ which validates a stretched exponential behavior in agreement to recent results and theories [@Stanley05]. The corresponding relaxation times $\tau$ for each concentration are presented in Fig. \[fig4\]. We clearly observe an increase of the relaxation times in presence of hydroxyectoine which can be related to diminished entropic contributions. The direct connection between the entropy and the relaxation dynamics has been discussed in Ref. [@Bencivenga09] in which it has been shown that the relation $\tau = \tau_0\exp(-S/k_B)$ with the entropy $S$ is valid for several measurable relaxation times. Furthermore a plateau value for hydroxyectoine concentrations of $c\geq 0.074$ mol/L in agreement to the results of Fig. \[fig3\] can be also observed. Thus, a strong water structure influence effect can be observed in presence of hydroxyectoine.\
To summarize the results of this subsection, we have validated that hydroxyectoine is repelled from DPPC lipid bilayer surfaces in terms of a preferential exclusion behavior. This effect is in good agreement to recent theories concerning kosmotropic behavior for osmolytes. In a recent study [@Smiatek12] we were able to indicate a net accumulation of roughly 8-9 water molecules due to electrostatic interactions around hydroxyectoine. Combined with these results, the analysis of the hydrogen bond life times and the dipole reorientation times clearly reveals the kosmotropic behavior of hydroxyectoine. It has to be noted that the dynamic properties have been calculated by considering the complete number of water molecules, regardless if they interact with hydroxyectoine, DPPC or with themselves. We are therefore confident that the change of global water dynamics in presence of low hydroxyectoine concentrations is a significant effect.
DPPC lipid bilayer properties
-----------------------------
In recent experiments for DPPC monolayers in presence of an aqueous hydroxyectoine concentration [@Galla10; @Galla11; @Smiatek12], a broadening of the liquid expanded (LE) - liquid condensed (LC) phase transition was observed. This finding for hydroxyectoine concentrations around 0.1 mol/L has been validated in terms of surface pressure-area diagrams and has been compared to urea solutions where this behavior was found to be absent.\
In order to prove the experimental results, we have calculated the surface pressure for varying hydroxyectoine concentrations according to Eqns. \[eq:gamma\] and \[eq:pi\] for concentrations that are comparable to the experiments. The results for the surface pressure in presence of hydroxyectoine are shown in Fig. \[fig5\].
![Surface pressure calculated by the relation $\Pi = \gamma_0-\gamma$ with $\gamma_0 = 71.6$ mN/m. The presence of hydroxyectoine leads to a pressure difference of 6 mN/m for the highest concentration.[]{data-label="fig5"}](fig5.eps)
It becomes obvious that the presence of the osmolyte leads to a significant increase of the surface pressure [@Galla10]. The corresponding Pearson correlation coefficient is given by $r=0.93$ which validates a linear dependence between osmolyte concentration and surface pressure. Although we have studied bilayers, the evident properties of Fig. \[fig5\] are in good agreement to the experimental findings in Refs. [@Smiatek12; @Galla10; @Galla11] for DPPC monolayers. The direct connection between the surface pressure and the surface tension $\gamma$ in regards to Eqn. \[eq:gamma\] also allows to validate an identical behavior as it has been observed for trehalose-DPPC mixtures [@Pastor05]. In regards to the presence of unfavorable environmental conditions for extremophilic organisms, the observed effect is advantageous due to the fact that it leads to a fluidization of lipid membranes. It has been discussed in Ref. [@Driller08] that the enhanced flexibility of membranes facilitates cell repair and signal transport.\
As an additional property, we have calculated the solvent accessible surface area (SASA). It can be assumed that a less rigid bilayer coincides with an increasing solvent accessible surface area. The results are shown in Fig. \[fig6\]. It can be seen that a small increase of the total SASA $\Sigma_{tot}$ for higher hydroxyectoine concentrations is evident. Although the overall amount of this increase compared to the total area is small ($\approx 1.5$ nm$^2$ compared to $43$ nm$^{2}$), the general behavior follows a monotonous increase with a Pearson correlation coefficient of $r=0.97$. We have also calculated the ratio of the hydrophilic SASA $\sigma_{HL}$, where only the polar regions of the molecules are taken into account to the total SASA $\Sigma_{tot}$. The results are shown in the inset of Fig. \[fig6\]. A linear increase for this quantity can be also identified. The observed behavior can be brought into agreement with the increased surface pressure as shown in Fig. \[fig5\], due to the fact that the hydrophilic regions form the interface with the hydroxyectoine solution. We therefore propose that this effect accounts for an increased fluidization of the lipid bilayer in agreement to recent experiments [@Galla11].\
In contrast to the proposed decreased solvent accessible surface for polar protein surfaces in presence of kosmotropes [@Collins04], we have observed the opposite trend for DPPC bilayers. The difference between molecular complex surfaces like membranes with a more flexible surface area compared to a single molecular surface may be responsible for this observation.\
![Total solvent accessible surface area for the DPPC lipid bilayer in presence of an increasing hydroxyectoine concentration. [*Inset:*]{} Ratio of the hydrophilic $\sigma_{HL}$ to the total solvent accessible surface area $\Sigma_{tot}$.[]{data-label="fig6"}](fig6.eps)
Finally we have calculated the electrostatic potential for the DPPC lipid bilayer in presence of varying hydroxyectoine concentrations. The results are presented in Fig. \[fig7\]. It can be seen that for higher hydroxyectoine concentrations an increase of the DPPC electrostatic potential can be additionally observed. The general trend is obvious although a deviation for the concentration $c=0.111$ can be identified which can be related to a slight asymmetric potential distribution compared to the other concentrations. However, these findings are in agreement to a recent publication [@Cebers08], where it has been found that electrostatic contributions play a significant role for the internal organization of lipid bilayers. It can be assumed that the strong zwitterionic charges of hydroxyectoine interact with the nitrogen atoms and the phosphate groups in DPPC. Due to the fact that the electrostatic potential has been calculated by integrating over the charges within a slice, we propose that the accumulation of charged groups at the interface accounts for this observation. This reason has been also discussed in a recent publication [@Chachisvilis11], where the strong dependence of the electrostatic potential on the local ordering and the packing fraction has been pointed out. Although the authors have remarked, that the molecular origin for the varying potential remains controversial, a strong dependence between local ordering of the lipid molecules and the resulting electrostatic potential has been proposed. We are therefore confident that the observed behavior validates the ordering effect of lipid bilayers in presence of low hydroxyectoine concentrations. Hence, it has to be stated that only a combination of many effects may explain the observed characteristics as it has been also proposed in Ref. [@Hunenberger10].
![Electrostatic potential $\Psi$ for the DPPC lipid bilayer in presence of varying hydroxyectoine concentrations.[]{data-label="fig7"}](fig7.eps)
Summary and conclusion
======================
Molecular Dynamics simulations of DPPC lipid bilayers in presence of hydroxyectoine have been performed. We have analyzed the properties of the aqueous solution for low concentrations of hydroxyectoine. Our results have clearly revealed that the presence of hydroxyectoine results in a strengthening of the water hydrogen bond network due to increased forward life times. Due to the fact that the life times are related to the activation free energy barriers, we propose that the water hydrogen bond network is strengthened in presence of hydroxyectoine. Although we have simulated very low concentrations of hydroxyectoine, all observed effects are clearly visible. Thus, we were able to show that specific cosolute-water-solute effects can be even observed at low physiological concentrations in agreement to recent experimental findings. Furthermore we have shown that the presence of hydroxyectoine leads to a energetic decrease of the hydrogen bond network between DPPC and water molecules in terms of slightly decreased forward life times. These findings are in good agreement to previous literature results for kosmotropic properties in solution. Summarizing the results for the co-solute-solvent interactions, it can be concluded that hydroxyectoine can be interpreted as a typical kosmotropic osmolyte.\
As a further characteristic property, it has been often proposed that kosmotropic cosolutes are preferentially excluded from polar surfaces [@Collins04]. The preferential binding coefficient in our simulations has been calculated by the Kirkwood-Buff theory and indicates a preferential exclusion behavior for hydroxyectoine in presence of DPPC bilayers. Thus, our findings are in good agreement to recent theories. The distance between the hydrophilic head groups of DPPC and the carboxy group of hydroxyectoine has been determined to fluctuate around 0.5 nm. This value is in agreement to recent conclusions [@Smiatek12] and roughly corresponds to the position of the second hydration shell.\
In regards to the results for the DPPC lipid bilayer, we have indicated an increase of the surface pressure for higher hydroxyectoine concentrations. These results are also in good agreement to recent experimental findings for DPPC monolayers. We propose that the increase of the surface pressure in presence of hydroxyectoine which has been observed in our simulations is responsible for the experimentally observed broadening of the LE/LC phase transition in monolayers. The slight swelling of the DPPC bilayer has been also identified by an increased solvent accessible surface area. In terms of electrostatic interactions, we have observed that an increased electrostatic potential can be estimated for higher hydroxyectoine concentrations. The molecular origin of this effect is not clear, but it was proposed that the local ordering of the DPPC molecules contributes a significant amount to this observation [@Chachisvilis11]. It can be therefore assumed that further contributions may also contribute significantly to the fluidization of membranes in addition to pure hydration effects.\
In summary, we have shown that the usage of small concentrations of co-solutes in computer simulations will lead to observable effects in agreement to recent experiments. The presence of hydroxyectoine as a typical kosmotropic co-solute results in an increased surface pressure for DPPC bilayers which accounts for the experimentally observed broadening of the LE/LC phase transition.
Acknowledgments
===============
The authors thank Davit Hakobyan and Oliver Rubner for enlightening discussions and helpful remarks. Financial support by the Deutsche Forschungsgemeinschaft (DFG) through the SFB 858 and the transregional collaborative research center TRR 61 is gratefully acknowledged.
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abstract: 'Unmanned aerial vehicles provide new opportunities for performance improvements in future wireless communications systems. For example, they can act as relays that extend the range of a communication link and improve the capacity. Unlike conventional relays that are deployed at fixed locations, UAVs can change their positions to optimize the capacity or range on demand. In this paper, we consider using a swarm of UAVs as amplify-and-forward MIMO relays to provide connectivity between an obstructed multi-antenna equipped source and destination. We start by optimizing UAV placement for the single antenna case, and analyze its dependence on the noise introduced by the relay, its gain, and transmit power constraint. We extend our analysis for an arbitrary UAV swarm and show how the MIMO link capacity can be optimized by changing the distance of the swarm to the source and the destination. Then, we consider the effect of optimizing the positions of the UAVs within the swarm and derive an upper bound for the capacity at any given placement of the swarm. We also propose a simple near optimal approach to find the positions that optimize the capacity for the end-to-end link given that the source and the destination have uniform rectangular arrays.'
author:
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bibliography:
- 'references.bib'
title: |
UAV Swarms as Amplify-and-Forward MIMO Relays\
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Unmanned aerial vehicle (UAV), amplify and forward relay, MIMO capacity.
Introduction
============
Due to their mobility and low cost, unmanned aerial vehicles (UAVs) have found their way to many applications in recent years. Examples include package delivery, law enforcement, search and rescue, etc. Driven by this demand, UAVs are expected to become more prevalent, which will further drive the development of this technology and demand regulations that will allow for higher presence of UAVs in the low-altitude air space. Following this trend, UAVs are getting an increased attention in the telecommunications sector due to the multitude of opportunities they can provide [@zeng_opportunities_2016]. Using UAVs has recently emerged as an idea to respond to high localized traffic demands in next generation cellular networks [@li2017uav; @wu2018uav; @mozaffari2016efficient]. Beyond using UAVs as basestations, UAVs can be used as relays to extend the range of communication, boost capacity, or as a substitute for failed infrastructure in case of disasters.
Wireless relaying is one of the classical ways to improve data rates, while increasing reliability by combating shadowing. By using more than one relay along with multiple antennas at the source and the destination, multiple-input multiple-output (MIMO) relay networks are able to boost the capacity [@capacity_scaling_2006]. Traditionally, relaying approaches relied on using fixed relays. However, fixed relays, typically deployed on the ground, are unable to meet fluctuating demands or respond to failures in communications infrastructure. Deploying relays on UAVs provides new opportunity to exploit agility of motion of UAVs and the capacity improvements offered by wireless relays.
UAVs typically fly at altitudes of a few hundred feet above the ground, which could provide a line-of-sight (LOS) channel between an obstructed source and destination. By using multiple UAVs along with multiple antennas at the source and the destination, MIMO capacity gains can be leveraged. The MIMO capacity of this end-to-end link depends on the placement of UAVs, as they directly affect both the source-to-UAV and the UAV-to-destination channels. Hence, by optimizing the placement of individual UAVs within the swarm, capacity can be improved.
There is a significant interest in using UAVs as relays in recent literature. In [@chen_optimum_2018], the authors considered the optimal placement of a UAV relay that minimizes the outage probability. Other works have addressed the problem of finding the optimal trajectory and transmit power of a mobile relay [@zeng_throughput_2016; @jiang_power_2018]. In [@larsen_optimal_2017], the authors show that positioning a UAV asymmetrically between two ground nodes could result in better service than with a UAV placed at the center position when using stepwise adaptive modulation. All these works have only considered a single UAV relay and do not apply to MIMO links. In [@hanna2018distributed], algorithms for optimizing the placement of a UAV swarm were developed, but here only a single-hop link was considered. The placement of multiple UAVs as relays in double and multiple hop relay networks was optimized in [@chen_multiple_2018]. However, this work assumes that only a single UAV is transmitting at a time, thus full MIMO gains are not leveraged. In [@kalogerias2018spatially], the placement of multiple UAV relays in a dynamic channel is analyzed but the source and destination are assumed to have only single antennas.
In this work, we study the effect of changing the positions and the arrangement of a UAV swarm acting as amplify-and-forward relay cluster between a multiple antenna transmitter and receiver under an obstructed direct link. We also propose a method for optimizing the channel capacity by controlling the placement of the UAV relays. We start by considering the single antenna case and show how the UAV relay design parameters such as amplification gain and noise figure affect its position for maximized capacity. We derive an upper bound for the achievable capacity in case of a UAV swarm, and show that the single antenna analysis can be extended to the UAV swarm case. For the swarm, we demonstrate the gains of optimizing positions of the UAVs within the swarm and we propose a simple approach to find the positions that can attain the upper bound for a transmitter and receiver consisting of a uniform rectangular array for some separations between the swarm and the transmitter, while giving a better capacity than random placement on the average for all separations. The rest of the paper is organized as follows. We start by defining the system model used throughout this work in Section \[sec:system\_model\]. An analysis for the capacity that can be obtained by a single UAV or a UAV swarm is presented in Section \[sec:cap\], while Section \[sec:method\] proposes a method to find this placement for URA source and destination. Simulation results analyzing impact of the UAV swarm placement and configuration are presented in Section \[sec:simulations\]. Section \[sec:conclusion\] concludes the paper and presents directions for future research.
System Model {#sec:system_model}
============
![UAV swarm assisting obstructed MIMO link.[]{data-label="fig:setup"}](system_model){width="45.00000%"}
We consider a link between a transmitter with $ {N_{\mathrm{T}}}$ antennas and a receiver with $ {N_{\mathrm{R}}}$ antennas separated by a distance $ {R}$. We assume that direct communication between the transmitter and the receiver is not possible due to obstructions. We use ${N_{\mathrm{U}}}$ UAVs placed at distance ${R_1}$ from the transmitter, each equipped with a single-antenna, to enable this link and maximize its capacity. Each UAV acts as an amplify-and-forward relay which simply receives the signal, amplifies, and re-transmits it in a synchronized manner.
The channel between the transmitter and the UAVs is denoted by $ {{\boldsymbol{\mathrm{H}}}_1}\in {\mathbb{C}^{{N_{\mathrm{U}}}\times {N_{\mathrm{T}}}}} $, while the channel between the UAVs and the receiver is denoted by $ {{\boldsymbol{\mathrm{H}}}_2}\in {\mathbb{C}^{{N_{\mathrm{R}}}\times {N_{\mathrm{U}}}}} $. We assume that both channels are strong line-of-sight (LOS) channels where each element is given by $
\left[{{\boldsymbol{\mathrm{H}}}}\right]_{i,j} = \frac{\lambda}{4 \pi d_{i,j}} e^{\frac{j 2 \pi d_{i,j}}{\lambda}}
$ and $ \lambda $ is the wavelength of the signal and $ d_{i,j} $ is the distance between antenna $ i $ and $ j $ at transmitter, relay or receiver. The signal sent by the transmitter is given by $
{{\boldsymbol{\mathrm{x}}}_{{\mathrm{T}}}} = {\alpha_{{\mathrm{T}}}}{{\boldsymbol{\mathrm{s}}}}$ where $ {{\boldsymbol{\mathrm{s}}}}$ are the transmitted symbols having an identity covariance matrix, ${\alpha_{{\mathrm{T}}}} = \sqrt{{P_{\mathrm{T}}}}$, and ${P_{\mathrm{T}}}$ is the transmitted power. The signal received at the UAVs is given by $${{\boldsymbol{\mathrm{y}}}_{{\mathrm{U}}}} = {{\boldsymbol{\mathrm{H}}}_1}{{\boldsymbol{\mathrm{x}}}_{{\mathrm{T}}}} + {{\boldsymbol{\mathrm{n}}}_{{\mathrm{U}}}}$$ where ${{\boldsymbol{\mathrm{n}}}_{{\mathrm{U}}}}\in{\mathbb{C}^{{N_{\mathrm{U}}}}}$ is additive white Gaussian noise with covariance $ {\sigma^2_{{\mathrm{U}}}} {\boldsymbol{\mathrm{I}}}$. The signal received by the UAVs is amplified and transmitted as ${{\boldsymbol{\mathrm{x}}}_{{\mathrm{U}}}} = {{\boldsymbol{\mathrm{D}}}}{{\boldsymbol{\mathrm{y}}}_{{\mathrm{U}}}} $, where $ {{\boldsymbol{\mathrm{D}}}}$ is the amplification matrix which is defined as $ {{\boldsymbol{\mathrm{D}}}}= \text{diag}\{ {\alpha_{1}},\cdots,{\alpha_{{N_{\mathrm{U}}}}} \} $ where $ {\alpha_{i}} $ is the gain of the $ i $th UAV. The signal received at the destination is $${{\boldsymbol{\mathrm{y}}}_{{\mathrm{R}}}}= {\alpha_{{\mathrm{T}}}} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{D}}}}{{\boldsymbol{\mathrm{H}}}_1}{{\boldsymbol{\mathrm{s}}}}+ {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{D}}}}{{\boldsymbol{\mathrm{n}}}_{{\mathrm{U}}}} + {{\boldsymbol{\mathrm{n}}}_{{\mathrm{R}}}}$$ where $ {{\boldsymbol{\mathrm{n}}}_{{\mathrm{R}}}} $ is the additive white Gaussian noise with variance $ {\sigma^2_{{\mathrm{R}}}} $.
We assume that $ {\sigma^2_{{\mathrm{R}}}} = {\sigma^2_{}} $ where $ {\sigma^2_{}} $ is the thermal noise and $ {\sigma^2_{{\mathrm{U}}}} = {f_{{\mathrm{U}}}}{\sigma^2_{}} $ where $ {f_{{\mathrm{U}}}}$ is the noise figure of the relay (the ratio between the input and output SNR). The noise covariance matrix of the end-to-end channel is ${{\boldsymbol{\mathrm{\Sigma}}}_{}} = ( {\sigma^2_{{\mathrm{U}}}} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{D}}}}{{\boldsymbol{\mathrm{D}}}}^H {{\boldsymbol{\mathrm{H}}}_2}^H + {\sigma^2_{{\mathrm{R}}}}{\boldsymbol{\mathrm{I}}})$. The theoretical MIMO capacity of this end-to-end link is given by $$\label{eq:capacity_general}
C = \log_2(\det({\boldsymbol{\mathrm{I}}} + {\alpha_{{\mathrm{T}}}}^2 {{\boldsymbol{\mathrm{H}}}_1}^H {{\boldsymbol{\mathrm{D}}}}^H {{\boldsymbol{\mathrm{H}}}_2}^H {{\boldsymbol{\mathrm{\Sigma}}}_{}}^{-1} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{D}}}}{{\boldsymbol{\mathrm{H}}}_1}))$$
We consider two practical constraints on the relay amplifier. First, it has a maximum transmit power that it cannot exceed given by $ {P^{\max}_{{\mathrm{U}}}} $. Second, it has a maximum value of amplification gain given by $ {\alpha^{\max}_{{\mathrm{U}}}} $. This translates to the following constraints: ${\alpha_{i}}^2 |\left[{{\boldsymbol{\mathrm{y}}}_{{\mathrm{U}}}} \right]_i|^2 \leq {P^{\max}_{{\mathrm{U}}}}$ and $ {\alpha_{i}} \leq {\alpha^{\max}_{{\mathrm{U}}}} $.
The problem of interest is to find the placement of the UAVs which will maximize the capacity given by (\[eq:capacity\_general\]) while realizing the power and gain constraints. Optimizing power allocations for MIMO relays has been studied extensively in the literature (see [@relay_tutorial_2012] and the references within), so in this work we focus on UAV placement and use simple strategies to set the gains of the UAVs. Namely, we assume that all UAVs have the same gain ${{\boldsymbol{\mathrm{D}}}}= {\alpha_{U}} {\boldsymbol{\mathrm{I}}}$, which is of practical interest for simplified radio design in UAV relays and the reduced overhead of power allocation and control.
Capacity Analysis {#sec:cap}
=================
Single UAV relay {#subsec:one_uav}
----------------
We will start with the simple case of ${N_{\mathrm{T}}}={N_{\mathrm{R}}}={N_{\mathrm{U}}}=1$ in order to gain insights about optimal placements of an UAV relay based on gain and power constraints. We can identify two regions, the first where the UAV is closer to the transmitter and is limited by the maximum power it can transmit and the second where it is limited by the maximum gain. For the region of maximum gain, the expression for the capacity simplifies to $$\label{eq:single_uav}
C = \log_2 \left(1 + \frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 \lambda^4 }{ {\sigma^2_{}} {R_1}^2 ({f_{{\mathrm{U}}}}\lambda^2 {\alpha_{{\mathrm{U}}}}^2 (4\pi)^2 + (4 \pi)^4 ({R}-{R_1})^2)} \right)$$ Solving for the maximas we get[^2] $$\label{eq:roots_const_gain}
{R_1}= (1/4) ({3 {R}+ \sqrt{{R}^2-\frac{\lambda^2 {\alpha_{{\mathrm{U}}}}^2 {f_{{\mathrm{U}}}}}{2\pi^2}}})$$
As for the second solution $ {R_1}^{(2)} $, its existence in the feasible region $[0,{R}]$ requires that relay gain $ {\alpha_{{\mathrm{U}}}} $ and noise figure $ {f_{{\mathrm{U}}}}$ have low values. If these constants have a high value, i.e, the noise figure is too high or the amplification (which also amplifies the noise) is too high, there will be no feasible second root and the optimal capacity will be to move the relay closer to the transmitter.
Under maximum power constraint, we set ${\alpha_{{\mathrm{U}}}}^2 = \frac{{P^{\max}_{{\mathrm{U}}}}}{{\alpha_{{\mathrm{T}}}}^2|{h_1}|^2}$, and capacity becomes $$C = \log_2 \left( 1 + \frac{ {P^{\max}_{{\mathrm{U}}}}{\alpha_{{\mathrm{T}}}}^2 \lambda^2}{ (4\pi)^2 ({P^{\max}_{{\mathrm{U}}}} {f_{{\mathrm{U}}}}{\sigma^2_{}} {R_1}^2 + {\alpha_{{\mathrm{T}}}}^2 {\sigma^2_{}} ({R}- {R_1})^2) } \right)$$ The maximum capacity is achieved at $$\label{eq:roots_max_power}
{R_1}= {{\alpha_{{\mathrm{T}}}}^2 {R}}/({{P^{\max}_{{\mathrm{U}}}} {f_{{\mathrm{U}}}}+ {\alpha_{{\mathrm{T}}}}^2 })$$ We can see from this expression that the optimal ${R_1}$ that maximizes capacity gets closer to the transmitter as we increase the noise figure of the UAVs or increase its maximum transmit power (which would lead to noise amplification).
UAV Swarm Relay {#subsec:multiple_uav}
---------------
In this section, we derive an upper bound for the capacity of the UAV swarm MIMO relay, then we propose a method to achieve that capacity. Assuming all UAVs have the same gain, the capacity becomes $$C = \log_2(\det({\boldsymbol{\mathrm{I}}} + {\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2
{{\boldsymbol{\mathrm{H}}}_1}^H {{\boldsymbol{\mathrm{H}}}_2}^H ( {{\boldsymbol{\mathrm{\Sigma}}}_{}} )^{-1} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_1}))$$ where $ {{\boldsymbol{\mathrm{\Sigma}}}_{}} = {\sigma^2_{}} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_2}^H {f_{{\mathrm{U}}}}+ {\sigma^2_{}} {\boldsymbol{\mathrm{I}}}$. Finding the position of UAV swarm and placement of UAVs within the swarm that maximize this capacity is a challenging problem in general. The magnitude and phase of each element of both ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$ depend on distance between transmitter, UAV and receiver antennas. We start our analysis by observing that this capacity is equivalent to the capacity of the channel $ {\Tilde{{\boldsymbol{\mathrm{H}}}}}= ( {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_2}^H {f_{{\mathrm{U}}}}+ {\boldsymbol{\mathrm{I}}})^{-\frac{1}{2}} {{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_1}$. In the following, we derive the conditions on ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$ that maximize the capacity.
An upper bound on the capacity of the channel ${\Tilde{{\boldsymbol{\mathrm{H}}}}}$ is defined as $C\leq K\log(1+\frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 }{{\sigma^2_{}}K} \operatorname{Tr}({\Tilde{{\boldsymbol{\mathrm{H}}}}}{\Tilde{{\boldsymbol{\mathrm{H}}}}}^{H}))$, where $K= \min({N_{\mathrm{T}}},\ {N_{\mathrm{R}}},\ {N_{\mathrm{U}}})$, and it is reached when ${\Tilde{{\boldsymbol{\mathrm{H}}}}}$ has orthogonal columns. \[theorem:ubound1\]
The proof follows from the proof for a non-relay channel. The reader is referred to [@Tse_Wireless_2005 p. 295] for the details of the latter proof.
There are many candidates for the matrices $ {{\boldsymbol{\mathrm{H}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}$, and effectively positions of UAVs, that can realize this condition on $ {\Tilde{{\boldsymbol{\mathrm{H}}}}}$. From the expression for $ {\Tilde{{\boldsymbol{\mathrm{H}}}}}$, we can observe that one way to achieve this is to have $ {{\boldsymbol{\mathrm{H}}}_1}$ have orthogonal columns and $ {{\boldsymbol{\mathrm{H}}}_2}$ have orthogonal columns. We now derive a capacity equation for the case when this condition is satisfied.
\[thm2\] An upper bound on the capacity of the channel ${\Tilde{{\boldsymbol{\mathrm{H}}}}}$ is $C \leq K\log(1+\frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 }{{\sigma^2_{}}K} \sum_{i=1}^{K} \psi_{1,i}^{2} \frac{\psi_{2,i}^{2}}{1 + {f_{{\mathrm{U}}}}\psi_{2,i}^{2}})$, where ${\psi}_{1,i}$ and ${\psi}_{2,i}$ are the singular values of ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$ respectively, and $K = \min({N_{\mathrm{T}}},\ {N_{\mathrm{R}}},\ {N_{\mathrm{U}}})$. The upper bound is achieved when ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$ are orthogonal.
From the proof of Theorem \[theorem:ubound1\] we know that $ C \leq K\log(1+\frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 }{{\sigma^2_{}}K}\operatorname{Tr}({\Tilde{{\boldsymbol{\mathrm{H}}}}}{\Tilde{{\boldsymbol{\mathrm{H}}}}}^{H}))$. Furthermore, we can expand the trace term as $$\operatorname{Tr}({\Tilde{{\boldsymbol{\mathrm{H}}}}}{\Tilde{{\boldsymbol{\mathrm{H}}}}}^{H}) = \operatorname{Tr}(({f_{{\mathrm{U}}}}{{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_2}^{H}+I)^{-1}{{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_1}{{\boldsymbol{\mathrm{H}}}_1}^{H} {{\boldsymbol{\mathrm{H}}}_2}^{H})$$ Let ${{\boldsymbol{\mathrm{H}}}_2}= \boldsymbol{U}_{2}\boldsymbol{\Lambda}_{2}\boldsymbol{V}_{2}^{H}$ and ${{\boldsymbol{\mathrm{H}}}_1}= \boldsymbol{U}_{1}\boldsymbol{\Lambda}_{1}\boldsymbol{V}_{1}^{H}$, by the singular value decomposition, then $$\operatorname{Tr}({\Tilde{{\boldsymbol{\mathrm{H}}}}}{\Tilde{{\boldsymbol{\mathrm{H}}}}}^{H}) = \operatorname{Tr}(({f_{{\mathrm{U}}}}\boldsymbol{\Lambda}_{2}^{2}+I)^{-1}\boldsymbol{\Lambda}_{2}\mathbf{V}_{2}^{H}\boldsymbol{U}_{1}\boldsymbol{\Lambda}_{1}^{2}\mathbf{U}_{1}^{H}V_{2}\boldsymbol{\Lambda}_{2}^{T})$$ Using the fact that $\mathbf{U}_{1}^{H}\mathbf{V}_{2}$ is an orthogonal matrix, it can be shown that $$\label{eq:trace}
\operatorname{Tr}({\Tilde{{\boldsymbol{\mathrm{H}}}}}{\Tilde{{\boldsymbol{\mathrm{H}}}}}^{H}) \leq \sum_{i=1}^{K} \psi_{1,i}^{2} \frac{\psi_{2,i}^{2}}{1 + {f_{{\mathrm{U}}}}\psi_{2,i}^{2}}$$ and that this upper bound is satisfied in two cases. The first one is when $\mathbf{U}_{1}^{H}V_{2}$ is the identity matrix. The second case is when $\psi_{1,i}$ are all equal and $\psi_{2,i}$ are equal, i.e. $ {{\boldsymbol{\mathrm{H}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}$ have orthogonal columns. Finally, using (\[eq:trace\]) and Theorem \[theorem:ubound1\] we can establish that $$C \leq K\log_2 \left( 1+\frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 }{{\sigma^2_{}}K} \sum_{i=1}^{K} \psi_{1,i}^{2} \frac{\psi_{2,i}^{2}}{1 + {f_{{\mathrm{U}}}}\psi_{2,i}^{2}} \right)$$ and the bound is achieved when ${{\boldsymbol{\mathrm{H}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}$ both have orthogonal columns.
In the far field region, where the UAVs are at a large distance from the transmitter and the receiver compared to the relative size of the swarm, the magnitudes of channel coefficients in ${{\boldsymbol{\mathrm{H}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}$ depend on $ {R_1}$ only. We can rewrite as $ {{\boldsymbol{\mathrm{H}}}_1}\approx \frac{\lambda}{4\pi {R_1}} {{\bar{{\boldsymbol{\mathrm{H}}}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}\approx \frac{\lambda}{4\pi ({R}- {R_1})} {{\bar{{\boldsymbol{\mathrm{H}}}}}_2}$, where $ |\left[{{\bar{{\boldsymbol{\mathrm{H}}}}}_1}\right]_{i,j}| = |\left[{{\bar{{\boldsymbol{\mathrm{H}}}}}_2}\right]_{i,j}| = 1$. When ${{\boldsymbol{\mathrm{H}}}_1}$ and $ {{\boldsymbol{\mathrm{H}}}_2}$ both have orthogonal columns, i.e, $ {{\boldsymbol{\mathrm{H}}}_1}^{H}{{\boldsymbol{\mathrm{H}}}_1}= {N_{\mathrm{T}}}{N_{\mathrm{U}}}\mathbf{I}$ and $ {{\boldsymbol{\mathrm{H}}}_2}^{H}{{\boldsymbol{\mathrm{H}}}_2}= {N_{\mathrm{U}}}{N_{\mathrm{R}}}\mathbf{I}$, the capacity then becomes $$\label{eq:capSwarm}
C \leq K \log_2(1 + \frac{{\alpha_{{\mathrm{T}}}}^2 {\alpha_{{\mathrm{U}}}}^2 \lambda^4 \phi_1^2 \phi_2^2 }{ {\sigma^2_{}} {R_1}^2 ({f_{{\mathrm{U}}}}\lambda^2 {\alpha_{{\mathrm{U}}}}^2 (4\pi)^2 + {\psi_{2}}^2 (4 \pi)^4 ({R}-{R_1})^2)} )$$ where $\phi_1 = \max({N_{\mathrm{T}}},{N_{\mathrm{U}}})$ and $\phi_2=\max({N_{\mathrm{U}}},{N_{\mathrm{R}}})$. The optimization becomes similar to the single UAV relay case given in (\[eq:single\_uav\]). Note that this results holds only in the far field and the power received by each UAV and the destination antennas is almost equal. For small values of ${R_1}$ or values of ${R_1}$ close to ${R}$, this relation is a loose upper bound.
Proposed Method {#sec:method}
===============
In this section, we propose a method to find positions that optimize the channel capacity for a UAV relay swarm. For a single UAV, to find the optimal value of ${R_1}$, we only need to evaluate the capacity at the roots derived in (\[eq:roots\_max\_power\]) and (\[eq:roots\_const\_gain\]) within their respective regions and at the end points of the regions and then take the maximum point.
For the UAV swarm, the problem is more challenging since changing the position of one UAV affects both channel matrices. Prior works have considered the problem of antenna array design for LoS MIMO between one transmitter and one receiver. Unfortunately, the geometry of the optimal receiver array depends on the geometry of the transmitter array. For this work, we assume that both the source and destination antennas in our problem consist of uniform rectangular arrays (URAs). The transmit antennas consists of ${N_{\mathrm{T}}}={N_{\mathrm{T}}}^{(0)} \times {N_{\mathrm{T}}}^{(1)}$ antennas, where ${N_{\mathrm{T}}}^{({ \ensuremath{i} })}$ where ${ \ensuremath{i} }\in {0,1}$ corresponds to the dimension, having spacing ${ \ensuremath{d_{{\mathrm{T}}}} }^{({ \ensuremath{i} })}$. Similarly, the receivers consists of ${N_{\mathrm{R}}}={N_{\mathrm{R}}}^{(0)} \times {N_{\mathrm{R}}}^{(1)}$ and spacing ${ \ensuremath{d_{{\mathrm{R}}}} }^{({ \ensuremath{i} })}$.
If we only consider the first hop, the optimal capacity over this link occurs when the channel ${{\boldsymbol{\mathrm{H}}}_1}$ is orthogonal. This can be achieved when the UAVs are placed in a parallel URA having spacing [@bohagen_ura_2007] $$\label{eq:orth1}
{ \ensuremath{d_{{\mathrm{T}}}} }^{({ \ensuremath{i} })} { \ensuremath{d_{{\mathrm{U}}}} }^{({ \ensuremath{i} })} = \lambda {R_1}\left( { \ensuremath{m} }_{ \ensuremath{i} }+ 1 / \max{({N_{\mathrm{T}}}^{({ \ensuremath{i} })},{N_{\mathrm{U}}}^{({ \ensuremath{i} })})}\right)
\vspace{-1.5mm}$$
for some integer ${ \ensuremath{m} }_{ \ensuremath{i} }$ under the condition that the distance ${R_1}$ is much larger than the dimension of the antennas, i.e, in the far field. Similarly, for the second hop for some integer ${ \ensuremath{n} }_{ \ensuremath{i} }$ $$\label{eq:orth2}
{ \ensuremath{d_{{\mathrm{U}}}} }^{({ \ensuremath{i} })} { \ensuremath{d_{{\mathrm{R}}}} }^{({ \ensuremath{i} })} = \lambda ({R}- {R_1}) \left({ \ensuremath{n} }_{ \ensuremath{i} }+ 1/\max{({N_{\mathrm{U}}}^{({ \ensuremath{i} })},{N_{\mathrm{R}}}^{({ \ensuremath{i} })})}\right).
\vspace{-1.5mm}$$ If we find a value of ${ \ensuremath{d_{{\mathrm{U}}}} }$ for a given ${R_1}$, which satisfies both relations at the far field of both antennas, we can attain the capacity given in (\[eq:capSwarm\]). An example where this relation applies is when ${R_1}={R}/2$ and both the transmit and receive antennas have identical shapes (${N_{\mathrm{T}}}^{({ \ensuremath{i} })}={N_{\mathrm{R}}}^{({ \ensuremath{i} })}$, ${ \ensuremath{d_{{\mathrm{T}}}} }^{({ \ensuremath{i} })}={ \ensuremath{d_{{\mathrm{R}}}} }^{({ \ensuremath{i} })}$), then both equations (\[eq:orth1\]) and (\[eq:orth2\]) can be trivially realized. Unfortunately, such a value of ${ \ensuremath{d_{{\mathrm{U}}}} }$ does not exist for all ${R_1}$. Additionally, the value of ${R_1}$ that maximizes the capacity given by ($\ref{eq:capSwarm}$) depends on the value of gains and noise figure similar to the single UAV case. To address this we propose the following simple search algorithm.
We propose a two-step approach that first optimizes over ${R_1}$ and then finds spacing ${ \ensuremath{d_{{\mathrm{U}}}} }^{({ \ensuremath{i} })}$ that maximizes the capacity. To find the optimal ${R_1}$, we use the approach for a single UAV placement optimization. Then, to a find UAV spacing, we use the fact that the capacity is optimized by improving both links. Therefore, we search the values of spacings between the optimal ${ \ensuremath{d_{{\mathrm{U}}}} }^{({ \ensuremath{i} })}$ values of each link. This is done as follows: we use the relation (\[eq:orth1\]) to calculate ${ \ensuremath{d_{{\mathrm{U}}}} }^{1({ \ensuremath{i} })}$ and (\[eq:orth2\]) to calculate ${ \ensuremath{d_{{\mathrm{U}}}} }^{2({ \ensuremath{i} })}$. Then, for each of the dimensions, we scan the spacing of the UAVs between ${ \ensuremath{d_{{\mathrm{U}}}} }^{2({ \ensuremath{i} })}$ and ${ \ensuremath{d_{{\mathrm{U}}}} }^{1({ \ensuremath{i} })}$ and evaluate the capacity to find the spacing that results in a highest capacity.
Simulations {#sec:simulations}
===========
In this section, we aim to demonstrate the benefits of UAV placement optimization on the MIMO capacity and evaluate our proposed approach. The baseline scenario considered in the simulations assumes that the transmitter and receiver are $1$ km apart. They operate at carrier frequency of $5$ GHz over a narrowband channel. The transmitter power is set to $ {P_{\mathrm{T}}}= 12$ dBm, while the total power of the UAV swarm is $ {P^{\max}_{{\mathrm{U}}}} = 0 $ dBm and $ {\alpha^{\max}_{{\mathrm{U}}}}=45$ dB. The noise power corresponds to thermal noise with -174dBm/Hz over a bandwidth of 1 MHz. The UAVs are placed such that the lowest UAV has a height of 30 meters relative to the base of the transmit and receive antennas[^3].
![The capacity dependence on relay to transmitter distance for the single UAV case under maximum power constraint.[]{data-label="fig:max_power"}](one_relay0)
![The capacity obtained for 4 UAVs with NF=5dB. The green line is the theoretical upper bound. Dashed green line corresponds to near field. The orange line is the maximum capacity obtained using URA placement. The solid blue line shows the mean value of the achieved capacity using random placement. The dash-dotted and the dashed blue line represent the $5^{th}$ and $95^{th}$ percentile capacities respectively. Dotted blue curve represents the maximum capacity obtained via random placement. []{data-label="fig:swarm_rand2"}](multiple_relay_rand31)
![The capacity obtained for 4 UAVs with NF=12dB. The color and line mapping is the same as in Figure. \[fig:swarm\_rand2\][]{data-label="fig:swarm_rand1"}](multiple_relay_rand30)
We start by considering the single UAV relay scenario, where ${N_{\mathrm{T}}}={N_{\mathrm{R}}}={N_{\mathrm{U}}}=1$. Fig. \[fig:max\_power\] shows the capacity of a UAV as it moves between ${R_1}=10$m and ${R_1}=990$m. In that case, we can identify two regions. In the first region, when the UAV is closer to the transmitter, the UAV needs to use a value of gain lower than the maximum gain in order to avoid exceeding the maximum power. As it moves further, it increases $ {\alpha_{{\mathrm{U}}}} $, thus compensating for the decay of $ {h_1}$. In that case the maxima, if any, would be given by Eq.(\[eq:roots\_max\_power\]). In the second region, the UAV reaches the maximum possible gain and it cannot further increase its amplification. In that case, the maxima, if exists within the region where the relation applies, would be given by Eq. (\[eq:roots\_const\_gain\]).
For the UAV swarm relay, we consider the case where both the transmitter and receiver are $2\times2$ uniform rectangular arrays (URA) with 50 cm separation between antennas and a swarm consisting of 4 UAVs between the transmitter and receiver. Figures \[fig:swarm\_rand2\] and \[fig:swarm\_rand1\] show the capacities for different positions of the swarm and placements of UAVs within the swarm, assuming noise figures (NF) of 5 dB and 12 dB, respectively. In both figures, we show the upper bound given by (\[eq:capSwarm\]), in green color. It is interesting to note that capacities for single UAV relay and UAV swarm MIMO relay follow the same trend in the far-field regions. We also note that for small ${R_1}$, the swarm is in near-field where assumptions used for deriving capacity bound (\[eq:capSwarm\]) do not hold. Next, we evaluate whether random placement of UAVs, within a square area of 80m width, can achieve the capacity upper bound. Results in Figures \[fig:swarm\_rand2\] and \[fig:swarm\_rand1\] show maximum, average, $5^{th}$, and $95^{th}$ percentile over 5000 different placements at each ${R_1}$. These results show that there exists random placement that achieves the upper bound in the region where the UAVs are in the far-field of both antennas. For small values of ${R_1}$, when swarm is close to the transmitter, the gap between the upper bound and the obtained values is large. On the other hand, for large ${R_1}$, the noise covariance in (\[eq:capacity\_general\]) becomes ${{\boldsymbol{\mathrm{\Sigma}}}_{}} \approx ({{\boldsymbol{\mathrm{H}}}_2}{{\boldsymbol{\mathrm{H}}}_2}^H)^{-1}$ and the effect of ${{\boldsymbol{\mathrm{H}}}_2}$ cancels out, while conditions for far-field apply to ${{\boldsymbol{\mathrm{H}}}_1}$. As a result, in the region where swarm is closer to the receiver, the gap between achievable capacity and upper bound is smaller. We also evaluate whether proposed URA placement of UAVs can achieve the capacity upper bound. The results in Figures \[fig:swarm\_rand2\] and \[fig:swarm\_rand1\] show that there exists optimal URA placement that attains the upper bound around mid point, ${R_1}={R}/2$, as well as several values of ${R_1}$, where both channels can be made orthogonal. Although the proposed UAV placement based on URA geometry does not always attain the upper bound, it can always achieve a capacity better than 95% of random placements.
In Fig. \[fig:swarm\_4\_cap\] and \[fig:swarm\_4\_icn\] we investigate the orthogonality conditions of ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$ for different URA spacings. Fig. \[fig:swarm\_4\_cap\] shows the capacity and Fig. \[fig:swarm\_4\_icn\] shows the inverse condition number (ICN), ratio between the smallest and largest singular values, for both the matrices ${{\boldsymbol{\mathrm{H}}}_1}$ and ${{\boldsymbol{\mathrm{H}}}_2}$. The ICN is equal to 1 when a matrix has orthogonal columns. From Fig. \[fig:swarm\_4\_icn\], we can see that ${{\boldsymbol{\mathrm{H}}}_2}$ becomes orthogonal for two values of ${ \ensuremath{d_{{\mathrm{U}}}} }$ within that range (corresponding to two different values of ${ \ensuremath{n} }$ in (\[eq:orth2\])) and ${{\boldsymbol{\mathrm{H}}}_1}$ becomes orthogonal at only one value. By sweeping the spacing in the range of ${ \ensuremath{d_{{\mathrm{U}}}} }$’s that orthogonalize each channel, the algorithm can find the spacing that maximizes capacity.
Conclusion and Future Work {#sec:conclusion}
==========================
In this paper, we studied the capacity improvements of an obstructed MIMO link enabled by a UAV swarm acting as an amplify-and-forward MIMO relay under a line-of-sight channel. Our analysis revealed that significant gains can be achieved by optimizing the placement of the position of the entire swarm as well as the UAVs within the swarm. We derived an upper bound of the capacity for UAV swarm. An algorithm that approaches this upper bound is proposed for the URA transmitter and receiver cases. Further research is needed to find a tighter upper bound for a planar UAV swarm placement and algorithms to achieve it when the UAVs are closer to the transmitter or receiver. Additionally, joint optimization of power gains and locations is a promising approach to further improve capacity.
[^1]: This work was supported in part by the CONIX Research Center, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA.
[^2]: Setting $ {R_1}$ to be equal to zero, while maximizes the expression, is not practical and violates the underlying assumptions of the LOS model.
[^3]: As the height increases to values that are significant with respect to ${R_1}$, the assumption of having equal received power at the UAVs gets violated, which makes it not possible to achieve the upper bound.
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abstract: 'The gravitational wave signal from a compact object spiralling toward a massive black hole (MBH) is thought to be one of the most difficult sources to detect in the LISA data stream. Due to the large parameter space of possible signals and many orbital cycles spent in the sensitivity band of LISA, it has been estimated previously that of the order of $10^{35}$ templates would be required for a fully coherent search with a template grid, which is computationally impossible. Here we describe an algorithm based on a constrained Metropolis-Hastings stochastic search which allows us to find and accurately estimate parameters of isolated EMRI signals buried in Gaussian instrumental noise. We illustrate the effectiveness of the algorithm with results from searches of the Mock LISA Data Challenge round 1B data sets.'
author:
- Stanislav Babak
- 'Jonathan R. Gair'
- 'Edward K. Porter'
bibliography:
- 'EMRI1BSearch.bib'
title: An algorithm for detection of extreme mass ratio inspirals in LISA data
---
Introduction {#sec:intro}
============
Extreme mass ratio inspiral (EMRI) — a stellar mass compact object (CO) that is captured and spirals into a massive Black Hole (MBH) through the emission of gravitational radiation — is one of the most interesting sources for the future LISA mission [@LISA:1998]. The inspiral proceeds very slowly for these sources (the inspiral rate is proportional to the mass ratio which is typically $1:10^5$) so the CO spends a significant amount of time in the strong field close to the MBH. The GW signal contains information about the geometry of the central hole which, in the case of a strong signal, could be extracted to confirm or otherwise that the massive objects observed in galactic nuclei are indeed Kerr BHs, as we suppose [@ryan95; @collins2004; @GlamBab06; @gairbumpy; @AKtest]. EMRI observations may also be used to probe the stellar population in the central parsecs of galaxies, and measure the properties of astrophysical black holes to high precision [@gairLISA7]. This is possible because the signal is long lived and so a coherent phase integration should recover the parameters of the binary to high accuracy, much better than anything that will be available from electromagnetic observations. We refer the reader to the review article [@AmaroSeoane:2007aw] for more details on the astrophysics that will be possible with EMRIs.
In order to scope out issues associated with LISA data analysis for EMRIs, we require waveform models that are cheap and easy to generate but still capture the main features of true EMRI waveforms. One such model of the signal is the so-called analytic kludge waveform [@Barack:2003fp]. It is a phenomenological template, constructed by piecing together the most important physical elements: post-Newtonian expressions for the rate of change of the orbital parameters and frequencies (Peter-Mathews approach), periastron precession, and precession of the orbital plane around the spin axis of the MBH. While these waveforms are not faithful representations, they are nonetheless representative of the true signal. This means that they should be sufficient to answer questions about what accuracy we can achieve in estimating the source parameters and at what level the confusion noise from cosmological EMRIs will be [@Barack:2003fp; @Barack:2006pq]. Because these waveforms are simple and fast to generate they were chosen for use in the Mock LISA Data Challenge (MLDC). The MLDC was organized to stimulate the development of data analysis tools for LISA and to establish standard notations and conventions which allow comparison of different algorithms. There have been three challenges to date [@Arnaud:2006gm; @Arnaud:2006gn; @Arnaud:2007jy; @Arnaud:2007vr; @2007arXiv0711.2667B], which were aimed at different sources: SMBH binaries, Galactic white-dwarf binaries and EMRIs. For the two EMRI challenges five data sets were released, each containing a single EMRI signal buried in instrumental noise.
The five EMRI data sets had some parameters drawn from priors common to all data sets, which were: the mass of the CO $\mu \in U[9.5, 10.5]M_{\odot}$, the spin $S/M^2 \in U[0.5, 0.7]$, the plunge time $U[1,2]$ years and the eccentricity at plunge $e_{pl} \in U[0.15, 0.25]$; and some parameters drawn from priors that were different for each data set: MBH mass $M \in U[0.95, 1.05] \times 10^7 M_{\odot}$ (high mass binary) with SNR$\in U[40, 110]$ (1.3.1 data set), $M \in U[4.75, 5.25] \times 10^6 M_{\odot}$ (medium mass binary) with SNR$\in U[70, 110]$ (1.3.2 data set) and with SNR$\in U[40, 60]$ (1.3.3 data set), $M \in U[0.95, 1.05] \times10^6 M_{\odot}$ (low mass binary) with $SNR \in U[70, 110]$ (1.3.4 data set) and with $SNR \in U[40, 60]$ (1.3.5 data set). Note that the last two types of signal are considered to be the most likely EMRIs to be seen by LISA: a $\sim 10 M_{\odot}$ BH falling into $\sim 10^6 M_{\odot}$ MBH in the galactic center [@gairLISA7]. More details on the parameter sets can be found in [@Arnaud:2007jy]. MLDC round 1[^1] and round 1B had the same sets of priors for the five data sets. For both challenges, it has been shown that the signal can be easily detected with high confidence, but with parameters quite different from the true ones.
An important feature of EMRI signals is that they have many local maxima in the likelihood surface, which are quite well separated and can be as high as $75\%$ of the true maximum. Search algorithms have a tendency to find secondary maxima quickly and then get stuck there. This represents a true detection but with incorrect parameters (see [@2007arXiv0711.2667B; @Babak:2008sn] for results). However in this article we will only regard a “detection” as finding the global maxima in the likelihood, i.e., the true source parameters. Secondary maxima are the biggest problem for LISA data analysis. Some signals can be seen by eye in a spectrogram or in the power spectral density of the data — the main problem is to estimate the source parameters with the best possible accuracy. A grid based search (with a sufficiently fine grid) would be guaranteed to find the global maxima as it covers the whole parameter space, but the required number of templates is so high [@emrirate] that no one is presently considering doing it even with tight priors like in the MLDC. An alternative approach which has proven to be both efficient and accurate was first suggested in the context of LISA for non-spinning SMBH binary searches [@Cornish:2006dt; @Cornish:2006ms; @Cornish:2007jv]. This approach is the semi-stochastic Metropolis-Hastings Monte-Carlo (MHMC) method, where one constructs a search chain through the parameter space (these are not in general Markovian) and follow this up by a Markov chain Monte-Carlo (MCMC) to sample the posterior distribution function. We say this approach is ‘semi-stochastic’ since, although successive points in the chains are chosen at random, they are chosen from [*directed*]{} proposal distributions. This method involves generating templates as the chain moves, but its power lies in the fact that the number of points usually required to find the source is many fewer than in a full template grid. However, the chains can get stuck on local maxima. One needs to use the properties of the signal to make chains move off local maxima and explore the parameter space more widely.
By including tricks such as simulated annealing — “heating” the likelihood surface by scaling both the log-likelihood and the size of proposed jumps by a temperature factor — and frequency annealing — systematically increasing the range of frequencies included in the waveform template — MHMC has solved the search problem for non-spinning SMBH binaries [@Cornish:2006dt; @Cornish:2006ms; @Cornish:2007jv]. The full utility of MHMC for EMRI searches has so far not been demonstrated. In the round 1B MLDC release, one of the five EMRIs (of 1.3.1 type) was found by an MHMC technique [@Cornish:2008zd] that used simulated annealing and began with searches on sub-segments of the data that were combined before finally running a long chain on the whole data set. The authors [@Gair:2008zc] also attempted to search for EMRIs in the round 1B data using MHMC, and recovered close to the true parameters for one of the five signals (of 1.3.2 type) before the Challenge deadline. However, in both round 1 and round 1B, the best performing algorithm was not MHMC based, but a time-frequency analysis [@tf1; @tf1B]. Such searches are easier to implement and parameter estimation can be done if the EMRI is isolated and of sufficient brightness. However, the achievable accuracy of parameter estimation using such template-free techniques is not as good as template based methods, and the algorithm will suffer in the presence of multiple source confusion.
In this paper we describe for the first time a complete template-based EMRI search that is able to detect and recover accurate parameters for bright, isolated EMRI sources buried in instrumental noise, with parameters drawn from any of the five canonical MLDC EMRI source types. This search technique is based on our previous MHMC search, but with several improvements which we describe below.
The origin of all of the secondary maxima in the EMRI likelihood surface is in the characteristics of the signal. An EMRI signal is composed of many harmonics of the three fundamental orbital frequencies (of the radial $r$-motion, the azimuthal $\phi$-motion and the polar $\theta$-motion), which are evolving in time. These harmonics vary in strength (amplitude) and the local maxima arise from matching the phase of the strongest (or of several strong) harmonics for some period of time. It is possible for a signal with very different parameters to match the dominant harmonic very well for the whole duration of the signal but miss completely all the other harmonics. We have tried to exploit this property by using several chains to identify the dominant harmonic and then impose a constraint between the fundamental frequencies that fixes the frequency of the dominant harmonic at some reference time. The key idea of our search is to determine the frequency of the dominant harmonic/harmonics using several local maxima and this was used for the 1B submission [@Gair:2008zc]. However, since the round 1B MLDC deadline, we have improved our search technique in three important ways. We have changed the parametrization of the signal, so that it is specified by the three orbital frequencies at some reference time, $t_{ref}$, and we have changed the proposal distribution accordingly. We use two main proposal distributions: a normal multivariate in the eigendirections of the Fisher Matrix and a variation of the Metropolis random walk which we will describe later. The second important improvement was to release the constraint after a certain point and let the chains correct the frequency of the dominant harmonic at $t_{ref}$. Finally, we have also improved the efficiency of generation of the templates by a factor of 3–5 which has allowed us to implement an analytic maximization of the likelihood over the initial phases, which reduces the parameter space that must be searched. This is possible because we have developed a new type of template composed of $N$ independent harmonics with frequency evolution defined from the analytic kludge model. From this model we can construct an $N$-dimensional $F$-statistic [@Jaranowski:1998qm]. This will be described later. These improvements have led to the success of the algorithm. We have analysed the “blind” data sets (the data sets which MLDC participants were supposed to analyze and return results for) from MLDC round 1B to tune the algorithm, and then analysed two other data sets using the search pipeline in a blind analysis. We successfully found the signal and determined the true parameters of the source for each of the seven data sets. The results of our search are summarized in Tables \[results\] and \[resultsBlind\] to follow. In the following sections we give details of the search algorithm.
The paper is organized as follows. In section \[model\] we describe the signal model we have constructed for our search templates. The details of our search method are given in section \[search\]. We discuss the results of our search in section \[resSec\], before concluding with a summary in section \[sum\].
Waveform Model {#model}
==============
The analytic kludge model of EMRI signals, as used to generate the data sets for the MLDC is described in [@Barack:2003fp] and the particular implementation used for the MLDC can be found in [@Arnaud:2007jy]. For our search, we have simplified the model in order to reduce computational time. The signal can be described by harmonics of three fundamental orbital frequencies: $\nu, f_{\gamma} \equiv \dot{\tilde{\gamma}}/(2\pi), f_{\alpha} \equiv \dot{\alpha}/(2\pi)$, where a dot denotes a derivative with respect to time. The frequencies evolve according to the PN expressions $$\begin{aligned}
\frac{d\nu}{dt} &=&
\frac{96}{10\pi}(\mu/M^3)(2\pi M\nu)^{11/3}(1-e^2)^{-9/2}
\bigl\{
\left[1+(73/24)e^2+(37/96)e^4\right](1-e^2) \nonumber \\
&&+ (2\pi M\nu)^{2/3}\left[(1273/336)-(2561/224)e^2-(3885/128)e^4
-(13147/5376)e^6 \right] \nonumber \\
&&- (2\pi M\nu)(S/M^2)\cos\lambda (1-e^2)^{-1/2}\bigl[(73/12)
+ (1211/24)e^2 \nonumber \\
&&+(3143/96)e^4 +(65/64)e^6 \bigr]
\bigr\}, \label{nudot} \\
\frac{df_{\gamma}}{dt} &=& \left[(2\pi\nu M)^{2/3} (1-e^2)^{-1}
\left[5+\frac{7}{4}(2\pi\nu M)^{2/3} (1-e^2)^{-1}(26-15e^2)\right]
-12\cos\lambda (S/M^2) (2\pi M\nu)(1-e^2)^{-3/2}\right]\frac{d\nu}{dt} \nonumber \\
&&+ \left\{6\nu(2\pi\nu M)^{2/3} (1-e^2)^{-1}
\left[1+\frac{11}{2}(2\pi\nu M)^{2/3} (1-e^2)^{-1}\right] \right.\nonumber \\&&
\left. \qquad-18\nu\cos\lambda (S/M^2) (2\pi M\nu)(1-e^2)^{-3/2}\right\}\frac{e}{(1-e^2)}\frac{de}{dt},
\label{fgamdot} \\
\frac{df_{\alpha}}{dt} &=& 2\nu (S/M^2) (2\pi M\nu)(1-e^2)^{-3/2} \left(\frac{1}{\nu} \frac{d\nu}{dt} + \frac{3e}{(1-e^2)}\frac{de}{dt}\right),\label{falphadot} \\
\frac{de}{dt} &=& -\frac{e}{15}(\mu/M^2) (1-e^2)^{-7/2} (2\pi M\nu)^{8/3}
\bigl[(304+121e^2)(1-e^2)\bigl(1 + 12 (2\pi M\nu)^{2/3}\bigr) \, \nonumber \\
&&- \frac{1}{56}(2\pi M\nu)^{2/3}\bigl( (8)(16705) + (12)(9082)e^2 - 25211e^4
\bigr)\bigr]\,
\nonumber \\
&&+ e (\mu/M^2)(S/M^2)\cos\lambda\,(2\pi M\nu)^{11/3}(1-e^2)^{-4}
\, \bigl[(1364/5) + (5032/15)e^2 + (263/10)e^4\bigr] ,
\label{edot}\end{aligned}$$ The harmonic structure of the signal is best seen when using a static source frame defined by the spin of the MBH which is assumed to be constant in this model. The radiative frame is then constructed using the direction of propagation (or direction to the source from the solar system barycenter (SSB)) and the spin direction of the MBH. The advantage of those two frames is that they are static and all the time dependence is encoded in the amplitude and phases of the harmonics explicitly. In the original analytic kludge paper [@Barack:2003fp], the waveform was expressed relative to a precessing frame, tied to the orbital angular momentum, which makes it more complicated to compute the harmonic decomposition. In the static SSB frame, the signal takes the following form h \~(2M (t))\^[2/3]{} \_[l,n,m]{} A\_[l,n,m]{}(e(t)) e\^[i(n(t) + l(t) + m(t))]{} The amplitude of each harmonic depends on the source location (ecliptic coordinates), orientation of the spin and the orbital eccentricity. These expressions are known analytically, but are messy so we do not include them explicitly here. We have examined the amplitudes of the harmonics for a wide range of parameters and find that harmonics of the perihelion precession with $l\neq2$ are significantly suppressed. We can also neglect the contribution from harmonics of the orbital frequency with $n>5$ for orbital eccentricities less than $e \sim 0.65$. Moreover, by construction, the analytic kludge waveforms are quadrupolar and therefore only harmonics of the orbital plane precession frequency with $m\in[-2,2]$ are allowed. This will not be the case for real EMRI signals and more sophisticated models include higher multipoles [@NK; @genTB]. Knowledge of the analytic form of the harmonic amplitudes and the restriction of the number of harmonics, to as few as $\sim4$–8 dominant harmonics in most cases, allows us to simplify the template and make its generation more efficient. The amplitudes of the harmonics depend on Bessel functions, with argument, $ne(t)$, that is usually small, so a further simplification follows by expanding these as Taylor series and truncating at the desired level of accuracy.
The technique of Time-Delay Interferometry (TDI) [@lrr-2005-4] will be used in order to cancel the laser noise in the LISA data. The basics for this technique is to combine the data sent and received by different spacecraft with time delays chosen to cancel the common laser noise component. The LISA response function is consequently somewhat complicated, although it can be significantly simplified in the long wavelength limit $\omega_{GW} L \ll 1$ [@Cornish:2002rt] ($L$ is LISA’s arm length $\sim 16.7$s[^2] and $\omega_{GW}$ is the GW frequency). For an EMRI into a lower mass MBH (1.3.4/1.3.5 type source), the frequency of the GWs can be quite high and neither the long wavelength nor rigid adiabatic approximations [@Cornish:2002rt] are valid. Our code uses the full response but with time delays applied only to $h_{GW}^{SSB}$ and not to the LISA motion — treating LISA as a solid rotating triangle. This is overkill for the high mass MBH EMRIs (1.3.1 type) (and probably for the medium mass, 1.3.2 type, sources as well), but we decided to use the same codes for all searches. We have saved on computing time for the higher mass systems by using a lower sampling rate during the integration of the orbital motion and then up-sampling while generating the TDI streams. We also use linear interpolation to compute the time delayed data from a regularly space SSB time series, rather than a more complicated and expensive interpolation scheme.
We have verified our waveform templates against full analytic kludge templates generated using [*SyntheticLISA*]{} [@Vallisneri:2004bn], by computing the overlap, which is the inner product, (s|h) = 2 df, \[olp\] between two normalized $(s|s) = (h|h) =1$ signals. In the expression above, a tilde denotes the Fourier transform, and $S_h(f)$ is the one-sided noise power spectral density. We have found that the overlap between our approximate model and the accurately computed templates is in the range $[0.93-0.99]$ depending on the source parameters, in particular the mass of the MBH (the overlap is usually higher for high mass MBH EMRIs). The loss in overlap comes primarily from mismatches in the amplitude, while the phase is tracked very well. This is to be expected, as we do not make any approximations in our computation of the evolution of the orbital parameters and frequencies. The small mismatch between the template and the signal will lead to a bias in the parameters estimated for the signal. A mismatch in amplitude will primarily affect the estimated signal-to-noise ratio/luminosity distance for the source, while a phase error will lead to errors in all of the intrinsic parameters. We have found that our model is very faithful, with typical model-induced parameter errors for MLDC source types being $\sim1$–$2\sigma$, where $\sigma$ is the parameter error as estimated from the Fisher-Matrix. In other words, we expect the model-error to be of similar size, but no larger than the error in parameter recovery that arises from instrumental noise in the detector. This is confirmed by the results of our search summarised in Tables \[results\]–\[resultsBlind\]. We see that our parameter recovery was very good, except for the SNR which was as much as $\sim5\%$ different in two of the low mass MBH cases.
Search Algorithm {#search}
================
In this section we will describe the overall search algorithm. In practice, there were some differences between the searches for each source and we will discuss these source-specific details in the next section. Our search consists of three steps. In the first step we look for the “footprints” of the signal, and for points from which we can seed our subsequent chains. In the second step we construct chains using the identified properties of the signal, via a constrained Metropolis Monte-Carlo search on the half year long segments of data. The final step is to narrow down the parameters of the signals by extending the duration of the templates to the total length of observation (this is similar to what was done here [@Cornish:2008zd]). In the following subsections we will give details on the implementation of the three steps.
Uniform Jumps
-------------
As mentioned previously, the basis of our search method is to identify as many strong local maxima in the likelihood as possible and then use the information encoded in the points to direct the search toward the true solution. The first step is very simple and remarkably efficient. In spirit it is similar to using a random template bank, as used in [@Messenger:2008ta]. We generate template waveforms for the last half a year of inspiral before plunge by integrating the equations of motion backwards), and with parameters randomly chosen from within uniform priors. For greater efficiency we include maximization of the log likelihood over the distance, plunge time and three orbital phases at plunge. We maximize over the plunge time in the usual way, by computing the correlation of the template with the data (instead of the inner product). The maximized value of the plunge time is then used for constructing a new filter and we compute the likelihood maximized over phases and distance. The maximization over the distance is done in the usual way: the log-likelihood (up to a constant factor) is given by = -\_[I]{}(x\_I -h\_I | x\_I -h\_I ) \~\_I 2(x\_I | h\_I) - (h\_I|h\_I), where $I= \{A,E\}$ runs over orthogonal TDI streams [@lrr-2005-4] which play the role here of independent detectors, $x_I = n_I + s_I$ is the corresponding TDI data which is combined out of the noise $n_I$ and a signal $s_I$; $h_I$ is a template and the inner product is defined in Eq. (\[olp\]) above.
An amplitude factor, ${\cal A}$ (inversely proportional to the luminosity distance to the source) can be factored out $h_I = A \hat{h}$ and maximized over analytically. The maximum likelihood estimator for the amplitude is = . and then the maximized log of likelihood is \_[max ([A]{})]{} = \[maxL\] This value is sometimes referred to as SNR$^2$, since if $s_I = h_I$, it reduces to $\sum_I (h_I | h_I)$, which is the square of the matched-filtering signal-to-noise ratio. Note that Eq. (\[maxL\]) is not sensitive to the sign of the inner product.
Maximization over the three initial orbital phases is more involved. We consider a template which is constructed out of three bright harmonics only. The brightness of each $m$-harmonic for a given set of source parameters depends only on the inclination angle $\lambda$ and on the inclination of the MBH’s spin to the direction to the source from the SSB [@tf1B]. The prior range on plunge eccentricity ensures that we would usually have $n=2$ and/or $n=3$ as the dominant harmonics. This reasoning suggests we take the following harmonics $n_0 = n_1 = 2,\;\; m_0\ne m_1,\;\;\; n_2 = 3,\;\; m_2=m_0$, with $m_0$ the brightest of $m_0$, $m_1$. The three initial phases for harmonics $h^{(i)}$ are \_0\^i = n\_i\_0 + 2\_0 + m\_i\_0. Each harmonic is of the form $\cos(\Phi_0^i+ \tilde{\phi}^i(t))$ and hence may be decomposed as h\^[(i)]{} = A\^i\_0 \^[(i)]{}(0) - A\^i\_0 \^[(i)]{}(/2), here $\tilde{h}^{(i)}(0)$ means taken at zero initial phase. The three-harmonic template can therefore be written h\^c = h\^[(0)]{} + h\^[(1)]{} + h\^[(2)]{} = \_[j=0]{}\^5 a\_j h\^j. \[3harm\] Omitting all cross harmonic terms we can analytically maximize the likelihood of our template $h^c$ over all values of the constants $a_j$, in a similar way to the $F$-statistic, a\_[2i]{} = , a\_[2i+1]{} = , \[acoef\] where $i=0,1,2$. This leads to the following maximum likelihood estimators for the amplitude and phase of the harmonics: \_0\^j = , A\^j\_0 = \[physMax\] With the choice of harmonics given above, we can obtain the initial orbital phases from the maximum likelihood estimates of the harmonic amplitudes and phases: \_0 &=&\
\_0 &=& 1[2]{}\
\_0 &=& . After a new plunge time is determined from the correlation analysis, we estimate the initial phases using the above method. This guarantees that we have chosen the optimal phases if at least one of the harmonics in the template matches the signal. We then use the maximised phases to compute the log-likelihood, Eq. (\[maxL\]). For this stage of the search, the longer we run the more good points we get. We typically use about 100 - 200 CPUs for several days in this phase, and normally identify a few dozen distinct secondaries with SNR of about 20% - 40% of the maximum.
Search on subsets of data
-------------------------
The next stage is to split the data into half-year long segments and to run a Markov chain Monte Carlo (MCMC) using the Metropolis rejection/acceptance rule [@metrop]. The MCMC technique works as follows: given a data set $s(t)$ and a set of templates $h(t;\vec{x})$, we choose a starting point, $\vec{x}$, in the parameter space. We then propose a jump to another point, $\vec{y}$, in the space by drawing from a certain proposal distribution, $q(\vec{y}|\vec{x})$, and evaluate the Metropolis-Hastings ratio $$H = \frac{\pi(\vec{y})p(s|\vec{y})q(\vec{x}|\vec{y})}{\pi(\vec{x})p(s|\vec{x})q(\vec{y}|\vec{x})}.
\label{MHrat}$$ Here $\pi(\vec{x})$ are the priors of the parameters, which, in our analysis, were taken to be uniform distributions within the ranges allowed by the MLDC. The function $p(s|\vec{x})$ is the likelihood $$\label{eqn:likelihood}
p(s|\vec{x}) = C\,e^{-\left<s-h\left(\vec{x}\right)|s-h\left(\vec{x}\right)\right>/\Theta},$$ where $C$ is a normalization constant and $\Theta=2$ without annealing. This jump is then accepted with probability $\alpha = \min(1,H)$, otherwise the chain stays at $\vec{x}$. In our search, we use the Metropolis rejection/acceptance rule which simplifies the above by assuming the proposal $q(\vec{y}|\vec{x})$ is symmetric, so the ratio (\[MHrat\]) is just the product of the likelihood ratio with the prior ratio. In this stage we also include simulated annealing, which means that $\Theta$ is allowed to vary from $2$. This has the effect of smoothing and flattening the likelihood surface, which makes it easier for the chain to move around and climb up the surface to the maximum. The idea is to have a high heat initially, to encourage the chain to explore widely and find the global maximum, then cool the surface so the chain locks into the vicinity of the maximum. We vary the temperature as the chain advances according to a schedule of the form $$\Theta = 2 \times \left\{ \begin{array}{ll} \left({\rm SNR_0}/{\rm SNR}\right)^3
& {\rm SNR} \le {\rm SNR_0}\\
1 & {\rm SNR} > {\rm SNR_0}
\end{array}\right.,
\label{heatsch}$$ where $SNR_0$ is typically $6-7$. This annealing scheme is used at the beginning of the search and helps to find the trace of the signal quickly by exploring widely in the large parameter space. At later stages of the search we also made use of thermostated annealing, as described in [@Cornish:2006ms; @Cornish:2007jv], which encourages the search chains to explore the vicinity of identified maxima.
At this stage of the search, we employ a different parametrization of the template, by prescribing the three orbital frequencies at some reference time $t_{ref}$ (usually chosen in the middle of the segment) instead of the mass and spin of the MBH. Then we start $p$ chains, where $p$ is the number of interesting points found in step one.The second step is based on the assumption that the points found in the first stage are not far in the parameter space from maxima (local/secondary or global/primary) on the likelihood surface. The MCMC is efficient at finding the maxima. We have been unsuccessful in attempts to make the chains efficiently jump between local maxima until they find the global one. Instead we will show how we can use the information stored in each maximum to guide the search in the right direction.
We use two main proposals, $q(\vec{y}|\vec{x})$, in the MCMC: (i) jumps within the scaled ambiguity ellipsoid (defined by the eigenvectors and eigenvalues of the variance-covariance matrix); (ii) jumps which (almost) preserve the frequency of the dominant harmonic. Let us give some more details. Following [@Balasubramanian:1995bm; @Owen:1998dk] we introduce the metric on the parameter space: ds\^2 = g\_ dx\^dx\^, g\_ = (h\_[,]{} | h\_[,]{}), where $h_{,\mu} = \partial h/\partial x^{\mu}$ and $x^{\mu}$ are parameters of the template. We can determine from the metric its eigenvectors, ${\bf V}_i$, and eigenvalues, $\lambda_i$, and hence write $g = V^T L V, $, where $V$ is the matrix of eigenvectors and $L$ is the diagonal matrix of eigenvalues. We introduce a new parametrization: $$ds^2 = dW^T dW, \;\;\;\; dW_i = dY_i \sqrt{\lambda_i}\;\;\;\;
dY = V^T dX$$ where $dX$ is a vector of parameters. We choose a value of $ds^2$ according to a gaussian distribution, $|\mathcal{N}(0,1)|$, and choose the direction vector $dW$ randomly oriented on the hyper-sphere with radius $ds$. In other words this proposes jumps on the surface of the ambiguity ellipsoid scaled according to the chosen $ds$. Another similar proposal which was used for the search of some data sets makes normal jumps in the eigendirections. This latter proposal was first suggested in [@Cornish:2006ms]. Note that for both proposals, the jumps are further scaled by the temperature when using the simulated annealing scheme, to ensure larger jumps when the surface is “hot”.
The second proposal which we have found to be efficient at the beginning of the chain is based on an estimation of the frequency of the dominant harmonic. For each of the high SNR points identified in the first step we can compute the frequency of all harmonics at the reference time, $t_{ref}$. In general, all of the points agree on the frequencies of the dominant harmonics, with a small dispersion, $\sigma$. An example from our blind search is shown in Figure \[harms\]. One can clearly see that all the points agreed about the frequency of the harmonic $l=2,\;\; m=2$. The scatter reflects the relative amplitude of the harmonics: the weakest will have the largest dispersion. One notices that some points managed to match the $m=2$ harmonic of the signal with $m=1, \;\; m=0$ or even $m=-1$ harmonic of the template (with completely wrong parameters).
![Harmonics $l=2$, $m=[-2,2]$ for the best points identified in the first stage of the search. The frequencies are computed at $t_{ref}$ in the middle of the third half year. The dominant harmonic is $m=2$ (triangles down), the one with the smallest dispersion.[]{data-label="harms"}](Harmonics){height="0.45\textheight"}
The idea of this proposal is to choose $F_{l=2, m=2}$ according to a distribution $N(\overline{F_{l=2, m=2}}, \sigma_{2,2})$, where $\overline{F_{l=2, m=2}}$ and $\sigma_{2,2}$ are the mean value and the standard deviation estimated from the points found in the first stage. Values for $f_{\alpha}$ and $f_{\gamma}$ are chosen from normal distributions in the same way. The value for $\nu$ is then defined from the constrain: $\nu = F_{l=2, m=2}/2 - f_{\gamma} - f_{\alpha}$. This proposal works well in the beginning of the chains to refine the frequencies given other parameters.
There are two reasons why we carry out the search on short duration segments at first: (i) it is much faster to generate small templates, so we can have longer chains, (ii) the accuracy of estimating the eigenvalues and eigenvectors of the Fisher Matrix in our main proposal drops with an increase in the duration of the template.
Once the search reaches a static state, when the chains explores the posterior distribution around the local maximum, we stop the chains. The next step is to understand what we have detected. For this purpose we change the full template to a phenomenological template which consists of $N$ independent harmonics similar to what we have used for the phase maximization (\[3harm\]). We assume that the inner product between harmonics is zero, which is not really true, however it is a reasonable approximation given the simplification that it allows. Each harmonic can be maximized over its amplitude and phase the same way as in (\[acoef\]), (\[physMax\]). For the physical template described in section \[model\], the amplitudes are functions of the spin orientation and the distance, so the amplitudes and phases are not independent. But, using this template we maximize over the amplitude and the phase for each harmonic independently (we call this a generalised $F$-statistic). This process tells us which harmonics of the template are actually detecting part of the signal. We claim a detection with a harmonic if the $SNR \ge 5$. Different chains detect different harmonics, although in the majority of cases they detect the dominant one. Usually it is easy to identify which harmonics have been detected by plotting a figure similar to Figure \[harms\], although frequently the indices of the harmonic in the template do not correspond to the indices of the harmonic of the signal that it has matched. In some cases, the chains do not agree on the identification of the harmonics and we cannot tell, for instance, whether the dominant harmonic of the signal is $m=2$ or $m=1$. In this case we run further analysis for both possibilities. Once the harmonic index of the detected harmonics have been inferred, we apply a least squares fit to determine the three fundamental orbital frequencies. Harmonics with high SNR do not always lie closer to the true frequencies than lower SNR points. We see sometimes that lower SNR points matches the frequency of the harmonics very well, but fail to fit the other waveform parameters, so the derivatives of frequency do not match.
For the next step we force the orbital frequencies to be fixed at the values estimated from all the chains. We do this by turning off the jumps in the orbital frequency until the SNR has reached a value which is better than the best attained by any of the chains prior to fixing the orbital frequencies. We then release the constraint. The aim of this procedure is to first find a good guess for the frequencies, then refine the other waveform parameters (hopefully close to the true ones), then refine our estimate of the frequencies again and so on. If the initial guess is not too bad, a high SNR is achieved quite quickly when re-adjusting the other parameters. We repeat the procedure“determine frequencies – fix them – release” several times if required, but this iteration need not usually be repeated more than three times. In the figure \[segmSearch\] we show the final result of a run on the data from the low mass binary (1.3.4 type). In this example, the signal was found in the first and third half years of the data after only one iteration.
![Non-blind search for the low-mass binary. Results are shown for the search of the first (top) and third (bottom) half-year long segments. We show SNR versus parameter value for all chains for two cases: the inclination angle, $\lambda$, (left) and the MBH spin parameter (right). The vertical line in each plot is the true value of the parameter.[]{data-label="segmSearch"}](lamspin){height="0.45\textheight"}
Before we conclude this subsection we should reiterate that we need to detect at least three harmonics in order to be able to estimate the orbital frequencies. Moreover two of these must have different $n$-number and two must have different $m$ and the same $n$. If the harmonics detected are not sufficient to determine the orbital frequencies, one can just use the frequency-refining proposal described above or use an $N-k$-harmonic template with the $k$ already identified harmonics excluded. This forces the search to look for other harmonics.
### Parameter finalization
A good indication that we have found the signal is that the highest SNR chains cluster in parameter space. In other words, if all of the chains with SNR close (say $\gtrsim90\%$) to the maximum SNR found across all chains, have similar parameters. A further indication is that these “best” parameters are approximately the same for the different data sub-segments. A good example was shown in Figure \[segmSearch\] above, where we see that the first half-year and third half-year searches are producing similar results.
At this stage, the accuracy of the recovered parameters is relatively poor because we have been searching shorter data segments — the total SNR is therefore lower and there is greater degeneracy between waveform parameters over a short duration of signal (one can accurately fit a small number of cycles in many distinct ways). The final stage of the search involves first reparameterizing the templates from all the chains in the different segments of data by their frequencies etc. at a common reference time (usually $t_{ref}=0$) and then increase the duration of the template first to one year and then to two. We then run an MCMC search with these longer waveforms, with chains starting at each of the high SNR points identified in the previous stage of the search. In Figure \[improv\] we show how the parameter estimation improves as we increase the template duration from one half-year to two years.
![Improvement in parameter estimation in the blind search for the low mass MBH EMRI (1.3.4 type) as we increase the duration of the template from half a year (black points) to one year (red points) to two years (green points).[]{data-label="improv"}](improv){height="0.4\textheight"}
In this stage, we again use the generalised F-statistic with our $N$-harmonic template to begin with. This reduces the parameter space as it does the maximization over spin orientation analytically and so it is more efficient to use than the physical model. There are two caveats to using this template, however. (i) If the number of harmonics included is large it is very slow, as we need to maximize the likelihood for each harmonic. However, 5-8 harmonics are usually enough to build up an SNR that is comparable to the full 25 harmonic SNR. (ii) The maximization leads to a smoother but larger ambiguity (error) ellipsoid in parameter space. The smoothness helps the chains to reach the global maxima, but the fact that the maximum is quite flat gives a larger error in parameter estimation. This is indicated in Figure \[improv\] by the wide spread in parameter values in the search with half-year long templates. To finally improve the parameter estimation we need to finish the search by using the full physical template after we have found global maxima using the $N$-harmonic template. To seed this final analysis, we need an estimate of the spin orientation. This can be obtained from the estimated amplitudes of the harmonics, but the analytic expressions are so complicated that it would have to be done numerically. Instead, we compute the likelihood for various spin orientations and take the one giving the highest value. However, there is a four-fold degeneracy in the angle $\phi_K$, and a complete degeneracy in $\theta_K$. This is shown in Figure \[degen\] for the blind search for the high mass MBH EMRI. This plot is colour-coded by SNR, and the points were chosen randomly from a uniform distribution over the $\theta_K - \phi_K$ plane.
![Degeneracy in determination of the orientation of the MBH spin for the blind search of the high mass binary (1.3.1 type).[]{data-label="degen"}](trythKphK){height="0.4\textheight"}
One can see that it is hard to distinguish between four different values of $\phi_K$ and it is almost flat in $\theta_K$. To deal with this, we ran chains for several different choices of $\theta_K,\;\; \phi_K$ and in the end took the chain with highest SNR, which did find the correct values.
Results {#resSec}
=======
In this section we will describe some source-specific peculiarities encountered while we were analyzing the data sets. In each case, the signal was detected at a different stage of the search algorithm described in the previous section, and we will attempt to explain why this was the case. This section will be divided into two subsections. The first one is dedicated to the non-blind searches which were the basis for the algorithm development and tuning. The second subsection gives details of the “blind” searches we did to test the search pipeline.
Round 1B analysis
-----------------
While developing the algorithm, we analyzed the five “challenge” data sets that were released within the MLDC round 1B. The bulk of the search tuning and development was done after the submission deadline, so we knew parameters of the signals. We used our knowledge of the parameters only to identify when the search was going in the wrong direction so that we could try other techniques and to identify when we had detected the signal. The results of these searches, for the intrinsic source parameters, are summarized in Table \[results\].
The signal was detected at different stages of the search in the various cases. The easiest signals for this algorithm to find appear to be those from medium mass MBHs (data sets 1.3.2 and 1.3.3). We believe the reason for this is that the frequency evolution of the harmonics is not so great that the harmonics are hard to detect (which is true for the low mass MBH case), but it is sufficient that the parameter space is not too degenerate (unlike the high mass MBH case which is very degenerate). The second signal in Table \[results\], 1.3.2, was detected without any iterations in the “determine frequencies – fix them – release” phase of the search. The second medium mass binary, 1.3.3, required more work (one iteration), primarily because of the lower SNR of this source.
The high mass and low mass MBH systems are more difficult to detect. The signal from the high mass MBH EMRI has a lot of secondary maxima which are quite strong compared to the primary. These arise because the evolution in these systems is very slow, so it is easy to match harmonics for long periods of time with very different parameters. These secondary maxima are well separated and lie all over the parameter space. The analyzed signal was even worse than usual, because the inclination of the spin of the black hole to our line of sight was such that almost all of the signal power was concentrated in a single $m$-harmonic for each $n$ (we encountered a similar case in our blind search and will discuss this later).
The low mass MBH EMRI is our canonical EMRI, as we expect these systems to dominate the event rate [@emrirate; @gairLISA7]. For these systems, the harmonic frequencies evolve rather significantly over the inspiral and so a template needs to match both the frequency and frequency derivative of a harmonic rather accurately in order to get high SNR. This means there are fewer secondaries, but, at the same time, it also implies the global maximum is rather ‘sharp’ in parameter space, which is reflected in the better accuracy of recovered parameters. Usually these signals require more time on the first stage of the search (or a larger number of CPUs). The loudest signal (1.3.4 — fourth in the table) was more difficult to detect because of its orientation ($\lambda$ is close to $\pi/2$). This caused us to make an incorrect guess of the dominant harmonic when doing the phase maximization. We had to use an $N$-harmonic template in the search with $N=9$. The second signal (1.3.5) was not peculiar, but we had to do two iterations of the “determine – fix – release” part of the search since our first guess of the orbital frequencies was not very good due to the lower signal SNR.
type $\nu$ (mHz) $\mu/M_{\odot}$ $M/M_{\odot}$ $e_0$ $\theta_S$ $\phi_S$ $\lambda$ $a/M^2$ SNR
------- ------------- ----------------- --------------- --------- ------------ ---------- ----------- --------- -------
True 0.1920421 10.296 9517952 0.21438 1.018 4.910 0.4394 0.69816 120.5
Found 0.1920437 10.288 9520796 0.21411 1.027 4.932 0.4384 0.69823 118.1
True 0.34227777 9.771 5215577 0.20791 1.211 4.6826 1.4358 0.63796 132.9
Found 0.34227742 9.769 5214091 0.20818 1.172 4.6822 1.4364 0.63804 132.8
True 0.3425731 9.697 5219668 0.19927 0.589 0.710 0.9282 0.53326 79.5
Found 0.3425712 9.694 5216925 0.19979 0.573 0.713 0.9298 0.53337 79.7
True 0.8514396 10.105 955795 0.45058 2.551 0.979 1.6707 0.62514 101.6
Found 0.8514390 10.106 955544 0.45053 2.565 1.012 1.6719 0.62534 96.0
True 0.8321840 9.790 1033413 0.42691 2.680 1.088 2.3196 0.65829 55.3
Found 0.8321846 9.787 1034208 0.42701 2.687 1.053 2.3153 0.65770 55.6
: \[results\] Results of the analysis of the five “blind” data sets used in MLDC Challenge 1B.3. These are 1B.3.1–1B.3.5 going from top to bottom. The analysis of these data sets was not blind, as it was mostly finished after the parameters were released.
Blind tests
-----------
Since the high and low mass MBH EMRIs were the hardest to find, we decided to test the algorithm pipeline by performing blind tests on one data set containing a high mass MBH EMRI, and one data set containing a low mass MBH EMRI. We use the MLDC round 1B “training” data sets 1.3.1 and 1.3.4 for this analysis, although we did not consult the parameter key until we had finished the search. The results are presented in Table \[resultsBlind\]. We used two criteria to determine the end-point of the search and claim a detection: (i) several chains converged to the same result; (ii) the SNR of all harmonics in these best chains was comparable to and no less than the best SNR of the corresponding harmonic found in all the other chains.
For the high mass MBH signal we did not encounter any difficulties. The search with the $N$-harmonic template resulted in quite a large error bar on the parameters because of the degeneracies in the parameter space and because of the relatively low SNR of this source. The degeneracy in $\theta_K,\;\; \phi_K$ for this source is illustrated in Figure \[degen\]. The signal was found after one iteration of the “determine – fix – release” stage.
The signal from the low mass MBH EMRI proved much more interesting and difficult. Almost all the power was concentrated in the $m=2$ harmonic (with $n=2,3,4$). The $F$-statistic for 25 harmonics ($l=1,...,5$, $m=-2,...,2$) is shown for each harmonic in the matrix below
$$F = \left( \begin{array}{cccccc} m= & -2 & -1 & 0 & 1 & 2 \\
l=1 & 1.98 & 1.38 & 1.52 & 5.94 & 205.53\\
l=2 & 2.14 & 0.66 & 2.75 & 178.61 & 4677.62 \\
l=3 & 1.06 & 3.13 & 2.39 & 103.74 & 2109.76 \\
l=4 & 5.22 & 1.70 & 0.78 & 13.35 & 576.45 \\
l=5 & 2.85 & 1.61 & 3.27 & 4.73 & 145.50
\end{array}\right).$$
We were only able to detect three $m=2$ harmonics ($n=2,3,4$) with the chains, but a second $m$-harmonic is required in order to estimate the three frequencies. To achieve this, we used the method mentioned above: we constructed an $N$-harmonic template that did not include the harmonics which had already been identified in the search. This allowed us to find the $n=2, \;\; m=1$ harmonic and hence make a preliminary estimation of all three orbital frequencies. We then needed three iterations of the “determine – fix – release” search to reach the final answer. At this stage, we were confident about the quality of the detection and, when we compared to the true parameters, we had indeed reached a very high accuracy for all parameters.
type $\nu$ (mHz) $\mu/M_{\odot}$ $M/M_{\odot}$ $e_0$ $\theta_S$ $\phi_S$ $\lambda$ $a/M^2$ SNR
------- ------------- ----------------- --------------- ---------- ------------ ---------- ----------- --------- -------
True 0.1674472 10.131 10397935 0.25240 2.985 4.894 1.2056 0.65101 52.0
Found 0.1674462 10.111 10375301 0.25419 3.023 4.857 1.2097 0.65148 51.7
True 0.9997627 9.7478 975650 0.360970 1.453 4.95326 0.5110 0.65005 122.9
Found 0.9997626 9.7479 975610 0.360966 1.422 4.95339 0.5113 0.65007 116.0
: \[resultsBlind\] Results of the blind analysis of two data sets. For this analysis we used the MLDC 1B.3.1 and 1B.3.4 “training” data sets. Our analysis was blind in the sense that the search was run end-to-end without reference to the parameters used to generate the data sets. Our results were then compared to the known parameters at the end.
Discussion {#sum}
==========
We have described an algorithm for the detection of EMRI signals in LISA data. This algorithm is based on running multiple Markov Chain Monte Carlo search chains simultaneously. However, it relies on the key refinement that the properties of the secondary solutions that all of the chains have identified are used together to constrain further movement of the chains. It is clear from the results in Tables \[results\]–\[resultsBlind\] that the algorithm is able to robustly and accurately find the true solution, in the simplified situation that we are searching for a single, bright EMRI source buried in gaussian instrumental noise. We have successfully found the source in seven out of seven mock data streams, even when the parameters were such that the waveform had unusual features, e.g., orbital inclination close to $\lambda=\pi/2$. This gives us reason to expect that the algorithm will work equally well in all comparable situations, independent of the parameters of the source.
This is the first algorithm to be published in the literature that has been demonstrated to be able to detect and determine parameters for a “typical” EMRI signal — a $10M_{\odot}$ black hole falling into a $10^6M_{\odot}$ black hole — which we expect to dominate the LISA event rate [@emrirate; @gairLISA7]. It is particularly gratifying that our parameter recovery is now reaching the theoretical level that was estimated from Fisher Matrix analyses [@Barack:2003fp] – MBH mass and spin determinations at the level of $10^{-4}$, and sky position accuracies of $10^{-3}$. While we would expect the Fisher Matrix to accurately represent the shape of the global maximum for these high SNR sources, it is a purely local analysis and therefore does not account for the presence of secondary maxima. The fact that our algorithm can now find the global maxima from among the bright secondaries bodes well for using EMRI sources for high precision astrophysical observations [@AmaroSeoane:2007aw; @gairLISA7].
The algorithm can still be improved further. The main problem at present is that the identification of secondaries in order to determine constraints on the orbital frequencies is done by hand. We stop the search chains after stage 1, and then by hand examine the results in order to estimate suitable proposal distributions for the frequencies in the second stage. In principle this could all be done automatically, although it would require communication between different chains. If each chain had information about the best points found by all the other chains, then an adaptive proposal could be constructed from this information. The resulting search would be more like a population MCMC search. The other area of the search that could potentially be improved is the way in which we use annealing. The annealing scheme was borrowed almost verbatim from SMBH searches [@Cornish:2006dt] and we have not attempted to optimize it for the EMRI problem. While this is clearly not affecting the ultimate convergence of our search, the convergence speed might be improved by modifying the annealing scheme. Other MCMC variants, for instance parallel tempering, might also improve the efficiency of the search. Parallel tempering has been demonstrated in LISA searches to great effect [@KeyCornish], but we have no immediate plans to include it in the search pipeline.
The algorithm as described here has only been demonstrated for a significantly simplified scenario — detection of a single, bright EMRI buried in purely instrumental gaussian noise. The real LISA data stream will be very different, and is expected to contain many thousands of resolvable signals which will be overlapping in time and frequency, in addition to a noise foreground from galactic compact binaries and non-gaussian instrumental artefacts etc. It is not clear how well this search will perform under those circumstances, since it relies on being able to identify all the secondary peaks in the likelihood surface that are associated with the same signal. The relative SNRs and track shapes will provide powerful discriminators for this purpose, but there will inevitably be problems distinguishing a dim sideband harmonic of a bright source from the dominant harmonic of a similar but much more distant source. The best way to explore these complications is to attempt to analyse more realistic data sets. The next round of the MLDC, Challenge 3, includes an EMRI data set that contains five overlapping signals of low SNR. We will begin to explore source confusion by using that data set as a test case. As future MLDC releases become increasingly realistic, we will analyse them in order to demarcate where this algorithm fails and how it can be improved to cope with this greater realism.
The work of SB and EKP was supported in part by DFG grant SFB/TR 7 “Gravitational Wave Astronomy” and by DLR (Deutsches Zentrum für Luft- und Raumfahrt). JG acknowledges support from the Royal Society and thanks the Albert Einstein Institute for hospitality and support while this work was being completed.
[^1]: The “round 1” EMRI data set was released as part of round 2 of the MLDC.
[^2]: We are working in geometrical units $G=c=1$
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